--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Jun 26 23:12:39 2017 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Jun 26 23:12:43 2017 +0200
@@ -9,6 +9,10 @@
Lebesgue_Measure Tagged_Division
begin
+(*FIXME DELETE*)
+lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
+lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
+
(* try instead structured proofs below *)
lemma norm_diff2: "\<lbrakk>y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \<le> e1; norm(y2 - x2) \<le> e2\<rbrakk>
\<Longrightarrow> norm(y - x) \<le> e"
@@ -890,103 +894,103 @@
subsection \<open>Cauchy-type criterion for integrability.\<close>
-(* XXXXXXX *)
-lemma integrable_cauchy:
+lemma 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>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>
- norm ((\<Sum>(x,k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x,k)\<in>p2. content k *\<^sub>R f x)) < e))"
- (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
-proof
+ (\<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))"
+ (is "?l = (\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>)")
+proof (intro iffI allI impI)
assume ?l
- then guess y unfolding integrable_on_def has_integral .. note y=this
- show "\<forall>e>0. \<exists>d. ?P e d"
- proof (clarify, goal_cases)
- case (1 e)
- then have "e/2 > 0" by auto
- then guess d
- apply -
- apply (drule y[rule_format])
- apply (elim exE conjE)
- done
- note d=this[rule_format]
- show ?case
- proof (rule_tac x=d in exI, clarsimp simp: d)
- fix p1 p2
- assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
- "p2 tagged_division_of (cbox a b)" "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"
- apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
- using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
+ 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)"
+ 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"
+ by meson
+ show ?thesis
+ apply (rule_tac x=\<gamma> in exI, clarsimp simp: \<open>gauge \<gamma>\<close>)
+ by (blast intro!: \<gamma> dist_triangle_half_l[where y=y,unfolded dist_norm])
+ qed
+next
+ assume "\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>"
+ then have "\<forall>n::nat. \<exists>\<gamma>. ?P (1 / (n + 1)) \<gamma>"
+ 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))
+ < 1 / (m + 1)"
+ by metis
+ have "\<And>n. gauge (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}})"
+ apply (rule gauge_Inter)
+ using \<gamma> by auto
+ then have "\<forall>n. \<exists>p. p tagged_division_of (cbox a b) \<and> (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}}) fine p"
+ by (meson fine_division_exists)
+ then obtain p where p: "\<And>z. p z tagged_division_of cbox a b"
+ "\<And>z. (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..z}}) fine p z"
+ by meson
+ have dp: "\<And>i n. i\<le>n \<Longrightarrow> \<gamma> i fine p n"
+ using p unfolding fine_Inter
+ using atLeastAtMost_iff by blast
+ have "Cauchy (\<lambda>n. sum (\<lambda>(x,K). content K *\<^sub>R (f x)) (p n))"
+ proof (rule CauchyI)
+ fix e::real
+ assume "0 < e"
+ then obtain N where "N \<noteq> 0" and N: "inverse (real N) < e"
+ using real_arch_inverse[of e] by blast
+ show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e"
+ proof (intro exI allI impI)
+ fix m n
+ assume mn: "N \<le> m" "N \<le> n"
+ have "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < 1 / (real N + 1)"
+ by (simp add: p(1) dp mn \<gamma>)
+ also have "... < e"
+ using N \<open>N \<noteq> 0\<close> \<open>0 < e\<close> by (auto simp: field_simps)
+ finally show "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e" .
