author paulson Wed, 21 Jun 2017 17:13:55 +0100 changeset 66154 bc5e6461f759 parent 66153 236339f97a88 child 66155 2463cba9f18f
Tidying up integration theory and some new theorems
```--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -122,7 +122,7 @@
by (auto intro: less_le_trans)
define d' where "d' x = d x \<inter> ball x (1 / (3 * Suc m))" for x
have "gauge d'"
-      unfolding d'_def by (intro gauge_inter \<open>gauge d\<close> gauge_ball) auto
+      unfolding d'_def by (intro gauge_Int \<open>gauge d\<close> gauge_ball) auto
then obtain p where p: "p tagged_division_of cbox x y" "d' fine p"
by (rule fine_division_exists)
then have "d fine p"
@@ -2184,7 +2184,7 @@
(\<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_inter [OF d1(1) d2(1)], of a b])
+          by (rule fine_division_exists [OF gauge_Int [OF d1(1) d2(1)], of a b])
have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e / 2 \<longrightarrow>
\<bar>sf' - di\<bar> < e / 2 \<longrightarrow> di < ?S + e"```
```--- a/src/HOL/Analysis/Euclidean_Space.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Analysis/Euclidean_Space.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -129,6 +129,25 @@

+lemma sum_if_inner [simp]:
+  assumes "i \<in> Basis" "j \<in> Basis"
+    shows "inner (\<Sum>k\<in>Basis. if k = i then f i *\<^sub>R i else g k *\<^sub>R k) j = (if j=i then f j else g j)"
+proof (cases "i=j")
+  case True
+  with assms show ?thesis
+    by (auto simp: inner_sum_left if_distrib [of "\<lambda>x. inner x j"] inner_Basis cong: if_cong)
+next
+  case False
+  have "(\<Sum>k\<in>Basis. inner (if k = i then f i *\<^sub>R i else g k *\<^sub>R k) j) =
+        (\<Sum>k\<in>Basis. if k = j then g k else 0)"
+    apply (rule sum.cong)
+    using False assms by (auto simp: inner_Basis)
+  also have "... = g j"
+    using assms by auto
+  finally show ?thesis
+    using False by (auto simp: inner_sum_left)
+qed
+
subsection \<open>Subclass relationships\<close>

