--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Wed Aug 23 00:38:53 2017 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Wed Aug 23 19:54:11 2017 +0100
@@ -245,26 +245,25 @@
lemma has_integral:
"(f has_integral y) (cbox a b) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d. gauge d \<and>
- (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
- norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+ (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
+ (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
+ norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)
lemma has_integral_real:
"(f has_integral y) {a..b::real} \<longleftrightarrow>
- (\<forall>e>0. \<exists>d. gauge d \<and>
- (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
- norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
- unfolding box_real[symmetric]
- by (rule has_integral)
+ (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
+ (\<forall>\<D>. \<D> tagged_division_of {a..b} \<and> \<gamma> fine \<D> \<longrightarrow>
+ norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
+ unfolding box_real[symmetric] by (rule has_integral)
lemma has_integralD[dest]:
assumes "(f has_integral y) (cbox a b)"
and "e > 0"
- obtains d
- where "gauge d"
- and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
- norm ((\<Sum>(x,k)\<in>p. content k *\<^sub>R f x) - y) < e"
+ obtains \<gamma>
+ where "gauge \<gamma>"
+ and "\<And>\<D>. \<D> tagged_division_of (cbox a b) \<Longrightarrow> \<gamma> fine \<D> \<Longrightarrow>
+ norm ((\<Sum>(x,k)\<in>\<D>. content k *\<^sub>R f x) - y) < e"
using assms unfolding has_integral by auto
lemma has_integral_alt:
@@ -906,28 +905,28 @@
subsection \<open>Cauchy-type criterion for integrability.\<close>
-lemma integrable_Cauchy:
+proposition integrable_Cauchy:
fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
shows "f integrable_on cbox a b \<longleftrightarrow>
(\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
- (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<and> \<gamma> fine p1 \<and>
- p2 tagged_division_of (cbox a b) \<and> \<gamma> fine p2 \<longrightarrow>
- norm ((\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x)) < e))"
+ (\<forall>\<D>1 \<D>2. \<D>1 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>1 \<and>
+ \<D>2 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>2 \<longrightarrow>
+ norm ((\<Sum>(x,K)\<in>\<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>\<D>2. content K *\<^sub>R f x)) < e))"
(is "?l = (\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>)")
proof (intro iffI allI impI)
assume ?l
then obtain y
where y: "\<And>e. e > 0 \<Longrightarrow>
\<exists>\<gamma>. gauge \<gamma> \<and>
- (\<forall>p. p tagged_division_of cbox a b \<and> \<gamma> fine p \<longrightarrow>
- norm ((\<Sum>(x,K) \<in> p. content K *\<^sub>R f x) - y) < e)"
+ (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
+ norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - y) < e)"
by (auto simp: integrable_on_def has_integral)
show "\<exists>\<gamma>. ?P e \<gamma>" if "e > 0" for e
proof -
have "e/2 > 0" using that by auto
with y obtain \<gamma> where "gauge \<gamma>"
- and \<gamma>: "\<And>p. p tagged_division_of cbox a b \<and> \<gamma> fine p \<Longrightarrow>
- norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f x) - y) < e/2"
+ and \<gamma>: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<Longrightarrow>
+ norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f x) - y) < e/2"
by meson
show ?thesis
apply (rule_tac x=\<gamma> in exI, clarsimp simp: \<open>gauge \<gamma>\<close>)
@@ -939,9 +938,9 @@
by auto
then obtain \<gamma> :: "nat \<Rightarrow> 'n \<Rightarrow> 'n set" where \<gamma>:
"\<And>m. gauge (\<gamma> m)"
- "\<And>m p1 p2. \<lbrakk>p1 tagged_division_of cbox a b;
