Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
--- a/src/HOL/Analysis/Bounded_Continuous_Function.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Bounded_Continuous_Function.thy Tue Feb 21 15:04:01 2017 +0000
@@ -190,9 +190,9 @@
have "g \<in> bcontfun" \<comment> \<open>The limit function is bounded and continuous\<close>
proof (intro bcontfunI)
show "continuous_on UNIV g"
- using bcontfunE[OF Rep_bcontfun] limit_function
- by (intro continuous_uniform_limit[where f="\<lambda>n. Rep_bcontfun (f n)" and F="sequentially"])
- (auto simp add: eventually_sequentially trivial_limit_def dist_norm)
+ apply (rule bcontfunE[OF Rep_bcontfun])
+ using limit_function
+ by (auto simp add: uniform_limit_sequentially_iff intro: uniform_limit_theorem[where f="\<lambda>n. Rep_bcontfun (f n)" and F="sequentially"])
next
fix x
from fg_dist have "dist (g x) (Rep_bcontfun (f N) x) < 1"
--- 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)
--- a/src/HOL/Analysis/Complex_Transcendental.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Tue Feb 21 15:04:01 2017 +0000
@@ -1386,7 +1386,7 @@
else Ln(z) - \<i> * of_real(3 * pi/2))"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z] Re_Ln_pos_le [of z]
- by (auto simp: Ln_times)
+ by (simp add: Ln_times) auto
lemma Ln_of_nat: "0 < n \<Longrightarrow> Ln (of_nat n) = of_real (ln (of_nat n))"
by (subst of_real_of_nat_eq[symmetric], subst Ln_of_real[symmetric]) simp_all
--- a/src/HOL/Analysis/Conformal_Mappings.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Conformal_Mappings.thy Tue Feb 21 15:04:01 2017 +0000
@@ -14,8 +14,8 @@
lemma Cauchy_higher_deriv_bound:
assumes holf: "f holomorphic_on (ball z r)"
and contf: "continuous_on (cball z r) f"
+ and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
and "0 < r" and "0 < n"
- and fin : "\<And>w. w \<in> ball z r \<Longrightarrow> f w \<in> ball y B0"
shows "norm ((deriv ^^ n) f z) \<le> (fact n) * B0 / r^n"
proof -
have "0 < B0" using \<open>0 < r\<close> fin [of z]
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Tue Feb 21 15:04:01 2017 +0000
@@ -1369,19 +1369,6 @@
by (auto simp add:norm_minus_commute)
qed
-lemma norm_minus_eqI: "x = - y \<Longrightarrow> norm x = norm y" by auto
-
-lemma Min_grI:
- assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a"
- shows "x < Min A"
- unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
-
-lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
- unfolding norm_eq_sqrt_inner by simp
-
-lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
- unfolding norm_eq_sqrt_inner by simp
-
subsection \<open>Affine set and affine hull\<close>
@@ -8474,19 +8461,8 @@
assume as: "\<forall>i\<in>Basis. 0 < x \<bullet> i" "sum (op \<bullet> x) Basis < 1"
obtain a :: 'b where "a \<in> UNIV" using UNIV_witness ..
let ?d = "(1 - sum (op \<bullet> x) Basis) / real (DIM('a))"
- have "Min ((op \<bullet> x) ` Basis) > 0"
- apply (rule Min_grI)
- using as(1)
- apply auto
- done
- moreover have "?d > 0"
- using as(2) by (auto simp: Suc_le_eq DIM_positive)
- ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
- apply (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI)
- apply rule
- defer
- apply (rule, rule)
- proof -
+ show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
+ proof (rule_tac x="min (Min ((op \<bullet> x) ` Basis)) D" for D in exI, intro conjI impI allI)
fix y
assume y: "dist x y < min (Min (op \<bullet> x ` Basis)) ?d"
have "sum (op \<bullet> y) Basis \<le> sum (\<lambda>i. x\<bullet>i + ?d) Basis"
@@ -8505,7 +8481,8 @@
also have "\<dots> \<le> 1"
unfolding sum.distrib sum_constant
by (auto simp add: Suc_le_eq)
- finally show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i) \<and> sum (op \<bullet> y) Basis \<le> 1"
+ finally show "sum (op \<bullet> y) Basis \<le> 1" .
