--- a/src/HOL/Analysis/Gamma_Function.thy Mon Aug 28 13:40:41 2017 +0100
+++ b/src/HOL/Analysis/Gamma_Function.thy Mon Aug 28 13:41:03 2017 +0100
@@ -2697,6 +2697,75 @@
subsubsection \<open>Integral form\<close>
+lemma integrable_on_powr_from_0':
+ assumes a: "a > (-1::real)" and c: "c \<ge> 0"
+ shows "(\<lambda>x. x powr a) integrable_on {0<..c}"
+proof -
+ from c have *: "{0<..c} - {0..c} = {}" "{0..c} - {0<..c} = {0}" by auto
+ show ?thesis
+ by (rule integrable_spike_set [OF integrable_on_powr_from_0[OF a c]]) (simp_all add: *)
+qed
+
+lemma absolutely_integrable_Gamma_integral:
+ assumes "Re z > 0" "a > 0"
+ shows "(\<lambda>t. complex_of_real t powr (z - 1) / of_real (exp (a * t)))
+ absolutely_integrable_on {0<..}" (is "?f absolutely_integrable_on _")
+proof -
+ have "((\<lambda>x. (Re z - 1) * (ln x / x)) \<longlongrightarrow> (Re z - 1) * 0) at_top"
+ by (intro tendsto_intros ln_x_over_x_tendsto_0)
+ hence "((\<lambda>x. ((Re z - 1) * ln x) / x) \<longlongrightarrow> 0) at_top" by simp
+ from order_tendstoD(2)[OF this, of "a/2"] and \<open>a > 0\<close>
+ have "eventually (\<lambda>x. (Re z - 1) * ln x / x < a/2) at_top" by simp
+ from eventually_conj[OF this eventually_gt_at_top[of 0]]
+ obtain x0 where "\<forall>x\<ge>x0. (Re z - 1) * ln x / x < a/2 \<and> x > 0"
+ by (auto simp: eventually_at_top_linorder)
+ hence "x0 > 0" by simp
+ have "x powr (Re z - 1) / exp (a * x) < exp (-(a/2) * x)" if "x \<ge> x0" for x
+ proof -
+ from that and \<open>\<forall>x\<ge>x0. _\<close> have x: "(Re z - 1) * ln x / x < a / 2" "x > 0" by auto
+ have "x powr (Re z - 1) = exp ((Re z - 1) * ln x)"
+ using \<open>x > 0\<close> by (simp add: powr_def)
+ also from x have "(Re z - 1) * ln x < (a * x) / 2" by (simp add: field_simps)
+ finally show ?thesis by (simp add: field_simps exp_add [symmetric])
+ qed
+ note x0 = \<open>x0 > 0\<close> this
+
+ have "?f absolutely_integrable_on ({0<..x0} \<union> {x0..})"
+ proof (rule set_integrable_Un)
+ show "?f absolutely_integrable_on {0<..x0}"
+ proof (rule Bochner_Integration.integrable_bound [OF _ _ AE_I2])
+ show "set_integrable lebesgue {0<..x0} (\<lambda>x. x powr (Re z - 1))" using x0(1) assms
+ by (intro nonnegative_absolutely_integrable_1 integrable_on_powr_from_0') auto
+ show "set_borel_measurable lebesgue {0<..x0}
+ (\<lambda>x. complex_of_real x powr (z - 1) / complex_of_real (exp (a * x)))"
+ by (intro measurable_completion)
+ (auto intro!: borel_measurable_continuous_on_indicator continuous_intros)
+ fix x :: real
+ have "x powr (Re z - 1) / exp (a * x) \<le> x powr (Re z - 1) / 1" if "x \<ge> 0"
+ using that assms by (intro divide_left_mono) auto
+ thus "norm (indicator {0<..x0} x *\<^sub>R ?f x) \<le>
+ norm (indicator {0<..x0} x *\<^sub>R x powr (Re z - 1))"
+ by (simp_all add: norm_divide norm_powr_real_powr indicator_def)
+ qed
+ next
+ show "?f absolutely_integrable_on {x0..}"
+ proof (rule Bochner_Integration.integrable_bound [OF _ _ AE_I2])
+ show "set_integrable lebesgue {x0..} (\<lambda>x. exp (-(a/2) * x))" using assms
+ by (intro nonnegative_absolutely_integrable_1 integrable_on_exp_minus_to_infinity) auto
+ show "set_borel_measurable lebesgue {x0..}
+ (\<lambda>x. complex_of_real x powr (z - 1) / complex_of_real (exp (a * x)))" using x0(1)
+ by (intro measurable_completion)
+ (auto intro!: borel_measurable_continuous_on_indicator continuous_intros)
+ fix x :: real
+ show "norm (indicator {x0..} x *\<^sub>R ?f x) \<le>
+ norm (indicator {x0..} x *\<^sub>R exp (-(a/2) * x))" using x0
+ by (auto simp: norm_divide norm_powr_real_powr indicator_def less_imp_le)
+ qed
+ qed auto
+ also have "{0<..x0} \<union> {x0..} = {0<..}" using x0(1) by auto
+ finally show ?thesis .
