src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy

Substantial new material for multivariate analysis. Also removal of some duplicates.

section \<open>Complex path integrals and Cauchy's integral theorem\<close>

text\<open>By John Harrison et al.  Ported from HOL Light by L C Paulson (2015)\<close>

theory Cauchy_Integral_Thm
imports Complex_Transcendental Weierstrass Ordered_Euclidean_Space
begin

subsection \<open>Piecewise differentiable functions\<close>

definition piecewise_differentiable_on
           (infixr "piecewise'_differentiable'_on" 50)
  where "f piecewise_differentiable_on i  \<equiv>
           continuous_on i f \<and>
           (\<exists>s. finite s \<and> (\<forall>x \<in> i - s. f differentiable (at x within i)))"

lemma piecewise_differentiable_on_imp_continuous_on:
    "f piecewise_differentiable_on s \<Longrightarrow> continuous_on s f"
by (simp add: piecewise_differentiable_on_def)

lemma piecewise_differentiable_on_subset:
    "f piecewise_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_differentiable_on t"
  using continuous_on_subset
  unfolding piecewise_differentiable_on_def
  apply safe
  apply (blast intro: elim: continuous_on_subset)
  by (meson Diff_iff differentiable_within_subset subsetCE)

lemma differentiable_on_imp_piecewise_differentiable:
  fixes a:: "'a::{linorder_topology,real_normed_vector}"
  shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
  apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
  apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
  done

lemma differentiable_imp_piecewise_differentiable:
    "(\<And>x. x \<in> s \<Longrightarrow> f differentiable (at x within s))
         \<Longrightarrow> f piecewise_differentiable_on s"
by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
         intro: differentiable_within_subset)

lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on s"
  by (simp add: differentiable_imp_piecewise_differentiable)

lemma piecewise_differentiable_compose:
    "\<lbrakk>f piecewise_differentiable_on s; g piecewise_differentiable_on (f ` s);
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
      \<Longrightarrow> (g o f) piecewise_differentiable_on s"
  apply (simp add: piecewise_differentiable_on_def, safe)
  apply (blast intro: continuous_on_compose2)
  apply (rename_tac A B)
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
  apply (blast intro: differentiable_chain_within)
  done

lemma piecewise_differentiable_affine:
  fixes m::real
  assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` s)"
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on s"
proof (cases "m = 0")
  case True
  then show ?thesis
    unfolding o_def
    by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
next
  case False
  show ?thesis
    apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
    apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
    done
qed

lemma piecewise_differentiable_cases:
  fixes c::real
  assumes "f piecewise_differentiable_on {a..c}"
          "g piecewise_differentiable_on {c..b}"
           "a \<le> c" "c \<le> b" "f c = g c"
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
proof -
  obtain s t where st: "finite s" "finite t"
                       "\<forall>x\<in>{a..c} - s. f differentiable at x within {a..c}"
                       "\<forall>x\<in>{c..b} - t. g differentiable at x within {c..b}"
    using assms
    by (auto simp: piecewise_differentiable_on_def)
  have finabc: "finite ({a,b,c} \<union> (s \<union> t))"
    by (metis \<open>finite s\<close> \<open>finite t\<close> finite_Un finite_insert finite.emptyI)
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
    using assms piecewise_differentiable_on_def by auto
  then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                               OF closed_real_atLeastAtMost [of c b],
                               of f g "\<lambda>x. x\<le>c"]  assms
    by (force simp: ivl_disj_un_two_touch)
  moreover
  { fix x
    assume x: "x \<in> {a..b} - ({a,b,c} \<union> (s \<union> t))"
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
    proof (cases x c rule: le_cases)
      case le show ?diff_fg
        apply (rule differentiable_transform_within [where d = "dist x c" and f = f])
        using x le st
        apply (simp_all add: dist_real_def)
        apply (rule differentiable_at_withinI)
        apply (rule differentiable_within_open [where s = "{a<..<c} - s", THEN iffD1], simp_all)
        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
        apply (force elim!: differentiable_subset)+
        done
    next
      case ge show ?diff_fg
        apply (rule differentiable_transform_within [where d = "dist x c" and f = g])
        using x ge st
        apply (simp_all add: dist_real_def)
        apply (rule differentiable_at_withinI)
        apply (rule differentiable_within_open [where s = "{c<..<b} - t", THEN iffD1], simp_all)
        apply (blast intro: open_greaterThanLessThan finite_imp_closed)
        apply (force elim!: differentiable_subset)+
        done
    qed
  }
  then have "\<exists>s. finite s \<and>
                 (\<forall>x\<in>{a..b} - s. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
    by (meson finabc)
  ultimately show ?thesis
    by (simp add: piecewise_differentiable_on_def)
qed

lemma piecewise_differentiable_neg:
    "f piecewise_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on s"
  by (auto simp: piecewise_differentiable_on_def continuous_on_minus)

lemma piecewise_differentiable_add:
  assumes "f piecewise_differentiable_on i"
          "g piecewise_differentiable_on i"
    shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
proof -
  obtain s t where st: "finite s" "finite t"
                       "\<forall>x\<in>i - s. f differentiable at x within i"
                       "\<forall>x\<in>i - t. g differentiable at x within i"
    using assms by (auto simp: piecewise_differentiable_on_def)
  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)"
    by auto
  moreover have "continuous_on i f" "continuous_on i g"
    using assms piecewise_differentiable_on_def by auto
  ultimately show ?thesis
    by (auto simp: piecewise_differentiable_on_def continuous_on_add)
qed

lemma piecewise_differentiable_diff:
    "\<lbrakk>f piecewise_differentiable_on s;  g piecewise_differentiable_on s\<rbrakk>
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on s"
  unfolding diff_conv_add_uminus
  by (metis piecewise_differentiable_add piecewise_differentiable_neg)

lemma continuous_on_joinpaths_D1:
    "continuous_on {0..1} (g1 +++ g2) \<Longrightarrow> continuous_on {0..1} g1"
  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (op*(inverse 2))"])
  apply (rule continuous_intros | simp)+
  apply (auto elim!: continuous_on_subset simp: joinpaths_def)
  done

lemma continuous_on_joinpaths_D2:
    "\<lbrakk>continuous_on {0..1} (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> continuous_on {0..1} g2"
  apply (rule continuous_on_eq [of _ "(g1 +++ g2) o (\<lambda>x. inverse 2*x + 1/2)"])
  apply (rule continuous_intros | simp)+
  apply (auto elim!: continuous_on_subset simp add: joinpaths_def pathfinish_def pathstart_def Ball_def)
  done

lemma piecewise_differentiable_D1:
    "(g1 +++ g2) piecewise_differentiable_on {0..1} \<Longrightarrow> g1 piecewise_differentiable_on {0..1}"
  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D1)
  apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
  apply simp
  apply (intro ballI)
  apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))"
       in differentiable_transform_within)
  apply (auto simp: dist_real_def joinpaths_def)
  apply (rule differentiable_chain_within derivative_intros | simp)+
  apply (rule differentiable_subset)
  apply (force simp:)+
  done

lemma piecewise_differentiable_D2:
    "\<lbrakk>(g1 +++ g2) piecewise_differentiable_on {0..1}; pathfinish g1 = pathstart g2\<rbrakk>
    \<Longrightarrow> g2 piecewise_differentiable_on {0..1}"
  apply (clarsimp simp add: piecewise_differentiable_on_def dest!: continuous_on_joinpaths_D2)
  apply (rule_tac x="insert 0 ((\<lambda>x. 2*x-1)`s)" in exI)
  apply simp
  apply (intro ballI)
  apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)"
          in differentiable_transform_within)
  apply (auto simp: dist_real_def joinpaths_def abs_if field_simps split: split_if_asm)
  apply (rule differentiable_chain_within derivative_intros | simp)+
  apply (rule differentiable_subset)
  apply (force simp: divide_simps)+
  done


subsubsection\<open>The concept of continuously differentiable\<close>

definition C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
           (infix "C1'_differentiable'_on" 50)
  where
  "f C1_differentiable_on s \<longleftrightarrow>
   (\<exists>D. (\<forall>x \<in> s. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on s D)"

lemma C1_differentiable_on_eq:
    "f C1_differentiable_on s \<longleftrightarrow>
     (\<forall>x \<in> s. f differentiable at x) \<and> continuous_on s (\<lambda>x. vector_derivative f (at x))"
  unfolding C1_differentiable_on_def
  apply safe
  using differentiable_def has_vector_derivative_def apply blast
  apply (erule continuous_on_eq)
  using vector_derivative_at apply fastforce
  using vector_derivative_works apply fastforce
  done

lemma C1_differentiable_on_subset:
  "f C1_differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f C1_differentiable_on s"
  unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
  by (blast intro:  continuous_within_subset)

lemma C1_differentiable_compose:
    "\<lbrakk>f C1_differentiable_on s; g C1_differentiable_on (f ` s);
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
      \<Longrightarrow> (g o f) C1_differentiable_on s"
  apply (simp add: C1_differentiable_on_eq, safe)
   using differentiable_chain_at apply blast
  apply (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
   apply (rule Limits.continuous_on_scaleR, assumption)
   apply (metis (mono_tags, lifting) continuous_on_eq continuous_at_imp_continuous_on continuous_on_compose differentiable_imp_continuous_within o_def)
  by (simp add: vector_derivative_chain_at)

lemma C1_diff_imp_diff: "f C1_differentiable_on s \<Longrightarrow> f differentiable_on s"
  by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)

lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on s"
  by (auto simp: C1_differentiable_on_eq continuous_on_const)

lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on s"
  by (auto simp: C1_differentiable_on_eq continuous_on_const)

lemma C1_differentiable_on_add [simp, derivative_intros]:
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on s"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_minus [simp, derivative_intros]:
  "f C1_differentiable_on s \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on s"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_diff [simp, derivative_intros]:
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on s"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_mult [simp, derivative_intros]:
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
  shows "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on s"
  unfolding C1_differentiable_on_eq
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)

lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
  "f C1_differentiable_on s \<Longrightarrow> g C1_differentiable_on s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on s"
  unfolding C1_differentiable_on_eq
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+


definition piecewise_C1_differentiable_on
           (infixr "piecewise'_C1'_differentiable'_on" 50)
  where "f piecewise_C1_differentiable_on i  \<equiv>
           continuous_on i f \<and>
           (\<exists>s. finite s \<and> (f C1_differentiable_on (i - s)))"

lemma C1_differentiable_imp_piecewise:
    "f C1_differentiable_on s \<Longrightarrow> f piecewise_C1_differentiable_on s"
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)

lemma piecewise_C1_imp_differentiable:
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
           intro: has_derivative_at_within)

lemma piecewise_C1_differentiable_compose:
    "\<lbrakk>f piecewise_C1_differentiable_on s; g piecewise_C1_differentiable_on (f ` s);
      \<And>x. finite (s \<inter> f-`{x})\<rbrakk>
      \<Longrightarrow> (g o f) piecewise_C1_differentiable_on s"
  apply (simp add: piecewise_C1_differentiable_on_def, safe)
  apply (blast intro: continuous_on_compose2)
  apply (rename_tac A B)
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. s \<inter> f-`{x})" in exI)
  apply (rule conjI, blast)
  apply (rule C1_differentiable_compose)
  apply (blast intro: C1_differentiable_on_subset)
  apply (blast intro: C1_differentiable_on_subset)
  by (simp add: Diff_Int_distrib2)

lemma piecewise_C1_differentiable_on_subset:
    "f piecewise_C1_differentiable_on s \<Longrightarrow> t \<le> s \<Longrightarrow> f piecewise_C1_differentiable_on t"
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)

lemma C1_differentiable_imp_continuous_on:
  "f C1_differentiable_on s \<Longrightarrow> continuous_on s f"
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
  using differentiable_at_withinI differentiable_imp_continuous_within by blast

lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
  unfolding C1_differentiable_on_def
  by auto

lemma piecewise_C1_differentiable_affine:
  fixes m::real
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` s)"
  shows "(f o (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on s"
proof (cases "m = 0")
  case True
  then show ?thesis
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def continuous_on_const)
next
  case False
  show ?thesis
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
    apply (rule assms derivative_intros | simp add: False vimage_def)+
    using real_vector_affinity_eq [OF False, where c=c, unfolded scaleR_conv_of_real]
    apply simp
    done
qed

lemma piecewise_C1_differentiable_cases:
  fixes c::real
  assumes "f piecewise_C1_differentiable_on {a..c}"
          "g piecewise_C1_differentiable_on {c..b}"
           "a \<le> c" "c \<le> b" "f c = g c"
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
proof -
  obtain s t where st: "f C1_differentiable_on ({a..c} - s)"
                       "g C1_differentiable_on ({c..b} - t)"
                       "finite s" "finite t"
    using assms
    by (force simp: piecewise_C1_differentiable_on_def)
  then have f_diff: "f differentiable_on {a..<c} - s"
        and g_diff: "g differentiable_on {c<..b} - t"
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
    using assms piecewise_C1_differentiable_on_def by auto
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                               OF closed_real_atLeastAtMost [of c b],
                               of f g "\<lambda>x. x\<le>c"]  assms
    by (force simp: ivl_disj_un_two_touch)
  { fix x
    assume x: "x \<in> {a..b} - insert c (s \<union> t)"
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
    proof (cases x c rule: le_cases)
      case le show ?diff_fg
        apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
    next
      case ge show ?diff_fg
        apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
    qed
  }
  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)"
    by auto
  moreover
  { assume fcon: "continuous_on ({a<..<c} - s) (\<lambda>x. vector_derivative f (at x))"
       and gcon: "continuous_on ({c<..<b} - t) (\<lambda>x. vector_derivative g (at x))"
    have "open ({a<..<c} - s)"  "open ({c<..<b} - t)"
      using st by (simp_all add: open_Diff finite_imp_closed)
    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))"
      apply (rule continuous_on_eq [OF fcon])
      apply (simp add:)
      apply (rule vector_derivative_at [symmetric])
      apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
      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))
      apply auto
      done
    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))"
      apply (rule continuous_on_eq [OF gcon])
      apply (simp add:)
      apply (rule vector_derivative_at [symmetric])
      apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
      apply (simp_all add: dist_norm vector_derivative_works [symmetric])
      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))
      apply auto
      done
    ultimately 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))"
      apply (rule continuous_on_subset [OF continuous_on_open_Un], auto)
      done
  } note * = this
  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))"
    using st
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
  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)"
    apply (rule_tac x="{a,b,c} \<union> s \<union> t" in exI)
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
  with cab show ?thesis
    by (simp add: piecewise_C1_differentiable_on_def)
qed

lemma piecewise_C1_differentiable_neg:
    "f piecewise_C1_differentiable_on s \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on s"
  unfolding piecewise_C1_differentiable_on_def
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)

lemma piecewise_C1_differentiable_add:
  assumes "f piecewise_C1_differentiable_on i"
          "g piecewise_C1_differentiable_on i"
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
proof -
  obtain s t where st: "finite s" "finite t"
                       "f C1_differentiable_on (i-s)"
                       "g C1_differentiable_on (i-t)"
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
  then have "finite (s \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (s \<union> t)"
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
  moreover have "continuous_on i f" "continuous_on i g"
    using assms piecewise_C1_differentiable_on_def by auto
  ultimately show ?thesis
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
qed

lemma piecewise_C1_differentiable_diff:
    "\<lbrakk>f piecewise_C1_differentiable_on s;  g piecewise_C1_differentiable_on s\<rbrakk>
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on s"
  unfolding diff_conv_add_uminus
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)

lemma piecewise_C1_differentiable_D1:
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}"
    shows "g1 piecewise_C1_differentiable_on {0..1}"
proof -
  obtain s where "finite s"
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  then have g1D: "g1 differentiable at x" if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
    apply (rule_tac d="dist (x/2) (1/2)" and f = "(g1 +++ g2) o (op*(inverse 2))" in differentiable_transform_within)
    using that
    apply (simp_all add: dist_real_def joinpaths_def)
    apply (rule differentiable_chain_at derivative_intros | force)+
    done
  have [simp]: "vector_derivative (g1 \<circ> op * 2) (at (x/2)) = 2 *\<^sub>R vector_derivative g1 (at x)"
               if "x \<in> {0..1} - insert 1 (op * 2 ` s)" for x
    apply (subst vector_derivative_chain_at)
    using that
    apply (rule derivative_eq_intros g1D | simp)+
    done
  have "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
    using co12 by (rule continuous_on_subset) force
  then have coDhalf: "continuous_on ({0..1/2} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 o op*2) (at x))"
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
    apply (rule_tac f="g1 o op*2" and d="dist x (1/2)" in has_vector_derivative_transform_within)
    apply (simp_all add: dist_norm joinpaths_def vector_derivative_works [symmetric])
    apply (force intro: g1D differentiable_chain_at)
    apply auto
    done
  have "continuous_on ({0..1} - insert 1 (op * 2 ` s))
                      ((\<lambda>x. 1/2 * vector_derivative (g1 o op*2) (at x)) o op*(1/2))"
    apply (rule continuous_intros)+
    using coDhalf
    apply (simp add: scaleR_conv_of_real image_set_diff image_image)
    done
  then have con_g1: "continuous_on ({0..1} - insert 1 (op * 2 ` s)) (\<lambda>x. vector_derivative g1 (at x))"
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
  have "continuous_on {0..1} g1"
    using continuous_on_joinpaths_D1 assms piecewise_C1_differentiable_on_def by blast
  with \<open>finite s\<close> show ?thesis
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
    apply (rule_tac x="insert 1 ((op*2)`s)" in exI)
    apply (simp add: g1D con_g1)
  done
qed

lemma piecewise_C1_differentiable_D2:
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
  assumes "(g1 +++ g2) piecewise_C1_differentiable_on {0..1}" "pathfinish g1 = pathstart g2"
    shows "g2 piecewise_C1_differentiable_on {0..1}"
proof -
  obtain s where "finite s"
             and co12: "continuous_on ({0..1} - s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
             and g12D: "\<forall>x\<in>{0..1} - s. g1 +++ g2 differentiable at x"
    using assms  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  then have g2D: "g2 differentiable at x" if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
    apply (rule_tac d="dist ((x+1)/2) (1/2)" and f = "(g1 +++ g2) o (\<lambda>x. (x+1)/2)" in differentiable_transform_within)
    using that
    apply (simp_all add: dist_real_def joinpaths_def)
    apply (auto simp: dist_real_def joinpaths_def field_simps)
    apply (rule differentiable_chain_at derivative_intros | force)+
    apply (drule_tac x= "(x + 1) / 2" in bspec, force simp: divide_simps)
    apply assumption
    done
  have [simp]: "vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at ((x+1)/2)) = 2 *\<^sub>R vector_derivative g2 (at x)"
               if "x \<in> {0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)" for x
    using that  by (auto simp: vector_derivative_chain_at divide_simps g2D)
  have "continuous_on ({1/2..1} - insert (1/2) s) (\<lambda>x. vector_derivative (g1 +++ g2) (at x))"
    using co12 by (rule continuous_on_subset) force
  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))"
    apply (rule continuous_on_eq [OF _ vector_derivative_at])
    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)
    apply (auto simp: dist_real_def field_simps joinpaths_def vector_derivative_works [symmetric]
                intro!: g2D differentiable_chain_at)
    done
  have [simp]: "((\<lambda>x. (x + 1) / 2) ` ({0..1} - insert 0 ((\<lambda>x. 2 * x - 1) ` s))) = ({1/2..1} - insert (1/2) s)"
    apply (simp add: image_set_diff inj_on_def image_image)
    apply (auto simp: image_affinity_atLeastAtMost_div add_divide_distrib)
    done
  have "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s))
                      ((\<lambda>x. 1/2 * vector_derivative (g2 \<circ> (\<lambda>x. 2*x-1)) (at x)) o (\<lambda>x. (x+1)/2))"
    by (rule continuous_intros | simp add:  coDhalf)+
  then have con_g2: "continuous_on ({0..1} - insert 0 ((\<lambda>x. 2*x-1) ` s)) (\<lambda>x. vector_derivative g2 (at x))"
    by (rule continuous_on_eq) (simp add: scaleR_conv_of_real)
  have "continuous_on {0..1} g2"
    using continuous_on_joinpaths_D2 assms piecewise_C1_differentiable_on_def by blast
  with \<open>finite s\<close> show ?thesis
    apply (clarsimp simp add: piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
    apply (rule_tac x="insert 0 ((\<lambda>x. 2 * x - 1) ` s)" in exI)
    apply (simp add: g2D con_g2)
  done
qed

subsection \<open>Valid paths, and their start and finish\<close>

lemma Diff_Un_eq: "A - (B \<union> C) = A - B - C"
  by blast

definition valid_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
  where "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1::real}"

definition closed_path :: "(real \<Rightarrow> 'a :: real_normed_vector) \<Rightarrow> bool"
  where "closed_path g \<equiv> g 0 = g 1"

subsubsection\<open>In particular, all results for paths apply\<close>

lemma valid_path_imp_path: "valid_path g \<Longrightarrow> path g"
by (simp add: path_def piecewise_C1_differentiable_on_def valid_path_def)

lemma connected_valid_path_image: "valid_path g \<Longrightarrow> connected(path_image g)"
  by (metis connected_path_image valid_path_imp_path)

lemma compact_valid_path_image: "valid_path g \<Longrightarrow> compact(path_image g)"
  by (metis compact_path_image valid_path_imp_path)

lemma bounded_valid_path_image: "valid_path g \<Longrightarrow> bounded(path_image g)"
  by (metis bounded_path_image valid_path_imp_path)

lemma closed_valid_path_image: "valid_path g \<Longrightarrow> closed(path_image g)"
  by (metis closed_path_image valid_path_imp_path)


subsection\<open>Contour Integrals along a path\<close>

text\<open>This definition is for complex numbers only, and does not generalise to line integrals in a vector field\<close>

text\<open>piecewise differentiable function on [0,1]\<close>

definition has_contour_integral :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool"
           (infixr "has'_contour'_integral" 50)
  where "(f has_contour_integral i) g \<equiv>
           ((\<lambda>x. f(g x) * vector_derivative g (at x within {0..1}))
            has_integral i) {0..1}"

definition contour_integrable_on
           (infixr "contour'_integrable'_on" 50)
  where "f contour_integrable_on g \<equiv> \<exists>i. (f has_contour_integral i) g"

definition contour_integral
  where "contour_integral g f \<equiv> @i. (f has_contour_integral i) g"

lemma contour_integral_unique: "(f has_contour_integral i)  g \<Longrightarrow> contour_integral g f = i"
  by (auto simp: contour_integral_def has_contour_integral_def integral_def [symmetric])

corollary has_contour_integral_eqpath:
     "\<lbrakk>(f has_contour_integral y) p; f contour_integrable_on \<gamma>;
       contour_integral p f = contour_integral \<gamma> f\<rbrakk>
      \<Longrightarrow> (f has_contour_integral y) \<gamma>"
using contour_integrable_on_def contour_integral_unique by auto

lemma has_contour_integral_integral:
    "f contour_integrable_on i \<Longrightarrow> (f has_contour_integral (contour_integral i f)) i"
  by (metis contour_integral_unique contour_integrable_on_def)

lemma has_contour_integral_unique:
    "(f has_contour_integral i) g \<Longrightarrow> (f has_contour_integral j) g \<Longrightarrow> i = j"
  using has_integral_unique
  by (auto simp: has_contour_integral_def)

lemma has_contour_integral_integrable: "(f has_contour_integral i) g \<Longrightarrow> f contour_integrable_on g"
  using contour_integrable_on_def by blast

(* Show that we can forget about the localized derivative.*)

lemma vector_derivative_within_interior:
     "\<lbrakk>x \<in> interior s; NO_MATCH UNIV s\<rbrakk>
      \<Longrightarrow> vector_derivative f (at x within s) = vector_derivative f (at x)"
  apply (simp add: vector_derivative_def has_vector_derivative_def has_derivative_def netlimit_within_interior)
  apply (subst lim_within_interior, auto)
  done

lemma has_integral_localized_vector_derivative:
    "((\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) has_integral i) {a..b} \<longleftrightarrow>
     ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {a..b}"
proof -
  have "{a..b} - {a,b} = interior {a..b}"
    by (simp add: atLeastAtMost_diff_ends)
  show ?thesis
    apply (rule has_integral_spike_eq [of "{a,b}"])
    apply (auto simp: vector_derivative_within_interior)
    done
qed

lemma integrable_on_localized_vector_derivative:
    "(\<lambda>x. f (g x) * vector_derivative g (at x within {a..b})) integrable_on {a..b} \<longleftrightarrow>
     (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {a..b}"
  by (simp add: integrable_on_def has_integral_localized_vector_derivative)

lemma has_contour_integral:
     "(f has_contour_integral i) g \<longleftrightarrow>
      ((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
  by (simp add: has_integral_localized_vector_derivative has_contour_integral_def)

lemma contour_integrable_on:
     "f contour_integrable_on g \<longleftrightarrow>
      (\<lambda>t. f(g t) * vector_derivative g (at t)) integrable_on {0..1}"
  by (simp add: has_contour_integral integrable_on_def contour_integrable_on_def)

subsection\<open>Reversing a path\<close>

lemma valid_path_imp_reverse:
  assumes "valid_path g"
    shows "valid_path(reversepath g)"
proof -
  obtain s where "finite s" "g C1_differentiable_on ({0..1} - s)"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
  then have "finite (op - 1 ` s)" "(reversepath g C1_differentiable_on ({0..1} - op - 1 ` s))"
    apply (auto simp: reversepath_def)
    apply (rule C1_differentiable_compose [of "\<lambda>x::real. 1-x" _ g, unfolded o_def])
    apply (auto simp: C1_differentiable_on_eq)
    apply (rule continuous_intros, force)
    apply (force elim!: continuous_on_subset)
    apply (simp add: finite_vimageI inj_on_def)
    done
  then show ?thesis using assms
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def path_def [symmetric])
qed

lemma valid_path_reversepath: "valid_path(reversepath g) \<longleftrightarrow> valid_path g"
  using valid_path_imp_reverse by force

lemma has_contour_integral_reversepath:
  assumes "valid_path g" "(f has_contour_integral i) g"
    shows "(f has_contour_integral (-i)) (reversepath g)"
proof -
  { fix s x
    assume xs: "g C1_differentiable_on ({0..1} - s)" "x \<notin> op - 1 ` s" "0 \<le> x" "x \<le> 1"
      have "vector_derivative (\<lambda>x. g (1 - x)) (at x within {0..1}) =
            - vector_derivative g (at (1 - x) within {0..1})"
      proof -
        obtain f' where f': "(g has_vector_derivative f') (at (1 - x))"
          using xs
          by (force simp: has_vector_derivative_def C1_differentiable_on_def)
        have "(g o (\<lambda>x. 1 - x) has_vector_derivative -1 *\<^sub>R f') (at x)"
          apply (rule vector_diff_chain_within)
          apply (intro vector_diff_chain_within derivative_eq_intros | simp)+
          apply (rule has_vector_derivative_at_within [OF f'])
          done
        then have mf': "((\<lambda>x. g (1 - x)) has_vector_derivative -f') (at x)"
          by (simp add: o_def)
        show ?thesis
          using xs
          by (auto simp: vector_derivative_at_within_ivl [OF mf'] vector_derivative_at_within_ivl [OF f'])
      qed
  } note * = this
  have 01: "{0..1::real} = cbox 0 1"
    by simp
  show ?thesis using assms
    apply (auto simp: has_contour_integral_def)
    apply (drule has_integral_affinity01 [where m= "-1" and c=1])
    apply (auto simp: reversepath_def valid_path_def piecewise_C1_differentiable_on_def)
    apply (drule has_integral_neg)
    apply (rule_tac s = "(\<lambda>x. 1 - x) ` s" in has_integral_spike_finite)
    apply (auto simp: *)
    done
qed

lemma contour_integrable_reversepath:
    "valid_path g \<Longrightarrow> f contour_integrable_on g \<Longrightarrow> f contour_integrable_on (reversepath g)"
  using has_contour_integral_reversepath contour_integrable_on_def by blast

lemma contour_integrable_reversepath_eq:
    "valid_path g \<Longrightarrow> (f contour_integrable_on (reversepath g) \<longleftrightarrow> f contour_integrable_on g)"
  using contour_integrable_reversepath valid_path_reversepath by fastforce

lemma contour_integral_reversepath:
    "\<lbrakk>valid_path g; f contour_integrable_on g\<rbrakk> \<Longrightarrow> contour_integral (reversepath g) f = -(contour_integral g f)"
  using has_contour_integral_reversepath contour_integrable_on_def contour_integral_unique by blast


subsection\<open>Joining two paths together\<close>

lemma valid_path_join:
  assumes "valid_path g1" "valid_path g2" "pathfinish g1 = pathstart g2"
    shows "valid_path(g1 +++ g2)"
proof -
  have "g1 1 = g2 0"
    using assms by (auto simp: pathfinish_def pathstart_def)
  moreover have "(g1 o (\<lambda>x. 2*x)) piecewise_C1_differentiable_on {0..1/2}"
    apply (rule piecewise_C1_differentiable_compose)
    using assms
    apply (auto simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_joinpaths)
    apply (rule continuous_intros | simp)+
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
    done
  moreover have "(g2 o (\<lambda>x. 2*x-1)) piecewise_C1_differentiable_on {1/2..1}"
    apply (rule piecewise_C1_differentiable_compose)
    using assms unfolding valid_path_def piecewise_C1_differentiable_on_def
    by (auto intro!: continuous_intros finite_vimageI [where h = "(\<lambda>x. 2*x - 1)"] inj_onI
             simp: image_affinity_atLeastAtMost_diff continuous_on_joinpaths)
  ultimately show ?thesis
    apply (simp only: valid_path_def continuous_on_joinpaths joinpaths_def)
    apply (rule piecewise_C1_differentiable_cases)
    apply (auto simp: o_def)
    done
qed

lemma valid_path_join_D1:
  fixes g1 :: "real \<Rightarrow> 'a::real_normed_field"
  shows "valid_path (g1 +++ g2) \<Longrightarrow> valid_path g1"
  unfolding valid_path_def
  by (rule piecewise_C1_differentiable_D1)

lemma valid_path_join_D2:
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
  shows "\<lbrakk>valid_path (g1 +++ g2); pathfinish g1 = pathstart g2\<rbrakk> \<Longrightarrow> valid_path g2"
  unfolding valid_path_def
  by (rule piecewise_C1_differentiable_D2)

lemma valid_path_join_eq [simp]:
  fixes g2 :: "real \<Rightarrow> 'a::real_normed_field"
  shows "pathfinish g1 = pathstart g2 \<Longrightarrow> (valid_path(g1 +++ g2) \<longleftrightarrow> valid_path g1 \<and> valid_path g2)"
  using valid_path_join_D1 valid_path_join_D2 valid_path_join by blast

lemma has_contour_integral_join:
  assumes "(f has_contour_integral i1) g1" "(f has_contour_integral i2) g2"
          "valid_path g1" "valid_path g2"
    shows "(f has_contour_integral (i1 + i2)) (g1 +++ g2)"
proof -
  obtain s1 s2
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
      and s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
    using assms
    by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  have 1: "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
   and 2: "((\<lambda>x. f (g2 x) * vector_derivative g2 (at x)) has_integral i2) {0..1}"
    using assms
    by (auto simp: has_contour_integral)
  have i1: "((\<lambda>x. (2*f (g1 (2*x))) * vector_derivative g1 (at (2*x))) has_integral i1) {0..1/2}"
   and i2: "((\<lambda>x. (2*f (g2 (2*x - 1))) * vector_derivative g2 (at (2*x - 1))) has_integral i2) {1/2..1}"
    using has_integral_affinity01 [OF 1, where m= 2 and c=0, THEN has_integral_cmul [where c=2]]
          has_integral_affinity01 [OF 2, where m= 2 and c="-1", THEN has_integral_cmul [where c=2]]
    by (simp_all only: image_affinity_atLeastAtMost_div_diff, simp_all add: scaleR_conv_of_real mult_ac)
  have g1: "\<lbrakk>0 \<le> z; z*2 < 1; z*2 \<notin> s1\<rbrakk> \<Longrightarrow>
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
            2 *\<^sub>R vector_derivative g1 (at (z*2))" for z
    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>"]])
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x" 2 _ g1, simplified o_def])
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
    using s1
    apply (auto simp: algebra_simps vector_derivative_works)
    done
  have g2: "\<lbrakk>1 < z*2; z \<le> 1; z*2 - 1 \<notin> s2\<rbrakk> \<Longrightarrow>
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at z) =
            2 *\<^sub>R vector_derivative g2 (at (z*2 - 1))" for z
    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>"]])
    apply (simp_all add: dist_real_def abs_if split: split_if_asm)
    apply (rule vector_diff_chain_at [of "\<lambda>x. 2*x - 1" 2 _ g2, simplified o_def])
    apply (simp add: has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
    using s2
    apply (auto simp: algebra_simps vector_derivative_works)
    done
  have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i1) {0..1/2}"
    apply (rule has_integral_spike_finite [OF _ _ i1, of "insert (1/2) (op*2 -` s1)"])
    using s1
    apply (force intro: finite_vimageI [where h = "op*2"] inj_onI)
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g1)
    done
  moreover have "((\<lambda>x. f ((g1 +++ g2) x) * vector_derivative (g1 +++ g2) (at x)) has_integral i2) {1/2..1}"
    apply (rule has_integral_spike_finite [OF _ _ i2, of "insert (1/2) ((\<lambda>x. 2*x-1) -` s2)"])
    using s2
    apply (force intro: finite_vimageI [where h = "\<lambda>x. 2*x-1"] inj_onI)
    apply (clarsimp simp add: joinpaths_def scaleR_conv_of_real mult_ac g2)
    done
  ultimately
  show ?thesis
    apply (simp add: has_contour_integral)
    apply (rule has_integral_combine [where c = "1/2"], auto)
    done
qed

lemma contour_integrable_joinI:
  assumes "f contour_integrable_on g1" "f contour_integrable_on g2"
          "valid_path g1" "valid_path g2"
    shows "f contour_integrable_on (g1 +++ g2)"
  using assms
  by (meson has_contour_integral_join contour_integrable_on_def)

lemma contour_integrable_joinD1:
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g1"
    shows "f contour_integrable_on g1"
proof -
  obtain s1
    where s1: "finite s1" "\<forall>x\<in>{0..1} - s1. g1 differentiable at x"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  have "(\<lambda>x. f ((g1 +++ g2) (x/2)) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
    using assms
    apply (auto simp: contour_integrable_on)
    apply (drule integrable_on_subcbox [where a=0 and b="1/2"])
    apply (auto intro: integrable_affinity [of _ 0 "1/2::real" "1/2" 0, simplified])
    done
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2))/2) * vector_derivative (g1 +++ g2) (at (x/2))) integrable_on {0..1}"
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
  have g1: "\<lbrakk>0 < z; z < 1; z \<notin> s1\<rbrakk> \<Longrightarrow>
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2)) =
            2 *\<^sub>R vector_derivative g1 (at z)"  for z
    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>"]])
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2" 2 _ g1, simplified o_def])
    using s1
    apply (auto simp: vector_derivative_works has_vector_derivative_def has_derivative_def bounded_linear_mult_left)
    done
  show ?thesis
    using s1
    apply (auto simp: contour_integrable_on)
    apply (rule integrable_spike_finite [of "{0,1} \<union> s1", OF _ _ *])
    apply (auto simp: joinpaths_def scaleR_conv_of_real g1)
    done
qed

lemma contour_integrable_joinD2:
  assumes "f contour_integrable_on (g1 +++ g2)" "valid_path g2"
    shows "f contour_integrable_on g2"
proof -
  obtain s2
    where s2: "finite s2" "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  have "(\<lambda>x. f ((g1 +++ g2) (x/2 + 1/2)) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2))) integrable_on {0..1}"
    using assms
    apply (auto simp: contour_integrable_on)
    apply (drule integrable_on_subcbox [where a="1/2" and b=1], auto)
    apply (drule integrable_affinity [of _ "1/2::real" 1 "1/2" "1/2", simplified])
    apply (simp add: image_affinity_atLeastAtMost_diff)
    done
  then have *: "(\<lambda>x. (f ((g1 +++ g2) (x/2 + 1/2))/2) * vector_derivative (g1 +++ g2) (at (x/2 + 1/2)))
                integrable_on {0..1}"
    by (auto dest: integrable_cmul [where c="1/2"] simp: scaleR_conv_of_real)
  have g2: "\<lbrakk>0 < z; z < 1; z \<notin> s2\<rbrakk> \<Longrightarrow>
            vector_derivative (\<lambda>x. if x*2 \<le> 1 then g1 (2*x) else g2 (2*x - 1)) (at (z/2+1/2)) =
            2 *\<^sub>R vector_derivative g2 (at z)" for z
    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>"]])
    apply (simp_all add: field_simps dist_real_def abs_if split: split_if_asm)
    apply (rule vector_diff_chain_at [of "\<lambda>x. x*2-1" 2 _ g2, simplified o_def])
    using s2
    apply (auto simp: has_vector_derivative_def has_derivative_def bounded_linear_mult_left
                      vector_derivative_works add_divide_distrib)
    done
  show ?thesis
    using s2
    apply (auto simp: contour_integrable_on)
    apply (rule integrable_spike_finite [of "{0,1} \<union> s2", OF _ _ *])
    apply (auto simp: joinpaths_def scaleR_conv_of_real g2)
    done
qed

lemma contour_integrable_join [simp]:
  shows
    "\<lbrakk>valid_path g1; valid_path g2\<rbrakk>
     \<Longrightarrow> f contour_integrable_on (g1 +++ g2) \<longleftrightarrow> f contour_integrable_on g1 \<and> f contour_integrable_on g2"
using contour_integrable_joinD1 contour_integrable_joinD2 contour_integrable_joinI by blast

lemma contour_integral_join [simp]:
  shows
    "\<lbrakk>f contour_integrable_on g1; f contour_integrable_on g2; valid_path g1; valid_path g2\<rbrakk>
        \<Longrightarrow> contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
  by (simp add: has_contour_integral_integral has_contour_integral_join contour_integral_unique)


subsection\<open>Shifting the starting point of a (closed) path\<close>

lemma shiftpath_alt_def: "shiftpath a f = (\<lambda>x. if x \<le> 1-a then f (a + x) else f (a + x - 1))"
  by (auto simp: shiftpath_def)

lemma valid_path_shiftpath [intro]:
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
    shows "valid_path(shiftpath a g)"
  using assms
  apply (auto simp: valid_path_def shiftpath_alt_def)
  apply (rule piecewise_C1_differentiable_cases)
  apply (auto simp: algebra_simps)
  apply (rule piecewise_C1_differentiable_affine [of g 1 a, simplified o_def scaleR_one])
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
  apply (rule piecewise_C1_differentiable_affine [of g 1 "a-1", simplified o_def scaleR_one algebra_simps])
  apply (auto simp: pathfinish_def pathstart_def elim: piecewise_C1_differentiable_on_subset)
  done

lemma has_contour_integral_shiftpath:
  assumes f: "(f has_contour_integral i) g" "valid_path g"
      and a: "a \<in> {0..1}"
    shows "(f has_contour_integral i) (shiftpath a g)"
proof -
  obtain s
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  have *: "((\<lambda>x. f (g x) * vector_derivative g (at x)) has_integral i) {0..1}"
    using assms by (auto simp: has_contour_integral)
  then have i: "i = integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)) +
                    integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x))"
    apply (rule has_integral_unique)
    apply (subst add.commute)
    apply (subst Integration.integral_combine)
    using assms * integral_unique by auto
  { fix x
    have "0 \<le> x \<Longrightarrow> x + a < 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a) ` s \<Longrightarrow>
         vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a))"
      unfolding shiftpath_def
      apply (rule vector_derivative_at [OF has_vector_derivative_transform_within [where f = "(\<lambda>x. g(a+x))" and d = "dist(1-a) x"]])
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a" 1 _ g, simplified o_def scaleR_one])
       apply (intro derivative_eq_intros | simp)+
      using g
       apply (drule_tac x="x+a" in bspec)
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
      done
  } note vd1 = this
  { fix x
    have "1 < x + a \<Longrightarrow> x \<le> 1 \<Longrightarrow> x \<notin> (\<lambda>x. x - a + 1) ` s \<Longrightarrow>
          vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
      unfolding shiftpath_def
      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"]])
        apply (auto simp: field_simps dist_real_def abs_if split: split_if_asm)
      apply (rule vector_diff_chain_at [of "\<lambda>x. x+a-1" 1 _ g, simplified o_def scaleR_one])
       apply (intro derivative_eq_intros | simp)+
      using g
      apply (drule_tac x="x+a-1" in bspec)
      using a apply (auto simp: has_vector_derivative_def vector_derivative_works image_def add.commute)
      done
  } note vd2 = this
  have va1: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({a..1})"
    using * a   by (fastforce intro: integrable_subinterval_real)
  have v0a: "(\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on ({0..a})"
    apply (rule integrable_subinterval_real)
    using * a by auto
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
        has_integral  integral {a..1} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {0..1 - a}"
    apply (rule has_integral_spike_finite
             [where s = "{1-a} \<union> (\<lambda>x. x-a) ` s" and f = "\<lambda>x. f(g(a+x)) * vector_derivative g (at(a+x))"])
      using s apply blast
     using a apply (auto simp: algebra_simps vd1)
     apply (force simp: shiftpath_def add.commute)
    using has_integral_affinity [where m=1 and c=a, simplified, OF integrable_integral [OF va1]]
    apply (simp add: image_affinity_atLeastAtMost_diff [where m=1 and c=a, simplified] add.commute)
    done
  moreover
  have "((\<lambda>x. f (shiftpath a g x) * vector_derivative (shiftpath a g) (at x))
        has_integral  integral {0..a} (\<lambda>x. f (g x) * vector_derivative g (at x)))  {1 - a..1}"
    apply (rule has_integral_spike_finite
             [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))"])
      using s apply blast
     using a apply (auto simp: algebra_simps vd2)
     apply (force simp: shiftpath_def add.commute)
    using has_integral_affinity [where m=1 and c="a-1", simplified, OF integrable_integral [OF v0a]]
    apply (simp add: image_affinity_atLeastAtMost [where m=1 and c="1-a", simplified])
    apply (simp add: algebra_simps)
    done
  ultimately show ?thesis
    using a
    by (auto simp: i has_contour_integral intro: has_integral_combine [where c = "1-a"])
qed

lemma has_contour_integral_shiftpath_D:
  assumes "(f has_contour_integral i) (shiftpath a g)"
          "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
    shows "(f has_contour_integral i) g"
proof -
  obtain s
    where s: "finite s" and g: "\<forall>x\<in>{0..1} - s. g differentiable at x"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq)
  { fix x
    assume x: "0 < x" "x < 1" "x \<notin> s"
    then have gx: "g differentiable at x"
      using g by auto
    have "vector_derivative g (at x within {0..1}) =
          vector_derivative (shiftpath (1 - a) (shiftpath a g)) (at x within {0..1})"
      apply (rule vector_derivative_at_within_ivl
                  [OF has_vector_derivative_transform_within_open
                      [where f = "(shiftpath (1 - a) (shiftpath a g))" and s = "{0<..<1}-s"]])
      using s g assms x
      apply (auto simp: finite_imp_closed open_Diff shiftpath_shiftpath
                        vector_derivative_within_interior vector_derivative_works [symmetric])
      apply (rule differentiable_transform_within [OF gx, of "min x (1-x)"])
      apply (auto simp: dist_real_def shiftpath_shiftpath abs_if split: split_if_asm)
      done
  } note vd = this
  have fi: "(f has_contour_integral i) (shiftpath (1 - a) (shiftpath a g))"
    using assms  by (auto intro!: has_contour_integral_shiftpath)
  show ?thesis
    apply (simp add: has_contour_integral_def)
    apply (rule has_integral_spike_finite [of "{0,1} \<union> s", OF _ _  fi [unfolded has_contour_integral_def]])
    using s assms vd
    apply (auto simp: Path_Connected.shiftpath_shiftpath)
    done
qed

lemma has_contour_integral_shiftpath_eq:
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
    shows "(f has_contour_integral i) (shiftpath a g) \<longleftrightarrow> (f has_contour_integral i) g"
  using assms has_contour_integral_shiftpath has_contour_integral_shiftpath_D by blast

lemma contour_integral_shiftpath:
  assumes "valid_path g" "pathfinish g = pathstart g" "a \<in> {0..1}"
    shows "contour_integral (shiftpath a g) f = contour_integral g f"
   using assms by (simp add: contour_integral_def has_contour_integral_shiftpath_eq)


subsection\<open>More about straight-line paths\<close>

lemma has_vector_derivative_linepath_within:
    "(linepath a b has_vector_derivative (b - a)) (at x within s)"
apply (simp add: linepath_def has_vector_derivative_def algebra_simps)
apply (rule derivative_eq_intros | simp)+
done

lemma vector_derivative_linepath_within:
    "x \<in> {0..1} \<Longrightarrow> vector_derivative (linepath a b) (at x within {0..1}) = b - a"
  apply (rule vector_derivative_within_closed_interval [of 0 "1::real", simplified])
  apply (auto simp: has_vector_derivative_linepath_within)
  done

lemma vector_derivative_linepath_at [simp]: "vector_derivative (linepath a b) (at x) = b - a"
  by (simp add: has_vector_derivative_linepath_within vector_derivative_at)

lemma valid_path_linepath [iff]: "valid_path (linepath a b)"
  apply (simp add: valid_path_def piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_on_linepath)
  apply (rule_tac x="{}" in exI)
  apply (simp add: differentiable_on_def differentiable_def)
  using has_vector_derivative_def has_vector_derivative_linepath_within
  apply (fastforce simp add: continuous_on_eq_continuous_within)
  done

lemma has_contour_integral_linepath:
  shows "(f has_contour_integral i) (linepath a b) \<longleftrightarrow>
         ((\<lambda>x. f(linepath a b x) * (b - a)) has_integral i) {0..1}"
  by (simp add: has_contour_integral vector_derivative_linepath_at)

