author paulson Mon, 11 Jun 2018 22:43:33 +0100 changeset 68420 529d6b132c27 parent 68410 4e27f5c361d2 child 68421 e082a36dc35d
tidier Cauchy proofs
```--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Sat Jun 09 21:52:16 2018 +0200
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy	Mon Jun 11 22:43:33 2018 +0100
@@ -6338,12 +6338,10 @@
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
-    done
+  proof (rule Liouville_weak_inverse [OF 1])
+    show "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (\<Sum>i\<le>n. a i * x ^ i)" for B
+      using i polyfun_extremal nz by force
+  qed (use that in auto)
qed

@@ -6358,14 +6356,15 @@
case less then show ?thesis by (force simp: ball_empty less_imp_le continuous_on_def holomorphic_on_def intro: that)
next
case equal then show ?thesis
-    by (force simp: holomorphic_on_def continuous_on_sing intro: that)
+    by (force simp: holomorphic_on_def intro: that)
next
case greater
have contg: "continuous_on (cball z r) g"
using cont uniform_limit_theorem [OF eventually_mono ulim F]  by blast
-  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])+
+  have "path_image (circlepath z r) \<subseteq> cball z r"
using \<open>0 < r\<close> by auto
+  then have 1: "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
+    by (intro continuous_intros continuous_on_subset [OF contg])
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 -
@@ -6389,18 +6388,16 @@
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 ul_less F \<open>0 < r\<close>])
have f_tends_cig: "((\<lambda>n. 2 * of_real pi * \<i> * 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 uniform_limitD [OF ulim]], assumption)
-      using w
-      apply (force simp: dist_norm)
-      done
+    proof (rule Lim_transform_eventually)
+      show "\<forall>\<^sub>F x in F. contour_integral (circlepath z r) (\<lambda>u. f x u / (u - w))
+                     = 2 * of_real pi * \<i> * f x w"
+        apply (rule eventually_mono [OF cont contour_integral_unique [OF Cauchy_integral_circlepath]])
+        using w\<open>0 < d\<close> d_def by auto
+    qed (auto simp: cif_tends_cig)
+    have "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. dist (f n w) (g w) < e"
+      by (rule eventually_mono [OF uniform_limitD [OF ulim]]) (use w in auto)
+    then have "((\<lambda>n. 2 * of_real pi * \<i> * f n w) \<longlongrightarrow> 2 * of_real pi * \<i> * g w) F"
+      by (rule tendsto_mult_left [OF tendstoI])
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: dist_norm)
@@ -6455,12 +6452,17 @@
define d where "d = (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 diff_ge_0_iff_ge dist_commute dist_norm less_eq_real_def mem_ball w)
-      done
+    have dle: "d \<le> cmod ((y - w)\<^sup>2)" if "r = cmod (z - y)" for y
+    proof -
+      have "w \<in> ball z (cmod (z - y))"
+        using that w by fastforce
+      then have "cmod (w - z) \<le> cmod (z - y)"
+        by (simp add: dist_complex_def norm_minus_commute)
+      moreover have "cmod (z - y) - cmod (w - z) \<le> cmod (y - w)"
+      ultimately show ?thesis
+        using that by (simp add: d_def norm_power power_mono)
+    qed
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: holomorphic_on_open intro: w Cauchy_derivative_integral_circlepath eventually_mono [OF cont])
have 2: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)\<^sup>2) (\<lambda>x. g x / (x - w)\<^sup>2) F"
@@ -6468,9 +6470,8 @@
proof clarify
fix e::real
assume "0 < e"
-      with  \<open>r > 0\<close>
-      show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
-        apply (simp add: diff_divide_distrib [symmetric] norm_divide divide_simps sphere_def dist_norm)
+      with \<open>r > 0\<close> show "\<forall>\<^sub>F n in F. \<forall>x\<in>sphere z r. dist (f n x / (x - w)\<^sup>2) (g x / (x - w)\<^sup>2) < e"
+        apply (simp add: norm_divide divide_simps sphere_def dist_norm)
apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
apply (simp add: \<open>0 < d\<close>)
apply (force simp: dist_norm dle intro: less_le_trans)
@@ -6508,10 +6509,12 @@
and ul: "uniform_limit (cball z r) f g sequentially"
using ulim_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
+    proof (intro eventuallyI conjI)
+      show "continuous_on (cball z r) (f x)" for x
+        using hol_fn holomorphic_on_imp_continuous_on holomorphic_on_subset r by blast
+      show "f x holomorphic_on ball z r" for x
+        by (metis hol_fn holomorphic_on_subset interior_cball interior_subset r)
