--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Sat Aug 05 16:18:35 2017 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Sat Aug 05 18:16:35 2017 +0200
@@ -1,30 +1,13 @@
(* Title: HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
+ Huge cleanup by LCP
*)
theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
begin
-lemma finite_product_dependent: (*FIXME DELETE*)
- assumes "finite s"
- and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
- shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
- using assms
-proof induct
- case (insert x s)
- have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
- (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
- show ?case
- unfolding *
- apply (rule finite_UnI)
- using insert
- apply auto
- done
-qed auto
-
-
lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
by (auto intro: order_trans)
@@ -90,7 +73,7 @@
obtain d
where "gauge d"
and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R f x) - I) < e"
using \<open>0<e\<close> f unfolding has_integral by auto
define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
@@ -194,14 +177,14 @@
then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
using p(1) by auto
qed
- ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e"
+ ultimately have I: "norm ((\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) - I) < e"
using integral_f by auto
- have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
+ have "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)"
using p(1)[THEN tagged_division_ofD(1)]
by (safe intro!: sum.union_inter_neutral) (auto simp: s_def T_def)
- also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
+ also have "(\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k))"
proof (subst sum.reindex_nontrivial, safe)
fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
@@ -209,8 +192,8 @@
show "x1 = x2"
by (auto simp: content_eq_0_interior)
qed (use p in \<open>auto intro!: sum.cong\<close>)
- finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
+ finally have eq: "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)" .
have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
using in_s[of x k] by (auto simp: C_def)
@@ -224,7 +207,7 @@
have [simp]: "finite p"
using tagged_division_ofD(1)[OF p(1)] .
- have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)"
+ have "(M - 3*e) * (b - a) \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k) * (b - a)"
proof (intro mult_right_mono)
have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
@@ -287,15 +270,15 @@
finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
using \<open>0 < e\<close> by (simp add: split_beta)
qed (use \<open>a < b\<close> in auto)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * (b - a))"
by (simp add: sum_distrib_right split_beta')
- also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
+ also have "\<dots> \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
using parts(3) by (auto intro!: sum_mono mult_left_mono diff_mono)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?E k))"
by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
- also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
+ also have "\<dots> = (\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x)"
by (subst (1 2) parts) auto
- also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
+ also have "\<dots> \<le> norm ((\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x))"
by auto
also have "\<dots> \<le> e + e"
using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
@@ -1633,7 +1616,7 @@
have *: "\<exists>d. gauge d \<and>
(\<forall>p. p tagged_division_of cbox a b \<and>
d fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e)"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e)"
if e: "e > 0" for e
proof -
have "?S - e/2 < ?S" using \<open>e > 0\<close> by simp
@@ -1664,7 +1647,7 @@
with henstock_lemma[OF f]
obtain g where g: "gauge g"
"\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; g fine p\<rbrakk>
- \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
+ \<Longrightarrow> (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
by (metis (no_types, lifting))
let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
show ?thesis
@@ -1689,6 +1672,7 @@
show "finite ((\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p))"
by (simp add: d'(1) p'(1))
qed
+ next
fix x k
assume "(x, k) \<in> p'"
then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
@@ -1752,7 +1736,7 @@
qed
qed
- then have sum_less_e2: "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e/2"
+ then have sum_less_e2: "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x - integral k f)) < e/2"
using g(2) gp' tagged_division_of_def by blast
have p'alt: "p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}}"
@@ -1780,26 +1764,27 @@
then show "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
using p'(2) by fastforce
qed
- have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
+ have sum_p': "(\<Sum>(x,k) \<in> p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
apply (auto intro: integral_null simp: content_eq_0_interior)
done
note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
+ have fin_d_sndp: "finite (d \<times> snd ` p)"
+ by (simp add: d'(1) snd_p(1))
have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e/2; ?S - e/2 < sni; sni' \<le> ?S;
sni \<le> sni'; sf' = sf\<rbrakk> \<Longrightarrow> \<bar>sf - ?S\<bar> < e"
by arith
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e"
+ show "norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e"
unfolding real_norm_def
proof (rule *)
show "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e/2"
using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
- show "(\<Sum>(x, k)\<in>p'. norm (integral k f)) \<le>?S"
+ show "(\<Sum>(x,k) \<in> p'. norm (integral k f)) \<le>?S"
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
- show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x, k)\<in>p'. norm (integral k f))"
+ show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
- have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
- (\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}"
+ have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = (\<lambda>(k,l). k \<inter> l) ` (d \<times> snd ` p)"
by auto
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
proof (rule sum_mono, goal_cases)
@@ -1858,10 +1843,10 @@
qed
finally show ?case .
