src/HOL/Analysis/Cauchy_Integral_Theorem.thy
 author paulson Tue Apr 25 16:39:54 2017 +0100 (2017-04-25) changeset 65578 e4997c181cce parent 65037 2cf841ff23be child 65587 16a8991ab398 permissions -rw-r--r--
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
```     1 section \<open>Complex path integrals and Cauchy's integral theorem\<close>
```
```     2
```
```     3 text\<open>By John Harrison et al.  Ported from HOL Light by L C Paulson (2015)\<close>
```
```     4
```
```     5 theory Cauchy_Integral_Theorem
```
```     6 imports Complex_Transcendental Weierstrass_Theorems Ordered_Euclidean_Space
```
```     7 begin
```
```     8
```
```     9 subsection\<open>Homeomorphisms of arc images\<close>
```
```    10
```
```    11 lemma homeomorphism_arc:
```
```    12   fixes g :: "real \<Rightarrow> 'a::t2_space"
```
```    13   assumes "arc g"
```
```    14   obtains h where "homeomorphism {0..1} (path_image g) g h"
```
```    15 using assms by (force simp add: arc_def homeomorphism_compact path_def path_image_def)
```
```    16
```
```    17 lemma homeomorphic_arc_image_interval:
```
```    18   fixes g :: "real \<Rightarrow> 'a::t2_space" and a::real
```
```    19   assumes "arc g" "a < b"
```
```    20   shows "(path_image g) homeomorphic {a..b}"
```
```    21 proof -
```
```    22   have "(path_image g) homeomorphic {0..1::real}"
```
```    23     by (meson assms(1) homeomorphic_def homeomorphic_sym homeomorphism_arc)
```
```    24   also have "... homeomorphic {a..b}"
```
```    25     using assms by (force intro: homeomorphic_closed_intervals_real)
```
```    26   finally show ?thesis .
```
```    27 qed
```
```    28
```
```    29 lemma homeomorphic_arc_images:
```
```    30   fixes g :: "real \<Rightarrow> 'a::t2_space" and h :: "real \<Rightarrow> 'b::t2_space"
```
```    31   assumes "arc g" "arc h"
```
```    32   shows "(path_image g) homeomorphic (path_image h)"
```
```    33 proof -
```
```    34   have "(path_image g) homeomorphic {0..1::real}"
```
```    35     by (meson assms homeomorphic_def homeomorphic_sym homeomorphism_arc)
```
```    36   also have "... homeomorphic (path_image h)"
```
```    37     by (meson assms homeomorphic_def homeomorphism_arc)
```
```    38   finally show ?thesis .
```
```    39 qed
```
```    40
```
```    41 lemma path_connected_arc_complement:
```
```    42   fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
```
```    43   assumes "arc \<gamma>" "2 \<le> DIM('a)"
```
```    44   shows "path_connected(- path_image \<gamma>)"
```
```    45 proof -
```
```    46   have "path_image \<gamma> homeomorphic {0..1::real}"
```
```    47     by (simp add: assms homeomorphic_arc_image_interval)
```
```    48   then
```
```    49   show ?thesis
```
```    50     apply (rule path_connected_complement_homeomorphic_convex_compact)
```
```    51       apply (auto simp: assms)
```
```    52     done
```
```    53 qed
```
```    54
```
```    55 lemma connected_arc_complement:
```
```    56   fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
```
```    57   assumes "arc \<gamma>" "2 \<le> DIM('a)"
```
```    58   shows "connected(- path_image \<gamma>)"
```
```    59   by (simp add: assms path_connected_arc_complement path_connected_imp_connected)
```
```    60
```
```    61 lemma inside_arc_empty:
```
```    62   fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
```
```    63   assumes "arc \<gamma>"
```
```    64     shows "inside(path_image \<gamma>) = {}"
```
```    65 proof (cases "DIM('a) = 1")
```
```    66   case True
```
```    67   then show ?thesis
```
```    68     using assms connected_arc_image connected_convex_1_gen inside_convex by blast
```
```    69 next
```
```    70   case False
```
```    71   show ?thesis
```
```    72   proof (rule inside_bounded_complement_connected_empty)
```
```    73     show "connected (- path_image \<gamma>)"
```
```    74       apply (rule connected_arc_complement [OF assms])
```
```    75       using False
```
```    76       by (metis DIM_ge_Suc0 One_nat_def Suc_1 not_less_eq_eq order_class.order.antisym)
```
```    77     show "bounded (path_image \<gamma>)"
```
```    78       by (simp add: assms bounded_arc_image)
```
```    79   qed
```
```    80 qed
```
```    81
```
```    82 lemma inside_simple_curve_imp_closed:
```
```    83   fixes \<gamma> :: "real \<Rightarrow> 'a::euclidean_space"
```
```    84     shows "\<lbrakk>simple_path \<gamma>; x \<in> inside(path_image \<gamma>)\<rbrakk> \<Longrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
```
```    85   using arc_simple_path  inside_arc_empty by blast
```
```    86
```
```    87
```
```    88 subsection \<open>Piecewise differentiable functions\<close>
```
```    89
```
```    90 definition piecewise_differentiable_on
```
```    91            (infixr "piecewise'_differentiable'_on" 50)
```
```    92   where "f piecewise_differentiable_on i  \<equiv>
```
```    93            continuous_on i f \<and>
```
```    94            (\<exists>s. finite s \<and> (\<forall>x \<in> i - s. f differentiable (at x within i)))"
```
```    95
```
```    96 lemma piecewise_differentiable_on_imp_continuous_on:
```
```    97     "f piecewise_differentiable_on s \<Longrightarrow> continuous_on s f"
```
```    98 by (simp add: piecewise_differentiable_on_def)
```
```    99
```
```   100 lemma piecewise_differentiable_on_subset:
```
```   101     "f piecewise_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_differentiable_on t"
```
```   102   using continuous_on_subset
```
```   103   unfolding piecewise_differentiable_on_def
```
```   104   apply safe
```
```   105   apply (blast intro: elim: continuous_on_subset)
```
```   106   by (meson Diff_iff differentiable_within_subset subsetCE)
```
```   107
```
```   108 lemma differentiable_on_imp_piecewise_differentiable:
```
```   109   fixes a:: "'a::{linorder_topology,real_normed_vector}"
```
```   110   shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
```
```   111   apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
```
```   112   apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
```
```   113   done
```
```   114
```
```   115 lemma differentiable_imp_piecewise_differentiable:
```
```   116     "(\<And>x. x \<in> s \<Longrightarrow> f differentiable (at x within s))
```
```   117          \<Longrightarrow> f piecewise_differentiable_on s"
```
```   118 by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
```
```   119          intro: differentiable_within_subset)
```
```   120
```
```   121 lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on s"
```
```   122   by (simp add: differentiable_imp_piecewise_differentiable)
```
```   123
```
```   124 lemma piecewise_differentiable_compose:
```
```   125     "\<lbrakk>f piecewise_differentiable_on s; g piecewise_differentiable_on (f ` s);
```
```   126       \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
```
```   127       \<Longrightarrow> (g o f) piecewise_differentiable_on s"
```
```   128   apply (simp add: piecewise_differentiable_on_def, safe)
```
```   129   apply (blast intro: continuous_on_compose2)
```
```   130   apply (rename_tac A B)
```
```   131   apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
```
```   132   apply (blast intro!: differentiable_chain_within)
```
```   133   done
```
```   134
```
```   135 lemma piecewise_differentiable_affine:
```
```   136   fixes m::real
```
```   137   assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` s)"
```
```   138   shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on s"
```
```   139 proof (cases "m = 0")
```
```   140   case True
```
```   141   then show ?thesis
```
```   142     unfolding o_def
```
```   143     by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
```
```   144 next
```
```   145   case False
```
```   146   show ?thesis
```
```   147     apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
```
```   148     apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
```
```   149     done
```
```   150 qed
```
```   151
```
```   152 lemma piecewise_differentiable_cases:
```
```   153   fixes c::real
```
```   154   assumes "f piecewise_differentiable_on {a..c}"
```
```   155           "g piecewise_differentiable_on {c..b}"
```
```   156            "a \<le> c" "c \<le> b" "f c = g c"
```
```   157   shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
```
```   158 proof -
```
```   159   obtain s t where st: "finite s" "finite t"
```
```   160                        "\<forall>x\<in>{a..c} - s. f differentiable at x within {a..c}"
```
```   161                        "\<forall>x\<in>{c..b} - t. g differentiable at x within {c..b}"
```
```   162     using assms
```
```   163     by (auto simp: piecewise_differentiable_on_def)
```
```   164   have finabc: "finite ({a,b,c} \<union> (s \<union> t))"
```
```   165     by (metis \<open>finite s\<close> \<open>finite t\<close> finite_Un finite_insert finite.emptyI)
```
```   166   have "continuous_on {a..c} f" "continuous_on {c..b} g"
```
```   167     using assms piecewise_differentiable_on_def by auto
```
```   168   then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
```
```   169     using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
```
```   170                                OF closed_real_atLeastAtMost [of c b],
```
```   171                                of f g "\<lambda>x. x\<le>c"]  assms
```
```   172     by (force simp: ivl_disj_un_two_touch)
```
```   173   moreover
```
```   174   { fix x
```
```   175     assume x: "x \<in> {a..b} - ({a,b,c} \<union> (s \<union> t))"
```
```   176     have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
```
```   177     proof (cases x c rule: le_cases)
```
```   178       case le show ?diff_fg
```
```   179       proof (rule differentiable_transform_within [where d = "dist x c" and f = f])
```
```   180         have "f differentiable at x within ({a<..<c} - s)"
```
```   181           apply (rule differentiable_at_withinI)
```
```   182           using x le st
```
```   183           by (metis (no_types, lifting) DiffD1 DiffD2 DiffI UnCI atLeastAtMost_diff_ends atLeastAtMost_iff at_within_interior insert_iff interior_atLeastAtMost le st(3) x)
```
```   184         moreover have "open ({a<..<c} - s)"
```
```   185           by (blast intro: open_greaterThanLessThan \<open>finite s\<close> finite_imp_closed)
```
```   186         ultimately show "f differentiable at x within {a..b}"
```
```   187           using x le by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI)
```
```   188       qed (use x le st dist_real_def in auto)
```
```   189     next
```
```   190       case ge show ?diff_fg
```
```   191       proof (rule differentiable_transform_within [where d = "dist x c" and f = g])
```
```   192         have "g differentiable at x within ({c<..<b} - t)"
```
```   193           apply (rule differentiable_at_withinI)
```
```   194           using x ge st
```
```   195           by (metis DiffD1 DiffD2 DiffI UnCI atLeastAtMost_diff_ends atLeastAtMost_iff at_within_interior insert_iff interior_atLeastAtMost)
```
```   196         moreover have "open ({c<..<b} - t)"
```
```   197           by (blast intro: open_greaterThanLessThan \<open>finite t\<close> finite_imp_closed)
```
```   198         ultimately show "g differentiable at x within {a..b}"
```
```   199           using x ge by (auto simp add: at_within_open_NO_MATCH differentiable_at_withinI)
```
```   200       qed (use x ge st dist_real_def in auto)
```
```   201     qed
```
```   202   }
```
```   203   then have "\<exists>s. finite s \<and>
```
```   204                  (\<forall>x\<in>{a..b} - s. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
```
```   205     by (meson finabc)
```
```   206   ultimately show ?thesis
```
```   207     by (simp add: piecewise_differentiable_on_def)
```
```   208 qed
```
```   209
```
```   210 lemma piecewise_differentiable_neg:
```
```   211     "f piecewise_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on s"
```
```   212   by (auto simp: piecewise_differentiable_on_def continuous_on_minus)
```
```   213
```
```   214 lemma piecewise_differentiable_add:
```
```   215   assumes "f piecewise_differentiable_on i"
```
```   216           "g piecewise_differentiable_on i"
```
```   217     shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
```
```   218 proof -
```
```   219   obtain s t where st: "finite s" "finite t"
```
```   220                        "\<forall>x\<in>i - s. f differentiable at x within i"
```
```   221                        "\<forall>x\<in>i - t. g differentiable at x within i"
```
```   222     using assms by (auto simp: piecewise_differentiable_on_def)
```
```   223   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)"
```
```   224     by auto
```
```   225   moreover have "continuous_on i f" "continuous_on i g"
```
```   226     using assms piecewise_differentiable_on_def by auto
```
```   227   ultimately show ?thesis
```
```   228     by (auto simp: piecewise_differentiable_on_def continuous_on_add)
```
```   229 qed
```
```   230
```
```   231 lemma piecewise_differentiable_diff:
```
```   232     "\<lbrakk>f piecewise_differentiable_on s;  g piecewise_differentiable_on s\<rbrakk>
```
```   233      \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on s"
```
```   234   unfolding diff_conv_add_uminus
```
```   235   by (metis piecewise_differentiable_add piecewise_differentiable_neg)
```
```   236
```
```   237 lemma continuous_on_joinpaths_D1:
```
```   238     "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
```
```   239   apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"])
```
```   240   apply (rule continuous_intros | simp)+
```
```   241   apply (auto elim!: continuous_on_subset simp: joinpaths_def)
```
```   242   done
```
```   243
```
```   244 lemma continuous_on_joinpaths_D2:
```
```   245     "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
```
```   246   apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (\<lambda>x. inverse 2*x + 1/2)"])
```
```   247   apply (rule continuous_intros | simp)+
```
```   248   apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
```
```   249   done
```
```   250
```
```   251 lemma piecewise_differentiable_D1:
```
```   252     "(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
```
```   253   apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
```
```   254   apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
```
```   255   apply simp
```
```   256   apply (intro ballI)
```
```   257   apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
```
```   258        in differentiable_transform_within)
```
```   259   apply (auto simp: dist_real_def joinpaths_def)
```
```   260   apply (rule differentiable_chain_within derivative_intros | simp)+
```
```   261   apply (rule differentiable_subset)
```
```   262   apply (force simp:)+
```
```   263   done
```
```   264
```
```   265 lemma piecewise_differentiable_D2:
```
```   266     "\<lbrakk>(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\<rbrakk>
```
```   267     \<Longrightarrow> g2 piecewise_differentiable_on {0..1}"
```
```   268   apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2)
```
```   269   apply (rule_tac x="insert 0 ((\<lambda>x. 2*x-1)`s)" in exI)
```
```   270   apply simp
```
```   271   apply (intro ballI)
```
```   272   apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)"
```
```   273           in differentiable_transform_within)
```
```   274   apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: if_split_asm)
```
```   275   apply (rule differentiable_chain_within derivative_intros | simp)+
```
```   276   apply (rule differentiable_subset)
```
```   277   apply (force simp: divide_simps)+
```
```   278   done
```
```   279
```
```   280
```
```   281 subsubsection\<open>The concept of continuously differentiable\<close>
```
```   282
```
```   283 text \<open>
```
```   284 John Harrison writes as follows:
```
```   285
```
```   286 ``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
```
```   287 continuously differentiable, which ensures that the path integral exists at least for any continuous
```
```   288 f, since all piecewise continuous functions are integrable. However, our notion of validity is
```
```   289 weaker, just piecewise differentiability... [namely] continuity plus differentiability except on a
```
```   290 ﬁnite set ... [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
```
```   291 the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
```
```   292 can integrate all derivatives.''
```
```   293
```
```   294 "Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
```
```   295 Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.
```
```   296
```
```   297 And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
```
```   298 difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
```
```   299 asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>
```
```   300
```
```   301 definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
```
```   302            (infix "C1'_differentiable'_on" 50)
```
```   303   where
```
```   304   "f C1_differentiable_on s \<longleftrightarrow>
```
```   305    (\<exists>D. (\<forall>x \<in> s. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on s D)"
```
```   306
```
```   307 lemma C1_differentiable_on_eq:
```
```   308     "f C1_differentiable_on s \<longleftrightarrow>
```
```   309      (\<forall>x \<in> s. f differentiable at x) \<and> continuous_on s (\<lambda>x. vector_derivative f (at x))"
```
```   310   unfolding C1_differentiable_on_def
```
```   311   apply safe
```
```   312   using differentiable_def has_vector_derivative_def apply blast
```
```   313   apply (erule continuous_on_eq)
```
```   314   using vector_derivative_at apply fastforce
```
```   315   using vector_derivative_works apply fastforce
```
```   316   done
```
```   317
```
```   318 lemma C1_differentiable_on_subset:
```
```   319   "f C1_differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f C1_differentiable_on s"
```
```   320   unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
```
```   321   by (blast intro:  continuous_within_subset)
```
```   322
```
```   323 lemma C1_differentiable_compose:
```
```   324     "\<lbrakk>f C1_differentiable_on s; g C1_differentiable_on (f ` s);
```
```   325       \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
```
```   326       \<Longrightarrow> (g o f) C1_differentiable_on s"
```
```   327   apply (simp add: C1_differentiable_on_eq, safe)
```
```   328    using differentiable_chain_at apply blast
```
```   329   apply (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
```
```   330    apply (rule Limits.continuous_on_scaleR, assumption)
```
```   331    apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def)
```
```   332   by (simp add: vector_derivative_chain_at)
```
```   333
```
```   334 lemma C1_diff_imp_diff: "f C1_differentiable_on s \<Longrightarrow> f differentiable_on s"
```
```   335   by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)
```
```   336
```
```   337 lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on s"
```
```   338   by (auto simp: C1_differentiable_on_eq continuous_on_const)
```
```   339
```
```   340 lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on s"
```
```   341   by (auto simp: C1_differentiable_on_eq continuous_on_const)
```
```   342
```
```   343 lemma C1_differentiable_on_add [simp, derivative_intros]:
```
```   344   "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
```
```   345   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
```
```   346
```
```   347 lemma C1_differentiable_on_minus [simp, derivative_intros]:
```
```   348   "f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
```
```   349   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
```
```   350
```
```   351 lemma C1_differentiable_on_diff [simp, derivative_intros]:
```
```   352   "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
```
```   353   unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)
```
```   354
```
```   355 lemma C1_differentiable_on_mult [simp, derivative_intros]:
```
```   356   fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
```
```   357   shows "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on s"
```
```   358   unfolding C1_differentiable_on_eq
```
```   359   by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)
```
```   360
```
```   361 lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
```
```   362   "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on s"
```
```   363   unfolding C1_differentiable_on_eq
```
```   364   by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+
```
```   365
```
```   366
```
```   367 definition piecewise_C1_differentiable_on
```
```   368            (infixr "piecewise'_C1'_differentiable'_on" 50)
```
```   369   where "f piecewise_C1_differentiable_on i  \<equiv>
```
```   370            continuous_on i f \<and>
```
```   371            (\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"
```
```   372
```
```   373 lemma C1_differentiable_imp_piecewise:
```
```   374     "f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
```
```   375   by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)
```
```   376
```
```   377 lemma piecewise_C1_imp_differentiable:
```
```   378     "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
```
```   379   by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
```
```   380            C1_differentiable_on_def differentiable_def has_vector_derivative_def
```
```   381            intro: has_derivative_at_within)
```
```   382
```
```   383 lemma piecewise_C1_differentiable_compose:
```
```   384     "\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
```
```   385       \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
```
```   386       \<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
```
```   387   apply (simp add: piecewise_C1_differentiable_on_def, safe)
```
```   388   apply (blast intro: continuous_on_compose2)
```
```   389   apply (rename_tac A B)
```
```   390   apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
```
```   391   apply (rule conjI, blast)
```
```   392   apply (rule C1_differentiable_compose)
```
```   393   apply (blast intro: C1_differentiable_on_subset)
```
```   394   apply (blast intro: C1_differentiable_on_subset)
```
```   395   by (simp add: Diff_Int_distrib2)
```
```   396
```
```   397 lemma piecewise_C1_differentiable_on_subset:
```
```   398     "f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
```
```   399   by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)
```
```   400
```
```   401 lemma C1_differentiable_imp_continuous_on:
```
```   402   "f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
```
```   403   unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
```
```   404   using differentiable_at_withinI differentiable_imp_continuous_within by blast
```
```   405
```
```   406 lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
```
```   407   unfolding C1_differentiable_on_def
```
```   408   by auto
```
```   409
```
```   410 lemma piecewise_C1_differentiable_affine:
```
```   411   fixes m::real
```
```   412   assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
```
```   413   shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
```
```   414 proof (cases "m = 0")
```
```   415   case True
```
```   416   then show ?thesis
```
```   417     unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
```
```   418 next
```
```   419   case False
```
```   420   show ?thesis
```
```   421     apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
```
```   422     apply (rule assms derivative_intros | simp add: False vimage_def)+
```
```   423     using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
```
```   424     apply simp
```
```   425     done
```
```   426 qed
```
```   427
```
```   428 lemma piecewise_C1_differentiable_cases:
```
```   429   fixes c::real
```
```   430   assumes "f piecewise_C1_differentiable_on {a..c}"
```
```   431           "g piecewise_C1_differentiable_on {c..b}"
```
```   432            "a \<le> c" "c \<le> b" "f c = g c"
```
```   433   shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
```
```   434 proof -
```
```   435   obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
```
```   436                        "g C1_differentiable_on ({c..b} - t)"
```
```   437                        "finite s" "finite t"
```
```   438     using assms
```
```   439     by (force simp: piecewise_C1_differentiable_on_def)
```
```   440   then have f_diff: "f differentiable_on {a..<c} - s"
```
```   441         and g_diff: "g differentiable_on {c<..b} - t"
```
```   442     by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
```
```   443   have "continuous_on {a..c} f" "continuous_on {c..b} g"
```
```   444     using assms piecewise_C1_differentiable_on_def by auto
```
```   445   then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
```
```   446     using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
```
```   447                                OF closed_real_atLeastAtMost [of c b],
```
```   448                                of f g "\<lambda>x. x\<le>c"]  assms
```
```   449     by (force simp: ivl_disj_un_two_touch)
```
```   450   { fix x
```
```   451     assume x: "x \<in> {a..b} - insert c (s \<union> t)"
```
```   452     have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
```
```   453     proof (cases x c rule: le_cases)
```
```   454       case le show ?diff_fg
```
```   455         apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
```
```   456         using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
```
```   457     next
```
```   458       case ge show ?diff_fg
```
```   459         apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
```
```   460         using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
```
```   461     qed
```
```   462   }
```
```   463   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)"
```
```   464     by auto
```
```   465   moreover
```
```   466   { assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
```
```   467        and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
```
```   468     have "open ({a<..<c} - s)"  "open ({c<..<b} - t)"
```
```   469       using st by (simp_all add: open_Diff finite_imp_closed)
```
```   470     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))"
```
```   471       apply (rule continuous_on_eq [OF fcon])
```
```   472       apply (simp add:)
```
```   473       apply (rule vector_derivative_at [symmetric])
```
```   474       apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
```
```   475       apply (simp_all add: dist_norm vector_derivative_works [symmetric])
```
```   476       apply (metis (full_types) C1_differentiable_on_eq Diff_iff Groups.add_ac(2) add_mono_thms_linordered_field(5) atLeastAtMost_iff linorder_not_le order_less_irrefl st(1))
```
```   477       apply auto
```
```   478       done
```
```   479     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))"
```
```   480       apply (rule continuous_on_eq [OF gcon])
```
```   481       apply (simp add:)
```
```   482       apply (rule vector_derivative_at [symmetric])
```
```   483       apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
```
```   484       apply (simp_all add: dist_norm vector_derivative_works [symmetric])
```
```   485       apply (metis (full_types) C1_differentiable_on_eq Diff_iff Groups.add_ac(2) add_mono_thms_linordered_field(5) atLeastAtMost_iff less_irrefl not_le st(2))
```
```   486       apply auto
```
```   487       done
```
```   488     ultimately have "continuous_on ({a<..<b} - insert c (s \<union> t))
```
```   489         (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
```
```   490       apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
```
```   491       done
```
```   492   } note * = this
```
```   493   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))"
```
```   494     using st
```
```   495     by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
```
```   496   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)"
```
```   497     apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
```
```   498     using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
```
```   499   with cab show ?thesis
```
```   500     by (simp add: piecewise_C1_differentiable_on_def)
```
```   501 qed
```
```   502
```
```   503 lemma piecewise_C1_differentiable_neg:
```
```   504     "f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
```
```   505   unfolding piecewise_C1_differentiable_on_def
```
```   506   by (auto intro!: continuous_on_minus C1_differentiable_on_minus)
```
```   507
```
```   508 lemma piecewise_C1_differentiable_add:
```
```   509   assumes "f piecewise_C1_differentiable_on i"
```
```   510           "g piecewise_C1_differentiable_on i"
```
```   511     shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
```
```   512 proof -
```
```   513   obtain s t where st: "finite s" "finite t"
```
```   514                        "f C1_differentiable_on (i-s)"
```
```   515                        "g C1_differentiable_on (i-t)"
```
```   516     using assms by (auto simp: piecewise_C1_differentiable_on_def)
```
```   517   then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
```
```   518     by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
```
```   519   moreover have "continuous_on i f" "continuous_on i g"
```
```   520     using assms piecewise_C1_differentiable_on_def by auto
```
```   521   ultimately show ?thesis
```
```   522     by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
```
```   523 qed
```
```   524
```
```   525 lemma piecewise_C1_differentiable_diff:
```
```   526     "\<lbrakk>f piecewise_C1_differentiable_on s;  g piecewise_C1_differentiable_on s\<rbrakk>
```
```   527      \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
```
```   528   unfolding diff_conv_add_uminus
```
```   529   by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)
```
```   530
```
```   531 lemma piecewise_C1_differentiable_D1:
```
```   532   fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
```
```   533   assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
```
```   534     shows "g1 piecewise_C1_differentiable_on {0..1}"
```
```   535 proof -
```
```   536   obtain s where "finite s"
```
```   537              and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
```
```   538              and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
```
```   539     using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   540   then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
```
```   541     apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_within)
```
```   542     using that
```
```   543     apply (simp_all add: dist_real_def joinpaths_def)
```
```   544     apply (rule differentiable_chain_at derivative_intros | force)+
```
```   545     done
```
```   546   have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
```
```   547                if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
```
```   548     apply (subst vector_derivative_chain_at)
```
```   549     using that
```
```   550     apply (rule derivative_eq_intros g1D | simp)+
```
```   551     done
```
```   552   have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
```
```   553     using co12 by (rule continuous_on_subset) force
```
```   554   then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
```
```   555     apply (rule continuous_on_eq [OF _ vector_derivative_at])
```
```   556     apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_within)
```
```   557     apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
```
```   558     apply (force intro: g1D differentiable_chain_at)
```
```   559     apply auto
```
```   560     done
```
```   561   have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
```
```   562                       ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
```
```   563     apply (rule continuous_intros)+
```
```   564     using coDhalf
```
```   565     apply (simp add: scaleR_conv_of_real image_set_diff image_image)
```
```   566     done
```
```   567   then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
```
```   568     by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
```
```   569   have "continuous_on {0..1} g1"
```
```   570     using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
```
```   571   with \<open>finite s\<close> show ?thesis
```
```   572     apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   573     apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
```
```   574     apply (simp add: g1D con_g1)
```
```   575   done
```
```   576 qed
```
```   577
```
```   578 lemma piecewise_C1_differentiable_D2:
```
```   579   fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
```
```   580   assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
```
```   581     shows "g2 piecewise_C1_differentiable_on {0..1}"
```
```   582 proof -
```
```   583   obtain s where "finite s"
```
```   584              and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
```
```   585              and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
```
```   586     using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   587   then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
```
```   588     apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_within)
```
```   589     using that
```
```   590     apply (simp_all add: dist_real_def joinpaths_def)
```
```   591     apply (auto simp: dist_real_def joinpaths_def field_simps)
```
```   592     apply (rule differentiable_chain_at derivative_intros | force)+
```
```   593     apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
```
```   594     apply assumption
```
```   595     done
```
```   596   have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
```
```   597                if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
```
```   598     using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
```
```   599   have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
```
```   600     using co12 by (rule continuous_on_subset) force
```
```   601   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))"
```
```   602     apply (rule continuous_on_eq [OF _ vector_derivative_at])
```
```   603     apply (rule_tac f="g2 o (\<lambda>x. 2*x-1)" and d="dist (3/4) ((x+1)/2)" in has_vector_derivative_transform_within)
```
```   604     apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
```
```   605                 intro!: g2D differentiable_chain_at)
```
```   606     done
```
```   607   have [simp]: "((\<lambda>x. (x + 1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` s))) = ({1/2..1} - insert (1/2) s)"
```
```   608     apply (simp add: image_set_diff inj_on_def image_image)
```
```   609     apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
```
```   610     done
```
```   611   have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
```
```   612                       ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
```
```   613     by (rule continuous_intros | simp add:  coDhalf)+
```
```   614   then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)) (\<lambda>x. vector_derivative g2 (at x))"
```
```   615     by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
```
```   616   have "continuous_on {0..1} g2"
```
```   617     using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
```
```   618   with \<open>finite s\<close> show ?thesis
```
```   619     apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   620     apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
```
```   621     apply (simp add: g2D con_g2)
```
```   622   done
```
```   623 qed
```
```   624
```
```   625 subsection \<open>Valid paths, and their start and finish\<close>
```
```   626
```
```   627 definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
```
```   628   where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"
```
```   629
```
```   630 definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
```
```   631   where "closed_path g \<equiv> g 0 = g 1"
```
```   632
```
```   633 subsubsection\<open>In particular, all results for paths apply\<close>
```
```   634
```
```   635 lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
```
```   636 by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)
```
```   637
```
```   638 lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
```
```   639   by (metis connected_path_image valid_path_imp_path)
```
```   640
```
```   641 lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
```
```   642   by (metis compact_path_image valid_path_imp_path)
```
```   643
```
```   644 lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
```
```   645   by (metis bounded_path_image valid_path_imp_path)
```
```   646
```
```   647 lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
```
```   648   by (metis closed_path_image valid_path_imp_path)
```
```   649
```
```   650 proposition valid_path_compose:
```
```   651   assumes "valid_path g"
```
```   652       and der: "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable (at x)"
```
```   653       and con: "continuous_on (path_image g) (deriv f)"
```
```   654     shows "valid_path (f o g)"
```
```   655 proof -
```
```   656   obtain s where "finite s" and g_diff: "g C1_differentiable_on {0..1} - s"
```
```   657     using \<open>valid_path g\<close> unfolding valid_path_def piecewise_C1_differentiable_on_def by auto
```
```   658   have "f \<circ> g differentiable at t" when "t\<in>{0..1} - s" for t
```
```   659     proof (rule differentiable_chain_at)
```
```   660       show "g differentiable at t" using \<open>valid_path g\<close>
```
```   661         by (meson C1_differentiable_on_eq \<open>g C1_differentiable_on {0..1} - s\<close> that)
```
```   662     next
```
```   663       have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
```
```   664       then show "f differentiable at (g t)"
```
```   665         using der[THEN field_differentiable_imp_differentiable] by auto
```
```   666     qed
```
```   667   moreover have "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (f \<circ> g) (at x))"
```
```   668     proof (rule continuous_on_eq [where f = "\<lambda>x. vector_derivative g (at x) * deriv f (g x)"],
```
```   669         rule continuous_intros)
```
```   670       show "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative g (at x))"
```
```   671         using g_diff C1_differentiable_on_eq by auto
```
```   672     next
```
```   673       have "continuous_on {0..1} (\<lambda>x. deriv f (g x))"
```
```   674         using continuous_on_compose[OF _ con[unfolded path_image_def],unfolded comp_def]
```
```   675           \<open>valid_path g\<close> piecewise_C1_differentiable_on_def valid_path_def
```
```   676         by blast
```
```   677       then show "continuous_on ({0..1} - s) (\<lambda>x. deriv f (g x))"
```
```   678         using continuous_on_subset by blast
```
```   679     next
```
```   680       show "vector_derivative g (at t) * deriv f (g t) = vector_derivative (f \<circ> g) (at t)"
```
```   681           when "t \<in> {0..1} - s" for t
```
```   682         proof (rule vector_derivative_chain_at_general[symmetric])
```
```   683           show "g differentiable at t" by (meson C1_differentiable_on_eq g_diff that)
```
```   684         next
```
```   685           have "g t\<in>path_image g" using that DiffD1 image_eqI path_image_def by metis
```
```   686           then show "f field_differentiable at (g t)" using der by auto
```
```   687         qed
```
```   688     qed
```
```   689   ultimately have "f o g C1_differentiable_on {0..1} - s"
```
```   690     using C1_differentiable_on_eq by blast
```
```   691   moreover have "path (f \<circ> g)"
```
```   692     apply (rule path_continuous_image[OF valid_path_imp_path[OF \<open>valid_path g\<close>]])
```
```   693     using der
```
```   694     by (simp add: continuous_at_imp_continuous_on field_differentiable_imp_continuous_at)
```
```   695   ultimately show ?thesis unfolding valid_path_def piecewise_C1_differentiable_on_def path_def
```
```   696     using \<open>finite s\<close> by auto
```
```   697 qed
```
```   698
```
```   699
```
```   700 subsection\<open>Contour Integrals along a path\<close>
```
```   701
```
```   702 text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>
```
```   703
```
```   704 text\<open>piecewise differentiable function on [0,1]\<close>
```
```   705
```
```   706 definition has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
```