lemma linepath_in_path:
  shows "x \<in> {0..1} \<Longrightarrow> linepath a b x \<in> closed_segment a b"
  by (auto simp: segment linepath_def)

lemma linepath_image_01: "linepath a b ` {0..1} = closed_segment a b"
  by (auto simp: segment linepath_def)

lemma linepath_in_convex_hull:
    fixes x::real
    assumes a: "a \<in> convex hull s"
        and b: "b \<in> convex hull s"
        and x: "0\<le>x" "x\<le>1"
       shows "linepath a b x \<in> convex hull s"
  apply (rule closed_segment_subset_convex_hull [OF a b, THEN subsetD])
  using x
  apply (auto simp: linepath_image_01 [symmetric])
  done

lemma Re_linepath: "Re(linepath (of_real a) (of_real b) x) = (1 - x)*a + x*b"
  by (simp add: linepath_def)

lemma Im_linepath: "Im(linepath (of_real a) (of_real b) x) = 0"
  by (simp add: linepath_def)

lemma linepath_of_real: "(linepath (of_real a) (of_real b) x) = of_real ((1 - x)*a + x*b)"
  by (simp add: scaleR_conv_of_real linepath_def)

lemma of_real_linepath: "of_real (linepath a b x) = linepath (of_real a) (of_real b) x"
  by (metis linepath_of_real mult.right_neutral of_real_def real_scaleR_def)

lemma has_contour_integral_trivial [iff]: "(f has_contour_integral 0) (linepath a a)"
  by (simp add: has_contour_integral_linepath)

lemma contour_integral_trivial [simp]: "contour_integral (linepath a a) f = 0"
  using has_contour_integral_trivial contour_integral_unique by blast


subsection\<open>Relation to subpath construction\<close>

lemma valid_path_subpath:
  fixes g :: "real \<Rightarrow> 'a :: real_normed_vector"
  assumes "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
    shows "valid_path(subpath u v g)"
proof (cases "v=u")
  case True
  then show ?thesis
    unfolding valid_path_def subpath_def
    by (force intro: C1_differentiable_on_const C1_differentiable_imp_piecewise)
next
  case False
  have "(g o (\<lambda>x. ((v-u) * x + u))) piecewise_C1_differentiable_on {0..1}"
    apply (rule piecewise_C1_differentiable_compose)
    apply (simp add: C1_differentiable_imp_piecewise)
     apply (simp add: image_affinity_atLeastAtMost)
    using assms False
    apply (auto simp: algebra_simps valid_path_def piecewise_C1_differentiable_on_subset)
    apply (subst Int_commute)
    apply (auto simp: inj_on_def algebra_simps crossproduct_eq finite_vimage_IntI)
    done
  then show ?thesis
    by (auto simp: o_def valid_path_def subpath_def)
qed

lemma has_contour_integral_subpath_refl [iff]: "(f has_contour_integral 0) (subpath u u g)"
  by (simp add: has_contour_integral subpath_def)

lemma contour_integrable_subpath_refl [iff]: "f contour_integrable_on (subpath u u g)"
  using has_contour_integral_subpath_refl contour_integrable_on_def by blast

lemma contour_integral_subpath_refl [simp]: "contour_integral (subpath u u g) f = 0"
  by (simp add: has_contour_integral_subpath_refl contour_integral_unique)

lemma has_contour_integral_subpath:
  assumes f: "f contour_integrable_on g" and g: "valid_path g"
      and uv: "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
    shows "(f has_contour_integral  integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x)))
           (subpath u v g)"
proof (cases "v=u")
  case True
  then show ?thesis
    using f   by (simp add: contour_integrable_on_def subpath_def has_contour_integral)
next
  case False
  obtain s where s: "\<And>x. x \<in> {0..1} - s \<Longrightarrow> g differentiable at x" and fs: "finite s"
    using g unfolding piecewise_C1_differentiable_on_def C1_differentiable_on_eq valid_path_def by blast
  have *: "((\<lambda>x. f (g ((v - u) * x + u)) * vector_derivative g (at ((v - u) * x + u)))
            has_integral (1 / (v - u)) * integral {u..v} (\<lambda>t. f (g t) * vector_derivative g (at t)))
           {0..1}"
    using f uv
    apply (simp add: contour_integrable_on subpath_def has_contour_integral)
    apply (drule integrable_on_subcbox [where a=u and b=v, simplified])
    apply (simp_all add: has_integral_integral)
    apply (drule has_integral_affinity [where m="v-u" and c=u, simplified])
    apply (simp_all add: False image_affinity_atLeastAtMost_div_diff scaleR_conv_of_real)
    apply (simp add: divide_simps False)
    done
  { fix x
    have "x \<in> {0..1} \<Longrightarrow>
           x \<notin> (\<lambda>t. (v-u) *\<^sub>R t + u) -` s \<Longrightarrow>
           vector_derivative (\<lambda>x. g ((v-u) * x + u)) (at x) = (v-u) *\<^sub>R vector_derivative g (at ((v-u) * x + u))"
      apply (rule vector_derivative_at [OF vector_diff_chain_at [simplified o_def]])
      apply (intro derivative_eq_intros | simp)+
      apply (cut_tac s [of "(v - u) * x + u"])
      using uv mult_left_le [of x "v-u"]
      apply (auto simp:  vector_derivative_works)
      done
  } note vd = this
  show ?thesis
    apply (cut_tac has_integral_cmul [OF *, where c = "v-u"])
    using fs assms
    apply (simp add: False subpath_def has_contour_integral)
    apply (rule_tac s = "(\<lambda>t. ((v-u) *\<^sub>R t + u)) -` s" in has_integral_spike_finite)
    apply (auto simp: inj_on_def False finite_vimageI vd scaleR_conv_of_real)
    done
qed

lemma contour_integrable_subpath:
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}"
    shows "f contour_integrable_on (subpath u v g)"
  apply (cases u v rule: linorder_class.le_cases)
   apply (metis contour_integrable_on_def has_contour_integral_subpath [OF assms])
  apply (subst reversepath_subpath [symmetric])
  apply (rule contour_integrable_reversepath)
   using assms apply (blast intro: valid_path_subpath)
  apply (simp add: contour_integrable_on_def)
  using assms apply (blast intro: has_contour_integral_subpath)
  done

lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
  by blast

lemma has_integral_contour_integral_subpath:
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
    shows "(((\<lambda>x. f(g x) * vector_derivative g (at x)))
            has_integral  contour_integral (subpath u v g) f) {u..v}"
  using assms
  apply (auto simp: has_integral_integrable_integral)
  apply (rule integrable_on_subcbox [where a=u and b=v and s = "{0..1}", simplified])
  apply (auto simp: contour_integral_unique [OF has_contour_integral_subpath] contour_integrable_on)
  done

lemma contour_integral_subcontour_integral:
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "u \<le> v"
    shows "contour_integral (subpath u v g) f =
           integral {u..v} (\<lambda>x. f(g x) * vector_derivative g (at x))"
  using assms has_contour_integral_subpath contour_integral_unique by blast

lemma contour_integral_subpath_combine_less:
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
          "u<v" "v<w"
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
           contour_integral (subpath u w g) f"
  using assms apply (auto simp: contour_integral_subcontour_integral)
  apply (rule integral_combine, auto)
  apply (rule integrable_on_subcbox [where a=u and b=w and s = "{0..1}", simplified])
  apply (auto simp: contour_integrable_on)
  done

lemma contour_integral_subpath_combine:
  assumes "f contour_integrable_on g" "valid_path g" "u \<in> {0..1}" "v \<in> {0..1}" "w \<in> {0..1}"
    shows "contour_integral (subpath u v g) f + contour_integral (subpath v w g) f =
           contour_integral (subpath u w g) f"
proof (cases "u\<noteq>v \<and> v\<noteq>w \<and> u\<noteq>w")
  case True
    have *: "subpath v u g = reversepath(subpath u v g) \<and>
             subpath w u g = reversepath(subpath u w g) \<and>
             subpath w v g = reversepath(subpath v w g)"
      by (auto simp: reversepath_subpath)
    have "u < v \<and> v < w \<or>
          u < w \<and> w < v \<or>
          v < u \<and> u < w \<or>
          v < w \<and> w < u \<or>
          w < u \<and> u < v \<or>
          w < v \<and> v < u"
      using True assms by linarith
    with assms show ?thesis
      using contour_integral_subpath_combine_less [of f g u v w]
            contour_integral_subpath_combine_less [of f g u w v]
            contour_integral_subpath_combine_less [of f g v u w]
            contour_integral_subpath_combine_less [of f g v w u]
            contour_integral_subpath_combine_less [of f g w u v]
            contour_integral_subpath_combine_less [of f g w v u]
      apply simp
      apply (elim disjE)
      apply (auto simp: * contour_integral_reversepath contour_integrable_subpath
                   valid_path_reversepath valid_path_subpath algebra_simps)
      done
next
  case False
  then show ?thesis
    apply (auto simp: contour_integral_subpath_refl)
    using assms
    by (metis eq_neg_iff_add_eq_0 contour_integrable_subpath contour_integral_reversepath reversepath_subpath valid_path_subpath)
qed

lemma contour_integral_integral:
  shows "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
  by (simp add: contour_integral_def integral_def has_contour_integral)


subsection\<open>Segments via convex hulls\<close>

lemma segments_subset_convex_hull:
    "closed_segment a b \<subseteq> (convex hull {a,b,c})"
    "closed_segment a c \<subseteq> (convex hull {a,b,c})"
    "closed_segment b c \<subseteq> (convex hull {a,b,c})"
    "closed_segment b a \<subseteq> (convex hull {a,b,c})"
    "closed_segment c a \<subseteq> (convex hull {a,b,c})"
    "closed_segment c b \<subseteq> (convex hull {a,b,c})"
by (auto simp: segment_convex_hull linepath_of_real  elim!: rev_subsetD [OF _ hull_mono])

lemma midpoints_in_convex_hull:
  assumes "x \<in> convex hull s" "y \<in> convex hull s"
    shows "midpoint x y \<in> convex hull s"
proof -
  have "(1 - inverse(2)) *\<^sub>R x + inverse(2) *\<^sub>R y \<in> convex hull s"
    apply (rule convexD_alt)
    using assms
    apply (auto simp: convex_convex_hull)
    done
  then show ?thesis
    by (simp add: midpoint_def algebra_simps)
qed

lemma convex_hull_subset:
    "s \<subseteq> convex hull t \<Longrightarrow> convex hull s \<subseteq> convex hull t"
  by (simp add: convex_convex_hull subset_hull)

lemma not_in_interior_convex_hull_3:
  fixes a :: "complex"
  shows "a \<notin> interior(convex hull {a,b,c})"
        "b \<notin> interior(convex hull {a,b,c})"
        "c \<notin> interior(convex hull {a,b,c})"
  by (auto simp: card_insert_le_m1 not_in_interior_convex_hull)


text\<open>Cauchy's theorem where there's a primitive\<close>

lemma contour_integral_primitive_lemma:
  fixes f :: "complex \<Rightarrow> complex" and g :: "real \<Rightarrow> complex"
  assumes "a \<le> b"
      and "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
      and "g piecewise_differentiable_on {a..b}"  "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
    shows "((\<lambda>x. f'(g x) * vector_derivative g (at x within {a..b}))
             has_integral (f(g b) - f(g a))) {a..b}"
proof -
  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"
    using assms by (auto simp: piecewise_differentiable_on_def)
  have cfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
    apply (rule continuous_on_compose [OF cg, unfolded o_def])
    using assms
    apply (metis complex_differentiable_def complex_differentiable_imp_continuous_at continuous_on_eq_continuous_within continuous_on_subset image_subset_iff)
    done
  { fix x::real
    assume a: "a < x" and b: "x < b" and xk: "x \<notin> k"
    then have "g differentiable at x within {a..b}"
      using k by (simp add: differentiable_at_withinI)
    then have "(g has_vector_derivative vector_derivative g (at x within {a..b})) (at x within {a..b})"
      by (simp add: vector_derivative_works has_field_derivative_def scaleR_conv_of_real)
    then have gdiff: "(g has_derivative (\<lambda>u. u * vector_derivative g (at x within {a..b}))) (at x within {a..b})"
      by (simp add: has_vector_derivative_def scaleR_conv_of_real)
    have "(f has_field_derivative (f' (g x))) (at (g x) within g ` {a..b})"
      using assms by (metis a atLeastAtMost_iff b DERIV_subset image_subset_iff less_eq_real_def)
    then have fdiff: "(f has_derivative op * (f' (g x))) (at (g x) within g ` {a..b})"
      by (simp add: has_field_derivative_def)
    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})"
      using diff_chain_within [OF gdiff fdiff]
      by (simp add: has_vector_derivative_def scaleR_conv_of_real o_def mult_ac)
  } note * = this
  show ?thesis
    apply (rule fundamental_theorem_of_calculus_interior_strong)
    using k assms cfg *
    apply (auto simp: at_within_closed_interval)
    done
qed

lemma contour_integral_primitive:
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
      and "valid_path g" "path_image g \<subseteq> s"
    shows "(f' has_contour_integral (f(pathfinish g) - f(pathstart g))) g"
  using assms
  apply (simp add: valid_path_def path_image_def pathfinish_def pathstart_def has_contour_integral_def)
  apply (auto intro!: piecewise_C1_imp_differentiable contour_integral_primitive_lemma [of 0 1 s])
  done

corollary Cauchy_theorem_primitive:
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s)"
      and "valid_path g"  "path_image g \<subseteq> s" "pathfinish g = pathstart g"
    shows "(f' has_contour_integral 0) g"
  using assms
  by (metis diff_self contour_integral_primitive)


text\<open>Existence of path integral for continuous function\<close>
lemma contour_integrable_continuous_linepath:
  assumes "continuous_on (closed_segment a b) f"
  shows "f contour_integrable_on (linepath a b)"
proof -
  have "continuous_on {0..1} ((\<lambda>x. f x * (b - a)) o linepath a b)"
    apply (rule continuous_on_compose [OF continuous_on_linepath], simp add: linepath_image_01)
    apply (rule continuous_intros | simp add: assms)+
    done
  then show ?thesis
    apply (simp add: contour_integrable_on_def has_contour_integral_def integrable_on_def [symmetric])
    apply (rule integrable_continuous [of 0 "1::real", simplified])
    apply (rule continuous_on_eq [where f = "\<lambda>x. f(linepath a b x)*(b - a)"])
    apply (auto simp: vector_derivative_linepath_within)
    done
qed

lemma has_field_der_id: "((\<lambda>x. x\<^sup>2 / 2) has_field_derivative x) (at x)"
  by (rule has_derivative_imp_has_field_derivative)
     (rule derivative_intros | simp)+

lemma contour_integral_id [simp]: "contour_integral (linepath a b) (\<lambda>y. y) = (b^2 - a^2)/2"
  apply (rule contour_integral_unique)
  using contour_integral_primitive [of UNIV "\<lambda>x. x^2/2" "\<lambda>x. x" "linepath a b"]
  apply (auto simp: field_simps has_field_der_id)
  done

lemma contour_integrable_on_const [iff]: "(\<lambda>x. c) contour_integrable_on (linepath a b)"
  by (simp add: continuous_on_const contour_integrable_continuous_linepath)

lemma contour_integrable_on_id [iff]: "(\<lambda>x. x) contour_integrable_on (linepath a b)"
  by (simp add: continuous_on_id contour_integrable_continuous_linepath)


subsection\<open>Arithmetical combining theorems\<close>

lemma has_contour_integral_neg:
    "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. -(f x)) has_contour_integral (-i)) g"
  by (simp add: has_integral_neg has_contour_integral_def)

lemma has_contour_integral_add:
    "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
     \<Longrightarrow> ((\<lambda>x. f1 x + f2 x) has_contour_integral (i1 + i2)) g"
  by (simp add: has_integral_add has_contour_integral_def algebra_simps)

lemma has_contour_integral_diff:
  "\<lbrakk>(f1 has_contour_integral i1) g; (f2 has_contour_integral i2) g\<rbrakk>
         \<Longrightarrow> ((\<lambda>x. f1 x - f2 x) has_contour_integral (i1 - i2)) g"
  by (simp add: has_integral_sub has_contour_integral_def algebra_simps)

lemma has_contour_integral_lmul:
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. c * (f x)) has_contour_integral (c*i)) g"
apply (simp add: has_contour_integral_def)
apply (drule has_integral_mult_right)
apply (simp add: algebra_simps)
done

lemma has_contour_integral_rmul:
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. (f x) * c) has_contour_integral (i*c)) g"
apply (drule has_contour_integral_lmul)
apply (simp add: mult.commute)
done

lemma has_contour_integral_div:
  "(f has_contour_integral i) g \<Longrightarrow> ((\<lambda>x. f x/c) has_contour_integral (i/c)) g"
  by (simp add: field_class.field_divide_inverse) (metis has_contour_integral_rmul)

lemma has_contour_integral_eq:
    "\<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"
apply (simp add: path_image_def has_contour_integral_def)
by (metis (no_types, lifting) image_eqI has_integral_eq)

lemma has_contour_integral_bound_linepath:
  assumes "(f has_contour_integral i) (linepath a b)"
          "0 \<le> B" "\<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B"
    shows "norm i \<le> B * norm(b - a)"
proof -
  { fix x::real
    assume x: "0 \<le> x" "x \<le> 1"
  have "norm (f (linepath a b x)) *
        norm (vector_derivative (linepath a b) (at x within {0..1})) \<le> B * norm (b - a)"
    by (auto intro: mult_mono simp: assms linepath_in_path of_real_linepath vector_derivative_linepath_within x)
  } note * = this
  have "norm i \<le> (B * norm (b - a)) * content (cbox 0 (1::real))"
    apply (rule has_integral_bound
       [of _ "\<lambda>x. f (linepath a b x) * vector_derivative (linepath a b) (at x within {0..1})"])
    using assms * unfolding has_contour_integral_def
    apply (auto simp: norm_mult)
    done
  then show ?thesis
    by (auto simp: content_real)
qed

(*UNUSED
lemma has_contour_integral_bound_linepath_strong:
  fixes a :: real and f :: "complex \<Rightarrow> real"
  assumes "(f has_contour_integral i) (linepath a b)"
          "finite k"
          "0 \<le> B" "\<And>x::real. x \<in> closed_segment a b - k \<Longrightarrow> norm(f x) \<le> B"
    shows "norm i \<le> B*norm(b - a)"
*)

lemma has_contour_integral_const_linepath: "((\<lambda>x. c) has_contour_integral c*(b - a))(linepath a b)"
  unfolding has_contour_integral_linepath
  by (metis content_real diff_0_right has_integral_const_real lambda_one of_real_1 scaleR_conv_of_real zero_le_one)

lemma has_contour_integral_0: "((\<lambda>x. 0) has_contour_integral 0) g"
  by (simp add: has_contour_integral_def)

lemma has_contour_integral_is_0:
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> (f has_contour_integral 0) g"
  by (rule has_contour_integral_eq [OF has_contour_integral_0]) auto

lemma has_contour_integral_setsum:
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p\<rbrakk>
     \<Longrightarrow> ((\<lambda>x. setsum (\<lambda>a. f a x) s) has_contour_integral setsum i s) p"
  by (induction s rule: finite_induct) (auto simp: has_contour_integral_0 has_contour_integral_add)


subsection \<open>Operations on path integrals\<close>

lemma contour_integral_const_linepath [simp]: "contour_integral (linepath a b) (\<lambda>x. c) = c*(b - a)"
  by (rule contour_integral_unique [OF has_contour_integral_const_linepath])

lemma contour_integral_neg:
    "f contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. -(f x)) = -(contour_integral g f)"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_neg)

lemma contour_integral_add:
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x + f2 x) =
                contour_integral g f1 + contour_integral g f2"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_add)

lemma contour_integral_diff:
    "f1 contour_integrable_on g \<Longrightarrow> f2 contour_integrable_on g \<Longrightarrow> contour_integral g (\<lambda>x. f1 x - f2 x) =
                contour_integral g f1 - contour_integral g f2"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_diff)

lemma contour_integral_lmul:
  shows "f contour_integrable_on g
           \<Longrightarrow> contour_integral g (\<lambda>x. c * f x) = c*contour_integral g f"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_lmul)

lemma contour_integral_rmul:
  shows "f contour_integrable_on g
        \<Longrightarrow> contour_integral g (\<lambda>x. f x * c) = contour_integral g f * c"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_rmul)

lemma contour_integral_div:
  shows "f contour_integrable_on g
        \<Longrightarrow> contour_integral g (\<lambda>x. f x / c) = contour_integral g f / c"
  by (simp add: contour_integral_unique has_contour_integral_integral has_contour_integral_div)

lemma contour_integral_eq:
    "(\<And>x. x \<in> path_image p \<Longrightarrow> f x = g x) \<Longrightarrow> contour_integral p f = contour_integral p g"
  by (simp add: contour_integral_def) (metis has_contour_integral_eq)

lemma contour_integral_eq_0:
    "(\<And>z. z \<in> path_image g \<Longrightarrow> f z = 0) \<Longrightarrow> contour_integral g f = 0"
  by (simp add: has_contour_integral_is_0 contour_integral_unique)

lemma contour_integral_bound_linepath:
  shows
    "\<lbrakk>f contour_integrable_on (linepath a b);
      0 \<le> B; \<And>x. x \<in> closed_segment a b \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
     \<Longrightarrow> norm(contour_integral (linepath a b) f) \<le> B*norm(b - a)"
  apply (rule has_contour_integral_bound_linepath [of f])
  apply (auto simp: has_contour_integral_integral)
  done

lemma contour_integral_0 [simp]: "contour_integral g (\<lambda>x. 0) = 0"
  by (simp add: contour_integral_unique has_contour_integral_0)

lemma contour_integral_setsum:
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
     \<Longrightarrow> contour_integral p (\<lambda>x. setsum (\<lambda>a. f a x) s) = setsum (\<lambda>a. contour_integral p (f a)) s"
  by (auto simp: contour_integral_unique has_contour_integral_setsum has_contour_integral_integral)

lemma contour_integrable_eq:
    "\<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"
  unfolding contour_integrable_on_def
  by (metis has_contour_integral_eq)


subsection \<open>Arithmetic theorems for path integrability\<close>

lemma contour_integrable_neg:
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. -(f x)) contour_integrable_on g"
  using has_contour_integral_neg contour_integrable_on_def by blast

lemma contour_integrable_add:
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x + f2 x) contour_integrable_on g"
  using has_contour_integral_add contour_integrable_on_def
  by fastforce

lemma contour_integrable_diff:
    "\<lbrakk>f1 contour_integrable_on g; f2 contour_integrable_on g\<rbrakk> \<Longrightarrow> (\<lambda>x. f1 x - f2 x) contour_integrable_on g"
  using has_contour_integral_diff contour_integrable_on_def
  by fastforce

lemma contour_integrable_lmul:
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. c * f x) contour_integrable_on g"
  using has_contour_integral_lmul contour_integrable_on_def
  by fastforce

lemma contour_integrable_rmul:
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x * c) contour_integrable_on g"
  using has_contour_integral_rmul contour_integrable_on_def
  by fastforce

lemma contour_integrable_div:
    "f contour_integrable_on g \<Longrightarrow> (\<lambda>x. f x / c) contour_integrable_on g"
  using has_contour_integral_div contour_integrable_on_def
  by fastforce

lemma contour_integrable_setsum:
    "\<lbrakk>finite s; \<And>a. a \<in> s \<Longrightarrow> (f a) contour_integrable_on p\<rbrakk>
     \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) s) contour_integrable_on p"
   unfolding contour_integrable_on_def
   by (metis has_contour_integral_setsum)


subsection\<open>Reversing a path integral\<close>

lemma has_contour_integral_reverse_linepath:
    "(f has_contour_integral i) (linepath a b)
     \<Longrightarrow> (f has_contour_integral (-i)) (linepath b a)"
  using has_contour_integral_reversepath valid_path_linepath by fastforce

lemma contour_integral_reverse_linepath:
    "continuous_on (closed_segment a b) f
     \<Longrightarrow> contour_integral (linepath a b) f = - (contour_integral(linepath b a) f)"
apply (rule contour_integral_unique)
apply (rule has_contour_integral_reverse_linepath)
by (simp add: closed_segment_commute contour_integrable_continuous_linepath has_contour_integral_integral)


(* Splitting a path integral in a flat way.*)

lemma has_contour_integral_split:
  assumes f: "(f has_contour_integral i) (linepath a c)" "(f has_contour_integral j) (linepath c b)"
      and k: "0 \<le> k" "k \<le> 1"
      and c: "c - a = k *\<^sub>R (b - a)"
    shows "(f has_contour_integral (i + j)) (linepath a b)"
proof (cases "k = 0 \<or> k = 1")
  case True
  then show ?thesis
    using assms
    apply auto
    apply (metis add.left_neutral has_contour_integral_trivial has_contour_integral_unique)
    apply (metis add.right_neutral has_contour_integral_trivial has_contour_integral_unique)
    done
next
  case False
  then have k: "0 < k" "k < 1" "complex_of_real k \<noteq> 1"
    using assms apply auto
    using of_real_eq_iff by fastforce
  have c': "c = k *\<^sub>R (b - a) + a"
    by (metis diff_add_cancel c)
  have bc: "(b - c) = (1 - k) *\<^sub>R (b - a)"
    by (simp add: algebra_simps c')
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R c) * (c - a)) has_integral i) {0..1}"
    have **: "\<And>x. ((k - x) / k) *\<^sub>R a + (x / k) *\<^sub>R c = (1 - x) *\<^sub>R a + x *\<^sub>R b"
      using False
      apply (simp add: c' algebra_simps)
      apply (simp add: real_vector.scale_left_distrib [symmetric] divide_simps)
      done
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral i) {0..k}"
      using * k
      apply -
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse k" "0", simplified])
      apply (simp_all add: divide_simps mult.commute [of _ "k"] image_affinity_atLeastAtMost ** c)
      apply (drule Integration.has_integral_cmul [where c = "inverse k"])
      apply (simp add: Integration.has_integral_cmul)
      done
  } note fi = this
  { assume *: "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}"
    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)"
      using k
      apply (simp add: c' field_simps)
      apply (simp add: scaleR_conv_of_real divide_simps)
      apply (simp add: field_simps)
      done
    have "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
      using * k
      apply -
      apply (drule has_integral_affinity [of _ _ 0 "1::real" "inverse(1 - k)" "-(k/(1 - k))", simplified])
      apply (simp_all add: divide_simps mult.commute [of _ "1-k"] image_affinity_atLeastAtMost ** bc)
      apply (drule Integration.has_integral_cmul [where k = "(1 - k) *\<^sub>R j" and c = "inverse (1 - k)"])
      apply (simp add: Integration.has_integral_cmul)
      done
  } note fj = this
  show ?thesis
    using f k
    apply (simp add: has_contour_integral_linepath)
    apply (simp add: linepath_def)
    apply (rule has_integral_combine [OF _ _ fi fj], simp_all)
    done
qed

lemma continuous_on_closed_segment_transform:
  assumes f: "continuous_on (closed_segment a b) f"
      and k: "0 \<le> k" "k \<le> 1"
      and c: "c - a = k *\<^sub>R (b - a)"
    shows "continuous_on (closed_segment a c) f"
proof -
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
    using c by (simp add: algebra_simps)
  show "continuous_on (closed_segment a c) f"
    apply (rule continuous_on_subset [OF f])
    apply (simp add: segment_convex_hull)
    apply (rule convex_hull_subset)
    using assms
    apply (auto simp: hull_inc c' Convex.convexD_alt)
    done
qed

lemma contour_integral_split:
  assumes f: "continuous_on (closed_segment a b) f"
      and k: "0 \<le> k" "k \<le> 1"
      and c: "c - a = k *\<^sub>R (b - a)"
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
proof -
  have c': "c = (1 - k) *\<^sub>R a + k *\<^sub>R b"
    using c by (simp add: algebra_simps)
  have *: "continuous_on (closed_segment a c) f" "continuous_on (closed_segment c b) f"
    apply (rule_tac [!] continuous_on_subset [OF f])
    apply (simp_all add: segment_convex_hull)
    apply (rule_tac [!] convex_hull_subset)
    using assms
    apply (auto simp: hull_inc c' Convex.convexD_alt)
    done
  show ?thesis
    apply (rule contour_integral_unique)
    apply (rule has_contour_integral_split [OF has_contour_integral_integral has_contour_integral_integral k c])
    apply (rule contour_integrable_continuous_linepath *)+
    done
qed

lemma contour_integral_split_linepath:
  assumes f: "continuous_on (closed_segment a b) f"
      and c: "c \<in> closed_segment a b"
    shows "contour_integral(linepath a b) f = contour_integral(linepath a c) f + contour_integral(linepath c b) f"
  using c
  by (auto simp: closed_segment_def algebra_simps intro!: contour_integral_split [OF f])

(* The special case of midpoints used in the main quadrisection.*)

lemma has_contour_integral_midpoint:
  assumes "(f has_contour_integral i) (linepath a (midpoint a b))"
          "(f has_contour_integral j) (linepath (midpoint a b) b)"
    shows "(f has_contour_integral (i + j)) (linepath a b)"
  apply (rule has_contour_integral_split [where c = "midpoint a b" and k = "1/2"])
  using assms
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
  done

lemma contour_integral_midpoint:
   "continuous_on (closed_segment a b) f
    \<Longrightarrow> contour_integral (linepath a b) f =
        contour_integral (linepath a (midpoint a b)) f + contour_integral (linepath (midpoint a b) b) f"
  apply (rule contour_integral_split [where c = "midpoint a b" and k = "1/2"])
  using assms
  apply (auto simp: midpoint_def algebra_simps scaleR_conv_of_real)
  done


text\<open>A couple of special case lemmas that are useful below\<close>

lemma triangle_linear_has_chain_integral:
    "((\<lambda>x. m*x + d) has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
  apply (rule Cauchy_theorem_primitive [of UNIV "\<lambda>x. m/2 * x^2 + d*x"])
  apply (auto intro!: derivative_eq_intros)
  done

lemma has_chain_integral_chain_integral3:
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d)
      \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c d) f = i"
  apply (subst contour_integral_unique [symmetric], assumption)
  apply (drule has_contour_integral_integrable)
  apply (simp add: valid_path_join)
  done

lemma has_chain_integral_chain_integral4:
     "(f has_contour_integral i) (linepath a b +++ linepath b c +++ linepath c d +++ linepath d e)
      \<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"
  apply (subst contour_integral_unique [symmetric], assumption)
  apply (drule has_contour_integral_integrable)
  apply (simp add: valid_path_join)
  done

subsection\<open>Reversing the order in a double path integral\<close>

text\<open>The condition is stronger than needed but it's often true in typical situations\<close>

lemma fst_im_cbox [simp]: "cbox c d \<noteq> {} \<Longrightarrow> (fst ` cbox (a,c) (b,d)) = cbox a b"
  by (auto simp: cbox_Pair_eq)

lemma snd_im_cbox [simp]: "cbox a b \<noteq> {} \<Longrightarrow> (snd ` cbox (a,c) (b,d)) = cbox c d"
  by (auto simp: cbox_Pair_eq)

lemma contour_integral_swap:
  assumes fcon:  "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1,y2). f y1 y2)"
      and vp:    "valid_path g" "valid_path h"
      and gvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"
      and hvcon: "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))"
  shows "contour_integral g (\<lambda>w. contour_integral h (f w)) =
         contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
proof -
  have gcon: "continuous_on {0..1} g" and hcon: "continuous_on {0..1} h"
    using assms by (auto simp: valid_path_def piecewise_C1_differentiable_on_def)
  have fgh1: "\<And>x. (\<lambda>t. f (g x) (h t)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g x, h t))"
    by (rule ext) simp
  have fgh2: "\<And>x. (\<lambda>t. f (g t) (h x)) = (\<lambda>(y1,y2). f y1 y2) o (\<lambda>t. (g t, h x))"
    by (rule ext) simp
  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)"
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
  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)"
    by (rule continuous_on_subset [OF fcon]) (auto simp: path_image_def)
  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}"
    apply (rule integrable_continuous_real)
    apply (rule continuous_on_mult [OF _ gvcon])
    apply (subst fgh2)
    apply (rule fcon_im2 gcon continuous_intros | simp)+
    done
  have "(\<lambda>z. vector_derivative g (at (fst z))) = (\<lambda>x. vector_derivative g (at x)) o fst"
    by auto
  then have gvcon': "continuous_on (cbox (0, 0) (1, 1::real)) (\<lambda>x. vector_derivative g (at (fst x)))"
    apply (rule ssubst)
    apply (rule continuous_intros | simp add: gvcon)+
    done
  have "(\<lambda>z. vector_derivative h (at (snd z))) = (\<lambda>x. vector_derivative h (at x)) o snd"
    by auto
  then have hvcon': "continuous_on (cbox (0, 0) (1::real, 1)) (\<lambda>x. vector_derivative h (at (snd x)))"
    apply (rule ssubst)
    apply (rule continuous_intros | simp add: hvcon)+
    done
  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))"
    by auto
  then have fgh: "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
    apply (rule ssubst)
    apply (rule gcon hcon continuous_intros | simp)+
    apply (auto simp: path_image_def intro: continuous_on_subset [OF fcon])
    done
  have "integral {0..1} (\<lambda>x. contour_integral h (f (g x)) * vector_derivative g (at x)) =
        integral {0..1} (\<lambda>x. contour_integral h (\<lambda>y. f (g x) y * vector_derivative g (at x)))"
    apply (rule integral_cong [OF contour_integral_rmul [symmetric]])
    apply (clarsimp simp: contour_integrable_on)
    apply (rule integrable_continuous_real)
    apply (rule continuous_on_mult [OF _ hvcon])
    apply (subst fgh1)
    apply (rule fcon_im1 hcon continuous_intros | simp)+
    done
  also have "... = integral {0..1}
                     (\<lambda>y. contour_integral g (\<lambda>x. f x (h y) * vector_derivative h (at y)))"
    apply (simp add: contour_integral_integral)
    apply (subst integral_swap_continuous [where 'a = real and 'b = real, of 0 0 1 1, simplified])
    apply (rule fgh gvcon' hvcon' continuous_intros | simp add: split_def)+
    apply (simp add: algebra_simps)
    done
  also have "... = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
    apply (simp add: contour_integral_integral)
    apply (rule integral_cong)
    apply (subst integral_mult_left [symmetric])
    apply (blast intro: vdg)
    apply (simp add: algebra_simps)
    done
  finally show ?thesis
    by (simp add: contour_integral_integral)
qed


subsection\<open>The key quadrisection step\<close>

lemma norm_sum_half:
  assumes "norm(a + b) >= e"
    shows "norm a >= e/2 \<or> norm b >= e/2"
proof -
  have "e \<le> norm (- a - b)"
    by (simp add: add.commute assms norm_minus_commute)
  thus ?thesis
    using norm_triangle_ineq4 order_trans by fastforce
qed

lemma norm_sum_lemma:
  assumes "e \<le> norm (a + b + c + d)"
    shows "e / 4 \<le> norm a \<or> e / 4 \<le> norm b \<or> e / 4 \<le> norm c \<or> e / 4 \<le> norm d"
proof -
  have "e \<le> norm ((a + b) + (c + d))" using assms
    by (simp add: algebra_simps)
  then show ?thesis
    by (auto dest!: norm_sum_half)
qed

lemma Cauchy_theorem_quadrisection:
  assumes f: "continuous_on (convex hull {a,b,c}) f"
      and dist: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
      and e: "e * K^2 \<le>
              norm (contour_integral(linepath a b) f + contour_integral(linepath b c) f + contour_integral(linepath c a) f)"
  shows "\<exists>a' b' c'.
           a' \<in> convex hull {a,b,c} \<and> b' \<in> convex hull {a,b,c} \<and> c' \<in> convex hull {a,b,c} \<and>
           dist a' b' \<le> K/2  \<and>  dist b' c' \<le> K/2  \<and>  dist c' a' \<le> K/2  \<and>
           e * (K/2)^2 \<le> norm(contour_integral(linepath a' b') f + contour_integral(linepath b' c') f + contour_integral(linepath c' a') f)"
proof -
  note divide_le_eq_numeral1 [simp del]
  def a' \<equiv> "midpoint b c"
  def b' \<equiv> "midpoint c a"
  def c' \<equiv> "midpoint a b"
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
    using f continuous_on_subset segments_subset_convex_hull by metis+
  have fcont': "continuous_on (closed_segment c' b') f"
               "continuous_on (closed_segment a' c') f"
               "continuous_on (closed_segment b' a') f"
    unfolding a'_def b'_def c'_def
    apply (rule continuous_on_subset [OF f],
           metis midpoints_in_convex_hull convex_hull_subset hull_subset insert_subset segment_convex_hull)+
    done
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
  have *: "?pathint a b + ?pathint b c + ?pathint c a =
          (?pathint a c' + ?pathint c' b' + ?pathint b' a) +
          (?pathint a' c' + ?pathint c' b + ?pathint b a') +
          (?pathint a' c + ?pathint c b' + ?pathint b' a') +
          (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
    apply (simp add: fcont' contour_integral_reverse_linepath)
    apply (simp add: a'_def b'_def c'_def contour_integral_midpoint fabc)
    done
  have [simp]: "\<And>x y. cmod (x * 2 - y * 2) = cmod (x - y) * 2"
    by (metis left_diff_distrib mult.commute norm_mult_numeral1)
  have [simp]: "\<And>x y. cmod (x - y) = cmod (y - x)"
    by (simp add: norm_minus_commute)
  consider "e * K\<^sup>2 / 4 \<le> cmod (?pathint a c' + ?pathint c' b' + ?pathint b' a)" |
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c' + ?pathint c' b + ?pathint b a')" |
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' c + ?pathint c b' + ?pathint b' a')" |
           "e * K\<^sup>2 / 4 \<le> cmod (?pathint a' b' + ?pathint b' c' + ?pathint c' a')"
    using assms
    apply (simp only: *)
    apply (blast intro: that dest!: norm_sum_lemma)
    done
  then show ?thesis
  proof cases
    case 1 then show ?thesis
      apply (rule_tac x=a in exI)
      apply (rule exI [where x=c'])
      apply (rule exI [where x=b'])
      using assms
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
      done
  next
    case 2 then show ?thesis
      apply (rule_tac x=a' in exI)
      apply (rule exI [where x=c'])
      apply (rule exI [where x=b])
      using assms
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
      done
  next
    case 3 then show ?thesis
      apply (rule_tac x=a' in exI)
      apply (rule exI [where x=c])
      apply (rule exI [where x=b'])
      using assms
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
      done
  next
    case 4 then show ?thesis
      apply (rule_tac x=a' in exI)
      apply (rule exI [where x=b'])
      apply (rule exI [where x=c'])
      using assms
      apply (auto simp: a'_def c'_def b'_def midpoints_in_convex_hull hull_subset [THEN subsetD])
      apply (auto simp: midpoint_def dist_norm scaleR_conv_of_real divide_simps)
      done
  qed
qed

subsection\<open>Cauchy's theorem for triangles\<close>

lemma triangle_points_closer:
  fixes a::complex
  shows "\<lbrakk>x \<in> convex hull {a,b,c};  y \<in> convex hull {a,b,c}\<rbrakk>
         \<Longrightarrow> norm(x - y) \<le> norm(a - b) \<or>
             norm(x - y) \<le> norm(b - c) \<or>
             norm(x - y) \<le> norm(c - a)"
  using simplex_extremal_le [of "{a,b,c}"]
  by (auto simp: norm_minus_commute)

lemma holomorphic_point_small_triangle:
  assumes x: "x \<in> s"
      and f: "continuous_on s f"
      and cd: "f complex_differentiable (at x within s)"
      and e: "0 < e"
    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>
              x \<in> convex hull {a,b,c} \<and> convex hull {a,b,c} \<subseteq> s
              \<longrightarrow> norm(contour_integral(linepath a b) f + contour_integral(linepath b c) f +
                       contour_integral(linepath c a) f)
                  \<le> e*(dist a b + dist b c + dist c a)^2"
           (is "\<exists>k>0. \<forall>a b c. _ \<longrightarrow> ?normle a b c")
proof -
  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>
                     \<Longrightarrow> a \<le> e*(x + y + z)^2"
    by (simp add: algebra_simps power2_eq_square)
  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"
             for x::real and a b c
    by linarith
  have fabc: "f contour_integrable_on linepath a b" "f contour_integrable_on linepath b c" "f contour_integrable_on linepath c a"
              if "convex hull {a, b, c} \<subseteq> s" for a b c
    using segments_subset_convex_hull that
    by (metis continuous_on_subset f contour_integrable_continuous_linepath)+
  note path_bound = has_contour_integral_bound_linepath [simplified norm_minus_commute, OF has_contour_integral_integral]
  { fix f' a b c d
    assume d: "0 < d"
       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)"
       and le: "cmod (a - b) \<le> d" "cmod (b - c) \<le> d" "cmod (c - a) \<le> d"
       and xc: "x \<in> convex hull {a, b, c}"
       and s: "convex hull {a, b, c} \<subseteq> s"
    have pa: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f =
              contour_integral (linepath a b) (\<lambda>y. f y - f x - f'*(y - x)) +
              contour_integral (linepath b c) (\<lambda>y. f y - f x - f'*(y - x)) +
              contour_integral (linepath c a) (\<lambda>y. f y - f x - f'*(y - x))"
      apply (simp add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc [OF s])
      apply (simp add: field_simps)
      done
    { fix y
      assume yc: "y \<in> convex hull {a,b,c}"
      have "cmod (f y - f x - f' * (y - x)) \<le> e*norm(y - x)"
        apply (rule f')
        apply (metis triangle_points_closer [OF xc yc] le norm_minus_commute order_trans)
        using s yc by blast
      also have "... \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))"
        by (simp add: yc e xc disj_le [OF triangle_points_closer])
      finally have "cmod (f y - f x - f' * (y - x)) \<le> e * (cmod (a - b) + cmod (b - c) + cmod (c - a))" .
    } note cm_le = this
    have "?normle a b c"
      apply (simp add: dist_norm pa)
      apply (rule le_of_3)
      using f' xc s e
      apply simp_all
      apply (intro norm_triangle_le add_mono path_bound)
      apply (simp_all add: contour_integral_diff contour_integral_lmul contour_integrable_lmul contour_integrable_diff fabc)
      apply (blast intro: cm_le elim: dest: segments_subset_convex_hull [THEN subsetD])+
      done
  } note * = this
  show ?thesis
    using cd e
    apply (simp add: complex_differentiable_def has_field_derivative_def has_derivative_within_alt approachable_lt_le2 Ball_def)
    apply (clarify dest!: spec mp)
    using *
    apply (simp add: dist_norm, blast)
    done
qed