- \<gamma> m fine p1; p2 tagged_division_of cbox a b; \<gamma> m fine p2\<rbrakk>
- \<Longrightarrow> norm ((\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x))
+ "\<And>m \<D>1 \<D>2. \<lbrakk>\<D>1 tagged_division_of cbox a b;
+ \<gamma> m fine \<D>1; \<D>2 tagged_division_of cbox a b; \<gamma> m fine \<D>2\<rbrakk>
+ \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> \<D>2. content K *\<^sub>R f x))
< 1 / (m + 1)"
by metis
have "\<And>n. gauge (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}})"
@@ -985,8 +984,8 @@
obtain N2::nat where N2: "\<And>n. n \<ge> N2 \<Longrightarrow> norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < e/2"
using y[OF e2] by metis
show "\<exists>\<gamma>. gauge \<gamma> \<and>
- (\<forall>p. p tagged_division_of (cbox a b) \<and> \<gamma> fine p \<longrightarrow>
- norm ((\<Sum>(x,K) \<in> p. content K *\<^sub>R f x) - y) < e)"
+ (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
+ norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - y) < e)"
proof (intro exI conjI allI impI)
show "gauge (\<gamma> (N1+N2))"
using \<gamma> by auto
@@ -1051,15 +1050,15 @@
by auto
obtain \<gamma>1 where \<gamma>1: "gauge \<gamma>1"
and \<gamma>1norm:
- "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; \<gamma>1 fine p\<rbrakk>
- \<Longrightarrow> norm ((\<Sum>(x,K) \<in> p. content K *\<^sub>R f x) - i) < e/2"
+ "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; \<gamma>1 fine \<D>\<rbrakk>
+ \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - i) < e/2"
apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
obtain \<gamma>2 where \<gamma>2: "gauge \<gamma>2"
and \<gamma>2norm:
- "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; \<gamma>2 fine p\<rbrakk>
- \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - j) < e/2"
+ "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; \<gamma>2 fine \<D>\<rbrakk>
+ \<Longrightarrow> norm ((\<Sum>(x, k) \<in> \<D>. content k *\<^sub>R f x) - j) < e/2"
apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
@@ -1067,8 +1066,8 @@
have "gauge ?\<gamma>"
using \<gamma>1 \<gamma>2 unfolding gauge_def by auto
then show "\<exists>\<gamma>. gauge \<gamma> \<and>
- (\<forall>p. p tagged_division_of cbox a b \<and> \<gamma> fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e)"
+ (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
+ norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - (i + j)) < e)"
proof (rule_tac x="?\<gamma>" in exI, safe)
fix p
assume p: "p tagged_division_of (cbox a b)" and "?\<gamma> fine p"
@@ -4741,9 +4740,9 @@
lemma has_integral_le:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
- assumes "(f has_integral i) s"
- and "(g has_integral j) s"
- and "\<forall>x\<in>s. f x \<le> g x"
+ assumes "(f has_integral i) S"
+ and "(g has_integral j) S"
+ and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
shows "i \<le> j"
using has_integral_component_le[OF _ assms(1-2), of 1]
using assms(3)
@@ -4751,27 +4750,27 @@
lemma integral_le:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
- assumes "f integrable_on s"
- and "g integrable_on s"
- and "\<forall>x\<in>s. f x \<le> g x"
- shows "integral s f \<le> integral s g"
+ assumes "f integrable_on S"
+ and "g integrable_on S"
+ and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
+ shows "integral S f \<le> integral S g"
by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])
lemma has_integral_nonneg:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
- assumes "(f has_integral i) s"
- and "\<forall>x\<in>s. 0 \<le> f x"
+ assumes "(f has_integral i) S"
+ and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
shows "0 \<le> i"
- using has_integral_component_nonneg[of 1 f i s]
+ using has_integral_component_nonneg[of 1 f i S]
unfolding o_def
using assms
by auto
lemma integral_nonneg:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