+ show "(\<forall>i\<in>Basis. 0 \<le> y \<bullet> i)"
proof safe
fix i :: 'a
assume i: "i \<in> Basis"
@@ -8519,7 +8496,14 @@
using Basis_le_norm[OF i, of "x - y"] and as(1)[rule_format, OF i]
by (auto simp: inner_simps)
qed
- qed auto
+ next
+ have "Min ((op \<bullet> x) ` Basis) > 0"
+ using as by simp
+ moreover have "?d > 0"
+ using as by (auto simp: Suc_le_eq)
+ ultimately show "0 < min (Min (op \<bullet> x ` Basis)) ((1 - sum (op \<bullet> x) Basis) / real DIM('a))"
+ by linarith
+ qed
qed
lemma interior_std_simplex_nonempty:
@@ -8655,7 +8639,7 @@
have "0 < card d" using \<open>d \<noteq> {}\<close> \<open>finite d\<close>
by (simp add: card_gt_0_iff)
have "Min ((op \<bullet> x) ` d) > 0"
- using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_grI)
+ using as \<open>d \<noteq> {}\<close> \<open>finite d\<close> by (simp add: Min_gr_iff)
moreover have "?d > 0" using as using \<open>0 < card d\<close> by auto
ultimately have h3: "min (Min ((op \<bullet> x) ` d)) ?d > 0"
by auto
--- a/src/HOL/Analysis/Finite_Product_Measure.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Finite_Product_Measure.thy Tue Feb 21 15:04:01 2017 +0000
@@ -1135,7 +1135,7 @@
proof (intro PiM_eqI)
fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)"
then have "(\<lambda>x. \<lambda>i\<in>{i}. x) -` Pi\<^sub>E {i} A \<inter> space (M i) = A i"
- by (auto dest: sets.sets_into_space)
+ by (fastforce dest: sets.sets_into_space)
with A show "emeasure (distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x)) (Pi\<^sub>E {i} A) = (\<Prod>i\<in>{i}. emeasure (M i) (A i))"
by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict)
qed simp_all
--- a/src/HOL/Analysis/Function_Topology.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Function_Topology.thy Tue Feb 21 15:04:01 2017 +0000
@@ -967,7 +967,7 @@
by blast
define m where "m = Min {(1/2)^(to_nat i) * e i|i. i \<in> I}"
have "m > 0" if "I\<noteq>{}"
- unfolding m_def apply (rule Min_grI) using \<open>finite I\<close> \<open>I \<noteq> {}\<close> \<open>\<And>i. e i > 0\<close> by auto
+ unfolding m_def Min_gr_iff using \<open>finite I\<close> \<open>I \<noteq> {}\<close> \<open>\<And>i. e i > 0\<close> by auto
moreover have "{y. dist x y < m} \<subseteq> U"
proof (auto)
fix y assume "dist x y < m"
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Tue Feb 21 15:04:01 2017 +0000
@@ -9,10 +9,7 @@
Lebesgue_Measure Tagged_Division
begin
-(* BEGIN MOVE *)
-lemma norm_minus2: "norm (x1-x2, y1-y2) = norm (x2-x1, y2-y1)"
- by (simp add: norm_minus_eqI)
-
+(* try instead structured proofs below *)
lemma norm_diff2: "\<lbrakk>y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \<le> e1; norm(y2 - x2) \<le> e2\<rbrakk>
\<Longrightarrow> norm(y - x) \<le> e"
using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
@@ -7696,10 +7693,12 @@
assume xuvwz: "x1 \<in> cbox u v" "x2 \<in> cbox w z"
then have x: "x1 \<in> cbox a b" "x2 \<in> cbox c d"
using uwvz_sub by auto
- have "norm (x1 - t1, x2 - t2) < k"
+ have "norm (x1 - t1, x2 - t2) = norm (t1 - x1, t2 - x2)"
+ by (simp add: norm_Pair norm_minus_commute)
+ also have "norm (t1 - x1, t2 - x2) < k"
using xuvwz ls uwvz_sub unfolding ball_def
- by (force simp add: cbox_Pair_eq dist_norm norm_minus2)
- then have "norm (f (x1,x2) - f (t1,t2)) \<le> e / content ?CBOX / 2"
+ by (force simp add: cbox_Pair_eq dist_norm )
+ finally have "norm (f (x1,x2) - f (t1,t2)) \<le> e / content ?CBOX / 2"
using nf [OF t x] by simp
} note nf' = this
have f_int_uwvz: "f integrable_on cbox (u,w) (v,z)"
@@ -7852,7 +7851,7 @@
finally show ?thesis .