instance euclidean_space \<subseteq> perfect_space```
```--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -182,6 +182,21 @@
"d tagged_division_of (cbox a b) \<Longrightarrow> sum (\<lambda>(x,l). content l) d = content (cbox a b)"
unfolding sum.operative_tagged_division[OF operative_content, symmetric] by blast

+  assumes "\<D> division_of S" "S \<subseteq> cbox a b"
+  shows "sum content \<D> \<le> content(cbox a b)"
+proof -
+  have "\<D> division_of \<Union>\<D>" "\<Union>\<D> \<subseteq> cbox a b"
+    using assms by auto
+  then obtain \<D>' where "\<D> \<subseteq> \<D>'" "\<D>' division_of cbox a b"
+    using partial_division_extend_interval by metis
+  then have "sum content \<D> \<le> sum content \<D>'"
+    using sum_mono2 by blast
+  also have "... \<le> content(cbox a b)"
+  finally show ?thesis .
+qed
+
lemma content_real_eq_0: "content {a .. b::real} = 0 \<longleftrightarrow> a \<ge> b"
by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)

@@ -451,8 +466,8 @@
using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)

lemma integrable_on_scaleR_left:
-  assumes "f integrable_on A"
-  shows "(\<lambda>x. f x *\<^sub>R y) integrable_on A"
+  assumes "f integrable_on A"
+  shows "(\<lambda>x. f x *\<^sub>R y) integrable_on A"
using assms has_integral_scaleR_left unfolding integrable_on_def by blast

lemma has_integral_mult_left:
@@ -1294,59 +1309,81 @@
by (auto intro: that[of d] d elim: )
qed

-lemma integrable_split[intro]:
+lemma integrable_split [intro]:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
-  assumes "f integrable_on cbox a b"
-    and k: "k \<in> Basis"
-  shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})" (is ?t1)
-    and "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
+  assumes f: "f integrable_on cbox a b"
+      and k: "k \<in> Basis"
+    shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})" (is ?thesis1)
+    and   "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"   (is ?thesis2)
proof -
-  guess y using assms(1) unfolding integrable_on_def .. note y=this
-  define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
+  obtain y where y: "(f has_integral y) (cbox a b)"
+    using f by blast
define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
-  show ?t1 ?t2
-    unfolding interval_split[OF k] integrable_cauchy
-    unfolding interval_split[symmetric,OF k]
-  proof (rule_tac[!] allI impI)+
-    fix e :: real
-    assume "e > 0"
-    then have "e/2>0"
-      by auto
-    from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
-    let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<inter> A \<and> d fine p1 \<and>
-      p2 tagged_division_of (cbox a b) \<inter> A \<and> d 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)"
-    show "?P {x. x \<bullet> k \<le> c}"
-    proof (rule_tac x=d in exI, clarsimp simp add: d)
+  define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
+  have "\<exists>d. gauge d \<and>
+            (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1 \<and>
+                     p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d 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)"
+    if "e > 0" for e
+  proof -
+    have "e/2 > 0" using that by auto
+  with has_integral_separate_sides[OF y this k, of c]
+  obtain d
+    where "gauge d"
+         and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
+                          p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d 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) - y) < e/2"
+    by metis
+  show ?thesis
+    proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
"p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
-      proof (rule fine_division_exists[OF d(1), of a' b] )
+      proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a' b])
fix p
assume "p tagged_division_of cbox a' b" "d fine p"
then show ?thesis
-          using as norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
+          using as norm_triangle_half_l[OF d[of p1 p] d[of p2 p]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
qed
qed
-    show "?P {x. x \<bullet> k \<ge> c}"
-    proof (rule_tac x=d in exI, clarsimp simp add: d)
+  qed
+  with f show ?thesis1
+    by (simp add: interval_split[OF k] integrable_cauchy)
+  have "\<exists>d. gauge d \<and>
+            (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1 \<and>
+                     p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d 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)"
+    if "e > 0" for e
+  proof -
+    have "e/2 > 0" using that by auto
+  with has_integral_separate_sides[OF y this k, of c]
+  obtain d
+    where "gauge d"
+         and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
+                          p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d 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) - y) < e/2"
+    by metis
+  show ?thesis
+    proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