- assumes "f integrable_on s"
- and "\<forall>x\<in>s. 0 \<le> f x"
- shows "0 \<le> integral s f"
+ assumes "f integrable_on S"
+ and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
+ shows "0 \<le> integral S f"
by (rule has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)])
@@ -5897,8 +5896,8 @@
lemma henstock_lemma_part2:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes fed: "f integrable_on cbox a b" "e > 0" "gauge d"
- and less_e: "\<And>p. \<lbrakk>p tagged_division_of (cbox a b); d fine p\<rbrakk> \<Longrightarrow>
- norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral (cbox a b) f) < e"
+ and less_e: "\<And>\<D>. \<lbrakk>\<D> tagged_division_of (cbox a b); d fine \<D>\<rbrakk> \<Longrightarrow>
+ norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) \<D> - integral (cbox a b) f) < e"
and tag: "p tagged_partial_division_of (cbox a b)"
and "d fine p"
shows "sum (\<lambda>(x,k). norm (content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
@@ -5930,8 +5929,8 @@
have *: "e/(2 * (real DIM('n) + 1)) > 0" using \<open>e > 0\<close> by simp
with integrable_integral[OF intf, unfolded has_integral]
obtain \<gamma> where "gauge \<gamma>"
- and \<gamma>: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk> \<Longrightarrow>
- norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f x) - integral (cbox a b) f)
+ and \<gamma>: "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow>
+ norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f x) - integral (cbox a b) f)
< e/(2 * (real DIM('n) + 1))"
by metis
show thesis
@@ -5951,14 +5950,12 @@
subsection \<open>Monotone convergence (bounded interval first)\<close>
-(* TODO: is this lemma necessary? *)
lemma bounded_increasing_convergent:
fixes f :: "nat \<Rightarrow> real"
shows "\<lbrakk>bounded (range f); \<And>n. f n \<le> f (Suc n)\<rbrakk> \<Longrightarrow> \<exists>l. f \<longlonglongrightarrow> l"
using Bseq_mono_convergent[of f] incseq_Suc_iff[of f]
by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
-(*FIXME: why so much " \<bullet> 1"? *)
lemma monotone_convergence_interval:
fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
assumes intf: "\<And>k. (f k) integrable_on cbox a b"
@@ -5967,57 +5964,53 @@
and bou: "bounded (range (\<lambda>k. integral (cbox a b) (f k)))"
shows "g integrable_on cbox a b \<and> ((\<lambda>k. integral (cbox a b) (f k)) \<longlongrightarrow> integral (cbox a b) g) sequentially"
proof (cases "content (cbox a b) = 0")
- case True
- then show ?thesis
+ case True then show ?thesis
by auto
next
case False
- have fg1: "(f k x) \<bullet> 1 \<le> (g x) \<bullet> 1" if x: "x \<in> cbox a b" for x k
+ have fg1: "(f k x) \<le> (g x)" if x: "x \<in> cbox a b" for x k
proof -
- have "\<forall>\<^sub>F xa in sequentially. f k x \<bullet> 1 \<le> f xa x \<bullet> 1"
- unfolding eventually_sequentially
- apply (rule_tac x=k in exI)
+ have "\<forall>\<^sub>F j in sequentially. f k x \<le> f j x"
+ apply (rule eventually_sequentiallyI [of k])
using le x apply (force intro: transitive_stepwise_le)
done
- with Lim_component_ge [OF fg] x
- show "f k x \<bullet> 1 \<le> g x \<bullet> 1"
- using trivial_limit_sequentially by blast
+ then show "f k x \<le> g x"
+ using tendsto_lowerbound [OF fg] x trivial_limit_sequentially by blast
qed
have int_inc: "\<And>n. integral (cbox a b) (f n) \<le> integral (cbox a b) (f (Suc n))"
- by (metis integral_le assms(1-2))
+ by (metis integral_le intf le)
then obtain i where i: "(\<lambda>k. integral (cbox a b) (f k)) \<longlonglongrightarrow> i"
using bounded_increasing_convergent bou by blast
- have "\<And>k. \<forall>\<^sub>F x in sequentially. integral (cbox a b) (f k) \<le> integral (cbox a b) (f x) \<bullet> 1"
+ have "\<And>k. \<forall>\<^sub>F x in sequentially. integral (cbox a b) (f k) \<le> integral (cbox a b) (f x)"
unfolding eventually_sequentially