+qed
+
lemma integrable_Gamma_integral_bound:
fixes a c :: real
assumes a: "a > -1" and c: "c \<ge> 0"
@@ -2898,6 +2967,25 @@
from has_integral_linear[OF this bounded_linear_Re] show ?thesis by (simp add: o_def)
qed
+lemma absolutely_integrable_Gamma_integral':
+ assumes "Re z > 0"
+ shows "(\<lambda>t. complex_of_real t powr (z - 1) / of_real (exp t)) absolutely_integrable_on {0<..}"
+ using absolutely_integrable_Gamma_integral [OF assms zero_less_one] by simp
+
+lemma Gamma_integral_complex':
+ assumes z: "Re z > 0"
+ shows "((\<lambda>t. of_real t powr (z - 1) / of_real (exp t)) has_integral Gamma z) {0<..}"
+proof -
+ have "((\<lambda>t. of_real t powr (z - 1) / of_real (exp t)) has_integral Gamma z) {0..}"
+ by (rule Gamma_integral_complex) fact+
+ hence "((\<lambda>t. if t \<in> {0<..} then of_real t powr (z - 1) / of_real (exp t) else 0)
+ has_integral Gamma z) {0..}"
+ by (rule has_integral_spike [of "{0}", rotated 2]) auto
+ also have "?this = ?thesis"
+ by (subst has_integral_restrict) auto
+ finally show ?thesis .
+qed
+
subsection \<open>The Weierstraß product formula for the sine\<close>
--- a/src/HOL/Analysis/Infinite_Set_Sum.thy Mon Aug 28 13:40:41 2017 +0100
+++ b/src/HOL/Analysis/Infinite_Set_Sum.thy Mon Aug 28 13:41:03 2017 +0100
@@ -108,6 +108,15 @@
"\<Sum>\<^sub>ai\<in>A. b" \<rightleftharpoons> "CONST infsetsum (\<lambda>i. b) A"
syntax (ASCII)
+ "_uinfsetsum" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b::{banach, second_countable_topology}"
+ ("(3INFSETSUM _:_./ _)" [0, 51, 10] 10)
+syntax
+ "_uinfsetsum" :: "pttrn \<Rightarrow> 'b \<Rightarrow> 'b::{banach, second_countable_topology}"
+ ("(2\<Sum>\<^sub>a_./ _)" [0, 10] 10)
+translations \<comment> \<open>Beware of argument permutation!\<close>
+ "\<Sum>\<^sub>ai. b" \<rightleftharpoons> "CONST infsetsum (\<lambda>i. b) (CONST UNIV)"
+
+syntax (ASCII)
"_qinfsetsum" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> 'a::{banach, second_countable_topology}"
("(3INFSETSUM _ |/ _./ _)" [0, 0, 10] 10)
syntax
@@ -159,6 +168,31 @@
"A \<subseteq> B \<Longrightarrow> f abs_summable_on A \<longleftrightarrow> set_integrable (count_space B) A f"
by (subst abs_summable_on_restrict[of _ B]) (auto simp: abs_summable_on_def)
+lemma abs_summable_on_norm_iff [simp]:
+ "(\<lambda>x. norm (f x)) abs_summable_on A \<longleftrightarrow> f abs_summable_on A"
+ by (simp add: abs_summable_on_def integrable_norm_iff)
+
+lemma abs_summable_on_normI: "f abs_summable_on A \<Longrightarrow> (\<lambda>x. norm (f x)) abs_summable_on A"
+ by simp
+
+lemma abs_summable_on_comparison_test:
+ assumes "g abs_summable_on A"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> norm (g x)"
+ shows "f abs_summable_on A"
+ using assms Bochner_Integration.integrable_bound[of "count_space A" g f]
+ unfolding abs_summable_on_def by (auto simp: AE_count_space)
+
+lemma abs_summable_on_comparison_test':
+ assumes "g abs_summable_on A"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> g x"
+ shows "f abs_summable_on A"
+proof (rule abs_summable_on_comparison_test[OF assms(1), of f])
+ fix x assume "x \<in> A"
+ with assms(2) have "norm (f x) \<le> g x" .