```   707            (infixr "has'_contour'_integral" 50)
```
```   708   where "(f has_contour_integral i) g \<equiv>
```
```   709            ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
```
```   710             has_integral i) {0..1}"
```
```   711
```
```   712 definition contour_integrable_on
```
```   713            (infixr "contour'_integrable'_on" 50)
```
```   714   where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"
```
```   715
```
```   716 definition contour_integral
```
```   717   where "contour_integral g f \<equiv> @i. (f has_contour_integral i) g \<or> ~ f contour_integrable_on g \<and> i=0"
```
```   718
```
```   719 lemma not_integrable_contour_integral: "~ f contour_integrable_on g \<Longrightarrow> contour_integral g f = 0"
```
```   720   unfolding contour_integrable_on_def contour_integral_def by blast
```
```   721
```
```   722 lemma contour_integral_unique: "(f has_contour_integral i) g \<Longrightarrow> contour_integral g f = i"
```
```   723   apply (simp add: contour_integral_def has_contour_integral_def contour_integrable_on_def)
```
```   724   using has_integral_unique by blast
```
```   725
```
```   726 corollary has_contour_integral_eqpath:
```
```   727      "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
```
```   728        contour_integral p f = contour_integral \<gamma> f\<rbrakk>
```
```   729       \<Longrightarrow> (f has_contour_integral y) \<gamma>"
```
```   730 using contour_integrable_on_def contour_integral_unique by auto
```
```   731
```
```   732 lemma has_contour_integral_integral:
```
```   733     "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
```
```   734   by (metis contour_integral_unique contour_integrable_on_def)
```
```   735
```
```   736 lemma has_contour_integral_unique:
```
```   737     "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
```
```   738   using has_integral_unique
```
```   739   by (auto simp: has_contour_integral_def)
```
```   740
```
```   741 lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
```
```   742   using contour_integrable_on_def by blast
```
```   743
```
```   744 (* Show that we can forget about the localized derivative.*)
```
```   745
```
```   746 lemma vector_derivative_within_interior:
```
```   747      "\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
```
```   748       \<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
```
```   749   apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
```
```   750   apply (subst lim_within_interior, auto)
```
```   751   done
```
```   752
```
```   753 lemma has_integral_localized_vector_derivative:
```
```   754     "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
```
```   755      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
```
```   756 proof -
```
```   757   have "{a..b} - {a,b} = interior {a..b}"
```
```   758     by (simp add: atLeastAtMost_diff_ends)
```
```   759   show ?thesis
```
```   760     apply (rule has_integral_spike_eq [of "{a,b}"])
```
```   761     apply (auto simp: vector_derivative_within_interior)
```
```   762     done
```
```   763 qed
```
```   764
```
```   765 lemma integrable_on_localized_vector_derivative:
```
```   766     "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
```
```   767      (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
```
```   768   by (simp add: integrable_on_def has_integral_localized_vector_derivative)
```
```   769
```
```   770 lemma has_contour_integral:
```
```   771      "(f has_contour_integral i) g \<longleftrightarrow>
```
```   772       ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
```
```   773   by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)
```
```   774
```
```   775 lemma contour_integrable_on:
```
```   776      "f contour_integrable_on g \<longleftrightarrow>
```
```   777       (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
```
```   778   by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)
```
```   779
```
```   780 subsection\<open>Reversing a path\<close>
```
```   781
```
```   782 lemma valid_path_imp_reverse:
```
```   783   assumes "valid_path g"
```
```   784     shows "valid_path(reversepath g)"
```
```   785 proof -
```
```   786   obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
```
```   787     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
```
```   788   then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
```
```   789     apply (auto simp: reversepath_def)
```
```   790     apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
```
```   791     apply (auto simp: C1_differentiable_on_eq)
```
```   792     apply (rule continuous_intros, force)
```
```   793     apply (force elim!: continuous_on_subset)
```
```   794     apply (simp add: finite_vimageI inj_on_def)
```
```   795     done
```
```   796   then show ?thesis using assms
```
```   797     by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
```
```   798 qed
```
```   799
```
```   800 lemma valid_path_reversepath [simp]: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
```
```   801   using valid_path_imp_reverse by force
```
```   802
```
```   803 lemma has_contour_integral_reversepath:
```
```   804   assumes "valid_path g" "(f has_contour_integral i) g"
```
```   805     shows "(f has_contour_integral (-i)) (reversepath g)"
```
```   806 proof -
```
```   807   { fix s x
```
```   808     assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
```
```   809       have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
```
```   810             - vector_derivative g (at (1 - x) within {0..1})"
```
```   811       proof -
```
```   812         obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
```
```   813           using xs
```
```   814           by (force simp: has_vector_derivative_def C1_differentiable_on_def)
```
```   815         have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
```
```   816           apply (rule vector_diff_chain_within)
```
```   817           apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
```
```   818           apply (rule has_vector_derivative_at_within [OF f'])
```
```   819           done
```
```   820         then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
```
```   821           by (simp add: o_def)
```
```   822         show ?thesis
```
```   823           using xs
```
```   824           by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
```
```   825       qed
```
```   826   } note * = this
```
```   827   have 01: "{0..1::real} = cbox 0 1"
```
```   828     by simp
```
```   829   show ?thesis using assms
```
```   830     apply (auto simp: has_contour_integral_def)
```
```   831     apply (drule has_integral_affinity01 [where m= "-1" and c=1])
```
```   832     apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
```
```   833     apply (drule has_integral_neg)
```
```   834     apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
```
```   835     apply (auto simp: *)
```
```   836     done
```
```   837 qed
```
```   838
```
```   839 lemma contour_integrable_reversepath:
```
```   840     "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
```
```   841   using has_contour_integral_reversepath contour_integrable_on_def by blast
```
```   842
```
```   843 lemma contour_integrable_reversepath_eq:
```
```   844     "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
```
```   845   using contour_integrable_reversepath valid_path_reversepath by fastforce
```
```   846
```
```   847 lemma contour_integral_reversepath:
```
```   848   assumes "valid_path g"
```
```   849     shows "contour_integral (reversepath g) f = - (contour_integral g f)"
```
```   850 proof (cases "f contour_integrable_on g")
```
```   851   case True then show ?thesis
```
```   852     by (simp add: assms contour_integral_unique has_contour_integral_integral has_contour_integral_reversepath)
```
```   853 next
```
```   854   case False then have "~ f contour_integrable_on (reversepath g)"
```
```   855     by (simp add: assms contour_integrable_reversepath_eq)
```
```   856   with False show ?thesis by (simp add: not_integrable_contour_integral)
```
```   857 qed
```
```   858
```
```   859
```
```   860 subsection\<open>Joining two paths together\<close>
```
```   861
```
```   862 lemma valid_path_join:
```
```   863   assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
```
```   864     shows "valid_path(g1 +++ g2)"
```
```   865 proof -
```
```   866   have "g1 1 = g2 0"
```
```   867     using assms by (auto simp: pathfinish_def pathstart_def)
```
```   868   moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
```
```   869     apply (rule piecewise_C1_differentiable_compose)
```
```   870     using assms
```
```   871     apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
```
```   872     apply (rule continuous_intros | simp)+
```
```   873     apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
```
```   874     done
```
```   875   moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
```
```   876     apply (rule piecewise_C1_differentiable_compose)
```
```   877     using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
```
```   878     by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
```
```   879              simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
```
```   880   ultimately show ?thesis
```
```   881     apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
```
```   882     apply (rule piecewise_C1_differentiable_cases)
```
```   883     apply (auto simp: o_def)
```
```   884     done
```
```   885 qed
```
```   886
```
```   887 lemma valid_path_join_D1:
```
```   888   fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
```
```   889   shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
```
```   890   unfolding valid_path_def
```
```   891   by (rule piecewise_C1_differentiable_D1)
```
```   892
```
```   893 lemma valid_path_join_D2:
```
```   894   fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
```
```   895   shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
```
```   896   unfolding valid_path_def
```
```   897   by (rule piecewise_C1_differentiable_D2)
```
```   898
```
```   899 lemma valid_path_join_eq [simp]:
```
```   900   fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
```
```   901   shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
```
```   902   using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast
```
```   903
```
```   904 lemma has_contour_integral_join:
```
```   905   assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
```
```   906           "valid_path g1" "valid_path g2"
```
```   907     shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
```
```   908 proof -
```
```   909   obtain s1 s2
```
```   910     where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
```
```   911       and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
```
```   912     using assms
```
```   913     by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   914   have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
```
```   915    and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
```
```   916     using assms
```
```   917     by (auto simp: has_contour_integral)
```
```   918   have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
```
```   919    and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
```
```   920     using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
```
```   921           has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
```
```   922     by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
```
```   923   have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
```
```   924             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
```
```   925             2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
```
```   926     apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>z - 1/2\<bar>"]])
```
```   927     apply (simp_all add: dist_real_def abs_if split: if_split_asm)
```
```   928     apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
```
```   929     apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
```
```   930     using s1
```
```   931     apply (auto simp: algebra_simps vector_derivative_works)
```
```   932     done
```
```   933   have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
```
```   934             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
```
```   935             2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
```
```   936     apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2 (2*x - 1))" and d = "\<bar>z - 1/2\<bar>"]])
```
```   937     apply (simp_all add: dist_real_def abs_if split: if_split_asm)
```
```   938     apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
```
```   939     apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
```
```   940     using s2
```
```   941     apply (auto simp: algebra_simps vector_derivative_works)
```
```   942     done
```
```   943   have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
```
```   944     apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
```
```   945     using s1
```
```   946     apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
```
```   947     apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
```
```   948     done
```
```   949   moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
```
```   950     apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
```
```   951     using s2
```
```   952     apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
```
```   953     apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
```
```   954     done
```
```   955   ultimately
```
```   956   show ?thesis
```
```   957     apply (simp add: has_contour_integral)
```
```   958     apply (rule has_integral_combine [where c = "1/2"], auto)
```
```   959     done
```
```   960 qed
```
```   961
```
```   962 lemma contour_integrable_joinI:
```
```   963   assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
```
```   964           "valid_path g1" "valid_path g2"
```
```   965     shows "f contour_integrable_on (g1 +++ g2)"
```
```   966   using assms
```
```   967   by (meson has_contour_integral_join contour_integrable_on_def)
```
```   968
```
```   969 lemma contour_integrable_joinD1:
```
```   970   assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
```
```   971     shows "f contour_integrable_on g1"
```
```   972 proof -
```
```   973   obtain s1
```
```   974     where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
```
```   975     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```   976   have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
```
```   977     using assms
```
```   978     apply (auto simp: contour_integrable_on)
```
```   979     apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
```
```   980     apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
```
```   981     done
```
```   982   then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
```
```   983     by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
```
```   984   have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
```
```   985             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
```
```   986             2 *\<^sub>R vector_derivative g1 (at z)"  for z
```
```   987     apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g1(2*x))" and d = "\<bar>(z-1)/2\<bar>"]])
```
```   988     apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
```
```   989     apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
```
```   990     using s1
```
```   991     apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
```
```   992     done
```
```   993   show ?thesis
```
```   994     using s1
```
```   995     apply (auto simp: contour_integrable_on)
```
```   996     apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
```
```   997     apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
```
```   998     done
```
```   999 qed
```
```  1000
```
```  1001 lemma contour_integrable_joinD2:
```
```  1002   assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
```
```  1003     shows "f contour_integrable_on g2"
```
```  1004 proof -
```
```  1005   obtain s2
```
```  1006     where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
```
```  1007     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```  1008   have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
```
```  1009     using assms
```
```  1010     apply (auto simp: contour_integrable_on)
```
```  1011     apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
```
```  1012     apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
```
```  1013     apply (simp add: image_affinity_atLeastAtMost_diff)
```
```  1014     done
```
```  1015   then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
```
```  1016                 integrable_on {0..1}"
```
```  1017     by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
```
```  1018   have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
```
```  1019             vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
```
```  1020             2 *\<^sub>R vector_derivative g2 (at z)" for z
```
```  1021     apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g2(2*x-1))" and d = "\<bar>z/2\<bar>"]])
```
```  1022     apply (simp_all add: field_simps dist_real_def abs_if split: if_split_asm)
```
```  1023     apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
```
```  1024     using s2
```
```  1025     apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
```
```  1026                       vector_derivative_works add_divide_distrib)
```
```  1027     done
```
```  1028   show ?thesis
```
```  1029     using s2
```
```  1030     apply (auto simp: contour_integrable_on)
```
```  1031     apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
```
```  1032     apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
```
```  1033     done
```
```  1034 qed
```
```  1035
```
```  1036 lemma contour_integrable_join [simp]:
```
```  1037   shows
```
```  1038     "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
```
```  1039      \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
```
```  1040 using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast
```
```  1041
```
```  1042 lemma contour_integral_join [simp]:
```
```  1043   shows
```
```  1044     "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
```
```  1045         \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
```
```  1046   by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)
```
```  1047
```
```  1048
```
```  1049 subsection\<open>Shifting the starting point of a (closed) path\<close>
```
```  1050
```
```  1051 lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
```
```  1052   by (auto simp: shiftpath_def)
```
```  1053
```
```  1054 lemma valid_path_shiftpath [intro]:
```
```  1055   assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```  1056     shows "valid_path(shiftpath a g)"
```
```  1057   using assms
```
```  1058   apply (auto simp: valid_path_def shiftpath_alt_def)
```
```  1059   apply (rule piecewise_C1_differentiable_cases)
```
```  1060   apply (auto simp: algebra_simps)
```
```  1061   apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
```
```  1062   apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
```
```  1063   apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
```
```  1064   apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
```
```  1065   done
```
```  1066
```
```  1067 lemma has_contour_integral_shiftpath:
```
```  1068   assumes f: "(f has_contour_integral i) g" "valid_path g"
```
```  1069       and a: "a \<in> {0..1}"
```
```  1070     shows "(f has_contour_integral i) (shiftpath a g)"
```
```  1071 proof -
```
```  1072   obtain s
```
```  1073     where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
```
```  1074     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```  1075   have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
```
```  1076     using assms by (auto simp: has_contour_integral)
```
```  1077   then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
```
```  1078                     integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
```
```  1079     apply (rule has_integral_unique)
```
```  1080     apply (subst add.commute)
```
```  1081     apply (subst integral_combine)
```
```  1082     using assms * integral_unique by auto
```
```  1083   { fix x
```
```  1084     have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
```
```  1085          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
```
```  1086       unfolding shiftpath_def
```
```  1087       apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
```
```  1088         apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
```
```  1089       apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
```
```  1090        apply (intro derivative_eq_intros | simp)+
```
```  1091       using g
```
```  1092        apply (drule_tac x="x+a" in bspec)
```
```  1093       using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
```
```  1094       done
```
```  1095   } note vd1 = this
```
```  1096   { fix x
```
```  1097     have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
```
```  1098           vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
```
```  1099       unfolding shiftpath_def
```
```  1100       apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x-1))" and d = "dist (1-a) x"]])
```
```  1101         apply (auto simp: field_simps dist_real_def abs_if split: if_split_asm)
```
```  1102       apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
```
```  1103        apply (intro derivative_eq_intros | simp)+
```
```  1104       using g
```
```  1105       apply (drule_tac x="x+a-1" in bspec)
```
```  1106       using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
```
```  1107       done
```
```  1108   } note vd2 = this
```
```  1109   have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
```
```  1110     using * a   by (fastforce intro: integrable_subinterval_real)
```
```  1111   have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
```
```  1112     apply (rule integrable_subinterval_real)
```
```  1113     using * a by auto
```
```  1114   have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
```
```  1115         has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
```
```  1116     apply (rule has_integral_spike_finite
```
```  1117              [where s = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
```
```  1118       using s apply blast
```
```  1119      using a apply (auto simp: algebra_simps vd1)
```
```  1120      apply (force simp: shiftpath_def add.commute)
```
```  1121     using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
```
```  1122     apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
```
```  1123     done
```
```  1124   moreover
```
```  1125   have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
```
```  1126         has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
```
```  1127     apply (rule has_integral_spike_finite
```
```  1128              [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))"])
```
```  1129       using s apply blast
```
```  1130      using a apply (auto simp: algebra_simps vd2)
```
```  1131      apply (force simp: shiftpath_def add.commute)
```
```  1132     using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
```
```  1133     apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
```
```  1134     apply (simp add: algebra_simps)
```
```  1135     done
```
```  1136   ultimately show ?thesis
```
```  1137     using a
```
```  1138     by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
```
```  1139 qed
```
```  1140
```
```  1141 lemma has_contour_integral_shiftpath_D:
```
```  1142   assumes "(f has_contour_integral i) (shiftpath a g)"
```
```  1143           "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```  1144     shows "(f has_contour_integral i) g"
```
```  1145 proof -
```
```  1146   obtain s
```
```  1147     where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
```
```  1148     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
```
```  1149   { fix x
```
```  1150     assume x: "0 < x" "x < 1" "x \<notin> s"
```
```  1151     then have gx: "g differentiable at x"
```
```  1152       using g by auto
```
```  1153     have "vector_derivative g (at x within {0..1}) =
```
```  1154           vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
```
```  1155       apply (rule vector_derivative_at_within_ivl
```
```  1156                   [OF has_vector_derivative_transform_within_open
```
```  1157                       [where f = "(shiftpath (1 - a) (shiftpath a g))" and s = "{0<..<1}-s"]])
```
```  1158       using s g assms x
```
```  1159       apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
```
```  1160                         vector_derivative_within_interior vector_derivative_works [symmetric])
```
```  1161       apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
```
```  1162       apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: if_split_asm)
```
```  1163       done
```
```  1164   } note vd = this
```
```  1165   have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
```
```  1166     using assms  by (auto intro!: has_contour_integral_shiftpath)
```
```  1167   show ?thesis
```
```  1168     apply (simp add: has_contour_integral_def)
```
```  1169     apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
```
```  1170     using s assms vd
```
```  1171     apply (auto simp: Path_Connected.shiftpath_shiftpath)
```
```  1172     done
```
```  1173 qed
```
```  1174
```
```  1175 lemma has_contour_integral_shiftpath_eq:
```
```  1176   assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```  1177     shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
```
```  1178   using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast
```
```  1179
```
```  1180 lemma contour_integrable_on_shiftpath_eq:
```
```  1181   assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```  1182     shows "f contour_integrable_on (shiftpath a g) \<longleftrightarrow> f contour_integrable_on g"
```
```  1183 using assms contour_integrable_on_def has_contour_integral_shiftpath_eq by auto
```
```  1184
```
```  1185 lemma contour_integral_shiftpath:
```
```  1186   assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
```
```  1187     shows "contour_integral (shiftpath a g) f = contour_integral g f"
```
```  1188    using assms
```
```  1189    by (simp add: contour_integral_def contour_integrable_on_def has_contour_integral_shiftpath_eq)
```
```  1190
```
```  1191
```
```  1192 subsection\<open>More about straight-line paths\<close>
```
```  1193
```
```  1194 lemma has_vector_derivative_linepath_within:
```
```  1195     "(linepath a b has_vector_derivative (b - a)) (at x within s)"
```
```  1196 apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
```
```  1197 apply (rule derivative_eq_intros | simp)+
```
```  1198 done
```
```  1199
```
```  1200 lemma vector_derivative_linepath_within:
```
```  1201     "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
```
```  1202   apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
```
```  1203   apply (auto simp: has_vector_derivative_linepath_within)
```
```  1204   done
```
```  1205
```
```  1206 lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
```
```  1207   by (simp add: has_vector_derivative_linepath_within vector_derivative_at)
```
```  1208
```
```  1209 lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
```
```  1210   apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
```
```  1211   apply (rule_tac x="{}" in exI)
```
```  1212   apply (simp add: differentiable_on_def differentiable_def)
```
```  1213   using has_vector_derivative_def has_vector_derivative_linepath_within
```
```  1214   apply (fastforce simp add: continuous_on_eq_continuous_within)
```
```  1215   done
```
```  1216
```
```  1217 lemma has_contour_integral_linepath:
```
```  1218   shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
```
```  1219          ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
```
```  1220   by (simp add: has_contour_integral vector_derivative_linepath_at)
```
```  1221
```
```  1222 lemma linepath_in_path:
```
```  1223   shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
```
```  1224   by (auto simp: segment linepath_def)
```
```  1225
```
```  1226 lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
```
```  1227   by (auto simp: segment linepath_def)
```
```  1228
```
```  1229 lemma linepath_in_convex_hull:
```
```  1230     fixes x::real
```
```  1231     assumes a: "a \<in> convex hull s"
```
```  1232         and b: "b \<in> convex hull s"
```
```  1233         and x: "0\<le>x" "x\<le>1"
```
```  1234        shows "linepath a b x \<in> convex hull s"
```
```  1235   apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
```
```  1236   using x
```
```  1237   apply (auto simp: linepath_image_01 [symmetric])
```
```  1238   done
```
```  1239
```
```  1240 lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
```
```  1241   by (simp add: linepath_def)
```
```  1242
```
```  1243 lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
```
```  1244   by (simp add: linepath_def)
```
```  1245
```
```  1246 lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
```
```  1247   by (simp add: has_contour_integral_linepath)
```
```  1248
```
```  1249 lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
```
```  1250   using has_contour_integral_trivial contour_integral_unique by blast
```
```  1251
```
```  1252
```
```  1253 subsection\<open>Relation to subpath construction\<close>
```
```  1254
```
```  1255 lemma valid_path_subpath:
```
```  1256   fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
```
```  1257   assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
```
```  1258     shows "valid_path(subpath u v g)"
```
```  1259 proof (cases "v=u")
```
```  1260   case True
```
```  1261   then show ?thesis
```
```  1262     unfolding valid_path_def subpath_def
```
```  1263     by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
```
```  1264 next
```
```  1265   case False
```
```  1266   have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
```
```  1267     apply (rule piecewise_C1_differentiable_compose)
```
```  1268     apply (simp add: C1_differentiable_imp_piecewise)
```
```  1269      apply (simp add: image_affinity_atLeastAtMost)
```
```  1270     using assms False
```
```  1271     apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
```
```  1272     apply (subst Int_commute)
```
```  1273     apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
```
```  1274     done
```
```  1275   then show ?thesis
```
```  1276     by (auto simp: o_def valid_path_def subpath_def)
```
```  1277 qed
```
```  1278
```
```  1279 lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
```
```  1280   by (simp add: has_contour_integral subpath_def)
```
```  1281
```
```  1282 lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
```
```  1283   using has_contour_integral_subpath_refl contour_integrable_on_def by blast
```
```  1284
```
```  1285 lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
```
```  1286   by (simp add: has_contour_integral_subpath_refl contour_integral_unique)
```
```  1287
```
```  1288 lemma has_contour_integral_subpath:
```
```  1289   assumes f: "f contour_integrable_on g" and g: "valid_path g"
```
```  1290       and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
```
```  1291     shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
```
```  1292            (subpath u v g)"
```
```  1293 proof (cases "v=u")
```
```  1294   case True
```
```  1295   then show ?thesis
```
```  1296     using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
```
```  1297 next
```
```  1298   case False
```
```  1299   obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
```
```  1300     using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
```
```  1301   have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
```
```  1302             has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
```
```  1303            {0..1}"
```
```  1304     using f uv
```
```  1305     apply (simp add: contour_integrable_on subpath_def has_contour_integral)
```
```  1306     apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
```
```  1307     apply (simp_all add: has_integral_integral)
```
```  1308     apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
```
```  1309     apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
```
```  1310     apply (simp add: divide_simps False)
```
```  1311     done
```
```  1312   { fix x
```
```  1313     have "x \<in> {0..1} \<Longrightarrow>
```
```  1314            x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
```
```  1315            vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
```
```  1316       apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
```
```  1317       apply (intro derivative_eq_intros | simp)+
```
```  1318       apply (cut_tac s [of "(v - u) * x + u"])
```
```  1319       using uv mult_left_le [of x "v-u"]
```
```  1320       apply (auto simp:  vector_derivative_works)
```
```  1321       done
```
```  1322   } note vd = this
```
```  1323   show ?thesis
```
```  1324     apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
```
```  1325     using fs assms
```
```  1326     apply (simp add: False subpath_def has_contour_integral)
```
```  1327     apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
```
```  1328     apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
```
```  1329     done
```
```  1330 qed
```
```  1331
```
```  1332 lemma contour_integrable_subpath:
```
```  1333   assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
```
```  1334     shows "f contour_integrable_on (subpath u v g)"
```
```  1335   apply (cases u v rule: linorder_class.le_cases)
```
```  1336    apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
```
```  1337   apply (subst reversepath_subpath [symmetric])
```
```  1338   apply (rule contour_integrable_reversepath)
```
```  1339    using assms apply (blast intro: valid_path_subpath)
```
```  1340   apply (simp add: contour_integrable_on_def)
```
```  1341   using assms apply (blast intro: has_contour_integral_subpath)
```
```  1342   done
```
```  1343
```
```  1344 lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
```
```  1345   by blast
```
```  1346
```
```  1347 lemma has_integral_contour_integral_subpath:
```
```  1348   assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
```
```  1349     shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
```
```  1350             has_integral  contour_integral (subpath u v g) f) {u..v}"
```
```  1351   using assms
```
```  1352   apply (auto simp: has_integral_integrable_integral)
```
```  1353   apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
```
```  1354   apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
```
```  1355   done
```
```  1356
```
```  1357 lemma contour_integral_subcontour_integral:
```
```  1358   assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
```
```  1359     shows "contour_integral (subpath u v g) f =
```
```  1360            integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
```
```  1361   using assms has_contour_integral_subpath contour_integral_unique by blast
```
```  1362
```
```  1363 lemma contour_integral_subpath_combine_less:
```
```  1364   assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
```
```  1365           "u<v" "v<w"
```
```  1366     shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
```
```  1367            contour_integral (subpath u w g) f"
```
```  1368   using assms apply (auto simp: contour_integral_subcontour_integral)
```
```  1369   apply (rule integral_combine, auto)
```
```  1370   apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
```
```  1371   apply (auto simp: contour_integrable_on)
```
```  1372   done
```
```  1373
```
```  1374 lemma contour_integral_subpath_combine:
```
```  1375   assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
```
```  1376     shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
```
```  1377            contour_integral (subpath u w g) f"
```
```  1378 proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
```
```  1379   case True
```
```  1380     have *: "subpath v u g = reversepath(subpath u v g) \<and>
```
```  1381              subpath w u g = reversepath(subpath u w g) \<and>
```
```  1382              subpath w v g = reversepath(subpath v w g)"
```
```  1383       by (auto simp: reversepath_subpath)
```
```  1384     have "u < v \<and> v < w \<or>
```
```  1385           u < w \<and> w < v \<or>
```
```  1386           v < u \<and> u < w \<or>
```
```  1387           v < w \<and> w < u \<or>
```
```  1388           w < u \<and> u < v \<or>
```
```  1389           w < v \<and> v < u"
```
```  1390       using True assms by linarith
```
```  1391     with assms show ?thesis
```
```  1392       using contour_integral_subpath_combine_less [of f g u v w]
```
```  1393             contour_integral_subpath_combine_less [of f g u w v]
```
```  1394             contour_integral_subpath_combine_less [of f g v u w]
```
```  1395             contour_integral_subpath_combine_less [of f g v w u]
```
```  1396             contour_integral_subpath_combine_less [of f g w u v]
```
```  1397             contour_integral_subpath_combine_less [of f g w v u]
```
```  1398       apply simp
```
```  1399       apply (elim disjE)
```
```  1400       apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
```
```  1401                    valid_path_reversepath valid_path_subpath algebra_simps)
```
```  1402       done
```
```  1403 next
```
```  1404   case False
```
```  1405   then show ?thesis
```
```  1406     apply (auto simp: contour_integral_subpath_refl)
```
```  1407     using assms
```
```  1408     by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath)
```
```  1409 qed
```
```  1410
```
```  1411 lemma contour_integral_integral:
```
```  1412      "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
```
```  1413   by (simp add: contour_integral_def integral_def has_contour_integral contour_integrable_on)
```
```  1414
```
```  1415
```
```  1416 text\<open>Cauchy's theorem where there's a primitive\<close>
```
```  1417
```
```  1418 lemma contour_integral_primitive_lemma:
```
```  1419   fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
```
```  1420   assumes "a \<le> b"
```
```  1421       and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
```
```  1422       and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
```
```  1423     shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
```
```  1424              has_integral (f(g b) - f(g a))) {a..b}"
```
```  1425 proof -
```
```  1426   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"
```
```  1427     using assms by (auto simp: piecewise_differentiable_on_def)
```
```  1428   have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
```
```  1429     apply (rule continuous_on_compose [OF cg, unfolded o_def])
```
```  1430     using assms
```
```  1431     apply (metis field_differentiable_def field_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
```
```  1432     done
```
```  1433   { fix x::real
```
```  1434     assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
```
```  1435     then have "g differentiable at x within {a..b}"
```
```  1436       using k by (simp add: differentiable_at_withinI)
```
```  1437     then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
```
```  1438       by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
```
```  1439     then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
```
```  1440       by (simp add: has_vector_derivative_def scaleR_conv_of_real)
```
```  1441     have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
```
```  1442       using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
```
```  1443     then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
```
```  1444       by (simp add: has_field_derivative_def)
```
```  1445     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})"
```
```  1446       using diff_chain_within [OF gdiff fdiff]
```
```  1447       by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
```
```  1448   } note * = this
```
```  1449   show ?thesis
```
```  1450     apply (rule fundamental_theorem_of_calculus_interior_strong)
```
```  1451     using k assms cfg *
```
```  1452     apply (auto simp: at_within_closed_interval)
```
```  1453     done