qed (insert a, simp_all add: integral_f)
thus "bounded {integral {c..} (f k) |k. True}"
- by (intro bounded_realI[of _ "exp (-a*c)/a"]) auto
+ by (intro boundedI[of _ "exp (-a*c)/a"]) auto
qed (auto simp: f_def)
from eventually_gt_at_top[of "nat \<lceil>c\<rceil>"] have "eventually (\<lambda>k. of_nat k > c) sequentially"
@@ -7945,7 +7944,7 @@
finally have "abs (F k) \<le> c powr (a + 1) / (a + 1)" .
}
thus "bounded {integral {0..c} (f k) |k. True}"
- by (intro bounded_realI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f)
+ by (intro boundedI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f)
qed
from False c have "c > 0" by simp
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue Feb 21 15:04:01 2017 +0000
@@ -92,12 +92,6 @@
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
-lemma continuous_on_cases:
- "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
- \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
- continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
- by (rule continuous_on_If) auto
-
lemma open_sums:
fixes T :: "('b::real_normed_vector) set"
assumes "open S \<or> open T"
@@ -3946,11 +3940,10 @@
lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
by (simp add: bounded_iff bdd_above_def)
-lemma bounded_realI:
- assumes "\<forall>x\<in>s. \<bar>x::real\<bar> \<le> B"
- shows "bounded s"
- unfolding bounded_def dist_real_def
- by (metis abs_minus_commute assms diff_0_right)
+lemma boundedI:
+ assumes "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
+ shows "bounded S"
+ using assms bounded_iff by blast
lemma bounded_empty [simp]: "bounded {}"
by (simp add: bounded_def)
@@ -5100,7 +5093,7 @@
"compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
-lemma compact_def:
+lemma compact_def: --\<open>this is the definition of compactness in HOL Light\<close>
"compact (S :: 'a::metric_space set) \<longleftrightarrow>
(\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
unfolding compact_eq_seq_compact_metric seq_compact_def by auto
@@ -7576,48 +7569,6 @@
qed (insert D, auto)
qed auto
-text \<open>A uniformly convergent limit of continuous functions is continuous.\<close>
-
-lemma continuous_uniform_limit:
- fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
- assumes "\<not> trivial_limit F"
- and "eventually (\<lambda>n. continuous_on s (f n)) F"
- and "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
- shows "continuous_on s g"
-proof -
- {
- fix x and e :: real
- assume "x\<in>s" "e>0"
- have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
- using \<open>e>0\<close> assms(3)[THEN spec[where x="e/3"]] by auto
- from eventually_happens [OF eventually_conj [OF this assms(2)]]
- obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3" "continuous_on s (f n)"
- using assms(1) by blast
- have "e / 3 > 0" using \<open>e>0\<close> by auto
- then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
- using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF \<open>x\<in>s\<close>, THEN spec[where x="e/3"]] by blast
- {
- fix y
- assume "y \<in> s" and "dist y x < d"
- then have "dist (f n y) (f n x) < e / 3"
- by (rule d [rule_format])
- then have "dist (f n y) (g x) < 2 * e / 3"