qed
- also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> (snd ` p). norm (integral (i\<inter>l) f))"
+ also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
by (simp add: sum.cartesian_product)
- also have "\<dots> = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (case_prod op \<inter> x) f))"
- by (force simp: split_def Sigma_def intro!: sum.cong)
+ also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod op \<inter> x) f))"
+ by (force simp: split_def intro!: sum.cong)
also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
proof -
have eq0: " (integral (l1 \<inter> k1) f) = 0"
@@ -1884,16 +1869,14 @@
qed
show ?thesis
unfolding *
- apply (rule sum.reindex_nontrivial [symmetric, unfolded o_def])
- apply (rule finite_product_dependent [OF \<open>finite d\<close>])
- apply (rule finite_imageI [OF \<open>finite p\<close>])
+ apply (rule sum.reindex_nontrivial [OF fin_d_sndp, symmetric, unfolded o_def])
apply clarsimp
by (metis eq0 fst_conv snd_conv)
qed
- also have "\<dots> = (\<Sum>(x, k)\<in>p'. norm (integral k f))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
have 0: "integral (ia \<inter> snd (a, b)) f = 0"
- if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
+ if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
proof -
have "ia \<inter> b = {}"
using that
@@ -1914,33 +1897,28 @@
show ?thesis
unfolding sum_p'
apply (rule sum.mono_neutral_right)
- apply (subst *)
- apply (rule finite_imageI[OF finite_product_dependent])
- apply fact
- apply (rule finite_imageI[OF p'(1)])
+ apply (metis * finite_imageI[OF fin_d_sndp])
using 0 1 by auto
qed
finally show ?thesis .
qed
- show "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
+ show "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
by force
- have pdfin: "finite (p \<times> d)"
+ have fin_pd: "finite (p \<times> d)"
using finite_cartesian_product[OF p'(1) d'(1)] by metis
- have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
+ have "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> ?S. \<bar>content k\<bar> * norm (f x))"
unfolding norm_scaleR
apply (rule sum.mono_neutral_left)
apply (subst *)
- apply (rule finite_imageI)
- apply fact
- unfolding p'alt apply blast
+ apply (rule finite_imageI [OF fin_pd])
+ unfolding p'alt apply auto
by fastforce
also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
unfolding *
- apply (subst sum.reindex_nontrivial)
- apply fact
+ apply (subst sum.reindex_nontrivial [OF fin_pd])
unfolding split_paired_all
unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff prod.inject
apply (elim conjE)
@@ -1952,16 +1930,8 @@
from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
by auto
then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
- apply -
- apply (erule disjE)
- apply (rule disjI2)
- defer
- apply (rule disjI1)
- apply (rule d'(5)[OF as(3-4)])
- apply assumption
- apply (rule p'(5)[OF as(1-2)])
- apply auto
- done
+ apply (rule disjE)
+ using as(1) as(2) p'(5) as(3) as(4) d'(5) by auto
moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
unfolding as ..
