--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Tue Feb 21 15:04:01 2017 +0000
@@ -82,7 +82,7 @@
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)
+ apply (blast intro!: differentiable_chain_within)
done
lemma piecewise_differentiable_affine:
@@ -5172,7 +5172,7 @@
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 ul_f: "uniform_limit (path_image \<gamma>) f l 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)"
@@ -5181,10 +5181,13 @@
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
+ then have "0 < e / (\<bar>B\<bar> + 1)" by simp
+ then have "\<forall>\<^sub>F n in F. \<forall>x\<in>path_image \<gamma>. cmod (f n x - l x) < e / (\<bar>B\<bar> + 1)"
+ using ul_f [unfolded uniform_limit_iff dist_norm] by auto
+ with ev_fint
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]]
+ using eventually_happens [OF eventually_conj]
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)
@@ -5209,7 +5212,8 @@
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)
+ using ul_f [unfolded uniform_limit_iff dist_norm, rule_format, 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
@@ -5235,12 +5239,13 @@
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)
+corollary contour_integral_uniform_limit_circlepath:
+ assumes "\<forall>\<^sub>F n::'a in F. (f n) contour_integrable_on (circlepath z r)"
+ and "uniform_limit (sphere z r) f l F"
+ and "~ (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"
+ using assms by (auto simp: vector_derivative_circlepath norm_mult intro!: contour_integral_uniform_limit)
subsection\<open> General stepping result for derivative formulas.\<close>
@@ -5371,11 +5376,11 @@
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 2: "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
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) -
@@ -5402,8 +5407,10 @@
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
+ show "\<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"
+ using twom [OF divide_pos_pos [OF \<open>0 < e\<close> \<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>"
@@ -6017,10 +6024,11 @@
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 ul: "uniform_limit (sphere z r) (\<lambda>n x. (\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k))) (\<lambda>x. f x / (x - w)) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix e::real
+ assume "0 < e"
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
@@ -6061,7 +6069,8 @@
finally show ?thesis
by (simp add: divide_simps norm_divide del: power_Suc)
qed
- with \<open>0 < r\<close> show ?thesis
+ with \<open>0 < r\<close> show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>sphere z r.
+ norm ((\<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) - f x / (x - w)) < e"
by (auto simp: mult_ac less_imp_le eventually_sequentially Ball_def)
qed
have eq: "\<forall>\<^sub>F x in sequentially.
@@ -6076,7 +6085,7 @@
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_integral_uniform_limit_circlepath [OF eventuallyI ul])
apply (rule contour_integrable_sum, simp)
apply (rule contour_integrable_lmul)
apply (rule Cauchy_higher_derivative_integral_circlepath [OF contf holf])
@@ -6189,7 +6198,7 @@
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 ulim: "uniform_limit (cball z r) f g 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)
@@ -6200,8 +6209,7 @@
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])
+ 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])+
using \<open>0 < r\<close> by auto
@@ -6217,17 +6225,16 @@
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)+
+ have ul_less: "uniform_limit (sphere z r) (\<lambda>n x. f n x / (x - w)) (\<lambda>x. g x / (x - w)) F"
+ using greater \<open>0 < d\<close>
+ apply (clarsimp simp add: uniform_limit_iff dist_norm norm_divide diff_divide_distrib [symmetric] divide_simps)
+ apply (rule_tac e1="e * d" in eventually_mono [OF uniform_limitD [OF ulim]])
+ 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>])
+ by (rule contour_integral_uniform_limit_circlepath [OF ev_int ul_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>])
+ 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])
@@ -6237,7 +6244,7 @@
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)
+ apply (rule eventually_mono [OF uniform_limitD [OF ulim]], assumption)
using w
apply (force simp add: dist_norm)
done
@@ -6262,7 +6269,7 @@
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 ulim: "uniform_limit (cball z r) f g F"
and F: "~ trivial_limit F" and "0 < r"
obtains g' where
"continuous_on (cball z r) g"
@@ -6270,7 +6277,7 @@
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];
+ by (rule holomorphic_uniform_limit [OF eventually_mono [OF cont] ulim F];
auto simp: holomorphic_on_open field_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
@@ -6303,13 +6310,19 @@
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 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"
+ unfolding uniform_limit_iff
+ 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)
+ apply (rule eventually_mono [OF uniform_limitD [OF ulim], of "e*d"])
+ apply (simp add: \<open>0 < d\<close>)
+ apply (force simp add: dist_norm dle intro: less_le_trans)
+ done
+ qed
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 contour_integral_uniform_limit_circlepath [OF 1 2 F \<open>0 < r\<close>])
@@ -6331,18 +6344,16 @@
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"
+ assumes S: "open S"
+ and hol_fn: "\<And>n. (f n) holomorphic_on S"
+ and ulim_g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g sequentially"
+ shows "g holomorphic_on S"
proof -
- have "\<exists>f'. (g has_field_derivative f') (at z)" if "z \<in> s" for z
+ 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
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ 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
@@ -6350,37 +6361,36 @@
done
show ?thesis
apply (rule holomorphic_uniform_limit [OF *])
- using \<open>0 < r\<close> centre_in_ball fg