+    qed
show ?thesis
apply (rule holomorphic_uniform_limit [OF *])
using \<open>0 < r\<close> centre_in_ball ul
@@ -6537,15 +6540,14 @@
using ulim_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
+    proof (intro eventuallyI conjI ballI)
+      show "continuous_on (cball z r) (f x)" for x
+        by (meson S continuous_on_subset hfd holomorphic_on_imp_continuous_on holomorphic_on_open r)
+      show "w \<in> ball z r \<Longrightarrow> (f x has_field_derivative f' x w) (at w)" for w x
+        using ball_subset_cball hfd r by blast
+    qed
show ?thesis
-      apply (rule has_complex_derivative_uniform_limit [OF *, of g])
-      using \<open>0 < r\<close> centre_in_ball ul
-      apply force+
-      done
+      by (rule has_complex_derivative_uniform_limit [OF *, of g]) (use \<open>0 < r\<close> ul in \<open>force+\<close>)
qed
show ?thesis
by (rule bchoice) (blast intro: y)
@@ -6569,11 +6571,11 @@
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)
-        using g uniform_limit_on_subset
-        apply (force simp: dist_norm eventually_sequentially)
-          done
+    show ?thesis
+    proof (intro conjI exI)
+      show "uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+        using d g uniform_limit_on_subset by (force simp: dist_norm eventually_sequentially)
+    qed (use \<open>d > 0\<close> d in auto)
qed
have "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i x) \<longlonglongrightarrow> g x"
by (metis tendsto_uniform_limitI [OF g])
@@ -6612,14 +6614,14 @@
using  summable_sums centre_in_ball \<open>0 < d\<close> \<open>summable h\<close> le_h
by (metis (full_types) Int_iff gg' summable_def that)
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
+    proof (rule DERIV_transform_at)
+      show "\<And>x. dist x z < r \<Longrightarrow> g x = (\<Sum>n. f n x)"
+        by (metis subsetD dist_commute gg' mem_ball r sums_unique)
+    qed (use \<open>0 < r\<close> gg' \<open>z \<in> S\<close> \<open>0 < d\<close> in auto)
ultimately show ?thesis by auto
qed
then show ?thesis
-    by (rule_tac x="\<lambda>x. suminf  (\<lambda>n. f n x)" in exI) meson
+    by (rule_tac x="\<lambda>x. suminf (\<lambda>n. f n x)" in exI) meson
qed

@@ -6682,12 +6684,9 @@
(\<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 (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 eventually_sequentially norm_mult norm_divide norm_power)
-      apply (rule_tac x=0 in exI)
-      apply (force simp: dist_norm intro!: mult_left_mono power_mono y_le)
+      apply (auto simp: sum eventually_sequentially norm_mult norm_power dist_norm intro!: mult_left_mono power_mono y_le)
done
then show ?thesis
@@ -6742,12 +6741,10 @@
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
+    have "(f has_field_derivative g' w) (at w)"
+      by (rule DERIV_transform_at [where d="(r - norm(z - w))/2"])
+      (use w gg' [of w] in \<open>(force simp: dist_norm)+\<close>)
+    then show ?thesis ..
qed
then show ?thesis by (simp add: holomorphic_on_open)
qed
@@ -6755,10 +6752,8 @@
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
+  apply (intro iffI ballI holomorphic_power_series, assumption+)
+  apply (force intro: power_series_holomorphic [where a = "\<lambda>n. (deriv ^^ n) f z / (fact n)"])
done

corollary power_series_analytic:
@@ -6791,102 +6786,104 @@
done

lemma holomorphic_fun_eq_0_on_connected:
-  assumes holf: "f holomorphic_on s" and "open s"
-      and cons: "connected s"
+  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"
+      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)
-  have 1: "openin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
+  have *: "ball x e \<subseteq> (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
+    if "\<forall>u. (deriv ^^ u) f x = 0" "ball x e \<subseteq> S" for x e
+  proof -
+    have "\<And>x' n. dist x x' < e \<Longrightarrow> (deriv ^^ n) f x' = 0"
+      apply (rule holomorphic_fun_eq_0_on_ball [OF holomorphic_higher_deriv])
+         apply (rule holomorphic_on_subset [OF holf])
+      using that apply simp_all
+    with that show ?thesis by auto
+  qed
+  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>
+    using \<open>open S\<close>
apply (erule all_forward)
using "*" by auto blast+
-  have 2: "closedin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
+  have 2: "closedin (subtopology euclidean S) (\<Inter>n. {w \<in> S. (deriv ^^ n) f w = 0})"
using assms
by (auto intro: continuous_closedin_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>] .