by (force intro: transitive_stepwise_le int_inc)
- then have i': "\<And>k. (integral(cbox a b) (f k)) \<le> i\<bullet>1"
- using Lim_component_ge [OF i] trivial_limit_sequentially by blast
+ then have i': "\<And>k. (integral(cbox a b) (f k)) \<le> i"
+ using tendsto_le [OF trivial_limit_sequentially i] by blast
have "(g has_integral i) (cbox a b)"
unfolding has_integral real_norm_def
proof clarify
fix e::real
assume e: "e > 0"
- have "\<And>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
- abs ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))"
+ have "\<And>k. (\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
+ abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))"
using intf e by (auto simp: has_integral_integral has_integral)
- then obtain c where c:
- "\<And>x. gauge (c x)"
- "\<And>x p. \<lbrakk>p tagged_division_of cbox a b; c x fine p\<rbrakk> \<Longrightarrow>
- abs ((\<Sum>(u, ka)\<in>p. content ka *\<^sub>R f x u) - integral (cbox a b) (f x))
+ then obtain c where c: "\<And>x. gauge (c x)"
+ "\<And>x \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; c x fine \<D>\<rbrakk> \<Longrightarrow>
+ abs ((\<Sum>(u,K)\<in>\<D>. content K *\<^sub>R f x u) - integral (cbox a b) (f x))
< e/2 ^ (x + 2)"
by metis
- have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i\<bullet>1 - (integral (cbox a b) (f k)) \<and> i\<bullet>1 - (integral (cbox a b) (f k)) < e/4"
+ have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i - (integral (cbox a b) (f k)) \<and> i - (integral (cbox a b) (f k)) < e/4"
proof -
have "e/4 > 0"
using e by auto
show ?thesis
using LIMSEQ_D [OF i \<open>e/4 > 0\<close>] i' by auto
qed
- then obtain r where r: "\<And>k. r \<le> k \<Longrightarrow> 0 \<le> i \<bullet> 1 - integral (cbox a b) (f k)"
- "\<And>k. r \<le> k \<Longrightarrow> i \<bullet> 1 - integral (cbox a b) (f k) < e/4"
+ then obtain r where r: "\<And>k. r \<le> k \<Longrightarrow> 0 \<le> i - integral (cbox a b) (f k)"
+ "\<And>k. r \<le> k \<Longrightarrow> i - integral (cbox a b) (f k) < e/4"
by metis
- have "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)\<bullet>1 - (f k x)\<bullet>1 \<and> (g x)\<bullet>1 - (f k x)\<bullet>1 < e/(4 * content(cbox a b))"
+ have "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x) - (f k x) \<and> (g x) - (f k x) < e/(4 * content(cbox a b))"
if "x \<in> cbox a b" for x
proof -
have "e/(4 * content (cbox a b)) > 0"
@@ -6027,8 +6020,8 @@
by metis
then show "\<exists>n\<ge>r.
\<forall>k\<ge>n.
- 0 \<le> g x \<bullet> 1 - f k x \<bullet> 1 \<and>
- g x \<bullet> 1 - f k x \<bullet> 1
+ 0 \<le> g x - f k x \<and>
+ g x - f k x
< e/(4 * content (cbox a b))"
apply (rule_tac x="N + r" in exI)
using fg1[OF that] apply (auto simp add: field_simps)
@@ -6036,30 +6029,29 @@
qed
then obtain m where r_le_m: "\<And>x. x \<in> cbox a b \<Longrightarrow> r \<le> m x"
and m: "\<And>x k. \<lbrakk>x \<in> cbox a b; m x \<le> k\<rbrakk>
- \<Longrightarrow> 0 \<le> g x \<bullet> 1 - f k x \<bullet> 1 \<and> g x \<bullet> 1 - f k x \<bullet> 1 < e/(4 * content (cbox a b))"
+ \<Longrightarrow> 0 \<le> g x - f k x \<and> g x - f k x < e/(4 * content (cbox a b))"
by metis
define d where "d x = c (m x) x" for x
show "\<exists>\<gamma>. gauge \<gamma> \<and>
- (\<forall>p. p tagged_division_of cbox a b \<and>
- \<gamma> fine p \<longrightarrow> abs ((\<Sum>(x,K)\<in>p. content K *\<^sub>R g x) - i) < e)"
+ (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and>
+ \<gamma> fine \<D> \<longrightarrow> abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - i) < e)"
proof (rule exI, safe)
show "gauge d"
using c(1) unfolding gauge_def d_def by auto
next
- fix p
- assume p: "p tagged_division_of (cbox a b)" "d fine p"
+ fix \<D>
+ assume p: "\<D> tagged_division_of (cbox a b)" "d fine \<D>"
note p'=tagged_division_ofD[OF p(1)]
- obtain s where s: "\<And>x. x \<in> p \<Longrightarrow> m (fst x) \<le> s"