```
```  1454 qed
```
```  1455
```
```  1456 lemma contour_integral_primitive:
```
```  1457   assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
```
```  1458       and "valid_path g" "path_image g \<subseteq> s"
```
```  1459     shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
```
```  1460   using assms
```
```  1461   apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
```
```  1462   apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
```
```  1463   done
```
```  1464
```
```  1465 corollary Cauchy_theorem_primitive:
```
```  1466   assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
```
```  1467       and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
```
```  1468     shows "(f' has_contour_integral 0) g"
```
```  1469   using assms
```
```  1470   by (metis diff_self contour_integral_primitive)
```
```  1471
```
```  1472
```
```  1473 text\<open>Existence of path integral for continuous function\<close>
```
```  1474 lemma contour_integrable_continuous_linepath:
```
```  1475   assumes "continuous_on (closed_segment a b) f"
```
```  1476   shows "f contour_integrable_on (linepath a b)"
```
```  1477 proof -
```
```  1478   have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
```
```  1479     apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
```
```  1480     apply (rule continuous_intros | simp add: assms)+
```
```  1481     done
```
```  1482   then show ?thesis
```
```  1483     apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
```
```  1484     apply (rule integrable_continuous [of 0 "1::real", simplified])
```
```  1485     apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
```
```  1486     apply (auto simp: vector_derivative_linepath_within)
```
```  1487     done
```
```  1488 qed
```
```  1489
```
```  1490 lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
```
```  1491   by (rule has_derivative_imp_has_field_derivative)
```
```  1492      (rule derivative_intros | simp)+
```
```  1493
```
```  1494 lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
```
```  1495   apply (rule contour_integral_unique)
```
```  1496   using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
```
```  1497   apply (auto simp: field_simps has_field_der_id)
```
```  1498   done
```
```  1499
```
```  1500 lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
```
```  1501   by (simp add: continuous_on_const contour_integrable_continuous_linepath)
```
```  1502
```
```  1503 lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
```
```  1504   by (simp add: continuous_on_id contour_integrable_continuous_linepath)
```
```  1505
```
```  1506
```
```  1507 subsection\<open>Arithmetical combining theorems\<close>
```
```  1508
```
```  1509 lemma has_contour_integral_neg:
```
```  1510     "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
```
```  1511   by (simp add: has_integral_neg has_contour_integral_def)
```
```  1512
```
```  1513 lemma has_contour_integral_add:
```
```  1514     "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
```
```  1515      \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
```
```  1516   by (simp add: has_integral_add has_contour_integral_def algebra_simps)
```
```  1517
```
```  1518 lemma has_contour_integral_diff:
```
```  1519   "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
```
```  1520          \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
```
```  1521   by (simp add: has_integral_sub has_contour_integral_def algebra_simps)
```
```  1522
```
```  1523 lemma has_contour_integral_lmul:
```
```  1524   "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
```
```  1525 apply (simp add: has_contour_integral_def)
```
```  1526 apply (drule has_integral_mult_right)
```
```  1527 apply (simp add: algebra_simps)
```
```  1528 done
```
```  1529
```
```  1530 lemma has_contour_integral_rmul:
```
```  1531   "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
```
```  1532 apply (drule has_contour_integral_lmul)
```
```  1533 apply (simp add: mult.commute)
```
```  1534 done
```
```  1535
```
```  1536 lemma has_contour_integral_div:
```
```  1537   "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
```
```  1538   by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)
```
```  1539
```
```  1540 lemma has_contour_integral_eq:
```
```  1541     "\<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"
```
```  1542 apply (simp add: path_image_def has_contour_integral_def)
```
```  1543 by (metis (no_types, lifting) image_eqI has_integral_eq)
```
```  1544
```
```  1545 lemma has_contour_integral_bound_linepath:
```
```  1546   assumes "(f has_contour_integral i) (linepath a b)"
```
```  1547           "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
```
```  1548     shows "norm i \<le> B * norm(b - a)"
```
```  1549 proof -
```
```  1550   { fix x::real
```
```  1551     assume x: "0 \<le> x" "x \<le> 1"
```
```  1552   have "norm (f (linepath a b x)) *
```
```  1553         norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
```
```  1554     by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
```
```  1555   } note * = this
```
```  1556   have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
```
```  1557     apply (rule has_integral_bound
```
```  1558        [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
```
```  1559     using assms * unfolding has_contour_integral_def
```
```  1560     apply (auto simp: norm_mult)
```
```  1561     done
```
```  1562   then show ?thesis
```
```  1563     by (auto simp: content_real)
```
```  1564 qed
```
```  1565
```
```  1566 (*UNUSED
```
```  1567 lemma has_contour_integral_bound_linepath_strong:
```
```  1568   fixes a :: real and f :: "complex \<Rightarrow> real"
```
```  1569   assumes "(f has_contour_integral i) (linepath a b)"
```
```  1570           "finite k"
```
```  1571           "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
```
```  1572     shows "norm i \<le> B*norm(b - a)"
```
```  1573 *)
```
```  1574
```
```  1575 lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
```
```  1576   unfolding has_contour_integral_linepath
```
```  1577   by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)
```
```  1578
```
```  1579 lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
```
```  1580   by (simp add: has_contour_integral_def)
```
```  1581
```
```  1582 lemma has_contour_integral_is_0:
```
```  1583     "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
```
```  1584   by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto
```
```  1585
```
```  1586 lemma has_contour_integral_sum:
```
```  1587     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
```
```  1588      \<Longrightarrow> ((\<lambda>x. sum (\<lambda>a. f a x) s) has_contour_integral sum i s) p"
```
```  1589   by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)
```
```  1590
```
```  1591
```
```  1592 subsection \<open>Operations on path integrals\<close>
```
```  1593
```
```  1594 lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
```
```  1595   by (rule contour_integral_unique [OF has_contour_integral_const_linepath])
```
```  1596
```
```  1597 lemma contour_integral_neg:
```
```  1598     "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
```
```  1599   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)
```
```  1600
```
```  1601 lemma contour_integral_add:
```
```  1602     "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
```
```  1603                 contour_integral g f1 + contour_integral g f2"
```
```  1604   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)
```
```  1605
```
```  1606 lemma contour_integral_diff:
```
```  1607     "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
```
```  1608                 contour_integral g f1 - contour_integral g f2"
```
```  1609   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)
```
```  1610
```
```  1611 lemma contour_integral_lmul:
```
```  1612   shows "f contour_integrable_on g
```
```  1613            \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
```
```  1614   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)
```
```  1615
```
```  1616 lemma contour_integral_rmul:
```
```  1617   shows "f contour_integrable_on g
```
```  1618         \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
```
```  1619   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)
```
```  1620
```
```  1621 lemma contour_integral_div:
```
```  1622   shows "f contour_integrable_on g
```
```  1623         \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
```
```  1624   by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)
```
```  1625
```
```  1626 lemma contour_integral_eq:
```
```  1627     "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
```
```  1628   apply (simp add: contour_integral_def)
```
```  1629   using has_contour_integral_eq
```
```  1630   by (metis contour_integral_unique has_contour_integral_integrable has_contour_integral_integral)
```
```  1631
```
```  1632 lemma contour_integral_eq_0:
```
```  1633     "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
```
```  1634   by (simp add: has_contour_integral_is_0 contour_integral_unique)
```
```  1635
```
```  1636 lemma contour_integral_bound_linepath:
```
```  1637   shows
```
```  1638     "\<lbrakk>f contour_integrable_on (linepath a b);
```
```  1639       0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
```
```  1640      \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
```
```  1641   apply (rule has_contour_integral_bound_linepath [of f])
```
```  1642   apply (auto simp: has_contour_integral_integral)
```
```  1643   done
```
```  1644
```
```  1645 lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
```
```  1646   by (simp add: contour_integral_unique has_contour_integral_0)
```
```  1647
```
```  1648 lemma contour_integral_sum:
```
```  1649     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
```
```  1650      \<Longrightarrow> contour_integral p (\<lambda>x. sum (\<lambda>a. f a x) s) = sum (\<lambda>a. contour_integral p (f a)) s"
```
```  1651   by (auto simp: contour_integral_unique has_contour_integral_sum has_contour_integral_integral)
```
```  1652
```
```  1653 lemma contour_integrable_eq:
```
```  1654     "\<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"
```
```  1655   unfolding contour_integrable_on_def
```
```  1656   by (metis has_contour_integral_eq)
```
```  1657
```
```  1658
```
```  1659 subsection \<open>Arithmetic theorems for path integrability\<close>
```
```  1660
```
```  1661 lemma contour_integrable_neg:
```
```  1662     "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
```
```  1663   using has_contour_integral_neg contour_integrable_on_def by blast
```
```  1664
```
```  1665 lemma contour_integrable_add:
```
```  1666     "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
```
```  1667   using has_contour_integral_add contour_integrable_on_def
```
```  1668   by fastforce
```
```  1669
```
```  1670 lemma contour_integrable_diff:
```
```  1671     "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
```
```  1672   using has_contour_integral_diff contour_integrable_on_def
```
```  1673   by fastforce
```
```  1674
```
```  1675 lemma contour_integrable_lmul:
```
```  1676     "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
```
```  1677   using has_contour_integral_lmul contour_integrable_on_def
```
```  1678   by fastforce
```
```  1679
```
```  1680 lemma contour_integrable_rmul:
```
```  1681     "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
```
```  1682   using has_contour_integral_rmul contour_integrable_on_def
```
```  1683   by fastforce
```
```  1684
```
```  1685 lemma contour_integrable_div:
```
```  1686     "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
```
```  1687   using has_contour_integral_div contour_integrable_on_def
```
```  1688   by fastforce
```
```  1689
```
```  1690 lemma contour_integrable_sum:
```
```  1691     "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
```
```  1692      \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) s) contour_integrable_on p"
```
```  1693    unfolding contour_integrable_on_def
```
```  1694    by (metis has_contour_integral_sum)
```
```  1695
```
```  1696
```
```  1697 subsection\<open>Reversing a path integral\<close>
```
```  1698
```
```  1699 lemma has_contour_integral_reverse_linepath:
```
```  1700     "(f has_contour_integral i) (linepath a b)
```
```  1701      \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
```
```  1702   using has_contour_integral_reversepath valid_path_linepath by fastforce
```
```  1703
```
```  1704 lemma contour_integral_reverse_linepath:
```
```  1705     "continuous_on (closed_segment a b) f
```
```  1706      \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
```
```  1707 apply (rule contour_integral_unique)
```
```  1708 apply (rule has_contour_integral_reverse_linepath)
```
```  1709 by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)
```
```  1710
```
```  1711
```
```  1712 (* Splitting a path integral in a flat way.*)
```
```  1713
```
```  1714 lemma has_contour_integral_split:
```
```  1715   assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
```
```  1716       and k: "0 \<le> k" "k \<le> 1"
```
```  1717       and c: "c - a = k *\<^sub>R (b - a)"
```
```  1718     shows "(f has_contour_integral (i + j)) (linepath a b)"
```
```  1719 proof (cases "k = 0 \<or> k = 1")
```
```  1720   case True
```
```  1721   then show ?thesis
```
```  1722     using assms
```
```  1723     apply auto
```
```  1724     apply (metis add.left_neutral has_contour_integral_trivial has_contour_integral_unique)
```
```  1725     apply (metis add.right_neutral has_contour_integral_trivial has_contour_integral_unique)
```
```  1726     done
```
```  1727 next
```
```  1728   case False
```
```  1729   then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
```
```  1730     using assms by auto
```
```  1731   have c': "c = k *\<^sub>R (b - a) + a"
```
```  1732     by (metis diff_add_cancel c)
```
```  1733   have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
```
```  1734     by (simp add: algebra_simps c')
```
```  1735   { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
```
```  1736     have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
```
```  1737       using False
```
```  1738       apply (simp add: c' algebra_simps)
```
```  1739       apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
```
```  1740       done
```
```  1741     have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
```
```  1742       using * k
```
```  1743       apply -
```
```  1744       apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
```
```  1745       apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
```
```  1746       apply (drule has_integral_cmul [where c = "inverse k"])
```
```  1747       apply (simp add: has_integral_cmul)
```
```  1748       done
```
```  1749   } note fi = this
```
```  1750   { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
```
```  1751     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)"
```
```  1752       using k
```
```  1753       apply (simp add: c' field_simps)
```
```  1754       apply (simp add: scaleR_conv_of_real divide_simps)
```
```  1755       apply (simp add: field_simps)
```
```  1756       done
```
```  1757     have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
```
```  1758       using * k
```
```  1759       apply -
```
```  1760       apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
```
```  1761       apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
```
```  1762       apply (drule has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
```
```  1763       apply (simp add: has_integral_cmul)
```
```  1764       done
```
```  1765   } note fj = this
```
```  1766   show ?thesis
```
```  1767     using f k
```
```  1768     apply (simp add: has_contour_integral_linepath)
```
```  1769     apply (simp add: linepath_def)
```
```  1770     apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
```
```  1771     done
```
```  1772 qed
```
```  1773
```
```  1774 lemma continuous_on_closed_segment_transform:
```
```  1775   assumes f: "continuous_on (closed_segment a b) f"
```
```  1776       and k: "0 \<le> k" "k \<le> 1"
```
```  1777       and c: "c - a = k *\<^sub>R (b - a)"
```
```  1778     shows "continuous_on (closed_segment a c) f"
```
```  1779 proof -
```
```  1780   have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
```
```  1781     using c by (simp add: algebra_simps)
```
```  1782   show "continuous_on (closed_segment a c) f"
```
```  1783     apply (rule continuous_on_subset [OF f])
```
```  1784     apply (simp add: segment_convex_hull)
```
```  1785     apply (rule convex_hull_subset)
```
```  1786     using assms
```
```  1787     apply (auto simp: hull_inc c' convexD_alt)
```
```  1788     done
```
```  1789 qed
```
```  1790
```
```  1791 lemma contour_integral_split:
```
```  1792   assumes f: "continuous_on (closed_segment a b) f"
```
```  1793       and k: "0 \<le> k" "k \<le> 1"
```
```  1794       and c: "c - a = k *\<^sub>R (b - a)"
```
```  1795     shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
```
```  1796 proof -
```
```  1797   have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
```
```  1798     using c by (simp add: algebra_simps)
```
```  1799   have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
```
```  1800     apply (rule_tac [!] continuous_on_subset [OF f])
```
```  1801     apply (simp_all add: segment_convex_hull)
```
```  1802     apply (rule_tac [!] convex_hull_subset)
```
```  1803     using assms
```
```  1804     apply (auto simp: hull_inc c' convexD_alt)
```
```  1805     done
```
```  1806   show ?thesis
```
```  1807     apply (rule contour_integral_unique)
```
```  1808     apply (rule has_contour_integral_split [OF has_contour_integral_integral has_contour_integral_integral k c])
```
```  1809     apply (rule contour_integrable_continuous_linepath *)+
```
```  1810     done
```
```  1811 qed
```
```  1812
```
```  1813 lemma contour_integral_split_linepath:
```
```  1814   assumes f: "continuous_on (closed_segment a b) f"
```
```  1815       and c: "c \<in> closed_segment a b"
```
```  1816     shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
```
```  1817   using c
```
```  1818   by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])
```
```  1819
```
```  1820 (* The special case of midpoints used in the main quadrisection.*)
```
```  1821
```
```  1822 lemma has_contour_integral_midpoint:
```
```  1823   assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
```
```  1824           "(f has_contour_integral j) (linepath (midpoint a b) b)"
```
```  1825     shows "(f has_contour_integral (i + j)) (linepath a b)"
```
```  1826   apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
```
```  1827   using assms
```
```  1828   apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
```
```  1829   done
```
```  1830
```
```  1831 lemma contour_integral_midpoint:
```
```  1832    "continuous_on (closed_segment a b) f
```
```  1833     \<Longrightarrow> contour_integral (linepath a b) f =
```
```  1834         contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
```
```  1835   apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
```
```  1836   apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
```
```  1837   done
```
```  1838
```
```  1839
```
```  1840 text\<open>A couple of special case lemmas that are useful below\<close>
```
```  1841
```
```  1842 lemma triangle_linear_has_chain_integral:
```
```  1843     "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  1844   apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
```
```  1845   apply (auto intro!: derivative_eq_intros)
```
```  1846   done
```
```  1847
```
```  1848 lemma has_chain_integral_chain_integral3:
```
```  1849      "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
```
```  1850       \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
```
```  1851   apply (subst contour_integral_unique [symmetric], assumption)
```
```  1852   apply (drule has_contour_integral_integrable)
```
```  1853   apply (simp add: valid_path_join)
```
```  1854   done
```
```  1855
```
```  1856 lemma has_chain_integral_chain_integral4:
```
```  1857      "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
```
```  1858       \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f + contour_integral (linepath d e) f = i"
```
```  1859   apply (subst contour_integral_unique [symmetric], assumption)
```
```  1860   apply (drule has_contour_integral_integrable)
```
```  1861   apply (simp add: valid_path_join)
```
```  1862   done
```
```  1863
```
```  1864 subsection\<open>Reversing the order in a double path integral\<close>
```
```  1865
```
```  1866 text\<open>The condition is stronger than needed but it's often true in typical situations\<close>
```
```  1867
```
```  1868 lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
```
```  1869   by (auto simp: cbox_Pair_eq)
```
```  1870
```
```  1871 lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
```
```  1872   by (auto simp: cbox_Pair_eq)
```
```  1873
```
```  1874 lemma contour_integral_swap:
```
```  1875   assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
```
```  1876       and vp:    "valid_path g" "valid_path h"
```
```  1877       and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
```
```  1878       and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
```
```  1879   shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
```
```  1880          contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
```
```  1881 proof -
```
```  1882   have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
```
```  1883     using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
```
```  1884   have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g x, h t))"
```
```  1885     by (rule ext) simp
```
```  1886   have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g t, h x))"
```
```  1887     by (rule ext) simp
```
```  1888   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)"
```
```  1889     by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
```
```  1890   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)"
```
```  1891     by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
```
```  1892   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}"
```
```  1893     apply (rule integrable_continuous_real)
```
```  1894     apply (rule continuous_on_mult [OF _ gvcon])
```
```  1895     apply (subst fgh2)
```
```  1896     apply (rule fcon_im2 gcon continuous_intros | simp)+
```
```  1897     done
```
```  1898   have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
```
```  1899     by auto
```
```  1900   then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
```
```  1901     apply (rule ssubst)
```
```  1902     apply (rule continuous_intros | simp add: gvcon)+
```
```  1903     done
```
```  1904   have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
```
```  1905     by auto
```
```  1906   then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
```
```  1907     apply (rule ssubst)
```
```  1908     apply (rule continuous_intros | simp add: hvcon)+
```
```  1909     done
```
```  1910   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))"
```
```  1911     by auto
```
```  1912   then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
```
```  1913     apply (rule ssubst)
```
```  1914     apply (rule gcon hcon continuous_intros | simp)+
```
```  1915     apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
```
```  1916     done
```
```  1917   have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
```
```  1918         integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
```
```  1919     apply (rule integral_cong [OF contour_integral_rmul [symmetric]])
```
```  1920     apply (clarsimp simp: contour_integrable_on)
```
```  1921     apply (rule integrable_continuous_real)
```
```  1922     apply (rule continuous_on_mult [OF _ hvcon])
```
```  1923     apply (subst fgh1)
```
```  1924     apply (rule fcon_im1 hcon continuous_intros | simp)+
```
```  1925     done
```
```  1926   also have "... = integral {0..1}
```
```  1927                      (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
```
```  1928     apply (simp only: contour_integral_integral)
```
```  1929     apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
```
```  1930      apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
```
```  1931     unfolding integral_mult_left [symmetric]
```
```  1932     apply (simp only: mult_ac)
```
```  1933     done
```
```  1934   also have "... = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
```
```  1935     apply (simp add: contour_integral_integral)
```
```  1936     apply (rule integral_cong)
```
```  1937     unfolding integral_mult_left [symmetric]
```
```  1938     apply (simp add: algebra_simps)
```
```  1939     done
```
```  1940   finally show ?thesis
```
```  1941     by (simp add: contour_integral_integral)
```
```  1942 qed
```
```  1943
```
```  1944
```
```  1945 subsection\<open>The key quadrisection step\<close>
```
```  1946
```
```  1947 lemma norm_sum_half:
```
```  1948   assumes "norm(a + b) >= e"
```
```  1949     shows "norm a >= e/2 \<or> norm b >= e/2"
```
```  1950 proof -
```
```  1951   have "e \<le> norm (- a - b)"
```
```  1952     by (simp add: add.commute assms norm_minus_commute)
```
```  1953   thus ?thesis
```
```  1954     using norm_triangle_ineq4 order_trans by fastforce
```
```  1955 qed
```
```  1956
```
```  1957 lemma norm_sum_lemma:
```
```  1958   assumes "e \<le> norm (a + b + c + d)"
```
```  1959     shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
```
```  1960 proof -
```
```  1961   have "e \<le> norm ((a + b) + (c + d))" using assms
```
```  1962     by (simp add: algebra_simps)
```
```  1963   then show ?thesis
```
```  1964     by (auto dest!: norm_sum_half)
```
```  1965 qed
```
```  1966
```
```  1967 lemma Cauchy_theorem_quadrisection:
```
```  1968   assumes f: "continuous_on (convex hull {a,b,c}) f"
```
```  1969       and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
```
```  1970       and e: "e * K^2 \<le>
```
```  1971               norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
```
```  1972   shows "\<exists>a' b' c'.
```
```  1973            a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
```
```  1974            dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
```
```  1975            e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
```
```  1976 proof -
```
```  1977   note divide_le_eq_numeral1 [simp del]
```
```  1978   define a' where "a' = midpoint b c"
```
```  1979   define b' where "b' = midpoint c a"
```
```  1980   define c' where "c' = midpoint a b"
```
```  1981   have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
```
```  1982     using f continuous_on_subset segments_subset_convex_hull by metis+
```
```  1983   have fcont': "continuous_on (closed_segment c' b') f"
```
```  1984                "continuous_on (closed_segment a' c') f"
```
```  1985                "continuous_on (closed_segment b' a') f"
```
```  1986     unfolding a'_def b'_def c'_def
```
```  1987     apply (rule continuous_on_subset [OF f],
```
```  1988            metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
```
```  1989     done
```
```  1990   let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
```
```  1991   have *: "?pathint a b + ?pathint b c + ?pathint c a =
```
```  1992           (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
```
```  1993           (?pathint a' c' + ?pathint c' b + ?pathint b a') +
```
```  1994           (?pathint a' c + ?pathint c b' + ?pathint b' a') +
```
```  1995           (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
```
```  1996     apply (simp add: fcont' contour_integral_reverse_linepath)
```
```  1997     apply (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
```
```  1998     done
```
```  1999   have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
```
```  2000     by (metis left_diff_distrib mult.commute norm_mult_numeral1)
```
```  2001   have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
```
```  2002     by (simp add: norm_minus_commute)
```
```  2003   consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
```
```  2004            "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
```
```  2005            "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
```
```  2006            "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
```
```  2007     using assms
```
```  2008     apply (simp only: *)
```
```  2009     apply (blast intro: that dest!: norm_sum_lemma)
```
```  2010     done
```
```  2011   then show ?thesis
```
```  2012   proof cases
```
```  2013     case 1 then show ?thesis
```
```  2014       apply (rule_tac x=a in exI)
```
```  2015       apply (rule exI [where x=c'])
```
```  2016       apply (rule exI [where x=b'])
```
```  2017       using assms
```
```  2018       apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
```
```  2019       apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
```
```  2020       done
```
```  2021   next
```
```  2022     case 2 then show ?thesis
```
```  2023       apply (rule_tac x=a' in exI)
```
```  2024       apply (rule exI [where x=c'])
```
```  2025       apply (rule exI [where x=b])
```
```  2026       using assms
```
```  2027       apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
```
```  2028       apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
```
```  2029       done
```
```  2030   next
```
```  2031     case 3 then show ?thesis
```
```  2032       apply (rule_tac x=a' in exI)
```
```  2033       apply (rule exI [where x=c])
```
```  2034       apply (rule exI [where x=b'])
```
```  2035       using assms
```
```  2036       apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
```
```  2037       apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
```
```  2038       done
```
```  2039   next
```
```  2040     case 4 then show ?thesis
```
```  2041       apply (rule_tac x=a' in exI)
```
```  2042       apply (rule exI [where x=b'])
```
```  2043       apply (rule exI [where x=c'])
```
```  2044       using assms
```
```  2045       apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
```
```  2046       apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
```
```  2047       done
```
```  2048   qed
```
```  2049 qed
```
```  2050
```
```  2051 subsection\<open>Cauchy's theorem for triangles\<close>
```
```  2052
```
```  2053 lemma triangle_points_closer:
```
```  2054   fixes a::complex
```
```  2055   shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
```
```  2056          \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
```
```  2057              norm(x - y) \<le> norm(b - c) \<or>
```
```  2058              norm(x - y) \<le> norm(c - a)"
```
```  2059   using simplex_extremal_le [of "{a,b,c}"]
```
```  2060   by (auto simp: norm_minus_commute)
```
```  2061
```
```  2062 lemma holomorphic_point_small_triangle:
```
```  2063   assumes x: "x \<in> s"
```
```  2064       and f: "continuous_on s f"
```
```  2065       and cd: "f field_differentiable (at x within s)"
```
```  2066       and e: "0 < e"
```
```  2067     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>
```
```  2068               x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
```
```  2069               \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
```
```  2070                        contour_integral(linepath c a) f)
```
```  2071                   \<le> e*(dist a b + dist b c + dist c a)^2"
```
```  2072            (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
```
```  2073 proof -
```
```  2074   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>
```
```  2075                      \<Longrightarrow> a \<le> e*(x + y + z)^2"
```
```  2076     by (simp add: algebra_simps power2_eq_square)
```
```  2077   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"
```
```  2078              for x::real and a b c
```
```  2079     by linarith
```
```  2080   have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
```
```  2081               if "convex hull {a, b, c} \<subseteq> s" for a b c
```
```  2082     using segments_subset_convex_hull that
```
```  2083     by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
```
```  2084   note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
```
```  2085   { fix f' a b c d
```
```  2086     assume d: "0 < d"
```
```  2087        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)"
```
```  2088        and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
```
```  2089        and xc: "x \<in> convex hull {a, b, c}"
```
```  2090        and s: "convex hull {a, b, c} \<subseteq> s"
```
```  2091     have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
```
```  2092               contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
```
```  2093               contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
```
```  2094               contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
```
```  2095       apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF s])
```
```  2096       apply (simp add: field_simps)
```
```  2097       done
```
```  2098     { fix y
```
```  2099       assume yc: "y \<in> convex hull {a,b,c}"
```
```  2100       have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
```
```  2101         apply (rule f')
```
```  2102         apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
```
```  2103         using s yc by blast
```
```  2104       also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
```
```  2105         by (simp add: yc e xc disj_le [OF triangle_points_closer])
```
```  2106       finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
```
```  2107     } note cm_le = this
```
```  2108     have "?normle a b c"
```
```  2109       apply (simp add: dist_norm pa)
```
```  2110       apply (rule le_of_3)
```
```  2111       using f' xc s e
```
```  2112       apply simp_all
```
```  2113       apply (intro norm_triangle_le add_mono path_bound)
```
```  2114       apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
```
```  2115       apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
```
```  2116       done
```
```  2117   } note * = this
```
```  2118   show ?thesis
```
```  2119     using cd e
```
```  2120     apply (simp add: field_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
```
```  2121     apply (clarify dest!: spec mp)
```
```  2122     using *
```
```  2123     apply (simp add: dist_norm, blast)
```
```  2124     done
```
```  2125 qed
```
```  2126
```
```  2127
```
```  2128 (* Hence the most basic theorem for a triangle.*)
```
```  2129 locale Chain =
```
```  2130   fixes x0 At Follows
```
```  2131   assumes At0: "At x0 0"
```
```  2132       and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
```
```  2133 begin
```
```  2134   primrec f where
```
```  2135     "f 0 = x0"
```
```  2136   | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"
```
```  2137
```
```  2138   lemma At: "At (f n) n"
```
```  2139   proof (induct n)
```
```  2140     case 0 show ?case
```
```  2141       by (simp add: At0)
```
```  2142   next
```
```  2143     case (Suc n) show ?case
```
```  2144       by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
```
```  2145   qed
```
```  2146
```
```  2147   lemma Follows: "Follows (f(Suc n)) (f n)"
```
```  2148     by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)
```
```  2149
```
```  2150   declare f.simps(2) [simp del]
```
```  2151 end
```
```  2152
```
```  2153 lemma Chain3:
```
```  2154   assumes At0: "At x0 y0 z0 0"
```
```  2155       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"
```
```  2156   obtains f g h where
```
```  2157     "f 0 = x0" "g 0 = y0" "h 0 = z0"
```
```  2158                       "\<And>n. At (f n) (g n) (h n) n"
```
```  2159                        "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
```
```  2160 proof -
```
```  2161   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"
```
```  2162     apply unfold_locales
```
```  2163     using At0 AtSuc by auto
```
```  2164   show ?thesis
```
```  2165   apply (rule that [of "\<lambda>n. fst (three.f n)"  "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
```
```  2166   apply simp_all
```
```  2167   using three.At three.Follows
```
```  2168   apply (simp_all add: split_beta')
```
```  2169   done
```
```  2170 qed
```
```  2171
```
```  2172 lemma Cauchy_theorem_triangle:
```
```  2173   assumes "f holomorphic_on (convex hull {a,b,c})"
```
```  2174     shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2175 proof -
```
```  2176   have contf: "continuous_on (convex hull {a,b,c}) f"
```
```  2177     by (metis assms holomorphic_on_imp_continuous_on)
```
```  2178   let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
```
```  2179   { fix y::complex
```
```  2180     assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2181        and ynz: "y \<noteq> 0"
```
```  2182     define K where "K = 1 + max (dist a b) (max (dist b c) (dist c a))"
```
```  2183     define e where "e = norm y / K^2"
```
```  2184     have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
```
```  2185     then have K: "K > 0" by linarith
```
```  2186     have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
```
```  2187       by (simp_all add: K_def)
```
```  2188     have e: "e > 0"
```
```  2189       unfolding e_def using ynz K1 by simp
```
```  2190     define At where "At x y z n \<longleftrightarrow>
```
```  2191         convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
```
```  2192         dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
```
```  2193         norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
```
```  2194       for x y z n
```
```  2195     have At0: "At a b c 0"
```
```  2196       using fy
```
```  2197       by (simp add: At_def e_def has_chain_integral_chain_integral3)
```
```  2198     { fix x y z n
```
```  2199       assume At: "At x y z n"
```
```  2200       then have contf': "continuous_on (convex hull {x,y,z}) f"
```
```  2201         using contf At_def continuous_on_subset by metis
```
```  2202       have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
```
```  2203         using At
```
```  2204         apply (simp add: At_def)
```
```  2205         using  Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
```
```  2206         apply clarsimp
```
```  2207         apply (rule_tac x="a'" in exI)
```
```  2208         apply (rule_tac x="b'" in exI)
```
```  2209         apply (rule_tac x="c'" in exI)
```
```  2210         apply (simp add: algebra_simps)
```
```  2211         apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
```
```  2212         done
```
```  2213     } note AtSuc = this
```
```  2214     obtain fa fb fc
```
```  2215       where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
```
```  2216         and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
```
```  2217         and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
```
```  2218                   "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
```
```  2219                   "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
```
```  2220         and no: "\<And>n. norm(?pathint (fa n) (fb n) +
```
```  2221                            ?pathint (fb n) (fc n) +
```
```  2222                            ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
```
```  2223         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}"
```
```  2224       apply (rule Chain3 [of At, OF At0 AtSuc])
```
```  2225       apply (auto simp: At_def)
```
```  2226       done
```
```  2227     have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
```
```  2228       apply (rule bounded_closed_nest)
```
```  2229       apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
```
```  2230       apply (rule allI)
```
```  2231       apply (rule transitive_stepwise_le)
```
```  2232       apply (auto simp: conv_le)
```
```  2233       done
```
```  2234     then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
```
```  2235     then have xin: "x \<in> convex hull {a,b,c}"
```
```  2236       using assms f0 by blast
```
```  2237     then have fx: "f field_differentiable at x within (convex hull {a,b,c})"
```
```  2238       using assms holomorphic_on_def by blast
```
```  2239     { fix k n
```
```  2240       assume k: "0 < k"
```
```  2241          and le:
```
```  2242             "\<And>x' y' z'.