+  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"
+  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>
+    using cons der \<open>z \<in> S\<close>
-    apply (drule_tac x="\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0}" in spec)
+    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"
+  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"
+      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
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>x. f x - g x" S z, simplified])
+  show "(\<lambda>x. f x - g x) holomorphic_on S"
+    by (intro assms holomorphic_intros)
+  show "\<And>n. (deriv ^^ n) (\<lambda>x. f x - g x) z = 0"
+    using assms higher_deriv_diff by auto
+qed (use assms in auto)

lemma holomorphic_fun_eq_const_on_connected:
-  assumes holf: "f holomorphic_on s" and "open s"
-      and cons: "connected s"
+  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"
+      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
-
+proof (rule holomorphic_fun_eq_0_on_connected [of "\<lambda>w. f w - f z" S z, simplified])
+  show "(\<lambda>w. f w - f z) holomorphic_on S"
+    by (intro assms holomorphic_intros)
+  show "\<And>n. (deriv ^^ n) (\<lambda>w. f w - f z) z = 0"
+    by (subst higher_deriv_diff) (use assms in \<open>auto intro: holomorphic_intros\<close>)
+qed (use assms in auto)

lemma pole_lemma:
-  assumes holf: "f holomorphic_on s" and a: "a \<in> interior s"
+  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")
+                 else (f z - f a) / (z - a)) holomorphic_on S" (is "?F holomorphic_on S")
proof -
-  have F1: "?F field_differentiable (at u within s)" if "u \<in> s" "u \<noteq> a" for u
+  have F1: "?F field_differentiable (at u within S)" if "u \<in> S" "u \<noteq> a" for u
proof -
-    have fcd: "f field_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)) field_differentiable at u within s"
+    have fcd: "f field_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)) field_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 field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> s\<close>)
+      by (rule field_differentiable_transform_within [OF _ _ _ cd]) (auto simp: \<open>u \<in> S\<close>)
qed
-  have F2: "?F field_differentiable at a" if "0 < e" "ball a e \<subseteq> s" for e
+  have F2: "?F field_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>])
+      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 field_differentiable_def [symmetric])
+      apply (simp add: holomorphic_on_def flip: field_differentiable_def)
using mem_ball that
apply (auto intro: F1 field_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 field_differentiable_def [symmetric])
-      apply (drule_tac x=a in bspec)
-      apply (auto simp: DERIV_deriv_iff_field_differentiable [symmetric] continuous_at DERIV_iff2
+      case True
+      then have "f field_differentiable at a"
+        using holfb \<open>0 < e\<close> holomorphic_on_imp_differentiable_at by auto
+      with True show ?thesis
+        by (auto simp: continuous_at DERIV_iff2 simp flip: DERIV_deriv_iff_field_differentiable
elim: rev_iffD1 [OF _ LIM_equal])
-      done
next
case False with 2 that show ?thesis
by (force simp: holomorphic_on_open open_Diff field_differentiable_def [symmetric] field_differentiable_imp_continuous_at)
@@ -6901,29 +6898,29 @@
qed
show ?thesis
proof
-    fix x assume "x \<in> s" show "?F field_differentiable at x within s"
+    fix x assume "x \<in> S" show "?F field_differentiable at x within S"
proof (cases "x=a")
case True then show ?thesis
using a by (auto simp: mem_interior intro: field_differentiable_at_within F2)
next
-      case False with F1 \<open>x \<in> s\<close>
+      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"
+  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"
+                 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")
+  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
@@ -6935,48 +6932,53 @@
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"
+  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]
+                 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"
+  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"
+    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"
+  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]
+    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
+  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 (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
+    shows "(\<lambda>z. if z = a then deriv g a else f z - g a/(z - a)) analytic_on S" (is "?F analytic_on S")
+  unfolding analytic_on_def
+proof
+  fix x
+  assume "x \<in> S"
+  with g obtain e where "0 < e" and e: "g holomorphic_on ball x e"