+ obtain s where s: "\<And>x. x \<in> \<D> \<Longrightarrow> m (fst x) \<le> s"
by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
have *: "\<bar>a - d\<bar> < e" if "\<bar>a - b\<bar> \<le> e/4" "\<bar>b - c\<bar> < e/2" "\<bar>c - d\<bar> < e/4" for a b c d
using that norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
norm_triangle_lt[of "a - b + (b - c)" "c - d" e]
by (auto simp add: algebra_simps)
- show "abs ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e"
+ show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - i\<bar> < e"
proof (rule *)
- show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - (\<Sum>(x, K)\<in>p. content K *\<^sub>R f (m x) x)\<bar>
- \<le> e/4"
- apply (rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e/(4 * content (cbox a b)))"])
+ show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> \<le> e/4"
+ apply (rule order_trans[of _ "\<Sum>(x, k)\<in>\<D>. content k * (e/(4 * content (cbox a b)))"])
unfolding sum_subtractf[symmetric]
apply (rule order_trans)
apply (rule sum_abs)
@@ -6069,94 +6061,85 @@
unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
proof -
fix x K
- assume xk: "(x, K) \<in> p"
- then have x: "x \<in> cbox a b"
- using p'(2-3)[OF xk] by auto
- with p'(4)[OF xk] obtain u v where "K = cbox u v" by metis
+ assume xk: "(x, K) \<in> \<D>"
+ with p(1) have x: "x \<in> cbox a b"
+ by blast
then show "abs (content K *\<^sub>R (g x - f (m x) x)) \<le> content K * (e/(4 * content (cbox a b)))"
- unfolding abs_scaleR using m[OF x]
- by (metis real_inner_1_right real_scaleR_def abs_of_nonneg inner_real_def less_eq_real_def measure_nonneg mult_left_mono order_refl)
+ unfolding abs_scaleR using m[OF x] p'(4)[OF xk]
+ by (metis real_scaleR_def abs_of_nonneg less_eq_real_def measure_nonneg mult_left_mono order_refl)
qed (insert False, auto)
-
next
- have "norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f (m x) x) - (\<Sum>(x, k)\<in>p. integral k (f (m x))))
- \<le> norm
- (\<Sum>j = 0..s.
- \<Sum>(x, K)\<in>{xk \<in> p. m (fst xk) = j}.
- content K *\<^sub>R f (m x) x - integral K (f (m x)))"
+ have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x))))
+ \<le> norm (\<Sum>j = 0..s. \<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))"
apply (subst sum_group)
using s by (auto simp: sum_subtractf split_def p'(1))
also have "\<dots> < e/2"
proof -
- have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))
+ have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))
\<le> (\<Sum>i = 0..s. e/2 ^ (i + 2))"
proof (rule sum_norm_le)
fix t
assume "t \<in> {0..s}"
- have "norm (\<Sum>(x,k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x - integral k (f (m x))) =
- norm (\<Sum>(x,k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f t x - integral k (f t))"
+ have "norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f (m x) x - integral k (f (m x))) =
+ norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f t x - integral k (f t))"
by (force intro!: sum.cong arg_cong[where f=norm])
also have "... \<le> e/2 ^ (t + 2)"
proof (rule henstock_lemma_part1 [OF intf])
- show "{xk \<in> p. m (fst xk) = t} tagged_partial_division_of cbox a b"
- apply (rule tagged_partial_division_subset[of p])
+ show "{xk \<in> \<D>. m (fst xk) = t} tagged_partial_division_of cbox a b"
+ apply (rule tagged_partial_division_subset[of \<D>])
using p by (auto simp: tagged_division_of_def)
- show "c t fine {xk \<in> p. m (fst xk) = t}"
+ show "c t fine {xk \<in> \<D>. m (fst xk) = t}"
using p(2) by (auto simp: fine_def d_def)
qed (use c e in auto)
- finally show "norm (\<Sum>(x,K)\<in>{xk \<in> p. m (fst xk) = t}. content K *\<^sub>R f (m x) x -
+ finally show "norm (\<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = t}. content K *\<^sub>R f (m x) x -
integral K (f (m x))) \<le> e/2 ^ (t + 2)" .