"p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
-      proof (rule fine_division_exists[OF d(1), of a b'] )
+      proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a b'])
fix p
assume "p tagged_division_of cbox a b'" "d fine p"
then show ?thesis
-          using as norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
+          using as norm_triangle_half_l[OF d[of p p1] d[of p p2]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
qed
qed
-  qed
+  qed
+  with f show ?thesis2
+    by (simp add: interval_split[OF k] integrable_cauchy)
qed

lemma operative_integral:
@@ -1544,7 +1581,7 @@
guess d1 using f_i[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
guess d2 using g_j[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
obtain p where p: "p tagged_division_of cbox a b" "d1 fine p" "d2 fine p"
-       using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter
+       using fine_division_exists[OF gauge_Int[OF d1(1) d2(1)], of a b] unfolding fine_inter
by metis
note le_less_trans[OF Basis_le_norm[OF k]]
then have "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
@@ -1733,7 +1770,7 @@
note * = i[unfolded has_integral,rule_format,OF this]
from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
-      from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b]
+      from fine_division_exists[OF gauge_Int[OF gm(1) gn(1)], of a b]
obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine p"
by auto
{ fix s1 s2 i1 and i2::'b
@@ -5454,67 +5491,74 @@

fixes f :: "'n::euclidean_space \<Rightarrow> real"
-  assumes "\<forall>e>0. \<exists>g  h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and>
-    norm (i - j) < e \<and> (\<forall>x\<in>cbox a b. (g x) \<le> f x \<and> f x \<le> h x)"
+  assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and>
+                            \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>cbox a b. (g x) \<le> f x \<and> f x \<le> h x)"
shows "f integrable_on cbox a b"
-proof (subst integrable_cauchy, safe, goal_cases)
-  case (1 e)
-  then have e: "e/3 > 0"
-    by auto
-  note assms[rule_format,OF this]
-  then guess g h i j by (elim exE conjE) note obt = this
-  from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
-  from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
-  show ?case
-    apply (rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI)
-    apply (rule conjI gauge_inter d1 d2)+
-    unfolding fine_inter
-  proof (safe, goal_cases)
-    have **: "\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
-      \<bar>i - j\<bar> < e / 3 \<Longrightarrow> \<bar>g2 - i\<bar> < e / 3 \<Longrightarrow> \<bar>g1 - i\<bar> < e / 3 \<Longrightarrow>
-      \<bar>h2 - j\<bar> < e / 3 \<Longrightarrow> \<bar>h1 - j\<bar> < e / 3 \<Longrightarrow> \<bar>f1 - f2\<bar> < e"
-    using \<open>e > 0\<close> by arith
-    case prems: (1 p1 p2)
-    note tagged_division_ofD(2-4) note * = this[OF prems(1)] this[OF prems(4)]
-
-    have "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
-      and "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
-      and "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
-      and "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
-      unfolding sum_subtractf[symmetric]
-      apply -
-      apply (rule_tac[!] sum_nonneg)
-      apply safe
-      unfolding real_scaleR_def right_diff_distrib[symmetric]
-         apply (rule_tac[!] mult_nonneg_nonneg)
-        apply simp_all
-      using obt(4) prems(1) prems(4) apply (blast intro:  elim: dest!: bspec)+
+proof -
+  have "\<exists>d. gauge d \<and>
+            (\<forall>p1 p2. p1 tagged_division_of cbox a b \<and> d fine p1 \<and>
+                     p2 tagged_division_of cbox a b \<and> d fine p2 \<longrightarrow>
+                     \<bar>(\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x)\<bar> < e)"
+    if "e > 0" for e
+  proof -
+    have e: "e/3 > 0"
+      using that by auto
+    then obtain g h i j where ij: "\<bar>i - j\<bar> < e/3"
+            and "(g has_integral i) (cbox a b)"
+            and "(h has_integral j) (cbox a b)"
+            and fgh: "\<And>x. x \<in> cbox a b \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
+      using assms real_norm_def by metis
+    then obtain d1 d2 where "gauge d1" "gauge d2"
+            and d1: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d1 fine p\<rbrakk> \<Longrightarrow>
+                          \<bar>(\<Sum>(x,K)\<in>p. content K *\<^sub>R g x) - i\<bar> < e/3"
+            and d2: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d2 fine p\<rbrakk> \<Longrightarrow>
+                          \<bar>(\<Sum>(x,K) \<in> p. content K *\<^sub>R h x) - j\<bar> < e/3"
+      by (metis e has_integral real_norm_def)
+    have "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)\<bar> < e"
+      if p1: "p1 tagged_division_of cbox a b" and 11: "d1 fine p1" and 21: "d2 fine p1"
+       and p2: "p2 tagged_division_of cbox a b" and 12: "d1 fine p2" and 22: "d2 fine p2" for p1 p2