+    by (auto simp add: analytic_on_def)
+  obtain d where "0 < d" and d: "\<And>w. w \<in> ball x d - {a} \<Longrightarrow> g w = (w - a) * f w"
+    using \<open>x \<in> S\<close> eq by blast
+  have "?F holomorphic_on ball x (min d e)"
+    using d e \<open>x \<in> S\<close> by (fastforce simp: holomorphic_on_subset subset_ball intro!: pole_theorem_open)
+  then show "\<exists>e>0. ?F holomorphic_on ball x e"
+    using \<open>0 < d\<close> \<open>0 < e\<close> not_le by fastforce
+qed

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)"
+  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"
+    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
@@ -6985,22 +6987,27 @@
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"
+  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
+                 else f z - g a/(z - a)) analytic_on S"
+proof (rule pole_theorem_analytic [OF g])
+  fix z
+  assume "z \<in> S"
+  then obtain e where "0 < e" and e: "ball z e \<subseteq> T"
+    using assms openE by blast
+  then show "\<exists>d>0. \<forall>w\<in>ball z d - {a}. g w = (w - a) * f w"
+    using eq by auto
+qed

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"
+  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"
+    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"
+  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
@@ -7011,24 +7018,25 @@
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))"
+  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>
+  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>
+          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}"
+    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 = "cball w \<delta>  \<times> 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
+    proof (rule compact_uniformly_continuous)
+      show "continuous_on ?TZ (\<lambda>(x,y). F x y)"
+        by (rule continuous_on_subset[OF cond_uu]) (use SigmaE \<delta> abu in blast)
+      show "compact ?TZ"
+    qed
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)"
@@ -7040,13 +7048,13 @@
for x1 x2 x1' x2'
using \<eta> [of "(x1,x2)" "(x1',x2')"] by (force simp: 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'
+                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)"
+      have "(\<lambda>x. F x' x - F w x) contour_integrable_on linepath a b"
+        by (simp add: \<open>w \<in> U\<close> cont_dw contour_integrable_diff that)
+      then 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)
+        using \<open>0 < \<epsilon>\<close> \<open>0 < \<delta>\<close> that apply (auto simp: norm_minus_commute)
done
also have "\<dots> = \<epsilon>" using \<open>a \<noteq> b\<close> by simp
finally show ?thesis .
@@ -7054,22 +7062,26 @@
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)
+      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
+  proof (cases "a=b")
+    case True
+    then show ?thesis by simp
+  next
+    case False
+    show ?thesis
+      by (rule continuous_onI) (use False in \<open>auto intro: *\<close>)
+  qed
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"
+  assumes "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 * \<i> * 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)"
@@ -7084,46 +7096,50 @@
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"
+  have uv_Un: "U \<union> v = UNIV"
using pasz zero by (auto simp: v_def)
-  have conf: "continuous_on u f"
+  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
+  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)
+    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 \<open>open U\<close>)
then have "isCont (\<lambda>x. if x = y then deriv f y else (f x - f y) / (x - y)) y"
-      using at_within_open field_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 at_within_open field_differentiable_imp_continuous_at holomorphic_on_def that \<open>open U\<close> by fastforce
+    then have "continuous_on U (d y)"
+      apply (simp add: d_def continuous_on_eq_continuous_at \<open>open U\<close>, clarify)
using * holomorphic_on_def
-      by (meson field_differentiable_within_open field_differentiable_imp_continuous_at u)
-    moreover have "d y holomorphic_on u - {y}"
-      apply (simp add: d_def holomorphic_on_open u open_delete field_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 field_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
+      by (meson field_differentiable_within_open field_differentiable_imp_continuous_at \<open>open U\<close>)
+    moreover have "d y holomorphic_on U - {y}"
+    proof -
+      have "\<And>w. w \<in> U - {y} \<Longrightarrow>
+                 (\<lambda>w. if w = y then deriv f y else (f w - f y) / (w - y)) field_differentiable at w"