qed
also have "... = (e/2/2) * (\<Sum>i = 0..s. (1/2) ^ i)"
by (simp add: sum_distrib_left field_simps)
also have "\<dots> < e/2"
by (simp add: sum_gp mult_strict_left_mono[OF _ e])
- finally show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p.
+ finally show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>.
m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e/2" .
qed
- finally show "\<bar>(\<Sum>(x,K)\<in>p. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>p. integral K (f (m x)))\<bar> < e/2"
+ finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x)))\<bar> < e/2"
by simp
next
- note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
- have *: "\<And>sr sx ss ks kr. \<lbrakk>kr = sr; ks = ss;
- ks \<le> i; sr \<le> sx; sx \<le> ss; 0 \<le> i\<bullet>1 - kr\<bullet>1; i\<bullet>1 - kr\<bullet>1 < e/4\<rbrakk> \<Longrightarrow> \<bar>sx - i\<bar> < e/4"
- by auto
- show "\<bar>(\<Sum>(x, k)\<in>p. integral k (f (m x))) - i\<bar> < e/4"
- unfolding real_norm_def
- apply (rule *)
- apply (rule comb[of r])
- apply (rule comb[of s])
- apply (rule i'[unfolded real_inner_1_right])
- apply (rule_tac[1-2] sum_mono)
- unfolding split_paired_all split_conv
- apply (rule_tac[1-2] integral_le[OF ])
- proof safe
- show "0 \<le> i\<bullet>1 - (integral (cbox a b) (f r))\<bullet>1" "i\<bullet>1 - (integral (cbox a b) (f r))\<bullet>1 < e/4"
- using r by auto
+ have comb: "integral (cbox a b) (f y) = (\<Sum>(x, k)\<in>\<D>. integral k (f y))" for y
+ using integral_combine_tagged_division_topdown[OF intf p(1)] by metis
+ have f_le: "\<And>y m n. \<lbrakk>y \<in> cbox a b; n\<ge>m\<rbrakk> \<Longrightarrow> f m y \<le> f n y"
+ using le by (auto intro: transitive_stepwise_le)
+ have "(\<Sum>(x, k)\<in>\<D>. integral k (f r)) \<le> (\<Sum>(x, K)\<in>\<D>. integral K (f (m x)))"
+ proof (rule sum_mono, simp add: split_paired_all)
fix x K
- assume xk: "(x, K) \<in> p"
- from p'(4)[OF this] guess u v by (elim exE) note uv=this
- show "f r integrable_on K"
- and "f s integrable_on K"
- and "f (m x) integrable_on K"
- and "f (m x) integrable_on K"
- unfolding uv
- apply (rule_tac[!] integrable_on_subcbox[OF assms(1)[rule_format]])
- using p'(3)[OF xk]
- unfolding uv
- apply auto
- done
- fix y
- assume "y \<in> K"
- then have "y \<in> cbox a b"
- using p'(3)[OF xk] by auto
- then have *: "\<And>m n. n\<ge>m \<Longrightarrow> f m y \<le> f n y"
- using assms(2) by (auto intro: transitive_stepwise_le)
- show "f r y \<le> f (m x) y" "f (m x) y \<le> f s y"
- using s[rule_format,OF xk] r_le_m[of x] p'(2-3)[OF xk]
- apply (auto intro!: *)
- done
+ assume xK: "(x, K) \<in> \<D>"
+ show "integral K (f r) \<le> integral K (f (m x))"
+ proof (rule integral_le)
+ show "f r integrable_on K"
+ by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
+ show "f (m x) integrable_on K"
+ by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
+ show "f r y \<le> f (m x) y" if "y \<in> K" for y
+ using that r_le_m[of x] p'(2-3)[OF xK] f_le by auto
+ qed
qed
+ moreover have "(\<Sum>(x, K)\<in>\<D>. integral K (f (m x))) \<le> (\<Sum>(x, k)\<in>\<D>. integral k (f s))"
+ proof (rule sum_mono, simp add: split_paired_all)
+ fix x K
+ assume xK: "(x, K) \<in> \<D>"