(* Hence the most basic theorem for a triangle.*)
locale Chain =
  fixes x0 At Follows
  assumes At0: "At x0 0"
      and AtSuc: "\<And>x n. At x n \<Longrightarrow> \<exists>x'. At x' (Suc n) \<and> Follows x' x"
begin
  primrec f where
    "f 0 = x0"
  | "f (Suc n) = (SOME x. At x (Suc n) \<and> Follows x (f n))"

  lemma At: "At (f n) n"
  proof (induct n)
    case 0 show ?case
      by (simp add: At0)
  next
    case (Suc n) show ?case
      by (metis (no_types, lifting) AtSuc [OF Suc] f.simps(2) someI_ex)
  qed

  lemma Follows: "Follows (f(Suc n)) (f n)"
    by (metis (no_types, lifting) AtSuc [OF At [of n]] f.simps(2) someI_ex)

  declare f.simps(2) [simp del]
end

lemma Chain3:
  assumes At0: "At x0 y0 z0 0"
      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"
  obtains f g h where
    "f 0 = x0" "g 0 = y0" "h 0 = z0"
                      "\<And>n. At (f n) (g n) (h n) n"
                       "\<And>n. Follows (f(Suc n)) (g(Suc n)) (h(Suc n)) (f n) (g n) (h n)"
proof -
  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"
    apply unfold_locales
    using At0 AtSuc by auto
  show ?thesis
  apply (rule that [of "\<lambda>n. fst (three.f n)"  "\<lambda>n. fst (snd (three.f n))" "\<lambda>n. snd (snd (three.f n))"])
  apply simp_all
  using three.At three.Follows
  apply (simp_all add: split_beta')
  done
qed

lemma Cauchy_theorem_triangle:
  assumes "f holomorphic_on (convex hull {a,b,c})"
    shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
proof -
  have contf: "continuous_on (convex hull {a,b,c}) f"
    by (metis assms holomorphic_on_imp_continuous_on)
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
  { fix y::complex
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
       and ynz: "y \<noteq> 0"
    def K \<equiv> "1 + max (dist a b) (max (dist b c) (dist c a))"
    def e \<equiv> "norm y / K^2"
    have K1: "K \<ge> 1"  by (simp add: K_def max.coboundedI1)
    then have K: "K > 0" by linarith
    have [iff]: "dist a b \<le> K" "dist b c \<le> K" "dist c a \<le> K"
      by (simp_all add: K_def)
    have e: "e > 0"
      unfolding e_def using ynz K1 by simp
    def At \<equiv> "\<lambda>x y z n. convex hull {x,y,z} \<subseteq> convex hull {a,b,c} \<and>
                         dist x y \<le> K/2^n \<and> dist y z \<le> K/2^n \<and> dist z x \<le> K/2^n \<and>
                         norm(?pathint x y + ?pathint y z + ?pathint z x) \<ge> e*(K/2^n)^2"
    have At0: "At a b c 0"
      using fy
      by (simp add: At_def e_def has_chain_integral_chain_integral3)
    { fix x y z n
      assume At: "At x y z n"
      then have contf': "continuous_on (convex hull {x,y,z}) f"
        using contf At_def continuous_on_subset by blast
      have "\<exists>x' y' z'. At x' y' z' (Suc n) \<and> convex hull {x',y',z'} \<subseteq> convex hull {x,y,z}"
        using At
        apply (simp add: At_def)
        using  Cauchy_theorem_quadrisection [OF contf', of "K/2^n" e]
        apply clarsimp
        apply (rule_tac x="a'" in exI)
        apply (rule_tac x="b'" in exI)
        apply (rule_tac x="c'" in exI)
        apply (simp add: algebra_simps)
        apply (meson convex_hull_subset empty_subsetI insert_subset subsetCE)
        done
    } note AtSuc = this
    obtain fa fb fc
      where f0 [simp]: "fa 0 = a" "fb 0 = b" "fc 0 = c"
        and cosb: "\<And>n. convex hull {fa n, fb n, fc n} \<subseteq> convex hull {a,b,c}"
        and dist: "\<And>n. dist (fa n) (fb n) \<le> K/2^n"
                  "\<And>n. dist (fb n) (fc n) \<le> K/2^n"
                  "\<And>n. dist (fc n) (fa n) \<le> K/2^n"
        and no: "\<And>n. norm(?pathint (fa n) (fb n) +
                           ?pathint (fb n) (fc n) +
                           ?pathint (fc n) (fa n)) \<ge> e * (K/2^n)^2"
        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}"
      apply (rule Chain3 [of At, OF At0 AtSuc])
      apply (auto simp: At_def)
      done
    have "\<exists>x. \<forall>n. x \<in> convex hull {fa n, fb n, fc n}"
      apply (rule bounded_closed_nest)
      apply (simp_all add: compact_imp_closed finite_imp_compact_convex_hull finite_imp_bounded_convex_hull)
      apply (rule allI)
      apply (rule transitive_stepwise_le)
      apply (auto simp: conv_le)
      done
    then obtain x where x: "\<And>n. x \<in> convex hull {fa n, fb n, fc n}" by auto
    then have xin: "x \<in> convex hull {a,b,c}"
      using assms f0 by blast
    then have fx: "f complex_differentiable at x within (convex hull {a,b,c})"
      using assms holomorphic_on_def by blast
    { fix k n
      assume k: "0 < k"
         and le:
            "\<And>x' y' z'.
               \<lbrakk>dist x' y' \<le> k; dist y' z' \<le> k; dist z' x' \<le> k;
                x \<in> convex hull {x',y',z'};
                convex hull {x',y',z'} \<subseteq> convex hull {a,b,c}\<rbrakk>
               \<Longrightarrow>
               cmod (?pathint x' y' + ?pathint y' z' + ?pathint z' x') * 10
                     \<le> e * (dist x' y' + dist y' z' + dist z' x')\<^sup>2"
         and Kk: "K / k < 2 ^ n"
      have "K / 2 ^ n < k" using Kk k
        by (auto simp: field_simps)
      then have DD: "dist (fa n) (fb n) \<le> k" "dist (fb n) (fc n) \<le> k" "dist (fc n) (fa n) \<le> k"
        using dist [of n]  k
        by linarith+
      have dle: "(dist (fa n) (fb n) + dist (fb n) (fc n) + dist (fc n) (fa n))\<^sup>2
               \<le> (3 * K / 2 ^ n)\<^sup>2"
        using dist [of n] e K
        by (simp add: abs_le_square_iff [symmetric])
      have less10: "\<And>x y::real. 0 < x \<Longrightarrow> y \<le> 9*x \<Longrightarrow> y < x*10"
        by linarith
      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"
        using ynz dle e mult_le_cancel_left_pos by blast
      also have "... <
          cmod (?pathint (fa n) (fb n) + ?pathint (fb n) (fc n) + ?pathint (fc n) (fa n)) * 10"
        using no [of n] e K
        apply (simp add: e_def field_simps)
        apply (simp only: zero_less_norm_iff [symmetric])
        done
      finally have False
        using le [OF DD x cosb] by auto
    } then
    have ?thesis
      using holomorphic_point_small_triangle [OF xin contf fx, of "e/10"] e
      apply clarsimp
      apply (rule_tac x1="K/k" in exE [OF real_arch_pow2], blast)
      done
  }
  moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
    by simp (meson contf continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(1)
                   segments_subset_convex_hull(3) segments_subset_convex_hull(5))
  ultimately show ?thesis
    using has_contour_integral_integral by fastforce
qed


subsection\<open>Version needing function holomorphic in interior only\<close>

lemma Cauchy_theorem_flat_lemma:
  assumes f: "continuous_on (convex hull {a,b,c}) f"
      and c: "c - a = k *\<^sub>R (b - a)"
      and k: "0 \<le> k"
    shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f +
          contour_integral (linepath c a) f = 0"
proof -
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
    using f continuous_on_subset segments_subset_convex_hull by metis+
  show ?thesis
  proof (cases "k \<le> 1")
    case True show ?thesis
      by (simp add: contour_integral_split [OF fabc(1) k True c] contour_integral_reverse_linepath fabc)
  next
    case False then show ?thesis
      using fabc c
      apply (subst contour_integral_split [of a c f "1/k" b, symmetric])
      apply (metis closed_segment_commute fabc(3))
      apply (auto simp: k contour_integral_reverse_linepath)
      done
  qed
qed

lemma Cauchy_theorem_flat:
  assumes f: "continuous_on (convex hull {a,b,c}) f"
      and c: "c - a = k *\<^sub>R (b - a)"
    shows "contour_integral (linepath a b) f +
           contour_integral (linepath b c) f +
           contour_integral (linepath c a) f = 0"
proof (cases "0 \<le> k")
  case True with assms show ?thesis
    by (blast intro: Cauchy_theorem_flat_lemma)
next
  case False
  have "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
    using f continuous_on_subset segments_subset_convex_hull by metis+
  moreover have "contour_integral (linepath b a) f + contour_integral (linepath a c) f +
        contour_integral (linepath c b) f = 0"
    apply (rule Cauchy_theorem_flat_lemma [of b a c f "1-k"])
    using False c
    apply (auto simp: f insert_commute scaleR_conv_of_real algebra_simps)
    done
  ultimately show ?thesis
    apply (auto simp: contour_integral_reverse_linepath)
    using add_eq_0_iff by force
qed


lemma Cauchy_theorem_triangle_interior:
  assumes contf: "continuous_on (convex hull {a,b,c}) f"
      and holf:  "f holomorphic_on interior (convex hull {a,b,c})"
     shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
proof -
  have fabc: "continuous_on (closed_segment a b) f" "continuous_on (closed_segment b c) f" "continuous_on (closed_segment c a) f"
    using contf continuous_on_subset segments_subset_convex_hull by metis+
  have "bounded (f ` (convex hull {a,b,c}))"
    by (simp add: compact_continuous_image compact_convex_hull compact_imp_bounded contf)
  then obtain B where "0 < B" and Bnf: "\<And>x. x \<in> convex hull {a,b,c} \<Longrightarrow> norm (f x) \<le> B"
     by (auto simp: dest!: bounded_pos [THEN iffD1])
  have "bounded (convex hull {a,b,c})"
    by (simp add: bounded_convex_hull)
  then obtain C where C: "0 < C" and Cno: "\<And>y. y \<in> convex hull {a,b,c} \<Longrightarrow> norm y < C"
    using bounded_pos_less by blast
  then have diff_2C: "norm(x - y) \<le> 2*C"
           if x: "x \<in> convex hull {a, b, c}" and y: "y \<in> convex hull {a, b, c}" for x y
  proof -
    have "cmod x \<le> C"
      using x by (meson Cno not_le not_less_iff_gr_or_eq)
    hence "cmod (x - y) \<le> C + C"
      using y by (meson Cno add_mono_thms_linordered_field(4) less_eq_real_def norm_triangle_ineq4 order_trans)
    thus "cmod (x - y) \<le> 2 * C"
      by (metis mult_2)
  qed
  have contf': "continuous_on (convex hull {b,a,c}) f"
    using contf by (simp add: insert_commute)
  { fix y::complex
    assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
       and ynz: "y \<noteq> 0"
    have pi_eq_y: "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = y"
      by (rule has_chain_integral_chain_integral3 [OF fy])
    have ?thesis
    proof (cases "c=a \<or> a=b \<or> b=c")
      case True then show ?thesis
        using Cauchy_theorem_flat [OF contf, of 0]
        using has_chain_integral_chain_integral3 [OF fy] ynz
        by (force simp: fabc contour_integral_reverse_linepath)
    next
      case False
      then have car3: "card {a, b, c} = Suc (DIM(complex))"
        by auto
      { assume "interior(convex hull {a,b,c}) = {}"
        then have "collinear{a,b,c}"
          using interior_convex_hull_eq_empty [OF car3]
          by (simp add: collinear_3_eq_affine_dependent)
        then have "False"
          using False
          apply (clarsimp simp add: collinear_3 collinear_lemma)
          apply (drule Cauchy_theorem_flat [OF contf'])
          using pi_eq_y ynz
          apply (simp add: fabc add_eq_0_iff contour_integral_reverse_linepath)
          done
      }
      then obtain d where d: "d \<in> interior (convex hull {a, b, c})"
        by blast
      { fix d1
        assume d1_pos: "0 < d1"
           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>
                           \<Longrightarrow> cmod (f x' - f x) < cmod y / (24 * C)"
        def e      \<equiv> "min 1 (min (d1/(4*C)) ((norm y / 24 / C) / B))"
        def shrink \<equiv> "\<lambda>x. x - e *\<^sub>R (x - d)"
        let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
        have e: "0 < e" "e \<le> 1" "e \<le> d1 / (4 * C)" "e \<le> cmod y / 24 / C / B"
          using d1_pos \<open>C>0\<close> \<open>B>0\<close> ynz by (simp_all add: e_def)
        then have eCB: "24 * e * C * B \<le> cmod y"
          using \<open>C>0\<close> \<open>B>0\<close>  by (simp add: field_simps)
        have e_le_d1: "e * (4 * C) \<le> d1"
          using e \<open>C>0\<close> by (simp add: field_simps)
        have "shrink a \<in> interior(convex hull {a,b,c})"
             "shrink b \<in> interior(convex hull {a,b,c})"
             "shrink c \<in> interior(convex hull {a,b,c})"
          using d e by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
        then have fhp0: "(f has_contour_integral 0)
                (linepath (shrink a) (shrink b) +++ linepath (shrink b) (shrink c) +++ linepath (shrink c) (shrink a))"
          by (simp add: Cauchy_theorem_triangle holomorphic_on_subset [OF holf] hull_minimal convex_interior)
        then have f_0_shrink: "?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a) = 0"
          by (simp add: has_chain_integral_chain_integral3)
        have fpi_abc: "f contour_integrable_on linepath (shrink a) (shrink b)"
                      "f contour_integrable_on linepath (shrink b) (shrink c)"
                      "f contour_integrable_on linepath (shrink c) (shrink a)"
          using fhp0  by (auto simp: valid_path_join dest: has_contour_integral_integrable)
        have cmod_shr: "\<And>x y. cmod (shrink y - shrink x - (y - x)) = e * cmod (x - y)"
          using e by (simp add: shrink_def real_vector.scale_right_diff_distrib [symmetric])
        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)"
          by (simp add: algebra_simps)
        have "cmod y / (24 * C) \<le> cmod y / cmod (b - a) / 12"
          using False \<open>C>0\<close> diff_2C [of b a] ynz
          by (auto simp: divide_simps hull_inc)
        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
          apply (cases "x=0", simp add: \<open>0<C\<close>)
          using Cno [of u] mult_left_le_one_le [of "cmod u" x] le_less_trans norm_ge_zero by blast
        { fix u v
          assume uv: "u \<in> convex hull {a, b, c}" "v \<in> convex hull {a, b, c}" "u\<noteq>v"
             and fpi_uv: "f contour_integrable_on linepath (shrink u) (shrink v)"
          have shr_uv: "shrink u \<in> interior(convex hull {a,b,c})"
                       "shrink v \<in> interior(convex hull {a,b,c})"
            using d e uv
            by (auto simp: hull_inc mem_interior_convex_shrink shrink_def)
          have cmod_fuv: "\<And>x. 0\<le>x \<Longrightarrow> x\<le>1 \<Longrightarrow> cmod (f (linepath (shrink u) (shrink v) x)) \<le> B"
            using shr_uv by (blast intro: Bnf linepath_in_convex_hull interior_subset [THEN subsetD])
          have By_uv: "B * (12 * (e * cmod (u - v))) \<le> cmod y"
            apply (rule order_trans [OF _ eCB])
            using e \<open>B>0\<close> diff_2C [of u v] uv
            by (auto simp: field_simps)
          { fix x::real   assume x: "0\<le>x" "x\<le>1"
            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)"
              apply (rule add_strict_mono; rule norm_triangle_half_l [of _ 0])
              using uv x d interior_subset
              apply (auto simp: hull_inc intro!: less_C)
              done
            have ll: "linepath (shrink u) (shrink v) x - linepath u v x = -e * ((1 - x) *\<^sub>R (u - d) + x *\<^sub>R (v - d))"
              by (simp add: linepath_def shrink_def algebra_simps scaleR_conv_of_real)
            have cmod_less_dt: "cmod (linepath (shrink u) (shrink v) x - linepath u v x) < d1"
              using \<open>e>0\<close>
              apply (simp add: ll norm_mult scaleR_diff_right)
              apply (rule less_le_trans [OF _ e_le_d1])
              using cmod_less_4C
              apply (force intro: norm_triangle_lt)
              done
            have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) < cmod y / (24 * C)"
              using x uv shr_uv cmod_less_dt
              by (auto simp: hull_inc intro: d1 interior_subset [THEN subsetD] linepath_in_convex_hull)
            also have "... \<le> cmod y / cmod (v - u) / 12"
              using False uv \<open>C>0\<close> diff_2C [of v u] ynz
              by (auto simp: divide_simps hull_inc)
            finally have "cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x)) \<le> cmod y / cmod (v - u) / 12"
              by simp
            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"
              using uv False by (auto simp: field_simps)
            have "cmod (f (linepath (shrink u) (shrink v) x)) * cmod (shrink v - shrink u - (v - u)) +
                  cmod (v - u) * cmod (f (linepath (shrink u) (shrink v) x) - f (linepath u v x))
                  \<le> cmod y / 6"
              apply (rule order_trans [of _ "B*((norm y / 24 / C / B)*2*C) + (2*C)*(norm y /24 / C)"])
              apply (rule add_mono [OF mult_mono])
              using By_uv e \<open>0 < B\<close> \<open>0 < C\<close> x ynz
              apply (simp_all add: cmod_fuv cmod_shr cmod_12_le hull_inc)
              apply (simp add: field_simps)
              done
          } note cmod_diff_le = this
          have f_uv: "continuous_on (closed_segment u v) f"
            by (blast intro: uv continuous_on_subset [OF contf closed_segment_subset_convex_hull])
          have **: "\<And>f' x' f x::complex. f'*x' - f*x = f'*(x' - x) + x*(f' - f)"
            by (simp add: algebra_simps)
          have "norm (?pathint (shrink u) (shrink v) - ?pathint u v) \<le> norm y / 6"
            apply (rule order_trans)
            apply (rule has_integral_bound
                    [of "B*(norm y /24/C/B)*2*C + (2*C)*(norm y/24/C)"
                        "\<lambda>x. f(linepath (shrink u) (shrink v) x) * (shrink v - shrink u) - f(linepath u v x)*(v - u)"
                        _ 0 1 ])
            using ynz \<open>0 < B\<close> \<open>0 < C\<close>
            apply (simp_all del: le_divide_eq_numeral1)
            apply (simp add: has_integral_sub has_contour_integral_linepath [symmetric] has_contour_integral_integral
                             fpi_uv f_uv contour_integrable_continuous_linepath, clarify)
            apply (simp only: **)
            apply (simp add: norm_triangle_le norm_mult cmod_diff_le del: le_divide_eq_numeral1)
            done
          } note * = this
          have "norm (?pathint (shrink a) (shrink b) - ?pathint a b) \<le> norm y / 6"
            using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
          moreover
          have "norm (?pathint (shrink b) (shrink c) - ?pathint b c) \<le> norm y / 6"
            using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
          moreover
          have "norm (?pathint (shrink c) (shrink a) - ?pathint c a) \<le> norm y / 6"
            using False fpi_abc by (rule_tac *) (auto simp: hull_inc)
          ultimately
          have "norm((?pathint (shrink a) (shrink b) - ?pathint a b) +
                     (?pathint (shrink b) (shrink c) - ?pathint b c) + (?pathint (shrink c) (shrink a) - ?pathint c a))
                \<le> norm y / 6 + norm y / 6 + norm y / 6"
            by (metis norm_triangle_le add_mono)
          also have "... = norm y / 2"
            by simp
          finally have "norm((?pathint (shrink a) (shrink b) + ?pathint (shrink b) (shrink c) + ?pathint (shrink c) (shrink a)) -
                          (?pathint a b + ?pathint b c + ?pathint c a))
                \<le> norm y / 2"
            by (simp add: algebra_simps)
          then
          have "norm(?pathint a b + ?pathint b c + ?pathint c a) \<le> norm y / 2"
            by (simp add: f_0_shrink) (metis (mono_tags) add.commute minus_add_distrib norm_minus_cancel uminus_add_conv_diff)
          then have "False"
            using pi_eq_y ynz by auto
        }
        moreover have "uniformly_continuous_on (convex hull {a,b,c}) f"
          by (simp add: contf compact_convex_hull compact_uniformly_continuous)
        ultimately have "False"
          unfolding uniformly_continuous_on_def
          by (force simp: ynz \<open>0 < C\<close> dist_norm)
        then show ?thesis ..
      qed
  }
  moreover have "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
    using fabc contour_integrable_continuous_linepath by auto
  ultimately show ?thesis
    using has_contour_integral_integral by fastforce
qed


subsection\<open>Version allowing finite number of exceptional points\<close>

lemma Cauchy_theorem_triangle_cofinite:
  assumes "continuous_on (convex hull {a,b,c}) f"
      and "finite s"
      and "(\<And>x. x \<in> interior(convex hull {a,b,c}) - s \<Longrightarrow> f complex_differentiable (at x))"
     shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
using assms
proof (induction "card s" arbitrary: a b c s rule: less_induct)
  case (less s a b c)
  show ?case
  proof (cases "s={}")
    case True with less show ?thesis
      by (fastforce simp: holomorphic_on_def complex_differentiable_at_within
                    Cauchy_theorem_triangle_interior)
  next
    case False
    then obtain d s' where d: "s = insert d s'" "d \<notin> s'"
      by (meson Set.set_insert all_not_in_conv)
    then show ?thesis
    proof (cases "d \<in> convex hull {a,b,c}")
      case False
      show "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
        apply (rule less.hyps [of "s'"])
        using False d \<open>finite s\<close> interior_subset
        apply (auto intro!: less.prems)
        done
    next
      case True
      have *: "convex hull {a, b, d} \<subseteq> convex hull {a, b, c}"
        by (meson True hull_subset insert_subset convex_hull_subset)
      have abd: "(f has_contour_integral 0) (linepath a b +++ linepath b d +++ linepath d a)"
        apply (rule less.hyps [of "s'"])
        using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
        apply (auto intro!: less.prems continuous_on_subset [OF  _ *])
        apply (metis * insert_absorb insert_subset interior_mono)
        done
      have *: "convex hull {b, c, d} \<subseteq> convex hull {a, b, c}"
        by (meson True hull_subset insert_subset convex_hull_subset)
      have bcd: "(f has_contour_integral 0) (linepath b c +++ linepath c d +++ linepath d b)"
        apply (rule less.hyps [of "s'"])
        using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
        apply (auto intro!: less.prems continuous_on_subset [OF _ *])
        apply (metis * insert_absorb insert_subset interior_mono)
        done
      have *: "convex hull {c, a, d} \<subseteq> convex hull {a, b, c}"
        by (meson True hull_subset insert_subset convex_hull_subset)
      have cad: "(f has_contour_integral 0) (linepath c a +++ linepath a d +++ linepath d c)"
        apply (rule less.hyps [of "s'"])
        using True d  \<open>finite s\<close> not_in_interior_convex_hull_3
        apply (auto intro!: less.prems continuous_on_subset [OF _ *])
        apply (metis * insert_absorb insert_subset interior_mono)
        done
      have "f contour_integrable_on linepath a b"
        using less.prems
        by (metis continuous_on_subset insert_commute contour_integrable_continuous_linepath segments_subset_convex_hull(3))
      moreover have "f contour_integrable_on linepath b c"
        using less.prems
        by (metis continuous_on_subset contour_integrable_continuous_linepath segments_subset_convex_hull(3))
      moreover have "f contour_integrable_on linepath c a"
        using less.prems
        by (metis continuous_on_subset insert_commute contour_integrable_continuous_linepath segments_subset_convex_hull(3))
      ultimately have fpi: "f contour_integrable_on (linepath a b +++ linepath b c +++ linepath c a)"
        by auto
      { fix y::complex
        assume fy: "(f has_contour_integral y) (linepath a b +++ linepath b c +++ linepath c a)"
           and ynz: "y \<noteq> 0"
        have cont_ad: "continuous_on (closed_segment a d) f"
          by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(3))
        have cont_bd: "continuous_on (closed_segment b d) f"
          by (meson True closed_segment_subset_convex_hull continuous_on_subset hull_subset insert_subset less.prems(1))
        have cont_cd: "continuous_on (closed_segment c d) f"
          by (meson "*" continuous_on_subset less.prems(1) segments_subset_convex_hull(2))
        have "contour_integral  (linepath a b) f = - (contour_integral (linepath b d) f + (contour_integral (linepath d a) f))"
                "contour_integral  (linepath b c) f = - (contour_integral (linepath c d) f + (contour_integral (linepath d b) f))"
                "contour_integral  (linepath c a) f = - (contour_integral (linepath a d) f + contour_integral (linepath d c) f)"
            using has_chain_integral_chain_integral3 [OF abd]
                  has_chain_integral_chain_integral3 [OF bcd]
                  has_chain_integral_chain_integral3 [OF cad]
            by (simp_all add: algebra_simps add_eq_0_iff)
        then have ?thesis
          using cont_ad cont_bd cont_cd fy has_chain_integral_chain_integral3 contour_integral_reverse_linepath by fastforce
      }
      then show ?thesis
        using fpi contour_integrable_on_def by blast
    qed
  qed
qed


subsection\<open>Cauchy's theorem for an open starlike set\<close>

lemma starlike_convex_subset:
  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"
    shows "convex hull {a,b,c} \<subseteq> s"
      using s
      apply (clarsimp simp add: convex_hull_insert [of "{b,c}" a] segment_convex_hull)
      apply (meson subs convexD convex_closed_segment ends_in_segment(1) ends_in_segment(2) subsetCE)
      done

lemma triangle_contour_integrals_starlike_primitive:
  assumes contf: "continuous_on s f"
      and s: "a \<in> s" "open s"
      and x: "x \<in> s"
      and subs: "\<And>y. y \<in> s \<Longrightarrow> closed_segment a y \<subseteq> s"
      and zer: "\<And>b c. closed_segment b c \<subseteq> s
                   \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
                       contour_integral (linepath c a) f = 0"
    shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x)"
proof -
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
  { fix e y
    assume e: "0 < e" and bxe: "ball x e \<subseteq> s" and close: "cmod (y - x) < e"
    have y: "y \<in> s"
      using bxe close  by (force simp: dist_norm norm_minus_commute)
    have cont_ayf: "continuous_on (closed_segment a y) f"
      using contf continuous_on_subset subs y by blast
    have xys: "closed_segment x y \<subseteq> s"
      apply (rule order_trans [OF _ bxe])
      using close
      by (auto simp: dist_norm ball_def norm_minus_commute dest: segment_bound)
    have "?pathint a y - ?pathint a x = ?pathint x y"
      using zer [OF xys]  contour_integral_reverse_linepath [OF cont_ayf]  add_eq_0_iff by force
  } note [simp] = this
  { fix e::real
    assume e: "0 < e"
    have cont_atx: "continuous (at x) f"
      using x s contf continuous_on_eq_continuous_at by blast
    then obtain d1 where d1: "d1>0" and d1_less: "\<And>y. cmod (y - x) < d1 \<Longrightarrow> cmod (f y - f x) < e/2"
      unfolding continuous_at Lim_at dist_norm  using e
      by (drule_tac x="e/2" in spec) force
    obtain d2 where d2: "d2>0" "ball x d2 \<subseteq> s" using  \<open>open s\<close> x
      by (auto simp: open_contains_ball)
    have dpos: "min d1 d2 > 0" using d1 d2 by simp
    { fix y
      assume yx: "y \<noteq> x" and close: "cmod (y - x) < min d1 d2"
      have y: "y \<in> s"
        using d2 close  by (force simp: dist_norm norm_minus_commute)
      have fxy: "f contour_integrable_on linepath x y"
        apply (rule contour_integrable_continuous_linepath)
        apply (rule continuous_on_subset [OF contf])
        using close d2
        apply (auto simp: dist_norm norm_minus_commute dest!: segment_bound(1))
        done
      then obtain i where i: "(f has_contour_integral i) (linepath x y)"
        by (auto simp: contour_integrable_on_def)
      then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
        by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
      then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
        apply (rule has_contour_integral_bound_linepath [where B = "e/2"])
        using e apply simp
        apply (rule d1_less [THEN less_imp_le])
        using close segment_bound
        apply force
        done
      also have "... < e * cmod (y - x)"
        by (simp add: e yx)
      finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
        using i yx  by (simp add: contour_integral_unique divide_less_eq)
    }
    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"
      using dpos by blast
  }
  then have *: "(\<lambda>y. (?pathint x y - f x * (y - x)) /\<^sub>R cmod (y - x)) \<midarrow>x\<rightarrow> 0"
    by (simp add: Lim_at dist_norm inverse_eq_divide)
  show ?thesis
    apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right)
    apply (rule Lim_transform [OF * Lim_eventually])
    apply (simp add: inverse_eq_divide [symmetric] eventually_at)
    using \<open>open s\<close> x
    apply (force simp: dist_norm open_contains_ball)
    done
qed

(** Existence of a primitive.*)

lemma holomorphic_starlike_primitive:
  assumes contf: "continuous_on s f"
      and s: "starlike s" and os: "open s"
      and k: "finite k"
      and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x"
    shows "\<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x)"
proof -
  obtain a where a: "a\<in>s" and a_cs: "\<And>x. x\<in>s \<Longrightarrow> closed_segment a x \<subseteq> s"
    using s by (auto simp: starlike_def)
  { fix x b c
    assume "x \<in> s" "closed_segment b c \<subseteq> s"
    then have abcs: "convex hull {a, b, c} \<subseteq> s"
      by (simp add: a a_cs starlike_convex_subset)
    then have *: "continuous_on (convex hull {a, b, c}) f"
      by (simp add: continuous_on_subset [OF contf])
    have "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
      apply (rule Cauchy_theorem_triangle_cofinite [OF _ k])
      using abcs apply (simp add: continuous_on_subset [OF contf])
      using * abcs interior_subset apply (auto intro: fcd)
      done
  } note 0 = this
  show ?thesis
    apply (intro exI ballI)
    apply (rule triangle_contour_integrals_starlike_primitive [OF contf a os], assumption)
    apply (metis a_cs)
    apply (metis has_chain_integral_chain_integral3 0)
    done
qed

lemma Cauchy_theorem_starlike:
 "\<lbrakk>open s; starlike s; finite k; continuous_on s f;
   \<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x;
   valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
   \<Longrightarrow> (f has_contour_integral 0)  g"
  by (metis holomorphic_starlike_primitive Cauchy_theorem_primitive at_within_open)

lemma Cauchy_theorem_starlike_simple:
  "\<lbrakk>open s; starlike s; f holomorphic_on s; valid_path g; path_image g \<subseteq> s; pathfinish g = pathstart g\<rbrakk>
   \<Longrightarrow> (f has_contour_integral 0) g"
apply (rule Cauchy_theorem_starlike [OF _ _ finite.emptyI])
apply (simp_all add: holomorphic_on_imp_continuous_on)
apply (metis at_within_open holomorphic_on_def)
done


subsection\<open>Cauchy's theorem for a convex set\<close>

text\<open>For a convex set we can avoid assuming openness and boundary analyticity\<close>

lemma triangle_contour_integrals_convex_primitive:
  assumes contf: "continuous_on s f"
      and s: "a \<in> s" "convex s"
      and x: "x \<in> s"
      and zer: "\<And>b c. \<lbrakk>b \<in> s; c \<in> s\<rbrakk>
                   \<Longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f +
                       contour_integral (linepath c a) f = 0"
    shows "((\<lambda>x. contour_integral(linepath a x) f) has_field_derivative f x) (at x within s)"
proof -
  let ?pathint = "\<lambda>x y. contour_integral(linepath x y) f"
  { fix y
    assume y: "y \<in> s"
    have cont_ayf: "continuous_on (closed_segment a y) f"
      using s y  by (meson contf continuous_on_subset convex_contains_segment)
    have xys: "closed_segment x y \<subseteq> s"  (*?*)
      using convex_contains_segment s x y by auto
    have "?pathint a y - ?pathint a x = ?pathint x y"
      using zer [OF x y]  contour_integral_reverse_linepath [OF cont_ayf]  add_eq_0_iff by force
  } note [simp] = this
  { fix e::real
    assume e: "0 < e"
    have cont_atx: "continuous (at x within s) f"
      using x s contf  by (simp add: continuous_on_eq_continuous_within)
    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"
      unfolding continuous_within Lim_within dist_norm using e
      by (drule_tac x="e/2" in spec) force
    { fix y
      assume yx: "y \<noteq> x" and close: "cmod (y - x) < d1" and y: "y \<in> s"
      have fxy: "f contour_integrable_on linepath x y"
        using convex_contains_segment s x y
        by (blast intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
      then obtain i where i: "(f has_contour_integral i) (linepath x y)"
        by (auto simp: contour_integrable_on_def)
      then have "((\<lambda>w. f w - f x) has_contour_integral (i - f x * (y - x))) (linepath x y)"
        by (rule has_contour_integral_diff [OF _ has_contour_integral_const_linepath])
      then have "cmod (i - f x * (y - x)) \<le> e / 2 * cmod (y - x)"
        apply (rule has_contour_integral_bound_linepath [where B = "e/2"])
        using e apply simp
        apply (rule d1_less [THEN less_imp_le])
        using convex_contains_segment s(2) x y apply blast
        using close segment_bound(1) apply fastforce
        done
      also have "... < e * cmod (y - x)"
        by (simp add: e yx)
      finally have "cmod (?pathint x y - f x * (y-x)) / cmod (y-x) < e"
        using i yx  by (simp add: contour_integral_unique divide_less_eq)
    }
    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"
      using d1 by blast
  }
  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)"
    by (simp add: Lim_within dist_norm inverse_eq_divide)
  show ?thesis
    apply (simp add: has_field_derivative_def has_derivative_within bounded_linear_mult_right)
    apply (rule Lim_transform [OF * Lim_eventually])
    using linordered_field_no_ub
    apply (force simp: inverse_eq_divide [symmetric] eventually_at)
    done
qed

lemma contour_integral_convex_primitive:
  "\<lbrakk>convex s; continuous_on s f;
    \<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>
         \<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
  apply (cases "s={}")
  apply (simp_all add: ex_in_conv [symmetric])
  apply (blast intro: triangle_contour_integrals_convex_primitive has_chain_integral_chain_integral3)
  done

lemma holomorphic_convex_primitive:
  "\<lbrakk>convex s; finite k; continuous_on s f;
    \<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x\<rbrakk>
   \<Longrightarrow> \<exists>g. \<forall>x \<in> s. (g has_field_derivative f x) (at x within s)"
apply (rule contour_integral_convex_primitive [OF _ _ Cauchy_theorem_triangle_cofinite])
prefer 3
apply (erule continuous_on_subset)
apply (simp add: subset_hull continuous_on_subset, assumption+)
by (metis Diff_iff convex_contains_segment insert_absorb insert_subset interior_mono segment_convex_hull subset_hull)

lemma Cauchy_theorem_convex:
    "\<lbrakk>continuous_on s f; convex s; finite k;
      \<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x;
     valid_path g; path_image g \<subseteq> s;
     pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
  by (metis holomorphic_convex_primitive Cauchy_theorem_primitive)

lemma Cauchy_theorem_convex_simple:
    "\<lbrakk>f holomorphic_on s; convex s;
     valid_path g; path_image g \<subseteq> s;
     pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
  apply (rule Cauchy_theorem_convex)
  apply (simp_all add: holomorphic_on_imp_continuous_on)
  apply (rule finite.emptyI)
  using at_within_interior holomorphic_on_def interior_subset by fastforce


text\<open>In particular for a disc\<close>
lemma Cauchy_theorem_disc:
    "\<lbrakk>finite k; continuous_on (cball a e) f;
      \<And>x. x \<in> ball a e - k \<Longrightarrow> f complex_differentiable at x;
     valid_path g; path_image g \<subseteq> cball a e;
     pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
  apply (rule Cauchy_theorem_convex)
  apply (auto simp: convex_cball interior_cball)
  done

lemma Cauchy_theorem_disc_simple:
    "\<lbrakk>f holomorphic_on (ball a e); valid_path g; path_image g \<subseteq> ball a e;
     pathfinish g = pathstart g\<rbrakk> \<Longrightarrow> (f has_contour_integral 0) g"
by (simp add: Cauchy_theorem_convex_simple)


subsection\<open>Generalize integrability to local primitives\<close>

lemma contour_integral_local_primitive_lemma:
  fixes f :: "complex\<Rightarrow>complex"
  shows
    "\<lbrakk>g piecewise_differentiable_on {a..b};
      \<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative f' x) (at x within s);
      \<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s\<rbrakk>
     \<Longrightarrow> (\<lambda>x. f' (g x) * vector_derivative g (at x within {a..b}))
            integrable_on {a..b}"
  apply (cases "cbox a b = {}", force)
  apply (simp add: integrable_on_def)
  apply (rule exI)
  apply (rule contour_integral_primitive_lemma, assumption+)
  using atLeastAtMost_iff by blast

lemma contour_integral_local_primitive_any:
  fixes f :: "complex \<Rightarrow> complex"
  assumes gpd: "g piecewise_differentiable_on {a..b}"
      and dh: "\<And>x. x \<in> s
               \<Longrightarrow> \<exists>d h. 0 < d \<and>
                         (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
      and gs: "\<And>x. x \<in> {a..b} \<Longrightarrow> g x \<in> s"
  shows "(\<lambda>x. f(g x) * vector_derivative g (at x)) integrable_on {a..b}"
proof -
  { fix x
    assume x: "a \<le> x" "x \<le> b"
    obtain d h where d: "0 < d"
               and h: "(\<And>y. norm(y - g x) < d \<Longrightarrow> (h has_field_derivative f y) (at y within s))"
      using x gs dh by (metis atLeastAtMost_iff)
    have "continuous_on {a..b} g" using gpd piecewise_differentiable_on_def by blast
    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"
      using x d
      apply (auto simp: dist_norm continuous_on_iff)
      apply (drule_tac x=x in bspec)
      using x apply simp
      apply (drule_tac x=d in spec, auto)
      done
    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>
                          (\<lambda>x. f (g x) * vector_derivative g (at x)) integrable_on {u..v}"
      apply (rule_tac x=e in exI)
      using e
      apply (simp add: integrable_on_localized_vector_derivative [symmetric], clarify)
      apply (rule_tac f = h and s = "g ` {u..v}" in contour_integral_local_primitive_lemma)
        apply (meson atLeastatMost_subset_iff gpd piecewise_differentiable_on_subset)
       apply (force simp: ball_def dist_norm intro: lessd gs DERIV_subset [OF h], force)
      done
  } then
  show ?thesis
    by (force simp: intro!: integrable_on_little_subintervals [of a b, simplified])
qed

lemma contour_integral_local_primitive:
  fixes f :: "complex \<Rightarrow> complex"
  assumes g: "valid_path g" "path_image g \<subseteq> s"
      and dh: "\<And>x. x \<in> s
               \<Longrightarrow> \<exists>d h. 0 < d \<and>
                         (\<forall>y. norm(y - x) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
  shows "f contour_integrable_on g"
  using g
  apply (simp add: valid_path_def path_image_def contour_integrable_on_def has_contour_integral_def
            has_integral_localized_vector_derivative integrable_on_def [symmetric])
  using contour_integral_local_primitive_any [OF _ dh]
  by (meson image_subset_iff piecewise_C1_imp_differentiable)


text\<open>In particular if a function is holomorphic\<close>

lemma contour_integrable_holomorphic:
  assumes contf: "continuous_on s f"
      and os: "open s"
      and k: "finite k"
      and g: "valid_path g" "path_image g \<subseteq> s"
      and fcd: "\<And>x. x \<in> s - k \<Longrightarrow> f complex_differentiable at x"
    shows "f contour_integrable_on g"
proof -
  { fix z
    assume z: "z \<in> s"
    obtain d where d: "d>0" "ball z d \<subseteq> s" using  \<open>open s\<close> z
      by (auto simp: open_contains_ball)
    then have contfb: "continuous_on (ball z d) f"
      using contf continuous_on_subset by blast
    obtain h where "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within ball z d)"
      using holomorphic_convex_primitive [OF convex_ball k contfb fcd] d
            interior_subset by force
    then have "\<forall>y\<in>ball z d. (h has_field_derivative f y) (at y within s)"
      by (metis Topology_Euclidean_Space.open_ball at_within_open d(2) os subsetCE)
    then have "\<exists>h. (\<forall>y. cmod (y - z) < d \<longrightarrow> (h has_field_derivative f y) (at y within s))"
      by (force simp: dist_norm norm_minus_commute)
    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))"
      using d by blast
  }
  then show ?thesis
    by (rule contour_integral_local_primitive [OF g])
qed

lemma contour_integrable_holomorphic_simple:
  assumes fh: "f holomorphic_on s"
      and os: "open s"
      and g: "valid_path g" "path_image g \<subseteq> s"
    shows "f contour_integrable_on g"
  apply (rule contour_integrable_holomorphic [OF _ os Finite_Set.finite.emptyI g])
  apply (simp add: fh holomorphic_on_imp_continuous_on)
  using fh  by (simp add: complex_differentiable_def holomorphic_on_open os)

lemma continuous_on_inversediff:
  fixes z:: "'a::real_normed_field" shows "z \<notin> s \<Longrightarrow> continuous_on s (\<lambda>w. 1 / (w - z))"
  by (rule continuous_intros | force)+

corollary contour_integrable_inversediff:
    "\<lbrakk>valid_path g; z \<notin> path_image g\<rbrakk> \<Longrightarrow> (\<lambda>w. 1 / (w-z)) contour_integrable_on g"
apply (rule contour_integrable_holomorphic_simple [of _ "UNIV-{z}"])
apply (auto simp: holomorphic_on_open open_delete intro!: derivative_eq_intros)
done

text\<open>Key fact that path integral is the same for a "nearby" path. This is the
 main lemma for the homotopy form of Cauchy's theorem and is also useful
 if we want "without loss of generality" to assume some nice properties of a
 path (e.g. smoothness). It can also be used to define the integrals of
 analytic functions over arbitrary continuous paths. This is just done for
 winding numbers now.
\<close>

text\<open>A technical definition to avoid duplication of similar proofs,
     for paths joined at the ends versus looping paths\<close>
definition linked_paths :: "bool \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> (real \<Rightarrow> 'a::topological_space) \<Rightarrow> bool"
  where "linked_paths atends g h ==
        (if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g
                   else pathfinish g = pathstart g \<and> pathfinish h = pathstart h)"

text\<open>This formulation covers two cases: @{term g} and @{term h} share their
      start and end points; @{term g} and @{term h} both loop upon themselves.\<close>
lemma contour_integral_nearby:
  assumes os: "open s" and p: "path p" "path_image p \<subseteq> s"
    shows
       "\<exists>d. 0 < d \<and>
            (\<forall>g h. valid_path g \<and> valid_path h \<and>
                  (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
                  linked_paths atends g h
                  \<longrightarrow> 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))"
proof -
  have "\<forall>z. \<exists>e. z \<in> path_image p \<longrightarrow> 0 < e \<and> ball z e \<subseteq> s"
    using open_contains_ball os p(2) by blast
  then obtain ee where ee: "\<And>z. z \<in> path_image p \<Longrightarrow> 0 < ee z \<and> ball z (ee z) \<subseteq> s"
    by metis
  def cover \<equiv> "(\<lambda>z. ball z (ee z/3)) ` (path_image p)"
  have "compact (path_image p)"
    by (metis p(1) compact_path_image)
  moreover have "path_image p \<subseteq> (\<Union>c\<in>path_image p. ball c (ee c / 3))"
    using ee by auto
  ultimately have "\<exists>D \<subseteq> cover. finite D \<and> path_image p \<subseteq> \<Union>D"
    by (simp add: compact_eq_heine_borel cover_def)
  then obtain D where D: "D \<subseteq> cover" "finite D" "path_image p \<subseteq> \<Union>D"
    by blast
  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"
    apply (simp add: cover_def path_image_def image_comp)
    apply (blast dest!: finite_subset_image [OF \<open>finite D\<close>])
    done
  then have kne: "k \<noteq> {}"
    using D by auto
  have pi: "\<And>i. i \<in> k \<Longrightarrow> p i \<in> path_image p"
    using k  by (auto simp: path_image_def)
  then have eepi: "\<And>i. i \<in> k \<Longrightarrow> 0 < ee((p i))"
    by (metis ee)
  def e \<equiv> "Min((ee o p) ` k)"
  have fin_eep: "finite ((ee o p) ` k)"
    using k  by blast
  have enz: "0 < e"
    using ee k  by (simp add: kne e_def Min_gr_iff [OF fin_eep] eepi)
  have "uniformly_continuous_on {0..1} p"
    using p  by (simp add: path_def compact_uniformly_continuous)
  then obtain d::real where d: "d>0"
          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"
    unfolding uniformly_continuous_on_def dist_norm real_norm_def
    by (metis divide_pos_pos enz zero_less_numeral)
  then obtain N::nat where N: "N>0" "inverse N < d"
    using real_arch_inv [of d]   by auto
  { fix g h
    assume g: "valid_path g" and gp: "\<forall>t\<in>{0..1}. cmod (g t - p t) < e / 3"
       and h: "valid_path h" and hp: "\<forall>t\<in>{0..1}. cmod (h t - p t) < e / 3"
       and joins: "linked_paths atends g h"
    { fix t::real
      assume t: "0 \<le> t" "t \<le> 1"
      then obtain u where u: "u \<in> k" and ptu: "p t \<in> ball(p u) (ee(p u) / 3)"
        using \<open>path_image p \<subseteq> \<Union>D\<close> D_eq by (force simp: path_image_def)
      then have ele: "e \<le> ee (p u)" using fin_eep
        by (simp add: e_def)
      have "cmod (g t - p t) < e / 3" "cmod (h t - p t) < e / 3"
        using gp hp t by auto
      with ele have "cmod (g t - p t) < ee (p u) / 3"
                    "cmod (h t - p t) < ee (p u) / 3"
        by linarith+
      then have "g t \<in> ball(p u) (ee(p u))"  "h t \<in> ball(p u) (ee(p u))"
        using norm_diff_triangle_ineq [of "g t" "p t" "p t" "p u"]
              norm_diff_triangle_ineq [of "h t" "p t" "p t" "p u"] ptu eepi u
        by (force simp: dist_norm ball_def norm_minus_commute)+
      then have "g t \<in> s" "h t \<in> s" using ee u k
        by (auto simp: path_image_def ball_def)
    }
    then have ghs: "path_image g \<subseteq> s" "path_image h \<subseteq> s"
      by (auto simp: path_image_def)
    moreover
    { fix f
      assume fhols: "f holomorphic_on s"
      then have fpa: "f contour_integrable_on g"  "f contour_integrable_on h"
        using g ghs h holomorphic_on_imp_continuous_on os contour_integrable_holomorphic_simple
        by blast+
      have contf: "continuous_on s f"
        by (simp add: fhols holomorphic_on_imp_continuous_on)
      { fix z
        assume z: "z \<in> path_image p"
        have "f holomorphic_on ball z (ee z)"
          using fhols ee z holomorphic_on_subset by blast
        then have "\<exists>ff. (\<forall>w \<in> ball z (ee z). (ff has_field_derivative f w) (at w))"
          using holomorphic_convex_primitive [of "ball z (ee z)" "{}" f, simplified]
          by (metis open_ball at_within_open holomorphic_on_def holomorphic_on_imp_continuous_on mem_ball)
      }
      then obtain ff where ff:
            "\<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)"
        by metis
      { fix n
        assume n: "n \<le> N"
        then have "contour_integral(subpath 0 (n/N) h) f - contour_integral(subpath 0 (n/N) g) f =
                   contour_integral(linepath (g(n/N)) (h(n/N))) f - contour_integral(linepath (g 0) (h 0)) f"
        proof (induct n)
          case 0 show ?case by simp
        next
          case (Suc n)
          obtain t where t: "t \<in> k" and "p (n/N) \<in> ball(p t) (ee(p t) / 3)"
            using \<open>path_image p \<subseteq> \<Union>D\<close> [THEN subsetD, where c="p (n/N)"] D_eq N Suc.prems
            by (force simp: path_image_def)
          then have ptu: "cmod (p t - p (n/N)) < ee (p t) / 3"
            by (simp add: dist_norm)
          have e3le: "e/3 \<le> ee (p t) / 3"  using fin_eep t
            by (simp add: e_def)
          { fix x
            assume x: "n/N \<le> x" "x \<le> (1 + n)/N"
            then have nN01: "0 \<le> n/N" "(1 + n)/N \<le> 1"
              using Suc.prems by auto
            then have x01: "0 \<le> x" "x \<le> 1"
              using x by linarith+
            have "cmod (p t - p x)  < ee (p t) / 3 + e/3"
              apply (rule norm_diff_triangle_less [OF ptu de])
              using x N x01 Suc.prems
              apply (auto simp: field_simps)
              done
            then have ptx: "cmod (p t - p x) < 2*ee (p t)/3"
              using e3le eepi [OF t] by simp
            have "cmod (p t - g x) < 2*ee (p t)/3 + e/3 "
              apply (rule norm_diff_triangle_less [OF ptx])
              using gp x01 by (simp add: norm_minus_commute)
            also have "... \<le> ee (p t)"
              using e3le eepi [OF t] by simp
            finally have gg: "cmod (p t - g x) < ee (p t)" .
            have "cmod (p t - h x) < 2*ee (p t)/3 + e/3 "
              apply (rule norm_diff_triangle_less [OF ptx])
              using hp x01 by (simp add: norm_minus_commute)
            also have "... \<le> ee (p t)"
              using e3le eepi [OF t] by simp
            finally have "cmod (p t - g x) < ee (p t)"
                         "cmod (p t - h x) < ee (p t)"
              using gg by auto
          } note ptgh_ee = this
          have pi_hgn: "path_image (linepath (h (n/N)) (g (n/N))) \<subseteq> ball (p t) (ee (p t))"
            using ptgh_ee [of "n/N"] Suc.prems
            by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
          then have gh_ns: "closed_segment (g (n/N)) (h (n/N)) \<subseteq> s"
            using \<open>N>0\<close> Suc.prems
            apply (simp add: path_image_join field_simps closed_segment_commute)
            apply (erule order_trans)
            apply (simp add: ee pi t)
            done
          have pi_ghn': "path_image (linepath (g ((1 + n) / N)) (h ((1 + n) / N)))
                  \<subseteq> ball (p t) (ee (p t))"
            using ptgh_ee [of "(1+n)/N"] Suc.prems
            by (auto simp: field_simps dist_norm dest: segment_furthest_le [where y="p t"])
          then have gh_n's: "closed_segment (g ((1 + n) / N)) (h ((1 + n) / N)) \<subseteq> s"
            using \<open>N>0\<close> Suc.prems ee pi t
            by (auto simp: Path_Connected.path_image_join field_simps)
          have pi_subset_ball:
                "path_image (subpath (n/N) ((1+n) / N) g +++ linepath (g ((1+n) / N)) (h ((1+n) / N)) +++
                             subpath ((1+n) / N) (n/N) h +++ linepath (h (n/N)) (g (n/N)))
                 \<subseteq> ball (p t) (ee (p t))"
            apply (intro subset_path_image_join pi_hgn pi_ghn')
            using \<open>N>0\<close> Suc.prems
            apply (auto simp: path_image_subpath dist_norm field_simps closed_segment_eq_real_ivl ptgh_ee)
            done
          have pi0: "(f has_contour_integral 0)
                       (subpath (n/ N) ((Suc n)/N) g +++ linepath(g ((Suc n) / N)) (h((Suc n) / N)) +++
                        subpath ((Suc n) / N) (n/N) h +++ linepath(h (n/N)) (g (n/N)))"
            apply (rule Cauchy_theorem_primitive [of "ball(p t) (ee(p t))" "ff (p t)" "f"])
            apply (metis ff open_ball at_within_open pi t)
            apply (intro valid_path_join)
            using Suc.prems pi_subset_ball apply (simp_all add: valid_path_subpath g h)
            done
          have fpa1: "f contour_integrable_on subpath (real n / real N) (real (Suc n) / real N) g"
            using Suc.prems by (simp add: contour_integrable_subpath g fpa)
          have fpa2: "f contour_integrable_on linepath (g (real (Suc n) / real N)) (h (real (Suc n) / real N))"
            using gh_n's
            by (auto intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
          have fpa3: "f contour_integrable_on linepath (h (real n / real N)) (g (real n / real N))"
            using gh_ns
            by (auto simp: closed_segment_commute intro!: contour_integrable_continuous_linepath continuous_on_subset [OF contf])
          have eq0: "contour_integral (subpath (n/N) ((Suc n) / real N) g) f +
                     contour_integral (linepath (g ((Suc n) / N)) (h ((Suc n) / N))) f +
                     contour_integral (subpath ((Suc n) / N) (n/N) h) f +
                     contour_integral (linepath (h (n/N)) (g (n/N))) f = 0"
            using contour_integral_unique [OF pi0] Suc.prems
            by (simp add: g h fpa valid_path_subpath contour_integrable_subpath
                          fpa1 fpa2 fpa3 algebra_simps del: of_nat_Suc)
          have *: "\<And>hn he hn' gn gd gn' hgn ghn gh0 ghn'.
                    \<lbrakk>hn - gn = ghn - gh0;
                     gd + ghn' + he + hgn = (0::complex);
                     hn - he = hn'; gn + gd = gn'; hgn = -ghn\<rbrakk> \<Longrightarrow> hn' - gn' = ghn' - gh0"
            by (auto simp: algebra_simps)
          have "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
                contour_integral (subpath 0 (n/N) h) f + contour_integral (subpath (n/N) ((Suc n) / N) h) f"
            unfolding reversepath_subpath [symmetric, of "((Suc n) / N)"]
            using Suc.prems by (simp add: h fpa contour_integral_reversepath valid_path_subpath contour_integrable_subpath)
          also have "... = contour_integral (subpath 0 ((Suc n) / N) h) f"
            using Suc.prems by (simp add: contour_integral_subpath_combine h fpa)
          finally have pi0_eq:
               "contour_integral (subpath 0 (n/N) h) f - contour_integral (subpath ((Suc n) / N) (n/N) h) f =
                contour_integral (subpath 0 ((Suc n) / N) h) f" .
          show ?case
            apply (rule * [OF Suc.hyps eq0 pi0_eq])
            using Suc.prems
            apply (simp_all add: g h fpa contour_integral_subpath_combine
                     contour_integral_reversepath [symmetric] contour_integrable_continuous_linepath
                     continuous_on_subset [OF contf gh_ns])
            done
      qed
      } note ind = this
      have "contour_integral h f = contour_integral g f"
        using ind [OF order_refl] N joins
        by (simp add: linked_paths_def pathstart_def pathfinish_def split: split_if_asm)
    }
    ultimately
    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)"
      by metis
  } note * = this
  show ?thesis
    apply (rule_tac x="e/3" in exI)
    apply (rule conjI)
    using enz apply simp
    apply (clarsimp simp only: ball_conj_distrib)
    apply (rule *; assumption)
    done
qed