+ also have "\<dots> \<le> norm (g x)" by simp
+ finally show "norm (f x) \<le> norm (g x)" .
+qed
+
lemma abs_summable_on_cong [cong]:
"(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> A = B \<Longrightarrow> (f abs_summable_on A) \<longleftrightarrow> (g abs_summable_on B)"
unfolding abs_summable_on_def by (intro integrable_cong) auto
@@ -210,6 +244,18 @@
shows "f abs_summable_on (A \<union> B)"
using assms unfolding abs_summable_on_altdef by (intro set_integrable_Un) auto
+lemma abs_summable_on_insert_iff [simp]:
+ "f abs_summable_on insert x A \<longleftrightarrow> f abs_summable_on A"
+proof safe
+ assume "f abs_summable_on insert x A"
+ thus "f abs_summable_on A"
+ by (rule abs_summable_on_subset) auto
+next
+ assume "f abs_summable_on A"
+ from abs_summable_on_union[OF this, of "{x}"]
+ show "f abs_summable_on insert x A" by simp
+qed
+
lemma abs_summable_on_reindex_bij_betw:
assumes "bij_betw g A B"
shows "(\<lambda>x. f (g x)) abs_summable_on A \<longleftrightarrow> f abs_summable_on B"
@@ -235,11 +281,11 @@
finally show ?thesis .
qed
-lemma abs_summable_reindex_iff:
+lemma abs_summable_on_reindex_iff:
"inj_on g A \<Longrightarrow> (\<lambda>x. f (g x)) abs_summable_on A \<longleftrightarrow> f abs_summable_on (g ` A)"
by (intro abs_summable_on_reindex_bij_betw inj_on_imp_bij_betw)
-lemma abs_summable_on_Sigma_project:
+lemma abs_summable_on_Sigma_project2:
fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
assumes "f abs_summable_on (Sigma A B)" "x \<in> A"
shows "(\<lambda>y. f (x, y)) abs_summable_on (B x)"
@@ -425,6 +471,46 @@
by (intro Bochner_Integration.integral_cong refl)
(auto simp: indicator_def split: if_splits)
+lemma infsetsum_mono_neutral:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "f abs_summable_on A" and "g abs_summable_on B"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
+ assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x \<le> 0"
+ assumes "\<And>x. x \<in> B - A \<Longrightarrow> g x \<ge> 0"
+ shows "infsetsum f A \<le> infsetsum g B"
+ using assms unfolding infsetsum_altdef abs_summable_on_altdef
+ by (intro Bochner_Integration.integral_mono) (auto simp: indicator_def)
+
+lemma infsetsum_mono_neutral_left:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "f abs_summable_on A" and "g abs_summable_on B"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
+ assumes "A \<subseteq> B"
+ assumes "\<And>x. x \<in> B - A \<Longrightarrow> g x \<ge> 0"
+ shows "infsetsum f A \<le> infsetsum g B"
+ using \<open>A \<subseteq> B\<close> by (intro infsetsum_mono_neutral assms) auto
+
+lemma infsetsum_mono_neutral_right:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "f abs_summable_on A" and "g abs_summable_on B"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
+ assumes "B \<subseteq> A"
+ assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x \<le> 0"
+ shows "infsetsum f A \<le> infsetsum g B"
+ using \<open>B \<subseteq> A\<close> by (intro infsetsum_mono_neutral assms) auto
+
+lemma infsetsum_mono:
+ fixes f g :: "'a \<Rightarrow> real"
+ assumes "f abs_summable_on A" and "g abs_summable_on A"
+ assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"
+ shows "infsetsum f A \<le> infsetsum g A"
+ by (intro infsetsum_mono_neutral assms) auto
+
+lemma norm_infsetsum_bound:
+ "norm (infsetsum f A) \<le> infsetsum (\<lambda>x. norm (f x)) A"
+ unfolding abs_summable_on_def infsetsum_def
+ by (rule Bochner_Integration.integral_norm_bound)
+
lemma infsetsum_Sigma:
fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
assumes [simp]: "countable A" and "\<And>i. countable (B i)"
@@ -460,6 +546,13 @@
finally show ?thesis ..