+        apply (rule_tac d="dist w y" and f = "\<lambda>w. (f w - f y)/(w - y)" in field_differentiable_transform_within)
+           apply (auto simp: dist_pos_lt dist_commute intro!: derivative_intros)
+        using \<open>open U\<close> holf holomorphic_on_imp_differentiable_at by blast
+      then show ?thesis
+        unfolding field_differentiable_def by (simp add: d_def holomorphic_on_open \<open>open U\<close> open_delete)
+    qed
ultimately show ?thesis
-      by (rule no_isolated_singularity) (auto simp: u)
+      by (rule no_isolated_singularity) (auto simp: \<open>open U\<close>)
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: that)+
-    done
+  proof (rule contour_integrable_holomorphic_simple [where S = "U-{y}"])
+    show "(\<lambda>x. (f x - f y) / (x - y)) holomorphic_on U - {y}"
+      by (force intro: holomorphic_intros holomorphic_on_subset [OF holf])
+    show "path_image \<gamma> \<subseteq> U - {y}"
+      using pasz that by blast
+  qed (auto simp: \<open>open U\<close> open_delete \<open>valid_path \<gamma>\<close>)
define h where
-    "h z = (if z \<in> u then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
-  have U: "\<And>z. z \<in> u \<Longrightarrow> ((d z) has_contour_integral h z) \<gamma>"
+    "h z = (if z \<in> U then contour_integral \<gamma> (d z) else contour_integral \<gamma> (\<lambda>w. f w/(w - z)))" for z
+  have U: "((d z) has_contour_integral h z) \<gamma>" if "z \<in> U" for z
+  proof -
+    have "d z holomorphic_on U"
+      by (simp add: hol_d that)
+    with that show ?thesis
-    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
+      by (meson Diff_subset \<open>open U\<close> \<open>valid_path \<gamma>\<close> contour_integrable_holomorphic_simple has_contour_integral_integral pasz subset_trans)
+  qed
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"
@@ -7142,24 +7158,24 @@
ultimately have *: "((\<lambda>x. f z / (x - z) + (f x - f z) / (x - z)) has_contour_integral (0 + contour_integral \<gamma> (d z))) \<gamma>"
have "((\<lambda>w. f w / (w - z)) has_contour_integral contour_integral \<gamma> (d z)) \<gamma>"
-            if  "z \<in> u"
+            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
+            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)+
+       apply (rule continuous_intros conf holomorphic_intros holf assms | 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
+  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 \<open>open U\<close>
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"
+  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)
@@ -7174,27 +7190,27 @@
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"
+  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>
+           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"
+  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
+  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
+    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"
@@ -7204,12 +7220,12 @@
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"
+    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
+      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
+      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
@@ -7238,32 +7254,33 @@
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
+                 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 add: split_def continuous_on_eq_continuous_at \<open>open U\<close>)
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)"
+    have open_uu_Id: "open (U \<times> U - Id)"
apply (rule open_Diff)
-      apply (simp add: open_Times u)
+      apply (simp add: open_Times \<open>open U\<close>)
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 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 \<open>open U\<close>)
+      by (simp add: interior_open that \<open>open U\<close>)
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
+                      (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>]]
+        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>)
+      obtain k2 where "k2>0" and k2: "ball x k2 \<subseteq> U"
+        by (blast intro: openE [OF \<open>open U\<close>] \<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'
@@ -7273,9 +7290,9 @@
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_field_differentiable at_within_open holf holomorphic_on_def u)
-        have "closed_segment x' z' \<subseteq> u"
+        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_field_differentiable at_within_open holf holomorphic_on_def \<open>open U\<close>)
+        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
@@ -7290,7 +7307,7 @@
also have "\<dots> \<le> e" using \<open>0 < e\<close> by simp
finally show ?thesis .
qed
-      show "\<exists>d>0. \<forall>xa\<in>u \<times> u.