- using dist_triangle [of "f n y" "g x" "f n x"]
- using n(1)[THEN bspec[where x=x], OF \<open>x\<in>s\<close>]
- by auto
- then have "dist (g y) (g x) < e"
- using n(1)[THEN bspec[where x=y], OF \<open>y\<in>s\<close>]
- using dist_triangle3 [of "g y" "g x" "f n y"]
- by auto
- }
- then have "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
- using \<open>d>0\<close> by auto
- }
- then show ?thesis
- unfolding continuous_on_iff by auto
-qed
-
-
subsection \<open>Topological stuff about the set of Reals\<close>
lemma open_real:
--- a/src/HOL/Analysis/Uniform_Limit.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Analysis/Uniform_Limit.thy Tue Feb 21 15:04:01 2017 +0000
@@ -472,6 +472,73 @@
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_mult]
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_scaleR]
+lemma uniform_lim_mult:
+ fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_algebra"
+ assumes f: "uniform_limit S f l F"
+ and g: "uniform_limit S g m F"
+ and l: "bounded (l ` S)"
+ and m: "bounded (m ` S)"
+ shows "uniform_limit S (\<lambda>a b. f a b * g a b) (\<lambda>a. l a * m a) F"
+ by (intro bounded_bilinear_bounded_uniform_limit_intros assms)
+
+lemma uniform_lim_inverse:
+ fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_field"
+ assumes f: "uniform_limit S f l F"
+ and b: "\<And>x. x \<in> S \<Longrightarrow> b \<le> norm(l x)"
+ and "b > 0"
+ shows "uniform_limit S (\<lambda>x y. inverse (f x y)) (inverse \<circ> l) F"
+proof (rule uniform_limitI)
+ fix e::real
+ assume "e > 0"
+ have lte: "dist (inverse (f x y)) ((inverse \<circ> l) y) < e"
+ if "b/2 \<le> norm (f x y)" "norm (f x y - l y) < e * b\<^sup>2 / 2" "y \<in> S"
+ for x y
+ proof -
+ have [simp]: "l y \<noteq> 0" "f x y \<noteq> 0"
+ using \<open>b > 0\<close> that b [OF \<open>y \<in> S\<close>] by fastforce+
+ have "norm (l y - f x y) < e * b\<^sup>2 / 2"
+ by (metis norm_minus_commute that(2))
+ also have "... \<le> e * (norm (f x y) * norm (l y))"
+ using \<open>e > 0\<close> that b [OF \<open>y \<in> S\<close>] apply (simp add: power2_eq_square)
+ by (metis \<open>b > 0\<close> less_eq_real_def mult.left_commute mult_mono')
+ finally show ?thesis
+ by (auto simp: dist_norm divide_simps norm_mult norm_divide)
+ qed
+ have "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < b/2"
+ using uniform_limitD [OF f, of "b/2"] by (simp add: \<open>b > 0\<close>)
+ then have "\<forall>\<^sub>F x in F. \<forall>y\<in>S. b/2 \<le> norm (f x y)"
+ apply (rule eventually_mono)
+ using b apply (simp only: dist_norm)
+ by (metis (no_types, hide_lams) diff_zero dist_commute dist_norm norm_triangle_half_l not_less)
+ then have "\<forall>\<^sub>F x in F. \<forall>y\<in>S. b/2 \<le> norm (f x y) \<and> norm (f x y - l y) < e * b\<^sup>2 / 2"
+ apply (simp only: ball_conj_distrib dist_norm [symmetric])
+ apply (rule eventually_conj, assumption)
+ apply (rule uniform_limitD [OF f, of "e * b ^ 2 / 2"])
+ using \<open>b > 0\<close> \<open>e > 0\<close> by auto
+ then show "\<forall>\<^sub>F x in F. \<forall>y\<in>S. dist (inverse (f x y)) ((inverse \<circ> l) y) < e"
+ using lte by (force intro: eventually_mono)
+qed
+
+lemma uniform_lim_div:
+ fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_field"