lemma
  assumes "open s" "path p" "path_image p \<subseteq> s"
    shows contour_integral_nearby_ends:
      "\<exists>d. 0 < d \<and>
              (\<forall>g h. valid_path g \<and> valid_path h \<and>
                    (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
                    pathstart h = pathstart g \<and> pathfinish h = pathfinish g
                    \<longrightarrow> 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))"
    and contour_integral_nearby_loops:
      "\<exists>d. 0 < d \<and>
              (\<forall>g h. valid_path g \<and> valid_path h \<and>
                    (\<forall>t \<in> {0..1}. norm(g t - p t) < d \<and> norm(h t - p t) < d) \<and>
                    pathfinish g = pathstart g \<and> pathfinish h = pathstart h
                    \<longrightarrow> 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))"
  using contour_integral_nearby [OF assms, where atends=True]
  using contour_integral_nearby [OF assms, where atends=False]
  unfolding linked_paths_def by simp_all

corollary differentiable_polynomial_function:
  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
  shows "polynomial_function p \<Longrightarrow> p differentiable_on s"
by (meson has_vector_derivative_polynomial_function differentiable_at_imp_differentiable_on differentiable_def has_vector_derivative_def)

lemma C1_differentiable_polynomial_function:
  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
  shows "polynomial_function p \<Longrightarrow> p C1_differentiable_on s"
  by (metis continuous_on_polymonial_function C1_differentiable_on_def  has_vector_derivative_polynomial_function)

lemma valid_path_polynomial_function:
  fixes p :: "real \<Rightarrow> 'a::euclidean_space"
  shows "polynomial_function p \<Longrightarrow> valid_path p"
by (force simp: valid_path_def piecewise_C1_differentiable_on_def continuous_on_polymonial_function C1_differentiable_polynomial_function)

lemma valid_path_subpath_trivial [simp]:
    fixes g :: "real \<Rightarrow> 'a::euclidean_space"
    shows "z \<noteq> g x \<Longrightarrow> valid_path (subpath x x g)"
  by (simp add: subpath_def valid_path_polynomial_function)

lemma contour_integral_bound_exists:
assumes s: "open s"
    and g: "valid_path g"
    and pag: "path_image g \<subseteq> s"
  shows "\<exists>L. 0 < L \<and>
       (\<forall>f B. f holomorphic_on s \<and> (\<forall>z \<in> s. norm(f z) \<le> B)
         \<longrightarrow> norm(contour_integral g f) \<le> L*B)"
proof -
have "path g" using g
  by (simp add: valid_path_imp_path)
then obtain d::real and p
  where d: "0 < d"
    and p: "polynomial_function p" "path_image p \<subseteq> s"
    and pi: "\<And>f. f holomorphic_on s \<Longrightarrow> contour_integral g f = contour_integral p f"
  using contour_integral_nearby_ends [OF s \<open>path g\<close> pag]
  apply clarify
  apply (drule_tac x=g in spec)
  apply (simp only: assms)
  apply (force simp: valid_path_polynomial_function dest: path_approx_polynomial_function)
  done
then obtain p' where p': "polynomial_function p'"
         "\<And>x. (p has_vector_derivative (p' x)) (at x)"
  using has_vector_derivative_polynomial_function by force
then have "bounded(p' ` {0..1})"
  using continuous_on_polymonial_function
  by (force simp: intro!: compact_imp_bounded compact_continuous_image)
then obtain L where L: "L>0" and nop': "\<And>x. x \<in> {0..1} \<Longrightarrow> norm (p' x) \<le> L"
  by (force simp: bounded_pos)
{ fix f B
  assume f: "f holomorphic_on s"
     and B: "\<And>z. z\<in>s \<Longrightarrow> cmod (f z) \<le> B"
  then have "f contour_integrable_on p \<and> valid_path p"
    using p s
    by (blast intro: valid_path_polynomial_function contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on)
  moreover have "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (vector_derivative p (at x)) * cmod (f (p x)) \<le> L * B"
    apply (rule mult_mono)
    apply (subst Derivative.vector_derivative_at; force intro: p' nop')
    using L B p
    apply (auto simp: path_image_def image_subset_iff)
    done
  ultimately have "cmod (contour_integral g f) \<le> L * B"
    apply (simp add: pi [OF f])
    apply (simp add: contour_integral_integral)
    apply (rule order_trans [OF integral_norm_bound_integral])
    apply (auto simp: mult.commute integral_norm_bound_integral contour_integrable_on [symmetric] norm_mult)
    done
} then
show ?thesis
  by (force simp: L contour_integral_integral)
qed

subsection\<open>Constancy of a function from a connected set into a finite, disconnected or discrete set\<close>

text\<open>Still missing: versions for a set that is smaller than R, or countable.\<close>

lemma continuous_disconnected_range_constant:
  assumes s: "connected s"
      and conf: "continuous_on s f"
      and fim: "f ` s \<subseteq> t"
      and cct: "\<And>y. y \<in> t \<Longrightarrow> connected_component_set t y = {y}"
    shows "\<exists>a. \<forall>x \<in> s. f x = a"
proof (cases "s = {}")
  case True then show ?thesis by force
next
  case False
  { fix x assume "x \<in> s"
    then have "f ` s \<subseteq> {f x}"
    by (metis connected_continuous_image conf connected_component_maximal fim image_subset_iff rev_image_eqI s cct)
  }
  with False show ?thesis
    by blast
qed

lemma discrete_subset_disconnected:
  fixes s :: "'a::topological_space set"
  fixes t :: "'b::real_normed_vector set"
  assumes conf: "continuous_on s f"
      and no: "\<And>x. x \<in> s \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> s \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
   shows "f ` s \<subseteq> {y. connected_component_set (f ` s) y = {y}}"
proof -
  { fix x assume x: "x \<in> s"
    then obtain e where "e>0" and ele: "\<And>y. \<lbrakk>y \<in> s; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
      using conf no [OF x] by auto
    then have e2: "0 \<le> e / 2"
      by simp
    have "f y = f x" if "y \<in> s" and ccs: "f y \<in> connected_component_set (f ` s) (f x)" for y
      apply (rule ccontr)
      using connected_closed [of "connected_component_set (f ` s) (f x)"] \<open>e>0\<close>
      apply (simp add: del: ex_simps)
      apply (drule spec [where x="cball (f x) (e / 2)"])
      apply (drule spec [where x="- ball(f x) e"])
      apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)
        apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
       using centre_in_cball connected_component_refl_eq e2 x apply blast
      using ccs
      apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF \<open>y \<in> s\<close>])
      done
    moreover have "connected_component_set (f ` s) (f x) \<subseteq> f ` s"
      by (auto simp: connected_component_in)
    ultimately have "connected_component_set (f ` s) (f x) = {f x}"
      by (auto simp: x)
  }
  with assms show ?thesis
    by blast
qed

lemma finite_implies_discrete:
  fixes s :: "'a::topological_space set"
  assumes "finite (f ` s)"
  shows "(\<forall>x \<in> s. \<exists>e>0. \<forall>y. y \<in> s \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x))"
proof -
  have "\<exists>e>0. \<forall>y. y \<in> s \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" if "x \<in> s" for x
  proof (cases "f ` s - {f x} = {}")
    case True
    with zero_less_numeral show ?thesis
      by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
  next
    case False
    then obtain z where z: "z \<in> s" "f z \<noteq> f x"
      by blast
    have finn: "finite {norm (z - f x) |z. z \<in> f ` s - {f x}}"
      using assms by simp
    then have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` s - {f x}}"
      apply (rule finite_imp_less_Inf)
      using z apply force+
      done
    show ?thesis
      by (force intro!: * cInf_le_finite [OF finn])
  qed
  with assms show ?thesis
    by blast
qed

text\<open>This proof requires the existence of two separate values of the range type.\<close>
lemma finite_range_constant_imp_connected:
  assumes "\<And>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
              \<lbrakk>continuous_on s f; finite(f ` s)\<rbrakk> \<Longrightarrow> \<exists>a. \<forall>x \<in> s. f x = a"
    shows "connected s"
proof -
  { fix t u
    assume clt: "closedin (subtopology euclidean s) t"
       and clu: "closedin (subtopology euclidean s) u"
       and tue: "t \<inter> u = {}" and tus: "t \<union> u = s"
    have conif: "continuous_on s (\<lambda>x. if x \<in> t then 0 else 1)"
      apply (subst tus [symmetric])
      apply (rule continuous_on_cases_local)
      using clt clu tue
      apply (auto simp: tus continuous_on_const)
      done
    have fi: "finite ((\<lambda>x. if x \<in> t then 0 else 1) ` s)"
      by (rule finite_subset [of _ "{0,1}"]) auto
    have "t = {} \<or> u = {}"
      using assms [OF conif fi] tus [symmetric]
      by (auto simp: Ball_def) (metis IntI empty_iff one_neq_zero tue)
  }
  then show ?thesis
    by (simp add: connected_closed_in_eq)
qed

lemma continuous_disconnected_range_constant_eq:
      "(connected s \<longleftrightarrow>
           (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
            \<forall>t. continuous_on s f \<and> f ` s \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})
            \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> s. f x = a)))" (is ?thesis1)
  and continuous_discrete_range_constant_eq:
      "(connected s \<longleftrightarrow>
         (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
          continuous_on s f \<and>
          (\<forall>x \<in> s. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> s \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))
          \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> s. f x = a)))" (is ?thesis2)
  and continuous_finite_range_constant_eq:
      "(connected s \<longleftrightarrow>
         (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
          continuous_on s f \<and> finite (f ` s)
          \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> s. f x = a)))" (is ?thesis3)
proof -
  have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>
    \<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"
    by blast
  have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"
    apply (rule *)
    using continuous_disconnected_range_constant apply metis
    apply clarify
    apply (frule discrete_subset_disconnected; blast)
    apply (blast dest: finite_implies_discrete)
    apply (blast intro!: finite_range_constant_imp_connected)
    done
  then show ?thesis1 ?thesis2 ?thesis3
    by blast+
qed

lemma continuous_discrete_range_constant:
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
  assumes s: "connected s"
      and "continuous_on s f"
      and "\<And>x. x \<in> s \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> s \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
    shows "\<exists>a. \<forall>x \<in> s. f x = a"
  using continuous_discrete_range_constant_eq [THEN iffD1, OF s] assms
  by blast

lemma continuous_finite_range_constant:
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
  assumes "connected s"
      and "continuous_on s f"
      and "finite (f ` s)"
    shows "\<exists>a. \<forall>x \<in> s. f x = a"
  using assms continuous_finite_range_constant_eq
  by blast


text\<open>We can treat even non-rectifiable paths as having a "length" for bounds on analytic functions in open sets.\<close>

subsection\<open>Winding Numbers\<close>

definition winding_number:: "[real \<Rightarrow> complex, complex] \<Rightarrow> complex" where
  "winding_number \<gamma> z \<equiv>
    @n. \<forall>e > 0. \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
                    pathstart p = pathstart \<gamma> \<and>
                    pathfinish p = pathfinish \<gamma> \<and>
                    (\<forall>t \<in> {0..1}. norm(\<gamma> t - p t) < e) \<and>
                    contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * ii * n"

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

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

lemma winding_number_unique_loop:
  assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
      and loop: "pathfinish \<gamma> = pathstart \<gamma>"
      and pi:
        "\<And>e. e>0 \<Longrightarrow> \<exists>p. valid_path p \<and> z \<notin> path_image p \<and>
                           pathfinish p = pathstart p \<and>
                           (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
                           contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * ii * n"
   shows "winding_number \<gamma> z = n"
proof -
  have "path_image \<gamma> \<subseteq> UNIV - {z}"
    using assms by blast
  then obtain e
    where e: "e>0"
      and pi_eq: "\<And>h1 h2 f. \<lbrakk>valid_path h1; valid_path h2;
                    (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < e \<and> cmod (h2 t - \<gamma> t) < e);
                    pathfinish h1 = pathstart h1; pathfinish h2 = pathstart h2; f holomorphic_on UNIV - {z}\<rbrakk> \<Longrightarrow>
                    contour_integral h2 f = contour_integral h1 f"
    using contour_integral_nearby_loops [of "UNIV - {z}" \<gamma>] assms  by (auto simp: open_delete)
  obtain p where p:
     "valid_path p \<and> z \<notin> path_image p \<and>
      pathfinish p = pathstart p \<and>
      (\<forall>t \<in> {0..1}. norm (\<gamma> t - p t) < e) \<and>
      contour_integral p (\<lambda>w. 1/(w - z)) = 2 * pi * ii * n"
    using pi [OF e] by blast
  obtain q where q:
     "valid_path q \<and> z \<notin> path_image q \<and>
      pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma> \<and>
      (\<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"
    using winding_number [OF \<gamma> e] by blast
  have "2 * complex_of_real pi * \<i> * n = contour_integral p (\<lambda>w. 1 / (w - z))"
    using p by auto
  also have "... = contour_integral q (\<lambda>w. 1 / (w - z))"
    apply (rule pi_eq)
    using p q loop
    by (auto simp: valid_path_polynomial_function norm_minus_commute intro!: holomorphic_intros)
  also have "... = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z"
    using q by auto
  finally have "2 * complex_of_real pi * \<i> * n = 2 * complex_of_real pi * \<i> * winding_number \<gamma> z" .
  then show ?thesis
    by simp
qed

lemma winding_number_valid_path:
  assumes "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
    shows "winding_number \<gamma> z = 1/(2*pi*ii) * contour_integral \<gamma> (\<lambda>w. 1/(w - z))"
  using assms by (auto simp: valid_path_imp_path intro!: winding_number_unique)

lemma has_contour_integral_winding_number:
  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
    shows "((\<lambda>w. 1/(w - z)) has_contour_integral (2*pi*ii*winding_number \<gamma> z)) \<gamma>"
by (simp add: winding_number_valid_path has_contour_integral_integral contour_integrable_inversediff assms)

lemma winding_number_trivial [simp]: "z \<noteq> a \<Longrightarrow> winding_number(linepath a a) z = 0"
  by (simp add: winding_number_valid_path)

lemma winding_number_subpath_trivial [simp]: "z \<noteq> g x \<Longrightarrow> winding_number (subpath x x g) z = 0"
  by (simp add: path_image_subpath winding_number_valid_path)

lemma winding_number_join:
  assumes g1: "path g1" "z \<notin> path_image g1"
      and g2: "path g2" "z \<notin> path_image g2"
      and "pathfinish g1 = pathstart g2"
    shows "winding_number(g1 +++ g2) z = winding_number g1 z + winding_number g2 z"
  apply (rule winding_number_unique)
  using assms apply (simp_all add: not_in_path_image_join)
  apply (frule winding_number [OF g2])
  apply (frule winding_number [OF g1], clarify)
  apply (rename_tac p2 p1)
  apply (rule_tac x="p1+++p2" in exI)
  apply (simp add: not_in_path_image_join contour_integrable_inversediff algebra_simps)
  apply (auto simp: joinpaths_def)
  done

lemma winding_number_reversepath:
  assumes "path \<gamma>" "z \<notin> path_image \<gamma>"
    shows "winding_number(reversepath \<gamma>) z = - (winding_number \<gamma> z)"
  apply (rule winding_number_unique)
  using assms
  apply simp_all
  apply (frule winding_number [OF assms], clarify)
  apply (rule_tac x="reversepath p" in exI)
  apply (simp add: contour_integral_reversepath contour_integrable_inversediff valid_path_imp_reverse)
  apply (auto simp: reversepath_def)
  done

lemma winding_number_shiftpath:
  assumes \<gamma>: "path \<gamma>" "z \<notin> path_image \<gamma>"
      and "pathfinish \<gamma> = pathstart \<gamma>" "a \<in> {0..1}"
    shows "winding_number(shiftpath a \<gamma>) z = winding_number \<gamma> z"
  apply (rule winding_number_unique_loop)
  using assms
  apply (simp_all add: path_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath)
  apply (frule winding_number [OF \<gamma>], clarify)
  apply (rule_tac x="shiftpath a p" in exI)
  apply (simp add: contour_integral_shiftpath path_image_shiftpath pathfinish_shiftpath pathstart_shiftpath valid_path_shiftpath)
  apply (auto simp: shiftpath_def)
  done

lemma winding_number_split_linepath:
  assumes "c \<in> closed_segment a b" "z \<notin> closed_segment a b"
    shows "winding_number(linepath a b) z = winding_number(linepath a c) z + winding_number(linepath c b) z"
proof -
  have "z \<notin> closed_segment a c" "z \<notin> closed_segment c b"
    using assms  apply (meson convex_contains_segment convex_segment ends_in_segment(1) subsetCE)
    using assms  by (meson convex_contains_segment convex_segment ends_in_segment(2) subsetCE)
  then show ?thesis
    using assms
    by (simp add: winding_number_valid_path contour_integral_split_linepath [symmetric] continuous_on_inversediff field_simps)
qed

lemma winding_number_cong:
   "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> p t = q t) \<Longrightarrow> winding_number p z = winding_number q z"
  by (simp add: winding_number_def pathstart_def pathfinish_def)

lemma winding_number_offset: "winding_number p z = winding_number (\<lambda>w. p w - z) 0"
  apply (simp add: winding_number_def contour_integral_integral path_image_def valid_path_def pathstart_def pathfinish_def)
  apply (intro ext arg_cong [where f = Eps] arg_cong [where f = All] imp_cong refl, safe)
  apply (rename_tac g)
  apply (rule_tac x="\<lambda>t. g t - z" in exI)
  apply (force simp: vector_derivative_def has_vector_derivative_diff_const piecewise_C1_differentiable_diff C1_differentiable_imp_piecewise)
  apply (rename_tac g)
  apply (rule_tac x="\<lambda>t. g t + z" in exI)
  apply (simp add: piecewise_C1_differentiable_add vector_derivative_def has_vector_derivative_add_const C1_differentiable_imp_piecewise)
  apply (force simp: algebra_simps)
  done

(* A combined theorem deducing several things piecewise.*)
lemma winding_number_join_pos_combined:
     "\<lbrakk>valid_path \<gamma>1; z \<notin> path_image \<gamma>1; 0 < Re(winding_number \<gamma>1 z);
       valid_path \<gamma>2; z \<notin> path_image \<gamma>2; 0 < Re(winding_number \<gamma>2 z); pathfinish \<gamma>1 = pathstart \<gamma>2\<rbrakk>
      \<Longrightarrow> valid_path(\<gamma>1 +++ \<gamma>2) \<and> z \<notin> path_image(\<gamma>1 +++ \<gamma>2) \<and> 0 < Re(winding_number(\<gamma>1 +++ \<gamma>2) z)"
  by (simp add: valid_path_join path_image_join winding_number_join valid_path_imp_path)


(* Useful sufficient conditions for the winding number to be positive etc.*)

lemma Re_winding_number:
    "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>\<rbrakk>
     \<Longrightarrow> Re(winding_number \<gamma> z) = Im(contour_integral \<gamma> (\<lambda>w. 1/(w - z))) / (2*pi)"
by (simp add: winding_number_valid_path field_simps Re_divide power2_eq_square)

lemma winding_number_pos_le:
  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
    shows "0 \<le> Re(winding_number \<gamma> z)"
proof -
  have *: "0 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))" if x: "0 < x" "x < 1" for x
    using ge by (simp add: Complex.Im_divide algebra_simps x)
  show ?thesis
    apply (simp add: Re_winding_number [OF \<gamma>] field_simps)
    apply (rule has_integral_component_nonneg
             [of ii "\<lambda>x. if x \<in> {0<..<1}
                         then 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x) else 0", simplified])
      prefer 3 apply (force simp: *)
     apply (simp add: Basis_complex_def)
    apply (rule has_integral_spike_interior [of 0 1 _ "\<lambda>x. 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x)"])
    apply simp
    apply (simp only: box_real)
    apply (subst has_contour_integral [symmetric])
    using \<gamma>
    apply (simp add: contour_integrable_inversediff has_contour_integral_integral)
    done
qed

lemma winding_number_pos_lt_lemma:
  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
      and e: "0 < e"
      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
    shows "0 < Re(winding_number \<gamma> z)"
proof -
  have "e \<le> Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z)))"
    apply (rule has_integral_component_le
             [of ii "\<lambda>x. ii*e" "ii*e" "{0..1}"
                    "\<lambda>x. if x \<in> {0<..<1} then 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x) else ii*e"
                    "contour_integral \<gamma> (\<lambda>w. 1/(w - z))", simplified])
    using e
    apply (simp_all add: Basis_complex_def)
    using has_integral_const_real [of _ 0 1] apply force
    apply (rule has_integral_spike_interior [of 0 1 _ "\<lambda>x. 1/(\<gamma> x - z) * vector_derivative \<gamma> (at x)", simplified box_real])
    apply simp
    apply (subst has_contour_integral [symmetric])
    using \<gamma>
    apply (simp_all add: contour_integrable_inversediff has_contour_integral_integral ge)
    done
  with e show ?thesis
    by (simp add: Re_winding_number [OF \<gamma>] field_simps)
qed

lemma winding_number_pos_lt:
  assumes \<gamma>: "valid_path \<gamma>" "z \<notin> path_image \<gamma>"
      and e: "0 < e"
      and ge: "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj(\<gamma> x - z))"
    shows "0 < Re (winding_number \<gamma> z)"
proof -
  have bm: "bounded ((\<lambda>w. w - z) ` (path_image \<gamma>))"
    using bounded_translation [of _ "-z"] \<gamma> by (simp add: bounded_valid_path_image)
  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"
    using bounded_pos [THEN iffD1, OF bm] by blast
  { fix x::real  assume x: "0 < x" "x < 1"
    then have B2: "cmod (\<gamma> x - z)^2 \<le> B^2" using Bno [of "\<gamma> x - z"]
      by (simp add: path_image_def power2_eq_square mult_mono')
    with x have "\<gamma> x \<noteq> z" using \<gamma>
      using path_image_def by fastforce
    then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z)) / (cmod (\<gamma> x - z))\<^sup>2"
      using B ge [OF x] B2 e
      apply (rule_tac y="e / (cmod (\<gamma> x - z))\<^sup>2" in order_trans)
      apply (auto simp: divide_left_mono divide_right_mono)
      done
    then have "e / B\<^sup>2 \<le> Im (vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
      by (simp add: Im_divide_Reals complex_div_cnj [of _ "\<gamma> x - z" for x] del: complex_cnj_diff times_complex.sel)
  } note * = this
  show ?thesis
    using e B by (simp add: * winding_number_pos_lt_lemma [OF \<gamma>, of "e/B^2"])
qed

subsection\<open>The winding number is an integer\<close>

text\<open>Proof from the book Complex Analysis by Lars V. Ahlfors, Chapter 4, section 2.1,
     Also on page 134 of Serge Lang's book with the name title, etc.\<close>

lemma exp_fg:
  fixes z::complex
  assumes g: "(g has_vector_derivative g') (at x within s)"
      and f: "(f has_vector_derivative (g' / (g x - z))) (at x within s)"
      and z: "g x \<noteq> z"
    shows "((\<lambda>x. exp(-f x) * (g x - z)) has_vector_derivative 0) (at x within s)"
proof -
  have *: "(exp o (\<lambda>x. (- f x)) has_vector_derivative - (g' / (g x - z)) * exp (- f x)) (at x within s)"
    using assms unfolding has_vector_derivative_def scaleR_conv_of_real
    by (auto intro!: derivative_eq_intros)
  show ?thesis
    apply (rule has_vector_derivative_eq_rhs)
    apply (rule bounded_bilinear.has_vector_derivative [OF bounded_bilinear_mult])
    using z
    apply (auto simp: intro!: derivative_eq_intros * [unfolded o_def] g)
    done
qed

lemma winding_number_exp_integral:
  fixes z::complex
  assumes \<gamma>: "\<gamma> piecewise_C1_differentiable_on {a..b}"
      and ab: "a \<le> b"
      and z: "z \<notin> \<gamma> ` {a..b}"
    shows "(\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)) integrable_on {a..b}"
          (is "?thesis1")
          "exp (- (integral {a..b} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))) * (\<gamma> b - z) = \<gamma> a - z"
          (is "?thesis2")
proof -
  let ?D\<gamma> = "\<lambda>x. vector_derivative \<gamma> (at x)"
  have [simp]: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<gamma> x \<noteq> z"
    using z by force
  have cong: "continuous_on {a..b} \<gamma>"
    using \<gamma> by (simp add: piecewise_C1_differentiable_on_def)
  obtain k where fink: "finite k" and g_C1_diff: "\<gamma> C1_differentiable_on ({a..b} - k)"
    using \<gamma> by (force simp: piecewise_C1_differentiable_on_def)
  have o: "open ({a<..<b} - k)"
    using \<open>finite k\<close> by (simp add: finite_imp_closed open_Diff)
  moreover have "{a<..<b} - k \<subseteq> {a..b} - k"
    by force
  ultimately have g_diff_at: "\<And>x. \<lbrakk>x \<notin> k; x \<in> {a<..<b}\<rbrakk> \<Longrightarrow> \<gamma> differentiable at x"
    by (metis Diff_iff differentiable_on_subset C1_diff_imp_diff [OF g_C1_diff] differentiable_on_def differentiable_within_open)
  { fix w
    assume "w \<noteq> z"
    have "continuous_on (ball w (cmod (w - z))) (\<lambda>w. 1 / (w - z))"
      by (auto simp: dist_norm intro!: continuous_intros)
    moreover have "\<And>x. cmod (w - x) < cmod (w - z) \<Longrightarrow> \<exists>f'. ((\<lambda>w. 1 / (w - z)) has_field_derivative f') (at x)"
      by (auto simp: intro!: derivative_eq_intros)
    ultimately have "\<exists>h. \<forall>y. norm(y - w) < norm(w - z) \<longrightarrow> (h has_field_derivative 1/(y - z)) (at y)"
      using holomorphic_convex_primitive [of "ball w (norm(w - z))" "{}" "\<lambda>w. 1/(w - z)"]
      by (simp add: complex_differentiable_def Ball_def dist_norm at_within_open_NO_MATCH norm_minus_commute)
  }
  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)"
    by meson
  have exy: "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * ?D\<gamma> x) has_integral y) {a..b}"
    unfolding integrable_on_def [symmetric]
    apply (rule contour_integral_local_primitive_any [OF piecewise_C1_imp_differentiable [OF \<gamma>], of "-{z}"])
    apply (rename_tac w)
    apply (rule_tac x="norm(w - z)" in exI)
    apply (simp_all add: inverse_eq_divide)
    apply (metis has_field_derivative_at_within h)
    done
  have vg_int: "(\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)) integrable_on {a..b}"
    unfolding box_real [symmetric] divide_inverse_commute
    by (auto intro!: exy integrable_subinterval simp add: integrable_on_def ab)
  with ab show ?thesis1
    by (simp add: divide_inverse_commute integral_def integrable_on_def)
  { fix t
    assume t: "t \<in> {a..b}"
    have cball: "continuous_on (ball (\<gamma> t) (dist (\<gamma> t) z)) (\<lambda>x. inverse (x - z))"
        using z by (auto intro!: continuous_intros simp: dist_norm)
    have icd: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow> (\<lambda>w. inverse (w - z)) complex_differentiable at x"
      unfolding complex_differentiable_def by (force simp: intro!: derivative_eq_intros)
    obtain h where h: "\<And>x. cmod (\<gamma> t - x) < cmod (\<gamma> t - z) \<Longrightarrow>
                       (h has_field_derivative inverse (x - z)) (at x within {y. cmod (\<gamma> t - y) < cmod (\<gamma> t - z)})"
      using holomorphic_convex_primitive [where f = "\<lambda>w. inverse(w - z)", OF convex_ball finite.emptyI cball icd]
      by simp (auto simp: ball_def dist_norm that)
    { fix x D
      assume x: "x \<notin> k" "a < x" "x < b"
      then have "x \<in> interior ({a..b} - k)"
        using open_subset_interior [OF o] by fastforce
      then have con: "isCont (\<lambda>x. ?D\<gamma> x) x"
        using g_C1_diff x by (auto simp: C1_differentiable_on_eq intro: continuous_on_interior)
      then have con_vd: "continuous (at x within {a..b}) (\<lambda>x. ?D\<gamma> x)"
        by (rule continuous_at_imp_continuous_within)
      have gdx: "\<gamma> differentiable at x"
        using x by (simp add: g_diff_at)
      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))
          (at x within {a..b})"
        using x gdx t
        apply (clarsimp simp add: differentiable_iff_scaleR)
        apply (rule exp_fg [unfolded has_vector_derivative_def, simplified], blast intro: has_derivative_at_within)
        apply (simp_all add: has_vector_derivative_def [symmetric])
        apply (rule has_vector_derivative_eq_rhs [OF integral_has_vector_derivative_continuous_at])
        apply (rule con_vd continuous_intros cong vg_int | simp add: continuous_at_imp_continuous_within has_vector_derivative_continuous vector_derivative_at)+
        done
      } note * = this
    have "exp (- (integral {a..t} (\<lambda>x. ?D\<gamma> x / (\<gamma> x - z)))) * (\<gamma> t - z) =\<gamma> a - z"
      apply (rule has_derivative_zero_unique_strong_interval [of "{a,b} \<union> k" a b])
      using t
      apply (auto intro!: * continuous_intros fink cong indefinite_integral_continuous [OF vg_int]  simp add: ab)+
      done
   }
  with ab show ?thesis2
    by (simp add: divide_inverse_commute integral_def)
qed

corollary winding_number_exp_2pi:
    "\<lbrakk>path p; z \<notin> path_image p\<rbrakk>
     \<Longrightarrow> pathfinish p - z = exp (2 * pi * ii * winding_number p z) * (pathstart p - z)"
using winding_number [of p z 1] unfolding valid_path_def path_image_def pathstart_def pathfinish_def
  by (force dest: winding_number_exp_integral(2) [of _ 0 1 z] simp: field_simps contour_integral_integral exp_minus)


subsection\<open>The version with complex integers and equality\<close>

lemma integer_winding_number_eq:
  assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
  shows "winding_number \<gamma> z \<in> \<int> \<longleftrightarrow> pathfinish \<gamma> = pathstart \<gamma>"
proof -
  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)"
      by (simp add: field_simps) algebra
  obtain p where p: "valid_path p" "z \<notin> path_image p"
                    "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
                    "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
    using winding_number [OF assms, of 1] by auto
  have [simp]: "(winding_number \<gamma> z \<in> \<int>) = (exp (contour_integral p (\<lambda>w. 1 / (w - z))) = 1)"
      using p by (simp add: exp_eq_1 complex_is_Int_iff)
  have "winding_number p z \<in> \<int> \<longleftrightarrow> pathfinish p = pathstart p"
    using p z
    apply (simp add: winding_number_valid_path valid_path_def path_image_def pathstart_def pathfinish_def)
    using winding_number_exp_integral(2) [of p 0 1 z]
    apply (simp add: field_simps contour_integral_integral exp_minus)
    apply (rule *)
    apply (auto simp: path_image_def field_simps)
    done
  then show ?thesis using p
    by (auto simp: winding_number_valid_path)
qed

theorem integer_winding_number:
  "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> path_image \<gamma>\<rbrakk> \<Longrightarrow> winding_number \<gamma> z \<in> \<int>"
by (metis integer_winding_number_eq)


text\<open>If the winding number's magnitude is at least one, then the path must contain points in every direction.*)
   We can thus bound the winding number of a path that doesn't intersect a given ray. \<close>

lemma winding_number_pos_meets:
  fixes z::complex
  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and 1: "Re (winding_number \<gamma> z) \<ge> 1"
      and w: "w \<noteq> z"
  shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
proof -
  have [simp]: "\<And>x. 0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> \<gamma> x \<noteq> z"
    using z by (auto simp: path_image_def)
  have [simp]: "z \<notin> \<gamma> ` {0..1}"
    using path_image_def z by auto
  have gpd: "\<gamma> piecewise_C1_differentiable_on {0..1}"
    using \<gamma> valid_path_def by blast
  def r \<equiv> "(w - z) / (\<gamma> 0 - z)"
  have [simp]: "r \<noteq> 0"
    using w z by (auto simp: r_def)
  have "Arg r \<le> 2*pi"
    by (simp add: Arg less_eq_real_def)
  also have "... \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))"
    using 1
    apply (simp add: winding_number_valid_path [OF \<gamma> z] Cauchy_Integral_Thm.contour_integral_integral)
    apply (simp add: Complex.Re_divide field_simps power2_eq_square)
    done
  finally have "Arg r \<le> Im (integral {0..1} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z)))" .
  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"
    apply (simp add:)
    apply (rule Topological_Spaces.IVT')
    apply (simp_all add: Complex_Transcendental.Arg_ge_0)
    apply (intro continuous_intros indefinite_integral_continuous winding_number_exp_integral [OF gpd]; simp)
    done
  then obtain t where t:     "t \<in> {0..1}"
                  and eqArg: "Im (integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x)/(\<gamma> x - z))) = Arg r"
    by blast
  def i \<equiv> "integral {0..t} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))"
  have iArg: "Arg r = Im i"
    using eqArg by (simp add: i_def)
  have gpdt: "\<gamma> piecewise_C1_differentiable_on {0..t}"
    by (metis atLeastAtMost_iff atLeastatMost_subset_iff order_refl piecewise_C1_differentiable_on_subset gpd t)
  have "exp (- i) * (\<gamma> t - z) = \<gamma> 0 - z"
    unfolding i_def
    apply (rule winding_number_exp_integral [OF gpdt])
    using t z unfolding path_image_def
    apply force+
    done
  then have *: "\<gamma> t - z = exp i * (\<gamma> 0 - z)"
    by (simp add: exp_minus field_simps)
  then have "(w - z) = r * (\<gamma> 0 - z)"
    by (simp add: r_def)
  then have "z + complex_of_real (exp (Re i)) * (w - z) / complex_of_real (cmod r) = \<gamma> t"
    apply (simp add:)
    apply (subst Complex_Transcendental.Arg_eq [of r])
    apply (simp add: iArg)
    using *
    apply (simp add: exp_eq_polar field_simps)
    done
  with t show ?thesis
    by (rule_tac x="exp(Re i) / norm r" in exI) (auto simp: path_image_def)
qed

lemma winding_number_big_meets:
  fixes z::complex
  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "\<bar>Re (winding_number \<gamma> z)\<bar> \<ge> 1"
      and w: "w \<noteq> z"
  shows "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image \<gamma>"
proof -
  { assume "Re (winding_number \<gamma> z) \<le> - 1"
    then have "Re (winding_number (reversepath \<gamma>) z) \<ge> 1"
      by (simp add: \<gamma> valid_path_imp_path winding_number_reversepath z)
    moreover have "valid_path (reversepath \<gamma>)"
      using \<gamma> valid_path_imp_reverse by auto
    moreover have "z \<notin> path_image (reversepath \<gamma>)"
      by (simp add: z)
    ultimately have "\<exists>a::real. 0 < a \<and> z + a*(w - z) \<in> path_image (reversepath \<gamma>)"
      using winding_number_pos_meets w by blast
    then have ?thesis
      by simp
  }
  then show ?thesis
    using assms
    by (simp add: Groups.abs_if_class.abs_if winding_number_pos_meets split: split_if_asm)
qed

lemma winding_number_less_1:
  fixes z::complex
  shows
  "\<lbrakk>valid_path \<gamma>; z \<notin> path_image \<gamma>; w \<noteq> z;
    \<And>a::real. 0 < a \<Longrightarrow> z + a*(w - z) \<notin> path_image \<gamma>\<rbrakk>
   \<Longrightarrow> \<bar>Re(winding_number \<gamma> z)\<bar> < 1"
   by (auto simp: not_less dest: winding_number_big_meets)

text\<open>One way of proving that WN=1 for a loop.\<close>
lemma winding_number_eq_1:
  fixes z::complex
  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
      and 0: "0 < Re(winding_number \<gamma> z)" and 2: "Re(winding_number \<gamma> z) < 2"
  shows "winding_number \<gamma> z = 1"
proof -
  have "winding_number \<gamma> z \<in> Ints"
    by (simp add: \<gamma> integer_winding_number loop valid_path_imp_path z)
  then show ?thesis
    using 0 2 by (auto simp: Ints_def)
qed


subsection\<open>Continuity of winding number and invariance on connected sets.\<close>

lemma continuous_at_winding_number:
  fixes z::complex
  assumes \<gamma>: "path \<gamma>" and z: "z \<notin> path_image \<gamma>"
  shows "continuous (at z) (winding_number \<gamma>)"
proof -
  obtain e where "e>0" and cbg: "cball z e \<subseteq> - path_image \<gamma>"
    using open_contains_cball [of "- path_image \<gamma>"]  z
    by (force simp: closed_def [symmetric] closed_path_image [OF \<gamma>])
  then have ppag: "path_image \<gamma> \<subseteq> - cball z (e/2)"
    by (force simp: cball_def dist_norm)
  have oc: "open (- cball z (e / 2))"
    by (simp add: closed_def [symmetric])
  obtain d where "d>0" and pi_eq:
    "\<And>h1 h2. \<lbrakk>valid_path h1; valid_path h2;
              (\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d);
              pathstart h2 = pathstart h1; pathfinish h2 = pathfinish h1\<rbrakk>
             \<Longrightarrow>
               path_image h1 \<subseteq> - cball z (e / 2) \<and>
               path_image h2 \<subseteq> - cball z (e / 2) \<and>
               (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
    using contour_integral_nearby_ends [OF oc \<gamma> ppag] by metis
  obtain p where p: "valid_path p" "z \<notin> path_image p"
                    "pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma>"
              and pg: "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (\<gamma> t - p t) < min d e / 2"
              and pi: "contour_integral p (\<lambda>x. 1 / (x - z)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z"
    using winding_number [OF \<gamma> z, of "min d e / 2"] \<open>d>0\<close> \<open>e>0\<close> by auto
  { fix w
    assume d2: "cmod (w - z) < d/2" and e2: "cmod (w - z) < e/2"
    then have wnotp: "w \<notin> path_image p"
      using cbg \<open>d>0\<close> \<open>e>0\<close>
      apply (simp add: path_image_def cball_def dist_norm, clarify)
      apply (frule pg)
      apply (drule_tac c="\<gamma> x" in subsetD)
      apply (auto simp: less_eq_real_def norm_minus_commute norm_triangle_half_l)
      done
    have wnotg: "w \<notin> path_image \<gamma>"
      using cbg e2 \<open>e>0\<close> by (force simp: dist_norm norm_minus_commute)
    { fix k::real
      assume k: "k>0"
      then obtain q where q: "valid_path q" "w \<notin> path_image q"
                             "pathstart q = pathstart \<gamma> \<and> pathfinish q = pathfinish \<gamma>"
                    and qg: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (\<gamma> t - q t) < min k (min d e) / 2"
                    and qi: "contour_integral q (\<lambda>u. 1 / (u - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
        using winding_number [OF \<gamma> wnotg, of "min k (min d e) / 2"] \<open>d>0\<close> \<open>e>0\<close> k
        by (force simp: min_divide_distrib_right)
      have "contour_integral p (\<lambda>u. 1 / (u - w)) = contour_integral q (\<lambda>u. 1 / (u - w))"
        apply (rule pi_eq [OF \<open>valid_path q\<close> \<open>valid_path p\<close>, THEN conjunct2, THEN conjunct2, rule_format])
        apply (frule pg)
        apply (frule qg)
        using p q \<open>d>0\<close> e2
        apply (auto simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
        done
      then have "contour_integral p (\<lambda>x. 1 / (x - w)) = complex_of_real (2 * pi) * \<i> * winding_number \<gamma> w"
        by (simp add: pi qi)
    } note pip = this
    have "path p"
      using p by (simp add: valid_path_imp_path)
    then have "winding_number p w = winding_number \<gamma> w"
      apply (rule winding_number_unique [OF _ wnotp])
      apply (rule_tac x=p in exI)
      apply (simp add: p wnotp min_divide_distrib_right pip)
      done
  } note wnwn = this
  obtain pe where "pe>0" and cbp: "cball z (3 / 4 * pe) \<subseteq> - path_image p"
    using p open_contains_cball [of "- path_image p"]
    by (force simp: closed_def [symmetric] closed_path_image [OF valid_path_imp_path])
  obtain L
    where "L>0"
      and L: "\<And>f B. \<lbrakk>f holomorphic_on - cball z (3 / 4 * pe);
                      \<forall>z \<in> - cball z (3 / 4 * pe). cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
                      cmod (contour_integral p f) \<le> L * B"
    using contour_integral_bound_exists [of "- cball z (3/4*pe)" p] cbp \<open>valid_path p\<close> by force
  { fix e::real and w::complex
    assume e: "0 < e" and w: "cmod (w - z) < pe/4" "cmod (w - z) < e * pe\<^sup>2 / (8 * L)"
    then have [simp]: "w \<notin> path_image p"
      using cbp p(2) \<open>0 < pe\<close>
      by (force simp: dist_norm norm_minus_commute path_image_def cball_def)
    have [simp]: "contour_integral p (\<lambda>x. 1/(x - w)) - contour_integral p (\<lambda>x. 1/(x - z)) =
                  contour_integral p (\<lambda>x. 1/(x - w) - 1/(x - z))"
      by (simp add: p contour_integrable_inversediff contour_integral_diff)
    { fix x
      assume pe: "3/4 * pe < cmod (z - x)"
      have "cmod (w - x) < pe/4 + cmod (z - x)"
        by (meson add_less_cancel_right norm_diff_triangle_le order_refl order_trans_rules(21) w(1))
      then have wx: "cmod (w - x) < 4/3 * cmod (z - x)" using pe by simp
      have "cmod (z - x) \<le> cmod (z - w) + cmod (w - x)"
        using norm_diff_triangle_le by blast
      also have "... < pe/4 + cmod (w - x)"
        using w by (simp add: norm_minus_commute)
      finally have "pe/2 < cmod (w - x)"
        using pe by auto
      then have "(pe/2)^2 < cmod (w - x) ^ 2"
        apply (rule power_strict_mono)
        using \<open>pe>0\<close> by auto
      then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
        by (simp add: power_divide)
      have "8 * L * cmod (w - z) < e * pe\<^sup>2"
        using w \<open>L>0\<close> by (simp add: field_simps)
      also have "... < e * 4 * cmod (w - x) * cmod (w - x)"
        using pe2 \<open>e>0\<close> by (simp add: power2_eq_square)
      also have "... < e * 4 * cmod (w - x) * (4/3 * cmod (z - x))"
        using wx
        apply (rule mult_strict_left_mono)
        using pe2 e not_less_iff_gr_or_eq by fastforce
      finally have "L * cmod (w - z) < 2/3 * e * cmod (w - x) * cmod (z - x)"
        by simp
      also have "... \<le> e * cmod (w - x) * cmod (z - x)"
         using e by simp
      finally have Lwz: "L * cmod (w - z) < e * cmod (w - x) * cmod (z - x)" .
      have "L * cmod (1 / (x - w) - 1 / (x - z)) \<le> e"
        apply (cases "x=z \<or> x=w")
        using pe \<open>pe>0\<close> w \<open>L>0\<close>
        apply (force simp: norm_minus_commute)
        using wx w(2) \<open>L>0\<close> pe pe2 Lwz
        apply (auto simp: divide_simps mult_less_0_iff norm_minus_commute norm_divide norm_mult power2_eq_square)
        done
    } note L_cmod_le = this
    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)"
      apply (rule L)
      using \<open>pe>0\<close> w
      apply (force simp: dist_norm norm_minus_commute intro!: holomorphic_intros)
      using \<open>pe>0\<close> w \<open>L>0\<close>
      apply (auto simp: cball_def dist_norm field_simps L_cmod_le  simp del: less_divide_eq_numeral1 le_divide_eq_numeral1)
      done
    have "cmod (contour_integral p (\<lambda>x. 1 / (x - w)) - contour_integral p (\<lambda>x. 1 / (x - z))) < 2*e"
      apply (simp add:)
      apply (rule le_less_trans [OF *])
      using \<open>L>0\<close> e
      apply (force simp: field_simps)
      done
    then have "cmod (winding_number p w - winding_number p z) < e"
      using pi_ge_two e
      by (force simp: winding_number_valid_path p field_simps norm_divide norm_mult intro: less_le_trans)
  } note cmod_wn_diff = this
  then have "isCont (winding_number p) z"
    apply (simp add: continuous_at_eps_delta, clarify)
    apply (rule_tac x="min (pe/4) (e/2*pe^2/L/4)" in exI)
    using \<open>pe>0\<close> \<open>L>0\<close>
    apply (simp add: dist_norm cmod_wn_diff)
    done
  then show ?thesis
    apply (rule continuous_transform_within [where d = "min d e / 2"])
    apply (auto simp: \<open>d>0\<close> \<open>e>0\<close> dist_norm wnwn)
    done
qed