+ show "integral K (f (m x)) \<le> integral K (f s)"
+ proof (rule integral_le)
+ show "f (m x) integrable_on K"
+ by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
+ show "f s integrable_on K"
+ by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
+ show "f (m x) y \<le> f s y" if "y \<in> K" for y
+ using that s xK f_le p'(3) by fastforce
+ qed
+ qed
+ moreover have "0 \<le> i - integral (cbox a b) (f r)" "i - integral (cbox a b) (f r) < e / 4"
+ using r by auto
+ ultimately show "\<bar>(\<Sum>(x,K)\<in>\<D>. integral K (f (m x))) - i\<bar> < e/4"
+ using comb i'[of s] by auto
qed
qed
qed
@@ -6180,23 +6163,19 @@
and bou: "bounded (range(\<lambda>k. integral S (f k)))"
for f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real" and g S
proof -
- have fg: "(f k x)\<bullet>1 \<le> (g x)\<bullet>1" if "x \<in> S" for x k
- apply (rule Lim_component_ge [OF lim [OF that] trivial_limit_sequentially])
- unfolding eventually_sequentially
- apply (rule_tac x=k in exI)
- using le by (force intro: transitive_stepwise_le that)
-
+ have fg: "(f k x) \<le> (g x)" if "x \<in> S" for x k
+ apply (rule tendsto_lowerbound [OF lim [OF that]])
+ apply (rule eventually_sequentiallyI [of k])
+ using le by (force intro: transitive_stepwise_le that)+
obtain i where i: "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> i"
using bounded_increasing_convergent [OF bou] le int_f integral_le by blast
- have i': "(integral S (f k))\<bullet>1 \<le> i\<bullet>1" for k
+ have i': "(integral S (f k)) \<le> i" for k
proof -
have *: "\<And>k. \<forall>x\<in>S. \<forall>n\<ge>k. f k x \<le> f n x"
using le by (force intro: transitive_stepwise_le)
show ?thesis
- apply (rule Lim_component_ge [OF i trivial_limit_sequentially])
- unfolding eventually_sequentially
- apply (rule_tac x=k in exI)
- using * by (simp add: int_f integral_le)
+ using tendsto_lowerbound [OF i eventually_sequentiallyI trivial_limit_sequentially]
+ by (meson "*" int_f integral_le)
qed
let ?f = "(\<lambda>k x. if x \<in> S then f k x else 0)"
@@ -6256,7 +6235,7 @@
show "integral (cbox a b) (?f N) \<le> integral (cbox a b) (?f (M + N))"
proof (intro ballI integral_le[OF int int])
fix x assume "x \<in> cbox a b"
- have "(f m x)\<bullet>1 \<le> (f n x)\<bullet>1" if "x \<in> S" "n \<ge> m" for m n
+ have "(f m x) \<le> (f n x)" if "x \<in> S" "n \<ge> m" for m n
apply (rule transitive_stepwise_le [OF \<open>n \<ge> m\<close> order_refl])
using dual_order.trans apply blast
by (simp add: le \<open>x \<in> S\<close>)
@@ -6395,41 +6374,41 @@
by auto
with integrable_integral[OF f,unfolded has_integral[of f]]
obtain \<gamma> where \<gamma>: "gauge \<gamma>"
- "\<And>p. p tagged_division_of cbox a b \<and> \<gamma> fine p
- \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
+ "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D>
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
by meson
moreover
from integrable_integral[OF g,unfolded has_integral[of g]] e
obtain \<delta> where \<delta>: "gauge \<delta>"
- "\<And>p. p tagged_division_of cbox a b \<and> \<delta> fine p
- \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - integral (cbox a b) g) < e/2"
+ "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<delta> fine \<D>