+ assumes f: "uniform_limit S f l F"
+ and g: "uniform_limit S g m F"
+ and l: "bounded (l ` S)"
+ and b: "\<And>x. x \<in> S \<Longrightarrow> b \<le> norm(m x)"
+ and "b > 0"
+ shows "uniform_limit S (\<lambda>a b. f a b / g a b) (\<lambda>a. l a / m a) F"
+proof -
+ have m: "bounded ((inverse \<circ> m) ` S)"
+ using b \<open>b > 0\<close>
+ apply (simp add: bounded_iff)
+ by (metis le_imp_inverse_le norm_inverse)
+ have "uniform_limit S (\<lambda>a b. f a b * inverse (g a b))
+ (\<lambda>a. l a * (inverse \<circ> m) a) F"
+ by (rule uniform_lim_mult [OF f uniform_lim_inverse [OF g b \<open>b > 0\<close>] l m])
+ then show ?thesis
+ by (simp add: field_class.field_divide_inverse)
+qed
+
lemma uniform_limit_null_comparison:
assumes "\<forall>\<^sub>F x in F. \<forall>a\<in>S. norm (f x a) \<le> g x a"
assumes "uniform_limit S g (\<lambda>_. 0) F"
@@ -482,7 +549,7 @@
using assms(1) by (rule eventually_mono) (force simp add: dist_norm)
qed
-lemma uniform_limit_on_union:
+lemma uniform_limit_on_Un:
"uniform_limit I f g F \<Longrightarrow> uniform_limit J f g F \<Longrightarrow> uniform_limit (I \<union> J) f g F"
by (auto intro!: uniform_limitI dest!: uniform_limitD elim: eventually_elim2)
@@ -495,7 +562,7 @@
assumes "\<And>s. s \<in> S \<Longrightarrow> uniform_limit (h s) f g F"
shows "uniform_limit (UNION S h) f g F"
using assms
- by induct (auto intro: uniform_limit_on_empty uniform_limit_on_union)
+ by induct (auto intro: uniform_limit_on_empty uniform_limit_on_Un)
lemma uniform_limit_on_Union:
assumes "finite I"
@@ -523,7 +590,6 @@
unfolding uniformly_convergent_on_def
by (blast dest: bounded_linear_uniform_limit_intros(13))
-
subsection\<open>Power series and uniform convergence\<close>
proposition powser_uniformly_convergent:
--- a/src/HOL/Limits.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Limits.thy Tue Feb 21 15:04:01 2017 +0000
@@ -1696,7 +1696,7 @@
unfolding tendsto_def eventually_sequentially
by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
-lemma Bseq_inverse_lemma: "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
+lemma norm_inverse_le_norm: "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
for x :: "'a::real_normed_div_algebra"
apply (subst nonzero_norm_inverse, clarsimp)
apply (erule (1) le_imp_inverse_le)
--- a/src/HOL/Real_Vector_Spaces.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Real_Vector_Spaces.thy Tue Feb 21 15:04:01 2017 +0000
@@ -1128,6 +1128,18 @@
lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
by (rule dist_triangle_half_l) (simp_all add: dist_commute)
+lemma dist_triangle_third:
+ assumes "dist x1 x2 < e/3" "dist x2 x3 < e/3" "dist x3 x4 < e/3"
+ shows "dist x1 x4 < e"
+proof -
+ have "dist x1 x3 < e/3 + e/3"
+ by (metis assms(1) assms(2) dist_commute dist_triangle_less_add)
+ then have "dist x1 x4 < (e/3 + e/3) + e/3"
+ by (metis assms(3) dist_commute dist_triangle_less_add)
+ then show ?thesis
+ by simp
+qed
+
subclass uniform_space
proof
fix E x
--- a/src/HOL/Topological_Spaces.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Topological_Spaces.thy Tue Feb 21 15:04:01 2017 +0000
@@ -468,9 +468,9 @@
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