corollary continuous_on_winding_number:
    "path \<gamma> \<Longrightarrow> continuous_on (- path_image \<gamma>) (\<lambda>w. winding_number \<gamma> w)"
  by (simp add: continuous_at_imp_continuous_on continuous_at_winding_number)


subsection\<open>The winding number is constant on a connected region\<close>

lemma winding_number_constant:
  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and cs: "connected s" and sg: "s \<inter> path_image \<gamma> = {}"
    shows "\<exists>k. \<forall>z \<in> s. winding_number \<gamma> z = k"
proof -
  { fix y z
    assume ne: "winding_number \<gamma> y \<noteq> winding_number \<gamma> z"
    assume "y \<in> s" "z \<in> s"
    then have "winding_number \<gamma> y \<in> \<int>"  "winding_number \<gamma> z \<in>  \<int>"
      using integer_winding_number [OF \<gamma> loop] sg \<open>y \<in> s\<close> by auto
    with ne have "1 \<le> cmod (winding_number \<gamma> y - winding_number \<gamma> z)"
      by (auto simp: Ints_def of_int_diff [symmetric] simp del: of_int_diff)
  } note * = this
  show ?thesis
    apply (rule continuous_discrete_range_constant [OF cs])
    using continuous_on_winding_number [OF \<gamma>] sg
    apply (metis Diff_Compl Diff_eq_empty_iff continuous_on_subset)
    apply (rule_tac x=1 in exI)
    apply (auto simp: *)
    done
qed

lemma winding_number_eq:
     "\<lbrakk>path \<gamma>; pathfinish \<gamma> = pathstart \<gamma>; w \<in> s; z \<in> s; connected s; s \<inter> path_image \<gamma> = {}\<rbrakk>
      \<Longrightarrow> winding_number \<gamma> w = winding_number \<gamma> z"
using winding_number_constant by fastforce

lemma open_winding_number_levelsets:
  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
    shows "open {z. z \<notin> path_image \<gamma> \<and> winding_number \<gamma> z = k}"
proof -
  have op: "open (- path_image \<gamma>)"
    by (simp add: closed_path_image \<gamma> open_Compl)
  { fix z assume z: "z \<notin> path_image \<gamma>" and k: "k = winding_number \<gamma> z"
    obtain e where e: "e>0" "ball z e \<subseteq> - path_image \<gamma>"
      using open_contains_ball [of "- path_image \<gamma>"] op z
      by blast
    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"
      apply (rule_tac x=e in exI)
      using e apply (simp add: dist_norm ball_def norm_minus_commute)
      apply (auto simp: dist_norm norm_minus_commute intro!: winding_number_eq [OF assms, where s = "ball z e"])
      done
  } then
  show ?thesis
    by (auto simp: open_dist)
qed

subsection\<open>Winding number is zero "outside" a curve, in various senses\<close>

lemma winding_number_zero_in_outside:
  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>" and z: "z \<in> outside (path_image \<gamma>)"
    shows "winding_number \<gamma> z = 0"
proof -
  obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
    using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
  obtain w::complex where w: "w \<notin> ball 0 (B + 1)"
    by (metis abs_of_nonneg le_less less_irrefl mem_ball_0 norm_of_real)
  have "- ball 0 (B + 1) \<subseteq> outside (path_image \<gamma>)"
    apply (rule outside_subset_convex)
    using B subset_ball by auto
  then have wout: "w \<in> outside (path_image \<gamma>)"
    using w by blast
  moreover obtain k where "\<And>z. z \<in> outside (path_image \<gamma>) \<Longrightarrow> winding_number \<gamma> z = k"
    using winding_number_constant [OF \<gamma> loop, of "outside(path_image \<gamma>)"] connected_outside
    by (metis DIM_complex bounded_path_image dual_order.refl \<gamma> outside_no_overlap)
  ultimately have "winding_number \<gamma> z = winding_number \<gamma> w"
    using z by blast
  also have "... = 0"
  proof -
    have wnot: "w \<notin> path_image \<gamma>"  using wout by (simp add: outside_def)
    { fix e::real assume "0<e"
      obtain p where p: "polynomial_function p" "pathstart p = pathstart \<gamma>" "pathfinish p = pathfinish \<gamma>"
                 and pg1: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < 1)"
                 and pge: "(\<And>t. \<lbrakk>0 \<le> t; t \<le> 1\<rbrakk> \<Longrightarrow> cmod (p t - \<gamma> t) < e)"
        using path_approx_polynomial_function [OF \<gamma>, of "min 1 e"] \<open>e>0\<close> by force
      have pip: "path_image p \<subseteq> ball 0 (B + 1)"
        using B
        apply (clarsimp simp add: path_image_def dist_norm ball_def)
        apply (frule (1) pg1)
        apply (fastforce dest: norm_add_less)
        done
      then have "w \<notin> path_image p"  using w by blast
      then have "\<exists>p. valid_path p \<and> w \<notin> path_image p \<and>
                     pathstart p = pathstart \<gamma> \<and> pathfinish p = pathfinish \<gamma> \<and>
                     (\<forall>t\<in>{0..1}. cmod (\<gamma> t - p t) < e) \<and> contour_integral p (\<lambda>wa. 1 / (wa - w)) = 0"
        apply (rule_tac x=p in exI)
        apply (simp add: p valid_path_polynomial_function)
        apply (intro conjI)
        using pge apply (simp add: norm_minus_commute)
        apply (rule contour_integral_unique [OF Cauchy_theorem_convex_simple [OF _ convex_ball [of 0 "B+1"]]])
        apply (rule holomorphic_intros | simp add: dist_norm)+
        using mem_ball_0 w apply blast
        using p apply (simp_all add: valid_path_polynomial_function loop pip)
        done
    }
    then show ?thesis
      by (auto intro: winding_number_unique [OF \<gamma>] simp add: wnot)
  qed
  finally show ?thesis .
qed

lemma winding_number_zero_outside:
    "\<lbrakk>path \<gamma>; convex s; pathfinish \<gamma> = pathstart \<gamma>; z \<notin> s; path_image \<gamma> \<subseteq> s\<rbrakk> \<Longrightarrow> winding_number \<gamma> z = 0"
  by (meson convex_in_outside outside_mono subsetCE winding_number_zero_in_outside)

lemma winding_number_zero_at_infinity:
  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
    shows "\<exists>B. \<forall>z. B \<le> norm z \<longrightarrow> winding_number \<gamma> z = 0"
proof -
  obtain B::real where "0 < B" and B: "path_image \<gamma> \<subseteq> ball 0 B"
    using bounded_subset_ballD [OF bounded_path_image [OF \<gamma>]] by auto
  then show ?thesis
    apply (rule_tac x="B+1" in exI, clarify)
    apply (rule winding_number_zero_outside [OF \<gamma> convex_cball [of 0 B] loop])
    apply (meson less_add_one mem_cball_0 not_le order_trans)
    using ball_subset_cball by blast
qed

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


text\<open>If a path winds round a set, it winds rounds its inside.\<close>
lemma winding_number_around_inside:
  assumes \<gamma>: "path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
      and cls: "closed s" and cos: "connected s" and s_disj: "s \<inter> path_image \<gamma> = {}"
      and z: "z \<in> s" and wn_nz: "winding_number \<gamma> z \<noteq> 0" and w: "w \<in> s \<union> inside s"
    shows "winding_number \<gamma> w = winding_number \<gamma> z"
proof -
  have ssb: "s \<subseteq> inside(path_image \<gamma>)"
  proof
    fix x :: complex
    assume "x \<in> s"
    hence "x \<notin> path_image \<gamma>"
      by (meson disjoint_iff_not_equal s_disj)
    thus "x \<in> inside (path_image \<gamma>)"
      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)
qed
  show ?thesis
    apply (rule winding_number_eq [OF \<gamma> loop w])
    using z apply blast
    apply (simp add: cls connected_with_inside cos)
    apply (simp add: Int_Un_distrib2 s_disj, safe)
    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)
 qed


text\<open>Bounding a WN by 1/2 for a path and point in opposite halfspaces.\<close>
lemma winding_number_subpath_continuous:
  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
    shows "continuous_on {0..1} (\<lambda>x. winding_number(subpath 0 x \<gamma>) z)"
proof -
  have *: "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
         winding_number (subpath 0 x \<gamma>) z"
         if x: "0 \<le> x" "x \<le> 1" for x
  proof -
    have "integral {0..x} (\<lambda>t. vector_derivative \<gamma> (at t) / (\<gamma> t - z)) / (2 * of_real pi * \<i>) =
          1 / (2*pi*ii) * contour_integral (subpath 0 x \<gamma>) (\<lambda>w. 1/(w - z))"
      using assms x
      apply (simp add: contour_integral_subcontour_integral [OF contour_integrable_inversediff])
      done
    also have "... = winding_number (subpath 0 x \<gamma>) z"
      apply (subst winding_number_valid_path)
      using assms x
      apply (simp_all add: path_image_subpath valid_path_subpath)
      by (force simp: path_image_def)
    finally show ?thesis .
  qed
  show ?thesis
    apply (rule continuous_on_eq
                 [where f = "\<lambda>x. 1 / (2*pi*ii) *
                                 integral {0..x} (\<lambda>t. 1/(\<gamma> t - z) * vector_derivative \<gamma> (at t))"])
    apply (rule continuous_intros)+
    apply (rule indefinite_integral_continuous)
    apply (rule contour_integrable_inversediff [OF assms, unfolded contour_integrable_on])
      using assms
    apply (simp add: *)
    done
qed

lemma winding_number_ivt_pos:
    assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "0 \<le> w" "w \<le> Re(winding_number \<gamma> z)"
      shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
  apply (rule ivt_increasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right])
  apply (simp add:)
  apply (rule winding_number_subpath_continuous [OF \<gamma> z])
  using assms
  apply (auto simp: path_image_def image_def)
  done

lemma winding_number_ivt_neg:
    assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>" and "Re(winding_number \<gamma> z) \<le> w" "w \<le> 0"
      shows "\<exists>t \<in> {0..1}. Re(winding_number(subpath 0 t \<gamma>) z) = w"
  apply (rule ivt_decreasing_component_on_1 [of 0 1, where ?k = "1::complex", simplified complex_inner_1_right])
  apply (simp add:)
  apply (rule winding_number_subpath_continuous [OF \<gamma> z])
  using assms
  apply (auto simp: path_image_def image_def)
  done

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

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

lemma winding_number_lt_half:
  assumes "valid_path \<gamma>" "a \<bullet> z \<le> b" "path_image \<gamma> \<subseteq> {w. a \<bullet> w > b}"
    shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1/2"
proof -
  have "z \<notin> path_image \<gamma>" using assms by auto
  with assms show ?thesis
    apply (simp add: winding_number_lt_half_lemma abs_if del: less_divide_eq_numeral1)
    apply (metis complex_inner_1_right winding_number_lt_half_lemma [OF valid_path_imp_reverse, of \<gamma> z a b]
                 winding_number_reversepath valid_path_imp_path inner_minus_left path_image_reversepath)
    done
qed

lemma winding_number_le_half:
  assumes \<gamma>: "valid_path \<gamma>" and z: "z \<notin> path_image \<gamma>"
      and anz: "a \<noteq> 0" and azb: "a \<bullet> z \<le> b" and pag: "path_image \<gamma> \<subseteq> {w. a \<bullet> w \<ge> b}"
    shows "\<bar>Re (winding_number \<gamma> z)\<bar> \<le> 1/2"
proof -
  { assume wnz_12: "\<bar>Re (winding_number \<gamma> z)\<bar> > 1/2"
    have "isCont (winding_number \<gamma>) z"
      by (metis continuous_at_winding_number valid_path_imp_path \<gamma> z)
    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"
      using continuous_at_eps_delta wnz_12 diff_gt_0_iff_gt by blast
    def z' \<equiv> "z - (d / (2 * cmod a)) *\<^sub>R a"
    have *: "a \<bullet> z' \<le> b - d / 3 * cmod a"
      unfolding z'_def inner_mult_right' divide_inverse
      apply (simp add: divide_simps algebra_simps dot_square_norm power2_eq_square anz)
      apply (metis \<open>0 < d\<close> add_increasing azb less_eq_real_def mult_nonneg_nonneg mult_right_mono norm_ge_zero norm_numeral)
      done
    have "cmod (winding_number \<gamma> z' - winding_number \<gamma> z) < \<bar>Re (winding_number \<gamma> z)\<bar> - 1/2"
      using d [of z'] anz \<open>d>0\<close> by (simp add: dist_norm z'_def)
    then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - cmod (winding_number \<gamma> z' - winding_number \<gamma> z)"
      by simp
    then have "1/2 < \<bar>Re (winding_number \<gamma> z)\<bar> - \<bar>Re (winding_number \<gamma> z') - Re (winding_number \<gamma> z)\<bar>"
      using abs_Re_le_cmod [of "winding_number \<gamma> z' - winding_number \<gamma> z"] by simp
    then have wnz_12': "\<bar>Re (winding_number \<gamma> z')\<bar> > 1/2"
      by linarith
    moreover have "\<bar>Re (winding_number \<gamma> z')\<bar> < 1/2"
      apply (rule winding_number_lt_half [OF \<gamma> *])
      using azb \<open>d>0\<close> pag
      apply (auto simp: add_strict_increasing anz divide_simps algebra_simps dest!: subsetD)
      done
    ultimately have False
      by simp
  }
  then show ?thesis by force
qed

lemma winding_number_lt_half_linepath: "z \<notin> closed_segment a b \<Longrightarrow> \<bar>Re (winding_number (linepath a b) z)\<bar> < 1/2"
  using separating_hyperplane_closed_point [of "closed_segment a b" z]
  apply auto
  apply (simp add: closed_segment_def)
  apply (drule less_imp_le)
  apply (frule winding_number_lt_half [OF valid_path_linepath [of a b]])
  apply (auto simp: segment)
  done


text\<open> Positivity of WN for a linepath.\<close>
lemma winding_number_linepath_pos_lt:
    assumes "0 < Im ((b - a) * cnj (b - z))"
      shows "0 < Re(winding_number(linepath a b) z)"
proof -
  have z: "z \<notin> path_image (linepath a b)"
    using assms
    by (simp add: closed_segment_def) (force simp: algebra_simps)
  show ?thesis
    apply (rule winding_number_pos_lt [OF valid_path_linepath z assms])
    apply (simp add: linepath_def algebra_simps)
    done
qed


subsection\<open>Cauchy's integral formula, again for a convex enclosing set.\<close>

lemma Cauchy_integral_formula_weak:
    assumes s: "convex s" and "finite k" and conf: "continuous_on s f"
        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x)"
        and z: "z \<in> interior s - k" and vpg: "valid_path \<gamma>"
        and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
proof -
  obtain f' where f': "(f has_field_derivative f') (at z)"
    using fcd [OF z] by (auto simp: complex_differentiable_def)
  have pas: "path_image \<gamma> \<subseteq> s" and znotin: "z \<notin> path_image \<gamma>" using pasz by blast+
  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
  proof (cases "x = z")
    case True then show ?thesis
      apply (simp add: continuous_within)
      apply (rule Lim_transform_away_within [of _ "z+1" _ "\<lambda>w::complex. (f w - f z)/(w - z)"])
      using has_field_derivative_at_within DERIV_within_iff f'
      apply (fastforce simp add:)+
      done
  next
    case False
    then have dxz: "dist x z > 0" by auto
    have cf: "continuous (at x within s) f"
      using conf continuous_on_eq_continuous_within that by blast
    have "continuous (at x within s) (\<lambda>w. (f w - f z) / (w - z))"
      by (rule cf continuous_intros | simp add: False)+
    then show ?thesis
      apply (rule continuous_transform_within [OF _ dxz that, of "\<lambda>w::complex. (f w - f z)/(w - z)"])
      apply (force simp: dist_commute)
      done
  qed
  have fink': "finite (insert z k)" using \<open>finite k\<close> by blast
  have *: "((\<lambda>w. if w = z then f' else (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
    apply (rule Cauchy_theorem_convex [OF _ s fink' _ vpg pas loop])
    using c apply (force simp: continuous_on_eq_continuous_within)
    apply (rename_tac w)
    apply (rule_tac d="dist w z" and f = "\<lambda>w. (f w - f z)/(w - z)" in complex_differentiable_transform_within)
    apply (simp_all add: dist_pos_lt dist_commute)
    apply (metis less_irrefl)
    apply (rule derivative_intros fcd | simp)+
    done
  show ?thesis
    apply (rule has_contour_integral_eq)
    using znotin has_contour_integral_add [OF has_contour_integral_lmul [OF has_contour_integral_winding_number [OF vpg znotin], of "f z"] *]
    apply (auto simp: mult_ac divide_simps)
    done
qed

theorem Cauchy_integral_formula_convex_simple:
    "\<lbrakk>convex s; f holomorphic_on s; z \<in> interior s; valid_path \<gamma>; path_image \<gamma> \<subseteq> s - {z};
      pathfinish \<gamma> = pathstart \<gamma>\<rbrakk>
     \<Longrightarrow> ((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
  apply (rule Cauchy_integral_formula_weak [where k = "{}"])
  using holomorphic_on_imp_continuous_on
  by auto (metis at_within_interior holomorphic_on_def interiorE subsetCE)


subsection\<open>Homotopy forms of Cauchy's theorem\<close>

proposition Cauchy_theorem_homotopic:
    assumes hom: "if atends then homotopic_paths s g h else homotopic_loops s g h"
        and "open s" and f: "f holomorphic_on s"
        and vpg: "valid_path g" and vph: "valid_path h"
    shows "contour_integral g f = contour_integral h f"
proof -
  have pathsf: "linked_paths atends g h"
    using hom  by (auto simp: linked_paths_def homotopic_paths_imp_pathstart homotopic_paths_imp_pathfinish homotopic_loops_imp_loop)
  obtain k :: "real \<times> real \<Rightarrow> complex"
    where contk: "continuous_on ({0..1} \<times> {0..1}) k"
      and ks: "k ` ({0..1} \<times> {0..1}) \<subseteq> s"
      and k [simp]: "\<forall>x. k (0, x) = g x" "\<forall>x. k (1, x) = h x"
      and ksf: "\<forall>t\<in>{0..1}. linked_paths atends g (\<lambda>x. k (t, x))"
      using hom pathsf by (auto simp: linked_paths_def homotopic_paths_def homotopic_loops_def homotopic_with_def split: split_if_asm)
  have ucontk: "uniformly_continuous_on ({0..1} \<times> {0..1}) k"
    by (blast intro: compact_Times compact_uniformly_continuous [OF contk])
  { fix t::real assume t: "t \<in> {0..1}"
    have pak: "path (k o (\<lambda>u. (t, u)))"
      unfolding path_def
      apply (rule continuous_intros continuous_on_subset [OF contk])+
      using t by force
    have pik: "path_image (k \<circ> Pair t) \<subseteq> s"
      using ks t by (auto simp: path_image_def)
    obtain e where "e>0" and e:
         "\<And>g h. \<lbrakk>valid_path g; valid_path h;
                  \<forall>u\<in>{0..1}. cmod (g u - (k \<circ> Pair t) u) < e \<and> cmod (h u - (k \<circ> Pair t) u) < e;
                  linked_paths atends g h\<rbrakk>
                 \<Longrightarrow> contour_integral h f = contour_integral g f"
      using contour_integral_nearby [OF \<open>open s\<close> pak pik, of atends] f by metis
    obtain d where "d>0" and d:
        "\<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"
      by (rule uniformly_continuous_onE [OF ucontk, of "e/4"]) (auto simp: dist_norm \<open>e>0\<close>)
    { fix t1 t2
      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"
      have no2: "\<And>g1 k1 kt. \<lbrakk>norm(g1 - k1) < e/4; norm(k1 - kt) < e/4\<rbrakk> \<Longrightarrow> norm(g1 - kt) < e"
        using \<open>e > 0\<close>
        apply (rule_tac y = k1 in norm_triangle_half_l)
        apply (auto simp: norm_minus_commute intro: order_less_trans)
        done
      have "\<exists>d>0. \<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
                          (\<forall>u\<in>{0..1}. cmod (g1 u - k (t1, u)) < d \<and> cmod (g2 u - k (t2, u)) < d) \<and>
                          linked_paths atends g1 g2 \<longrightarrow>
                          contour_integral g2 f = contour_integral g1 f"
        apply (rule_tac x="e/4" in exI)
        using t t1 t2 ltd \<open>e > 0\<close>
        apply (auto intro!: e simp: d no2 simp del: less_divide_eq_numeral1)
        done
    }
    then have "\<exists>e. 0 < e \<and>
              (\<forall>t1 t2. t1 \<in> {0..1} \<and> t2 \<in> {0..1} \<and> \<bar>t1 - t\<bar> < e \<and> \<bar>t2 - t\<bar> < e
                \<longrightarrow> (\<exists>d. 0 < d \<and>
                     (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
                       (\<forall>u \<in> {0..1}.
                          norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
                          linked_paths atends g1 g2
                          \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
      by (rule_tac x=d in exI) (simp add: \<open>d > 0\<close>)
  }
  then obtain ee where ee:
       "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0 \<and>
          (\<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
            \<longrightarrow> (\<exists>d. 0 < d \<and>
                 (\<forall>g1 g2. valid_path g1 \<and> valid_path g2 \<and>
                   (\<forall>u \<in> {0..1}.
                      norm(g1 u - k((t1,u))) < d \<and> norm(g2 u - k((t2,u))) < d) \<and>
                      linked_paths atends g1 g2
                      \<longrightarrow> contour_integral g2 f = contour_integral g1 f)))"
    by metis
  note ee_rule = ee [THEN conjunct2, rule_format]
  def C \<equiv> "(\<lambda>t. ball t (ee t / 3)) ` {0..1}"
  have "\<forall>t \<in> C. open t" by (simp add: C_def)
  moreover have "{0..1} \<subseteq> \<Union>C"
    using ee [THEN conjunct1] by (auto simp: C_def dist_norm)
  ultimately obtain C' where C': "C' \<subseteq> C" "finite C'" and C'01: "{0..1} \<subseteq> \<Union>C'"
    by (rule compactE [OF compact_interval])
  def kk \<equiv> "{t \<in> {0..1}. ball t (ee t / 3) \<in> C'}"
  have kk01: "kk \<subseteq> {0..1}" by (auto simp: kk_def)
  def e \<equiv> "Min (ee ` kk)"
  have C'_eq: "C' = (\<lambda>t. ball t (ee t / 3)) ` kk"
    using C' by (auto simp: kk_def C_def)
  have ee_pos[simp]: "\<And>t. t \<in> {0..1} \<Longrightarrow> ee t > 0"
    by (simp add: kk_def ee)
  moreover have "finite kk"
    using \<open>finite C'\<close> kk01 by (force simp: C'_eq inj_on_def ball_eq_ball_iff dest: ee_pos finite_imageD)
  moreover have "kk \<noteq> {}" using \<open>{0..1} \<subseteq> \<Union>C'\<close> C'_eq by force
  ultimately have "e > 0"
    using finite_less_Inf_iff [of "ee ` kk" 0] kk01 by (force simp: e_def)
  then obtain N::nat where "N > 0" and N: "1/N < e/3"
    by (meson divide_pos_pos nat_approx_posE zero_less_Suc zero_less_numeral)
  have e_le_ee: "\<And>i. i \<in> kk \<Longrightarrow> e \<le> ee i"
    using \<open>finite kk\<close> by (simp add: e_def Min_le_iff [of "ee ` kk"])
  have plus: "\<exists>t \<in> kk. x \<in> ball t (ee t / 3)" if "x \<in> {0..1}" for x
    using C' subsetD [OF C'01 that]  unfolding C'_eq by blast
  have [OF order_refl]:
      "\<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
                        \<longrightarrow> contour_integral j f = contour_integral g f)"
       if "n \<le> N" for n
  using that
  proof (induct n)
    case 0 show ?case using ee_rule [of 0 0 0]
      apply clarsimp
      apply (rule_tac x=d in exI, safe)
      by (metis diff_self vpg norm_zero)
  next
    case (Suc n)
    then have N01: "n/N \<in> {0..1}" "(Suc n)/N \<in> {0..1}"  by auto
    then obtain t where t: "t \<in> kk" "n/N \<in> ball t (ee t / 3)"
      using plus [of "n/N"] by blast
    then have nN_less: "\<bar>n/N - t\<bar> < ee t"
      by (simp add: dist_norm del: less_divide_eq_numeral1)
    have n'N_less: "\<bar>real (Suc n) / real N - t\<bar> < ee t"
      using t N \<open>N > 0\<close> e_le_ee [of t]
      by (simp add: dist_norm add_divide_distrib abs_diff_less_iff del: less_divide_eq_numeral1) (simp add: field_simps)
    have t01: "t \<in> {0..1}" using \<open>kk \<subseteq> {0..1}\<close> \<open>t \<in> kk\<close> by blast
    obtain d1 where "d1 > 0" and d1:
        "\<And>g1 g2. \<lbrakk>valid_path g1; valid_path g2;
                   \<forall>u\<in>{0..1}. cmod (g1 u - k (n/N, u)) < d1 \<and> cmod (g2 u - k ((Suc n) / N, u)) < d1;
                   linked_paths atends g1 g2\<rbrakk>
                   \<Longrightarrow> contour_integral g2 f = contour_integral g1 f"
      using ee [THEN conjunct2, rule_format, OF t01 N01 nN_less n'N_less] by fastforce
    have "n \<le> N" using Suc.prems by auto
    with Suc.hyps
    obtain d2 where "d2 > 0"
      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>
                     \<Longrightarrow> contour_integral j f = contour_integral g f"
        by auto
    have "continuous_on {0..1} (k o (\<lambda>u. (n/N, u)))"
      apply (rule continuous_intros continuous_on_subset [OF contk])+
      using N01 by auto
    then have pkn: "path (\<lambda>u. k (n/N, u))"
      by (simp add: path_def)
    have min12: "min d1 d2 > 0" by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
    obtain p where "polynomial_function p"
        and psf: "pathstart p = pathstart (\<lambda>u. k (n/N, u))"
                 "pathfinish p = pathfinish (\<lambda>u. k (n/N, u))"
        and pk_le:  "\<And>t. t\<in>{0..1} \<Longrightarrow> cmod (p t - k (n/N, t)) < min d1 d2"
      using path_approx_polynomial_function [OF pkn min12] by blast
    then have vpp: "valid_path p" using valid_path_polynomial_function by blast
    have lpa: "linked_paths atends g p"
      by (metis (mono_tags, lifting) N01(1) ksf linked_paths_def pathfinish_def pathstart_def psf)
    show ?case
      apply (rule_tac x="min d1 d2" in exI)
      apply (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>, clarify)
      apply (rule_tac s="contour_integral p f" in trans)
      using pk_le N01(1) ksf pathfinish_def pathstart_def
      apply (force intro!: vpp d1 simp add: linked_paths_def psf ksf)
      using pk_le N01 apply (force intro!: vpp d2 lpa simp add: linked_paths_def psf ksf)
      done
  qed
  then obtain d where "0 < d"
                       "\<And>j. valid_path j \<and> (\<forall>u \<in> {0..1}. norm(j u - k (1,u)) < d) \<and>
                            linked_paths atends g j
                            \<Longrightarrow> contour_integral j f = contour_integral g f"
    using \<open>N>0\<close> by auto
  then have "linked_paths atends g h \<Longrightarrow> contour_integral h f = contour_integral g f"
    using \<open>N>0\<close> vph by fastforce
  then show ?thesis
    by (simp add: pathsf)
qed

proposition Cauchy_theorem_homotopic_paths:
    assumes hom: "homotopic_paths s g h"
        and "open s" and f: "f holomorphic_on s"
        and vpg: "valid_path g" and vph: "valid_path h"
    shows "contour_integral g f = contour_integral h f"
  using Cauchy_theorem_homotopic [of True s g h] assms by simp

proposition Cauchy_theorem_homotopic_loops:
    assumes hom: "homotopic_loops s g h"
        and "open s" and f: "f holomorphic_on s"
        and vpg: "valid_path g" and vph: "valid_path h"
    shows "contour_integral g f = contour_integral h f"
  using Cauchy_theorem_homotopic [of False s g h] assms by simp

lemma has_contour_integral_newpath:
    "\<lbrakk>(f has_contour_integral y) h; f contour_integrable_on g; contour_integral g f = contour_integral h f\<rbrakk>
     \<Longrightarrow> (f has_contour_integral y) g"
  using has_contour_integral_integral contour_integral_unique by auto

lemma Cauchy_theorem_null_homotopic:
     "\<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"
  apply (rule has_contour_integral_newpath [where h = "linepath a a"], simp)
  using contour_integrable_holomorphic_simple
    apply (blast dest: holomorphic_on_imp_continuous_on homotopic_loops_imp_subset)
  by (simp add: Cauchy_theorem_homotopic_loops)



subsection\<open>More winding number properties\<close>

text\<open>including the fact that it's +-1 inside a simple closed curve.\<close>

lemma winding_number_homotopic_paths:
    assumes "homotopic_paths (-{z}) g h"
      shows "winding_number g z = winding_number h z"
proof -
  have "path g" "path h" using homotopic_paths_imp_path [OF assms] by auto
  moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
    using homotopic_paths_imp_subset [OF assms] by auto
  ultimately obtain d e where "d > 0" "e > 0"
      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>
            \<Longrightarrow> homotopic_paths (-{z}) g p"
      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>
            \<Longrightarrow> homotopic_paths (-{z}) h q"
    using homotopic_nearby_paths [of g "-{z}"] homotopic_nearby_paths [of h "-{z}"] by force
  obtain p where p:
       "valid_path p" "z \<notin> path_image p"
       "pathstart p = pathstart g" "pathfinish p = pathfinish g"
       and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
       and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
    using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] by blast
  obtain q where q:
       "valid_path q" "z \<notin> path_image q"
       "pathstart q = pathstart h" "pathfinish q = pathfinish h"
       and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
       and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
    using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] by blast
  have gp: "homotopic_paths (- {z}) g p"
    by (simp add: d p valid_path_imp_path norm_minus_commute gp_less)
  have hq: "homotopic_paths (- {z}) h q"
    by (simp add: e q valid_path_imp_path norm_minus_commute hq_less)
  have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
    apply (rule Cauchy_theorem_homotopic_paths [of "-{z}"])
    apply (blast intro: homotopic_paths_trans homotopic_paths_sym gp hq assms)
    apply (auto intro!: holomorphic_intros simp: p q)
    done
  then show ?thesis
    by (simp add: pap paq)
qed

lemma winding_number_homotopic_loops:
    assumes "homotopic_loops (-{z}) g h"
      shows "winding_number g z = winding_number h z"
proof -
  have "path g" "path h" using homotopic_loops_imp_path [OF assms] by auto
  moreover have pag: "z \<notin> path_image g" and pah: "z \<notin> path_image h"
    using homotopic_loops_imp_subset [OF assms] by auto
  moreover have gloop: "pathfinish g = pathstart g" and hloop: "pathfinish h = pathstart h"
    using homotopic_loops_imp_loop [OF assms] by auto
  ultimately obtain d e where "d > 0" "e > 0"
      and d: "\<And>p. \<lbrakk>path p; pathfinish p = pathstart p; \<forall>t\<in>{0..1}. norm (p t - g t) < d\<rbrakk>
            \<Longrightarrow> homotopic_loops (-{z}) g p"
      and e: "\<And>q. \<lbrakk>path q; pathfinish q = pathstart q; \<forall>t\<in>{0..1}. norm (q t - h t) < e\<rbrakk>
            \<Longrightarrow> homotopic_loops (-{z}) h q"
    using homotopic_nearby_loops [of g "-{z}"] homotopic_nearby_loops [of h "-{z}"] by force
  obtain p where p:
       "valid_path p" "z \<notin> path_image p"
       "pathstart p = pathstart g" "pathfinish p = pathfinish g"
       and gp_less:"\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
       and pap: "contour_integral p (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number g z"
    using winding_number [OF \<open>path g\<close> pag \<open>0 < d\<close>] by blast
  obtain q where q:
       "valid_path q" "z \<notin> path_image q"
       "pathstart q = pathstart h" "pathfinish q = pathfinish h"
       and hq_less: "\<forall>t\<in>{0..1}. cmod (h t - q t) < e"
       and paq:  "contour_integral q (\<lambda>w. 1 / (w - z)) = complex_of_real (2 * pi) * \<i> * winding_number h z"
    using winding_number [OF \<open>path h\<close> pah \<open>0 < e\<close>] by blast
  have gp: "homotopic_loops (- {z}) g p"
    by (simp add: gloop d gp_less norm_minus_commute p valid_path_imp_path)
  have hq: "homotopic_loops (- {z}) h q"
    by (simp add: e hloop hq_less norm_minus_commute q valid_path_imp_path)
  have "contour_integral p (\<lambda>w. 1/(w - z)) = contour_integral q (\<lambda>w. 1/(w - z))"
    apply (rule Cauchy_theorem_homotopic_loops [of "-{z}"])
    apply (blast intro: homotopic_loops_trans homotopic_loops_sym gp hq assms)
    apply (auto intro!: holomorphic_intros simp: p q)
    done
  then show ?thesis
    by (simp add: pap paq)
qed

lemma winding_number_paths_linear_eq:
  "\<lbrakk>path g; path h; pathstart h = pathstart g; pathfinish h = pathfinish g;
    \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
        \<Longrightarrow> winding_number h z = winding_number g z"
  by (blast intro: sym homotopic_paths_linear winding_number_homotopic_paths elim: )

lemma winding_number_loops_linear_eq:
  "\<lbrakk>path g; path h; pathfinish g = pathstart g; pathfinish h = pathstart h;
    \<And>t. t \<in> {0..1} \<Longrightarrow> z \<notin> closed_segment (g t) (h t)\<rbrakk>
        \<Longrightarrow> winding_number h z = winding_number g z"
  by (blast intro: sym homotopic_loops_linear winding_number_homotopic_loops elim: )

lemma winding_number_nearby_paths_eq:
     "\<lbrakk>path g; path h;
      pathstart h = pathstart g; pathfinish h = pathfinish g;
      \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
      \<Longrightarrow> winding_number h z = winding_number g z"
  by (metis segment_bound(2) norm_minus_commute not_le winding_number_paths_linear_eq)

lemma winding_number_nearby_loops_eq:
     "\<lbrakk>path g; path h;
      pathfinish g = pathstart g;
        pathfinish h = pathstart h;
      \<And>t. t \<in> {0..1} \<Longrightarrow> norm(h t - g t) < norm(g t - z)\<rbrakk>
      \<Longrightarrow> winding_number h z = winding_number g z"
  by (metis segment_bound(2) norm_minus_commute not_le winding_number_loops_linear_eq)


proposition winding_number_subpath_combine:
    "\<lbrakk>path g; z \<notin> path_image g;
      u \<in> {0..1}; v \<in> {0..1}; w \<in> {0..1}\<rbrakk>
      \<Longrightarrow> winding_number (subpath u v g) z + winding_number (subpath v w g) z =
          winding_number (subpath u w g) z"
apply (rule trans [OF winding_number_join [THEN sym]
                      winding_number_homotopic_paths [OF homotopic_join_subpaths]])
apply (auto dest: path_image_subpath_subset)
done


subsection\<open>Partial circle path\<close>

definition part_circlepath :: "[complex, real, real, real, real] \<Rightarrow> complex"
  where "part_circlepath z r s t \<equiv> \<lambda>x. z + of_real r * exp (ii * of_real (linepath s t x))"

lemma pathstart_part_circlepath [simp]:
     "pathstart(part_circlepath z r s t) = z + r*exp(ii * s)"
by (metis part_circlepath_def pathstart_def pathstart_linepath)

lemma pathfinish_part_circlepath [simp]:
     "pathfinish(part_circlepath z r s t) = z + r*exp(ii*t)"
by (metis part_circlepath_def pathfinish_def pathfinish_linepath)

proposition has_vector_derivative_part_circlepath [derivative_intros]:
    "((part_circlepath z r s t) has_vector_derivative
      (ii * r * (of_real t - of_real s) * exp(ii * linepath s t x)))
     (at x within X)"
  apply (simp add: part_circlepath_def linepath_def scaleR_conv_of_real)
  apply (rule has_vector_derivative_real_complex)
  apply (rule derivative_eq_intros | simp)+
  done

corollary vector_derivative_part_circlepath:
    "vector_derivative (part_circlepath z r s t) (at x) =
       ii * r * (of_real t - of_real s) * exp(ii * linepath s t x)"
  using has_vector_derivative_part_circlepath vector_derivative_at by blast

corollary vector_derivative_part_circlepath01:
    "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
     \<Longrightarrow> vector_derivative (part_circlepath z r s t) (at x within {0..1}) =
          ii * r * (of_real t - of_real s) * exp(ii * linepath s t x)"
  using has_vector_derivative_part_circlepath
  by (auto simp: vector_derivative_at_within_ivl)

lemma valid_path_part_circlepath [simp]: "valid_path (part_circlepath z r s t)"
  apply (simp add: valid_path_def)
  apply (rule C1_differentiable_imp_piecewise)
  apply (auto simp: C1_differentiable_on_eq vector_derivative_works vector_derivative_part_circlepath has_vector_derivative_part_circlepath
              intro!: continuous_intros)
  done

lemma path_part_circlepath [simp]: "path (part_circlepath z r s t)"
  by (simp add: valid_path_imp_path)

proposition path_image_part_circlepath:
  assumes "s \<le> t"
    shows "path_image (part_circlepath z r s t) = {z + r * exp(ii * of_real x) | x. s \<le> x \<and> x \<le> t}"
proof -
  { fix z::real
    assume "0 \<le> z" "z \<le> 1"
    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"
      apply (rule_tac x="(1 - z) * s + z * t" in exI)
      apply (simp add: linepath_def scaleR_conv_of_real algebra_simps)
      apply (rule conjI)
      using mult_right_mono apply blast
      using affine_ineq  by (metis "mult.commute")
  }
  moreover
  { fix z
    assume "s \<le> z" "z \<le> t"
    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}"
      apply (rule_tac x="(z - s)/(t - s)" in image_eqI)
      apply (simp add: linepath_def scaleR_conv_of_real divide_simps exp_eq)
      apply (auto simp: algebra_simps divide_simps)
      done
  }
  ultimately show ?thesis
    by (fastforce simp add: path_image_def part_circlepath_def)
qed

corollary path_image_part_circlepath_subset:
    "\<lbrakk>s \<le> t; 0 \<le> r\<rbrakk> \<Longrightarrow> path_image(part_circlepath z r s t) \<subseteq> sphere z r"
by (auto simp: path_image_part_circlepath sphere_def dist_norm algebra_simps norm_mult)

proposition in_path_image_part_circlepath:
  assumes "w \<in> path_image(part_circlepath z r s t)" "s \<le> t" "0 \<le> r"
    shows "norm(w - z) = r"
proof -
  have "w \<in> {c. dist z c = r}"
    by (metis (no_types) path_image_part_circlepath_subset sphere_def subset_eq assms)
  thus ?thesis
    by (simp add: dist_norm norm_minus_commute)
qed

proposition finite_bounded_log: "finite {z::complex. norm z \<le> b \<and> exp z = w}"
proof (cases "w = 0")
  case True then show ?thesis by auto
next
  case False
  have *: "finite {x. cmod (complex_of_real (2 * real_of_int x * pi) * \<i>) \<le> b + cmod (Ln w)}"
    apply (simp add: norm_mult finite_int_iff_bounded_le)
    apply (rule_tac x="\<lfloor>(b + cmod (Ln w)) / (2*pi)\<rfloor>" in exI)
    apply (auto simp: divide_simps le_floor_iff)
    done
  have [simp]: "\<And>P f. {z. P z \<and> (\<exists>n. z = f n)} = f ` {n. P (f n)}"
    by blast
  show ?thesis
    apply (subst exp_Ln [OF False, symmetric])
    apply (simp add: exp_eq)
    using norm_add_leD apply (fastforce intro: finite_subset [OF _ *])
    done
qed

lemma finite_bounded_log2:
  fixes a::complex
    assumes "a \<noteq> 0"
    shows "finite {z. norm z \<le> b \<and> exp(a*z) = w}"
proof -
  have *: "finite ((\<lambda>z. z / a) ` {z. cmod z \<le> b * cmod a \<and> exp z = w})"
    by (rule finite_imageI [OF finite_bounded_log])
  show ?thesis
    by (rule finite_subset [OF _ *]) (force simp: assms norm_mult)
qed

proposition has_contour_integral_bound_part_circlepath_strong:
  assumes fi: "(f has_contour_integral i) (part_circlepath z r s t)"
      and "finite k" and le: "0 \<le> B" "0 < r" "s \<le> t"
      and B: "\<And>x. x \<in> path_image(part_circlepath z r s t) - k \<Longrightarrow> norm(f x) \<le> B"
    shows "cmod i \<le> B * r * (t - s)"
proof -
  consider "s = t" | "s < t" using \<open>s \<le> t\<close> by linarith
  then show ?thesis
  proof cases
    case 1 with fi [unfolded has_contour_integral]
    have "i = 0"  by (simp add: vector_derivative_part_circlepath)
    with assms show ?thesis by simp
  next
    case 2
    have [simp]: "\<bar>r\<bar> = r" using \<open>r > 0\<close> by linarith
    have [simp]: "cmod (complex_of_real t - complex_of_real s) = t-s"
      by (metis "2" abs_of_pos diff_gt_0_iff_gt norm_of_real of_real_diff)
    have "finite (part_circlepath z r s t -` {y} \<inter> {0..1})" if "y \<in> k" for y
    proof -
      def w \<equiv> "(y - z)/of_real r / exp(ii * of_real s)"
      have fin: "finite (of_real -` {z. cmod z \<le> 1 \<and> exp (\<i> * complex_of_real (t - s) * z) = w})"
        apply (rule finite_vimageI [OF finite_bounded_log2])
        using \<open>s < t\<close> apply (auto simp: inj_of_real)
        done
      show ?thesis
        apply (simp add: part_circlepath_def linepath_def vimage_def)
        apply (rule finite_subset [OF _ fin])
        using le
        apply (auto simp: w_def algebra_simps scaleR_conv_of_real exp_add exp_diff)
        done
    qed
    then have fin01: "finite ((part_circlepath z r s t) -` k \<inter> {0..1})"
      by (rule finite_finite_vimage_IntI [OF \<open>finite k\<close>])
    have **: "((\<lambda>x. if (part_circlepath z r s t x) \<in> k then 0
                    else f(part_circlepath z r s t x) *
                       vector_derivative (part_circlepath z r s t) (at x)) has_integral i)  {0..1}"
      apply (rule has_integral_spike
              [where f = "\<lambda>x. f(part_circlepath z r s t x) * vector_derivative (part_circlepath z r s t) (at x)"])
      apply (rule negligible_finite [OF fin01])
      using fi has_contour_integral
      apply auto
      done
    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"
      by (auto intro!: B [unfolded path_image_def image_def, simplified])
    show ?thesis
      apply (rule has_integral_bound [where 'a=real, simplified, OF _ **, simplified])
      using assms apply force
      apply (simp add: norm_mult vector_derivative_part_circlepath)
      using le * "2" \<open>r > 0\<close> by auto
  qed
qed