```
```  2243                \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
```
```  2244                 x \<in> convex hull {x',y',z'};
```
```  2245                 convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
```
```  2246                \<Longrightarrow>
```
```  2247                cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
```
```  2248                      \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
```
```  2249          and Kk: "K / k < 2 ^ n"
```
```  2250       have "K / 2 ^ n < k" using Kk k
```
```  2251         by (auto simp: field_simps)
```
```  2252       then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
```
```  2253         using dist [of n]  k
```
```  2254         by linarith+
```
```  2255       have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
```
```  2256                \<le> (3 * K / 2 ^ n)\<^sup>2"
```
```  2257         using dist [of n] e K
```
```  2258         by (simp add: abs_le_square_iff [symmetric])
```
```  2259       have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
```
```  2260         by linarith
```
```  2261       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"
```
```  2262         using ynz dle e mult_le_cancel_left_pos by blast
```
```  2263       also have "... <
```
```  2264           cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
```
```  2265         using no [of n] e K
```
```  2266         apply (simp add: e_def field_simps)
```
```  2267         apply (simp only: zero_less_norm_iff [symmetric])
```
```  2268         done
```
```  2269       finally have False
```
```  2270         using le [OF DD x cosb] by auto
```
```  2271     } then
```
```  2272     have ?thesis
```
```  2273       using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
```
```  2274       apply clarsimp
```
```  2275       apply (rule_tac y1="K/k" in exE [OF real_arch_pow[of 2]])
```
```  2276       apply force+
```
```  2277       done
```
```  2278   }
```
```  2279   moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2280     by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1)
```
```  2281                    segments_subset_convex_hull(3) segments_subset_convex_hull(5))
```
```  2282   ultimately show ?thesis
```
```  2283     using has_contour_integral_integral by fastforce
```
```  2284 qed
```
```  2285
```
```  2286
```
```  2287 subsection\<open>Version needing function holomorphic in interior only\<close>
```
```  2288
```
```  2289 lemma Cauchy_theorem_flat_lemma:
```
```  2290   assumes f: "continuous_on (convex hull {a,b,c}) f"
```
```  2291       and c: "c - a = k *\<^sub>R (b - a)"
```
```  2292       and k: "0 \<le> k"
```
```  2293     shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f +
```
```  2294           contour_integral (linepath c a) f = 0"
```
```  2295 proof -
```
```  2296   have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
```
```  2297     using f continuous_on_subset segments_subset_convex_hull by metis+
```
```  2298   show ?thesis
```
```  2299   proof (cases "k \<le> 1")
```
```  2300     case True show ?thesis
```
```  2301       by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc)
```
```  2302   next
```
```  2303     case False then show ?thesis
```
```  2304       using fabc c
```
```  2305       apply (subst contour_integral_split [of a c f "1/k" b, symmetric])
```
```  2306       apply (metis closed_segment_commute fabc(3))
```
```  2307       apply (auto simp: k contour_integral_reverse_linepath)
```
```  2308       done
```
```  2309   qed
```
```  2310 qed
```
```  2311
```
```  2312 lemma Cauchy_theorem_flat:
```
```  2313   assumes f: "continuous_on (convex hull {a,b,c}) f"
```
```  2314       and c: "c - a = k *\<^sub>R (b - a)"
```
```  2315     shows "contour_integral (linepath a b) f +
```
```  2316            contour_integral (linepath b c) f +
```
```  2317            contour_integral (linepath c a) f = 0"
```
```  2318 proof (cases "0 \<le> k")
```
```  2319   case True with assms show ?thesis
```
```  2320     by (blast intro: Cauchy_theorem_flat_lemma)
```
```  2321 next
```
```  2322   case False
```
```  2323   have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
```
```  2324     using f continuous_on_subset segments_subset_convex_hull by metis+
```
```  2325   moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f +
```
```  2326         contour_integral (linepath c b) f = 0"
```
```  2327     apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
```
```  2328     using False c
```
```  2329     apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
```
```  2330     done
```
```  2331   ultimately show ?thesis
```
```  2332     apply (auto simp: contour_integral_reverse_linepath)
```
```  2333     using add_eq_0_iff by force
```
```  2334 qed
```
```  2335
```
```  2336
```
```  2337 lemma Cauchy_theorem_triangle_interior:
```
```  2338   assumes contf: "continuous_on (convex hull {a,b,c}) f"
```
```  2339       and holf:  "f holomorphic_on interior (convex hull {a,b,c})"
```
```  2340      shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2341 proof -
```
```  2342   have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
```
```  2343     using contf continuous_on_subset segments_subset_convex_hull by metis+
```
```  2344   have "bounded (f ` (convex hull {a,b,c}))"
```
```  2345     by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
```
```  2346   then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
```
```  2347      by (auto simp: dest!: bounded_pos [THEN iffD1])
```
```  2348   have "bounded (convex hull {a,b,c})"
```
```  2349     by (simp add: bounded_convex_hull)
```
```  2350   then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
```
```  2351     using bounded_pos_less by blast
```
```  2352   then have diff_2C: "norm(x - y) \<le> 2*C"
```
```  2353            if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
```
```  2354   proof -
```
```  2355     have "cmod x \<le> C"
```
```  2356       using x by (meson Cno not_le not_less_iff_gr_or_eq)
```
```  2357     hence "cmod (x - y) \<le> C + C"
```
```  2358       using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
```
```  2359     thus "cmod (x - y) \<le> 2 * C"
```
```  2360       by (metis mult_2)
```
```  2361   qed
```
```  2362   have contf': "continuous_on (convex hull {b,a,c}) f"
```
```  2363     using contf by (simp add: insert_commute)
```
```  2364   { fix y::complex
```
```  2365     assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2366        and ynz: "y \<noteq> 0"
```
```  2367     have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y"
```
```  2368       by (rule has_chain_integral_chain_integral3 [OF fy])
```
```  2369     have ?thesis
```
```  2370     proof (cases "c=a \<or> a=b \<or> b=c")
```
```  2371       case True then show ?thesis
```
```  2372         using Cauchy_theorem_flat [OF contf, of 0]
```
```  2373         using has_chain_integral_chain_integral3 [OF fy] ynz
```
```  2374         by (force simp: fabc contour_integral_reverse_linepath)
```
```  2375     next
```
```  2376       case False
```
```  2377       then have car3: "card {a, b, c} = Suc (DIM(complex))"
```
```  2378         by auto
```
```  2379       { assume "interior(convex hull {a,b,c}) = {}"
```
```  2380         then have "collinear{a,b,c}"
```
```  2381           using interior_convex_hull_eq_empty [OF car3]
```
```  2382           by (simp add: collinear_3_eq_affine_dependent)
```
```  2383         then have "False"
```
```  2384           using False
```
```  2385           apply (clarsimp simp add: collinear_3 collinear_lemma)
```
```  2386           apply (drule Cauchy_theorem_flat [OF contf'])
```
```  2387           using pi_eq_y ynz
```
```  2388           apply (simp add: fabc add_eq_0_iff contour_integral_reverse_linepath)
```
```  2389           done
```
```  2390       }
```
```  2391       then obtain d where d: "d \<in> interior (convex hull {a, b, c})"
```
```  2392         by blast
```
```  2393       { fix d1
```
```  2394         assume d1_pos: "0 < d1"
```
```  2395            and d1: "\<And>x x'. \<lbrakk>x\<in>convex hull {a, b, c}; x'\<in>convex hull {a, b, c}; cmod (x' - x) < d1\<rbrakk>
```
```  2396                            \<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)"
```
```  2397         define e where "e = min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))"
```
```  2398         define shrink where "shrink x = x - e *\<^sub>R (x - d)" for x
```
```  2399         let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
```
```  2400         have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B"
```
```  2401           using d1_pos \<open>C>0\<close> \<open>B>0\<close> ynz by (simp_all add: e_def)
```
```  2402         then have eCB: "24 * e * C * B \<le> cmod y"
```
```  2403           using \<open>C>0\<close> \<open>B>0\<close>  by (simp add: field_simps)
```
```  2404         have e_le_d1: "e * (4 * C) \<le> d1"
```
```  2405           using e \<open>C>0\<close> by (simp add: field_simps)
```
```  2406         have "shrink a \<in> interior(convex hull {a,b,c})"
```
```  2407              "shrink b \<in> interior(convex hull {a,b,c})"
```
```  2408              "shrink c \<in> interior(convex hull {a,b,c})"
```
```  2409           using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
```
```  2410         then have fhp0: "(f has_contour_integral 0)
```
```  2411                 (linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))"
```
```  2412           by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal convex_interior)
```
```  2413         then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0"
```
```  2414           by (simp add: has_chain_integral_chain_integral3)
```
```  2415         have fpi_abc: "f contour_integrable_on linepath (shrink a) (shrink b)"
```
```  2416                       "f contour_integrable_on linepath (shrink b) (shrink c)"
```
```  2417                       "f contour_integrable_on linepath (shrink c) (shrink a)"
```
```  2418           using fhp0  by (auto simp: valid_path_join dest: has_contour_integral_integrable)
```
```  2419         have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)"
```
```  2420           using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric])
```
```  2421         have sh_eq: "\<And>a b d::complex. (b - e *\<^sub>R (b - d)) - (a - e *\<^sub>R (a - d)) - (b - a) = e *\<^sub>R (a - b)"
```
```  2422           by (simp add: algebra_simps)
```
```  2423         have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
```
```  2424           using False \<open>C>0\<close> diff_2C [of b a] ynz
```
```  2425           by (auto simp: divide_simps hull_inc)
```
```  2426         have less_C: "\<lbrakk>u \<in> convex hull {a, b, c}; 0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> x * cmod u < C" for x u
```
```  2427           apply (cases "x=0", simp add: \<open>0<C\<close>)
```
```  2428           using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
```
```  2429         { fix u v
```
```  2430           assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v"
```
```  2431              and fpi_uv: "f contour_integrable_on linepath (shrink u) (shrink v)"
```
```  2432           have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})"
```
```  2433                        "shrink v \<in> interior(convex hull {a,b,c})"
```
```  2434             using d e uv
```
```  2435             by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
```
```  2436           have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B"
```
```  2437             using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD])
```
```  2438           have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y"
```
```  2439             apply (rule order_trans [OF _ eCB])
```
```  2440             using e \<open>B>0\<close> diff_2C [of u v] uv
```
```  2441             by (auto simp: field_simps)
```
```  2442           { fix x::real   assume x: "0\<le>x" "x\<le>1"
```
```  2443             have cmod_less_4C: "cmod ((1 - x) *\<^sub>R u - (1 - x) *\<^sub>R d) + cmod (x *\<^sub>R v - x *\<^sub>R d) < (C+C) + (C+C)"
```
```  2444               apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0])
```
```  2445               using uv x d interior_subset
```
```  2446               apply (auto simp: hull_inc intro!: less_C)
```
```  2447               done
```
```  2448             have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))"
```
```  2449               by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real)
```
```  2450             have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1"
```
```  2451               using \<open>e>0\<close>
```
```  2452               apply (simp add: ll norm_mult scaleR_diff_right)
```
```  2453               apply (rule less_le_trans [OF _ e_le_d1])
```
```  2454               using cmod_less_4C
```
```  2455               apply (force intro: norm_triangle_lt)
```
```  2456               done
```
```  2457             have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)"
```
```  2458               using x uv shr_uv cmod_less_dt
```
```  2459               by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
```
```  2460             also have "... \<le> cmod y / cmod (v - u) / 12"
```
```  2461               using False uv \<open>C>0\<close> diff_2C [of v u] ynz
```
```  2462               by (auto simp: divide_simps hull_inc)
```
```  2463             finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
```
```  2464               by simp
```
```  2465             then have cmod_12_le: "cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) * 12 \<le> cmod y"
```
```  2466               using uv False by (auto simp: field_simps)
```
```  2467             have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
```
```  2468                   cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
```
```  2469                   \<le> cmod y / 6"
```
```  2470               apply (rule order_trans [of _ "B*((norm y / 24 / C / B)*2*C) + (2*C)*(norm y /24 / C)"])
```
```  2471               apply (rule add_mono [OF mult_mono])
```
```  2472               using By_uv e \<open>0 < B\<close> \<open>0 < C\<close> x ynz
```
```  2473               apply (simp_all add: cmod_fuv cmod_shr cmod_12_le hull_inc)
```
```  2474               apply (simp add: field_simps)
```
```  2475               done
```
```  2476           } note cmod_diff_le = this
```
```  2477           have f_uv: "continuous_on (closed_segment u v) f"
```
```  2478             by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull])
```
```  2479           have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)"
```
```  2480             by (simp add: algebra_simps)
```
```  2481           have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6"
```
```  2482             apply (rule order_trans)
```
```  2483             apply (rule has_integral_bound
```
```  2484                     [of "B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)"
```
```  2485                         "\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)"
```
```  2486                         _ 0 1 ])
```
```  2487             using ynz \<open>0 < B\<close> \<open>0 < C\<close>
```
```  2488             apply (simp_all del: le_divide_eq_numeral1)
```
```  2489             apply (simp add: has_integral_sub has_contour_integral_linepath [symmetric] has_contour_integral_integral
```
```  2490                              fpi_uv f_uv contour_integrable_continuous_linepath, clarify)
```
```  2491             apply (simp only: **)
```
```  2492             apply (simp add: norm_triangle_le norm_mult cmod_diff_le del: le_divide_eq_numeral1)
```
```  2493             done
```
```  2494           } note * = this
```
```  2495           have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6"
```
```  2496             using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
```
```  2497           moreover
```
```  2498           have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6"
```
```  2499             using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
```
```  2500           moreover
```
```  2501           have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6"
```
```  2502             using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
```
```  2503           ultimately
```
```  2504           have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) +
```
```  2505                      (?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a))
```
```  2506                 \<le> norm y / 6 + norm y / 6 + norm y / 6"
```
```  2507             by (metis norm_triangle_le add_mono)
```
```  2508           also have "... = norm y / 2"
```
```  2509             by simp
```
```  2510           finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) -
```
```  2511                           (?pathint a b + ?pathint b c + ?pathint c a))
```
```  2512                 \<le> norm y / 2"
```
```  2513             by (simp add: algebra_simps)
```
```  2514           then
```
```  2515           have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2"
```
```  2516             by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff)
```
```  2517           then have "False"
```
```  2518             using pi_eq_y ynz by auto
```
```  2519         }
```
```  2520         moreover have "uniformly_continuous_on (convex hull {a,b,c}) f"
```
```  2521           by (simp add: contf compact_convex_hull compact_uniformly_continuous)
```
```  2522         ultimately have "False"
```
```  2523           unfolding uniformly_continuous_on_def
```
```  2524           by (force simp: ynz \<open>0 < C\<close> dist_norm)
```
```  2525         then show ?thesis ..
```
```  2526       qed
```
```  2527   }
```
```  2528   moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2529     using fabc contour_integrable_continuous_linepath by auto
```
```  2530   ultimately show ?thesis
```
```  2531     using has_contour_integral_integral by fastforce
```
```  2532 qed
```
```  2533
```
```  2534
```
```  2535 subsection\<open>Version allowing finite number of exceptional points\<close>
```
```  2536
```
```  2537 lemma Cauchy_theorem_triangle_cofinite:
```
```  2538   assumes "continuous_on (convex hull {a,b,c}) f"
```
```  2539       and "finite s"
```
```  2540       and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - s \<Longrightarrow> f field_differentiable (at x))"
```
```  2541      shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2542 using assms
```
```  2543 proof (induction "card s" arbitrary: a b c s rule: less_induct)
```
```  2544   case (less s a b c)
```
```  2545   show ?case
```
```  2546   proof (cases "s={}")
```
```  2547     case True with less show ?thesis
```
```  2548       by (fastforce simp: holomorphic_on_def field_differentiable_at_within
```
```  2549                     Cauchy_theorem_triangle_interior)
```
```  2550   next
```
```  2551     case False
```
```  2552     then obtain d s' where d: "s = insert d s'" "d \<notin> s'"
```
```  2553       by (meson Set.set_insert all_not_in_conv)
```
```  2554     then show ?thesis
```
```  2555     proof (cases "d \<in> convex hull {a,b,c}")
```
```  2556       case False
```
```  2557       show "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2558         apply (rule less.hyps [of "s'"])
```
```  2559         using False d \<open>finite s\<close> interior_subset
```
```  2560         apply (auto intro!: less.prems)
```
```  2561         done
```
```  2562     next
```
```  2563       case True
```
```  2564       have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}"
```
```  2565         by (meson True hull_subset insert_subset convex_hull_subset)
```
```  2566       have abd: "(f has_contour_integral 0) (linepath a b +++ linepath b d +++ linepath d a)"
```
```  2567         apply (rule less.hyps [of "s'"])
```
```  2568         using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
```
```  2569         apply (auto intro!: less.prems continuous_on_subset [OF  _ *])
```
```  2570         apply (metis * insert_absorb insert_subset interior_mono)
```
```  2571         done
```
```  2572       have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}"
```
```  2573         by (meson True hull_subset insert_subset convex_hull_subset)
```
```  2574       have bcd: "(f has_contour_integral 0) (linepath b c +++ linepath c d +++ linepath d b)"
```
```  2575         apply (rule less.hyps [of "s'"])
```
```  2576         using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
```
```  2577         apply (auto intro!: less.prems continuous_on_subset [OF _ *])
```
```  2578         apply (metis * insert_absorb insert_subset interior_mono)
```
```  2579         done
```
```  2580       have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}"
```
```  2581         by (meson True hull_subset insert_subset convex_hull_subset)
```
```  2582       have cad: "(f has_contour_integral 0) (linepath c a +++ linepath a d +++ linepath d c)"
```
```  2583         apply (rule less.hyps [of "s'"])
```
```  2584         using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
```
```  2585         apply (auto intro!: less.prems continuous_on_subset [OF _ *])
```
```  2586         apply (metis * insert_absorb insert_subset interior_mono)
```
```  2587         done
```
```  2588       have "f contour_integrable_on linepath a b"
```
```  2589         using less.prems
```
```  2590         by (metis continuous_on_subset insert_commute contour_integrable_continuous_linepath segments_subset_convex_hull(3))
```
```  2591       moreover have "f contour_integrable_on linepath b c"
```
```  2592         using less.prems
```
```  2593         by (metis continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(3))
```
```  2594       moreover have "f contour_integrable_on linepath c a"
```
```  2595         using less.prems
```
```  2596         by (metis continuous_on_subset insert_commute contour_integrable_continuous_linepath segments_subset_convex_hull(3))
```
```  2597       ultimately have fpi: "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2598         by auto
```
```  2599       { fix y::complex
```
```  2600         assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2601            and ynz: "y \<noteq> 0"
```
```  2602         have cont_ad: "continuous_on (closed_segment a d) f"
```
```  2603           by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3))
```
```  2604         have cont_bd: "continuous_on (closed_segment b d) f"
```
```  2605           by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1))
```
```  2606         have cont_cd: "continuous_on (closed_segment c d) f"
```
```  2607           by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2))
```
```  2608         have "contour_integral  (linepath a b) f = - (contour_integral (linepath b d) f + (contour_integral (linepath d a) f))"
```
```  2609                 "contour_integral  (linepath b c) f = - (contour_integral (linepath c d) f + (contour_integral (linepath d b) f))"
```
```  2610                 "contour_integral  (linepath c a) f = - (contour_integral (linepath a d) f + contour_integral (linepath d c) f)"
```
```  2611             using has_chain_integral_chain_integral3 [OF abd]
```
```  2612                   has_chain_integral_chain_integral3 [OF bcd]
```
```  2613                   has_chain_integral_chain_integral3 [OF cad]
```
```  2614             by (simp_all add: algebra_simps add_eq_0_iff)
```
```  2615         then have ?thesis
```
```  2616           using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 contour_integral_reverse_linepath by fastforce
```
```  2617       }
```
```  2618       then show ?thesis
```
```  2619         using fpi contour_integrable_on_def by blast
```
```  2620     qed
```
```  2621   qed
```
```  2622 qed
```
```  2623
```
```  2624
```
```  2625 subsection\<open>Cauchy's theorem for an open starlike set\<close>
```
```  2626
```
```  2627 lemma starlike_convex_subset:
```
```  2628   assumes s: "a \<in> s" "closed_segment b c \<subseteq> s" and subs: "\<And>x. x \<in> s \<Longrightarrow> closed_segment a x \<subseteq> s"
```
```  2629     shows "convex hull {a,b,c} \<subseteq> s"
```
```  2630       using s
```
```  2631       apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull)
```
```  2632       apply (meson subs convexD convex_closed_segment ends_in_segment(1) ends_in_segment(2) subsetCE)
```
```  2633       done
```
```  2634
```
```  2635 lemma triangle_contour_integrals_starlike_primitive:
```
```  2636   assumes contf: "continuous_on s f"
```
```  2637       and s: "a \<in> s" "open s"
```
```  2638       and x: "x \<in> s"
```
```  2639       and subs: "\<And>y. y \<in> s \<Longrightarrow> closed_segment a y \<subseteq> s"
```
```  2640       and zer: "\<And>b c. closed_segment b c \<subseteq> s
```
```  2641                    \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
```
```  2642                        contour_integral (linepath c a) f = 0"
```
```  2643     shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x)"
```
```  2644 proof -
```
```  2645   let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
```
```  2646   { fix e y
```
```  2647     assume e: "0 < e" and bxe: "ball x e \<subseteq> s" and close: "cmod (y - x) < e"
```
```  2648     have y: "y \<in> s"
```
```  2649       using bxe close  by (force simp: dist_norm norm_minus_commute)
```
```  2650     have cont_ayf: "continuous_on (closed_segment a y) f"
```
```  2651       using contf continuous_on_subset subs y by blast
```
```  2652     have xys: "closed_segment x y \<subseteq> s"
```
```  2653       apply (rule order_trans [OF _ bxe])
```
```  2654       using close
```
```  2655       by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound)
```
```  2656     have "?pathint a y - ?pathint a x = ?pathint x y"
```
```  2657       using zer [OF xys]  contour_integral_reverse_linepath [OF cont_ayf]  add_eq_0_iff by force
```
```  2658   } note [simp] = this
```
```  2659   { fix e::real
```
```  2660     assume e: "0 < e"
```
```  2661     have cont_atx: "continuous (at x) f"
```
```  2662       using x s contf continuous_on_eq_continuous_at by blast
```
```  2663     then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2"
```
```  2664       unfolding continuous_at Lim_at dist_norm  using e
```
```  2665       by (drule_tac x="e/2" in spec) force
```
```  2666     obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> s" using  \<open>open s\<close> x
```
```  2667       by (auto simp: open_contains_ball)
```
```  2668     have dpos: "min d1 d2 > 0" using d1 d2 by simp
```
```  2669     { fix y
```
```  2670       assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2"
```
```  2671       have y: "y \<in> s"
```
```  2672         using d2 close  by (force simp: dist_norm norm_minus_commute)
```
```  2673       have fxy: "f contour_integrable_on linepath x y"
```
```  2674         apply (rule contour_integrable_continuous_linepath)
```
```  2675         apply (rule continuous_on_subset [OF contf])
```
```  2676         using close d2
```
```  2677         apply (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1))
```
```  2678         done
```
```  2679       then obtain i where i: "(f has_contour_integral i) (linepath x y)"
```
```  2680         by (auto simp: contour_integrable_on_def)
```
```  2681       then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
```
```  2682         by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
```
```  2683       then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
```
```  2684         apply (rule has_contour_integral_bound_linepath [where B = "e/2"])
```
```  2685         using e apply simp
```
```  2686         apply (rule d1_less [THEN less_imp_le])
```
```  2687         using close segment_bound
```
```  2688         apply force
```
```  2689         done
```
```  2690       also have "... < e * cmod (y - x)"
```
```  2691         by (simp add: e yx)
```
```  2692       finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
```
```  2693         using i yx  by (simp add: contour_integral_unique divide_less_eq)
```
```  2694     }
```
```  2695     then have "\<exists>d>0. \<forall>y. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
```
```  2696       using dpos by blast
```
```  2697   }
```
```  2698   then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) \<midarrow>x\<rightarrow> 0"
```
```  2699     by (simp add: Lim_at dist_norm inverse_eq_divide)
```
```  2700   show ?thesis
```
```  2701     apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right)
```
```  2702     apply (rule Lim_transform [OF * Lim_eventually])
```
```  2703     apply (simp add: inverse_eq_divide [symmetric] eventually_at)
```
```  2704     using \<open>open s\<close> x
```
```  2705     apply (force simp: dist_norm open_contains_ball)
```
```  2706     done
```
```  2707 qed
```
```  2708
```
```  2709 (** Existence of a primitive.*)
```
```  2710
```
```  2711 lemma holomorphic_starlike_primitive:
```
```  2712   fixes f :: "complex \<Rightarrow> complex"
```
```  2713   assumes contf: "continuous_on s f"
```
```  2714       and s: "starlike s" and os: "open s"
```
```  2715       and k: "finite k"
```
```  2716       and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x"
```
```  2717     shows "\<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x)"
```
```  2718 proof -
```
```  2719   obtain a where a: "a\<in>s" and a_cs: "\<And>x. x\<in>s \<Longrightarrow> closed_segment a x \<subseteq> s"
```
```  2720     using s by (auto simp: starlike_def)
```
```  2721   { fix x b c
```
```  2722     assume "x \<in> s" "closed_segment b c \<subseteq> s"
```
```  2723     then have abcs: "convex hull {a, b, c} \<subseteq> s"
```
```  2724       by (simp add: a a_cs starlike_convex_subset)
```
```  2725     then have *: "continuous_on (convex hull {a, b, c}) f"
```
```  2726       by (simp add: continuous_on_subset [OF contf])
```
```  2727     have "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
```
```  2728       apply (rule Cauchy_theorem_triangle_cofinite [OF _ k])
```
```  2729       using abcs apply (simp add: continuous_on_subset [OF contf])
```
```  2730       using * abcs interior_subset apply (auto intro: fcd)
```
```  2731       done
```
```  2732   } note 0 = this
```
```  2733   show ?thesis
```
```  2734     apply (intro exI ballI)
```
```  2735     apply (rule triangle_contour_integrals_starlike_primitive [OF contf a os], assumption)
```
```  2736     apply (metis a_cs)
```
```  2737     apply (metis has_chain_integral_chain_integral3 0)
```
```  2738     done
```
```  2739 qed
```
```  2740
```
```  2741 lemma Cauchy_theorem_starlike:
```
```  2742  "\<lbrakk>open s; starlike s; finite k; continuous_on s f;
```
```  2743    \<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x;
```
```  2744    valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
```
```  2745    \<Longrightarrow> (f has_contour_integral 0)  g"
```
```  2746   by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open)
```
```  2747
```
```  2748 lemma Cauchy_theorem_starlike_simple:
```
```  2749   "\<lbrakk>open s; starlike s; f holomorphic_on s; valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
```
```  2750    \<Longrightarrow> (f has_contour_integral 0) g"
```
```  2751 apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI])
```
```  2752 apply (simp_all add: holomorphic_on_imp_continuous_on)
```
```  2753 apply (metis at_within_open holomorphic_on_def)
```
```  2754 done
```
```  2755
```
```  2756
```
```  2757 subsection\<open>Cauchy's theorem for a convex set\<close>
```
```  2758
```
```  2759 text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close>
```
```  2760
```
```  2761 lemma triangle_contour_integrals_convex_primitive:
```
```  2762   assumes contf: "continuous_on s f"
```
```  2763       and s: "a \<in> s" "convex s"
```
```  2764       and x: "x \<in> s"
```
```  2765       and zer: "\<And>b c. \<lbrakk>b \<in> s; c \<in> s\<rbrakk>
```
```  2766                    \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
```
```  2767                        contour_integral (linepath c a) f = 0"
```
```  2768     shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x within s)"
```
```  2769 proof -
```
```  2770   let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
```
```  2771   { fix y
```
```  2772     assume y: "y \<in> s"
```
```  2773     have cont_ayf: "continuous_on (closed_segment a y) f"
```
```  2774       using s y  by (meson contf continuous_on_subset convex_contains_segment)
```
```  2775     have xys: "closed_segment x y \<subseteq> s"  (*?*)
```
```  2776       using convex_contains_segment s x y by auto
```
```  2777     have "?pathint a y - ?pathint a x = ?pathint x y"
```
```  2778       using zer [OF x y]  contour_integral_reverse_linepath [OF cont_ayf]  add_eq_0_iff by force
```
```  2779   } note [simp] = this
```
```  2780   { fix e::real
```
```  2781     assume e: "0 < e"
```
```  2782     have cont_atx: "continuous (at x within s) f"
```
```  2783       using x s contf  by (simp add: continuous_on_eq_continuous_within)
```
```  2784     then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. \<lbrakk>y \<in> s; cmod (y - x) < d1\<rbrakk> \<Longrightarrow> cmod (f y - f x) < e/2"
```
```  2785       unfolding continuous_within Lim_within dist_norm using e
```
```  2786       by (drule_tac x="e/2" in spec) force
```
```  2787     { fix y
```
```  2788       assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> s"
```
```  2789       have fxy: "f contour_integrable_on linepath x y"
```
```  2790         using convex_contains_segment s x y
```
```  2791         by (blast intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
```
```  2792       then obtain i where i: "(f has_contour_integral i) (linepath x y)"
```
```  2793         by (auto simp: contour_integrable_on_def)
```
```  2794       then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
```
```  2795         by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
```
```  2796       then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
```
```  2797         apply (rule has_contour_integral_bound_linepath [where B = "e/2"])
```
```  2798         using e apply simp
```
```  2799         apply (rule d1_less [THEN less_imp_le])
```
```  2800         using convex_contains_segment s(2) x y apply blast
```
```  2801         using close segment_bound(1) apply fastforce
```
```  2802         done
```
```  2803       also have "... < e * cmod (y - x)"
```
```  2804         by (simp add: e yx)
```
```  2805       finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
```
```  2806         using i yx  by (simp add: contour_integral_unique divide_less_eq)
```
```  2807     }
```
```  2808     then have "\<exists>d>0. \<forall>y\<in>s. y \<noteq> x \<and> cmod (y-x) < d \<longrightarrow> cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
```
```  2809       using d1 by blast
```
```  2810   }
```
```  2811   then have *: "((\<lambda>y. (contour_integral (linepath x y) f - f x * (y - x)) /\<^sub>R cmod (y - x)) \<longlongrightarrow> 0) (at x within s)"
```
```  2812     by (simp add: Lim_within dist_norm inverse_eq_divide)
```
```  2813   show ?thesis
```
```  2814     apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
```
```  2815     apply (rule Lim_transform [OF * Lim_eventually])
```
```  2816     using linordered_field_no_ub
```
```  2817     apply (force simp: inverse_eq_divide [symmetric] eventually_at)
```
```  2818     done
```
```  2819 qed
```
```  2820
```
```  2821 lemma contour_integral_convex_primitive:
```
```  2822   "\<lbrakk>convex s; continuous_on s f;
```
```  2823     \<And>a b c. \<lbrakk>a \<in> s; b \<in> s; c \<in> s\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)\<rbrakk>
```
```  2824          \<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
```
```  2825   apply (cases "s={}")
```
```  2826   apply (simp_all add: ex_in_conv [symmetric])
```
```  2827   apply (blast intro: triangle_contour_integrals_convex_primitive has_chain_integral_chain_integral3)
```
```  2828   done
```
```  2829
```
```  2830 lemma holomorphic_convex_primitive:
```
```  2831   fixes f :: "complex \<Rightarrow> complex"
```
```  2832   shows
```
```  2833   "\<lbrakk>convex s; finite k; continuous_on s f;
```
```  2834     \<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x\<rbrakk>
```
```  2835    \<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
```
```  2836 apply (rule contour_integral_convex_primitive [OF _ _ Cauchy_theorem_triangle_cofinite])
```
```  2837 prefer 3
```
```  2838 apply (erule continuous_on_subset)
```
```  2839 apply (simp add: subset_hull continuous_on_subset, assumption+)
```
```  2840 by (metis Diff_iff convex_contains_segment insert_absorb insert_subset interior_mono segment_convex_hull subset_hull)
```
```  2841
```
```  2842 lemma Cauchy_theorem_convex:
```
```  2843     "\<lbrakk>continuous_on s f; convex s; finite k;
```
```  2844       \<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x;
```
```  2845      valid_path g; path_image g \<subseteq> s;
```
```  2846      pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
```
```  2847   by (metis holomorphic_convex_primitive Cauchy_theorem_primitive)
```
```  2848
```
```  2849 lemma Cauchy_theorem_convex_simple:
```
```  2850     "\<lbrakk>f holomorphic_on s; convex s;
```
```  2851      valid_path g; path_image g \<subseteq> s;
```
```  2852      pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
```
```  2853   apply (rule Cauchy_theorem_convex)
```
```  2854   apply (simp_all add: holomorphic_on_imp_continuous_on)
```
```  2855   apply (rule finite.emptyI)
```
```  2856   using at_within_interior holomorphic_on_def interior_subset by fastforce
```
```  2857
```
```  2858
```
```  2859 text\<open>In particular for a disc\<close>
```
```  2860 lemma Cauchy_theorem_disc:
```
```  2861     "\<lbrakk>finite k; continuous_on (cball a e) f;
```
```  2862       \<And>x. x \<in> ball a e - k \<Longrightarrow> f field_differentiable at x;
```
```  2863      valid_path g; path_image g \<subseteq> cball a e;
```
```  2864      pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
```
```  2865   apply (rule Cauchy_theorem_convex)
```
```  2866   apply (auto simp: convex_cball interior_cball)
```
```  2867   done
```
```  2868
```
```  2869 lemma Cauchy_theorem_disc_simple:
```
```  2870     "\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e;
```
```  2871      pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
```
```  2872 by (simp add: Cauchy_theorem_convex_simple)
```
```  2873
```
```  2874
```
```  2875 subsection\<open>Generalize integrability to local primitives\<close>
```
```  2876
```
```  2877 lemma contour_integral_local_primitive_lemma:
```
```  2878   fixes f :: "complex\<Rightarrow>complex"
```
```  2879   shows
```
```  2880     "\<lbrakk>g piecewise_differentiable_on {a..b};
```
```  2881       \<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s);
```
```  2882       \<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk>
```
```  2883      \<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b}))
```
```  2884             integrable_on {a..b}"
```
```  2885   apply (cases "cbox a b = {}", force)
```
```  2886   apply (simp add: integrable_on_def)
```
```  2887   apply (rule exI)
```
```  2888   apply (rule contour_integral_primitive_lemma, assumption+)
```
```  2889   using atLeastAtMost_iff by blast
```
```  2890
```
```  2891 lemma contour_integral_local_primitive_any:
```
```  2892   fixes f :: "complex \<Rightarrow> complex"
```
```  2893   assumes gpd: "g piecewise_differentiable_on {a..b}"
```
```  2894       and dh: "\<And>x. x \<in> s
```
```  2895                \<Longrightarrow> \<exists>d h. 0 < d \<and>
```
```  2896                          (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
```
```  2897       and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
```
```  2898   shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}"
```
```  2899 proof -
```
```  2900   { fix x
```
```  2901     assume x: "a \<le> x" "x \<le> b"
```
```  2902     obtain d h where d: "0 < d"
```
```  2903                and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))"
```
```  2904       using x gs dh by (metis atLeastAtMost_iff)
```
```  2905     have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast
```
```  2906     then obtain e where e: "e>0" and lessd: "\<And>x'. x' \<in> {a..b} \<Longrightarrow> \<bar>x' - x\<bar> < e \<Longrightarrow> cmod (g x' - g x) < d"
```
```  2907       using x d
```
```  2908       apply (auto simp: dist_norm continuous_on_iff)
```
```  2909       apply (drule_tac x=x in bspec)
```
```  2910       using x apply simp
```
```  2911       apply (drule_tac x=d in spec, auto)
```
```  2912       done
```
```  2913     have "\<exists>d>0. \<forall>u v. u \<le> x \<and> x \<le> v \<and> {u..v} \<subseteq> ball x d \<and> (u \<le> v \<longrightarrow> a \<le> u \<and> v \<le> b) \<longrightarrow>
```
```  2914                           (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}"
```
```  2915       apply (rule_tac x=e in exI)
```
```  2916       using e
```
```  2917       apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify)
```
```  2918       apply (rule_tac f = h and s = "g ` {u..v}" in contour_integral_local_primitive_lemma)
```
```  2919         apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset)
```
```  2920        apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force)
```
```  2921       done
```
```  2922   } then
```
```  2923   show ?thesis
```
```  2924     by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified])
```
```  2925 qed
```
```  2926
```
```  2927 lemma contour_integral_local_primitive:
```
```  2928   fixes f :: "complex \<Rightarrow> complex"
```
```  2929   assumes g: "valid_path g" "path_image g \<subseteq> s"
```
```  2930       and dh: "\<And>x. x \<in> s
```
```  2931                \<Longrightarrow> \<exists>d h. 0 < d \<and>
```
```  2932                          (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
```
```  2933   shows "f contour_integrable_on g"
```
```  2934   using g
```
```  2935   apply (simp add: valid_path_def path_image_def contour_integrable_on_def has_contour_integral_def
```
```  2936             has_integral_localized_vector_derivative integrable_on_def [symmetric])
```
```  2937   using contour_integral_local_primitive_any [OF _ dh]
```
```  2938   by (meson image_subset_iff piecewise_C1_imp_differentiable)
```
```  2939
```
```  2940
```
```  2941 text\<open>In particular if a function is holomorphic\<close>
```
```  2942
```
```  2943 lemma contour_integrable_holomorphic:
```
```  2944   assumes contf: "continuous_on s f"
```
```  2945       and os: "open s"
```
```  2946       and k: "finite k"
```
```  2947       and g: "valid_path g" "path_image g \<subseteq> s"
```
```  2948       and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f field_differentiable at x"
```
```  2949     shows "f contour_integrable_on g"
```
```  2950 proof -
```
```  2951   { fix z
```
```  2952     assume z: "z \<in> s"
```
```  2953     obtain d where d: "d>0" "ball z d \<subseteq> s" using  \<open>open s\<close> z
```
```  2954       by (auto simp: open_contains_ball)
```
```  2955     then have contfb: "continuous_on (ball z d) f"
```
```  2956       using contf continuous_on_subset by blast
```
```  2957     obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)"
```
```  2958       using holomorphic_convex_primitive [OF convex_ball k contfb fcd] d
```
```  2959             interior_subset by force
```
```  2960     then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)"
```
```  2961       by (metis Topology_Euclidean_Space.open_ball at_within_open d(2) os subsetCE)
```
```  2962     then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
```
```  2963       by (force simp: dist_norm norm_minus_commute)
```
```  2964     then have "\<exists>d h. 0 < d \<and> (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
```
```  2965       using d by blast
```
```  2966   }
```
```  2967   then show ?thesis
```
```  2968     by (rule contour_integral_local_primitive [OF g])
```
```  2969 qed
```
```  2970
```
```  2971 lemma contour_integrable_holomorphic_simple:
```
```  2972   assumes fh: "f holomorphic_on s"
```
```  2973       and os: "open s"
```
```  2974       and g: "valid_path g" "path_image g \<subseteq> s"
```
```  2975     shows "f contour_integrable_on g"
```
```  2976   apply (rule contour_integrable_holomorphic [OF _ os Finite_Set.finite.emptyI g])
```
```  2977   apply (simp add: fh holomorphic_on_imp_continuous_on)
```
```  2978   using fh  by (simp add: field_differentiable_def holomorphic_on_open os)
```
```  2979
```
```  2980 lemma continuous_on_inversediff:
```
```  2981   fixes z:: "'a::real_normed_field" shows "z \<notin> s \<Longrightarrow> continuous_on s (\<lambda>w. 1 / (w - z))"
```
```  2982   by (rule continuous_intros | force)+
```
```  2983
```
```  2984 corollary contour_integrable_inversediff:
```
```  2985     "\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) contour_integrable_on g"
```
```  2986 apply (rule contour_integrable_holomorphic_simple [of _ "UNIV-{z}"])
```
```  2987 apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros)
```
```  2988 done
```
```  2989
```
```  2990 text\<open>Key fact that path integral is the same for a "nearby" path. This is the
```
```  2991  main lemma for the homotopy form of Cauchy's theorem and is also useful
```
```  2992  if we want "without loss of generality" to assume some nice properties of a
```
```  2993  path (e.g. smoothness). It can also be used to define the integrals of
```
```  2994  analytic functions over arbitrary continuous paths. This is just done for
```
```  2995  winding numbers now.