-lemma eventually_eventually:
+lemma eventually_eventually:
"eventually (\<lambda>y. eventually P (nhds y)) (nhds x) = eventually P (nhds x)"
- by (auto simp: eventually_nhds)
+ by (auto simp: eventually_nhds)
lemma (in topological_space) eventually_nhds_in_open:
"open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
@@ -1050,6 +1050,12 @@
definition subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool"
where "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
+lemma subseq_le_mono: "subseq r \<Longrightarrow> m \<le> n \<Longrightarrow> r m \<le> r n"
+ by (simp add: less_mono_imp_le_mono subseq_def)
+
+lemma subseq_id: "subseq id"
+ by (simp add: subseq_def)
+
lemma incseq_SucI: "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
using lift_Suc_mono_le[of X] by (auto simp: incseq_def)
@@ -1818,6 +1824,12 @@
by (rule continuous_on_closed_Un)
qed
+lemma continuous_on_cases:
+ "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
+ \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
+ continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
+ by (rule continuous_on_If) auto
+
lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"
unfolding continuous_on_def by fast
--- a/src/HOL/Transcendental.thy Sun Feb 19 11:58:51 2017 +0100
+++ b/src/HOL/Transcendental.thy Tue Feb 21 15:04:01 2017 +0000
@@ -3194,6 +3194,12 @@
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
by (simp add: diffs_def cos_coeff_Suc del: of_nat_Suc)
+lemma sin_int_times_real: "sin (of_int m * of_real x) = of_real (sin (of_int m * x))"
+ by (metis sin_of_real of_real_mult of_real_of_int_eq)
+
+lemma cos_int_times_real: "cos (of_int m * of_real x) = of_real (cos (of_int m * x))"
+ by (metis cos_of_real of_real_mult of_real_of_int_eq)
+
text \<open>Now at last we can get the derivatives of exp, sin and cos.\<close>
lemma DERIV_sin [simp]: "DERIV sin x :> cos x"
@@ -4077,6 +4083,9 @@
using dvd_triv_left apply fastforce
done
+lemma sin_npi_int [simp]: "sin (pi * of_int n) = 0"
+ by (simp add: sin_zero_iff_int2)
+
lemma cos_monotone_0_pi:
assumes "0 \<le> y" and "y < x" and "x \<le> pi"
shows "cos x < cos y"
@@ -4271,13 +4280,23 @@
by (metis cos_2npi cos_minus mult.assoc mult.left_commute)
qed
-lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)"
- apply auto (* FIXME simproc bug? *)
- apply (auto simp: cos_one_2pi)
- apply (metis of_int_of_nat_eq)
- apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
- apply (metis mult_minus_right of_int_of_nat)
- done
+lemma cos_one_2pi_int: "cos x = 1 \<longleftrightarrow> (\<exists>n::int. x = n * 2 * pi)" (is "?lhs = ?rhs")
+proof
+ assume "cos x = 1"
+ then show ?rhs
+ apply (auto simp: cos_one_2pi)
+ apply (metis of_int_of_nat_eq)
+ apply (metis mult_minus_right of_int_minus of_int_of_nat_eq)
+ done
+next
+ assume ?rhs
+ then show "cos x = 1"
+ by (clarsimp simp add: cos_one_2pi) (metis mult_minus_right of_int_of_nat)
+qed
+
+lemma cos_npi_int [simp]:
+ fixes n::int shows "cos (pi * of_int n) = (if even n then 1 else -1)"
+ by (auto simp: algebra_simps cos_one_2pi_int elim!: oddE evenE)
lemma sin_cos_sqrt: "0 \<le> sin x \<Longrightarrow> sin x = sqrt (1 - (cos(x) ^ 2))"
using sin_squared_eq real_sqrt_unique by fastforce