+    proof -
+      have *: "\<And>g1 g2 h1 h2 f1 f2.
+                  \<lbrakk>\<bar>g2 - i\<bar> < e/3; \<bar>g1 - i\<bar> < e/3; \<bar>h2 - j\<bar> < e/3; \<bar>h1 - j\<bar> < e/3;
+                   g1 - h2 \<le> f1 - f2; f1 - f2 \<le> h1 - g2\<rbrakk>
+                  \<Longrightarrow> \<bar>f1 - f2\<bar> < e"
+        using \<open>e > 0\<close> ij by arith
+      have 0: "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
+              "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
+              "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
+              "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
+        unfolding sum_subtractf[symmetric]
+           apply (auto intro!: sum_nonneg)
+           apply (meson fgh measure_nonneg mult_left_mono tag_in_interval that sum_nonneg)+
done
-    then show ?case
-      apply -
-      unfolding real_norm_def
-      apply (rule **)
-      defer
-      defer
-      unfolding real_norm_def[symmetric]
-      apply (rule obt(3))
-      apply (rule d1(2)[OF conjI[OF prems(4,5)]])
-      apply (rule d1(2)[OF conjI[OF prems(1,2)]])
-      apply (rule d2(2)[OF conjI[OF prems(4,6)]])
-      apply (rule d2(2)[OF conjI[OF prems(1,3)]])
-      apply auto
-      done
+      show ?thesis
+      proof (rule *)
+        show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R g x) - i\<bar> < e/3"
+          by (rule d1[OF p2 12])
+        show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R g x) - i\<bar> < e/3"
+          by (rule d1[OF p1 11])
+        show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R h x) - j\<bar> < e/3"
+          by (rule d2[OF p2 22])
+        show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R h x) - j\<bar> < e/3"
+          by (rule d2[OF p1 21])
+      qed (use 0 in auto)
+    qed
+    then show ?thesis
+      by (rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI)
+        (auto simp: fine_inter intro: \<open>gauge d1\<close> \<open>gauge d2\<close> d1 d2)
qed
+  then show ?thesis
qed

fixes f :: "'n::euclidean_space \<Rightarrow> real"
-  assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
-    norm (i - j) < e \<and> (\<forall>x\<in>s. g x \<le> f x \<and> f x \<le> h x)"
+  assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
+                     \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>s. g x \<le> f x \<and> f x \<le> h x)"
shows "f integrable_on s"
proof -
have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
-  proof (rule integrable_straddle_interval, safe, goal_cases)
case (1 a b e)
then have *: "e/4 > 0"
by auto
@@ -5543,7 +5587,7 @@
unfolding c_def d_def by auto
qed
have **: "\<And>ch cg ag ah::real. norm (ah - ag) \<le> norm (ch - cg) \<Longrightarrow> norm (cg - i) < e / 4 \<Longrightarrow>
-      norm (ch - j) < e / 4 \<Longrightarrow> norm (ag - ah) < e"
+      norm (ch - j) < e / 4 \<Longrightarrow> abs (ag - ah) < e"
using obt(3)
unfolding real_norm_def
by arith
@@ -5554,7 +5598,7 @@
apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then h x else 0)" in exI)
apply safe
apply (rule_tac[1-2] integrable_integral,rule g)
-      apply (rule h)
+         apply (rule h)
apply (rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
proof -
have *: "\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
@@ -5642,7 +5686,7 @@
apply (rule order_trans)
apply (rule **)
apply (rule as(2))
-        apply (rule obt)
+        using obt(3) apply auto[1]
apply (rule_tac[!] integral_le)
using obt
apply (auto intro!: h g interv)
@@ -5946,7 +5990,7 @@
done
note integrable_integral[OF this, unfolded has_integral[of f]]
from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
-    note gauge_inter[OF \<open>gauge d\<close> dd(1)]
+    note gauge_Int[OF \<open>gauge d\<close> dd(1)]
from fine_division_exists[OF this,of u v] guess qq .
then show ?case
apply (rule_tac x=qq in exI)
@@ -6838,7 +6882,7 @@
guess d1 .. note d1 = conjunctD2[OF this,rule_format]
from integrable_integral[OF that(2),unfolded has_integral[of g],rule_format,OF *]
guess d2 .. note d2 = conjunctD2[OF this,rule_format]
-    note gauge_inter[OF d1(1) d2(1)]
+    note gauge_Int[OF d1(1) d2(1)]
from fine_division_exists[OF this, of a b] guess p . note p=this
show "norm (integral (cbox a b) f) < integral (cbox a b) g + e"
apply (rule norm)
@@ -7066,7 +7110,7 @@
by simp
also have "\<dots> < e' * norm (x - x0)"
using \<open>e' > 0\<close>
-        apply (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
+        apply (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
apply  (auto simp: divide_simps e_def)
by (metis \<open>0 < e\<close> e_def order.asym zero_less_divide_iff)
finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .```
```--- a/src/HOL/Analysis/Tagged_Division.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Analysis/Tagged_Division.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -272,7 +272,7 @@
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
by (rule gauge_ball) auto

-lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
+lemma gauge_Int[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
unfolding gauge_def by auto

lemma gauge_inters:
@@ -667,7 +667,7 @@
qed
qed

-lemma partial_division_extend_interval:
+proposition partial_division_extend_interval:
assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
proof (cases "p = {}")```
```--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -1374,6 +1374,128 @@
then show ?thesis by (auto simp: I_def)
qed

+corollary open_countable_Union_open_box:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "open S"
+  obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" "\<Union>\<D> = S"
+proof -
+  let ?a = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
+  let ?b = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
+  let ?I = "{f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (?a f) (?b f) \<subseteq> S}"
+  let ?\<D> = "(\<lambda>f. box (?a f) (?b f)) ` ?I"
+  show ?thesis
+  proof
+    have "countable ?I"
+      by (simp add: countable_PiE countable_rat)
+    then show "countable ?\<D>"
+      by blast
+    show "\<Union>?\<D> = S"
+      using open_UNION_box [OF assms] by metis
+  qed auto
+qed
+
+lemma rational_cboxes:
+  fixes x :: "'a::euclidean_space"
+  assumes "e > 0"
+  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> cbox a b \<and> cbox a b \<subseteq> ball x e"
+proof -
+  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
+  then have e: "e' > 0"
+    using assms by auto
+  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
+  proof
+    fix i
+    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
+    show "?th i" by auto
+  qed
+  from choice[OF this] obtain a where
+    a: "\<forall>u. a u \<in> \<rat> \<and> a u < x \<bullet> u \<and> x \<bullet> u - a u < e'" ..
+  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
+  proof
+    fix i
+    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
+    show "?th i" by auto
+  qed
+  from choice[OF this] obtain b where
+    b: "\<forall>u. b u \<in> \<rat> \<and> x \<bullet> u < b u \<and> b u - x \<bullet> u < e'" ..
+  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
+  show ?thesis
+  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
+    fix y :: 'a
+    assume *: "y \<in> cbox ?a ?b"
+    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
+      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
+    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
+    proof (rule real_sqrt_less_mono, rule sum_strict_mono)
+      fix i :: "'a"
+      assume i: "i \<in> Basis"
+      have "a i \<le> y\<bullet>i \<and> y\<bullet>i \<le> b i"
+        using * i by (auto simp: cbox_def)
+      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
+        using a by auto
+      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
+        using b by auto
+      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
+        by auto
+      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
+        unfolding e'_def by (auto simp: dist_real_def)
+      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
+        by (rule power_strict_mono) auto
+      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
+    qed auto
+    also have "\<dots> = e"
+      using \<open>0 < e\<close> by simp
+    finally show "y \<in> ball x e"
+      by (auto simp: ball_def)
+  next
+    show "x \<in> cbox (\<Sum>i\<in>Basis. a i *\<^sub>R i) (\<Sum>i\<in>Basis. b i *\<^sub>R i)"
+      using a b less_imp_le by (auto simp: cbox_def)
+  qed (use a b cbox_def in auto)
+qed
+
+lemma open_UNION_cbox:
+  fixes M :: "'a::euclidean_space set"
+  assumes "open M"
+  defines "a' \<equiv> \<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
+  defines "b' \<equiv> \<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
+  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (a' f) (b' f) \<subseteq> M}"
+  shows "M = (\<Union>f\<in>I. cbox (a' f) (b' f))"
+proof -
+  have "x \<in> (\<Union>f\<in>I. cbox (a' f) (b' f))" if "x \<in> M" for x
+  proof -
+    obtain e where e: "e > 0" "ball x e \<subseteq> M"
+      using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
+    moreover obtain a b where ab: "x \<in> cbox a b" "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
+                                  "\<forall>i \<in> Basis. b \<bullet> i \<in> \<rat>" "cbox a b \<subseteq> ball x e"
+      using rational_cboxes[OF e(1)] by metis
+    ultimately show ?thesis
+       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
+          (auto simp: euclidean_representation I_def a'_def b'_def)
+  qed
+  then show ?thesis by (auto simp: I_def)
+qed
+
+corollary open_countable_Union_open_cbox:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "open S"
+  obtains \<D> where "countable \<D>" "\<D> \<subseteq> Pow S" "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" "\<Union>\<D> = S"
+proof -
+  let ?a = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
+  let ?b = "\<lambda>f. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
+  let ?I = "{f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. cbox (?a f) (?b f) \<subseteq> S}"
+  let ?\<D> = "(\<lambda>f. cbox (?a f) (?b f)) ` ?I"
+  show ?thesis
+  proof
+    have "countable ?I"
+      by (simp add: countable_PiE countable_rat)
+    then show "countable ?\<D>"
+      by blast
+    show "\<Union>?\<D> = S"
+      using open_UNION_cbox [OF assms] by metis
+  qed auto
+qed
+
lemma box_eq_empty:
fixes a :: "'a::euclidean_space"
shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
@@ -7398,6 +7520,24 @@
by (simp add: assms bij_is_surj open_surjective_linear_image)
qed

+corollary interior_bijective_linear_image:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes "linear f" "bij f"
+  shows "interior (f ` S) = f ` interior S"  (is "?lhs = ?rhs")
+proof safe
+  fix x
+  assume x: "x \<in> ?lhs"
+  then obtain T where "open T" and "x \<in> T" and "T \<subseteq> f ` S"
+    by (metis interiorE)
+  then show "x \<in> ?rhs"
+    by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
+next
+  fix x
+  assume x: "x \<in> interior S"
+  then show "f x \<in> interior (f ` S)"
+    by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
+qed
+
text \<open>Also bilinear functions, in composition form.\<close>

lemma bilinear_continuous_at_compose:```
```--- a/src/HOL/Archimedean_Field.thy	Wed Jun 21 15:04:26 2017 +0200
+++ b/src/HOL/Archimedean_Field.thy	Wed Jun 21 17:13:55 2017 +0100
@@ -339,6 +339,11 @@
lemma floor_less_neg_numeral [simp]: "\<lfloor>x\<rfloor> < - numeral v \<longleftrightarrow> x < - numeral v"