ultimately have "interior (l1 \<inter> k1) = {}"
@@ -1971,55 +1941,53 @@
unfolding content_eq_0_interior[symmetric]
by auto
qed safe
- also have "\<dots> = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
- apply (subst sum_Sigma_product[symmetric])
- apply (rule p')
- apply (rule d')
- apply (rule sum.cong)
- apply (rule refl)
- unfolding split_paired_all split_conv
+ also have "\<dots> = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
- fix x l
- assume as: "(x, l) \<in> p"
- note xl = p'(2-4)[OF this]
- from this(3) guess u v by (elim exE) note uv=this
- have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
- by (simp add: Int_commute uv)
- also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
+ have sumeq: "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ if "(x, l) \<in> p" for x l
proof -
- have eq0: "content (k \<inter> cbox u v) = 0"
- if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
+ note xl = p'(2-4)[OF that]
+ from this(3) guess u v by (elim exE) note uv=this
+ have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
+ by (simp add: Int_commute uv)
+ also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
proof -
- from d'(4)[OF that(1)] d'(4)[OF that(2)]
- obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
- by (meson Int_interval)
- have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
- by (simp add: d'(5) that)
- also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
- by auto
- also have "\<dots> = interior (k \<inter> cbox u v)"
- unfolding eq by auto
- finally show ?thesis
- unfolding \<alpha> content_eq_0_interior ..
+ have eq0: "content (k \<inter> cbox u v) = 0"
+ if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
+ proof -
+ from d'(4)[OF that(1)] d'(4)[OF that(2)]
+ obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
+ by (meson Int_interval)
+ have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
+ by (simp add: d'(5) that)
+ also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
+ by auto
+ also have "\<dots> = interior (k \<inter> cbox u v)"
+ unfolding eq by auto
+ finally show ?thesis
+ unfolding \<alpha> content_eq_0_interior ..
+ qed
+ then show ?thesis
+ unfolding Setcompr_eq_image
+ apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
+ by auto
+ qed
+ also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
+ apply (rule sum.mono_neutral_right)
+ unfolding Setcompr_eq_image
+ apply (rule finite_imageI [OF \<open>finite d\<close>])
+ apply (fastforce simp: inf.commute)+
+ done
+ finally show "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ unfolding sum_distrib_right[symmetric] real_scaleR_def
+ apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
+ using xl(2)[unfolded uv] unfolding uv apply auto
+ done
qed
- then show ?thesis
- unfolding Setcompr_eq_image
- apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
- by auto
+ show ?thesis
+ by (subst sum_Sigma_product[symmetric]) (auto intro!: sumeq sum.cong p' d')
qed
- also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
- apply (rule sum.mono_neutral_right)
- unfolding Setcompr_eq_image
- apply (rule finite_imageI [OF \<open>finite d\<close>])
- apply (fastforce simp: inf.commute)+
- done
- finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
- unfolding sum_distrib_right[symmetric] real_scaleR_def
- apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
- using xl(2)[unfolded uv] unfolding uv apply auto
- done
- qed
- finally show ?thesis .
+ finally show ?thesis .
qed
qed (rule d)
qed
@@ -2135,11 +2103,11 @@
using \<open>e > 0\<close> by auto
from * [OF this] obtain d1 where
d1: "gauge d1" "\<forall>p. p tagged_division_of (cbox a b) \<and> d1 fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e/2"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e/2"
by auto
from henstock_lemma [OF f(1) \<open>e/2>0\<close>] obtain d2 where
d2: "gauge d2" "\<forall>p. p tagged_partial_division_of (cbox a b) \<and> d2 fine p \<longrightarrow>
- (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e/2" .
+ (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2" .
obtain p where
p: "p tagged_division_of (cbox a b)" "d1 fine p" "d2 fine p"
by (rule fine_division_exists [OF gauge_Int [OF d1(1) d2(1)], of a b])
@@ -2159,10 +2127,10 @@
unfolding tagged_division_of_def split_def
apply auto
done
- show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e/2"
+ show "\<bar>(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e/2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]]
by (simp only: real_norm_def)
- show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
+ show "(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) = (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x))"
apply (rule sum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
@@ -2171,7 +2139,7 @@
apply (subst abs_of_nonneg)
apply auto
done
- show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
+ show "(\<Sum>(x,k) \<in> p. norm (integral k f)) \<le> ?S"
using partial_division_of_tagged_division[of p "cbox a b"] p(1)
apply (subst sum.over_tagged_division_lemma[OF p(1)])
apply (simp add: content_eq_0_interior)