corollary has_contour_integral_bound_part_circlepath:
      "\<lbrakk>(f has_contour_integral i) (part_circlepath z r s t);
        0 \<le> B; 0 < r; s \<le> t;
        \<And>x. x \<in> path_image(part_circlepath z r s t) \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
       \<Longrightarrow> norm i \<le> B*r*(t - s)"
  by (auto intro: has_contour_integral_bound_part_circlepath_strong)

proposition contour_integrable_continuous_part_circlepath:
     "continuous_on (path_image (part_circlepath z r s t)) f
      \<Longrightarrow> f contour_integrable_on (part_circlepath z r s t)"
  apply (simp add: contour_integrable_on has_contour_integral_def vector_derivative_part_circlepath path_image_def)
  apply (rule integrable_continuous_real)
  apply (fast intro: path_part_circlepath [unfolded path_def] continuous_intros continuous_on_compose2 [where g=f, OF _ _ order_refl])
  done

proposition winding_number_part_circlepath_pos_less:
  assumes "s < t" and no: "norm(w - z) < r"
    shows "0 < Re (winding_number(part_circlepath z r s t) w)"
proof -
  have "0 < r" by (meson no norm_not_less_zero not_le order.strict_trans2)
  note valid_path_part_circlepath
  moreover have " w \<notin> path_image (part_circlepath z r s t)"
    using assms by (auto simp: path_image_def image_def part_circlepath_def norm_mult linepath_def)
  moreover have "0 < r * (t - s) * (r - cmod (w - z))"
    using assms by (metis \<open>0 < r\<close> diff_gt_0_iff_gt mult_pos_pos)
  ultimately show ?thesis
    apply (rule winding_number_pos_lt [where e = "r*(t - s)*(r - norm(w - z))"])
    apply (simp add: vector_derivative_part_circlepath right_diff_distrib [symmetric] mult_ac)
    apply (rule mult_left_mono)+
    using Re_Im_le_cmod [of "w-z" "linepath s t x" for x]
    apply (simp add: exp_Euler cos_of_real sin_of_real part_circlepath_def algebra_simps cos_squared_eq [unfolded power2_eq_square])
    using assms \<open>0 < r\<close> by auto
qed

proposition simple_path_part_circlepath:
    "simple_path(part_circlepath z r s t) \<longleftrightarrow> (r \<noteq> 0 \<and> s \<noteq> t \<and> \<bar>s - t\<bar> \<le> 2*pi)"
proof (cases "r = 0 \<or> s = t")
  case True
  then show ?thesis
    apply (rule disjE)
    apply (force simp: part_circlepath_def simple_path_def intro: bexI [where x = "1/4"] bexI [where x = "1/3"])+
    done
next
  case False then have "r \<noteq> 0" "s \<noteq> t" by auto
  have *: "\<And>x y z s t. ii*((1 - x) * s + x * t) = ii*(((1 - y) * s + y * t)) + z  \<longleftrightarrow> ii*(x - y) * (t - s) = z"
    by (simp add: algebra_simps)
  have abs01: "\<And>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1
                      \<Longrightarrow> (x = y \<or> x = 0 \<and> y = 1 \<or> x = 1 \<and> y = 0 \<longleftrightarrow> \<bar>x - y\<bar> \<in> {0,1})"
    by auto
  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)"
    by force
  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>
                  (\<exists>n. \<bar>x - y\<bar> * (t - s) = 2 * (of_int n * pi))"
    by (force simp: algebra_simps abs_if dest: arg_cong [where f=Re] arg_cong [where f=complex_of_real]
                    intro: exI [where x = "-n" for n])
  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"
    apply (rule ccontr)
    apply (drule_tac x="2*pi / \<bar>t - s\<bar>" in spec)
    using False
    apply (simp add: abs_minus_commute divide_simps)
    apply (frule_tac x=1 in spec)
    apply (drule_tac x="-1" in spec)
    apply (simp add:)
    done
  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
  proof -
    have "t-s = 2 * (real_of_int n * pi)/x"
      using that by (simp add: field_simps)
    then show ?thesis by (metis abs_minus_commute)
  qed
  show ?thesis using False
    apply (simp add: simple_path_def path_part_circlepath)
    apply (simp add: part_circlepath_def linepath_def exp_eq  * ** abs01  del: Set.insert_iff)
    apply (subst abs_away)
    apply (auto simp: 1)
    apply (rule ccontr)
    apply (auto simp: 2 divide_simps abs_mult dest: of_int_leD)
    done
qed

proposition arc_part_circlepath:
  assumes "r \<noteq> 0" "s \<noteq> t" "\<bar>s - t\<bar> < 2*pi"
    shows "arc (part_circlepath z r s t)"
proof -
  have *: "x = y" if eq: "\<i> * (linepath s t x) = \<i> * (linepath s t y) + 2 * of_int n * complex_of_real pi * \<i>"
                  and x: "x \<in> {0..1}" and y: "y \<in> {0..1}" for x y n
    proof -
      have "(linepath s t x) = (linepath s t y) + 2 * of_int n * complex_of_real pi"
        by (metis add_divide_eq_iff complex_i_not_zero mult.commute nonzero_mult_divide_cancel_left eq)
      then have "s*y + t*x = s*x + (t*y + of_int n * (pi * 2))"
        by (force simp: algebra_simps linepath_def dest: arg_cong [where f=Re])
      then have st: "x \<noteq> y \<Longrightarrow> (s-t) = (of_int n * (pi * 2) / (y-x))"
        by (force simp: field_simps)
      show ?thesis
        apply (rule ccontr)
        using assms x y
        apply (simp add: st abs_mult field_simps)
        using st
        apply (auto simp: dest: of_int_lessD)
        done
    qed
  show ?thesis
    using assms
    apply (simp add: arc_def)
    apply (simp add: part_circlepath_def inj_on_def exp_eq)
    apply (blast intro: *)
    done
qed


subsection\<open>Special case of one complete circle\<close>

definition circlepath :: "[complex, real, real] \<Rightarrow> complex"
  where "circlepath z r \<equiv> part_circlepath z r 0 (2*pi)"

lemma circlepath: "circlepath z r = (\<lambda>x. z + r * exp(2 * of_real pi * ii * of_real x))"
  by (simp add: circlepath_def part_circlepath_def linepath_def algebra_simps)

lemma pathstart_circlepath [simp]: "pathstart (circlepath z r) = z + r"
  by (simp add: circlepath_def)

lemma pathfinish_circlepath [simp]: "pathfinish (circlepath z r) = z + r"
  by (simp add: circlepath_def) (metis exp_two_pi_i mult.commute)

lemma circlepath_minus: "circlepath z (-r) x = circlepath z r (x + 1/2)"
proof -
  have "z + of_real r * exp (2 * pi * \<i> * (x + 1 / 2)) =
        z + of_real r * exp (2 * pi * \<i> * x + pi * \<i>)"
    by (simp add: divide_simps) (simp add: algebra_simps)
  also have "... = z - r * exp (2 * pi * \<i> * x)"
    by (simp add: exp_add)
  finally show ?thesis
    by (simp add: circlepath path_image_def sphere_def dist_norm)
qed

lemma circlepath_add1: "circlepath z r (x+1) = circlepath z r x"
  using circlepath_minus [of z r "x+1/2"] circlepath_minus [of z "-r" x]
  by (simp add: add.commute)

lemma circlepath_add_half: "circlepath z r (x + 1/2) = circlepath z r (x - 1/2)"
  using circlepath_add1 [of z r "x-1/2"]
  by (simp add: add.commute)

lemma path_image_circlepath_minus_subset:
     "path_image (circlepath z (-r)) \<subseteq> path_image (circlepath z r)"
  apply (simp add: path_image_def image_def circlepath_minus, clarify)
  apply (case_tac "xa \<le> 1/2", force)
  apply (force simp add: circlepath_add_half)+
  done

lemma path_image_circlepath_minus: "path_image (circlepath z (-r)) = path_image (circlepath z r)"
  using path_image_circlepath_minus_subset by fastforce

proposition has_vector_derivative_circlepath [derivative_intros]:
 "((circlepath z r) has_vector_derivative (2 * pi * ii * r * exp (2 * of_real pi * ii * of_real x)))
   (at x within X)"
  apply (simp add: circlepath_def scaleR_conv_of_real)
  apply (rule derivative_eq_intros)
  apply (simp add: algebra_simps)
  done

corollary vector_derivative_circlepath:
   "vector_derivative (circlepath z r) (at x) =
    2 * pi * ii * r * exp(2 * of_real pi * ii * x)"
using has_vector_derivative_circlepath vector_derivative_at by blast

corollary vector_derivative_circlepath01:
    "\<lbrakk>0 \<le> x; x \<le> 1\<rbrakk>
     \<Longrightarrow> vector_derivative (circlepath z r) (at x within {0..1}) =
          2 * pi * ii * r * exp(2 * of_real pi * ii * x)"
  using has_vector_derivative_circlepath
  by (auto simp: vector_derivative_at_within_ivl)

lemma valid_path_circlepath [simp]: "valid_path (circlepath z r)"
  by (simp add: circlepath_def)

lemma path_circlepath [simp]: "path (circlepath z r)"
  by (simp add: valid_path_imp_path)

lemma path_image_circlepath_nonneg:
  assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
proof -
  have *: "x \<in> (\<lambda>u. z + (cmod (x - z)) * exp (\<i> * (of_real u * (of_real pi * 2)))) ` {0..1}" for x
  proof (cases "x = z")
    case True then show ?thesis by force
  next
    case False
    def w \<equiv> "x - z"
    then have "w \<noteq> 0" by (simp add: False)
    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)"
      using cis_conv_exp complex_eq_iff by auto
    show ?thesis
      apply (rule sincos_total_2pi [of "Re(w/of_real(norm w))" "Im(w/of_real(norm w))"])
      apply (simp add: divide_simps \<open>w \<noteq> 0\<close> cmod_power2 [symmetric])
      apply (rule_tac x="t / (2*pi)" in image_eqI)
      apply (simp add: divide_simps \<open>w \<noteq> 0\<close>)
      using False **
      apply (auto simp: w_def)
      done
  qed
  show ?thesis
    unfolding circlepath path_image_def sphere_def dist_norm
    by (force simp: assms algebra_simps norm_mult norm_minus_commute intro: *)
qed

proposition path_image_circlepath [simp]:
    "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
  using path_image_circlepath_minus
  by (force simp add: path_image_circlepath_nonneg abs_if)

lemma has_contour_integral_bound_circlepath_strong:
      "\<lbrakk>(f has_contour_integral i) (circlepath z r);
        finite k; 0 \<le> B; 0 < r;
        \<And>x. \<lbrakk>norm(x - z) = r; x \<notin> k\<rbrakk> \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
        \<Longrightarrow> norm i \<le> B*(2*pi*r)"
  unfolding circlepath_def
  by (auto simp: algebra_simps in_path_image_part_circlepath dest!: has_contour_integral_bound_part_circlepath_strong)

corollary has_contour_integral_bound_circlepath:
      "\<lbrakk>(f has_contour_integral i) (circlepath z r);
        0 \<le> B; 0 < r; \<And>x. norm(x - z) = r \<Longrightarrow> norm(f x) \<le> B\<rbrakk>
        \<Longrightarrow> norm i \<le> B*(2*pi*r)"
  by (auto intro: has_contour_integral_bound_circlepath_strong)

proposition contour_integrable_continuous_circlepath:
    "continuous_on (path_image (circlepath z r)) f
     \<Longrightarrow> f contour_integrable_on (circlepath z r)"
  by (simp add: circlepath_def contour_integrable_continuous_part_circlepath)

lemma simple_path_circlepath: "simple_path(circlepath z r) \<longleftrightarrow> (r \<noteq> 0)"
  by (simp add: circlepath_def simple_path_part_circlepath)

lemma notin_path_image_circlepath [simp]: "cmod (w - z) < r \<Longrightarrow> w \<notin> path_image (circlepath z r)"
  by (simp add: sphere_def dist_norm norm_minus_commute)

proposition contour_integral_circlepath:
     "0 < r \<Longrightarrow> contour_integral (circlepath z r) (\<lambda>w. 1 / (w - z)) = 2 * complex_of_real pi * \<i>"
  apply (rule contour_integral_unique)
  apply (simp add: has_contour_integral_def)
  apply (subst has_integral_cong)
  apply (simp add: vector_derivative_circlepath01)
  using has_integral_const_real [of _ 0 1]
  apply (force simp: circlepath)
  done

lemma winding_number_circlepath_centre: "0 < r \<Longrightarrow> winding_number (circlepath z r) z = 1"
  apply (rule winding_number_unique_loop)
  apply (simp_all add: sphere_def valid_path_imp_path)
  apply (rule_tac x="circlepath z r" in exI)
  apply (simp add: sphere_def contour_integral_circlepath)
  done

proposition winding_number_circlepath:
  assumes "norm(w - z) < r" shows "winding_number(circlepath z r) w = 1"
proof (cases "w = z")
  case True then show ?thesis
    using assms winding_number_circlepath_centre by auto
next
  case False
  have [simp]: "r > 0"
    using assms le_less_trans norm_ge_zero by blast
  def r' \<equiv> "norm(w - z)"
  have "r' < r"
    by (simp add: assms r'_def)
  have disjo: "cball z r' \<inter> sphere z r = {}"
    using \<open>r' < r\<close> by (force simp: cball_def sphere_def)
  have "winding_number(circlepath z r) w = winding_number(circlepath z r) z"
    apply (rule winding_number_around_inside [where s = "cball z r'"])
    apply (simp_all add: disjo order.strict_implies_order winding_number_circlepath_centre)
    apply (simp_all add: False r'_def dist_norm norm_minus_commute)
    done
  also have "... = 1"
    by (simp add: winding_number_circlepath_centre)
  finally show ?thesis .
qed


text\<open> Hence the Cauchy formula for points inside a circle.\<close>

theorem Cauchy_integral_circlepath:
  assumes "continuous_on (cball z r) f" "f holomorphic_on (ball z r)" "norm(w - z) < r"
  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * ii * f w))
         (circlepath z r)"
proof -
  have "r > 0"
    using assms le_less_trans norm_ge_zero by blast
  have "((\<lambda>u. f u / (u - w)) has_contour_integral (2 * pi) * \<i> * winding_number (circlepath z r) w * f w)
        (circlepath z r)"
    apply (rule Cauchy_integral_formula_weak [where s = "cball z r" and k = "{}"])
    using assms  \<open>r > 0\<close>
    apply (simp_all add: dist_norm norm_minus_commute)
    apply (metis at_within_interior dist_norm holomorphic_on_def interior_ball mem_ball norm_minus_commute)
    apply (simp add: cball_def sphere_def dist_norm, clarify)
    apply (simp add:)
    by (metis dist_commute dist_norm less_irrefl)
  then show ?thesis
    by (simp add: winding_number_circlepath assms)
qed

corollary Cauchy_integral_circlepath_simple:
  assumes "f holomorphic_on cball z r" "norm(w - z) < r"
  shows "((\<lambda>u. f u/(u - w)) has_contour_integral (2 * of_real pi * ii * f w))
         (circlepath z r)"
using assms by (force simp: holomorphic_on_imp_continuous_on holomorphic_on_subset Cauchy_integral_circlepath)


lemma no_bounded_connected_component_imp_winding_number_zero:
  assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
      and nb: "\<And>z. bounded (connected_component_set (- s) z) \<longrightarrow> z \<in> s"
  shows "winding_number g z = 0"
apply (rule winding_number_zero_in_outside)
apply (simp_all add: assms)
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)

lemma no_bounded_path_component_imp_winding_number_zero:
  assumes g: "path g" "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
      and nb: "\<And>z. bounded (path_component_set (- s) z) \<longrightarrow> z \<in> s"
  shows "winding_number g z = 0"
apply (rule no_bounded_connected_component_imp_winding_number_zero [OF g])
by (simp add: bounded_subset nb path_component_subset_connected_component)


subsection\<open> Uniform convergence of path integral\<close>

text\<open>Uniform convergence when the derivative of the path is bounded, and in particular for the special case of a circle.\<close>

proposition contour_integral_uniform_limit:
  assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on \<gamma>) F"
      and ev_no: "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>n. \<forall>x \<in> path_image \<gamma>. norm(f n x - l x) < e) F"
      and noleB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
      and \<gamma>: "valid_path \<gamma>"
      and [simp]: "~ (trivial_limit F)"
  shows "l contour_integrable_on \<gamma>" "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
proof -
  have "0 \<le> B" by (meson noleB [of 0] atLeastAtMost_iff norm_ge_zero order_refl order_trans zero_le_one)
  { fix e::real
    assume "0 < e"
    then have eB: "0 < e / (\<bar>B\<bar> + 1)" by simp
    obtain a where fga: "\<And>x. x \<in> {0..1} \<Longrightarrow> cmod (f a (\<gamma> x) - l (\<gamma> x)) < e / (\<bar>B\<bar> + 1)"
               and inta: "(\<lambda>t. f a (\<gamma> t) * vector_derivative \<gamma> (at t)) integrable_on {0..1}"
      using eventually_happens [OF eventually_conj [OF ev_no [OF eB] ev_fint]]
      by (fastforce simp: contour_integrable_on path_image_def)
    have Ble: "B * e / (\<bar>B\<bar> + 1) \<le> e"
      using \<open>0 \<le> B\<close>  \<open>0 < e\<close> by (simp add: divide_simps)
    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}"
      apply (rule_tac x="\<lambda>x. f (a::'a) (\<gamma> x) * vector_derivative \<gamma> (at x)" in exI)
      apply (intro inta conjI ballI)
      apply (rule order_trans [OF _ Ble])
      apply (frule noleB)
      apply (frule fga)
      using \<open>0 \<le> B\<close>  \<open>0 < e\<close>
      apply (simp add: norm_mult left_diff_distrib [symmetric] norm_minus_commute divide_simps)
      apply (drule (1) mult_mono [OF less_imp_le])
      apply (simp_all add: mult_ac)
      done
  }
  then show lintg: "l contour_integrable_on \<gamma>"
    apply (simp add: contour_integrable_on)
    apply (blast intro: integrable_uniform_limit_real)
    done
  { fix e::real
    def B' \<equiv> "B+1"
    have B': "B' > 0" "B' > B" using  \<open>0 \<le> B\<close> by (auto simp: B'_def)
    assume "0 < e"
    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'"
      using ev_no [of "e / B' / 2"] B' by (simp add: field_simps)
    have ie: "integral {0..1::real} (\<lambda>x. e / 2) < e" using \<open>0 < e\<close> by simp
    have *: "cmod (f x (\<gamma> t) * vector_derivative \<gamma> (at t) - l (\<gamma> t) * vector_derivative \<gamma> (at t)) \<le> e / 2"
             if t: "t\<in>{0..1}" and leB': "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) < e / B'" for x t
    proof -
      have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) \<le> e * (B/ B')"
        using mult_mono [OF less_imp_le [OF leB'] noleB] B' \<open>0 < e\<close> t by auto
      also have "... < e"
        by (simp add: B' \<open>0 < e\<close> mult_imp_div_pos_less)
      finally have "2 * cmod (f x (\<gamma> t) - l (\<gamma> t)) * cmod (vector_derivative \<gamma> (at t)) < e" .
      then show ?thesis
        by (simp add: left_diff_distrib [symmetric] norm_mult)
    qed
    have "\<forall>\<^sub>F x in F. dist (contour_integral \<gamma> (f x)) (contour_integral \<gamma> l) < e"
      apply (rule eventually_mono [OF eventually_conj [OF ev_no' ev_fint]])
      apply (simp add: dist_norm contour_integrable_on path_image_def contour_integral_integral)
      apply (simp add: lintg integral_diff [symmetric] contour_integrable_on [symmetric], clarify)
      apply (rule le_less_trans [OF integral_norm_bound_integral ie])
      apply (simp add: lintg integrable_diff contour_integrable_on [symmetric])
      apply (blast intro: *)+
      done
  }
  then show "((\<lambda>n. contour_integral \<gamma> (f n)) \<longlongrightarrow> contour_integral \<gamma> l) F"
    by (rule tendstoI)
qed

proposition contour_integral_uniform_limit_circlepath:
  assumes ev_fint: "eventually (\<lambda>n::'a. (f n) contour_integrable_on (circlepath z r)) F"
      and ev_no: "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>n. \<forall>x \<in> path_image (circlepath z r). norm(f n x - l x) < e) F"
      and [simp]: "~ (trivial_limit F)" "0 < r"
  shows "l contour_integrable_on (circlepath z r)" "((\<lambda>n. contour_integral (circlepath z r) (f n)) \<longlongrightarrow> contour_integral (circlepath z r) l) F"
by (auto simp: vector_derivative_circlepath norm_mult intro: contour_integral_uniform_limit assms)


subsection\<open> General stepping result for derivative formulas.\<close>

lemma sum_sqs_eq:
  fixes x::"'a::idom" shows "x * x + y * y = x * (y * 2) \<Longrightarrow> y = x"
  by algebra

proposition Cauchy_next_derivative:
  assumes "continuous_on (path_image \<gamma>) f'"
      and leB: "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (vector_derivative \<gamma> (at t)) \<le> B"
      and int: "\<And>w. w \<in> s - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f' u / (u - w)^k) has_contour_integral f w) \<gamma>"
      and k: "k \<noteq> 0"
      and "open s"
      and \<gamma>: "valid_path \<gamma>"
      and w: "w \<in> s - path_image \<gamma>"
    shows "(\<lambda>u. f' u / (u - w)^(Suc k)) contour_integrable_on \<gamma>"
      and "(f has_field_derivative (k * contour_integral \<gamma> (\<lambda>u. f' u/(u - w)^(Suc k))))
           (at w)"  (is "?thes2")
proof -
  have "open (s - path_image \<gamma>)" using \<open>open s\<close> closed_valid_path_image \<gamma> by blast
  then obtain d where "d>0" and d: "ball w d \<subseteq> s - path_image \<gamma>" using w
    using open_contains_ball by blast
  have [simp]: "\<And>n. cmod (1 + of_nat n) = 1 + of_nat n"
    by (metis norm_of_nat of_nat_Suc)
  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)
                         contour_integrable_on \<gamma>"
    apply (simp add: eventually_at)
    apply (rule_tac x=d in exI)
    apply (simp add: \<open>d > 0\<close> dist_norm field_simps, clarify)
    apply (rule contour_integrable_div [OF contour_integrable_diff])
    using int w d
    apply (force simp:  dist_norm norm_minus_commute intro!: has_contour_integral_integrable)+
    done
  have bim_g: "bounded (image f' (path_image \<gamma>))"
    by (simp add: compact_imp_bounded compact_continuous_image compact_valid_path_image assms)
  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"
    by (force simp: bounded_pos path_image_def)
  have twom: "\<forall>\<^sub>F n in at w.
               \<forall>x\<in>path_image \<gamma>.
                cmod ((inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k - inverse (x - w) ^ Suc k) < e"
         if "0 < e" for e
  proof -
    have *: "cmod ((inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k) - inverse (x - w) ^ Suc k)   < e"
            if x: "x \<in> path_image \<gamma>" and "u \<noteq> w" and uwd: "cmod (u - w) < d/2"
                and uw_less: "cmod (u - w) < e * (d / 2) ^ (k+2) / (1 + real k)"
            for u x
    proof -
      def ff \<equiv> "\<lambda>n::nat. \<lambda>w. if n = 0 then inverse(x - w)^k
                              else if n = 1 then k / (x - w)^(Suc k)
                              else (k * of_real(Suc k)) / (x - w)^(k + 2)"
      have km1: "\<And>z::complex. z \<noteq> 0 \<Longrightarrow> z ^ (k - Suc 0) = z ^ k / z"
        by (simp add: field_simps) (metis Suc_pred \<open>k \<noteq> 0\<close> neq0_conv power_Suc)
      have ff1: "(ff i has_field_derivative ff (Suc i) z) (at z within ball w (d / 2))"
              if "z \<in> ball w (d / 2)" "i \<le> 1" for i z
      proof -
        have "z \<notin> path_image \<gamma>"
          using \<open>x \<in> path_image \<gamma>\<close> d that ball_divide_subset_numeral by blast
        then have xz[simp]: "x \<noteq> z" using \<open>x \<in> path_image \<gamma>\<close> by blast
        then have neq: "x * x + z * z \<noteq> x * (z * 2)"
          by (blast intro: dest!: sum_sqs_eq)
        with xz have "\<And>v. v \<noteq> 0 \<Longrightarrow> (x * x + z * z) * v \<noteq> (x * (z * 2) * v)" by auto
        then have neqq: "\<And>v. v \<noteq> 0 \<Longrightarrow> x * (x * v) + z * (z * v) \<noteq> x * (z * (2 * v))"
          by (simp add: algebra_simps)
        show ?thesis using \<open>i \<le> 1\<close>
          apply (simp add: ff_def dist_norm Nat.le_Suc_eq km1, safe)
          apply (rule derivative_eq_intros | simp add: km1 | simp add: field_simps neq neqq)+
          done
      qed
      { fix a::real and b::real assume ab: "a > 0" "b > 0"
        then have "k * (1 + real k) * (1 / a) \<le> k * (1 + real k) * (4 / b) \<longleftrightarrow> b \<le> 4 * a"
          apply (subst mult_le_cancel_left_pos)
          using \<open>k \<noteq> 0\<close>
          apply (auto simp: divide_simps)
          done
        with ab have "real k * (1 + real k) / a \<le> (real k * 4 + real k * real k * 4) / b \<longleftrightarrow> b \<le> 4 * a"
          by (simp add: field_simps)
      } note canc = this
      have ff2: "cmod (ff (Suc 1) v) \<le> real (k * (k + 1)) / (d / 2) ^ (k + 2)"
                if "v \<in> ball w (d / 2)" for v
      proof -
        have "d/2 \<le> cmod (x - v)" using d x that
          apply (simp add: dist_norm path_image_def ball_def not_less [symmetric] del: divide_const_simps, clarify)
          apply (drule subsetD)
           prefer 2 apply blast
          apply (metis norm_minus_commute norm_triangle_half_r CollectI)
          done
        then have "d \<le> cmod (x - v) * 2"
          by (simp add: divide_simps)
        then have dpow_le: "d ^ (k+2) \<le> (cmod (x - v) * 2) ^ (k+2)"
          using \<open>0 < d\<close> order_less_imp_le power_mono by blast
        have "x \<noteq> v" using that
          using \<open>x \<in> path_image \<gamma>\<close> ball_divide_subset_numeral d by fastforce
        then show ?thesis
        using \<open>d > 0\<close>
        apply (simp add: ff_def norm_mult norm_divide norm_power dist_norm canc)
        using dpow_le
        apply (simp add: algebra_simps divide_simps mult_less_0_iff)
        done
      qed
      have ub: "u \<in> ball w (d / 2)"
        using uwd by (simp add: dist_commute dist_norm)
      have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
                  \<le> (real k * 4 + real k * real k * 4) * (cmod (u - w) * cmod (u - w)) / (d * (d * (d / 2) ^ k))"
        using complex_taylor [OF _ ff1 ff2 _ ub, of w, simplified]
        by (simp add: ff_def \<open>0 < d\<close>)
      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
                  \<le> (cmod (u - w) * real k) * (1 + real k) * cmod (u - w) / (d / 2) ^ (k+2)"
        by (simp add: field_simps)
      then have "cmod (inverse (x - u) ^ k - (inverse (x - w) ^ k + of_nat k * (u - w) / ((x - w) * (x - w) ^ k)))
                 / (cmod (u - w) * real k)
                  \<le> (1 + real k) * cmod (u - w) / (d / 2) ^ (k+2)"
        using \<open>k \<noteq> 0\<close> \<open>u \<noteq> w\<close> by (simp add: mult_ac zero_less_mult_iff pos_divide_le_eq)
      also have "... < e"
        using uw_less \<open>0 < d\<close> by (simp add: mult_ac divide_simps)
      finally have e: "cmod (inverse (x-u)^k - (inverse (x-w)^k + of_nat k * (u-w) / ((x-w) * (x-w)^k)))
                        / cmod ((u - w) * real k)   <   e"
        by (simp add: norm_mult)
      have "x \<noteq> u"
        using uwd \<open>0 < d\<close> x d by (force simp: dist_norm ball_def norm_minus_commute)
      show ?thesis
        apply (rule le_less_trans [OF _ e])
        using \<open>k \<noteq> 0\<close> \<open>x \<noteq> u\<close>  \<open>u \<noteq> w\<close>
        apply (simp add: field_simps norm_divide [symmetric])
        done
    qed
    show ?thesis
      unfolding eventually_at
      apply (rule_tac x = "min (d/2) ((e*(d/2)^(k + 2))/(Suc k))" in exI)
      apply (force simp: \<open>d > 0\<close> dist_norm that simp del: power_Suc intro: *)
      done
  qed
  have 2: "\<forall>\<^sub>F n in at w.
              \<forall>x\<in>path_image \<gamma>.
               cmod (f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k - f' x / (x - w) ^ Suc k) < e"
          if "0 < e" for e
  proof -
    have *: "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) < e"
              if ec: "cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
                      inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k) < e / C"
                 and x: "0 \<le> x" "x \<le> 1"
              for u x
    proof (cases "(f' (\<gamma> x)) = 0")
      case True then show ?thesis by (simp add: \<open>0 < e\<close>)
    next
      case False
      have "cmod (f' (\<gamma> x) * (inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
                        f' (\<gamma> x) / ((\<gamma> x - w) * (\<gamma> x - w) ^ k)) =
            cmod (f' (\<gamma> x) * ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k))"
        by (simp add: field_simps)
      also have "... = cmod (f' (\<gamma> x)) *
                       cmod ((inverse (\<gamma> x - u) ^ k - inverse (\<gamma> x - w) ^ k) / ((u - w) * k) -
                             inverse (\<gamma> x - w) * inverse (\<gamma> x - w) ^ k)"
        by (simp add: norm_mult)
      also have "... < cmod (f' (\<gamma> x)) * (e/C)"
        apply (rule mult_strict_left_mono [OF ec])
        using False by simp
      also have "... \<le> e" using C
        by (metis False \<open>0 < e\<close> frac_le less_eq_real_def mult.commute pos_le_divide_eq x zero_less_norm_iff)
      finally show ?thesis .
    qed
    show ?thesis
      using twom [OF divide_pos_pos [OF that \<open>C > 0\<close>]]   unfolding path_image_def
      by (force intro: * elim: eventually_mono)
  qed
  show "(\<lambda>u. f' u / (u - w) ^ (Suc k)) contour_integrable_on \<gamma>"
    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
  have *: "(\<lambda>n. contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / k))
           \<midarrow>w\<rightarrow> contour_integral \<gamma> (\<lambda>u. f' u / (u - w) ^ (Suc k))"
    by (rule contour_integral_uniform_limit [OF 1 2 leB \<gamma>]) auto
  have **: "contour_integral \<gamma> (\<lambda>x. f' x * (inverse (x - u) ^ k - inverse (x - w) ^ k) / ((u - w) * k)) =
              (f u - f w) / (u - w) / k"
           if "dist u w < d" for u
    apply (rule contour_integral_unique)
    apply (simp add: diff_divide_distrib algebra_simps)
    apply (rule has_contour_integral_diff; rule has_contour_integral_div; simp add: field_simps; rule int)
    apply (metis contra_subsetD d dist_commute mem_ball that)
    apply (rule w)
    done
  show ?thes2
    apply (simp add: DERIV_within_iff del: power_Suc)
    apply (rule Lim_transform_within [OF tendsto_mult_left [OF *] \<open>0 < d\<close> ])
    apply (simp add: \<open>k \<noteq> 0\<close> **)
    done
qed

corollary Cauchy_next_derivative_circlepath:
  assumes contf: "continuous_on (path_image (circlepath z r)) f"
      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)"
      and k: "k \<noteq> 0"
      and w: "w \<in> ball z r"
    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
           (is "?thes1")
      and "(g has_field_derivative (k * contour_integral (circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)))) (at w)"
           (is "?thes2")
proof -
  have "r > 0" using w
    using ball_eq_empty by fastforce
  have wim: "w \<in> ball z r - path_image (circlepath z r)"
    using w by (auto simp: dist_norm)
  show ?thes1 ?thes2
    by (rule Cauchy_next_derivative [OF contf _ int k open_ball valid_path_circlepath wim, where B = "2 * pi * \<bar>r\<bar>"];
        auto simp: vector_derivative_circlepath norm_mult)+
qed


text\<open> In particular, the first derivative formula.\<close>

proposition Cauchy_derivative_integral_circlepath:
  assumes contf: "continuous_on (cball z r) f"
      and holf: "f holomorphic_on ball z r"
      and w: "w \<in> ball z r"
    shows "(\<lambda>u. f u/(u - w)^2) contour_integrable_on (circlepath z r)"
           (is "?thes1")
      and "(f has_field_derivative (1 / (2 * of_real pi * ii) * contour_integral(circlepath z r) (\<lambda>u. f u / (u - w)^2))) (at w)"
           (is "?thes2")
proof -
  have [simp]: "r \<ge> 0" using w
    using ball_eq_empty by fastforce
  have f: "continuous_on (path_image (circlepath z r)) f"
    by (rule continuous_on_subset [OF contf]) (force simp add: cball_def sphere_def)
  have int: "\<And>w. dist z w < r \<Longrightarrow>
                 ((\<lambda>u. f u / (u - w)) has_contour_integral (\<lambda>x. 2 * of_real pi * ii * f x) w) (circlepath z r)"
    by (rule Cauchy_integral_circlepath [OF contf holf]) (simp add: dist_norm norm_minus_commute)
  show ?thes1
    apply (simp add: power2_eq_square)
    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1, simplified])
    apply (blast intro: int)
    done
  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)"
    apply (simp add: power2_eq_square)
    apply (rule Cauchy_next_derivative_circlepath [OF f _ _ w, where k=1 and g = "\<lambda>x. 2 * of_real pi * ii * f x", simplified])
    apply (blast intro: int)
    done
  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)"
    by (rule DERIV_cdivide [where f = "\<lambda>x. 2 * of_real pi * \<i> * f x" and c = "2 * of_real pi * \<i>", simplified])
  show ?thes2
    by simp (rule fder)
qed

subsection\<open> Existence of all higher derivatives.\<close>

proposition derivative_is_holomorphic:
  assumes "open s"
      and fder: "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z)"
    shows "f' holomorphic_on s"
proof -
  have *: "\<exists>h. (f' has_field_derivative h) (at z)" if "z \<in> s" for z
  proof -
    obtain r where "r > 0" and r: "cball z r \<subseteq> s"
      using open_contains_cball \<open>z \<in> s\<close> \<open>open s\<close> by blast
    then have holf_cball: "f holomorphic_on cball z r"
      apply (simp add: holomorphic_on_def)
      using complex_differentiable_at_within complex_differentiable_def fder by blast
    then have "continuous_on (path_image (circlepath z r)) f"
      using \<open>r > 0\<close> by (force elim: holomorphic_on_subset [THEN holomorphic_on_imp_continuous_on])
    then have contfpi: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1/(2 * of_real pi*ii) * f x)"
      by (auto intro: continuous_intros)+
    have contf_cball: "continuous_on (cball z r) f" using holf_cball
      by (simp add: holomorphic_on_imp_continuous_on holomorphic_on_subset)
    have holf_ball: "f holomorphic_on ball z r" using holf_cball
      using ball_subset_cball holomorphic_on_subset by blast
    { fix w  assume w: "w \<in> ball z r"
      have intf: "(\<lambda>u. f u / (u - w)\<^sup>2) contour_integrable_on circlepath z r"
        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
      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))
                  (at w)"
        by (blast intro: w Cauchy_derivative_integral_circlepath [OF contf_cball holf_ball])
      have f'_eq: "f' w = contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>)"
        using fder' ball_subset_cball r w by (force intro: DERIV_unique [OF fder])
      have "((\<lambda>u. f u / (u - w)\<^sup>2 / (2 * of_real pi * \<i>)) has_contour_integral
                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
                (circlepath z r)"
        by (rule Cauchy_Integral_Thm.has_contour_integral_div [OF has_contour_integral_integral [OF intf]])
      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral
                contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * of_real pi * \<i>))
                (circlepath z r)"
        by (simp add: algebra_simps)
      then have "((\<lambda>u. f u / (2 * of_real pi * \<i> * (u - w)\<^sup>2)) has_contour_integral f' w) (circlepath z r)"
        by (simp add: f'_eq)
    } note * = this
    show ?thesis
      apply (rule exI)
      apply (rule Cauchy_next_derivative_circlepath [OF contfpi, of 2 f', simplified])
      apply (simp_all add: \<open>0 < r\<close> * dist_norm)
      done
  qed
  show ?thesis
    by (simp add: holomorphic_on_open [OF \<open>open s\<close>] *)
qed

lemma holomorphic_deriv [holomorphic_intros]:
    "\<lbrakk>f holomorphic_on s; open s\<rbrakk> \<Longrightarrow> (deriv f) holomorphic_on s"
by (metis DERIV_deriv_iff_complex_differentiable at_within_open derivative_is_holomorphic holomorphic_on_def)

lemma analytic_deriv: "f analytic_on s \<Longrightarrow> (deriv f) analytic_on s"
  using analytic_on_holomorphic holomorphic_deriv by auto

lemma holomorphic_higher_deriv [holomorphic_intros]: "\<lbrakk>f holomorphic_on s; open s\<rbrakk> \<Longrightarrow> (deriv ^^ n) f holomorphic_on s"
  by (induction n) (auto simp: holomorphic_deriv)

lemma analytic_higher_deriv: "f analytic_on s \<Longrightarrow> (deriv ^^ n) f analytic_on s"
  unfolding analytic_on_def using holomorphic_higher_deriv by blast

lemma has_field_derivative_higher_deriv:
     "\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk>
      \<Longrightarrow> ((deriv ^^ n) f has_field_derivative (deriv ^^ (Suc n)) f x) (at x)"
by (metis (no_types, hide_lams) DERIV_deriv_iff_complex_differentiable at_within_open comp_apply
         funpow.simps(2) holomorphic_higher_deriv holomorphic_on_def)


subsection\<open> Morera's theorem.\<close>

lemma Morera_local_triangle_ball:
  assumes "\<And>z. z \<in> s
          \<Longrightarrow> \<exists>e a. 0 < e \<and> z \<in> ball a e \<and> continuous_on (ball a e) f \<and>
                    (\<forall>b c. closed_segment b c \<subseteq> ball a e
                           \<longrightarrow> contour_integral (linepath a b) f +
                               contour_integral (linepath b c) f +
                               contour_integral (linepath c a) f = 0)"
  shows "f analytic_on s"
proof -
  { fix z  assume "z \<in> s"
    with assms obtain e a where
            "0 < e" and z: "z \<in> ball a e" and contf: "continuous_on (ball a e) f"
        and 0: "\<And>b c. closed_segment b c \<subseteq> ball a e
                      \<Longrightarrow> contour_integral (linepath a b) f +
                          contour_integral (linepath b c) f +
                          contour_integral (linepath c a) f = 0"
      by fastforce
    have az: "dist a z < e" using mem_ball z by blast
    have sb_ball: "ball z (e - dist a z) \<subseteq> ball a e"
      by (simp add: dist_commute ball_subset_ball_iff)
    have "\<exists>e>0. f holomorphic_on ball z e"
      apply (rule_tac x="e - dist a z" in exI)
      apply (simp add: az)
      apply (rule holomorphic_on_subset [OF _ sb_ball])
      apply (rule derivative_is_holomorphic[OF open_ball])
      apply (rule triangle_contour_integrals_starlike_primitive [OF contf _ open_ball, of a])
         apply (simp_all add: 0 \<open>0 < e\<close>)
      apply (meson \<open>0 < e\<close> centre_in_ball convex_ball convex_contains_segment mem_ball)
      done
  }
  then show ?thesis
    by (simp add: analytic_on_def)
qed

lemma Morera_local_triangle:
  assumes "\<And>z. z \<in> s
          \<Longrightarrow> \<exists>t. open t \<and> z \<in> t \<and> continuous_on t f \<and>
                  (\<forall>a b c. convex hull {a,b,c} \<subseteq> t
                              \<longrightarrow> contour_integral (linepath a b) f +
                                  contour_integral (linepath b c) f +
                                  contour_integral (linepath c a) f = 0)"
  shows "f analytic_on s"
proof -
  { fix z  assume "z \<in> s"
    with assms obtain t where
            "open t" and z: "z \<in> t" and contf: "continuous_on t f"
        and 0: "\<And>a b c. convex hull {a,b,c} \<subseteq> t
                      \<Longrightarrow> contour_integral (linepath a b) f +
                          contour_integral (linepath b c) f +
                          contour_integral (linepath c a) f = 0"
      by force
    then obtain e where "e>0" and e: "ball z e \<subseteq> t"
      using open_contains_ball by blast
    have [simp]: "continuous_on (ball z e) f" using contf
      using continuous_on_subset e by blast
    have "\<exists>e a. 0 < e \<and>
               z \<in> ball a e \<and>
               continuous_on (ball a e) f \<and>
               (\<forall>b c. closed_segment b c \<subseteq> ball a e \<longrightarrow>
                      contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0)"
      apply (rule_tac x=e in exI)
      apply (rule_tac x=z in exI)
      apply (simp add: \<open>e > 0\<close>, clarify)
      apply (rule 0)
      apply (meson z \<open>0 < e\<close> centre_in_ball closed_segment_subset convex_ball dual_order.trans e starlike_convex_subset)
      done
  }
  then show ?thesis
    by (simp add: Morera_local_triangle_ball)
qed

proposition Morera_triangle:
    "\<lbrakk>continuous_on s f; open s;
      \<And>a b c. convex hull {a,b,c} \<subseteq> s
              \<longrightarrow> contour_integral (linepath a b) f +
                  contour_integral (linepath b c) f +
                  contour_integral (linepath c a) f = 0\<rbrakk>
     \<Longrightarrow> f analytic_on s"
  using Morera_local_triangle by blast



subsection\<open> Combining theorems for higher derivatives including Leibniz rule.\<close>

lemma higher_deriv_linear [simp]:
    "(deriv ^^ n) (\<lambda>w. c*w) = (\<lambda>z. if n = 0 then c*z else if n = 1 then c else 0)"
  by (induction n) (auto simp: deriv_const deriv_linear)

lemma higher_deriv_const [simp]: "(deriv ^^ n) (\<lambda>w. c) = (\<lambda>w. if n=0 then c else 0)"
  by (induction n) (auto simp: deriv_const)

lemma higher_deriv_ident [simp]:
     "(deriv ^^ n) (\<lambda>w. w) z = (if n = 0 then z else if n = 1 then 1 else 0)"
  apply (induction n, simp)
  apply (metis higher_deriv_linear lambda_one)
  done

corollary higher_deriv_id [simp]:
     "(deriv ^^ n) id z = (if n = 0 then z else if n = 1 then 1 else 0)"
  by (simp add: id_def)

lemma has_complex_derivative_funpow_1:
     "\<lbrakk>(f has_field_derivative 1) (at z); f z = z\<rbrakk> \<Longrightarrow> (f^^n has_field_derivative 1) (at z)"
  apply (induction n)
  apply auto
  apply (metis DERIV_ident DERIV_transform_at id_apply zero_less_one)
  by (metis DERIV_chain comp_funpow comp_id funpow_swap1 mult.right_neutral)

proposition higher_deriv_uminus:
  assumes "f holomorphic_on s" "open s" and z: "z \<in> s"
    shows "(deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
using z
proof (induction n arbitrary: z)
  case 0 then show ?case by simp
next
  case (Suc n z)
  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
    using Suc.prems assms has_field_derivative_higher_deriv by auto
  show ?case
    apply simp
    apply (rule DERIV_imp_deriv)
    apply (rule DERIV_transform_within_open [of "\<lambda>w. -((deriv ^^ n) f w)"])
    apply (rule derivative_eq_intros | rule * refl assms Suc)+
    apply (simp add: Suc)
    done
qed

proposition higher_deriv_add:
  fixes z::complex
  assumes "f holomorphic_on s" "g holomorphic_on s" "open s" and z: "z \<in> s"
    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
using z
proof (induction n arbitrary: z)
  case 0 then show ?case by simp
next
  case (Suc n z)
  have *: "((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
          "((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
    using Suc.prems assms has_field_derivative_higher_deriv by auto
  show ?case
    apply simp
    apply (rule DERIV_imp_deriv)
    apply (rule DERIV_transform_within_open [of "\<lambda>w. (deriv ^^ n) f w + (deriv ^^ n) g w"])
    apply (rule derivative_eq_intros | rule * refl assms Suc)+
    apply (simp add: Suc)
    done
qed