```
```  2996 \<close>
```
```  2997
```
```  2998 text\<open>A technical definition to avoid duplication of similar proofs,
```
```  2999      for paths joined at the ends versus looping paths\<close>
```
```  3000 definition linked_paths :: "bool \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
```
```  3001   where "linked_paths atends g h ==
```
```  3002         (if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
```
```  3003                    else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)"
```
```  3004
```
```  3005 text\<open>This formulation covers two cases: @{term g} and @{term h} share their
```
```  3006       start and end points; @{term g} and @{term h} both loop upon themselves.\<close>
```
```  3007 lemma contour_integral_nearby:
```
```  3008   assumes os: "open s" and p: "path p" "path_image p \<subseteq> s"
```
```  3009     shows
```
```  3010        "\<exists>d. 0 < d \<and>
```
```  3011             (\<forall>g h. valid_path g \<and> valid_path h \<and>
```
```  3012                   (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
```
```  3013                   linked_paths atends g h
```
```  3014                   \<longrightarrow> path_image g \<subseteq> s \<and> path_image h \<subseteq> s \<and>
```
```  3015                       (\<forall>f. f holomorphic_on s \<longrightarrow> contour_integral h f = contour_integral g f))"
```
```  3016 proof -
```
```  3017   have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> s"
```
```  3018     using open_contains_ball os p(2) by blast
```
```  3019   then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> s"
```
```  3020     by metis
```
```  3021   define cover where "cover = (\<lambda>z. ball z (ee z/3)) ` (path_image p)"
```
```  3022   have "compact (path_image p)"
```
```  3023     by (metis p(1) compact_path_image)
```
```  3024   moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))"
```
```  3025     using ee by auto
```
```  3026   ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D"
```
```  3027     by (simp add: compact_eq_heine_borel cover_def)
```
```  3028   then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D"
```
```  3029     by blast
```
```  3030   then obtain k where k: "k \<subseteq> {0..1}" "finite k" and D_eq: "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k"
```
```  3031     apply (simp add: cover_def path_image_def image_comp)
```
```  3032     apply (blast dest!: finite_subset_image [OF \<open>finite D\<close>])
```
```  3033     done
```
```  3034   then have kne: "k \<noteq> {}"
```
```  3035     using D by auto
```
```  3036   have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p"
```
```  3037     using k  by (auto simp: path_image_def)
```
```  3038   then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))"
```
```  3039     by (metis ee)
```
```  3040   define e where "e = Min((ee o p) ` k)"
```
```  3041   have fin_eep: "finite ((ee o p) ` k)"
```
```  3042     using k  by blast
```
```  3043   have enz: "0 < e"
```
```  3044     using ee k  by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi)
```
```  3045   have "uniformly_continuous_on {0..1} p"
```
```  3046     using p  by (simp add: path_def compact_uniformly_continuous)
```
```  3047   then obtain d::real where d: "d>0"
```
```  3048           and de: "\<And>x x'. \<bar>x' - x\<bar> < d \<Longrightarrow> x\<in>{0..1} \<Longrightarrow> x'\<in>{0..1} \<Longrightarrow> cmod (p x' - p x) < e/3"
```
```  3049     unfolding uniformly_continuous_on_def dist_norm real_norm_def
```
```  3050     by (metis divide_pos_pos enz zero_less_numeral)
```
```  3051   then obtain N::nat where N: "N>0" "inverse N < d"
```
```  3052     using real_arch_inverse [of d]   by auto
```
```  3053   { fix g h
```
```  3054     assume g: "valid_path g" and gp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3"
```
```  3055        and h: "valid_path h" and hp: "\<forall>t\<in>{0..1}. cmod (h t - p t) < e / 3"
```
```  3056        and joins: "linked_paths atends g h"
```
```  3057     { fix t::real
```
```  3058       assume t: "0 \<le> t" "t \<le> 1"
```
```  3059       then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)"
```
```  3060         using \<open>path_image p \<subseteq> \<Union>D\<close> D_eq by (force simp: path_image_def)
```
```  3061       then have ele: "e \<le> ee (p u)" using fin_eep
```
```  3062         by (simp add: e_def)
```
```  3063       have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3"
```
```  3064         using gp hp t by auto
```
```  3065       with ele have "cmod (g t - p t) < ee (p u) / 3"
```
```  3066                     "cmod (h t - p t) < ee (p u) / 3"
```
```  3067         by linarith+
```
```  3068       then have "g t \<in> ball(p u) (ee(p u))"  "h t \<in> ball(p u) (ee(p u))"
```
```  3069         using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"]
```
```  3070               norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u
```
```  3071         by (force simp: dist_norm ball_def norm_minus_commute)+
```
```  3072       then have "g t \<in> s" "h t \<in> s" using ee u k
```
```  3073         by (auto simp: path_image_def ball_def)
```
```  3074     }
```
```  3075     then have ghs: "path_image g \<subseteq> s" "path_image h \<subseteq> s"
```
```  3076       by (auto simp: path_image_def)
```
```  3077     moreover
```
```  3078     { fix f
```
```  3079       assume fhols: "f holomorphic_on s"
```
```  3080       then have fpa: "f contour_integrable_on g"  "f contour_integrable_on h"
```
```  3081         using g ghs h holomorphic_on_imp_continuous_on os contour_integrable_holomorphic_simple
```
```  3082         by blast+
```
```  3083       have contf: "continuous_on s f"
```
```  3084         by (simp add: fhols holomorphic_on_imp_continuous_on)
```
```  3085       { fix z
```
```  3086         assume z: "z \<in> path_image p"
```
```  3087         have "f holomorphic_on ball z (ee z)"
```
```  3088           using fhols ee z holomorphic_on_subset by blast
```
```  3089         then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))"
```
```  3090           using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified]
```
```  3091           by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball)
```
```  3092       }
```
```  3093       then obtain ff where ff:
```
```  3094             "\<And>z w. \<lbrakk>z \<in> path_image p; w \<in> ball z (ee z)\<rbrakk> \<Longrightarrow> (ff z has_field_derivative f w) (at w)"
```
```  3095         by metis
```
```  3096       { fix n
```
```  3097         assume n: "n \<le> N"
```
```  3098         then have "contour_integral(subpath 0 (n/N) h) f - contour_integral(subpath 0 (n/N) g) f =
```
```  3099                    contour_integral(linepath (g(n/N)) (h(n/N))) f - contour_integral(linepath (g 0) (h 0)) f"
```
```  3100         proof (induct n)
```
```  3101           case 0 show ?case by simp
```
```  3102         next
```
```  3103           case (Suc n)
```
```  3104           obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)"
```
```  3105             using \<open>path_image p \<subseteq> \<Union>D\<close> [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems
```
```  3106             by (force simp: path_image_def)
```
```  3107           then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3"
```
```  3108             by (simp add: dist_norm)
```
```  3109           have e3le: "e/3 \<le> ee (p t) / 3"  using fin_eep t
```
```  3110             by (simp add: e_def)
```
```  3111           { fix x
```
```  3112             assume x: "n/N \<le> x" "x \<le> (1 + n)/N"
```
```  3113             then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1"
```
```  3114               using Suc.prems by auto
```
```  3115             then have x01: "0 \<le> x" "x \<le> 1"
```
```  3116               using x by linarith+
```
```  3117             have "cmod (p t - p x)  < ee (p t) / 3 + e/3"
```
```  3118               apply (rule norm_diff_triangle_less [OF ptu de])
```
```  3119               using x N x01 Suc.prems
```
```  3120               apply (auto simp: field_simps)
```
```  3121               done
```
```  3122             then have ptx: "cmod (p t - p x) < 2*ee (p t)/3"
```
```  3123               using e3le eepi [OF t] by simp
```
```  3124             have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 "
```
```  3125               apply (rule norm_diff_triangle_less [OF ptx])
```
```  3126               using gp x01 by (simp add: norm_minus_commute)
```
```  3127             also have "... \<le> ee (p t)"
```
```  3128               using e3le eepi [OF t] by simp
```
```  3129             finally have gg: "cmod (p t - g x) < ee (p t)" .
```
```  3130             have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 "
```
```  3131               apply (rule norm_diff_triangle_less [OF ptx])
```
```  3132               using hp x01 by (simp add: norm_minus_commute)
```
```  3133             also have "... \<le> ee (p t)"
```
```  3134               using e3le eepi [OF t] by simp
```
```  3135             finally have "cmod (p t - g x) < ee (p t)"
```
```  3136                          "cmod (p t - h x) < ee (p t)"
```
```  3137               using gg by auto
```
```  3138           } note ptgh_ee = this
```
```  3139           have pi_hgn: "path_image (linepath (h (n/N)) (g (n/N))) \<subseteq> ball (p t) (ee (p t))"
```
```  3140             using ptgh_ee [of "n/N"] Suc.prems
```
```  3141             by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
```
```  3142           then have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> s"
```
```  3143             using \<open>N>0\<close> Suc.prems
```
```  3144             apply (simp add: path_image_join field_simps closed_segment_commute)
```
```  3145             apply (erule order_trans)
```
```  3146             apply (simp add: ee pi t)
```
```  3147             done
```
```  3148           have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N)))
```
```  3149                   \<subseteq> ball (p t) (ee (p t))"
```
```  3150             using ptgh_ee [of "(1+n)/N"] Suc.prems
```
```  3151             by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
```
```  3152           then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> s"
```
```  3153             using \<open>N>0\<close> Suc.prems ee pi t
```
```  3154             by (auto simp: Path_Connected.path_image_join field_simps)
```
```  3155           have pi_subset_ball:
```
```  3156                 "path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++
```
```  3157                              subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N)))
```
```  3158                  \<subseteq> ball (p t) (ee (p t))"
```
```  3159             apply (intro subset_path_image_join pi_hgn pi_ghn')
```
```  3160             using \<open>N>0\<close> Suc.prems
```
```  3161             apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
```
```  3162             done
```
```  3163           have pi0: "(f has_contour_integral 0)
```
```  3164                        (subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
```
```  3165                         subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))"
```
```  3166             apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"])
```
```  3167             apply (metis ff open_ball at_within_open pi t)
```
```  3168             apply (intro valid_path_join)
```
```  3169             using Suc.prems pi_subset_ball apply (simp_all add: valid_path_subpath g h)
```
```  3170             done
```
```  3171           have fpa1: "f contour_integrable_on subpath (real n / real N) (real (Suc n) / real N) g"
```
```  3172             using Suc.prems by (simp add: contour_integrable_subpath g fpa)
```
```  3173           have fpa2: "f contour_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))"
```
```  3174             using gh_n's
```
```  3175             by (auto intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
```
```  3176           have fpa3: "f contour_integrable_on linepath (h (real n / real N)) (g (real n / real N))"
```
```  3177             using gh_ns
```
```  3178             by (auto simp: closed_segment_commute intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
```
```  3179           have eq0: "contour_integral (subpath (n/N) ((Suc n) / real N) g) f +
```
```  3180                      contour_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f +
```
```  3181                      contour_integral (subpath ((Suc n) / N) (n/N) h) f +
```
```  3182                      contour_integral (linepath (h (n/N)) (g (n/N))) f = 0"
```
```  3183             using contour_integral_unique [OF pi0] Suc.prems
```
```  3184             by (simp add: g h fpa valid_path_subpath contour_integrable_subpath
```
```  3185                           fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc)
```
```  3186           have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
```
```  3187                     \<lbrakk>hn - gn = ghn - gh0;
```
```  3188                      gd + ghn' + he + hgn = (0::complex);
```
```  3189                      hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0"
```
```  3190             by (auto simp: algebra_simps)
```
```  3191           have "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
```
```  3192                 contour_integral (subpath 0 (n/N) h) f + contour_integral (subpath (n/N) ((Suc n) / N) h) f"
```
```  3193             unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"]
```
```  3194             using Suc.prems by (simp add: h fpa contour_integral_reversepath valid_path_subpath contour_integrable_subpath)
```
```  3195           also have "... = contour_integral (subpath 0 ((Suc n) / N) h) f"
```
```  3196             using Suc.prems by (simp add: contour_integral_subpath_combine h fpa)
```
```  3197           finally have pi0_eq:
```
```  3198                "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
```
```  3199                 contour_integral (subpath 0 ((Suc n) / N) h) f" .
```
```  3200           show ?case
```
```  3201             apply (rule * [OF Suc.hyps eq0 pi0_eq])
```
```  3202             using Suc.prems
```
```  3203             apply (simp_all add: g h fpa contour_integral_subpath_combine
```
```  3204                      contour_integral_reversepath [symmetric] contour_integrable_continuous_linepath
```
```  3205                      continuous_on_subset [OF contf gh_ns])
```
```  3206             done
```
```  3207       qed
```
```  3208       } note ind = this
```
```  3209       have "contour_integral h f = contour_integral g f"
```
```  3210         using ind [OF order_refl] N joins
```
```  3211         by (simp add: linked_paths_def pathstart_def pathfinish_def split: if_split_asm)
```
```  3212     }
```
```  3213     ultimately
```
```  3214     have "path_image g \<subseteq> s \<and> path_image h \<subseteq> s \<and> (\<forall>f. f holomorphic_on s \<longrightarrow> contour_integral h f = contour_integral g f)"
```
```  3215       by metis
```
```  3216   } note * = this
```
```  3217   show ?thesis
```
```  3218     apply (rule_tac x="e/3" in exI)
```
```  3219     apply (rule conjI)
```
```  3220     using enz apply simp
```
```  3221     apply (clarsimp simp only: ball_conj_distrib)
```
```  3222     apply (rule *; assumption)
```
```  3223     done
```
```  3224 qed
```
```  3225
```
```  3226
```
```  3227 lemma
```
```  3228   assumes "open s" "path p" "path_image p \<subseteq> s"
```
```  3229     shows contour_integral_nearby_ends:
```
```  3230       "\<exists>d. 0 < d \<and>
```
```  3231               (\<forall>g h. valid_path g \<and> valid_path h \<and>
```
```  3232                     (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
```
```  3233                     pathstart h = pathstart g \<and> pathfinish h = pathfinish g
```
```  3234                     \<longrightarrow> path_image g \<subseteq> s \<and>
```
```  3235                         path_image h \<subseteq> s \<and>
```
```  3236                         (\<forall>f. f holomorphic_on s
```
```  3237                             \<longrightarrow> contour_integral h f = contour_integral g f))"
```
```  3238     and contour_integral_nearby_loops:
```
```  3239       "\<exists>d. 0 < d \<and>
```
```  3240               (\<forall>g h. valid_path g \<and> valid_path h \<and>
```
```  3241                     (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
```
```  3242                     pathfinish g = pathstart g \<and> pathfinish h = pathstart h
```
```  3243                     \<longrightarrow> path_image g \<subseteq> s \<and>
```
```  3244                         path_image h \<subseteq> s \<and>
```
```  3245                         (\<forall>f. f holomorphic_on s
```
```  3246                             \<longrightarrow> contour_integral h f = contour_integral g f))"
```
```  3247   using contour_integral_nearby [OF assms, where atends=True]
```
```  3248   using contour_integral_nearby [OF assms, where atends=False]
```
```  3249   unfolding linked_paths_def by simp_all
```
```  3250
```
```  3251 lemma C1_differentiable_polynomial_function:
```
```  3252   fixes p :: "real \<Rightarrow> 'a::euclidean_space"
```
```  3253   shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on s"
```
```  3254   by (metis continuous_on_polymonial_function C1_differentiable_on_def  has_vector_derivative_polynomial_function)
```
```  3255
```
```  3256 lemma valid_path_polynomial_function:
```
```  3257   fixes p :: "real \<Rightarrow> 'a::euclidean_space"
```
```  3258   shows "polynomial_function p \<Longrightarrow> valid_path p"
```
```  3259 by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)
```
```  3260
```
```  3261 lemma valid_path_subpath_trivial [simp]:
```
```  3262     fixes g :: "real \<Rightarrow> 'a::euclidean_space"
```
```  3263     shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
```
```  3264   by (simp add: subpath_def valid_path_polynomial_function)
```
```  3265
```
```  3266 lemma contour_integral_bound_exists:
```
```  3267 assumes s: "open s"
```
```  3268     and g: "valid_path g"
```
```  3269     and pag: "path_image g \<subseteq> s"
```
```  3270   shows "\<exists>L. 0 < L \<and>
```
```  3271        (\<forall>f B. f holomorphic_on s \<and> (\<forall>z \<in> s. norm(f z) \<le> B)
```
```  3272          \<longrightarrow> norm(contour_integral g f) \<le> L*B)"
```
```  3273 proof -
```
```  3274 have "path g" using g
```
```  3275   by (simp add: valid_path_imp_path)
```
```  3276 then obtain d::real and p
```
```  3277   where d: "0 < d"
```
```  3278     and p: "polynomial_function p" "path_image p \<subseteq> s"
```
```  3279     and pi: "\<And>f. f holomorphic_on s \<Longrightarrow> contour_integral g f = contour_integral p f"
```
```  3280   using contour_integral_nearby_ends [OF s \<open>path g\<close> pag]
```
```  3281   apply clarify
```
```  3282   apply (drule_tac x=g in spec)
```
```  3283   apply (simp only: assms)
```
```  3284   apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function)
```
```  3285   done
```
```  3286 then obtain p' where p': "polynomial_function p'"
```
```  3287          "\<And>x. (p has_vector_derivative (p' x)) (at x)"
```
```  3288   by (blast intro: has_vector_derivative_polynomial_function that elim: )
```
```  3289 then have "bounded(p' ` {0..1})"
```
```  3290   using continuous_on_polymonial_function
```
```  3291   by (force simp: intro!: compact_imp_bounded compact_continuous_image)
```
```  3292 then obtain L where L: "L>0" and nop': "\<And>x. x \<in> {0..1} \<Longrightarrow> norm (p' x) \<le> L"
```
```  3293   by (force simp: bounded_pos)
```
```  3294 { fix f B
```
```  3295   assume f: "f holomorphic_on s"
```
```  3296      and B: "\<And>z. z\<in>s \<Longrightarrow> cmod (f z) \<le> B"
```
```  3297   then have "f contour_integrable_on p \<and> valid_path p"
```
```  3298     using p s
```
```  3299     by (blast intro: valid_path_polynomial_function contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on)
```
```  3300   moreover have "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B"
```
```  3301     apply (rule mult_mono)
```
```  3302     apply (subst Derivative.vector_derivative_at; force intro: p' nop')
```
```  3303     using L B p
```
```  3304     apply (auto simp: path_image_def image_subset_iff)
```
```  3305     done
```
```  3306   ultimately have "cmod (contour_integral g f) \<le> L * B"
```
```  3307     apply (simp add: pi [OF f])
```
```  3308     apply (simp add: contour_integral_integral)
```
```  3309     apply (rule order_trans [OF integral_norm_bound_integral])
```
```  3310     apply (auto simp: mult.commute integral_norm_bound_integral contour_integrable_on [symmetric] norm_mult)
```
```  3311     done
```
```  3312 } then
```
```  3313 show ?thesis
```
```  3314   by (force simp: L contour_integral_integral)
```
```  3315 qed
```
```  3316
```
```  3317 text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>
```
```  3318
```
```  3319 subsection\<open>Winding Numbers\<close>
```
```  3320
```
```  3321 definition winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
```
```  3322   "winding_number \<gamma> z \<equiv>
```
```  3323     @n. \<forall>e > 0. \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
```
```  3324                     pathstart p = pathstart \<gamma> \<and>
```
```  3325                     pathfinish p = pathfinish \<gamma> \<and>
```
```  3326                     (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
```
```  3327                     contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3328
```
```  3329 lemma winding_number:
```
```  3330   assumes "path \<gamma>" "z \<notin> path_image \<gamma>" "0 < e"
```
```  3331     shows "\<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
```
```  3332                pathstart p = pathstart \<gamma> \<and>
```
```  3333                pathfinish p = pathfinish \<gamma> \<and>
```
```  3334                (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
```
```  3335                contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * winding_number \<gamma> z"
```
```  3336 proof -
```
```  3337   have "path_image \<gamma> \<subseteq> UNIV - {z}"
```
```  3338     using assms by blast
```
```  3339   then obtain d
```
```  3340     where d: "d>0"
```
```  3341       and pi_eq: "\<And>h1 h2. valid_path h1 \<and> valid_path h2 \<and>
```
```  3342                     (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d) \<and>
```
```  3343                     pathstart h2 = pathstart h1 \<and> pathfinish h2 = pathfinish h1 \<longrightarrow>
```
```  3344                       path_image h1 \<subseteq> UNIV - {z} \<and> path_image h2 \<subseteq> UNIV - {z} \<and>
```
```  3345                       (\<forall>f. f holomorphic_on UNIV - {z} \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
```
```  3346     using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms by (auto simp: open_delete)
```
```  3347   then obtain h where h: "polynomial_function h \<and> pathstart h = pathstart \<gamma> \<and> pathfinish h = pathfinish \<gamma> \<and>
```
```  3348                           (\<forall>t \<in> {0..1}. norm(h t - \<gamma> t) < d/2)"
```
```  3349     using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "d/2"] d by auto
```
```  3350   define nn where "nn = 1/(2* pi*\<i>) * contour_integral h (\<lambda>w. 1/(w - z))"
```
```  3351   have "\<exists>n. \<forall>e > 0. \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
```
```  3352                         pathstart p = pathstart \<gamma> \<and>  pathfinish p = pathfinish \<gamma> \<and>
```
```  3353                         (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
```
```  3354                         contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3355                     (is "\<exists>n. \<forall>e > 0. ?PP e n")
```
```  3356     proof (rule_tac x=nn in exI, clarify)
```
```  3357       fix e::real
```
```  3358       assume e: "e>0"
```
```  3359       obtain p where p: "polynomial_function p \<and>
```
```  3360             pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and> (\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < min e (d / 2))"
```
```  3361         using path_approx_polynomial_function [OF \<open>path \<gamma>\<close>, of "min e (d/2)"] d \<open>0<e\<close> by auto
```
```  3362       have "(\<lambda>w. 1 / (w - z)) holomorphic_on UNIV - {z}"
```
```  3363         by (auto simp: intro!: holomorphic_intros)
```
```  3364       then show "?PP e nn"
```
```  3365         apply (rule_tac x=p in exI)
```
```  3366         using pi_eq [of h p] h p d
```
```  3367         apply (auto simp: valid_path_polynomial_function norm_minus_commute nn_def)
```
```  3368         done
```
```  3369     qed
```
```  3370   then show ?thesis
```
```  3371     unfolding winding_number_def
```
```  3372     apply (rule someI2_ex)
```
```  3373     apply (blast intro: \<open>0<e\<close>)
```
```  3374     done
```
```  3375 qed
```
```  3376
```
```  3377 lemma winding_number_unique:
```
```  3378   assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3379       and pi:
```
```  3380         "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
```
```  3381                           pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
```
```  3382                           (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
```
```  3383                           contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3384    shows "winding_number \<gamma> z = n"
```
```  3385 proof -
```
```  3386   have "path_image \<gamma> \<subseteq> UNIV - {z}"
```
```  3387     using assms by blast
```
```  3388   then obtain e
```
```  3389     where e: "e>0"
```
```  3390       and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
```
```  3391                     (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
```
```  3392                     pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
```
```  3393                     contour_integral h2 f = contour_integral h1 f"
```
```  3394     using contour_integral_nearby_ends [of "UNIV - {z}" \<gamma>] assms  by (auto simp: open_delete)
```
```  3395   obtain p where p:
```
```  3396      "valid_path p \<and> z \<notin> path_image p \<and>
```
```  3397       pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
```
```  3398       (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
```
```  3399       contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3400     using pi [OF e] by blast
```
```  3401   obtain q where q:
```
```  3402      "valid_path q \<and> z \<notin> path_image q \<and>
```
```  3403       pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma> \<and>
```
```  3404       (\<forall>t\<in>{0..1}. cmod (\<gamma> t - q t) < e) \<and> contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
```
```  3405     using winding_number [OF \<gamma> e] by blast
```
```  3406   have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
```
```  3407     using p by auto
```
```  3408   also have "... = contour_integral q (\<lambda>w. 1 / (w - z))"
```
```  3409     apply (rule pi_eq)
```
```  3410     using p q
```
```  3411     by (auto simp: valid_path_polynomial_function norm_minus_commute intro!: holomorphic_intros)
```
```  3412   also have "... = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
```
```  3413     using q by auto
```
```  3414   finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
```
```  3415   then show ?thesis
```
```  3416     by simp
```
```  3417 qed
```
```  3418
```
```  3419 lemma winding_number_unique_loop:
```
```  3420   assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3421       and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  3422       and pi:
```
```  3423         "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
```
```  3424                            pathfinish p = pathstart p \<and>
```
```  3425                            (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
```
```  3426                            contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3427    shows "winding_number \<gamma> z = n"
```
```  3428 proof -
```
```  3429   have "path_image \<gamma> \<subseteq> UNIV - {z}"
```
```  3430     using assms by blast
```
```  3431   then obtain e
```
```  3432     where e: "e>0"
```
```  3433       and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
```
```  3434                     (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
```
```  3435                     pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
```
```  3436                     contour_integral h2 f = contour_integral h1 f"
```
```  3437     using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms  by (auto simp: open_delete)
```
```  3438   obtain p where p:
```
```  3439      "valid_path p \<and> z \<notin> path_image p \<and>
```
```  3440       pathfinish p = pathstart p \<and>
```
```  3441       (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
```
```  3442       contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * \<i> * n"
```
```  3443     using pi [OF e] by blast
```
```  3444   obtain q where q:
```
```  3445      "valid_path q \<and> z \<notin> path_image q \<and>
```
```  3446       pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma> \<and>
```
```  3447       (\<forall>t\<in>{0..1}. cmod (\<gamma> t - q t) < e) \<and> contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
```
```  3448     using winding_number [OF \<gamma> e] by blast
```
```  3449   have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
```
```  3450     using p by auto
```
```  3451   also have "... = contour_integral q (\<lambda>w. 1 / (w - z))"
```
```  3452     apply (rule pi_eq)
```
```  3453     using p q loop
```
```  3454     by (auto simp: valid_path_polynomial_function norm_minus_commute intro!: holomorphic_intros)
```
```  3455   also have "... = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
```
```  3456     using q by auto
```
```  3457   finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
```
```  3458   then show ?thesis
```
```  3459     by simp
```
```  3460 qed
```
```  3461
```
```  3462 lemma winding_number_valid_path:
```
```  3463   assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3464     shows "winding_number \<gamma> z = 1/(2*pi*\<i>) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
```
```  3465   using assms by (auto simp: valid_path_imp_path intro!: winding_number_unique)
```
```  3466
```
```  3467 lemma has_contour_integral_winding_number:
```
```  3468   assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3469     shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*\<i>*winding_number \<gamma> z)) \<gamma>"
```
```  3470 by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)
```
```  3471
```
```  3472 lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
```
```  3473   by (simp add: winding_number_valid_path)
```
```  3474
```
```  3475 lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
```
```  3476   by (simp add: path_image_subpath winding_number_valid_path)
```
```  3477
```
```  3478 lemma winding_number_join:
```
```  3479   assumes g1: "path g1" "z \<notin> path_image g1"
```
```  3480       and g2: "path g2" "z \<notin> path_image g2"
```
```  3481       and "pathfinish g1 = pathstart g2"
```
```  3482     shows "winding_number(g1 +++ g2) z = winding_number g1 z + winding_number g2 z"
```
```  3483   apply (rule winding_number_unique)
```
```  3484   using assms apply (simp_all add: not_in_path_image_join)
```
```  3485   apply (frule winding_number [OF g2])
```
```  3486   apply (frule winding_number [OF g1], clarify)
```
```  3487   apply (rename_tac p2 p1)
```
```  3488   apply (rule_tac x="p1+++p2" in exI)
```
```  3489   apply (simp add: not_in_path_image_join contour_integrable_inversediff algebra_simps)
```
```  3490   apply (auto simp: joinpaths_def)
```
```  3491   done
```
```  3492
```
```  3493 lemma winding_number_reversepath:
```
```  3494   assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3495     shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
```
```  3496   apply (rule winding_number_unique)
```
```  3497   using assms
```
```  3498   apply simp_all
```
```  3499   apply (frule winding_number [OF assms], clarify)
```
```  3500   apply (rule_tac x="reversepath p" in exI)
```
```  3501   apply (simp add: contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
```
```  3502   apply (auto simp: reversepath_def)
```
```  3503   done
```
```  3504
```
```  3505 lemma winding_number_shiftpath:
```
```  3506   assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3507       and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
```
```  3508     shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
```
```  3509   apply (rule winding_number_unique_loop)
```
```  3510   using assms
```
```  3511   apply (simp_all add: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath)
```
```  3512   apply (frule winding_number [OF \<gamma>], clarify)
```
```  3513   apply (rule_tac x="shiftpath a p" in exI)
```
```  3514   apply (simp add: contour_integral_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath valid_path_shiftpath)
```
```  3515   apply (auto simp: shiftpath_def)
```
```  3516   done
```
```  3517
```
```  3518 lemma winding_number_split_linepath:
```
```  3519   assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
```
```  3520     shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
```
```  3521 proof -
```
```  3522   have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
```
```  3523     using assms  apply (meson convex_contains_segment convex_segment ends_in_segment(1) subsetCE)
```
```  3524     using assms  by (meson convex_contains_segment convex_segment ends_in_segment(2) subsetCE)
```
```  3525   then show ?thesis
```
```  3526     using assms
```
```  3527     by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
```
```  3528 qed
```
```  3529
```
```  3530 lemma winding_number_cong:
```
```  3531    "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
```
```  3532   by (simp add: winding_number_def pathstart_def pathfinish_def)
```
```  3533
```
```  3534 lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
```
```  3535   apply (simp add: winding_number_def contour_integral_integral path_image_def valid_path_def pathstart_def pathfinish_def)
```
```  3536   apply (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
```
```  3537   apply (rename_tac g)
```
```  3538   apply (rule_tac x="\<lambda>t. g t - z" in exI)
```
```  3539   apply (force simp: vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
```
```  3540   apply (rename_tac g)
```
```  3541   apply (rule_tac x="\<lambda>t. g t + z" in exI)
```
```  3542   apply (simp add: piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
```
```  3543   apply (force simp: algebra_simps)
```
```  3544   done
```
```  3545
```
```  3546 (* A combined theorem deducing several things piecewise.*)
```
```  3547 lemma winding_number_join_pos_combined:
```
```  3548      "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
```
```  3549        valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
```
```  3550       \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
```
```  3551   by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)
```
```  3552
```
```  3553
```
```  3554 (* Useful sufficient conditions for the winding number to be positive etc.*)
```
```  3555
```
```  3556 lemma Re_winding_number:
```
```  3557     "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
```
```  3558      \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
```
```  3559 by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)
```
```  3560
```
```  3561 lemma winding_number_pos_le:
```
```  3562   assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3563       and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
```
```  3564     shows "0 \<le> Re(winding_number \<gamma> z)"
```
```  3565 proof -
```
```  3566   have *: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
```
```  3567     using ge by (simp add: Complex.Im_divide algebra_simps x)
```
```  3568   show ?thesis
```
```  3569     apply (simp add: Re_winding_number [OF \<gamma>] field_simps)
```
```  3570     apply (rule has_integral_component_nonneg
```
```  3571              [of \<i> "\<lambda>x. if x \<in> {0<..<1}
```
```  3572                          then 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x) else 0", simplified])
```
```  3573       prefer 3 apply (force simp: *)
```
```  3574      apply (simp add: Basis_complex_def)
```
```  3575     apply (rule has_integral_spike_interior [of 0 1 _ "\<lambda>x. 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x)"])
```
```  3576     apply simp
```
```  3577     apply (simp only: box_real)
```
```  3578     apply (subst has_contour_integral [symmetric])
```
```  3579     using \<gamma>
```
```  3580     apply (simp add: contour_integrable_inversediff has_contour_integral_integral)