+      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)
@@ -7299,49 +7316,51 @@
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"
+      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>"
-    have cond_uu: "continuous_on (u \<times> u) (\<lambda>(x,y). d x y)"
+    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 (simp add: tendsto_within_open_NO_MATCH open_Times \<open>open U\<close>, 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
+    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) \<circ> (\<lambda>z. (w,z)))"
+      have "continuous_on U ((\<lambda>(x,y). d x y) \<circ> (\<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))"
+      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)) field_differentiable at x"
+      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)) field_differentiable at x"
apply (rule_tac f = "\<lambda>x. (f w - f x)/(w - x)" and d = "dist x w" in field_differentiable_transform_within)
-        apply (rule u derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
+        apply (rule \<open>open U\<close> derivative_intros holomorphic_on_imp_differentiable_at [OF holf] | force simp: dist_commute)+
done
show ?thesis
unfolding d_def
-        apply (rule no_isolated_singularity [OF * _ u, where K = "{w}"])
-        apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff u **)
+        apply (rule no_isolated_singularity [OF * _ \<open>open U\<close>, where K = "{w}"])
+        apply (auto simp: field_differentiable_def [symmetric] holomorphic_on_open open_Diff \<open>open U\<close> **)
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)"
+      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])
+      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 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)) \<circ> \<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: path_image_def)
-        done
+      proof (rule continuous_on_compose)
+        show "continuous_on {0..1} \<gamma>"
+          using \<open>path \<gamma>\<close> path_def by blast
+        show "continuous_on (\<gamma> ` {0..1}) (\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w))"
+          using pasz unfolding path_image_def
+          by (auto intro!: continuous_on_subset [OF cont_cint_d])
+      qed
have cint_cint: "(\<lambda>w. contour_integral (linepath a b) (\<lambda>z. d z w)) contour_integrable_on \<gamma>"
apply (rule integrable_continuous_real)
@@ -7361,13 +7380,13 @@
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"
+    have conthu: "continuous_on U h"
fix a x
-      assume x: "x \<in> u" and au: "\<forall>n. a n \<in> u" and ax: "a \<longlonglongrightarrow> 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
+      obtain dd where "dd>0" and dd: "cball x dd \<subseteq> U" using open_contains_cball \<open>open U\<close> x by force
have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
unfolding uniform_limit_iff dist_norm
proof clarify
@@ -7382,10 +7401,9 @@
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>
+            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
+            by (rule uniformly_continuous_onE [where e = ee]) (use \<open>0 < ee\<close> in 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: norm_minus_commute dist_norm)
@@ -7397,35 +7415,34 @@
done
qed
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
+      have "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> contour_integral \<gamma> (d x)"
+        by (rule contour_integral_uniform_limit [OF A1 A2 le_B]) (auto simp: \<open>valid_path \<gamma>\<close>)
+      then have tendsto_hx: "(\<lambda>n. contour_integral \<gamma> (d (a n))) \<longlonglongrightarrow> h x"
+        by (simp add: h_def x)
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 field_differentiable_def [symmetric], clarify)
fix z0
-      consider "z0 \<in> v" | "z0 \<in> u" using uv_Un by blast
+      consider "z0 \<in> v" | "z0 \<in> U" using uv_Un by blast
then show "h field_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: field_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])
+        assume "z0 \<in> U" then
+        obtain e where "e>0" and e: "ball z0 e \<subseteq> U" by (blast intro: openE [OF \<open>open U\<close>])
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
+          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 \<open>open U\<close>
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"
+          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"
+          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
@@ -7434,7 +7451,7 @@
(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
+            using abc pasz U
apply (subst contour_integral_join [symmetric], auto intro: eq0 *)+
done
then show ?thesis
@@ -7540,13 +7557,12 @@
"path_image g \<subseteq> S" "pathfinish g = pathstart g" "z \<notin> S"
shows "winding_number g z = 0"
proof -
-  have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
-    apply (rule winding_number_homotopic_paths)
-    apply (rule homotopic_loops_imp_homotopic_paths_null [where a = "pathstart g"])
-    apply (rule homotopic_loops_subset [of S])
-    using assms
-    apply (auto simp: homotopic_paths_imp_homotopic_loops path_defs simply_connected_eq_contractible_path)
-    done
+  have hom: "homotopic_loops S g (linepath (pathstart g) (pathstart g))"
+    by (meson assms homotopic_paths_imp_homotopic_loops pathfinish_linepath simply_connected_eq_contractible_path)
+  then have "homotopic_paths (- {z}) g (linepath (pathstart g) (pathstart g))"
+    by (meson \<open>z \<notin> S\<close> homotopic_loops_imp_homotopic_paths_null homotopic_paths_subset subset_Compl_singleton)
+  then have "winding_number g z = winding_number(linepath (pathstart g) (pathstart g)) z"
+    by (rule winding_number_homotopic_paths)
also have "\<dots> = 0"
using assms by (force intro: winding_number_trivial)
finally show ?thesis .
@@ -7562,7 +7578,7 @@
homotopic_paths_imp_homotopic_loops)
using valid_path_imp_path by blast

-lemma holomorphic_logarithm_exists:
+proposition holomorphic_logarithm_exists:
assumes A: "convex A" "open A"
and f: "f holomorphic_on A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
and z0: "z0 \<in> A"
@@ -7586,7 +7602,6 @@
from 2 and z0 and f show ?case
by (auto simp: h_def exp_diff field_simps intro!: derivative_eq_intros g f')
qed fact+
-
then obtain c where c: "\<And>x. x \<in> A \<Longrightarrow> f x / exp (h x) - 1 = c"
by blast
from c[OF z0] and z0 and f have "c = 0"```