+ using \<open>0 < r\<close> centre_in_ball ul
apply (auto simp: holomorphic_on_open)
done
qed
- with s show ?thesis
+ 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"
+ 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 ulim_g: "\<And>x. x \<in> S
+ \<Longrightarrow> \<exists>d. 0 < d \<and> cball x d \<subseteq> S \<and> uniform_limit (cball x d) f g 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
+ 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
+ obtain r where "0 < r" and r: "cball z r \<subseteq> S"
+ 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>
(\<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)
+ 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
+ using \<open>0 < r\<close> centre_in_ball ul
apply force+
done
qed
@@ -6390,67 +6400,67 @@
subsection\<open>On analytic functions defined by a series.\<close>
-
+
lemma series_and_derivative_comparison:
- fixes s :: "complex set"
- assumes s: "open s"
+ 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)"
+ and hfd: "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
+ and to_g: "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. 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
+ obtain g where g: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ using weierstrass_m_test_ev [OF to_g h] by force
+ have *: "\<exists>d>0. cball x d \<subseteq> S \<and> uniform_limit (cball x d) (\<lambda>n x. \<Sum>i<n. f i x) g sequentially"
+ 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
+ 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
+ using g uniform_limit_on_subset
+ apply (force simp: dist_norm eventually_sequentially)
+ 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_sum)+
+ 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])
+ 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_sum)+
ultimately show ?thesis
- by (force simp add: sums_def conj_commute intro: that)
+ by (metis sums_def 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)"
+ 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. 0 < d \<and> summable h \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. 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
+ 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
+ obtain d h where "0 < d" "summable h" and le_h: "\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball z d \<inter> S. 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)"
+ 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>
+ 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)
+ 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 (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
@@ -6463,21 +6473,16 @@
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)
+ 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. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. 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 (rule ex_forward [OF to_g], assumption)
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)
+apply (force simp add: summable_Re o_def nonneg_Reals_cmod_eq_Re image_subset_iff)
done
@@ -6512,12 +6517,12 @@
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 (auto simp: sum eventually_sequentially norm_mult norm_divide norm_power)
apply (rule_tac x=0 in exI)
- apply (force simp: norm_mult norm_divide norm_power intro!: mult_left_mono power_mono y_le)
+ apply (force simp: dist_norm intro!: mult_left_mono power_mono y_le)
done
then show ?thesis
- by (simp add: dist_0_norm ball_def)
+ by (simp add: ball_def)
next
case False then show ?thesis
apply (simp add: not_less)
@@ -6833,7 +6838,6 @@
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>
@@ -7196,28 +7200,34 @@
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"
+ have A2: "uniform_limit (path_image \<gamma>) (\<lambda>n. d (a n)) (d x) sequentially"
+ unfolding uniform_limit_iff dist_norm
+ proof clarify
+ fix ee::real
+ assume "0 < ee"
+ show "\<forall>\<^sub>F n in sequentially. \<forall>\<xi>\<in>path_image \<gamma>. cmod (d (a n) \<xi> - d x \<xi>) < 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
+ 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
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)
@@ -7285,87 +7295,87 @@
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"
+ 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 * \<i> * 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}"
+ 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)
+ 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
+ \<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)"
+ 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
+ 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}"
+ 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"
+ 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
+ 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"
+ 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"
+ 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>"
+ 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)
+ 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"
+ 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]
+ 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>)"
+ 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)
lemma simply_connected_imp_winding_number_zero:
- assumes "simply_connected s" "path g"
- "path_image g \<subseteq> s" "pathfinish g = pathstart g" "z \<notin> s"
+ assumes "simply_connected S" "path g"
+ "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])
+ 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
@@ -7375,8 +7385,8 @@
qed
lemma Cauchy_theorem_simply_connected:
- assumes "open s" "simply_connected s" "f holomorphic_on s" "valid_path g"
- "path_image g \<subseteq> s" "pathfinish g = pathstart g"
+ assumes "open S" "simply_connected S" "f holomorphic_on S" "valid_path g"
+ "path_image g \<subseteq> S" "pathfinish g = pathstart g"
shows "(f has_contour_integral 0) g"
using assms
apply (simp add: simply_connected_eq_contractible_path)