```
```  3581     done
```
```  3582 qed
```
```  3583
```
```  3584 lemma winding_number_pos_lt_lemma:
```
```  3585   assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3586       and e: "0 < e"
```
```  3587       and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
```
```  3588     shows "0 < Re(winding_number \<gamma> z)"
```
```  3589 proof -
```
```  3590   have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
```
```  3591     apply (rule has_integral_component_le
```
```  3592              [of \<i> "\<lambda>x. \<i>*e" "\<i>*e" "{0..1}"
```
```  3593                     "\<lambda>x. if x \<in> {0<..<1} then 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x) else \<i>*e"
```
```  3594                     "contour_integral \<gamma> (\<lambda>w. 1/(w - z))", simplified])
```
```  3595     using e
```
```  3596     apply (simp_all add: Basis_complex_def)
```
```  3597     using has_integral_const_real [of _ 0 1] apply force
```
```  3598     apply (rule has_integral_spike_interior [of 0 1 _ "\<lambda>x. 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x)", simplified box_real])
```
```  3599     apply simp
```
```  3600     apply (subst has_contour_integral [symmetric])
```
```  3601     using \<gamma>
```
```  3602     apply (simp_all add: contour_integrable_inversediff has_contour_integral_integral ge)
```
```  3603     done
```
```  3604   with e show ?thesis
```
```  3605     by (simp add: Re_winding_number [OF \<gamma>] field_simps)
```
```  3606 qed
```
```  3607
```
```  3608 lemma winding_number_pos_lt:
```
```  3609   assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
```
```  3610       and e: "0 < e"
```
```  3611       and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
```
```  3612     shows "0 < Re (winding_number \<gamma> z)"
```
```  3613 proof -
```
```  3614   have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
```
```  3615     using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
```
```  3616   then obtain B where B: "B > 0" and Bno: "\<And>x. x \<in> (\<lambda>w. w - z) ` (path_image \<gamma>) \<Longrightarrow> norm x \<le> B"
```
```  3617     using bounded_pos [THEN iffD1, OF bm] by blast
```
```  3618   { fix x::real  assume x: "0 < x" "x < 1"
```
```  3619     then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
```
```  3620       by (simp add: path_image_def power2_eq_square mult_mono')
```
```  3621     with x have "\<gamma> x \<noteq> z" using \<gamma>
```
```  3622       using path_image_def by fastforce
```
```  3623     then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
```
```  3624       using B ge [OF x] B2 e
```
```  3625       apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
```
```  3626       apply (auto simp: divide_left_mono divide_right_mono)
```
```  3627       done
```
```  3628     then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
```
```  3629       by (simp add: Im_divide_Reals complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
```
```  3630   } note * = this
```
```  3631   show ?thesis
```
```  3632     using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
```
```  3633 qed
```
```  3634
```
```  3635 subsection\<open>The winding number is an integer\<close>
```
```  3636
```
```  3637 text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
```
```  3638      Also on page 134 of Serge Lang's book with the name title, etc.\<close>
```
```  3639
```
```  3640 lemma exp_fg:
```
```  3641   fixes z::complex
```
```  3642   assumes g: "(g has_vector_derivative g') (at x within s)"
```
```  3643       and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
```
```  3644       and z: "g x \<noteq> z"
```
```  3645     shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
```
```  3646 proof -
```
```  3647   have *: "(exp o (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
```
```  3648     using assms unfolding has_vector_derivative_def scaleR_conv_of_real
```
```  3649     by (auto intro!: derivative_eq_intros)
```
```  3650   show ?thesis
```
```  3651     apply (rule has_vector_derivative_eq_rhs)
```
```  3652     apply (rule bounded_bilinear.has_vector_derivative [OF bounded_bilinear_mult])
```
```  3653     using z
```
```  3654     apply (auto simp: intro!: derivative_eq_intros * [unfolded o_def] g)
```
```  3655     done
```
```  3656 qed
```
```  3657
```
```  3658 lemma winding_number_exp_integral:
```
```  3659   fixes z::complex
```
```  3660   assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
```
```  3661       and ab: "a \<le> b"
```
```  3662       and z: "z \<notin> \<gamma> ` {a..b}"
```
```  3663     shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
```
```  3664           (is "?thesis1")
```
```  3665           "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
```
```  3666           (is "?thesis2")
```
```  3667 proof -
```
```  3668   let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
```
```  3669   have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
```
```  3670     using z by force
```
```  3671   have cong: "continuous_on {a..b} \<gamma>"
```
```  3672     using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
```
```  3673   obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
```
```  3674     using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
```
```  3675   have o: "open ({a<..<b} - k)"
```
```  3676     using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
```
```  3677   moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
```
```  3678     by force
```
```  3679   ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
```
```  3680     by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def at_within_open)
```
```  3681   { fix w
```
```  3682     assume "w \<noteq> z"
```
```  3683     have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
```
```  3684       by (auto simp: dist_norm intro!: continuous_intros)
```
```  3685     moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
```
```  3686       by (auto simp: intro!: derivative_eq_intros)
```
```  3687     ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
```
```  3688       using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
```
```  3689       by (simp add: field_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
```
```  3690   }
```
```  3691   then obtain h where h: "\<And>w y. w \<noteq> z \<Longrightarrow> norm(y - w) < norm(w - z) \<Longrightarrow> (h w has_field_derivative 1/(y - z)) (at y)"
```
```  3692     by meson
```
```  3693   have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
```
```  3694     unfolding integrable_on_def [symmetric]
```
```  3695     apply (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>], of "-{z}"])
```
```  3696     apply (rename_tac w)
```
```  3697     apply (rule_tac x="norm(w - z)" in exI)
```
```  3698     apply (simp_all add: inverse_eq_divide)
```
```  3699     apply (metis has_field_derivative_at_within h)
```
```  3700     done
```
```  3701   have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
```
```  3702     unfolding box_real [symmetric] divide_inverse_commute
```
```  3703     by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
```
```  3704   with ab show ?thesis1
```
```  3705     by (simp add: divide_inverse_commute integral_def integrable_on_def)
```
```  3706   { fix t
```
```  3707     assume t: "t \<in> {a..b}"
```
```  3708     have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
```
```  3709         using z by (auto intro!: continuous_intros simp: dist_norm)
```
```  3710     have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) field_differentiable at x"
```
```  3711       unfolding field_differentiable_def by (force simp: intro!: derivative_eq_intros)
```
```  3712     obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
```
```  3713                        (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
```
```  3714       using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
```
```  3715       by simp (auto simp: ball_def dist_norm that)
```
```  3716     { fix x D
```
```  3717       assume x: "x \<notin> k" "a < x" "x < b"
```
```  3718       then have "x \<in> interior ({a..b} - k)"
```
```  3719         using open_subset_interior [OF o] by fastforce
```
```  3720       then have con: "isCont (\<lambda>x. ?D\<gamma> x) x"
```
```  3721         using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
```
```  3722       then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
```
```  3723         by (rule continuous_at_imp_continuous_within)
```
```  3724       have gdx: "\<gamma> differentiable at x"
```
```  3725         using x by (simp add: g_diff_at)
```
```  3726       have "((\<lambda>c. exp (- integral {a..c} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))) * (\<gamma> c - z)) has_derivative (\<lambda>h. 0))
```
```  3727           (at x within {a..b})"
```
```  3728         using x gdx t
```
```  3729         apply (clarsimp simp add: differentiable_iff_scaleR)
```
```  3730         apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_within)
```
```  3731         apply (simp_all add: has_vector_derivative_def [symmetric])
```
```  3732         apply (rule has_vector_derivative_eq_rhs [OF integral_has_vector_derivative_continuous_at])
```
```  3733         apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
```
```  3734         done
```
```  3735       } note * = this
```
```  3736     have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
```
```  3737       apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
```
```  3738       using t
```
```  3739       apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous [OF vg_int]  simp add: ab)+
```
```  3740       done
```
```  3741    }
```
```  3742   with ab show ?thesis2
```
```  3743     by (simp add: divide_inverse_commute integral_def)
```
```  3744 qed
```
```  3745
```
```  3746 corollary winding_number_exp_2pi:
```
```  3747     "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
```
```  3748      \<Longrightarrow> pathfinish p - z = exp (2 * pi * \<i> * winding_number p z) * (pathstart p - z)"
```
```  3749 using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def
```
```  3750   by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)
```
```  3751
```
```  3752
```
```  3753 subsection\<open>The version with complex integers and equality\<close>
```
```  3754
```
```  3755 lemma integer_winding_number_eq:
```
```  3756   assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
```
```  3757   shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
```
```  3758 proof -
```
```  3759   have *: "\<And>i::complex. \<And>g0 g1. \<lbrakk>i \<noteq> 0; g0 \<noteq> z; (g1 - z) / i = g0 - z\<rbrakk> \<Longrightarrow> (i = 1 \<longleftrightarrow> g1 = g0)"
```
```  3760       by (simp add: field_simps) algebra
```
```  3761   obtain p where p: "valid_path p" "z \<notin> path_image p"
```
```  3762                     "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
```
```  3763                     "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
```
```  3764     using winding_number [OF assms, of 1] by auto
```
```  3765   have [simp]: "(winding_number \<gamma> z \<in> \<int>) = (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
```
```  3766       using p by (simp add: exp_eq_1 complex_is_Int_iff)
```
```  3767   have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
```
```  3768     using p z
```
```  3769     apply (simp add: winding_number_valid_path valid_path_def path_image_def pathstart_def pathfinish_def)
```
```  3770     using winding_number_exp_integral(2) [of p 0 1 z]
```
```  3771     apply (simp add: field_simps contour_integral_integral exp_minus)
```
```  3772     apply (rule *)
```
```  3773     apply (auto simp: path_image_def field_simps)
```
```  3774     done
```
```  3775   then show ?thesis using p
```
```  3776     by (auto simp: winding_number_valid_path)
```
```  3777 qed
```
```  3778
```
```  3779 theorem integer_winding_number:
```
```  3780   "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
```
```  3781 by (metis integer_winding_number_eq)
```
```  3782
```
```  3783
```
```  3784 text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
```
```  3785    We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>
```
```  3786
```
```  3787 lemma winding_number_pos_meets:
```
```  3788   fixes z::complex
```
```  3789   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
```
```  3790       and w: "w \<noteq> z"
```
```  3791   shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
```
```  3792 proof -
```
```  3793   have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
```
```  3794     using z by (auto simp: path_image_def)
```
```  3795   have [simp]: "z \<notin> \<gamma> ` {0..1}"
```
```  3796     using path_image_def z by auto
```
```  3797   have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
```
```  3798     using \<gamma> valid_path_def by blast
```
```  3799   define r where "r = (w - z) / (\<gamma> 0 - z)"
```
```  3800   have [simp]: "r \<noteq> 0"
```
```  3801     using w z by (auto simp: r_def)
```
```  3802   have "Arg r \<le> 2*pi"
```
```  3803     by (simp add: Arg less_eq_real_def)
```
```  3804   also have "... \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
```
```  3805     using 1
```
```  3806     apply (simp add: winding_number_valid_path [OF \<gamma> z] contour_integral_integral)
```
```  3807     apply (simp add: Complex.Re_divide field_simps power2_eq_square)
```
```  3808     done
```
```  3809   finally have "Arg r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
```
```  3810   then have "\<exists>t. t \<in> {0..1} \<and> Im(integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg r"
```
```  3811     apply (simp add:)
```
```  3812     apply (rule IVT')
```
```  3813     apply (simp_all add: Arg_ge_0)
```
```  3814     apply (intro continuous_intros indefinite_integral_continuous winding_number_exp_integral [OF gpd]; simp)
```
```  3815     done
```
```  3816   then obtain t where t:     "t \<in> {0..1}"
```
```  3817                   and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg r"
```
```  3818     by blast
```
```  3819   define i where "i = integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
```
```  3820   have iArg: "Arg r = Im i"
```
```  3821     using eqArg by (simp add: i_def)
```
```  3822   have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
```
```  3823     by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
```
```  3824   have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
```
```  3825     unfolding i_def
```
```  3826     apply (rule winding_number_exp_integral [OF gpdt])
```
```  3827     using t z unfolding path_image_def
```
```  3828     apply force+
```
```  3829     done
```
```  3830   then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
```
```  3831     by (simp add: exp_minus field_simps)
```
```  3832   then have "(w - z) = r * (\<gamma> 0 - z)"
```
```  3833     by (simp add: r_def)
```
```  3834   then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
```
```  3835     apply (simp add:)
```
```  3836     apply (subst Complex_Transcendental.Arg_eq [of r])
```
```  3837     apply (simp add: iArg)
```
```  3838     using *
```
```  3839     apply (simp add: exp_eq_polar field_simps)
```
```  3840     done
```
```  3841   with t show ?thesis
```
```  3842     by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
```
```  3843 qed
```
```  3844
```
```  3845 lemma winding_number_big_meets:
```
```  3846   fixes z::complex
```
```  3847   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
```
```  3848       and w: "w \<noteq> z"
```
```  3849   shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
```
```  3850 proof -
```
```  3851   { assume "Re (winding_number \<gamma> z) \<le> - 1"
```
```  3852     then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
```
```  3853       by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
```
```  3854     moreover have "valid_path (reversepath \<gamma>)"
```
```  3855       using \<gamma> valid_path_imp_reverse by auto
```
```  3856     moreover have "z \<notin> path_image (reversepath \<gamma>)"
```
```  3857       by (simp add: z)
```
```  3858     ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
```
```  3859       using winding_number_pos_meets w by blast
```
```  3860     then have ?thesis
```
```  3861       by simp
```
```  3862   }
```
```  3863   then show ?thesis
```
```  3864     using assms
```
```  3865     by (simp add: abs_if winding_number_pos_meets split: if_split_asm)
```
```  3866 qed
```
```  3867
```
```  3868 lemma winding_number_less_1:
```
```  3869   fixes z::complex
```
```  3870   shows
```
```  3871   "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
```
```  3872     \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
```
```  3873    \<Longrightarrow> Re(winding_number \<gamma> z) < 1"
```
```  3874    by (auto simp: not_less dest: winding_number_big_meets)
```
```  3875
```
```  3876 text\<open>One way of proving that WN=1 for a loop.\<close>
```
```  3877 lemma winding_number_eq_1:
```
```  3878   fixes z::complex
```
```  3879   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  3880       and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
```
```  3881   shows "winding_number \<gamma> z = 1"
```
```  3882 proof -
```
```  3883   have "winding_number \<gamma> z \<in> Ints"
```
```  3884     by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
```
```  3885   then show ?thesis
```
```  3886     using 0 2 by (auto simp: Ints_def)
```
```  3887 qed
```
```  3888
```
```  3889
```
```  3890 subsection\<open>Continuity of winding number and invariance on connected sets.\<close>
```
```  3891
```
```  3892 lemma continuous_at_winding_number:
```
```  3893   fixes z::complex
```
```  3894   assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
```
```  3895   shows "continuous (at z) (winding_number \<gamma>)"
```
```  3896 proof -
```
```  3897   obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
```
```  3898     using open_contains_cball [of "- path_image \<gamma>"]  z
```
```  3899     by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
```
```  3900   then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
```
```  3901     by (force simp: cball_def dist_norm)
```
```  3902   have oc: "open (- cball z (e / 2))"
```
```  3903     by (simp add: closed_def [symmetric])
```
```  3904   obtain d where "d>0" and pi_eq:
```
```  3905     "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
```
```  3906               (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
```
```  3907               pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
```
```  3908              \<Longrightarrow>
```
```  3909                path_image h1 \<subseteq> - cball z (e / 2) \<and>
```
```  3910                path_image h2 \<subseteq> - cball z (e / 2) \<and>
```
```  3911                (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
```
```  3912     using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
```
```  3913   obtain p where p: "valid_path p" "z \<notin> path_image p"
```
```  3914                     "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
```
```  3915               and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
```
```  3916               and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
```
```  3917     using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by auto
```
```  3918   { fix w
```
```  3919     assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
```
```  3920     then have wnotp: "w \<notin> path_image p"
```
```  3921       using cbg \<open>d>0\<close> \<open>e>0\<close>
```
```  3922       apply (simp add: path_image_def cball_def dist_norm, clarify)
```
```  3923       apply (frule pg)
```
```  3924       apply (drule_tac c="\<gamma> x" in subsetD)
```
```  3925       apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
```
```  3926       done
```
```  3927     have wnotg: "w \<notin> path_image \<gamma>"
```
```  3928       using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
```
```  3929     { fix k::real
```
```  3930       assume k: "k>0"
```
```  3931       then obtain q where q: "valid_path q" "w \<notin> path_image q"
```
```  3932                              "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
```
```  3933                     and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
```
```  3934                     and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
```
```  3935         using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
```
```  3936         by (force simp: min_divide_distrib_right)
```
```  3937       have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
```
```  3938         apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
```
```  3939         apply (frule pg)
```
```  3940         apply (frule qg)
```
```  3941         using p q \<open>d>0\<close> e2
```
```  3942         apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
```
```  3943         done
```
```  3944       then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
```
```  3945         by (simp add: pi qi)
```
```  3946     } note pip = this
```
```  3947     have "path p"
```
```  3948       using p by (simp add: valid_path_imp_path)
```
```  3949     then have "winding_number p w = winding_number \<gamma> w"
```
```  3950       apply (rule winding_number_unique [OF _ wnotp])
```
```  3951       apply (rule_tac x=p in exI)
```
```  3952       apply (simp add: p wnotp min_divide_distrib_right pip)
```
```  3953       done
```
```  3954   } note wnwn = this
```
```  3955   obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
```
```  3956     using p open_contains_cball [of "- path_image p"]
```
```  3957     by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
```
```  3958   obtain L
```
```  3959     where "L>0"
```
```  3960       and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
```
```  3961                       \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
```
```  3962                       cmod (contour_integral p f) \<le> L * B"
```
```  3963     using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by force
```
```  3964   { fix e::real and w::complex
```
```  3965     assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
```
```  3966     then have [simp]: "w \<notin> path_image p"
```
```  3967       using cbp p(2) \<open>0 < pe\<close>
```
```  3968       by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
```
```  3969     have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
```
```  3970                   contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
```
```  3971       by (simp add: p contour_integrable_inversediff contour_integral_diff)
```
```  3972     { fix x
```
```  3973       assume pe: "3/4 * pe < cmod (z - x)"
```
```  3974       have "cmod (w - x) < pe/4 + cmod (z - x)"
```
```  3975         by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
```
```  3976       then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
```
```  3977       have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
```
```  3978         using norm_diff_triangle_le by blast
```
```  3979       also have "... < pe/4 + cmod (w - x)"
```
```  3980         using w by (simp add: norm_minus_commute)
```
```  3981       finally have "pe/2 < cmod (w - x)"
```
```  3982         using pe by auto
```
```  3983       then have "(pe/2)^2 < cmod (w - x) ^ 2"
```
```  3984         apply (rule power_strict_mono)
```
```  3985         using \<open>pe>0\<close> by auto
```
```  3986       then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
```
```  3987         by (simp add: power_divide)
```
```  3988       have "8 * L * cmod (w - z) < e * pe\<^sup>2"
```
```  3989         using w \<open>L>0\<close> by (simp add: field_simps)
```
```  3990       also have "... < e * 4 * cmod (w - x) * cmod (w - x)"
```
```  3991         using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
```
```  3992       also have "... < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
```
```  3993         using wx
```
```  3994         apply (rule mult_strict_left_mono)
```
```  3995         using pe2 e not_less_iff_gr_or_eq by fastforce
```
```  3996       finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
```
```  3997         by simp
```
```  3998       also have "... \<le> e * cmod (w - x) * cmod (z - x)"
```
```  3999          using e by simp
```
```  4000       finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
```
```  4001       have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
```
```  4002         apply (cases "x=z \<or> x=w")
```
```  4003         using pe \<open>pe>0\<close> w \<open>L>0\<close>
```
```  4004         apply (force simp: norm_minus_commute)
```
```  4005         using wx w(2) \<open>L>0\<close> pe pe2 Lwz
```
```  4006         apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
```
```  4007         done
```
```  4008     } note L_cmod_le = this
```
```  4009     have *: "cmod (contour_integral p (\<lambda>x. 1 / (x - w) - 1 / (x - z))) \<le> L * (e * pe\<^sup>2 / L / 4 * (inverse (pe / 2))\<^sup>2)"
```
```  4010       apply (rule L)
```
```  4011       using \<open>pe>0\<close> w
```
```  4012       apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
```
```  4013       using \<open>pe>0\<close> w \<open>L>0\<close>
```
```  4014       apply (auto simp: cball_def dist_norm field_simps L_cmod_le  simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
```
```  4015       done
```
```  4016     have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
```
```  4017       apply (simp add:)
```
```  4018       apply (rule le_less_trans [OF *])
```
```  4019       using \<open>L>0\<close> e
```
```  4020       apply (force simp: field_simps)
```
```  4021       done
```
```  4022     then have "cmod (winding_number p w - winding_number p z) < e"
```
```  4023       using pi_ge_two e
```
```  4024       by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
```
```  4025   } note cmod_wn_diff = this
```
```  4026   then have "isCont (winding_number p) z"
```
```  4027     apply (simp add: continuous_at_eps_delta, clarify)
```
```  4028     apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
```
```  4029     using \<open>pe>0\<close> \<open>L>0\<close>
```
```  4030     apply (simp add: dist_norm cmod_wn_diff)
```
```  4031     done
```
```  4032   then show ?thesis
```
```  4033     apply (rule continuous_transform_within [where d = "min d e / 2"])
```
```  4034     apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
```
```  4035     done
```
```  4036 qed
```
```  4037
```
```  4038 corollary continuous_on_winding_number:
```
```  4039     "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
```
```  4040   by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)
```
```  4041
```
```  4042
```
```  4043 subsection\<open>The winding number is constant on a connected region\<close>
```
```  4044
```
```  4045 lemma winding_number_constant:
```
```  4046   assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected s" and sg: "s \<inter> path_image \<gamma> = {}"
```
```  4047   obtains k where "\<And>z. z \<in> s \<Longrightarrow> winding_number \<gamma> z = k"
```
```  4048 proof -
```
```  4049   have *: "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
```
```  4050       if ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z" and "y \<in> s" "z \<in> s" for y z
```
```  4051   proof -
```
```  4052     have "winding_number \<gamma> y \<in> \<int>"  "winding_number \<gamma> z \<in>  \<int>"
```
```  4053       using that integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> s\<close> by auto
```
```  4054     with ne show ?thesis
```
```  4055       by (auto simp: Ints_def of_int_diff [symmetric] simp del: of_int_diff)
```
```  4056   qed
```
```  4057   have cont: "continuous_on s (\<lambda>w. winding_number \<gamma> w)"
```
```  4058     using continuous_on_winding_number [OF \<gamma>] sg
```
```  4059     by (meson continuous_on_subset disjoint_eq_subset_Compl)
```
```  4060   show ?thesis
```
```  4061     apply (rule continuous_discrete_range_constant [OF cs cont])
```
```  4062     using "*" zero_less_one apply blast
```
```  4063     by (simp add: that)
```
```  4064 qed
```
```  4065
```
```  4066 lemma winding_number_eq:
```
```  4067      "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> s; z \<in> s; connected s; s \<inter> path_image \<gamma> = {}\<rbrakk>
```
```  4068       \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
```
```  4069   using winding_number_constant by blast
```
```  4070
```
```  4071 lemma open_winding_number_levelsets:
```
```  4072   assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  4073     shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
```
```  4074 proof -
```
```  4075   have op: "open (- path_image \<gamma>)"
```
```  4076     by (simp add: closed_path_image \<gamma> open_Compl)
```
```  4077   { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
```
```  4078     obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
```
```  4079       using open_contains_ball [of "- path_image \<gamma>"] op z
```
```  4080       by blast
```
```  4081     have "\<exists>e>0. \<forall>y. dist y z < e \<longrightarrow> y \<notin> path_image \<gamma> \<and> winding_number \<gamma> y = winding_number \<gamma> z"
```
```  4082       apply (rule_tac x=e in exI)
```
```  4083       using e apply (simp add: dist_norm ball_def norm_minus_commute)
```
```  4084       apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where s = "ball z e"])
```
```  4085       done
```
```  4086   } then
```
```  4087   show ?thesis
```
```  4088     by (auto simp: open_dist)
```
```  4089 qed
```
```  4090
```
```  4091 subsection\<open>Winding number is zero "outside" a curve, in various senses\<close>
```
```  4092
```
```  4093 lemma winding_number_zero_in_outside:
```
```  4094   assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
```
```  4095     shows "winding_number \<gamma> z = 0"
```
```  4096 proof -
```
```  4097   obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
```
```  4098     using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
```
```  4099   obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
```
```  4100     by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
```
```  4101   have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
```
```  4102     apply (rule outside_subset_convex)
```
```  4103     using B subset_ball by auto
```
```  4104   then have wout: "w \<in> outside (path_image \<gamma>)"
```
```  4105     using w by blast
```
```  4106   moreover obtain k where "\<And>z. z \<in> outside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = k"
```
```  4107     using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
```
```  4108     by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
```
```  4109   ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
```
```  4110     using z by blast
```
```  4111   also have "... = 0"
```
```  4112   proof -
```
```  4113     have wnot: "w \<notin> path_image \<gamma>"  using wout by (simp add: outside_def)
```
```  4114     { fix e::real assume "0<e"
```
```  4115       obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
```
```  4116                  and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
```
```  4117                  and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
```
```  4118         using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
```
```  4119       have pip: "path_image p \<subseteq> ball 0 (B + 1)"
```
```  4120         using B
```
```  4121         apply (clarsimp simp add: path_image_def dist_norm ball_def)
```
```  4122         apply (frule (1) pg1)
```
```  4123         apply (fastforce dest: norm_add_less)
```
```  4124         done
```
```  4125       then have "w \<notin> path_image p"  using w by blast
```
```  4126       then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
```
```  4127                      pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
```
```  4128                      (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
```
```  4129         apply (rule_tac x=p in exI)
```
```  4130         apply (simp add: p valid_path_polynomial_function)
```
```  4131         apply (intro conjI)
```
```  4132         using pge apply (simp add: norm_minus_commute)
```
```  4133         apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
```
```  4134         apply (rule holomorphic_intros | simp add: dist_norm)+
```
```  4135         using mem_ball_0 w apply blast
```
```  4136         using p apply (simp_all add: valid_path_polynomial_function loop pip)
```
```  4137         done
```
```  4138     }
```
```  4139     then show ?thesis
```
```  4140       by (auto intro: winding_number_unique [OF \<gamma>] simp add: wnot)
```
```  4141   qed
```
```  4142   finally show ?thesis .
```
```  4143 qed
```
```  4144
```
```  4145 lemma winding_number_zero_outside:
```
```  4146     "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
```
```  4147   by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)
```
```  4148
```
```  4149 lemma winding_number_zero_at_infinity:
```
```  4150   assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  4151     shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
```
```  4152 proof -
```
```  4153   obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
```
```  4154     using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
```
```  4155   then show ?thesis
```
```  4156     apply (rule_tac x="B+1" in exI, clarify)
```
```  4157     apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
```
```  4158     apply (meson less_add_one mem_cball_0 not_le order_trans)
```
```  4159     using ball_subset_cball by blast
```
```  4160 qed
```
```  4161
```
```  4162 lemma winding_number_zero_point:
```
```  4163     "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; open s; path_image \<gamma> \<subseteq> s\<rbrakk>
```
```  4164      \<Longrightarrow> \<exists>z. z \<in> s \<and> winding_number \<gamma> z = 0"
```
```  4165   using outside_compact_in_open [of "path_image \<gamma>" s] path_image_nonempty winding_number_zero_in_outside
```
```  4166   by (fastforce simp add: compact_path_image)
```
```  4167
```
```  4168
```
```  4169 text\<open>If a path winds round a set, it winds rounds its inside.\<close>
```
```  4170 lemma winding_number_around_inside:
```
```  4171   assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  4172       and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
```
```  4173       and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
```
```  4174     shows "winding_number \<gamma> w = winding_number \<gamma> z"
```
```  4175 proof -
```
```  4176   have ssb: "s \<subseteq> inside(path_image \<gamma>)"
```
```  4177   proof
```
```  4178     fix x :: complex
```
```  4179     assume "x \<in> s"
```
```  4180     hence "x \<notin> path_image \<gamma>"
```
```  4181       by (meson disjoint_iff_not_equal s_disj)
```
```  4182     thus "x \<in> inside (path_image \<gamma>)"
```
```  4183       using \<open>x \<in> s\<close> by (metis (no_types) ComplI UnE cos \<gamma> loop s_disj union_with_outside winding_number_eq winding_number_zero_in_outside wn_nz z)
```
```  4184 qed
```
```  4185   show ?thesis
```
```  4186     apply (rule winding_number_eq [OF \<gamma> loop w])
```
```  4187     using z apply blast
```
```  4188     apply (simp add: cls connected_with_inside cos)
```
```  4189     apply (simp add: Int_Un_distrib2 s_disj, safe)
```
```  4190     by (meson ssb inside_inside_compact_connected [OF cls, of "path_image \<gamma>"] compact_path_image connected_path_image contra_subsetD disjoint_iff_not_equal \<gamma> inside_no_overlap)
```
```  4191  qed
```
```  4192
```
```  4193
```
```  4194 text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
```
```  4195 lemma winding_number_subpath_continuous:
```
```  4196   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
```
```  4197     shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
```
```  4198 proof -
```
```  4199   have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
```
```  4200          winding_number (subpath 0 x \<gamma>) z"
```
```  4201          if x: "0 \<le> x" "x \<le> 1" for x
```
```  4202   proof -
```
```  4203     have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
```
```  4204           1 / (2*pi*\<i>) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
```
```  4205       using assms x
```
```  4206       apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
```
```  4207       done
```
```  4208     also have "... = winding_number (subpath 0 x \<gamma>) z"
```
```  4209       apply (subst winding_number_valid_path)
```
```  4210       using assms x
```
```  4211       apply (simp_all add: path_image_subpath valid_path_subpath)
```
```  4212       by (force simp: path_image_def)
```
```  4213     finally show ?thesis .