qed
+lemma infsetsum_Sigma':
+ fixes A :: "'a set" and B :: "'a \<Rightarrow> 'b set"
+ assumes [simp]: "countable A" and "\<And>i. countable (B i)"
+ assumes summable: "(\<lambda>(x,y). f x y) abs_summable_on (Sigma A B)"
+ shows "infsetsum (\<lambda>x. infsetsum (\<lambda>y. f x y) (B x)) A = infsetsum (\<lambda>(x,y). f x y) (Sigma A B)"
+ using assms by (subst infsetsum_Sigma) auto
+
lemma infsetsum_Times:
fixes A :: "'a set" and B :: "'b set"
assumes [simp]: "countable A" and "countable B"
@@ -496,6 +589,95 @@
finally show ?thesis .
qed
+lemma abs_summable_on_Sigma_iff:
+ assumes [simp]: "countable A" and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"
+ shows "f abs_summable_on Sigma A B \<longleftrightarrow>
+ (\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x) \<and>
+ ((\<lambda>x. infsetsum (\<lambda>y. norm (f (x, y))) (B x)) abs_summable_on A)"
+proof safe
+ define B' where "B' = (\<Union>x\<in>A. B x)"
+ have [simp]: "countable B'"
+ unfolding B'_def using assms by auto
+ interpret pair_sigma_finite "count_space A" "count_space B'"
+ by (intro pair_sigma_finite.intro sigma_finite_measure_count_space_countable) fact+
+
+ {
+ assume *: "f abs_summable_on Sigma A B"
+ thus "(\<lambda>y. f (x, y)) abs_summable_on B x" if "x \<in> A" for x
+ using that by (rule abs_summable_on_Sigma_project2)
+
+ have "set_integrable (count_space (A \<times> B')) (Sigma A B) (\<lambda>z. norm (f z))"
+ using abs_summable_on_normI[OF *]
+ by (subst abs_summable_on_altdef' [symmetric]) (auto simp: B'_def)
+ also have "count_space (A \<times> B') = count_space A \<Otimes>\<^sub>M count_space B'"
+ by (simp add: pair_measure_countable)
+ finally have "integrable (count_space A)
+ (\<lambda>x. lebesgue_integral (count_space B')
+ (\<lambda>y. indicator (Sigma A B) (x, y) *\<^sub>R norm (f (x, y))))"
+ by (rule integrable_fst')
+ also have "?this \<longleftrightarrow> integrable (count_space A)
+ (\<lambda>x. lebesgue_integral (count_space B')
+ (\<lambda>y. indicator (B x) y *\<^sub>R norm (f (x, y))))"
+ by (intro integrable_cong refl) (simp_all add: indicator_def)
+ also have "\<dots> \<longleftrightarrow> integrable (count_space A) (\<lambda>x. infsetsum (\<lambda>y. norm (f (x, y))) (B x))"
+ by (intro integrable_cong refl infsetsum_altdef' [symmetric]) (auto simp: B'_def)
+ also have "\<dots> \<longleftrightarrow> (\<lambda>x. infsetsum (\<lambda>y. norm (f (x, y))) (B x)) abs_summable_on A"
+ by (simp add: abs_summable_on_def)
+ finally show \<dots> .
+ }
+
+ {
+ assume *: "\<forall>x\<in>A. (\<lambda>y. f (x, y)) abs_summable_on B x"
+ assume "(\<lambda>x. \<Sum>\<^sub>ay\<in>B x. norm (f (x, y))) abs_summable_on A"
+ also have "?this \<longleftrightarrow> (\<lambda>x. \<integral>y\<in>B x. norm (f (x, y)) \<partial>count_space B') abs_summable_on A"
+ by (intro abs_summable_on_cong refl infsetsum_altdef') (auto simp: B'_def)
+ also have "\<dots> \<longleftrightarrow> (\<lambda>x. \<integral>y. indicator (Sigma A B) (x, y) *\<^sub>R norm (f (x, y)) \<partial>count_space B')
+ abs_summable_on A" (is "_ \<longleftrightarrow> ?h abs_summable_on _")
+ by (intro abs_summable_on_cong) (auto simp: indicator_def)
+ also have "\<dots> \<longleftrightarrow> integrable (count_space A) ?h"
+ by (simp add: abs_summable_on_def)
+ finally have **: \<dots> .