+lemma le_mult_floor_Ints:
+  assumes "0 \<le> a" "a \<in> Ints"
+  shows "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> (of_int\<lfloor>a * b\<rfloor> :: 'a :: linordered_idom)"
+  by (metis Ints_cases assms floor_less_iff floor_of_int linorder_not_less mult_left_mono of_int_floor_le of_int_less_iff of_int_mult)
+

text \<open>Addition and subtraction of integers.\<close>

@@ -492,6 +497,16 @@
lemma ceiling_add_le: "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>"

+lemma mult_ceiling_le:
+  assumes "0 \<le> a" and "0 \<le> b"
+  shows "\<lceil>a * b\<rceil> \<le> \<lceil>a\<rceil> * \<lceil>b\<rceil>"
+  by (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult)
+
+lemma mult_ceiling_le_Ints:
+  assumes "0 \<le> a" "a \<in> Ints"
+  shows "(of_int \<lceil>a * b\<rceil> :: 'a :: linordered_idom) \<le> of_int(\<lceil>a\<rceil> * \<lceil>b\<rceil>)"
+  by (metis Ints_cases assms ceiling_le_iff ceiling_of_int le_of_int_ceiling mult_left_mono of_int_le_iff of_int_mult)
+
lemma finite_int_segment:
fixes a :: "'a::floor_ceiling"
shows "finite {x \<in> \<int>. a \<le> x \<and> x \<le> b}"
@@ -504,6 +519,10 @@
by simp
qed

+corollary finite_abs_int_segment:
+  fixes a :: "'a::floor_ceiling"
+  shows "finite {k \<in> \<int>. \<bar>k\<bar> \<le> a}"
+  using finite_int_segment [of "-a" a] by (auto simp add: abs_le_iff conj_commute minus_le_iff)

text \<open>Ceiling with numerals.\<close>
```