+ \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g) < e/2"
by meson
ultimately have "gauge (\<lambda>x. \<gamma> x \<inter> \<delta> x)"
using gauge_Int by blast
- with fine_division_exists obtain p
- where p: "p tagged_division_of cbox a b" "(\<lambda>x. \<gamma> x \<inter> \<delta> x) fine p"
+ with fine_division_exists obtain \<D>
+ where p: "\<D> tagged_division_of cbox a b" "(\<lambda>x. \<gamma> x \<inter> \<delta> x) fine \<D>"
by metis
- have "\<gamma> fine p" "\<delta> fine p"
+ have "\<gamma> fine \<D>" "\<delta> fine \<D>"
using fine_Int p(2) by blast+
show "norm (integral (cbox a b) f) < integral (cbox a b) g + e"
proof (rule norm)
- have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if "(x, K) \<in> p" for x K
+ have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if "(x, K) \<in> \<D>" for x K
proof-
have K: "x \<in> K" "K \<subseteq> cbox a b"
- using \<open>(x, K) \<in> p\<close> p(1) by blast+
+ using \<open>(x, K) \<in> \<D>\<close> p(1) by blast+
obtain u v where "K = cbox u v"
- using \<open>(x, K) \<in> p\<close> p(1) by blast
+ using \<open>(x, K) \<in> \<D>\<close> p(1) by blast
moreover have "content K * norm (f x) \<le> content K * g x"
by (metis K subsetD dual_order.antisym measure_nonneg mult_zero_left nle not_le real_mult_le_cancel_iff2)
then show ?thesis
by simp
qed
- then show "norm (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) \<le> (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
+ then show "norm (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) \<le> (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x)"
by (simp add: sum_norm_le split_def)
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
- using \<open>\<gamma> fine p\<close> \<gamma> p(1) by simp
- show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - integral (cbox a b) g\<bar> < e/2"
- using \<open>\<delta> fine p\<close> \<delta> p(1) by simp
+ show "norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
+ using \<open>\<gamma> fine \<D>\<close> \<gamma> p(1) by simp
+ show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g\<bar> < e/2"
+ using \<open>\<delta> fine \<D>\<close> \<delta> p(1) by simp
qed
qed
show ?thesis
@@ -7144,13 +7123,13 @@
norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) \<le> e"
using opd1 [OF p] False by (simp add: comm_monoid_set_F_and)
} note op_acbd = this
- { fix k::real and p and u::'a and v w and z::'b and t1 t2 l
+ { fix k::real and \<D> and u::'a and v w and z::'b and t1 t2 l
assume k: "0 < k"
and nf: "\<And>x y u v.
\<lbrakk>x \<in> cbox a b; y \<in> cbox c d; u \<in> cbox a b; v\<in>cbox c d; norm (u-x, v-y) < k\<rbrakk>
\<Longrightarrow> norm (f(u,v) - f(x,y)) < e/(2 * (content ?CBOX))"
- and p_acbd: "p tagged_division_of cbox (a,c) (b,d)"
- and fine: "(\<lambda>x. ball x k) fine p" "((t1,t2), l) \<in> p"
+ and p_acbd: "\<D> tagged_division_of cbox (a,c) (b,d)"
+ and fine: "(\<lambda>x. ball x k) fine \<D>" "((t1,t2), l) \<in> \<D>"
and uwvz_sub: "cbox (u,w) (v,z) \<subseteq> l" "cbox (u,w) (v,z) \<subseteq> cbox (a,c) (b,d)"
have t: "t1 \<in> cbox a b" "t2 \<in> cbox c d"
by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+