+
+ have "integrable (count_space A \<Otimes>\<^sub>M count_space B') (\<lambda>z. indicator (Sigma A B) z *\<^sub>R f z)"
+ proof (rule Fubini_integrable, goal_cases)
+ case 3
+ {
+ fix x assume x: "x \<in> A"
+ with * have "(\<lambda>y. f (x, y)) abs_summable_on B x"
+ by blast
+ also have "?this \<longleftrightarrow> integrable (count_space B')
+ (\<lambda>y. indicator (B x) y *\<^sub>R f (x, y))"
+ using x by (intro abs_summable_on_altdef') (auto simp: B'_def)
+ also have "(\<lambda>y. indicator (B x) y *\<^sub>R f (x, y)) =
+ (\<lambda>y. indicator (Sigma A B) (x, y) *\<^sub>R f (x, y))"
+ using x by (auto simp: indicator_def)
+ finally have "integrable (count_space B')
+ (\<lambda>y. indicator (Sigma A B) (x, y) *\<^sub>R f (x, y))" .
+ }
+ thus ?case by (auto simp: AE_count_space)
+ qed (insert **, auto simp: pair_measure_countable)
+ also have "count_space A \<Otimes>\<^sub>M count_space B' = count_space (A \<times> B')"
+ by (simp add: pair_measure_countable)
+ also have "set_integrable (count_space (A \<times> B')) (Sigma A B) f \<longleftrightarrow>
+ f abs_summable_on Sigma A B"
+ by (rule abs_summable_on_altdef' [symmetric]) (auto simp: B'_def)
+ finally show \<dots> .
+ }
+qed
+
+lemma abs_summable_on_Sigma_project1:
+ assumes "(\<lambda>(x,y). f x y) abs_summable_on Sigma A B"
+ assumes [simp]: "countable A" and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"
+ shows "(\<lambda>x. infsetsum (\<lambda>y. norm (f x y)) (B x)) abs_summable_on A"
+ using assms by (subst (asm) abs_summable_on_Sigma_iff) auto
+
+lemma abs_summable_on_Sigma_project1':
+ assumes "(\<lambda>(x,y). f x y) abs_summable_on Sigma A B"
+ assumes [simp]: "countable A" and "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"
+ shows "(\<lambda>x. infsetsum (\<lambda>y. f x y) (B x)) abs_summable_on A"
+ by (intro abs_summable_on_comparison_test' [OF abs_summable_on_Sigma_project1[OF assms]]
+ norm_infsetsum_bound)
+
lemma infsetsum_prod_PiE:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c :: {real_normed_field,banach,second_countable_topology}"
assumes finite: "finite A" and countable: "\<And>x. x \<in> A \<Longrightarrow> countable (B x)"
@@ -565,6 +747,29 @@
using assms unfolding infsetsum_def abs_summable_on_def
by (rule Bochner_Integration.integral_mult_right)
+lemma infsetsum_cdiv:
+ fixes f :: "'a \<Rightarrow> 'b :: {banach, real_normed_field, second_countable_topology}"
+ assumes "c \<noteq> 0 \<Longrightarrow> f abs_summable_on A"
+ shows "infsetsum (\<lambda>x. f x / c) A = infsetsum f A / c"
+ using assms unfolding infsetsum_def abs_summable_on_def by auto
+
+
(* TODO Generalise with bounded_linear *)
+lemma
+ fixes f :: "'a \<Rightarrow> 'c :: {banach, real_normed_field, second_countable_topology}"
+ assumes [simp]: "countable A" and [simp]: "countable B"
+ assumes "f abs_summable_on A" and "g abs_summable_on B"
+ shows abs_summable_on_product: "(\<lambda>(x,y). f x * g y) abs_summable_on A \<times> B"
+ and infsetsum_product: "infsetsum (\<lambda>(x,y). f x * g y) (A \<times> B) =
+ infsetsum f A * infsetsum g B"
+proof -
+ from assms show "(\<lambda>(x,y). f x * g y) abs_summable_on A \<times> B"
+ by (subst abs_summable_on_Sigma_iff)
+ (auto intro!: abs_summable_on_cmult_right simp: norm_mult infsetsum_cmult_right)
+ with assms show "infsetsum (\<lambda>(x,y). f x * g y) (A \<times> B) = infsetsum f A * infsetsum g B"
+ by (subst infsetsum_Sigma)
+ (auto simp: infsetsum_cmult_left infsetsum_cmult_right)
+qed
+
end