```
```  4214   qed
```
```  4215   show ?thesis
```
```  4216     apply (rule continuous_on_eq
```
```  4217                  [where f = "\<lambda>x. 1 / (2*pi*\<i>) *
```
```  4218                                  integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
```
```  4219     apply (rule continuous_intros)+
```
```  4220     apply (rule indefinite_integral_continuous)
```
```  4221     apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
```
```  4222       using assms
```
```  4223     apply (simp add: *)
```
```  4224     done
```
```  4225 qed
```
```  4226
```
```  4227 lemma winding_number_ivt_pos:
```
```  4228     assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
```
```  4229       shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
```
```  4230   apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right])
```
```  4231   apply (simp add:)
```
```  4232   apply (rule winding_number_subpath_continuous [OF \<gamma> z])
```
```  4233   using assms
```
```  4234   apply (auto simp: path_image_def image_def)
```
```  4235   done
```
```  4236
```
```  4237 lemma winding_number_ivt_neg:
```
```  4238     assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
```
```  4239       shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
```
```  4240   apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right])
```
```  4241   apply (simp add:)
```
```  4242   apply (rule winding_number_subpath_continuous [OF \<gamma> z])
```
```  4243   using assms
```
```  4244   apply (auto simp: path_image_def image_def)
```
```  4245   done
```
```  4246
```
```  4247 lemma winding_number_ivt_abs:
```
```  4248     assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> \<bar>Re(winding_number \<gamma> z)\<bar>"
```
```  4249       shows "\<exists>t \<in> {0..1}. \<bar>Re (winding_number (subpath 0 t \<gamma>) z)\<bar> = w"
```
```  4250   using assms winding_number_ivt_pos [of \<gamma> z w] winding_number_ivt_neg [of \<gamma> z "-w"]
```
```  4251   by force
```
```  4252
```
```  4253 lemma winding_number_lt_half_lemma:
```
```  4254   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and az: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
```
```  4255     shows "Re(winding_number \<gamma> z) < 1/2"
```
```  4256 proof -
```
```  4257   { assume "Re(winding_number \<gamma> z) \<ge> 1/2"
```
```  4258     then obtain t::real where t: "0 \<le> t" "t \<le> 1" and sub12: "Re (winding_number (subpath 0 t \<gamma>) z) = 1/2"
```
```  4259       using winding_number_ivt_pos [OF \<gamma> z, of "1/2"] by auto
```
```  4260     have gt: "\<gamma> t - z = - (of_real (exp (- (2 * pi * Im (winding_number (subpath 0 t \<gamma>) z)))) * (\<gamma> 0 - z))"
```
```  4261       using winding_number_exp_2pi [of "subpath 0 t \<gamma>" z]
```
```  4262       apply (simp add: t \<gamma> valid_path_imp_path)
```
```  4263       using closed_segment_eq_real_ivl path_image_def t z by (fastforce simp: path_image_subpath Euler sub12)
```
```  4264     have "b < a \<bullet> \<gamma> 0"
```
```  4265     proof -
```
```  4266       have "\<gamma> 0 \<in> {c. b < a \<bullet> c}"
```
```  4267         by (metis (no_types) pag atLeastAtMost_iff image_subset_iff order_refl path_image_def zero_le_one)
```
```  4268       thus ?thesis
```
```  4269         by blast
```
```  4270     qed
```
```  4271     moreover have "b < a \<bullet> \<gamma> t"
```
```  4272     proof -
```
```  4273       have "\<gamma> t \<in> {c. b < a \<bullet> c}"
```
```  4274         by (metis (no_types) pag atLeastAtMost_iff image_subset_iff path_image_def t)
```
```  4275       thus ?thesis
```
```  4276         by blast
```
```  4277     qed
```
```  4278     ultimately have "0 < a \<bullet> (\<gamma> 0 - z)" "0 < a \<bullet> (\<gamma> t - z)" using az
```
```  4279       by (simp add: inner_diff_right)+
```
```  4280     then have False
```
```  4281       by (simp add: gt inner_mult_right mult_less_0_iff)
```
```  4282   }
```
```  4283   then show ?thesis by force
```
```  4284 qed
```
```  4285
```
```  4286 lemma winding_number_lt_half:
```
```  4287   assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
```
```  4288     shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
```
```  4289 proof -
```
```  4290   have "z \<notin> path_image \<gamma>" using assms by auto
```
```  4291   with assms show ?thesis
```
```  4292     apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
```
```  4293     apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
```
```  4294                  winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
```
```  4295     done
```
```  4296 qed
```
```  4297
```
```  4298 lemma winding_number_le_half:
```
```  4299   assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
```
```  4300       and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
```
```  4301     shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
```
```  4302 proof -
```
```  4303   { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
```
```  4304     have "isCont (winding_number \<gamma>) z"
```
```  4305       by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
```
```  4306     then obtain d where "d>0" and d: "\<And>x'. dist x' z < d \<Longrightarrow> dist (winding_number \<gamma> x') (winding_number \<gamma> z) < \<bar>Re(winding_number \<gamma> z)\<bar> - 1/2"
```
```  4307       using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
```
```  4308     define z' where "z' = z - (d / (2 * cmod a)) *\<^sub>R a"
```
```  4309     have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
```
```  4310       unfolding z'_def inner_mult_right' divide_inverse
```
```  4311       apply (simp add: divide_simps algebra_simps dot_square_norm power2_eq_square anz)
```
```  4312       apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
```
```  4313       done
```
```  4314     have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
```
```  4315       using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
```
```  4316     then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
```
```  4317       by simp
```
```  4318     then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
```
```  4319       using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
```
```  4320     then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
```
```  4321       by linarith
```
```  4322     moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
```
```  4323       apply (rule winding_number_lt_half [OF \<gamma> *])
```
```  4324       using azb \<open>d>0\<close> pag
```
```  4325       apply (auto simp: add_strict_increasing anz divide_simps algebra_simps dest!: subsetD)
```
```  4326       done
```
```  4327     ultimately have False
```
```  4328       by simp
```
```  4329   }
```
```  4330   then show ?thesis by force
```
```  4331 qed
```
```  4332
```
```  4333 lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
```
```  4334   using separating_hyperplane_closed_point [of "closed_segment a b" z]
```
```  4335   apply auto
```
```  4336   apply (simp add: closed_segment_def)
```
```  4337   apply (drule less_imp_le)
```
```  4338   apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
```
```  4339   apply (auto simp: segment)
```
```  4340   done
```
```  4341
```
```  4342
```
```  4343 text\<open> Positivity of WN for a linepath.\<close>
```
```  4344 lemma winding_number_linepath_pos_lt:
```
```  4345     assumes "0 < Im ((b - a) * cnj (b - z))"
```
```  4346       shows "0 < Re(winding_number(linepath a b) z)"
```
```  4347 proof -
```
```  4348   have z: "z \<notin> path_image (linepath a b)"
```
```  4349     using assms
```
```  4350     by (simp add: closed_segment_def) (force simp: algebra_simps)
```
```  4351   show ?thesis
```
```  4352     apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
```
```  4353     apply (simp add: linepath_def algebra_simps)
```
```  4354     done
```
```  4355 qed
```
```  4356
```
```  4357
```
```  4358 subsection\<open>Cauchy's integral formula, again for a convex enclosing set.\<close>
```
```  4359
```
```  4360 lemma Cauchy_integral_formula_weak:
```
```  4361     assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
```
```  4362         and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f field_differentiable at x)"
```
```  4363         and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
```
```  4364         and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
```
```  4365       shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
```
```  4366 proof -
```
```  4367   obtain f' where f': "(f has_field_derivative f') (at z)"
```
```  4368     using fcd [OF z] by (auto simp: field_differentiable_def)
```
```  4369   have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
```
```  4370   have c: "continuous (at x within s) (\<lambda>w. if w = z then f' else (f w - f z) / (w - z))" if "x \<in> s" for x
```
```  4371   proof (cases "x = z")
```
```  4372     case True then show ?thesis
```
```  4373       apply (simp add: continuous_within)
```
```  4374       apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
```
```  4375       using has_field_derivative_at_within DERIV_within_iff f'
```
```  4376       apply (fastforce simp add:)+
```
```  4377       done
```
```  4378   next
```
```  4379     case False
```
```  4380     then have dxz: "dist x z > 0" by auto
```
```  4381     have cf: "continuous (at x within s) f"
```
```  4382       using conf continuous_on_eq_continuous_within that by blast
```
```  4383     have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
```
```  4384       by (rule cf continuous_intros | simp add: False)+
```
```  4385     then show ?thesis
```
```  4386       apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
```
```  4387       apply (force simp: dist_commute)
```
```  4388       done
```
```  4389   qed
```
```  4390   have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
```
```  4391   have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
```
```  4392     apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
```
```  4393     using c apply (force simp: continuous_on_eq_continuous_within)
```
```  4394     apply (rename_tac w)
```
```  4395     apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in field_differentiable_transform_within)
```
```  4396     apply (simp_all add: dist_pos_lt dist_commute)
```
```  4397     apply (metis less_irrefl)
```
```  4398     apply (rule derivative_intros fcd | simp)+
```
```  4399     done
```
```  4400   show ?thesis
```
```  4401     apply (rule has_contour_integral_eq)
```
```  4402     using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
```
```  4403     apply (auto simp: mult_ac divide_simps)
```
```  4404     done
```
```  4405 qed
```
```  4406
```
```  4407 theorem Cauchy_integral_formula_convex_simple:
```
```  4408     "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
```
```  4409       pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
```
```  4410      \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * \<i> * winding_number \<gamma> z * f z)) \<gamma>"
```
```  4411   apply (rule Cauchy_integral_formula_weak [where k = "{}"])
```
```  4412   using holomorphic_on_imp_continuous_on
```
```  4413   by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)
```
```  4414
```
```  4415
```
```  4416 subsection\<open>Homotopy forms of Cauchy's theorem\<close>
```
```  4417
```
```  4418 proposition Cauchy_theorem_homotopic:
```
```  4419     assumes hom: "if atends then homotopic_paths s g h else homotopic_loops s g h"
```
```  4420         and "open s" and f: "f holomorphic_on s"
```
```  4421         and vpg: "valid_path g" and vph: "valid_path h"
```
```  4422     shows "contour_integral g f = contour_integral h f"
```
```  4423 proof -
```
```  4424   have pathsf: "linked_paths atends g h"
```
```  4425     using hom  by (auto simp: linked_paths_def homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish homotopic_loops_imp_loop)
```
```  4426   obtain k :: "real \<times> real \<Rightarrow> complex"
```
```  4427     where contk: "continuous_on ({0..1} \<times> {0..1}) k"
```
```  4428       and ks: "k ` ({0..1} \<times> {0..1}) \<subseteq> s"
```
```  4429       and k [simp]: "\<forall>x. k (0, x) = g x" "\<forall>x. k (1, x) = h x"
```
```  4430       and ksf: "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))"
```
```  4431       using hom pathsf by (auto simp: linked_paths_def homotopic_paths_def homotopic_loops_def homotopic_with_def split: if_split_asm)
```
```  4432   have ucontk: "uniformly_continuous_on ({0..1} \<times> {0..1}) k"
```
```  4433     by (blast intro: compact_Times compact_uniformly_continuous [OF contk])
```
```  4434   { fix t::real assume t: "t \<in> {0..1}"
```
```  4435     have pak: "path (k o (\<lambda>u. (t, u)))"
```
```  4436       unfolding path_def
```
```  4437       apply (rule continuous_intros continuous_on_subset [OF contk])+
```
```  4438       using t by force
```
```  4439     have pik: "path_image (k \<circ> Pair t) \<subseteq> s"
```
```  4440       using ks t by (auto simp: path_image_def)
```
```  4441     obtain e where "e>0" and e:
```
```  4442          "\<And>g h. \<lbrakk>valid_path g; valid_path h;
```
```  4443                   \<forall>u\<in>{0..1}. cmod (g u - (k \<circ> Pair t) u) < e \<and> cmod (h u - (k \<circ> Pair t) u) < e;
```
```  4444                   linked_paths atends g h\<rbrakk>
```
```  4445                  \<Longrightarrow> contour_integral h f = contour_integral g f"
```
```  4446       using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis
```
```  4447     obtain d where "d>0" and d:
```
```  4448         "\<And>x x'. \<lbrakk>x \<in> {0..1} \<times> {0..1}; x' \<in> {0..1} \<times> {0..1}; norm (x'-x) < d\<rbrakk> \<Longrightarrow> norm (k x' - k x) < e/4"
```
```  4449       by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>)
```
```  4450     { fix t1 t2
```
```  4451       assume t1: "0 \<le> t1" "t1 \<le> 1" and t2: "0 \<le> t2" "t2 \<le> 1" and ltd: "\<bar>t1 - t\<bar> < d" "\<bar>t2 - t\<bar> < d"
```
```  4452       have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e"
```
```  4453         using \<open>e > 0\<close>
```
```  4454         apply (rule_tac y = k1 in norm_triangle_half_l)
```
```  4455         apply (auto simp: norm_minus_commute intro: order_less_trans)
```
```  4456         done
```
```  4457       have "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
```
```  4458                           (\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and>
```
```  4459                           linked_paths atends g1 g2 \<longrightarrow>
```
```  4460                           contour_integral g2 f = contour_integral g1 f"
```
```  4461         apply (rule_tac x="e/4" in exI)
```
```  4462         using t t1 t2 ltd \<open>e > 0\<close>
```
```  4463         apply (auto intro!: e simp: d no2 simp del: less_divide_eq_numeral1)
```
```  4464         done
```
```  4465     }
```
```  4466     then have "\<exists>e. 0 < e \<and>
```
```  4467               (\<forall>t1 t2. t1 \<in> {0..1} \<and> t2 \<in> {0..1} \<and> \<bar>t1 - t\<bar> < e \<and> \<bar>t2 - t\<bar> < e
```
```  4468                 \<longrightarrow> (\<exists>d. 0 < d \<and>
```
```  4469                      (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
```
```  4470                        (\<forall>u \<in> {0..1}.
```
```  4471                           norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
```
```  4472                           linked_paths atends g1 g2
```
```  4473                           \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
```
```  4474       by (rule_tac x=d in exI) (simp add: \<open>d > 0\<close>)
```
```  4475   }
```
```  4476   then obtain ee where ee:
```
```  4477        "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0 \<and>
```
```  4478           (\<forall>t1 t2. t1 \<in> {0..1} \<longrightarrow> t2 \<in> {0..1} \<longrightarrow> \<bar>t1 - t\<bar> < ee t \<longrightarrow> \<bar>t2 - t\<bar> < ee t
```
```  4479             \<longrightarrow> (\<exists>d. 0 < d \<and>
```
```  4480                  (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
```
```  4481                    (\<forall>u \<in> {0..1}.
```
```  4482                       norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
```
```  4483                       linked_paths atends g1 g2
```
```  4484                       \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
```
```  4485     by metis
```
```  4486   note ee_rule = ee [THEN conjunct2, rule_format]
```
```  4487   define C where "C = (\<lambda>t. ball t (ee t / 3)) ` {0..1}"
```
```  4488   obtain C' where C': "C' \<subseteq> C" "finite C'" and C'01: "{0..1} \<subseteq> \<Union>C'"
```
```  4489   proof (rule compactE [OF compact_interval])
```
```  4490     show "{0..1} \<subseteq> \<Union>C"
```
```  4491       using ee [THEN conjunct1] by (auto simp: C_def dist_norm)
```
```  4492   qed (use C_def in auto)
```
```  4493   define kk where "kk = {t \<in> {0..1}. ball t (ee t / 3) \<in> C'}"
```
```  4494   have kk01: "kk \<subseteq> {0..1}" by (auto simp: kk_def)
```
```  4495   define e where "e = Min (ee ` kk)"
```
```  4496   have C'_eq: "C' = (\<lambda>t. ball t (ee t / 3)) ` kk"
```
```  4497     using C' by (auto simp: kk_def C_def)
```
```  4498   have ee_pos[simp]: "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0"
```
```  4499     by (simp add: kk_def ee)
```
```  4500   moreover have "finite kk"
```
```  4501     using \<open>finite C'\<close> kk01 by (force simp: C'_eq inj_on_def ball_eq_ball_iff dest: ee_pos finite_imageD)
```
```  4502   moreover have "kk \<noteq> {}" using \<open>{0..1} \<subseteq> \<Union>C'\<close> C'_eq by force
```
```  4503   ultimately have "e > 0"
```
```  4504     using finite_less_Inf_iff [of "ee ` kk" 0] kk01 by (force simp: e_def)
```
```  4505   then obtain N::nat where "N > 0" and N: "1/N < e/3"
```
```  4506     by (meson divide_pos_pos nat_approx_posE zero_less_Suc zero_less_numeral)
```
```  4507   have e_le_ee: "\<And>i. i \<in> kk \<Longrightarrow> e \<le> ee i"
```
```  4508     using \<open>finite kk\<close> by (simp add: e_def Min_le_iff [of "ee ` kk"])
```
```  4509   have plus: "\<exists>t \<in> kk. x \<in> ball t (ee t / 3)" if "x \<in> {0..1}" for x
```
```  4510     using C' subsetD [OF C'01 that]  unfolding C'_eq by blast
```
```  4511   have [OF order_refl]:
```
```  4512       "\<exists>d. 0 < d \<and> (\<forall>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (n/N, u)) < d) \<and> linked_paths atends g j
```
```  4513                         \<longrightarrow> contour_integral j f = contour_integral g f)"
```
```  4514        if "n \<le> N" for n
```
```  4515   using that
```
```  4516   proof (induct n)
```
```  4517     case 0 show ?case using ee_rule [of 0 0 0]
```
```  4518       apply clarsimp
```
```  4519       apply (rule_tac x=d in exI, safe)
```
```  4520       by (metis diff_self vpg norm_zero)
```
```  4521   next
```
```  4522     case (Suc n)
```
```  4523     then have N01: "n/N \<in> {0..1}" "(Suc n)/N \<in> {0..1}"  by auto
```
```  4524     then obtain t where t: "t \<in> kk" "n/N \<in> ball t (ee t / 3)"
```
```  4525       using plus [of "n/N"] by blast
```
```  4526     then have nN_less: "\<bar>n/N - t\<bar> < ee t"
```
```  4527       by (simp add: dist_norm del: less_divide_eq_numeral1)
```
```  4528     have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t"
```
```  4529       using t N \<open>N > 0\<close> e_le_ee [of t]
```
```  4530       by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps)
```
```  4531     have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast
```
```  4532     obtain d1 where "d1 > 0" and d1:
```
```  4533         "\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2;
```
```  4534                    \<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1;
```
```  4535                    linked_paths atends g1 g2\<rbrakk>
```
```  4536                    \<Longrightarrow> contour_integral g2 f = contour_integral g1 f"
```
```  4537       using ee [THEN conjunct2, rule_format, OF t01 N01 nN_less n'N_less] by fastforce
```
```  4538     have "n \<le> N" using Suc.prems by auto
```
```  4539     with Suc.hyps
```
```  4540     obtain d2 where "d2 > 0"
```
```  4541       and d2: "\<And>j. \<lbrakk>valid_path j; \<forall>u\<in>{0..1}. cmod (j u - k (n/N, u)) < d2; linked_paths atends g j\<rbrakk>
```
```  4542                      \<Longrightarrow> contour_integral j f = contour_integral g f"
```
```  4543         by auto
```
```  4544     have "continuous_on {0..1} (k o (\<lambda>u. (n/N, u)))"
```
```  4545       apply (rule continuous_intros continuous_on_subset [OF contk])+
```
```  4546       using N01 by auto
```
```  4547     then have pkn: "path (\<lambda>u. k (n/N, u))"
```
```  4548       by (simp add: path_def)
```
```  4549     have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
```
```  4550     obtain p where "polynomial_function p"
```
```  4551         and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))"
```
```  4552                  "pathfinish p = pathfinish (\<lambda>u. k (n/N, u))"
```
```  4553         and pk_le:  "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (p t - k (n/N, t)) < min d1 d2"
```
```  4554       using path_approx_polynomial_function [OF pkn min12] by blast
```
```  4555     then have vpp: "valid_path p" using valid_path_polynomial_function by blast
```
```  4556     have lpa: "linked_paths atends g p"
```
```  4557       by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf)
```
```  4558     show ?case
```
```  4559       apply (rule_tac x="min d1 d2" in exI)
```
```  4560       apply (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>, clarify)
```
```  4561       apply (rule_tac s="contour_integral p f" in trans)
```
```  4562       using pk_le N01(1) ksf pathfinish_def pathstart_def
```
```  4563       apply (force intro!: vpp d1 simp add: linked_paths_def psf ksf)
```
```  4564       using pk_le N01 apply (force intro!: vpp d2 lpa simp add: linked_paths_def psf ksf)
```
```  4565       done
```
```  4566   qed
```
```  4567   then obtain d where "0 < d"
```
```  4568                        "\<And>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (1,u)) < d) \<and>
```
```  4569                             linked_paths atends g j
```
```  4570                             \<Longrightarrow> contour_integral j f = contour_integral g f"
```
```  4571     using \<open>N>0\<close> by auto
```
```  4572   then have "linked_paths atends g h \<Longrightarrow> contour_integral h f = contour_integral g f"
```
```  4573     using \<open>N>0\<close> vph by fastforce
```
```  4574   then show ?thesis
```
```  4575     by (simp add: pathsf)
```
```  4576 qed
```
```  4577
```
```  4578 proposition Cauchy_theorem_homotopic_paths:
```
```  4579     assumes hom: "homotopic_paths s g h"
```
```  4580         and "open s" and f: "f holomorphic_on s"
```
```  4581         and vpg: "valid_path g" and vph: "valid_path h"
```
```  4582     shows "contour_integral g f = contour_integral h f"
```
```  4583   using Cauchy_theorem_homotopic [of True s g h] assms by simp
```
```  4584
```
```  4585 proposition Cauchy_theorem_homotopic_loops:
```
```  4586     assumes hom: "homotopic_loops s g h"
```
```  4587         and "open s" and f: "f holomorphic_on s"
```
```  4588         and vpg: "valid_path g" and vph: "valid_path h"
```
```  4589     shows "contour_integral g f = contour_integral h f"
```
```  4590   using Cauchy_theorem_homotopic [of False s g h] assms by simp
```
```  4591
```
```  4592 lemma has_contour_integral_newpath:
```
```  4593     "\<lbrakk>(f has_contour_integral y) h; f contour_integrable_on g; contour_integral g f = contour_integral h f\<rbrakk>
```
```  4594      \<Longrightarrow> (f has_contour_integral y) g"
```
```  4595   using has_contour_integral_integral contour_integral_unique by auto
```
```  4596
```
```  4597 lemma Cauchy_theorem_null_homotopic:
```
```  4598      "\<lbrakk>f holomorphic_on s; open s; valid_path g; homotopic_loops s g (linepath a a)\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
```
```  4599   apply (rule has_contour_integral_newpath [where h = "linepath a a"], simp)
```
```  4600   using contour_integrable_holomorphic_simple
```
```  4601     apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
```
```  4602   by (simp add: Cauchy_theorem_homotopic_loops)
```
```  4603
```
```  4604
```
```  4605
```
```  4606 subsection\<open>More winding number properties\<close>
```
```  4607
```
```  4608 text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>
```
```  4609
```
```  4610 lemma winding_number_homotopic_paths:
```
```  4611     assumes "homotopic_paths (-{z}) g h"
```
```  4612       shows "winding_number g z = winding_number h z"
```
```  4613 proof -
```
```  4614   have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
```
```  4615   moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
```
```  4616     using homotopic_paths_imp_subset [OF assms] by auto
```
```  4617   ultimately obtain d e where "d > 0" "e > 0"
```
```  4618       and d: "\<And>p. \<lbrakk>path p; pathstart p = pathstart g; pathfinish p = pathfinish g; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
```
```  4619             \<Longrightarrow> homotopic_paths (-{z}) g p"
```
```  4620       and e: "\<And>q. \<lbrakk>path q; pathstart q = pathstart h; pathfinish q = pathfinish h; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
```
```  4621             \<Longrightarrow> homotopic_paths (-{z}) h q"
```
```  4622     using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
```
```  4623   obtain p where p:
```
```  4624        "valid_path p" "z \<notin> path_image p"
```
```  4625        "pathstart p = pathstart g" "pathfinish p = pathfinish g"
```
```  4626        and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
```
```  4627        and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
```
```  4628     using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] by blast
```
```  4629   obtain q where q:
```
```  4630        "valid_path q" "z \<notin> path_image q"
```
```  4631        "pathstart q = pathstart h" "pathfinish q = pathfinish h"
```
```  4632        and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
```
```  4633        and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
```
```  4634     using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] by blast
```
```  4635   have gp: "homotopic_paths (- {z}) g p"
```
```  4636     by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
```
```  4637   have hq: "homotopic_paths (- {z}) h q"
```
```  4638     by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
```
```  4639   have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
```
```  4640     apply (rule Cauchy_theorem_homotopic_paths [of "-{z}"])
```
```  4641     apply (blast intro: homotopic_paths_trans homotopic_paths_sym gp hq assms)
```
```  4642     apply (auto intro!: holomorphic_intros simp: p q)
```
```  4643     done
```
```  4644   then show ?thesis
```
```  4645     by (simp add: pap paq)
```
```  4646 qed
```
```  4647
```
```  4648 lemma winding_number_homotopic_loops:
```
```  4649     assumes "homotopic_loops (-{z}) g h"
```
```  4650       shows "winding_number g z = winding_number h z"
```
```  4651 proof -
```
```  4652   have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
```
```  4653   moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
```
```  4654     using homotopic_loops_imp_subset [OF assms] by auto
```
```  4655   moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
```
```  4656     using homotopic_loops_imp_loop [OF assms] by auto
```
```  4657   ultimately obtain d e where "d > 0" "e > 0"
```
```  4658       and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
```
```  4659             \<Longrightarrow> homotopic_loops (-{z}) g p"
```
```  4660       and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
```
```  4661             \<Longrightarrow> homotopic_loops (-{z}) h q"
```
```  4662     using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
```
```  4663   obtain p where p:
```
```  4664        "valid_path p" "z \<notin> path_image p"
```
```  4665        "pathstart p = pathstart g" "pathfinish p = pathfinish g"
```
```  4666        and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
```
```  4667        and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
```
```  4668     using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] by blast
```
```  4669   obtain q where q:
```
```  4670        "valid_path q" "z \<notin> path_image q"
```
```  4671        "pathstart q = pathstart h" "pathfinish q = pathfinish h"
```
```  4672        and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
```
```  4673        and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
```
```  4674     using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] by blast
```
```  4675   have gp: "homotopic_loops (- {z}) g p"
```
```  4676     by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
```
```  4677   have hq: "homotopic_loops (- {z}) h q"
```
```  4678     by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
```
```  4679   have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
```
```  4680     apply (rule Cauchy_theorem_homotopic_loops [of "-{z}"])
```
```  4681     apply (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
```
```  4682     apply (auto intro!: holomorphic_intros simp: p q)
```
```  4683     done
```
```  4684   then show ?thesis
```
```  4685     by (simp add: pap paq)
```
```  4686 qed
```
```  4687
```
```  4688 lemma winding_number_paths_linear_eq:
```
```  4689   "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
```
```  4690     \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
```
```  4691         \<Longrightarrow> winding_number h z = winding_number g z"
```
```  4692   by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths elim: )
```
```  4693
```
```  4694 lemma winding_number_loops_linear_eq:
```
```  4695   "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
```
```  4696     \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
```
```  4697         \<Longrightarrow> winding_number h z = winding_number g z"
```
```  4698   by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops elim: )
```
```  4699
```
```  4700 lemma winding_number_nearby_paths_eq:
```
```  4701      "\<lbrakk>path g; path h;
```
```  4702       pathstart h = pathstart g; pathfinish h = pathfinish g;
```
```  4703       \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
```
```  4704       \<Longrightarrow> winding_number h z = winding_number g z"
```
```  4705   by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)
```
```  4706
```
```  4707 lemma winding_number_nearby_loops_eq:
```
```  4708      "\<lbrakk>path g; path h;
```
```  4709       pathfinish g = pathstart g;
```
```  4710         pathfinish h = pathstart h;
```
```  4711       \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
```
```  4712       \<Longrightarrow> winding_number h z = winding_number g z"
```
```  4713   by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)
```
```  4714
```
```  4715
```
```  4716 proposition winding_number_subpath_combine:
```
```  4717     "\<lbrakk>path g; z \<notin> path_image g;
```
```  4718       u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
```
```  4719       \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
```
```  4720           winding_number (subpath u w g) z"
```
```  4721 apply (rule trans [OF winding_number_join [THEN sym]
```
```  4722                       winding_number_homotopic_paths [OF homotopic_join_subpaths]])
```
```  4723 apply (auto dest: path_image_subpath_subset)
```
```  4724 done
```
```  4725
```
```  4726
```
```  4727 subsection\<open>Partial circle path\<close>
```
```  4728
```
```  4729 definition part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
```
```  4730   where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (\<i> * of_real (linepath s t x))"
```
```  4731
```
```  4732 lemma pathstart_part_circlepath [simp]:
```
```  4733      "pathstart(part_circlepath z r s t) = z + r*exp(\<i> * s)"
```
```  4734 by (metis part_circlepath_def pathstart_def pathstart_linepath)
```
```  4735
```
```  4736 lemma pathfinish_part_circlepath [simp]:
```
```  4737      "pathfinish(part_circlepath z r s t) = z + r*exp(\<i>*t)"
```
```  4738 by (metis part_circlepath_def pathfinish_def pathfinish_linepath)
```
```  4739
```
```  4740 proposition has_vector_derivative_part_circlepath [derivative_intros]:
```
```  4741     "((part_circlepath z r s t) has_vector_derivative
```
```  4742       (\<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)))
```
```  4743      (at x within X)"
```
```  4744   apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
```
```  4745   apply (rule has_vector_derivative_real_complex)
```
```  4746   apply (rule derivative_eq_intros | simp)+
```
```  4747   done
```
```  4748
```
```  4749 corollary vector_derivative_part_circlepath:
```
```  4750     "vector_derivative (part_circlepath z r s t) (at x) =
```
```  4751        \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
```
```  4752   using has_vector_derivative_part_circlepath vector_derivative_at by blast
```
```  4753
```
```  4754 corollary vector_derivative_part_circlepath01:
```
```  4755     "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
```
```  4756      \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
```
```  4757           \<i> * r * (of_real t - of_real s) * exp(\<i> * linepath s t x)"
```
```  4758   using has_vector_derivative_part_circlepath
```
```  4759   by (auto simp: vector_derivative_at_within_ivl)
```
```  4760
```
```  4761 lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
```
```  4762   apply (simp add: valid_path_def)
```
```  4763   apply (rule C1_differentiable_imp_piecewise)
```
```  4764   apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
```
```  4765               intro!: continuous_intros)
```
```  4766   done
```
```  4767
```
```  4768 lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
```
```  4769   by (simp add: valid_path_imp_path)
```
```  4770
```
```  4771 proposition path_image_part_circlepath:
```
```  4772   assumes "s \<le> t"
```
```  4773     shows "path_image (part_circlepath z r s t) = {z + r * exp(\<i> * of_real x) | x. s \<le> x \<and> x \<le> t}"
```
```  4774 proof -
```
```  4775   { fix z::real
```
```  4776     assume "0 \<le> z" "z \<le> 1"
```
```  4777     with \<open>s \<le> t\<close> have "\<exists>x. (exp (\<i> * linepath s t z) = exp (\<i> * of_real x)) \<and> s \<le> x \<and> x \<le> t"
```
```  4778       apply (rule_tac x="(1 - z) * s + z * t" in exI)
```
```  4779       apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
```
```  4780       apply (rule conjI)
```
```  4781       using mult_right_mono apply blast
```
```  4782       using affine_ineq  by (metis "mult.commute")
```
```  4783   }
```
```  4784   moreover
```
```  4785   { fix z
```
```  4786     assume "s \<le> z" "z \<le> t"
```
```  4787     then have "z + of_real r * exp (\<i> * of_real z) \<in> (\<lambda>x. z + of_real r * exp (\<i> * linepath s t x)) ` {0..1}"
```
```  4788       apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
```
```  4789       apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
```
```  4790       apply (auto simp: algebra_simps divide_simps)
```
```  4791       done
```
```  4792   }
```
```  4793   ultimately show ?thesis
```
```  4794     by (fastforce simp add: path_image_def part_circlepath_def)
```
```  4795 qed
```
```  4796
```
```  4797 corollary path_image_part_circlepath_subset:
```
```  4798     "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
```
```  4799 by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)
```
```  4800
```
```  4801 proposition in_path_image_part_circlepath:
```
```  4802   assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
```
```  4803     shows "norm(w - z) = r"
```
```  4804 proof -
```
```  4805   have "w \<in> {c. dist z c = r}"
```
```  4806     by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
```
```  4807   thus ?thesis
```
```  4808     by (simp add: dist_norm norm_minus_commute)
```
```  4809 qed
```
```  4810
```
```  4811 proposition finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
```
```  4812 proof (cases "w = 0")
```
```  4813   case True then show ?thesis by auto
```
```  4814 next
```
```  4815   case False
```
```  4816   have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
```
```  4817     apply (simp add: norm_mult finite_int_iff_bounded_le)
```
```  4818     apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
```
```  4819     apply (auto simp: divide_simps le_floor_iff)
```
```  4820     done
```
```  4821   have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
```
```  4822     by blast
```
```  4823   show ?thesis
```
```  4824     apply (subst exp_Ln [OF False, symmetric])
```
```  4825     apply (simp add: exp_eq)
```
```  4826     using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
```
```  4827     done
```
```  4828 qed
```
```  4829
```
```  4830 lemma finite_bounded_log2:
```
```  4831   fixes a::complex
```
```  4832     assumes "a \<noteq> 0"
```
```  4833     shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
```
```  4834 proof -
```
```  4835   have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
```
```  4836     by (rule finite_imageI [OF finite_bounded_log])
```
```  4837   show ?thesis
```
```  4838     by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
```
```  4839 qed
```
```  4840
```
```  4841 proposition has_contour_integral_bound_part_circlepath_strong:
```
```  4842   assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
```
```  4843       and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
```
```  4844       and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
```
```  4845     shows "cmod i \<le> B * r * (t - s)"
```
```  4846 proof -
```
```  4847   consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
```
```  4848   then show ?thesis
```
```  4849   proof cases
```
```  4850     case 1 with fi [unfolded has_contour_integral]
```
```  4851     have "i = 0"  by (simp add: vector_derivative_part_circlepath)
```
```  4852     with assms show ?thesis by simp
```
```  4853   next
```
```  4854     case 2
```
```  4855     have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
```
```  4856     have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
```
```  4857       by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
```
```  4858     have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
```
```  4859     proof -
```
```  4860       define w where "w = (y - z)/of_real r / exp(\<i> * of_real s)"
```
```  4861       have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
```
```  4862         apply (rule finite_vimageI [OF finite_bounded_log2])
```
```  4863         using \<open>s < t\<close> apply (auto simp: inj_of_real)
```
```  4864         done
```
```  4865       show ?thesis
```
```  4866         apply (simp add: part_circlepath_def linepath_def vimage_def)
```
```  4867         apply (rule finite_subset [OF _ fin])
```
```  4868         using le
```
```  4869         apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
```
```  4870         done
```
```  4871     qed
```
```  4872     then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
```
```  4873       by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
```
```  4874     have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
```
```  4875                     else f(part_circlepath z r s t x) *
```
```  4876                        vector_derivative (part_circlepath z r s t) (at x)) has_integral i)  {0..1}"
```
```  4877       apply (rule has_integral_spike
```
```  4878               [where f = "\<lambda>x. f(part_circlepath z r s t x) * vector_derivative (part_circlepath z r s t) (at x)"])
```
```  4879       apply (rule negligible_finite [OF fin01])
```
```  4880       using fi has_contour_integral
```
```  4881       apply auto
```
```  4882       done
```
```  4883     have *: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1; part_circlepath z r s t x \<notin> k\<rbrakk> \<Longrightarrow> cmod (f (part_circlepath z r s t x)) \<le> B"
```
```  4884       by (auto intro!: B [unfolded path_image_def image_def, simplified])
```
```  4885     show ?thesis
```
```  4886       apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
```
```  4887       using assms apply force
```
```  4888       apply (simp add: norm_mult vector_derivative_part_circlepath)
```
```  4889       using le * "2" \<open>r > 0\<close> by auto
```
```  4890   qed
```
```  4891 qed
```
```  4892
```
```  4893 corollary has_contour_integral_bound_part_circlepath:
```
```  4894       "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
```
```  4895         0 \<le> B; 0 < r; s \<le> t;
```
```  4896         \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
```
```  4897        \<Longrightarrow> norm i \<le> B*r*(t - s)"
```
```  4898   by (auto intro: has_contour_integral_bound_part_circlepath_strong)
```
```  4899
```
```  4900 proposition contour_integrable_continuous_part_circlepath:
```
```  4901      "continuous_on (path_image (part_circlepath z r s t)) f
```
```  4902       \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
```
```  4903   apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
```
```  4904   apply (rule integrable_continuous_real)
```
```  4905   apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
```
```  4906   done
```
```  4907
```
```  4908 proposition winding_number_part_circlepath_pos_less:
```
```  4909   assumes "s < t" and no: "norm(w - z) < r"
```
```  4910     shows "0 < Re (winding_number(part_circlepath z r s t) w)"
```
```  4911 proof -
```
```  4912   have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
```
```  4913   note valid_path_part_circlepath
```
```  4914   moreover have " w \<notin> path_image (part_circlepath z r s t)"
```
```  4915     using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
```
```  4916   moreover have "0 < r * (t - s) * (r - cmod (w - z))"
```
```  4917     using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
```
```  4918   ultimately show ?thesis