corollary higher_deriv_diff:
  fixes z::complex
  assumes "f holomorphic_on s" "g holomorphic_on s" "open s" and z: "z \<in> s"
    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
  apply (simp only: Groups.group_add_class.diff_conv_add_uminus higher_deriv_add)
  apply (subst higher_deriv_add)
  using assms holomorphic_on_minus apply (auto simp: higher_deriv_uminus)
  done


lemma bb: "Suc n choose k = (n choose k) + (if k = 0 then 0 else (n choose (k - 1)))"
  by (simp add: Binomial.binomial.simps)

proposition higher_deriv_mult:
  fixes z::complex
  assumes "f holomorphic_on s" "g holomorphic_on s" "open s" and z: "z \<in> s"
    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
using z
proof (induction n arbitrary: z)
  case 0 then show ?case by simp
next
  case (Suc n z)
  have *: "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) z) (at z)"
          "\<And>n. ((deriv ^^ n) g has_field_derivative deriv ((deriv ^^ n) g) z) (at z)"
    using Suc.prems assms has_field_derivative_higher_deriv by auto
  have sumeq: "(\<Sum>i = 0..n.
               of_nat (n choose i) * (deriv ((deriv ^^ i) f) z * (deriv ^^ (n - i)) g z + deriv ((deriv ^^ (n - i)) g) z * (deriv ^^ i) f z)) =
            g z * deriv ((deriv ^^ n) f) z + (\<Sum>i = 0..n. (deriv ^^ i) f z * (of_nat (Suc n choose i) * (deriv ^^ (Suc n - i)) g z))"
    apply (simp add: bb distrib_right algebra_simps setsum.distrib)
    apply (subst (4) setsum_Suc_reindex)
    apply (auto simp: algebra_simps Suc_diff_le intro: setsum.cong)
    done
  show ?case
    apply (simp only: funpow.simps o_apply)
    apply (rule DERIV_imp_deriv)
    apply (rule DERIV_transform_within_open
             [of "\<lambda>w. (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f w * (deriv ^^ (n - i)) g w)"])
    apply (simp add: algebra_simps)
    apply (rule DERIV_cong [OF DERIV_setsum])
    apply (rule DERIV_cmult)
    apply (auto simp: intro: DERIV_mult * sumeq \<open>open s\<close> Suc.prems Suc.IH [symmetric])
    done
qed


proposition higher_deriv_transform_within_open:
  fixes z::complex
  assumes "f holomorphic_on s" "g holomorphic_on s" "open s" and z: "z \<in> s"
      and fg: "\<And>w. w \<in> s \<Longrightarrow> f w = g w"
    shows "(deriv ^^ i) f z = (deriv ^^ i) g z"
using z
by (induction i arbitrary: z)
   (auto simp: fg intro: complex_derivative_transform_within_open holomorphic_higher_deriv assms)

proposition higher_deriv_compose_linear:
  fixes z::complex
  assumes f: "f holomorphic_on t" and s: "open s" and t: "open t" and z: "z \<in> s"
      and fg: "\<And>w. w \<in> s \<Longrightarrow> u * w \<in> t"
    shows "(deriv ^^ n) (\<lambda>w. f (u * w)) z = u^n * (deriv ^^ n) f (u * z)"
using z
proof (induction n arbitrary: z)
  case 0 then show ?case by simp
next
  case (Suc n z)
  have holo0: "f holomorphic_on op * u ` s"
    by (meson fg f holomorphic_on_subset image_subset_iff)
  have holo1: "(\<lambda>w. f (u * w)) holomorphic_on s"
    apply (rule holomorphic_on_compose [where g=f, unfolded o_def])
    apply (rule holo0 holomorphic_intros)+
    done
  have holo2: "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on s"
    apply (rule holomorphic_intros)+
    apply (rule holomorphic_on_compose [where g="(deriv ^^ n) f", unfolded o_def])
    apply (rule holomorphic_intros)
    apply (rule holomorphic_on_subset [where s=t])
    apply (rule holomorphic_intros assms)+
    apply (blast intro: fg)
    done
  have "deriv ((deriv ^^ n) (\<lambda>w. f (u * w))) z = deriv (\<lambda>z. u^n * (deriv ^^ n) f (u*z)) z"
    apply (rule complex_derivative_transform_within_open [OF _ holo2 s Suc.prems])
    apply (rule holomorphic_higher_deriv [OF holo1 s])
    apply (simp add: Suc.IH)
    done
  also have "... = u^n * deriv (\<lambda>z. (deriv ^^ n) f (u * z)) z"
    apply (rule complex_derivative_cmult)
    apply (rule analytic_on_imp_differentiable_at [OF _ Suc.prems])
    apply (rule analytic_on_compose_gen [where g="(deriv ^^ n) f" and t=t, unfolded o_def])
    apply (simp add: analytic_on_linear)
    apply (simp add: analytic_on_open f holomorphic_higher_deriv t)
    apply (blast intro: fg)
    done
  also have "... = u * u ^ n * deriv ((deriv ^^ n) f) (u * z)"
      apply (subst complex_derivative_chain [where g = "(deriv ^^ n) f" and f = "op*u", unfolded o_def])
      apply (rule derivative_intros)
      using Suc.prems complex_differentiable_def f fg has_field_derivative_higher_deriv t apply blast
      apply (simp add: deriv_linear)
      done
  finally show ?case
    by simp
qed

lemma higher_deriv_add_at:
  assumes "f analytic_on {z}" "g analytic_on {z}"
    shows "(deriv ^^ n) (\<lambda>w. f w + g w) z = (deriv ^^ n) f z + (deriv ^^ n) g z"
proof -
  have "f analytic_on {z} \<and> g analytic_on {z}"
    using assms by blast
  with higher_deriv_add show ?thesis
    by (auto simp: analytic_at_two)
qed

lemma higher_deriv_diff_at:
  assumes "f analytic_on {z}" "g analytic_on {z}"
    shows "(deriv ^^ n) (\<lambda>w. f w - g w) z = (deriv ^^ n) f z - (deriv ^^ n) g z"
proof -
  have "f analytic_on {z} \<and> g analytic_on {z}"
    using assms by blast
  with higher_deriv_diff show ?thesis
    by (auto simp: analytic_at_two)
qed

lemma higher_deriv_uminus_at:
   "f analytic_on {z}  \<Longrightarrow> (deriv ^^ n) (\<lambda>w. -(f w)) z = - ((deriv ^^ n) f z)"
  using higher_deriv_uminus
    by (auto simp: analytic_at)

lemma higher_deriv_mult_at:
  assumes "f analytic_on {z}" "g analytic_on {z}"
    shows "(deriv ^^ n) (\<lambda>w. f w * g w) z =
           (\<Sum>i = 0..n. of_nat (n choose i) * (deriv ^^ i) f z * (deriv ^^ (n - i)) g z)"
proof -
  have "f analytic_on {z} \<and> g analytic_on {z}"
    using assms by blast
  with higher_deriv_mult show ?thesis
    by (auto simp: analytic_at_two)
qed


text\<open> Nonexistence of isolated singularities and a stronger integral formula.\<close>

proposition no_isolated_singularity:
  fixes z::complex
  assumes f: "continuous_on s f" and holf: "f holomorphic_on (s - k)" and s: "open s" and k: "finite k"
    shows "f holomorphic_on s"
proof -
  { fix z
    assume "z \<in> s" and cdf: "\<And>x. x\<in>s - k \<Longrightarrow> f complex_differentiable at x"
    have "f complex_differentiable at z"
    proof (cases "z \<in> k")
      case False then show ?thesis by (blast intro: cdf \<open>z \<in> s\<close>)
    next
      case True
      with finite_set_avoid [OF k, of z]
      obtain d where "d>0" and d: "\<And>x. \<lbrakk>x\<in>k; x \<noteq> z\<rbrakk> \<Longrightarrow> d \<le> dist z x"
        by blast
      obtain e where "e>0" and e: "ball z e \<subseteq> s"
        using  s \<open>z \<in> s\<close> by (force simp add: open_contains_ball)
      have fde: "continuous_on (ball z (min d e)) f"
        by (metis Int_iff ball_min_Int continuous_on_subset e f subsetI)
      have "\<exists>g. \<forall>w \<in> ball z (min d e). (g has_field_derivative f w) (at w within ball z (min d e))"
        apply (rule contour_integral_convex_primitive [OF convex_ball fde])
        apply (rule Cauchy_theorem_triangle_cofinite [OF _ k])
         apply (metis continuous_on_subset [OF fde] closed_segment_subset convex_ball starlike_convex_subset)
        apply (rule cdf)
        apply (metis Diff_iff Int_iff ball_min_Int bot_least contra_subsetD convex_ball e insert_subset
               interior_mono interior_subset subset_hull)
        done
      then have "f holomorphic_on ball z (min d e)"
        by (metis open_ball at_within_open derivative_is_holomorphic)
      then show ?thesis
        unfolding holomorphic_on_def
        by (metis open_ball \<open>0 < d\<close> \<open>0 < e\<close> at_within_open centre_in_ball min_less_iff_conj)
    qed
  }
  with holf s k show ?thesis
    by (simp add: holomorphic_on_open open_Diff finite_imp_closed complex_differentiable_def [symmetric])
qed

proposition Cauchy_integral_formula_convex:
    assumes s: "convex s" and k: "finite k" and contf: "continuous_on s f"
        and fcd: "(\<And>x. x \<in> interior s - k \<Longrightarrow> f complex_differentiable at x)"
        and z: "z \<in> interior s" and vpg: "valid_path \<gamma>"
        and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
  apply (rule Cauchy_integral_formula_weak [OF s finite.emptyI contf])
  apply (simp add: holomorphic_on_open [symmetric] complex_differentiable_def)
  using no_isolated_singularity [where s = "interior s"]
  apply (metis k contf fcd holomorphic_on_open complex_differentiable_def continuous_on_subset
               has_field_derivative_at_within holomorphic_on_def interior_subset open_interior)
  using assms
  apply auto
  done


text\<open> Formula for higher derivatives.\<close>

proposition Cauchy_has_contour_integral_higher_derivative_circlepath:
  assumes contf: "continuous_on (cball z r) f"
      and holf: "f holomorphic_on ball z r"
      and w: "w \<in> ball z r"
    shows "((\<lambda>u. f u / (u - w) ^ (Suc k)) has_contour_integral ((2 * pi * ii) / (fact k) * (deriv ^^ k) f w))
           (circlepath z r)"
using w
proof (induction k arbitrary: w)
  case 0 then show ?case
    using assms by (auto simp: Cauchy_integral_circlepath dist_commute dist_norm)
next
  case (Suc k)
  have [simp]: "r > 0" using w
    using ball_eq_empty by fastforce
  have f: "continuous_on (path_image (circlepath z r)) f"
    by (rule continuous_on_subset [OF contf]) (force simp add: cball_def sphere_def less_imp_le)
  obtain X where X: "((\<lambda>u. f u / (u - w) ^ Suc (Suc k)) has_contour_integral X) (circlepath z r)"
    using Cauchy_next_derivative_circlepath(1) [OF f Suc.IH _ Suc.prems]
    by (auto simp: contour_integrable_on_def)
  then have con: "contour_integral (circlepath z r) ((\<lambda>u. f u / (u - w) ^ Suc (Suc k))) = X"
    by (rule contour_integral_unique)
  have "\<And>n. ((deriv ^^ n) f has_field_derivative deriv ((deriv ^^ n) f) w) (at w)"
    using Suc.prems assms has_field_derivative_higher_deriv by auto
  then have dnf_diff: "\<And>n. (deriv ^^ n) f complex_differentiable (at w)"
    by (force simp add: complex_differentiable_def)
  have "deriv (\<lambda>w. complex_of_real (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w) w =
          of_nat (Suc k) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w) ^ Suc (Suc k))"
    by (force intro!: DERIV_imp_deriv Cauchy_next_derivative_circlepath [OF f Suc.IH _ Suc.prems])
  also have "... = of_nat (Suc k) * X"
    by (simp only: con)
  finally have "deriv (\<lambda>w. ((2 * pi) * \<i> / (fact k)) * (deriv ^^ k) f w) w = of_nat (Suc k) * X" .
  then have "((2 * pi) * \<i> / (fact k)) * deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X"
    by (metis complex_derivative_cmult dnf_diff)
  then have "deriv (\<lambda>w. (deriv ^^ k) f w) w = of_nat (Suc k) * X / ((2 * pi) * \<i> / (fact k))"
    by (simp add: field_simps)
  then show ?case
  using of_nat_eq_0_iff X by fastforce
qed

proposition Cauchy_higher_derivative_integral_circlepath:
  assumes contf: "continuous_on (cball z r) f"
      and holf: "f holomorphic_on ball z r"
      and w: "w \<in> ball z r"
    shows "(\<lambda>u. f u / (u - w)^(Suc k)) contour_integrable_on (circlepath z r)"
           (is "?thes1")
      and "(deriv ^^ k) f w = (fact k) / (2 * pi * ii) * contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k))"
           (is "?thes2")
proof -
  have *: "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral (2 * pi) * \<i> / (fact k) * (deriv ^^ k) f w)
           (circlepath z r)"
    using Cauchy_has_contour_integral_higher_derivative_circlepath [OF assms]
    by simp
  show ?thes1 using *
    using contour_integrable_on_def by blast
  show ?thes2
    unfolding contour_integral_unique [OF *] by (simp add: divide_simps)
qed

corollary Cauchy_contour_integral_circlepath:
  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^(Suc k)) = (2 * pi * ii) * (deriv ^^ k) f w / (fact k)"
by (simp add: Cauchy_higher_derivative_integral_circlepath [OF assms])

corollary Cauchy_contour_integral_circlepath_2:
  assumes "continuous_on (cball z r) f" "f holomorphic_on ball z r" "w \<in> ball z r"
    shows "contour_integral(circlepath z r) (\<lambda>u. f u/(u - w)^2) = (2 * pi * ii) * deriv f w"
  using Cauchy_contour_integral_circlepath [OF assms, of 1]
  by (simp add: power2_eq_square)


subsection\<open>A holomorphic function is analytic, i.e. has local power series.\<close>

theorem holomorphic_power_series:
  assumes holf: "f holomorphic_on ball z r"
      and w: "w \<in> ball z r"
    shows "((\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
proof -
  have fh': "f holomorphic_on cball z ((r + dist w z) / 2)"
     apply (rule holomorphic_on_subset [OF holf])
     apply (clarsimp simp del: divide_const_simps)
     apply (metis add.commute dist_commute le_less_trans mem_ball real_gt_half_sum w)
     done
  \<comment>\<open>Replacing @{term r} and the original (weak) premises\<close>
  obtain r where "0 < r" and holfc: "f holomorphic_on cball z r" and w: "w \<in> ball z r"
    apply (rule that [of "(r + dist w z) / 2"])
      apply (simp_all add: fh')
     apply (metis add_0_iff ball_eq_empty dist_nz dist_self empty_iff not_less pos_add_strict w)
    apply (metis add_less_cancel_right dist_commute mem_ball mult_2_right w)
    done
  then have holf: "f holomorphic_on ball z r" and contf: "continuous_on (cball z r) f"
    using ball_subset_cball holomorphic_on_subset apply blast
    by (simp add: holfc holomorphic_on_imp_continuous_on)
  have cint: "\<And>k. (\<lambda>u. f u / (u - z) ^ Suc k) contour_integrable_on circlepath z r"
    apply (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf])
    apply (simp add: \<open>0 < r\<close>)
    done
  obtain B where "0 < B" and B: "\<And>u. u \<in> cball z r \<Longrightarrow> norm(f u) \<le> B"
    by (metis (no_types) bounded_pos compact_cball compact_continuous_image compact_imp_bounded contf image_eqI)
  obtain k where k: "0 < k" "k \<le> r" and wz_eq: "norm(w - z) = r - k"
             and kle: "\<And>u. norm(u - z) = r \<Longrightarrow> k \<le> norm(u - w)"
    apply (rule_tac k = "r - dist z w" in that)
    using w
    apply (auto simp: dist_norm norm_minus_commute)
    by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
  have *: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>path_image (circlepath z r).
                norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
          if "0 < e" for e
  proof -
    have rr: "0 \<le> (r - k) / r" "(r - k) / r < 1" using  k by auto
    obtain n where n: "((r - k) / r) ^ n < e / B * k"
      using real_arch_pow_inv [of "e/B*k" "(r - k)/r"] \<open>0 < e\<close> \<open>0 < B\<close> k by force
    have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) - f u / (u - w)) < e"
         if "n \<le> N" and r: "r = dist z u"  for N u
    proof -
      have N: "((r - k) / r) ^ N < e / B * k"
        apply (rule le_less_trans [OF power_decreasing n])
        using  \<open>n \<le> N\<close> k by auto
      have u [simp]: "(u \<noteq> z) \<and> (u \<noteq> w)"
        using \<open>0 < r\<close> r w by auto
      have wzu_not1: "(w - z) / (u - z) \<noteq> 1"
        by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
      have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
            = norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
        unfolding setsum_left_distrib setsum_divide_distrib power_divide by (simp add: algebra_simps)
      also have "... = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
        using \<open>0 < B\<close>
        apply (auto simp: geometric_sum [OF wzu_not1])
        apply (simp add: field_simps norm_mult [symmetric])
        done
      also have "... = norm ((u-z) ^ N * (w - u) - ((w - z) ^ N - (u-z) ^ N) * (u-w)) / (r ^ N * norm (u-w)) * norm (f u)"
        using \<open>0 < r\<close> r by (simp add: divide_simps norm_mult norm_divide norm_power dist_norm norm_minus_commute)
      also have "... = norm ((w - z) ^ N * (w - u)) / (r ^ N * norm (u - w)) * norm (f u)"
        by (simp add: algebra_simps)
      also have "... = norm (w - z) ^ N * norm (f u) / r ^ N"
        by (simp add: norm_mult norm_power norm_minus_commute)
      also have "... \<le> (((r - k)/r)^N) * B"
        using \<open>0 < r\<close> w k
        apply (simp add: divide_simps)
        apply (rule mult_mono [OF power_mono])
        apply (auto simp: norm_divide wz_eq norm_power dist_norm norm_minus_commute B r)
        done
      also have "... < e * k"
        using \<open>0 < B\<close> N by (simp add: divide_simps)
      also have "... \<le> e * norm (u - w)"
        using r kle \<open>0 < e\<close> by (simp add: dist_commute dist_norm)
      finally show ?thesis
        by (simp add: divide_simps norm_divide del: power_Suc)
    qed
    with \<open>0 < r\<close> show ?thesis
      by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
  qed
  have eq: "\<forall>\<^sub>F x in sequentially.
             contour_integral (circlepath z r) (\<lambda>u. \<Sum>k<x. (w - z) ^ k * (f u / (u - z) ^ Suc k)) =
             (\<Sum>k<x. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z) ^ k)"
    apply (rule eventuallyI)
    apply (subst contour_integral_setsum, simp)
    using contour_integrable_lmul [OF cint, of "(w - z) ^ a" for a] apply (simp add: field_simps)
    apply (simp only: contour_integral_lmul cint algebra_simps)
    done
  have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
        sums contour_integral (circlepath z r) (\<lambda>u. f u/(u - w))"
    unfolding sums_def
    apply (rule Lim_transform_eventually [OF eq])
    apply (rule contour_integral_uniform_limit_circlepath [OF eventuallyI *])
    apply (rule contour_integrable_setsum, simp)
    apply (rule contour_integrable_lmul)
    apply (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf])
    using \<open>0 < r\<close>
    apply auto
    done
  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u/(u - z)^(Suc k)) * (w - z)^k)
             sums (2 * of_real pi * ii * f w)"
    using w by (auto simp: dist_commute dist_norm contour_integral_unique [OF Cauchy_integral_circlepath_simple [OF holfc]])
  then have "(\<lambda>k. contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc k) * (w - z)^k / (\<i> * (of_real pi * 2)))
            sums ((2 * of_real pi * ii * f w) / (\<i> * (complex_of_real pi * 2)))"
    by (rule sums_divide)
  then have "(\<lambda>n. (w - z) ^ n * contour_integral (circlepath z r) (\<lambda>u. f u / (u - z) ^ Suc n) / (\<i> * (of_real pi * 2)))
            sums f w"
    by (simp add: field_simps)
  then show ?thesis
    by (simp add: field_simps \<open>0 < r\<close> Cauchy_higher_derivative_integral_circlepath [OF contf holf])
qed


subsection\<open>The Liouville theorem and the Fundamental Theorem of Algebra.\<close>

text\<open> These weak Liouville versions don't even need the derivative formula.\<close>

lemma Liouville_weak_0:
  assumes holf: "f holomorphic_on UNIV" and inf: "(f \<longlongrightarrow> 0) at_infinity"
    shows "f z = 0"
proof (rule ccontr)
  assume fz: "f z \<noteq> 0"
  with inf [unfolded Lim_at_infinity, rule_format, of "norm(f z)/2"]
  obtain B where B: "\<And>x. B \<le> cmod x \<Longrightarrow> norm (f x) * 2 < cmod (f z)"
    by (auto simp: dist_norm)
  def R \<equiv> "1 + \<bar>B\<bar> + norm z"
  have "R > 0" unfolding R_def by (meson abs_add_one_gt_zero le_less_trans less_add_same_cancel2 norm_ge_zero)
  have *: "((\<lambda>u. f u / (u - z)) has_contour_integral 2 * complex_of_real pi * \<i> * f z) (circlepath z R)"
    apply (rule Cauchy_integral_circlepath)
    using \<open>R > 0\<close> apply (auto intro: holomorphic_on_subset [OF holf] holomorphic_on_imp_continuous_on)+
    done
  have "cmod (x - z) = R \<Longrightarrow> cmod (f x) * 2 \<le> cmod (f z)" for x
    apply (simp add: R_def)
    apply (rule less_imp_le)
    apply (rule B)
    using norm_triangle_ineq4 [of x z]
    apply (auto simp:)
    done
  with  \<open>R > 0\<close> fz show False
    using has_contour_integral_bound_circlepath [OF *, of "norm(f z)/2/R"]
    by (auto simp: norm_mult norm_divide divide_simps)
qed

proposition Liouville_weak:
  assumes "f holomorphic_on UNIV" and "(f \<longlongrightarrow> l) at_infinity"
    shows "f z = l"
  using Liouville_weak_0 [of "\<lambda>z. f z - l"]
  by (simp add: assms holomorphic_on_const holomorphic_on_diff LIM_zero)


proposition Liouville_weak_inverse:
  assumes "f holomorphic_on UNIV" and unbounded: "\<And>B. eventually (\<lambda>x. norm (f x) \<ge> B) at_infinity"
    obtains z where "f z = 0"
proof -
  { assume f: "\<And>z. f z \<noteq> 0"
    have 1: "(\<lambda>x. 1 / f x) holomorphic_on UNIV"
      by (simp add: holomorphic_on_divide holomorphic_on_const assms f)
    have 2: "((\<lambda>x. 1 / f x) \<longlongrightarrow> 0) at_infinity"
      apply (rule tendstoI [OF eventually_mono])
      apply (rule_tac B="2/e" in unbounded)
      apply (simp add: dist_norm norm_divide divide_simps mult_ac)
      done
    have False
      using Liouville_weak_0 [OF 1 2] f by simp
  }
  then show ?thesis
    using that by blast
qed


text\<open> In particular we get the Fundamental Theorem of Algebra.\<close>

theorem fundamental_theorem_of_algebra:
    fixes a :: "nat \<Rightarrow> complex"
  assumes "a 0 = 0 \<or> (\<exists>i \<in> {1..n}. a i \<noteq> 0)"
  obtains z where "(\<Sum>i\<le>n. a i * z^i) = 0"
using assms
proof (elim disjE bexE)
  assume "a 0 = 0" then show ?thesis
    by (auto simp: that [of 0])
next
  fix i
  assume i: "i \<in> {1..n}" and nz: "a i \<noteq> 0"
  have 1: "(\<lambda>z. \<Sum>i\<le>n. a i * z^i) holomorphic_on UNIV"
    by (rule holomorphic_intros)+
  show ?thesis
    apply (rule Liouville_weak_inverse [OF 1])
    apply (rule polyfun_extremal)
    apply (rule nz)
    using i that
    apply (auto simp:)
    done
qed


subsection\<open> Weierstrass convergence theorem.\<close>

proposition holomorphic_uniform_limit:
  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and> (f n) holomorphic_on ball z r) F"
      and lim: "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>n. \<forall>x \<in> cball z r. norm(f n x - g x) < e) F"
      and F:  "~ trivial_limit F"
  obtains "continuous_on (cball z r) g" "g holomorphic_on ball z r"
proof (cases r "0::real" rule: linorder_cases)
  case less then show ?thesis by (force simp add: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
next
  case equal then show ?thesis
    by (force simp add: holomorphic_on_def continuous_on_sing intro: that)
next
  case greater
  have contg: "continuous_on (cball z r) g"
    using cont
    by (fastforce simp: eventually_conj_iff dist_norm intro: eventually_mono [OF lim] continuous_uniform_limit [OF F])
  have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
    apply (rule continuous_intros continuous_on_subset [OF contg])+
    using \<open>0 < r\<close> by auto
  have 2: "((\<lambda>u. 1 / (2 * of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
       if w: "w \<in> ball z r" for w
  proof -
    def d \<equiv> "(r - norm(w - z))"
    have "0 < d"  "d \<le> r" using w by (auto simp: norm_minus_commute d_def dist_norm)
    have dle: "\<And>u. cmod (z - u) = r \<Longrightarrow> d \<le> cmod (u - w)"
      unfolding d_def by (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
    have ev_int: "\<forall>\<^sub>F n in F. (\<lambda>u. f n u / (u - w)) contour_integrable_on circlepath z r"
      apply (rule eventually_mono [OF cont])
      using w
      apply (auto intro: Cauchy_higher_derivative_integral_circlepath [where k=0, simplified])
      done
    have ev_less: "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image (circlepath z r). cmod (f n x / (x - w) - g x / (x - w)) < e"
         if "e > 0" for e
      using greater \<open>0 < d\<close> \<open>0 < e\<close>
      apply (simp add: norm_divide diff_divide_distrib [symmetric] divide_simps)
      apply (rule_tac e1="e * d" in eventually_mono [OF lim])
      apply (force simp: dist_norm intro: dle mult_left_mono less_le_trans)+
      done
    have g_cint: "(\<lambda>u. g u/(u - w)) contour_integrable_on circlepath z r"
      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ev_less F \<open>0 < r\<close>])
    have cif_tends_cig: "((\<lambda>n. contour_integral(circlepath z r) (\<lambda>u. f n u / (u - w))) \<longlongrightarrow> contour_integral(circlepath z r) (\<lambda>u. g u/(u - w))) F"
      by (rule contour_integral_uniform_limit_circlepath [OF ev_int ev_less F \<open>0 < r\<close>])
    have f_tends_cig: "((\<lambda>n. 2 * of_real pi * ii * f n w) \<longlongrightarrow> contour_integral (circlepath z r) (\<lambda>u. g u / (u - w))) F"
      apply (rule Lim_transform_eventually [where f = "\<lambda>n. contour_integral (circlepath z r) (\<lambda>u. f n u/(u - w))"])
      apply (rule eventually_mono [OF cont])
      apply (rule contour_integral_unique [OF Cauchy_integral_circlepath])
      using w
      apply (auto simp: norm_minus_commute dist_norm cif_tends_cig)
      done
    have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
      apply (rule tendsto_mult_left [OF tendstoI])
      apply (rule eventually_mono [OF lim], assumption)
      using w
      apply (force simp add: dist_norm)
      done
    then have "((\<lambda>u. g u / (u - w)) has_contour_integral 2 * of_real pi * \<i> * g w) (circlepath z r)"
      using has_contour_integral_integral [OF g_cint] tendsto_unique [OF F f_tends_cig] w
      by (force simp add: dist_norm)
    then have "((\<lambda>u. g u / (2 * of_real pi * \<i> * (u - w))) has_contour_integral g w) (circlepath z r)"
      using has_contour_integral_div [where c = "2 * of_real pi * \<i>"]
      by (force simp add: field_simps)
    then show ?thesis
      by (simp add: dist_norm)
  qed
  show ?thesis
    using Cauchy_next_derivative_circlepath(2) [OF 1 2, simplified]
    by (fastforce simp add: holomorphic_on_open contg intro: that)
qed


text\<open> Version showing that the limit is the limit of the derivatives.\<close>

proposition has_complex_derivative_uniform_limit:
  fixes z::complex
  assumes cont: "eventually (\<lambda>n. continuous_on (cball z r) (f n) \<and>
                               (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))) F"
      and lim: "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>n. \<forall>x \<in> cball z r. norm(f n x - g x) < e) F"
      and F:  "~ trivial_limit F" and "0 < r"
  obtains g' where
      "continuous_on (cball z r) g"
      "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
proof -
  let ?conint = "contour_integral (circlepath z r)"
  have g: "continuous_on (cball z r) g" "g holomorphic_on ball z r"
    by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] lim F];
             auto simp: holomorphic_on_open complex_differentiable_def)+
  then obtain g' where g': "\<And>x. x \<in> ball z r \<Longrightarrow> (g has_field_derivative g' x) (at x)"
    using DERIV_deriv_iff_has_field_derivative
    by (fastforce simp add: holomorphic_on_open)
  then have derg: "\<And>x. x \<in> ball z r \<Longrightarrow> deriv g x = g' x"
    by (simp add: DERIV_imp_deriv)
  have tends_f'n_g': "((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F" if w: "w \<in> ball z r" for w
  proof -
    have eq_f': "?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2) = (f' n w - g' w) * (2 * of_real pi * \<i>)"
             if cont_fn: "continuous_on (cball z r) (f n)"
             and fnd: "\<And>w. w \<in> ball z r \<Longrightarrow> (f n has_field_derivative f' n w) (at w)" for n
    proof -
      have hol_fn: "f n holomorphic_on ball z r"
        using fnd by (force simp add: holomorphic_on_open)
      have "(f n has_field_derivative 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)) (at w)"
        by (rule Cauchy_derivative_integral_circlepath [OF cont_fn hol_fn w])
      then have f': "f' n w = 1 / (2 * of_real pi * \<i>) * ?conint (\<lambda>u. f n u / (u - w)\<^sup>2)"
        using DERIV_unique [OF fnd] w by blast
      show ?thesis
        by (simp add: f' Cauchy_contour_integral_circlepath_2 [OF g w] derg [OF w] divide_simps)
    qed
    def d \<equiv> "(r - norm(w - z))^2"
    have "d > 0"
      using w by (simp add: dist_commute dist_norm d_def)
    have dle: "\<And>y. r = cmod (z - y) \<Longrightarrow> d \<le> cmod ((y - w)\<^sup>2)"
      apply (simp add: d_def norm_power)
      apply (rule power_mono)
      apply (metis add_diff_eq diff_add_cancel norm_diff_ineq norm_minus_commute)
      apply (metis diff_ge_0_iff_ge dist_commute dist_norm less_eq_real_def mem_ball w)
      done
    have 1: "\<forall>\<^sub>F n in F. (\<lambda>x. f n x / (x - w)\<^sup>2) contour_integrable_on circlepath z r"
      by (force simp add: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
    have 2: "0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x \<in> path_image (circlepath z r). cmod (f n x / (x - w)\<^sup>2 - g x / (x - w)\<^sup>2) < e" for e
      using \<open>r > 0\<close>
      apply (simp add: diff_divide_distrib [symmetric] norm_divide divide_simps sphere_def)
      apply (rule eventually_mono [OF lim, of "e*d"])
      apply (simp add: \<open>0 < d\<close>)
      apply (force simp add: dist_norm dle intro: less_le_trans)
      done
    have "((\<lambda>n. contour_integral (circlepath z r) (\<lambda>x. f n x / (x - w)\<^sup>2))
             \<longlongrightarrow> contour_integral (circlepath z r) ((\<lambda>x. g x / (x - w)\<^sup>2))) F"
      by (rule Cauchy_Integral_Thm.contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
    then have tendsto_0: "((\<lambda>n. 1 / (2 * of_real pi * \<i>) * (?conint (\<lambda>x. f n x / (x - w)\<^sup>2) - ?conint (\<lambda>x. g x / (x - w)\<^sup>2))) \<longlongrightarrow> 0) F"
      using Lim_null by (force intro!: tendsto_mult_right_zero)
    have "((\<lambda>n. f' n w - g' w) \<longlongrightarrow> 0) F"
      apply (rule Lim_transform_eventually [OF _ tendsto_0])
      apply (force simp add: divide_simps intro: eq_f' eventually_mono [OF cont])
      done
    then show ?thesis using Lim_null by blast
  qed
  obtain g' where "\<And>w. w \<in> ball z r \<Longrightarrow> (g has_field_derivative (g' w)) (at w) \<and> ((\<lambda>n. f' n w) \<longlongrightarrow> g' w) F"
      by (blast intro: tends_f'n_g' g' )
  then show ?thesis using g
    using that by blast
qed


subsection\<open>Some more simple/convenient versions for applications.\<close>

lemma holomorphic_uniform_sequence:
  assumes s: "open s"
      and hol_fn: "\<And>n. (f n) holomorphic_on s"
      and to_g: "\<And>x. x \<in> s
                     \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> s \<and>
                             (\<forall>e. 0 < e \<longrightarrow> eventually (\<lambda>n. \<forall>y \<in> cball x d. norm(f n y - g y) < e) sequentially)"
  shows "g holomorphic_on s"
proof -
  have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> s" for z
  proof -
    obtain r where "0 < r" and r: "cball z r \<subseteq> s"
               and fg: "\<forall>e. 0 < e \<longrightarrow> eventually (\<lambda>n. \<forall>y \<in> cball z r. norm(f n y - g y) < e) sequentially"
      using to_g [OF \<open>z \<in> s\<close>] by blast
    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"
      apply (intro eventuallyI conjI)
      using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r apply blast
      apply (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
      done
    show ?thesis
      apply (rule holomorphic_uniform_limit [OF *])
      using \<open>0 < r\<close> centre_in_ball fg
      apply (auto simp: holomorphic_on_open)
      done
  qed
  with s show ?thesis
    by (simp add: holomorphic_on_open)
qed

lemma has_complex_derivative_uniform_sequence:
  fixes s :: "complex set"
  assumes s: "open s"
      and hfd: "\<And>n x. x \<in> s \<Longrightarrow> ((f n) has_field_derivative f' n x) (at x)"
      and to_g: "\<And>x. x \<in> s
             \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> s \<and>
                     (\<forall>e. 0 < e \<longrightarrow> eventually (\<lambda>n. \<forall>y \<in> cball x d. norm(f n y - g y) < e) sequentially)"
  shows "\<exists>g'. \<forall>x \<in> s. (g has_field_derivative g' x) (at x) \<and> ((\<lambda>n. f' n x) \<longlongrightarrow> g' x) sequentially"
proof -
  have y: "\<exists>y. (g has_field_derivative y) (at z) \<and> (\<lambda>n. f' n z) \<longlonglongrightarrow> y" if "z \<in> s" for z
  proof -
    obtain r where "0 < r" and r: "cball z r \<subseteq> s"
               and fg: "\<forall>e. 0 < e \<longrightarrow> eventually (\<lambda>n. \<forall>y \<in> cball z r. norm(f n y - g y) < e) sequentially"
      using to_g [OF \<open>z \<in> s\<close>] by blast
    have *: "\<forall>\<^sub>F n in sequentially. continuous_on (cball z r) (f n) \<and>
                                   (\<forall>w \<in> ball z r. ((f n) has_field_derivative (f' n w)) (at w))"
      apply (intro eventuallyI conjI)
      apply (meson hfd holomorphic_on_imp_continuous_on holomorphic_on_open holomorphic_on_subset r s)
      using ball_subset_cball hfd r apply blast
      done
    show ?thesis
      apply (rule has_complex_derivative_uniform_limit [OF *, of g])
      using \<open>0 < r\<close> centre_in_ball fg
      apply force+
      done
  qed
  show ?thesis
    by (rule bchoice) (blast intro: y)
qed


subsection\<open>On analytic functions defined by a series.\<close>

lemma series_and_derivative_comparison:
  fixes s :: "complex set"
  assumes s: "open s"
      and h: "summable h"
      and hfd: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
      and to_g: "\<And>n x. \<lbrakk>N \<le> n; x \<in> s\<rbrakk> \<Longrightarrow> norm(f n x) \<le> h n"
  obtains g g' where "\<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
proof -
  obtain g where g: "\<And>e. e>0 \<Longrightarrow> \<exists>N. \<forall>n x. N \<le> n \<and> x \<in> s \<longrightarrow> dist (\<Sum>n<n. f n x) (g x) < e"
    using series_comparison_uniform [OF h to_g, of N s] by force
  have *: "\<exists>d>0. cball x d \<subseteq> s \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>y\<in>cball x d. cmod ((\<Sum>a<n. f a y) - g y) < e)"
         if "x \<in> s" for x
  proof -
    obtain d where "d>0" and d: "cball x d \<subseteq> s"
      using open_contains_cball [of "s"] \<open>x \<in> s\<close> s by blast
    then show ?thesis
      apply (rule_tac x=d in exI)
      apply (auto simp: dist_norm eventually_sequentially)
      apply (metis g contra_subsetD dist_norm)
      done
  qed
  have "(\<forall>x\<in>s. (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x)"
    using g by (force simp add: lim_sequentially)
  moreover have "\<exists>g'. \<forall>x\<in>s. (g has_field_derivative g' x) (at x) \<and> (\<lambda>n. \<Sum>i<n. f' i x) \<longlonglongrightarrow> g' x"
    by (rule has_complex_derivative_uniform_sequence [OF s]) (auto intro: * hfd DERIV_setsum)+
  ultimately show ?thesis
    by (force simp add: sums_def  conj_commute intro: that)
qed

text\<open>A version where we only have local uniform/comparative convergence.\<close>

lemma series_and_derivative_comparison_local:
  fixes s :: "complex set"
  assumes s: "open s"
      and hfd: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
      and to_g: "\<And>x. x \<in> s \<Longrightarrow>
                      \<exists>d h N. 0 < d \<and> summable h \<and> (\<forall>n y. N \<le> n \<and> y \<in> ball x d \<longrightarrow> norm(f n y) \<le> h n)"
  shows "\<exists>g g'. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
proof -
  have "\<exists>y. (\<lambda>n. f n z) sums (\<Sum>n. f n z) \<and> (\<lambda>n. f' n z) sums y \<and> ((\<lambda>x. \<Sum>n. f n x) has_field_derivative y) (at z)"
       if "z \<in> s" for z
  proof -
    obtain d h N where "0 < d" "summable h" and le_h: "\<And>n y. \<lbrakk>N \<le> n; y \<in> ball z d\<rbrakk> \<Longrightarrow> norm(f n y) \<le> h n"
      using to_g \<open>z \<in> s\<close> by meson
    then obtain r where "r>0" and r: "ball z r \<subseteq> ball z d \<inter> s" using \<open>z \<in> s\<close> s
      by (metis Int_iff open_ball centre_in_ball open_Int open_contains_ball_eq)
    have 1: "open (ball z d \<inter> s)"
      by (simp add: open_Int s)
    have 2: "\<And>n x. x \<in> ball z d \<inter> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
      by (auto simp: hfd)
    obtain g g' where gg': "\<forall>x \<in> ball z d \<inter> s. ((\<lambda>n. f n x) sums g x) \<and>
                                    ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
      by (auto intro: le_h series_and_derivative_comparison [OF 1 \<open>summable h\<close> hfd])
    then have "(\<lambda>n. f' n z) sums g' z"
      by (meson \<open>0 < r\<close> centre_in_ball contra_subsetD r)
    moreover have "(\<lambda>n. f n z) sums (\<Sum>n. f n z)"
      by (metis summable_comparison_test' summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h)
    moreover have "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' z) (at z)"
      apply (rule_tac f=g in DERIV_transform_at [OF _ \<open>0 < r\<close>])
      apply (simp add: gg' \<open>z \<in> s\<close> \<open>0 < d\<close>)
      apply (metis (full_types) contra_subsetD dist_commute gg' mem_ball r sums_unique)
      done
    ultimately show ?thesis by auto
  qed
  then show ?thesis
    by (rule_tac x="\<lambda>x. suminf  (\<lambda>n. f n x)" in exI) meson
qed


text\<open>Sometimes convenient to compare with a complex series of positive reals. (?)\<close>

lemma series_and_derivative_comparison_complex:
  fixes s :: "complex set"
  assumes s: "open s"
      and hfd: "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
      and to_g: "\<And>x. x \<in> s \<Longrightarrow>
                      \<exists>d h N. 0 < d \<and> summable h \<and> range h \<subseteq> nonneg_Reals \<and> (\<forall>n y. N \<le> n \<and> y \<in> ball x d \<longrightarrow> cmod(f n y) \<le> cmod (h n))"
  shows "\<exists>g g'. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((\<lambda>n. f' n x) sums g' x) \<and> (g has_field_derivative g' x) (at x)"
apply (rule series_and_derivative_comparison_local [OF s hfd], assumption)
apply (frule to_g)
apply (erule ex_forward)
apply (erule exE)
apply (rule_tac x="Re o h" in exI)
apply (erule ex_forward)
apply (simp add: summable_Re o_def )
apply (elim conjE all_forward)
apply (simp add: nonneg_Reals_cmod_eq_Re image_subset_iff)
done


text\<open>In particular, a power series is analytic inside circle of convergence.\<close>

lemma power_series_and_derivative_0:
  fixes a :: "nat \<Rightarrow> complex" and r::real
  assumes "summable (\<lambda>n. a n * r^n)"
    shows "\<exists>g g'. \<forall>z. cmod z < r \<longrightarrow>
             ((\<lambda>n. a n * z^n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * z^(n - 1)) sums g' z) \<and> (g has_field_derivative g' z) (at z)"
proof (cases "0 < r")
  case True
    have der: "\<And>n z. ((\<lambda>x. a n * x ^ n) has_field_derivative of_nat n * a n * z ^ (n - 1)) (at z)"
      by (rule derivative_eq_intros | simp)+
    have y_le: "\<lbrakk>cmod (z - y) * 2 < r - cmod z\<rbrakk> \<Longrightarrow> cmod y \<le> cmod (of_real r + of_real (cmod z)) / 2" for z y
      using \<open>r > 0\<close>
      apply (auto simp: algebra_simps norm_mult norm_divide norm_power of_real_add [symmetric] simp del: of_real_add)
      using norm_triangle_ineq2 [of y z]
      apply (simp only: diff_le_eq norm_minus_commute mult_2)
      done
    have "summable (\<lambda>n. a n * complex_of_real r ^ n)"
      using assms \<open>r > 0\<close> by simp
    moreover have "\<And>z. cmod z < r \<Longrightarrow> cmod ((of_real r + of_real (cmod z)) / 2) < cmod (of_real r)"
      using \<open>r > 0\<close>
      by (simp add: of_real_add [symmetric] del: of_real_add)
    ultimately have sum: "\<And>z. cmod z < r \<Longrightarrow> summable (\<lambda>n. of_real (cmod (a n)) * ((of_real r + complex_of_real (cmod z)) / 2) ^ n)"
      by (rule power_series_conv_imp_absconv_weak)
    have "\<exists>g g'. \<forall>z \<in> ball 0 r. (\<lambda>n.  (a n) * z ^ n) sums g z \<and>
               (\<lambda>n. of_nat n * (a n) * z ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
      apply (rule series_and_derivative_comparison_complex [OF open_ball der])
      apply (rule_tac x="(r - norm z)/2" in exI)
      apply (simp add: dist_norm)
      apply (rule_tac x="\<lambda>n. of_real(norm(a n)*((r + norm z)/2)^n)" in exI)
      using \<open>r > 0\<close>
      apply (auto simp: sum nonneg_Reals_divide_I)
      apply (rule_tac x=0 in exI)
      apply (force simp: norm_mult norm_divide norm_power intro!: mult_left_mono power_mono y_le)
      done
  then show ?thesis
    by (simp add: dist_0_norm ball_def)
next
  case False then show ?thesis
    apply (simp add: not_less)
    using less_le_trans norm_not_less_zero by blast
qed

proposition power_series_and_derivative:
  fixes a :: "nat \<Rightarrow> complex" and r::real
  assumes "summable (\<lambda>n. a n * r^n)"
    obtains g g' where "\<forall>z \<in> ball w r.
             ((\<lambda>n. a n * (z - w) ^ n) sums g z) \<and> ((\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z) \<and>
              (g has_field_derivative g' z) (at z)"
  using power_series_and_derivative_0 [OF assms]
  apply clarify
  apply (rule_tac g="(\<lambda>z. g(z - w))" in that)
  using DERIV_shift [where z="-w"]
  apply (auto simp: norm_minus_commute Ball_def dist_norm)
  done

proposition power_series_holomorphic:
  assumes "\<And>w. w \<in> ball z r \<Longrightarrow> ((\<lambda>n. a n*(w - z)^n) sums f w)"
    shows "f holomorphic_on ball z r"
proof -
  have "\<exists>f'. (f has_field_derivative f') (at w)" if w: "dist z w < r" for w
  proof -
    have inb: "z + complex_of_real ((dist z w + r) / 2) \<in> ball z r"
    proof -
      have wz: "cmod (w - z) < r" using w
        by (auto simp: divide_simps dist_norm norm_minus_commute)
      then have "0 \<le> r"
        by (meson less_eq_real_def norm_ge_zero order_trans)
      show ?thesis
        using w by (simp add: dist_norm \<open>0\<le>r\<close> of_real_add [symmetric] del: of_real_add)
    qed
    have sum: "summable (\<lambda>n. a n * of_real (((cmod (z - w) + r) / 2) ^ n))"
      using assms [OF inb] by (force simp add: summable_def dist_norm)
    obtain g g' where gg': "\<And>u. u \<in> ball z ((cmod (z - w) + r) / 2) \<Longrightarrow>
                               (\<lambda>n. a n * (u - z) ^ n) sums g u \<and>
                               (\<lambda>n. of_nat n * a n * (u - z) ^ (n - 1)) sums g' u \<and> (g has_field_derivative g' u) (at u)"
      by (rule power_series_and_derivative [OF sum, of z]) fastforce
    have [simp]: "g u = f u" if "cmod (u - w) < (r - cmod (z - w)) / 2" for u
    proof -
      have less: "cmod (z - u) * 2 < cmod (z - w) + r"
        using that dist_triangle2 [of z u w]
        by (simp add: dist_norm [symmetric] algebra_simps)
      show ?thesis
        apply (rule sums_unique2 [of "\<lambda>n. a n*(u - z)^n"])
        using gg' [of u] less w
        apply (auto simp: assms dist_norm)
        done
    qed
    show ?thesis
      apply (rule_tac x="g' w" in exI)
      apply (rule DERIV_transform_at [where f=g and d="(r - norm(z - w))/2"])
      using w gg' [of w]
      apply (auto simp: dist_norm)
      done
  qed
  then show ?thesis by (simp add: holomorphic_on_open)
qed