```
```  4919     apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
```
```  4920     apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
```
```  4921     apply (rule mult_left_mono)+
```
```  4922     using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
```
```  4923     apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
```
```  4924     using assms \<open>0 < r\<close> by auto
```
```  4925 qed
```
```  4926
```
```  4927 proposition simple_path_part_circlepath:
```
```  4928     "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
```
```  4929 proof (cases "r = 0 \<or> s = t")
```
```  4930   case True
```
```  4931   then show ?thesis
```
```  4932     apply (rule disjE)
```
```  4933     apply (force simp: part_circlepath_def simple_path_def intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
```
```  4934     done
```
```  4935 next
```
```  4936   case False then have "r \<noteq> 0" "s \<noteq> t" by auto
```
```  4937   have *: "\<And>x y z s t. \<i>*((1 - x) * s + x * t) = \<i>*(((1 - y) * s + y * t)) + z  \<longleftrightarrow> \<i>*(x - y) * (t - s) = z"
```
```  4938     by (simp add: algebra_simps)
```
```  4939   have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
```
```  4940                       \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
```
```  4941     by auto
```
```  4942   have abs_away: "\<And>P. (\<forall>x\<in>{0..1}. \<forall>y\<in>{0..1}. P \<bar>x - y\<bar>) \<longleftrightarrow> (\<forall>x::real. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> P x)"
```
```  4943     by force
```
```  4944   have **: "\<And>x y. (\<exists>n. (complex_of_real x - of_real y) * (of_real t - of_real s) = 2 * (of_int n * of_real pi)) \<longleftrightarrow>
```
```  4945                   (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
```
```  4946     by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
```
```  4947                     intro: exI [where x = "-n" for n])
```
```  4948   have 1: "\<forall>x. 0 \<le> x \<and> x \<le> 1 \<longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1 \<Longrightarrow> \<bar>s - t\<bar> \<le> 2 * pi"
```
```  4949     apply (rule ccontr)
```
```  4950     apply (drule_tac x="2*pi / \<bar>t - s\<bar>" in spec)
```
```  4951     using False
```
```  4952     apply (simp add: abs_minus_commute divide_simps)
```
```  4953     apply (frule_tac x=1 in spec)
```
```  4954     apply (drule_tac x="-1" in spec)
```
```  4955     apply (simp add:)
```
```  4956     done
```
```  4957   have 2: "\<bar>s - t\<bar> = \<bar>2 * (real_of_int n * pi) / x\<bar>" if "x \<noteq> 0" "x * (t - s) = 2 * (real_of_int n * pi)" for x n
```
```  4958   proof -
```
```  4959     have "t-s = 2 * (real_of_int n * pi)/x"
```
```  4960       using that by (simp add: field_simps)
```
```  4961     then show ?thesis by (metis abs_minus_commute)
```
```  4962   qed
```
```  4963   show ?thesis using False
```
```  4964     apply (simp add: simple_path_def path_part_circlepath)
```
```  4965     apply (simp add: part_circlepath_def linepath_def exp_eq  * ** abs01  del: Set.insert_iff)
```
```  4966     apply (subst abs_away)
```
```  4967     apply (auto simp: 1)
```
```  4968     apply (rule ccontr)
```
```  4969     apply (auto simp: 2 divide_simps abs_mult dest: of_int_leD)
```
```  4970     done
```
```  4971 qed
```
```  4972
```
```  4973 proposition arc_part_circlepath:
```
```  4974   assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
```
```  4975     shows "arc (part_circlepath z r s t)"
```
```  4976 proof -
```
```  4977   have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
```
```  4978                   and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
```
```  4979     proof -
```
```  4980       have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
```
```  4981         by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_div_cancel_left eq)
```
```  4982       then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
```
```  4983         by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
```
```  4984       then have st: "x \<noteq> y \<Longrightarrow> (s-t) = (of_int n * (pi * 2) / (y-x))"
```
```  4985         by (force simp: field_simps)
```
```  4986       show ?thesis
```
```  4987         apply (rule ccontr)
```
```  4988         using assms x y
```
```  4989         apply (simp add: st abs_mult field_simps)
```
```  4990         using st
```
```  4991         apply (auto simp: dest: of_int_lessD)
```
```  4992         done
```
```  4993     qed
```
```  4994   show ?thesis
```
```  4995     using assms
```
```  4996     apply (simp add: arc_def)
```
```  4997     apply (simp add: part_circlepath_def inj_on_def exp_eq)
```
```  4998     apply (blast intro: *)
```
```  4999     done
```
```  5000 qed
```
```  5001
```
```  5002
```
```  5003 subsection\<open>Special case of one complete circle\<close>
```
```  5004
```
```  5005 definition circlepath :: "[complex, real, real] \<Rightarrow> complex"
```
```  5006   where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"
```
```  5007
```
```  5008 lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * \<i> * of_real x))"
```
```  5009   by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)
```
```  5010
```
```  5011 lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
```
```  5012   by (simp add: circlepath_def)
```
```  5013
```
```  5014 lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
```
```  5015   by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)
```
```  5016
```
```  5017 lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
```
```  5018 proof -
```
```  5019   have "z + of_real r * exp (2 * pi * \<i> * (x + 1 / 2)) =
```
```  5020         z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
```
```  5021     by (simp add: divide_simps) (simp add: algebra_simps)
```
```  5022   also have "... = z - r * exp (2 * pi * \<i> * x)"
```
```  5023     by (simp add: exp_add)
```
```  5024   finally show ?thesis
```
```  5025     by (simp add: circlepath path_image_def sphere_def dist_norm)
```
```  5026 qed
```
```  5027
```
```  5028 lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
```
```  5029   using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
```
```  5030   by (simp add: add.commute)
```
```  5031
```
```  5032 lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
```
```  5033   using circlepath_add1 [of z r "x-1/2"]
```
```  5034   by (simp add: add.commute)
```
```  5035
```
```  5036 lemma path_image_circlepath_minus_subset:
```
```  5037      "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
```
```  5038   apply (simp add: path_image_def image_def circlepath_minus, clarify)
```
```  5039   apply (case_tac "xa \<le> 1/2", force)
```
```  5040   apply (force simp add: circlepath_add_half)+
```
```  5041   done
```
```  5042
```
```  5043 lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
```
```  5044   using path_image_circlepath_minus_subset by fastforce
```
```  5045
```
```  5046 proposition has_vector_derivative_circlepath [derivative_intros]:
```
```  5047  "((circlepath z r) has_vector_derivative (2 * pi * \<i> * r * exp (2 * of_real pi * \<i> * of_real x)))
```
```  5048    (at x within X)"
```
```  5049   apply (simp add: circlepath_def scaleR_conv_of_real)
```
```  5050   apply (rule derivative_eq_intros)
```
```  5051   apply (simp add: algebra_simps)
```
```  5052   done
```
```  5053
```
```  5054 corollary vector_derivative_circlepath:
```
```  5055    "vector_derivative (circlepath z r) (at x) =
```
```  5056     2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
```
```  5057 using has_vector_derivative_circlepath vector_derivative_at by blast
```
```  5058
```
```  5059 corollary vector_derivative_circlepath01:
```
```  5060     "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
```
```  5061      \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
```
```  5062           2 * pi * \<i> * r * exp(2 * of_real pi * \<i> * x)"
```
```  5063   using has_vector_derivative_circlepath
```
```  5064   by (auto simp: vector_derivative_at_within_ivl)
```
```  5065
```
```  5066 lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
```
```  5067   by (simp add: circlepath_def)
```
```  5068
```
```  5069 lemma path_circlepath [simp]: "path (circlepath z r)"
```
```  5070   by (simp add: valid_path_imp_path)
```
```  5071
```
```  5072 lemma path_image_circlepath_nonneg:
```
```  5073   assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
```
```  5074 proof -
```
```  5075   have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
```
```  5076   proof (cases "x = z")
```
```  5077     case True then show ?thesis by force
```
```  5078   next
```
```  5079     case False
```
```  5080     define w where "w = x - z"
```
```  5081     then have "w \<noteq> 0" by (simp add: False)
```
```  5082     have **: "\<And>t. \<lbrakk>Re w = cos t * cmod w; Im w = sin t * cmod w\<rbrakk> \<Longrightarrow> w = of_real (cmod w) * exp (\<i> * t)"
```
```  5083       using cis_conv_exp complex_eq_iff by auto
```
```  5084     show ?thesis
```
```  5085       apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
```
```  5086       apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
```
```  5087       apply (rule_tac x="t / (2*pi)" in image_eqI)
```
```  5088       apply (simp add: divide_simps \<open>w \<noteq> 0\<close>)
```
```  5089       using False **
```
```  5090       apply (auto simp: w_def)
```
```  5091       done
```
```  5092   qed
```
```  5093   show ?thesis
```
```  5094     unfolding circlepath path_image_def sphere_def dist_norm
```
```  5095     by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
```
```  5096 qed
```
```  5097
```
```  5098 proposition path_image_circlepath [simp]:
```
```  5099     "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
```
```  5100   using path_image_circlepath_minus
```
```  5101   by (force simp add: path_image_circlepath_nonneg abs_if)
```
```  5102
```
```  5103 lemma has_contour_integral_bound_circlepath_strong:
```
```  5104       "\<lbrakk>(f has_contour_integral i) (circlepath z r);
```
```  5105         finite k; 0 \<le> B; 0 < r;
```
```  5106         \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
```
```  5107         \<Longrightarrow> norm i \<le> B*(2*pi*r)"
```
```  5108   unfolding circlepath_def
```
```  5109   by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)
```
```  5110
```
```  5111 corollary has_contour_integral_bound_circlepath:
```
```  5112       "\<lbrakk>(f has_contour_integral i) (circlepath z r);
```
```  5113         0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
```
```  5114         \<Longrightarrow> norm i \<le> B*(2*pi*r)"
```
```  5115   by (auto intro: has_contour_integral_bound_circlepath_strong)
```
```  5116
```
```  5117 proposition contour_integrable_continuous_circlepath:
```
```  5118     "continuous_on (path_image (circlepath z r)) f
```
```  5119      \<Longrightarrow> f contour_integrable_on (circlepath z r)"
```
```  5120   by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)
```
```  5121
```
```  5122 lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
```
```  5123   by (simp add: circlepath_def simple_path_part_circlepath)
```
```  5124
```
```  5125 lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
```
```  5126   by (simp add: sphere_def dist_norm norm_minus_commute)
```
```  5127
```
```  5128 proposition contour_integral_circlepath:
```
```  5129      "0 < r \<Longrightarrow> contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
```
```  5130   apply (rule contour_integral_unique)
```
```  5131   apply (simp add: has_contour_integral_def)
```
```  5132   apply (subst has_integral_cong)
```
```  5133   apply (simp add: vector_derivative_circlepath01)
```
```  5134   using has_integral_const_real [of _ 0 1]
```
```  5135   apply (force simp: circlepath)
```
```  5136   done
```
```  5137
```
```  5138 lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
```
```  5139   apply (rule winding_number_unique_loop)
```
```  5140   apply (simp_all add: sphere_def valid_path_imp_path)
```
```  5141   apply (rule_tac x="circlepath z r" in exI)
```
```  5142   apply (simp add: sphere_def contour_integral_circlepath)
```
```  5143   done
```
```  5144
```
```  5145 proposition winding_number_circlepath:
```
```  5146   assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
```
```  5147 proof (cases "w = z")
```
```  5148   case True then show ?thesis
```
```  5149     using assms winding_number_circlepath_centre by auto
```
```  5150 next
```
```  5151   case False
```
```  5152   have [simp]: "r > 0"
```
```  5153     using assms le_less_trans norm_ge_zero by blast
```
```  5154   define r' where "r' = norm(w - z)"
```
```  5155   have "r' < r"
```
```  5156     by (simp add: assms r'_def)
```
```  5157   have disjo: "cball z r' \<inter> sphere z r = {}"
```
```  5158     using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
```
```  5159   have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
```
```  5160     apply (rule winding_number_around_inside [where s = "cball z r'"])
```
```  5161     apply (simp_all add: disjo order.strict_implies_order winding_number_circlepath_centre)
```
```  5162     apply (simp_all add: False r'_def dist_norm norm_minus_commute)
```
```  5163     done
```
```  5164   also have "... = 1"
```
```  5165     by (simp add: winding_number_circlepath_centre)
```
```  5166   finally show ?thesis .
```
```  5167 qed
```
```  5168
```
```  5169
```
```  5170 text\<open> Hence the Cauchy formula for points inside a circle.\<close>
```
```  5171
```
```  5172 theorem Cauchy_integral_circlepath:
```
```  5173   assumes "continuous_on (cball z r) f" "f holomorphic_on (ball z r)" "norm(w - z) < r"
```
```  5174   shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
```
```  5175          (circlepath z r)"
```
```  5176 proof -
```
```  5177   have "r > 0"
```
```  5178     using assms le_less_trans norm_ge_zero by blast
```
```  5179   have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
```
```  5180         (circlepath z r)"
```
```  5181     apply (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
```
```  5182     using assms  \<open>r > 0\<close>
```
```  5183     apply (simp_all add: dist_norm norm_minus_commute)
```
```  5184     apply (metis at_within_interior dist_norm holomorphic_on_def interior_ball mem_ball norm_minus_commute)
```
```  5185     apply (simp add: cball_def sphere_def dist_norm, clarify)
```
```  5186     apply (simp add:)
```
```  5187     by (metis dist_commute dist_norm less_irrefl)
```
```  5188   then show ?thesis
```
```  5189     by (simp add: winding_number_circlepath assms)
```
```  5190 qed
```
```  5191
```
```  5192 corollary Cauchy_integral_circlepath_simple:
```
```  5193   assumes "f holomorphic_on cball z r" "norm(w - z) < r"
```
```  5194   shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * \<i> * f w))
```
```  5195          (circlepath z r)"
```
```  5196 using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)
```
```  5197
```
```  5198
```
```  5199 lemma no_bounded_connected_component_imp_winding_number_zero:
```
```  5200   assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
```
```  5201       and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
```
```  5202   shows "winding_number g z = 0"
```
```  5203 apply (rule winding_number_zero_in_outside)
```
```  5204 apply (simp_all add: assms)
```
```  5205 by (metis nb [of z] \<open>path_image g \<subseteq> s\<close> \<open>z \<notin> s\<close> contra_subsetD mem_Collect_eq outside outside_mono)
```
```  5206
```
```  5207 lemma no_bounded_path_component_imp_winding_number_zero:
```
```  5208   assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
```
```  5209       and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
```
```  5210   shows "winding_number g z = 0"
```
```  5211 apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
```
```  5212 by (simp add: bounded_subset nb path_component_subset_connected_component)
```
```  5213
```
```  5214
```
```  5215 subsection\<open> Uniform convergence of path integral\<close>
```
```  5216
```
```  5217 text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>
```
```  5218
```
```  5219 proposition contour_integral_uniform_limit:
```
```  5220   assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
```
```  5221       and ul_f: "uniform_limit (path_image \<gamma>) f l F"
```
```  5222       and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
```
```  5223       and \<gamma>: "valid_path \<gamma>"
```
```  5224       and [simp]: "~ (trivial_limit F)"
```
```  5225   shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
```
```  5226 proof -
```
```  5227   have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
```
```  5228   { fix e::real
```
```  5229     assume "0 < e"
```
```  5230     then have "0 < e / (\<bar>B\<bar> + 1)" by simp
```
```  5231     then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
```
```  5232       using ul_f [unfolded uniform_limit_iff dist_norm] by auto
```
```  5233     with ev_fint
```
```  5234     obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
```
```  5235                and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
```
```  5236       using eventually_happens [OF eventually_conj]
```
```  5237       by (fastforce simp: contour_integrable_on path_image_def)
```
```  5238     have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
```
```  5239       using \<open>0 \<le> B\<close>  \<open>0 < e\<close> by (simp add: divide_simps)
```
```  5240     have "\<exists>h. (\<forall>x\<in>{0..1}. cmod (l (\<gamma> x) * vector_derivative \<gamma> (at x) - h x) \<le> e) \<and> h integrable_on {0..1}"
```
```  5241       apply (rule_tac x="\<lambda>x. f (a::'a) (\<gamma> x) * vector_derivative \<gamma> (at x)" in exI)
```
```  5242       apply (intro inta conjI ballI)
```
```  5243       apply (rule order_trans [OF _ Ble])
```
```  5244       apply (frule noleB)
```
```  5245       apply (frule fga)
```
```  5246       using \<open>0 \<le> B\<close>  \<open>0 < e\<close>
```
```  5247       apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
```
```  5248       apply (drule (1) mult_mono [OF less_imp_le])
```
```  5249       apply (simp_all add: mult_ac)
```
```  5250       done
```
```  5251   }
```
```  5252   then show lintg: "l contour_integrable_on \<gamma>"
```
```  5253     apply (simp add: contour_integrable_on)
```
```  5254     apply (blast intro: integrable_uniform_limit_real)
```
```  5255     done
```
```  5256   { fix e::real
```
```  5257     define B' where "B' = B + 1"
```
```  5258     have B': "B' > 0" "B' > B" using  \<open>0 \<le> B\<close> by (auto simp: B'_def)
```
```  5259     assume "0 < e"
```
```  5260     then have ev_no': "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. 2 * cmod (f n x - l x) < e / B'"
```
```  5261       using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, of "e / B' / 2"] B'
```
```  5262         by (simp add: field_simps)
```
```  5263     have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
```
```  5264     have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
```
```  5265              if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
```
```  5266     proof -
```
```  5267       have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
```
```  5268         using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
```
```  5269       also have "... < e"
```
```  5270         by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
```
```  5271       finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
```
```  5272       then show ?thesis
```
```  5273         by (simp add: left_diff_distrib [symmetric] norm_mult)
```
```  5274     qed
```
```  5275     have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
```
```  5276       apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
```
```  5277       apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
```
```  5278       apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric], clarify)
```
```  5279       apply (rule le_less_trans [OF integral_norm_bound_integral ie])
```
```  5280       apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
```
```  5281       apply (blast intro: *)+
```
```  5282       done
```
```  5283   }
```
```  5284   then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
```
```  5285     by (rule tendstoI)
```
```  5286 qed
```
```  5287
```
```  5288 corollary contour_integral_uniform_limit_circlepath:
```
```  5289   assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
```
```  5290       and "uniform_limit (sphere z r) f l F"
```
```  5291       and "~ (trivial_limit F)" "0 < r"
```
```  5292     shows "l contour_integrable_on (circlepath z r)"
```
```  5293           "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
```
```  5294   using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
```
```  5295
```
```  5296
```
```  5297 subsection\<open> General stepping result for derivative formulas.\<close>
```
```  5298
```
```  5299 proposition Cauchy_next_derivative:
```
```  5300   assumes "continuous_on (path_image \<gamma>) f'"
```
```  5301       and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
```
```  5302       and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
```
```  5303       and k: "k \<noteq> 0"
```
```  5304       and "open s"
```
```  5305       and \<gamma>: "valid_path \<gamma>"
```
```  5306       and w: "w \<in> s - path_image \<gamma>"
```
```  5307     shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
```
```  5308       and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
```
```  5309            (at w)"  (is "?thes2")
```
```  5310 proof -
```
```  5311   have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
```
```  5312   then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
```
```  5313     using open_contains_ball by blast
```
```  5314   have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
```
```  5315     by (metis norm_of_nat of_nat_Suc)
```
```  5316   have 1: "\<forall>\<^sub>F n in at w. (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k)
```
```  5317                          contour_integrable_on \<gamma>"
```
```  5318     apply (simp add: eventually_at)
```
```  5319     apply (rule_tac x=d in exI)
```
```  5320     apply (simp add: \<open>d > 0\<close> dist_norm field_simps, clarify)
```
```  5321     apply (rule contour_integrable_div [OF contour_integrable_diff])
```
```  5322     using int w d
```
```  5323     apply (force simp:  dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
```
```  5324     done
```
```  5325   have bim_g: "bounded (image f' (path_image \<gamma>))"
```
```  5326     by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
```
```  5327   then obtain C where "C > 0" and C: "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cmod (f' (\<gamma> x)) \<le> C"
```
```  5328     by (force simp: bounded_pos path_image_def)
```
```  5329   have twom: "\<forall>\<^sub>F n in at w.
```
```  5330                \<forall>x\<in>path_image \<gamma>.
```
```  5331                 cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
```
```  5332          if "0 < e" for e
```
```  5333   proof -
```
```  5334     have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k)   < e"
```
```  5335             if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
```
```  5336                 and uw_less: "cmod (u - w) < e * (d / 2) ^ (k+2) / (1 + real k)"
```
```  5337             for u x
```
```  5338     proof -
```
```  5339       define ff where [abs_def]:
```
```  5340         "ff n w =
```
```  5341           (if n = 0 then inverse(x - w)^k
```
```  5342            else if n = 1 then k / (x - w)^(Suc k)
```
```  5343            else (k * of_real(Suc k)) / (x - w)^(k + 2))" for n :: nat and w
```
```  5344       have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
```
```  5345         by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
```
```  5346       have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d / 2))"
```
```  5347               if "z \<in> ball w (d / 2)" "i \<le> 1" for i z
```
```  5348       proof -
```
```  5349         have "z \<notin> path_image \<gamma>"
```
```  5350           using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
```
```  5351         then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
```
```  5352         then have neq: "x * x + z * z \<noteq> x * (z * 2)"
```
```  5353           by (blast intro: dest!: sum_sqs_eq)
```
```  5354         with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
```
```  5355         then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
```
```  5356           by (simp add: algebra_simps)
```
```  5357         show ?thesis using \<open>i \<le> 1\<close>
```
```  5358           apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
```
```  5359           apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
```
```  5360           done
```
```  5361       qed
```
```  5362       { fix a::real and b::real assume ab: "a > 0" "b > 0"
```
```  5363         then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
```
```  5364           apply (subst mult_le_cancel_left_pos)
```
```  5365           using \<open>k \<noteq> 0\<close>
```
```  5366           apply (auto simp: divide_simps)
```
```  5367           done
```
```  5368         with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
```
```  5369           by (simp add: field_simps)
```
```  5370       } note canc = this
```
```  5371       have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d / 2) ^ (k + 2)"
```
```  5372                 if "v \<in> ball w (d / 2)" for v
```
```  5373       proof -
```
```  5374         have "d/2 \<le> cmod (x - v)" using d x that
```
```  5375           apply (simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps, clarify)
```
```  5376           apply (drule subsetD)
```
```  5377            prefer 2 apply blast
```
```  5378           apply (metis norm_minus_commute norm_triangle_half_r CollectI)
```
```  5379           done
```
```  5380         then have "d \<le> cmod (x - v) * 2"
```
```  5381           by (simp add: divide_simps)
```
```  5382         then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
```
```  5383           using \<open>0 < d\<close> order_less_imp_le power_mono by blast
```
```  5384         have "x \<noteq> v" using that
```
```  5385           using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
```
```  5386         then show ?thesis
```
```  5387         using \<open>d > 0\<close>
```
```  5388         apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
```
```  5389         using dpow_le
```
```  5390         apply (simp add: algebra_simps divide_simps mult_less_0_iff)
```
```  5391         done
```
```  5392       qed
```
```  5393       have ub: "u \<in> ball w (d / 2)"
```
```  5394         using uwd by (simp add: dist_commute dist_norm)
```
```  5395       have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
```
```  5396                   \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d / 2) ^ k))"
```
```  5397         using complex_taylor [OF _ ff1 ff2 _ ub, of w, simplified]
```
```  5398         by (simp add: ff_def \<open>0 < d\<close>)
```
```  5399       then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
```
```  5400                   \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d / 2) ^ (k+2)"
```
```  5401         by (simp add: field_simps)
```
```  5402       then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
```
```  5403                  / (cmod (u - w) * real k)
```
```  5404                   \<le> (1 + real k) * cmod (u - w) / (d / 2) ^ (k+2)"
```
```  5405         using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
```
```  5406       also have "... < e"
```
```  5407         using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
```
```  5408       finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
```
```  5409                         / cmod ((u - w) * real k)   <   e"
```
```  5410         by (simp add: norm_mult)
```
```  5411       have "x \<noteq> u"
```
```  5412         using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
```
```  5413       show ?thesis
```
```  5414         apply (rule le_less_trans [OF _ e])
```
```  5415         using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close>  \<open>u \<noteq> w\<close>
```
```  5416         apply (simp add: field_simps norm_divide [symmetric])
```
```  5417         done
```
```  5418     qed
```
```  5419     show ?thesis
```
```  5420       unfolding eventually_at
```
```  5421       apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
```
```  5422       apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
```
```  5423       done
```
```  5424   qed
```
```  5425   have 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
```
```  5426     unfolding uniform_limit_iff dist_norm
```
```  5427   proof clarify
```
```  5428     fix e::real
```
```  5429     assume "0 < e"
```
```  5430     have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
```
```  5431                         f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
```
```  5432               if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
```
```  5433                       inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
```
```  5434                  and x: "0 \<le> x" "x \<le> 1"
```
```  5435               for u x
```
```  5436     proof (cases "(f' (\<gamma> x)) = 0")
```
```  5437       case True then show ?thesis by (simp add: \<open>0 < e\<close>)
```
```  5438     next
```
```  5439       case False
```
```  5440       have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
```
```  5441                         f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
```
```  5442             cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
```
```  5443                              inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
```
```  5444         by (simp add: field_simps)
```
```  5445       also have "... = cmod (f' (\<gamma> x)) *
```
```  5446                        cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
```
```  5447                              inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
```
```  5448         by (simp add: norm_mult)
```
```  5449       also have "... < cmod (f' (\<gamma> x)) * (e/C)"
```
```  5450         apply (rule mult_strict_left_mono [OF ec])
```
```  5451         using False by simp
```
```  5452       also have "... \<le> e" using C
```
```  5453         by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
```
```  5454       finally show ?thesis .
```
```  5455     qed
```
```  5456     show "\<forall>\<^sub>F n in at w.
```
```  5457               \<forall>x\<in>path_image \<gamma>.
```
```  5458                cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
```
```  5459       using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<open>C > 0\<close>]]   unfolding path_image_def
```
```  5460       by (force intro: * elim: eventually_mono)
```
```  5461   qed
```
```  5462   show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
```
```  5463     by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
```
```  5464   have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
```
```  5465            \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
```
```  5466     by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
```
```  5467   have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
```
```  5468               (f u - f w) / (u - w) / k"
```
```  5469            if "dist u w < d" for u
```
```  5470     apply (rule contour_integral_unique)
```
```  5471     apply (simp add: diff_divide_distrib algebra_simps)
```
```  5472     apply (rule has_contour_integral_diff; rule has_contour_integral_div; simp add: field_simps; rule int)
```
```  5473     apply (metis contra_subsetD d dist_commute mem_ball that)
```
```  5474     apply (rule w)
```
```  5475     done
```
```  5476   show ?thes2
```
```  5477     apply (simp add: DERIV_within_iff del: power_Suc)
```
```  5478     apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
```
```  5479     apply (simp add: \<open>k \<noteq> 0\<close> **)
```
```  5480     done
```
```  5481 qed
```
```  5482
```
```  5483 corollary Cauchy_next_derivative_circlepath:
```
```  5484   assumes contf: "continuous_on (path_image (circlepath z r)) f"
```
```  5485       and int: "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>u. f u / (u - w)^k) has_contour_integral g w) (circlepath z r)"
```
```  5486       and k: "k \<noteq> 0"
```
```  5487       and w: "w \<in> ball z r"
```
```  5488     shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
```
```  5489            (is "?thes1")
```
```  5490       and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
```
```  5491            (is "?thes2")
```
```  5492 proof -
```
```  5493   have "r > 0" using w
```
```  5494     using ball_eq_empty by fastforce
```
```  5495   have wim: "w \<in> ball z r - path_image (circlepath z r)"
```
```  5496     using w by (auto simp: dist_norm)
```
```  5497   show ?thes1 ?thes2
```
```  5498     by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
```
```  5499         auto simp: vector_derivative_circlepath norm_mult)+
```
```  5500 qed
```
```  5501
```
```  5502
```
```  5503 text\<open> In particular, the first derivative formula.\<close>
```
```  5504
```
```  5505 proposition Cauchy_derivative_integral_circlepath:
```
```  5506   assumes contf: "continuous_on (cball z r) f"
```
```  5507       and holf: "f holomorphic_on ball z r"
```
```  5508       and w: "w \<in> ball z r"
```
```  5509     shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
```
```  5510            (is "?thes1")
```
```  5511       and "(f has_field_derivative (1 / (2 * of_real pi * \<i>) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
```
```  5512            (is "?thes2")
```
```  5513 proof -
```
```  5514   have [simp]: "r \<ge> 0" using w
```
```  5515     using ball_eq_empty by fastforce
```
```  5516   have f: "continuous_on (path_image (circlepath z r)) f"
```
```  5517     by (rule continuous_on_subset [OF contf]) (force simp add: cball_def sphere_def)
```
```  5518   have int: "\<And>w. dist z w < r \<Longrightarrow>
```
```  5519                  ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * \<i> * f x) w) (circlepath z r)"
```
```  5520     by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
```
```  5521   show ?thes1
```
```  5522     apply (simp add: power2_eq_square)
```
```  5523     apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
```
```  5524     apply (blast intro: int)
```
```  5525     done
```
```  5526   have "((\<lambda>x. 2 * of_real pi * \<i> * f x) has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2)) (at w)"
```
```  5527     apply (simp add: power2_eq_square)
```
```  5528     apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * \<i> * f x", simplified])
```
```  5529     apply (blast intro: int)
```
```  5530     done
```
```  5531   then have fder: "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)^2) / (2 * of_real pi * \<i>)) (at w)"
```
```  5532     by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
```
```  5533   show ?thes2
```
```  5534     by simp (rule fder)
```
```  5535 qed
```
```  5536
```
```  5537 subsection\<open> Existence of all higher derivatives.\<close>
```
```  5538
```
```  5539 proposition derivative_is_holomorphic:
```
```  5540   assumes "open s"
```
```  5541       and fder: "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z)"
```
```  5542     shows "f' holomorphic_on s"
```
```  5543 proof -
```
```  5544   have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> s" for z
```
```  5545   proof -
```
```  5546     obtain r where "r > 0" and r: "cball z r \<subseteq> s"
```
```  5547       using open_contains_cball \<open>z \<in> s\<close> \<open>open s\<close> by blast
```
```  5548     then have holf_cball: "f holomorphic_on cball z r"
```
```  5549       apply (simp add: holomorphic_on_def)
```
```  5550       using field_differentiable_at_within field_differentiable_def fder by blast
```
```  5551     then have "continuous_on (path_image (circlepath z r)) f"
```
```  5552       using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
```
```  5553     then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*\<i>) * f x)"
```
```  5554       by (auto intro: continuous_intros)+
```
```  5555     have contf_cball: "continuous_on (cball z r) f" using holf_cball
```
```  5556       by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
```
```  5557     have holf_ball: "f holomorphic_on ball z r" using holf_cball
```
```  5558       using ball_subset_cball holomorphic_on_subset by blast
```
```  5559     { fix w  assume w: "w \<in> ball z r"
```
```  5560       have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
```
```  5561         by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
```
```  5562       have fder': "(f has_field_derivative 1 / (2 * of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2))
```
```  5563                   (at w)"
```
```  5564         by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
```
```  5565       have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
```
```  5566         using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
```
```  5567       have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
```
```  5568                 contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
```
```  5569                 (circlepath z r)"
```
```  5570         by (rule has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
```
```  5571       then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
```
```  5572                 contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
```
```  5573                 (circlepath z r)"
```
```  5574         by (simp add: algebra_simps)
```
```  5575       then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
```
```  5576         by (simp add: f'_eq)
```
```  5577     } note * = this
```
```  5578     show ?thesis
```
```  5579       apply (rule exI)
```
```  5580       apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
```
```  5581       apply (simp_all add: \<open>0 < r\<close> * dist_norm)
```
```  5582       done
```
```  5583   qed
```
```  5584   show ?thesis
```
```  5585     by (simp add: holomorphic_on_open [OF \<open>open s\<close>] *)
```
```  5586 qed
```
```  5587
```
```  5588 lemma holomorphic_deriv [holomorphic_intros]:
```
```  5589     "\<lbrakk>f holomorphic_on s; open s\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on s"
```
```  5590 by (metis DERIV_deriv_iff_field_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)
```
```  5591
```
```  5592 lemma analytic_deriv: "f analytic_on s \<Longrightarrow> (deriv f) analytic_on s"
```
```  5593   using analytic_on_holomorphic holomorphic_deriv by auto
```
```  5594
```
```  5595 lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on s; open s\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on s"
```
```  5596   by (induction n) (auto simp: holomorphic_deriv)
```
```  5597
```
```  5598 lemma analytic_higher_deriv: "f analytic_on s \<Longrightarrow> (deriv ^^ n) f analytic_on s"
```
```  5599   unfolding analytic_on_def using holomorphic_higher_deriv by blast
```
```  5600
```
```  5601 lemma has_field_derivative_higher_deriv:
```
```  5602      "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk>
```
```  5603       \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
```
```  5604 by (metis (no_types, hide_lams) DERIV_deriv_iff_field_differentiable at_within_open comp_apply
```
```  5605          funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)
```
```  5606
```
```  5607 lemma valid_path_compose_holomorphic:
```
```  5608   assumes "valid_path g" and holo:"f holomorphic_on s" and "open s" "path_image g \<subseteq> s"
```
```  5609   shows "valid_path (f o g)"
```
```  5610 proof (rule valid_path_compose[OF \<open>valid_path g\<close>])
```
```  5611   fix x assume "x \<in> path_image g"
```
```  5612   then show "f field_differentiable at x"
```
```  5613     using analytic_on_imp_differentiable_at analytic_on_open assms holo by blast
```
```  5614 next
```
```  5615   have "deriv f holomorphic_on s"
```
```  5616     using holomorphic_deriv holo \<open>open s\<close> by auto
```
```  5617   then show "continuous_on (path_image g) (deriv f)"
```
```  5618     using assms(4) holomorphic_on_imp_continuous_on holomorphic_on_subset by auto
```
```  5619 qed
```
```  5620
```
```  5621
```
```  5622 subsection\<open> Morera's theorem.\<close>
```
```  5623
```
```  5624 lemma Morera_local_triangle_ball:
```
```  5625   assumes "\<And>z. z \<in> s
```
```  5626           \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
```
```  5627                     (\<forall>b c. closed_segment b c \<subseteq> ball a e
```
```  5628                            \<longrightarrow> contour_integral (linepath a b) f +
```
```  5629                                contour_integral (linepath b c) f +
```
```  5630                                contour_integral (linepath c a) f = 0)"
```
```  5631   shows "f analytic_on s"
```
```  5632 proof -
```
```  5633   { fix z  assume "z \<in> s"
```
```  5634     with assms obtain e a where
```
```  5635             "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
```
```  5636         and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
```
```  5637                       \<Longrightarrow> contour_integral (linepath a b) f +
```
```  5638                           contour_integral (linepath b c) f +
```
```  5639                           contour_integral (linepath c a) f = 0"
```
```  5640       by fastforce
```
```  5641     have az: "dist a z < e" using mem_ball z by blast
```
```  5642     have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
```
```  5643       by (simp add: dist_commute ball_subset_ball_iff)
```
```  5644     have "\<exists>e>0. f holomorphic_on ball z e"
```
```  5645       apply (rule_tac x="e - dist a z" in exI)
```
```  5646       apply (simp add: az)
```
```  5647       apply (rule holomorphic_on_subset [OF _ sb_ball])
```