corollary holomorphic_iff_power_series:
     "f holomorphic_on ball z r \<longleftrightarrow>
      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
  apply (intro iffI ballI)
   using holomorphic_power_series  apply force
  apply (rule power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
  apply force
  done

corollary power_series_analytic:
     "(\<And>w. w \<in> ball z r \<Longrightarrow> (\<lambda>n. a n*(w - z)^n) sums f w) \<Longrightarrow> f analytic_on ball z r"
  by (force simp add: analytic_on_open intro!: power_series_holomorphic)

corollary analytic_iff_power_series:
     "f analytic_on ball z r \<longleftrightarrow>
      (\<forall>w \<in> ball z r. (\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n) sums f w)"
  by (simp add: analytic_on_open holomorphic_iff_power_series)


subsection\<open>Equality between holomorphic functions, on open ball then connected set.\<close>

lemma holomorphic_fun_eq_on_ball:
   "\<lbrakk>f holomorphic_on ball z r; g holomorphic_on ball z r;
     w \<in> ball z r;
     \<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z\<rbrakk>
     \<Longrightarrow> f w = g w"
  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
  apply (auto simp: holomorphic_iff_power_series)
  done

lemma holomorphic_fun_eq_0_on_ball:
   "\<lbrakk>f holomorphic_on ball z r;  w \<in> ball z r;
     \<And>n. (deriv ^^ n) f z = 0\<rbrakk>
     \<Longrightarrow> f w = 0"
  apply (rule sums_unique2 [of "\<lambda>n. (deriv ^^ n) f z / (fact n) * (w - z)^n"])
  apply (auto simp: holomorphic_iff_power_series)
  done

lemma holomorphic_fun_eq_0_on_connected:
  assumes holf: "f holomorphic_on s" and "open s"
      and cons: "connected s"
      and der: "\<And>n. (deriv ^^ n) f z = 0"
      and "z \<in> s" "w \<in> s"
    shows "f w = 0"
proof -
  have *: "\<And>x e. \<lbrakk> \<forall>xa. (deriv ^^ xa) f x = 0;  ball x e \<subseteq> s\<rbrakk>
           \<Longrightarrow> ball x e \<subseteq> (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
    apply auto
    apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
    apply (rule holomorphic_on_subset [OF holf], simp_all)
    by (metis funpow_add o_apply)
  have 1: "openin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
    apply (rule open_subset, force)
    using \<open>open s\<close>
    apply (simp add: open_contains_ball Ball_def)
    apply (erule all_forward)
    using "*" by auto blast+
  have 2: "closedin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
    using assms
    by (auto intro: continuous_closed_in_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
  obtain e where "e>0" and e: "ball w e \<subseteq> s" using openE [OF \<open>open s\<close> \<open>w \<in> s\<close>] .
  then have holfb: "f holomorphic_on ball w e"
    using holf holomorphic_on_subset by blast
  have 3: "(\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0}) = s \<Longrightarrow> f w = 0"
    using \<open>e>0\<close> e by (force intro: holomorphic_fun_eq_0_on_ball [OF holfb])
  show ?thesis
    using cons der \<open>z \<in> s\<close>
    apply (simp add: connected_clopen)
    apply (drule_tac x="\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0}" in spec)
    apply (auto simp: 1 2 3)
    done
qed

lemma holomorphic_fun_eq_on_connected:
  assumes "f holomorphic_on s" "g holomorphic_on s" and "open s"  "connected s"
      and "\<And>n. (deriv ^^ n) f z = (deriv ^^ n) g z"
      and "z \<in> s" "w \<in> s"
    shows "f w = g w"
  apply (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" s z, simplified])
  apply (intro assms holomorphic_intros)
  using assms apply simp_all
  apply (subst higher_deriv_diff, auto)
  done

lemma holomorphic_fun_eq_const_on_connected:
  assumes holf: "f holomorphic_on s" and "open s"
      and cons: "connected s"
      and der: "\<And>n. 0 < n \<Longrightarrow> (deriv ^^ n) f z = 0"
      and "z \<in> s" "w \<in> s"
    shows "f w = f z"
  apply (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" s z, simplified])
  apply (intro assms holomorphic_intros)
  using assms apply simp_all
  apply (subst higher_deriv_diff)
  apply (intro holomorphic_intros | simp)+
  done


subsection\<open>Some basic lemmas about poles/singularities.\<close>

lemma pole_lemma:
  assumes holf: "f holomorphic_on s" and a: "a \<in> interior s"
    shows "(\<lambda>z. if z = a then deriv f a
                 else (f z - f a) / (z - a)) holomorphic_on s" (is "?F holomorphic_on s")
proof -
  have F1: "?F complex_differentiable (at u within s)" if "u \<in> s" "u \<noteq> a" for u
  proof -
    have fcd: "f complex_differentiable at u within s"
      using holf holomorphic_on_def by (simp add: \<open>u \<in> s\<close>)
    have cd: "(\<lambda>z. (f z - f a) / (z - a)) complex_differentiable at u within s"
      by (rule fcd derivative_intros | simp add: that)+
    have "0 < dist a u" using that dist_nz by blast
    then show ?thesis
      by (rule complex_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> s\<close>)
  qed
  have F2: "?F complex_differentiable at a" if "0 < e" "ball a e \<subseteq> s" for e
  proof -
    have holfb: "f holomorphic_on ball a e"
      by (rule holomorphic_on_subset [OF holf \<open>ball a e \<subseteq> s\<close>])
    have 2: "?F holomorphic_on ball a e - {a}"
      apply (rule holomorphic_on_subset [where s = "s - {a}"])
      apply (simp add: holomorphic_on_def complex_differentiable_def [symmetric])
      using mem_ball that
      apply (auto intro: F1 complex_differentiable_within_subset)
      done
    have "isCont (\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) x"
            if "dist a x < e" for x
    proof (cases "x=a")
      case True then show ?thesis
      using holfb \<open>0 < e\<close>
      apply (simp add: holomorphic_on_open complex_differentiable_def [symmetric])
      apply (drule_tac x=a in bspec)
      apply (auto simp: DERIV_deriv_iff_complex_differentiable [symmetric] continuous_at DERIV_iff2
                elim: rev_iffD1 [OF _ LIM_equal])
      done
    next
      case False with 2 that show ?thesis
        by (force simp: holomorphic_on_open open_Diff complex_differentiable_def [symmetric] complex_differentiable_imp_continuous_at)
    qed
    then have 1: "continuous_on (ball a e) ?F"
      by (clarsimp simp:  continuous_on_eq_continuous_at)
    have "?F holomorphic_on ball a e"
      by (auto intro: no_isolated_singularity [OF 1 2])
    with that show ?thesis
      by (simp add: holomorphic_on_open complex_differentiable_def [symmetric]
                    complex_differentiable_at_within)
  qed
  show ?thesis
  proof
    fix x assume "x \<in> s" show "?F complex_differentiable at x within s"
    proof (cases "x=a")
      case True then show ?thesis
      using a by (auto simp: mem_interior intro: complex_differentiable_at_within F2)
    next
      case False with F1 \<open>x \<in> s\<close>
      show ?thesis by blast
    qed
  qed
qed

proposition pole_theorem:
  assumes holg: "g holomorphic_on s" and a: "a \<in> interior s"
      and eq: "\<And>z. z \<in> s - {a} \<Longrightarrow> g z = (z - a) * f z"
    shows "(\<lambda>z. if z = a then deriv g a
                 else f z - g a/(z - a)) holomorphic_on s"
  using pole_lemma [OF holg a]
  by (rule holomorphic_transform) (simp add: eq divide_simps)

lemma pole_lemma_open:
  assumes "f holomorphic_on s" "open s"
    shows "(\<lambda>z. if z = a then deriv f a else (f z - f a)/(z - a)) holomorphic_on s"
proof (cases "a \<in> s")
  case True with assms interior_eq pole_lemma
    show ?thesis by fastforce
next
  case False with assms show ?thesis
    apply (simp add: holomorphic_on_def complex_differentiable_def [symmetric], clarify)
    apply (rule complex_differentiable_transform_within [where f = "\<lambda>z. (f z - f a)/(z - a)" and d = 1])
    apply (rule derivative_intros | force)+
    done
qed

proposition pole_theorem_open:
  assumes holg: "g holomorphic_on s" and s: "open s"
      and eq: "\<And>z. z \<in> s - {a} \<Longrightarrow> g z = (z - a) * f z"
    shows "(\<lambda>z. if z = a then deriv g a
                 else f z - g a/(z - a)) holomorphic_on s"
  using pole_lemma_open [OF holg s]
  by (rule holomorphic_transform) (auto simp: eq divide_simps)

proposition pole_theorem_0:
  assumes holg: "g holomorphic_on s" and a: "a \<in> interior s"
      and eq: "\<And>z. z \<in> s - {a} \<Longrightarrow> g z = (z - a) * f z"
      and [simp]: "f a = deriv g a" "g a = 0"
    shows "f holomorphic_on s"
  using pole_theorem [OF holg a eq]
  by (rule holomorphic_transform) (auto simp: eq divide_simps)

proposition pole_theorem_open_0:
  assumes holg: "g holomorphic_on s" and s: "open s"
      and eq: "\<And>z. z \<in> s - {a} \<Longrightarrow> g z = (z - a) * f z"
      and [simp]: "f a = deriv g a" "g a = 0"
    shows "f holomorphic_on s"
  using pole_theorem_open [OF holg s eq]
  by (rule holomorphic_transform) (auto simp: eq divide_simps)

lemma pole_theorem_analytic:
  assumes g: "g analytic_on s"
      and eq: "\<And>z. z \<in> s
             \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
    shows "(\<lambda>z. if z = a then deriv g a
                 else f z - g a/(z - a)) analytic_on s"
using g
apply (simp add: analytic_on_def Ball_def)
apply (safe elim!: all_forward dest!: eq)
apply (rule_tac x="min d e" in exI, simp)
apply (rule pole_theorem_open)
apply (auto simp: holomorphic_on_subset subset_ball)
done

lemma pole_theorem_analytic_0:
  assumes g: "g analytic_on s"
      and eq: "\<And>z. z \<in> s \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>w \<in> ball z d - {a}. g w = (w - a) * f w)"
      and [simp]: "f a = deriv g a" "g a = 0"
    shows "f analytic_on s"
proof -
  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
    by auto
  show ?thesis
    using pole_theorem_analytic [OF g eq] by simp
qed

lemma pole_theorem_analytic_open_superset:
  assumes g: "g analytic_on s" and "s \<subseteq> t" "open t"
      and eq: "\<And>z. z \<in> t - {a} \<Longrightarrow> g z = (z - a) * f z"
    shows "(\<lambda>z. if z = a then deriv g a
                 else f z - g a/(z - a)) analytic_on s"
  apply (rule pole_theorem_analytic [OF g])
  apply (rule openE [OF \<open>open t\<close>])
  using assms eq by auto

lemma pole_theorem_analytic_open_superset_0:
  assumes g: "g analytic_on s" "s \<subseteq> t" "open t" "\<And>z. z \<in> t - {a} \<Longrightarrow> g z = (z - a) * f z"
      and [simp]: "f a = deriv g a" "g a = 0"
    shows "f analytic_on s"
proof -
  have [simp]: "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) = f"
    by auto
  have "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on s"
    by (rule pole_theorem_analytic_open_superset [OF g])
  then show ?thesis by simp
qed



subsection\<open>General, homology form of Cauchy's theorem.\<close>

text\<open>Proof is based on Dixon's, as presented in Lang's "Complex Analysis" book (page 147).\<close>

lemma contour_integral_continuous_on_linepath_2D:
  assumes "open u" and cont_dw: "\<And>w. w \<in> u \<Longrightarrow> F w contour_integrable_on (linepath a b)"
      and cond_uu: "continuous_on (u \<times> u) (\<lambda>(x,y). F x y)"
      and abu: "closed_segment a b \<subseteq> u"
    shows "continuous_on u (\<lambda>w. contour_integral (linepath a b) (F w))"
proof -
  have *: "\<exists>d>0. \<forall>x'\<in>u. dist x' w < d \<longrightarrow>
                         dist (contour_integral (linepath a b) (F x'))
                              (contour_integral (linepath a b) (F w)) \<le> \<epsilon>"
          if "w \<in> u" "0 < \<epsilon>" "a \<noteq> b" for w \<epsilon>
  proof -
    obtain \<delta> where "\<delta>>0" and \<delta>: "cball w \<delta> \<subseteq> u" using open_contains_cball \<open>open u\<close> \<open>w \<in> u\<close> by force
    let ?TZ = "{(t,z) |t z. t \<in> cball w \<delta> \<and> z \<in> closed_segment a b}"
    have "uniformly_continuous_on ?TZ (\<lambda>(x,y). F x y)"
      apply (rule compact_uniformly_continuous)
      apply (rule continuous_on_subset[OF cond_uu])
      using abu \<delta>
      apply (auto simp: compact_Times simp del: mem_cball)
      done
    then obtain \<eta> where "\<eta>>0"
        and \<eta>: "\<And>x x'. \<lbrakk>x\<in>?TZ; x'\<in>?TZ; dist x' x < \<eta>\<rbrakk> \<Longrightarrow>
                         dist ((\<lambda>(x,y). F x y) x') ((\<lambda>(x,y). F x y) x) < \<epsilon>/norm(b - a)"
      apply (rule uniformly_continuous_onE [where e = "\<epsilon>/norm(b - a)"])
      using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> by auto
    have \<eta>: "\<lbrakk>norm (w - x1) \<le> \<delta>;   x2 \<in> closed_segment a b;
              norm (w - x1') \<le> \<delta>;  x2' \<in> closed_segment a b; norm ((x1', x2') - (x1, x2)) < \<eta>\<rbrakk>
              \<Longrightarrow> norm (F x1' x2' - F x1 x2) \<le> \<epsilon> / cmod (b - a)"
             for x1 x2 x1' x2'
      using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp add: dist_norm)
    have le_ee: "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>"
                if "x' \<in> u" "cmod (x' - w) < \<delta>" "cmod (x' - w) < \<eta>"  for x'
    proof -
      have "cmod (contour_integral (linepath a b) (\<lambda>x. F x' x - F w x)) \<le> \<epsilon>/norm(b - a) * norm(b - a)"
        apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_integral _ \<eta>])
        apply (rule contour_integrable_diff [OF cont_dw cont_dw])
        using \<open>0 < \<epsilon>\<close> \<open>a \<noteq> b\<close> \<open>0 < \<delta>\<close> \<open>w \<in> u\<close> that
        apply (auto simp: norm_minus_commute)
        done
      also have "... = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
      finally show ?thesis .
    qed
    show ?thesis
      apply (rule_tac x="min \<delta> \<eta>" in exI)
      using \<open>0 < \<delta>\<close> \<open>0 < \<eta>\<close>
      apply (auto simp: dist_norm contour_integral_diff [OF cont_dw cont_dw, symmetric] \<open>w \<in> u\<close> intro: le_ee)
      done
  qed
  show ?thesis
    apply (rule continuous_onI)
    apply (cases "a=b")
    apply (auto intro: *)
    done
qed

text\<open>This version has @{term"polynomial_function \<gamma>"} as an additional assumption.\<close>
lemma Cauchy_integral_formula_global_weak:
    assumes u: "open u" and holf: "f holomorphic_on u"
        and z: "z \<in> u" and \<gamma>: "polynomial_function \<gamma>"
        and pasz: "path_image \<gamma> \<subseteq> u - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
        and zero: "\<And>w. w \<notin> u \<Longrightarrow> winding_number \<gamma> w = 0"
      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
proof -
  obtain \<gamma>' where pf\<gamma>': "polynomial_function \<gamma>'" and \<gamma>': "\<And>x. (\<gamma> has_vector_derivative (\<gamma>' x)) (at x)"
    using has_vector_derivative_polynomial_function [OF \<gamma>] by blast
  then have "bounded(path_image \<gamma>')"
    by (simp add: path_image_def compact_imp_bounded compact_continuous_image continuous_on_polymonial_function)
  then obtain B where "B>0" and B: "\<And>x. x \<in> path_image \<gamma>' \<Longrightarrow> norm x \<le> B"
    using bounded_pos by force
  def d \<equiv> "\<lambda>z w. if w = z then deriv f z else (f w - f z)/(w - z)"
  def v \<equiv> "{w. w \<notin> path_image \<gamma> \<and> winding_number \<gamma> w = 0}"
  have "path \<gamma>" "valid_path \<gamma>" using \<gamma>
    by (auto simp: path_polynomial_function valid_path_polynomial_function)
  then have ov: "open v"
    by (simp add: v_def open_winding_number_levelsets loop)
  have uv_Un: "u \<union> v = UNIV"
    using pasz zero by (auto simp: v_def)
  have conf: "continuous_on u f"
    by (metis holf holomorphic_on_imp_continuous_on)
  have hol_d: "(d y) holomorphic_on u" if "y \<in> u" for y
  proof -
    have *: "(\<lambda>c. if c = y then deriv f y else (f c - f y) / (c - y)) holomorphic_on u"
      by (simp add: holf pole_lemma_open u)
    then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
      using at_within_open complex_differentiable_imp_continuous_at holomorphic_on_def that u by fastforce
    then have "continuous_on u (d y)"
      apply (simp add: d_def continuous_on_eq_continuous_at u, clarify)
      using * holomorphic_on_def
      by (meson complex_differentiable_within_open complex_differentiable_imp_continuous_at u)
    moreover have "d y holomorphic_on u - {y}"
      apply (simp add: d_def holomorphic_on_open u open_delete complex_differentiable_def [symmetric])
      apply (intro ballI)
      apply (rename_tac w)
      apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in complex_differentiable_transform_within)
      apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
      using analytic_on_imp_differentiable_at analytic_on_open holf u apply blast
      done
    ultimately show ?thesis
      by (rule no_isolated_singularity) (auto simp: u)
  qed
  have cint_fxy: "(\<lambda>x. (f x - f y) / (x - y)) contour_integrable_on \<gamma>" if "y \<notin> path_image \<gamma>" for y
    apply (rule contour_integrable_holomorphic_simple [where s = "u-{y}"])
    using \<open>valid_path \<gamma>\<close> pasz
    apply (auto simp: u open_delete)
    apply (rule continuous_intros holomorphic_intros continuous_on_subset [OF conf] holomorphic_on_subset [OF holf] |
                force simp add: that)+
    done
  def h \<equiv> "\<lambda>z. if z \<in> u then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z))"
  have U: "\<And>z. z \<in> u \<Longrightarrow> ((d z) has_contour_integral h z) \<gamma>"
    apply (simp add: h_def)
    apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where s=u]])
    using u pasz \<open>valid_path \<gamma>\<close>
    apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
    done
  have V: "((\<lambda>w. f w / (w - z)) has_contour_integral h z) \<gamma>" if z: "z \<in> v" for z
  proof -
    have 0: "0 = (f z) * 2 * of_real (2 * pi) * \<i> * winding_number \<gamma> z"
      using v_def z by auto
    then have "((\<lambda>x. 1 / (x - z)) has_contour_integral 0) \<gamma>"
     using z v_def  has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close>] by fastforce
    then have "((\<lambda>x. f z * (1 / (x - z))) has_contour_integral 0) \<gamma>"
      using has_contour_integral_lmul by fastforce
    then have "((\<lambda>x. f z / (x - z)) has_contour_integral 0) \<gamma>"
      by (simp add: divide_simps)
    moreover have "((\<lambda>x. (f x - f z) / (x - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
      using z
      apply (auto simp: v_def)
      apply (metis (no_types, lifting) contour_integrable_eq d_def has_contour_integral_eq has_contour_integral_integral cint_fxy)
      done
    ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
      by (rule has_contour_integral_add)
    have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
            if  "z \<in> u"
      using * by (auto simp: divide_simps has_contour_integral_eq)
    moreover have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (\<lambda>w. f w / (w - z))) \<gamma>"
            if "z \<notin> u"
      apply (rule has_contour_integral_integral [OF contour_integrable_holomorphic_simple [where s=u]])
      using u pasz \<open>valid_path \<gamma>\<close> that
      apply (auto intro: holomorphic_on_imp_continuous_on hol_d)
      apply (rule continuous_intros conf holomorphic_intros holf | force)+
      done
    ultimately show ?thesis
      using z by (simp add: h_def)
  qed
  have znot: "z \<notin> path_image \<gamma>"
    using pasz by blast
  obtain d0 where "d0>0" and d0: "\<And>x y. x \<in> path_image \<gamma> \<Longrightarrow> y \<in> - u \<Longrightarrow> d0 \<le> dist x y"
    using separate_compact_closed [of "path_image \<gamma>" "-u"] pasz u
    by (fastforce simp add: \<open>path \<gamma>\<close> compact_path_image)
  obtain dd where "0 < dd" and dd: "{y + k | y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> u"
    apply (rule that [of "d0/2"])
    using \<open>0 < d0\<close>
    apply (auto simp: dist_norm dest: d0)
    done
  have "\<And>x x'. \<lbrakk>x \<in> path_image \<gamma>; dist x x' * 2 < dd\<rbrakk> \<Longrightarrow> \<exists>y k. x' = y + k \<and> y \<in> path_image \<gamma> \<and> dist 0 k * 2 \<le> dd"
    apply (rule_tac x=x in exI)
    apply (rule_tac x="x'-x" in exI)
    apply (force simp add: dist_norm)
    done
  then have 1: "path_image \<gamma> \<subseteq> interior {y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> cball 0 (dd / 2)}"
    apply (clarsimp simp add: mem_interior)
    using \<open>0 < dd\<close>
    apply (rule_tac x="dd/2" in exI, auto)
    done
  obtain t where "compact t" and subt: "path_image \<gamma> \<subseteq> interior t" and t: "t \<subseteq> u"
    apply (rule that [OF _ 1])
    apply (fastforce simp add: \<open>valid_path \<gamma>\<close> compact_valid_path_image intro!: compact_sums)
    apply (rule order_trans [OF _ dd])
    using \<open>0 < dd\<close> by fastforce
  obtain L where "L>0"
           and L: "\<And>f B. \<lbrakk>f holomorphic_on interior t; \<And>z. z\<in>interior t \<Longrightarrow> cmod (f z) \<le> B\<rbrakk> \<Longrightarrow>
                         cmod (contour_integral \<gamma> f) \<le> L * B"
      using contour_integral_bound_exists [OF open_interior \<open>valid_path \<gamma>\<close> subt]
      by blast
  have "bounded(f ` t)"
    by (meson \<open>compact t\<close> compact_continuous_image compact_imp_bounded conf continuous_on_subset t)
  then obtain D where "D>0" and D: "\<And>x. x \<in> t \<Longrightarrow> norm (f x) \<le> D"
    by (auto simp: bounded_pos)
  obtain C where "C>0" and C: "\<And>x. x \<in> t \<Longrightarrow> norm x \<le> C"
    using \<open>compact t\<close> bounded_pos compact_imp_bounded by force
  have "dist (h y) 0 \<le> e" if "0 < e" and le: "D * L / e + C \<le> cmod y" for e y
  proof -
    have "D * L / e > 0"  using \<open>D>0\<close> \<open>L>0\<close> \<open>e>0\<close> by simp
    with le have ybig: "norm y > C" by force
    with C have "y \<notin> t"  by force
    then have ynot: "y \<notin> path_image \<gamma>"
      using subt interior_subset by blast
    have [simp]: "winding_number \<gamma> y = 0"
      apply (rule winding_number_zero_outside [of _ "cball 0 C"])
      using ybig interior_subset subt
      apply (force simp add: loop \<open>path \<gamma>\<close> dist_norm intro!: C)+
      done
    have [simp]: "h y = contour_integral \<gamma> (\<lambda>w. f w/(w - y))"
      by (rule contour_integral_unique [symmetric]) (simp add: v_def ynot V)
    have holint: "(\<lambda>w. f w / (w - y)) holomorphic_on interior t"
      apply (rule holomorphic_on_divide)
      using holf holomorphic_on_subset interior_subset t apply blast
      apply (rule holomorphic_intros)+
      using \<open>y \<notin> t\<close> interior_subset by auto
    have leD: "cmod (f z / (z - y)) \<le> D * (e / L / D)" if z: "z \<in> interior t" for z
    proof -
      have "D * L / e + cmod z \<le> cmod y"
        using le C [of z] z using interior_subset by force
      then have DL2: "D * L / e \<le> cmod (z - y)"
        using norm_triangle_ineq2 [of y z] by (simp add: norm_minus_commute)
      have "cmod (f z / (z - y)) = cmod (f z) * inverse (cmod (z - y))"
        by (simp add: norm_mult norm_inverse Fields.field_class.field_divide_inverse)
      also have "... \<le> D * (e / L / D)"
        apply (rule mult_mono)
        using that D interior_subset apply blast
        using \<open>L>0\<close> \<open>e>0\<close> \<open>D>0\<close> DL2
        apply (auto simp: norm_divide divide_simps algebra_simps)
        done
      finally show ?thesis .
    qed
    have "dist (h y) 0 = cmod (contour_integral \<gamma> (\<lambda>w. f w / (w - y)))"
      by (simp add: dist_norm)
    also have "... \<le> L * (D * (e / L / D))"
      by (rule L [OF holint leD])
    also have "... = e"
      using  \<open>L>0\<close> \<open>0 < D\<close> by auto
    finally show ?thesis .
  qed
  then have "(h \<longlongrightarrow> 0) at_infinity"
    by (meson Lim_at_infinityI)
  moreover have "h holomorphic_on UNIV"
  proof -
    have con_ff: "continuous (at (x,z)) (\<lambda>(x,y). (f y - f x) / (y - x))"
                 if "x \<in> u" "z \<in> u" "x \<noteq> z" for x z
      using that conf
      apply (simp add: split_def continuous_on_eq_continuous_at u)
      apply (simp | rule continuous_intros continuous_within_compose2 [where g=f])+
      done
    have con_fstsnd: "continuous_on UNIV (\<lambda>x. (fst x - snd x) ::complex)"
      by (rule continuous_intros)+
    have open_uu_Id: "open (u \<times> u - Id)"
      apply (rule open_Diff)
      apply (simp add: open_Times u)
      using continuous_closed_preimage_constant [OF con_fstsnd closed_UNIV, of 0]
      apply (auto simp: Id_fstsnd_eq algebra_simps)
      done
    have con_derf: "continuous (at z) (deriv f)" if "z \<in> u" for z
      apply (rule continuous_on_interior [of u])
      apply (simp add: holf holomorphic_deriv holomorphic_on_imp_continuous_on u)
      by (simp add: interior_open that u)
    have tendsto_f': "((\<lambda>(x,y). if y = x then deriv f (x)
                                else (f (y) - f (x)) / (y - x)) \<longlongrightarrow> deriv f x)
                      (at (x, x) within u \<times> u)" if "x \<in> u" for x
    proof (rule Lim_withinI)
      fix e::real assume "0 < e"
      obtain k1 where "k1>0" and k1: "\<And>x'. norm (x' - x) \<le> k1 \<Longrightarrow> norm (deriv f x' - deriv f x) < e"
        using \<open>0 < e\<close> continuous_within_E [OF con_derf [OF \<open>x \<in> u\<close>]]
        by (metis UNIV_I dist_norm)
      obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> u" by (blast intro: openE [OF u] \<open>x \<in> u\<close>)
      have neq: "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e"
                    if "z' \<noteq> x'" and less_k1: "norm (x'-x, z'-x) < k1" and less_k2: "norm (x'-x, z'-x) < k2"
                 for x' z'
      proof -
        have cs_less: "w \<in> closed_segment x' z' \<Longrightarrow> cmod (w - x) \<le> norm (x'-x, z'-x)" for w
          apply (drule segment_furthest_le [where y=x])
          by (metis (no_types) dist_commute dist_norm norm_fst_le norm_snd_le order_trans)
        have derf_le: "w \<in> closed_segment x' z' \<Longrightarrow> z' \<noteq> x' \<Longrightarrow> cmod (deriv f w - deriv f x) \<le> e" for w
          by (blast intro: cs_less less_k1 k1 [unfolded divide_const_simps dist_norm] less_imp_le le_less_trans)
        have f_has_der: "\<And>x. x \<in> u \<Longrightarrow> (f has_field_derivative deriv f x) (at x within u)"
          by (metis DERIV_deriv_iff_complex_differentiable at_within_open holf holomorphic_on_def u)
        have "closed_segment x' z' \<subseteq> u"
          by (rule order_trans [OF _ k2]) (simp add: cs_less  le_less_trans [OF _ less_k2] dist_complex_def norm_minus_commute subset_iff)
        then have cint_derf: "(deriv f has_contour_integral f z' - f x') (linepath x' z')"
          using contour_integral_primitive [OF f_has_der valid_path_linepath] pasz  by simp
        then have *: "((\<lambda>x. deriv f x / (z' - x')) has_contour_integral (f z' - f x') / (z' - x')) (linepath x' z')"
          by (rule has_contour_integral_div)
        have "norm ((f z' - f x') / (z' - x') - deriv f x) \<le> e/norm(z' - x') * norm(z' - x')"
          apply (rule has_contour_integral_bound_linepath [OF has_contour_integral_diff [OF *]])
          using has_contour_integral_div [where c = "z' - x'", OF has_contour_integral_const_linepath [of "deriv f x" z' x']]
                 \<open>e > 0\<close>  \<open>z' \<noteq> x'\<close>
          apply (auto simp: norm_divide divide_simps derf_le)
          done
        also have "... \<le> e" using \<open>0 < e\<close> by simp
        finally show ?thesis .
      qed
      show "\<exists>d>0. \<forall>xa\<in>u \<times> u.
                  0 < dist xa (x, x) \<and> dist xa (x, x) < d \<longrightarrow>
                  dist (case xa of (x, y) \<Rightarrow> if y = x then deriv f x else (f y - f x) / (y - x)) (deriv f x) \<le> e"
        apply (rule_tac x="min k1 k2" in exI)
        using \<open>k1>0\<close> \<open>k2>0\<close> \<open>e>0\<close>
        apply (force simp: dist_norm neq intro: dual_order.strict_trans2 k1 less_imp_le norm_fst_le)
        done
    qed
    have con_pa_f: "continuous_on (path_image \<gamma>) f"
      by (meson holf holomorphic_on_imp_continuous_on holomorphic_on_subset interior_subset subt t)
    have le_B: "\<And>t. t \<in> {0..1} \<Longrightarrow> cmod (vector_derivative \<gamma> (at t)) \<le> B"
      apply (rule B)
      using \<gamma>' using path_image_def vector_derivative_at by fastforce
    have f_has_cint: "\<And>w. w \<in> v - path_image \<gamma> \<Longrightarrow> ((\<lambda>u. f u / (u - w) ^ 1) has_contour_integral h w) \<gamma>"
      by (simp add: V)
    have cond_uu: "continuous_on (u \<times> u) (\<lambda>(x,y). d x y)"
      apply (simp add: continuous_on_eq_continuous_within d_def continuous_within tendsto_f')
      apply (simp add: tendsto_within_open_NO_MATCH open_Times u, clarify)
      apply (rule Lim_transform_within_open [OF _ open_uu_Id, where f = "(\<lambda>(x,y). (f y - f x) / (y - x))"])
      using con_ff
      apply (auto simp: continuous_within)
      done
    have hol_dw: "(\<lambda>z. d z w) holomorphic_on u" if "w \<in> u" for w
    proof -
      have "continuous_on u ((\<lambda>(x,y). d x y) o (\<lambda>z. (w,z)))"
        by (rule continuous_on_compose continuous_intros continuous_on_subset [OF cond_uu] | force intro: that)+
      then have *: "continuous_on u (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z))"
        by (rule rev_iffD1 [OF _ continuous_on_cong [OF refl]]) (simp add: d_def field_simps)
      have **: "\<And>x. \<lbrakk>x \<in> u; x \<noteq> w\<rbrakk> \<Longrightarrow> (\<lambda>z. if w = z then deriv f z else (f w - f z) / (w - z)) complex_differentiable at x"
        apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in complex_differentiable_transform_within)
        apply (rule u derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp add: dist_commute)+
        done
      show ?thesis
        unfolding d_def
        apply (rule no_isolated_singularity [OF * _ u, where k = "{w}"])
        apply (auto simp: complex_differentiable_def [symmetric] holomorphic_on_open open_Diff u **)
        done
    qed
    { fix a b
      assume abu: "closed_segment a b \<subseteq> u"
      then have "\<And>w. w \<in> u \<Longrightarrow> (\<lambda>z. d z w) contour_integrable_on (linepath a b)"
        by (metis hol_dw continuous_on_subset contour_integrable_continuous_linepath holomorphic_on_imp_continuous_on)
      then have cont_cint_d: "continuous_on u (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
        apply (rule contour_integral_continuous_on_linepath_2D [OF \<open>open u\<close> _ _ abu])
        apply (auto simp: intro: continuous_on_swap_args cond_uu)
        done
      have cont_cint_d\<gamma>: "continuous_on {0..1} ((\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) o \<gamma>)"
        apply (rule continuous_on_compose)
        using \<open>path \<gamma>\<close> path_def pasz
        apply (auto intro!: continuous_on_subset [OF cont_cint_d])
        apply (force simp add: path_image_def)
        done
      have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
        apply (simp add: contour_integrable_on)
        apply (rule integrable_continuous_real)
        apply (rule continuous_on_mult [OF cont_cint_d\<gamma> [unfolded o_def]])
        using pf\<gamma>'
        by (simp add: continuous_on_polymonial_function vector_derivative_at [OF \<gamma>'])
      have "contour_integral (linepath a b) h = contour_integral (linepath a b) (\<lambda>z. contour_integral \<gamma> (d z))"
        using abu  by (force simp add: h_def intro: contour_integral_eq)
      also have "... =  contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
        apply (rule contour_integral_swap)
        apply (rule continuous_on_subset [OF cond_uu])
        using abu pasz \<open>valid_path \<gamma>\<close>
        apply (auto intro!: continuous_intros)
        by (metis \<gamma>' continuous_on_eq path_def path_polynomial_function pf\<gamma>' vector_derivative_at)
      finally have cint_h_eq:
          "contour_integral (linepath a b) h =
                    contour_integral \<gamma> (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))" .
      note cint_cint cint_h_eq
    } note cint_h = this
    have conthu: "continuous_on u h"
    proof (simp add: continuous_on_sequentially, clarify)
      fix a x
      assume x: "x \<in> u" and au: "\<forall>n. a n \<in> u" and ax: "a \<longlonglongrightarrow> x"
      then have A1: "\<forall>\<^sub>F n in sequentially. d (a n) contour_integrable_on \<gamma>"
        by (meson U contour_integrable_on_def eventuallyI)
      obtain dd where "dd>0" and dd: "cball x dd \<subseteq> u" using open_contains_cball u x by force
      have A2: "\<forall>\<^sub>F n in sequentially. \<forall>xa\<in>path_image \<gamma>. cmod (d (a n) xa - d x xa) < ee" if "0 < ee" for ee
      proof -
        let ?ddpa = "{(w,z) |w z. w \<in> cball x dd \<and> z \<in> path_image \<gamma>}"
        have "uniformly_continuous_on ?ddpa (\<lambda>(x,y). d x y)"
          apply (rule compact_uniformly_continuous [OF continuous_on_subset[OF cond_uu]])
          using dd pasz \<open>valid_path \<gamma>\<close>
          apply (auto simp: compact_Times compact_valid_path_image simp del: mem_cball)
          done
        then obtain kk where "kk>0"
            and kk: "\<And>x x'. \<lbrakk>x\<in>?ddpa; x'\<in>?ddpa; dist x' x < kk\<rbrakk> \<Longrightarrow>
                             dist ((\<lambda>(x,y). d x y) x') ((\<lambda>(x,y). d x y) x) < ee"
          apply (rule uniformly_continuous_onE [where e = ee])
          using \<open>0 < ee\<close> by auto
        have kk: "\<lbrakk>norm (w - x) \<le> dd; z \<in> path_image \<gamma>; norm ((w, z) - (x, z)) < kk\<rbrakk> \<Longrightarrow> norm (d w z - d x z) < ee"
                 for  w z
          using \<open>dd>0\<close> kk [of "(x,z)" "(w,z)"] by (force simp add: norm_minus_commute dist_norm)
        show ?thesis
          using ax unfolding lim_sequentially eventually_sequentially
          apply (drule_tac x="min dd kk" in spec)
          using \<open>dd > 0\<close> \<open>kk > 0\<close>
          apply (fastforce simp: kk dist_norm)
          done
      qed
      have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
        apply (simp add: contour_integral_unique [OF U, symmetric] x)
        apply (rule contour_integral_uniform_limit [OF A1 A2 le_B])
        apply (auto simp: \<open>valid_path \<gamma>\<close>)
        done
      then show "(h \<circ> a) \<longlonglongrightarrow> h x"
        by (simp add: h_def x au o_def)
    qed
    show ?thesis
    proof (simp add: holomorphic_on_open complex_differentiable_def [symmetric], clarify)
      fix z0
      consider "z0 \<in> v" | "z0 \<in> u" using uv_Un by blast
      then show "h complex_differentiable at z0"
      proof cases
        assume "z0 \<in> v" then show ?thesis
          using Cauchy_next_derivative [OF con_pa_f le_B f_has_cint _ ov] V f_has_cint \<open>valid_path \<gamma>\<close>
          by (auto simp: complex_differentiable_def v_def)
      next
        assume "z0 \<in> u" then
        obtain e where "e>0" and e: "ball z0 e \<subseteq> u" by (blast intro: openE [OF u])
        have *: "contour_integral (linepath a b) h + contour_integral (linepath b c) h + contour_integral (linepath c a) h = 0"
                if abc_subset: "convex hull {a, b, c} \<subseteq> ball z0 e"  for a b c
        proof -
          have *: "\<And>x1 x2 z. z \<in> u \<Longrightarrow> closed_segment x1 x2 \<subseteq> u \<Longrightarrow> (\<lambda>w. d w z) contour_integrable_on linepath x1 x2"
            using  hol_dw holomorphic_on_imp_continuous_on u
            by (auto intro!: contour_integrable_holomorphic_simple)
          have abc: "closed_segment a b \<subseteq> u"  "closed_segment b c \<subseteq> u"  "closed_segment c a \<subseteq> u"
            using that e segments_subset_convex_hull by fastforce+
          have eq0: "\<And>w. w \<in> u \<Longrightarrow> contour_integral (linepath a b +++ linepath b c +++ linepath c a) (\<lambda>z. d z w) = 0"
            apply (rule contour_integral_unique [OF Cauchy_theorem_triangle])
            apply (rule holomorphic_on_subset [OF hol_dw])
            using e abc_subset by auto
          have "contour_integral \<gamma>
                   (\<lambda>x. contour_integral (linepath a b) (\<lambda>z. d z x) +
                        (contour_integral (linepath b c) (\<lambda>z. d z x) +
                         contour_integral (linepath c a) (\<lambda>z. d z x)))  =  0"
            apply (rule contour_integral_eq_0)
            using abc pasz u
            apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
            done
          then show ?thesis
            by (simp add: cint_h abc contour_integrable_add contour_integral_add [symmetric] add_ac)
        qed
        show ?thesis
          using e \<open>e > 0\<close>
          by (auto intro!: holomorphic_on_imp_differentiable_at [OF _ open_ball] analytic_imp_holomorphic
                           Morera_triangle continuous_on_subset [OF conthu] *)
      qed
    qed
  qed
  ultimately have [simp]: "h z = 0" for z
    by (meson Liouville_weak)
  have "((\<lambda>w. 1 / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z) \<gamma>"
    by (rule has_contour_integral_winding_number [OF \<open>valid_path \<gamma>\<close> znot])
  then have "((\<lambda>w. f z * (1 / (w - z))) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
    by (metis mult.commute has_contour_integral_lmul)
  then have 1: "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
    by (simp add: divide_simps)
  moreover have 2: "((\<lambda>w. (f w - f z) / (w - z)) has_contour_integral 0) \<gamma>"
    using U [OF z] pasz d_def by (force elim: has_contour_integral_eq [where g = "\<lambda>w. (f w - f z)/(w - z)"])
  show ?thesis
    using has_contour_integral_add [OF 1 2]  by (simp add: diff_divide_distrib)
qed


theorem Cauchy_integral_formula_global:
    assumes s: "open s" and holf: "f holomorphic_on s"
        and z: "z \<in> s" and vpg: "valid_path \<gamma>"
        and pasz: "path_image \<gamma> \<subseteq> s - {z}" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
        and zero: "\<And>w. w \<notin> s \<Longrightarrow> winding_number \<gamma> w = 0"
      shows "((\<lambda>w. f w / (w - z)) has_contour_integral (2*pi * ii * winding_number \<gamma> z * f z)) \<gamma>"
proof -
  have "path \<gamma>" using vpg by (blast intro: valid_path_imp_path)
  have hols: "(\<lambda>w. f w / (w - z)) holomorphic_on s - {z}" "(\<lambda>w. 1 / (w - z)) holomorphic_on s - {z}"
    by (rule holomorphic_intros holomorphic_on_subset [OF holf] | force)+
  then have cint_fw: "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
    by (meson contour_integrable_holomorphic_simple holomorphic_on_imp_continuous_on open_delete s vpg pasz)
  obtain d where "d>0"
      and d: "\<And>g h. \<lbrakk>valid_path g; valid_path h; \<forall>t\<in>{0..1}. cmod (g t - \<gamma> t) < d \<and> cmod (h t - \<gamma> t) < d;
                     pathstart h = pathstart g \<and> pathfinish h = pathfinish g\<rbrakk>
                     \<Longrightarrow> path_image h \<subseteq> s - {z} \<and> (\<forall>f. f holomorphic_on s - {z} \<longrightarrow> contour_integral h f = contour_integral g f)"
    using contour_integral_nearby_ends [OF _ \<open>path \<gamma>\<close> pasz] s by (simp add: open_Diff) metis
  obtain p where polyp: "polynomial_function p"
             and ps: "pathstart p = pathstart \<gamma>" and pf: "pathfinish p = pathfinish \<gamma>" and led: "\<forall>t\<in>{0..1}. cmod (p t - \<gamma> t) < d"
    using path_approx_polynomial_function [OF \<open>path \<gamma>\<close> \<open>d > 0\<close>] by blast
  then have ploop: "pathfinish p = pathstart p" using loop by auto
  have vpp: "valid_path p"  using polyp valid_path_polynomial_function by blast
  have [simp]: "z \<notin> path_image \<gamma>" using pasz by blast
  have paps: "path_image p \<subseteq> s - {z}" and cint_eq: "(\<And>f. f holomorphic_on s - {z} \<Longrightarrow> contour_integral p f = contour_integral \<gamma> f)"
    using pf ps led d [OF vpg vpp] \<open>d > 0\<close> by auto
  have wn_eq: "winding_number p z = winding_number \<gamma> z"
    using vpp paps
    by (simp add: subset_Diff_insert vpg valid_path_polynomial_function winding_number_valid_path cint_eq hols)
  have "winding_number p w = winding_number \<gamma> w" if "w \<notin> s" for w
  proof -
    have hol: "(\<lambda>v. 1 / (v - w)) holomorphic_on s - {z}"
      using that by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
   have "w \<notin> path_image p" "w \<notin> path_image \<gamma>" using paps pasz that by auto
   then show ?thesis
    using vpp vpg by (simp add: subset_Diff_insert valid_path_polynomial_function winding_number_valid_path cint_eq [OF hol])
  qed
  then have wn0: "\<And>w. w \<notin> s \<Longrightarrow> winding_number p w = 0"
    by (simp add: zero)
  show ?thesis
    using Cauchy_integral_formula_global_weak [OF s holf z polyp paps ploop wn0] hols
    by (metis wn_eq cint_eq has_contour_integral_eqpath cint_fw cint_eq)
qed

theorem Cauchy_theorem_global:
    assumes s: "open s" and holf: "f holomorphic_on s"
        and vpg: "valid_path \<gamma>" and loop: "pathfinish \<gamma> = pathstart \<gamma>"
        and pas: "path_image \<gamma> \<subseteq> s"
        and zero: "\<And>w. w \<notin> s \<Longrightarrow> winding_number \<gamma> w = 0"
      shows "(f has_contour_integral 0) \<gamma>"
proof -
  obtain z where "z \<in> s" and znot: "z \<notin> path_image \<gamma>"
  proof -
    have "compact (path_image \<gamma>)"
      using compact_valid_path_image vpg by blast
    then have "path_image \<gamma> \<noteq> s"
      by (metis (no_types) compact_open path_image_nonempty s)
    with pas show ?thesis by (blast intro: that)
  qed
  then have pasz: "path_image \<gamma> \<subseteq> s - {z}" using pas by blast
  have hol: "(\<lambda>w. (w - z) * f w) holomorphic_on s"
    by (rule holomorphic_intros holf)+
  show ?thesis
    using Cauchy_integral_formula_global [OF s hol \<open>z \<in> s\<close> vpg pasz loop zero]
    by (auto simp: znot elim!: has_contour_integral_eq)
qed

corollary Cauchy_theorem_global_outside:
    assumes "open s" "f holomorphic_on s" "valid_path \<gamma>"  "pathfinish \<gamma> = pathstart \<gamma>" "path_image \<gamma> \<subseteq> s"
            "\<And>w. w \<notin> s \<Longrightarrow> w \<in> outside(path_image \<gamma>)"
      shows "(f has_contour_integral 0) \<gamma>"
by (metis Cauchy_theorem_global assms winding_number_zero_in_outside valid_path_imp_path)


end