qed
qed
-next
- assume "\<forall>e>0. \<exists>d. ?P e d"
- then have "\<forall>n::nat. \<exists>d. ?P (inverse(of_nat (n + 1))) d"
- by auto
- from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
- have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})"
- apply (rule gauge_Inter)
- using d(1)
- apply auto
- done
- then have "\<forall>n. \<exists>p. p tagged_division_of (cbox a b) \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p"
- by (meson fine_division_exists)
- from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
- have dp: "\<And>i n. i\<le>n \<Longrightarrow> d i fine p n"
- using p(2) unfolding fine_inters by auto
- have "Cauchy (\<lambda>n. sum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
- proof (rule CauchyI, goal_cases)
- case (1 e)
- then guess N unfolding real_arch_inverse[of e] .. note N=this
- show ?case
- apply (rule_tac x=N in exI)
- proof clarify
- fix m n
- assume mn: "N \<le> m" "N \<le> n"
- have *: "N = (N - 1) + 1" using N by auto
- show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
- apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
- apply(subst *)
- using dp p(1) mn d(2) by auto
- qed
- qed
- then guess y unfolding convergent_eq_Cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
+ then obtain y where y: "\<exists>no. \<forall>n\<ge>no. norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < r" if "r > 0" for r
+ by (auto simp: convergent_eq_Cauchy[symmetric] dest: LIMSEQ_D)
show ?l
unfolding integrable_on_def has_integral
proof (rule_tac x=y in exI, clarify)
fix e :: real
assume "e>0"
- then have *:"e/2 > 0" by auto
- then guess N1 unfolding real_arch_inverse[of "e/2"] .. note N1=this
- then have N1': "N1 = N1 - 1 + 1"
- by auto
- guess N2 using y[OF *] .. note N2=this
- have "gauge (d (N1 + N2))"
- using d by auto
- moreover
- {
- fix q
- assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
- have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
- apply (rule less_trans)
- using N1
- apply auto
- done
- have "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e"
- apply (rule norm_triangle_half_r)
- apply (rule less_trans[OF _ *])
- apply (subst N1', rule d(2)[of "p (N1+N2)"])
- using N1' as(1) as(2) dp
- apply (simp add: \<open>\<forall>x. p x tagged_division_of cbox a b \<and> (\<lambda>xa. \<Inter>{d i xa |i. i \<in> {0..x}}) fine p x\<close>)
- using N2 le_add2 by blast
- }
- ultimately show "\<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 f x) - y) < e)"
- by (rule_tac x="d (N1 + N2)" in exI) auto
+ then have e2: "e/2 > 0" by auto
+ then obtain N1::nat where N1: "N1 \<noteq> 0" "inverse (real N1) < e / 2"
+ using real_arch_inverse by blast
+ 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)"
+ proof (intro exI conjI allI impI)
+ show "gauge (\<gamma> (N1+N2))"
+ using \<gamma> by auto
+ show "norm ((\<Sum>(x,K) \<in> q. content K *\<^sub>R f x) - y) < e"
+ if "q tagged_division_of cbox a b \<and> \<gamma> (N1+N2) fine q" for q
+ proof (rule norm_triangle_half_r)
+ have "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x))
+ < 1 / (real (N1+N2) + 1)"
+ by (rule \<gamma>; simp add: dp p that)
+ also have "... < e/2"
+ using N1 \<open>0 < e\<close> by (auto simp: field_simps intro: less_le_trans)
+ finally show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x)) < e / 2" .
+ show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - y) < e/2"
+ using N2 le_add_same_cancel2 by blast
+ qed
+ qed
qed
qed
@@ -1021,221 +1025,220 @@
qed
-lemma has_integral_split:
+proposition has_integral_split:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
and k: "k \<in> Basis"
- shows "(f has_integral (i + j)) (cbox a b)"
-proof (unfold has_integral, rule, rule, goal_cases)
- case (1 e)
+shows "(f has_integral (i + j)) (cbox a b)"
+ unfolding has_integral
+proof clarify
+ fix e::real
+ assume "0 < e"
then have e: "e/2 > 0"
by auto
- obtain d1
- where d1: "gauge d1"
- and d1norm:
- "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c};
- d1 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - i) < e / 2"
+ 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"
apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
- done
- obtain d2
- where d2: "gauge d2"
- and d2norm:
- "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k};
- d2 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - j) < e / 2"
+ 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"
apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
- let ?d = "\<lambda>x. if x\<bullet>k = c then (d1 x \<inter> d2 x) else ball x \<bar>x\<bullet>k - c\<bar> \<inter> d1 x \<inter> d2 x"
- have "gauge ?d"
- using d1 d2 unfolding gauge_def by auto
- then show ?case
- proof (rule_tac x="?d" in exI, safe)
+ let ?\<gamma> = "\<lambda>x. if x\<bullet>k = c then (\<gamma>1 x \<inter> \<gamma>2 x) else ball x \<bar>x\<bullet>k - c\<bar> \<inter> \<gamma>1 x \<inter> \<gamma>2 x"
+ 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)"
+ proof (rule_tac x="?\<gamma>" in exI, safe)
fix p
- assume "p tagged_division_of (cbox a b)" "?d fine p"
- note p = this tagged_division_ofD[OF this(1)]
- have xk_le_c: "\<And>x kk. (x, kk) \<in> p \<Longrightarrow> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {} \<Longrightarrow> x\<bullet>k \<le> c"
- proof -
- fix x kk
- assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}"
- show "x\<bullet>k \<le> c"
- proof (rule ccontr)
- assume **: "\<not> ?thesis"
- from this[unfolded not_le]
- have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
- using p(2)[unfolded fine_def, rule_format,OF as] by auto
- with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
- by blast
- then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
- using Basis_le_norm[OF k, of "x - y"]
- by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
- with y show False
- using ** by (auto simp add: field_simps)
- qed
+ assume p: "p tagged_division_of (cbox a b)" and "?\<gamma> fine p"
+ have ab_eqp: "cbox a b = \<Union>{K. \<exists>x. (x, K) \<in> p}"
+ using p by blast
+ have xk_le_c: "x\<bullet>k \<le> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}" for x K
+ proof (rule ccontr)
+ assume **: "\<not> x \<bullet> k \<le> c"
+ then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
+ using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
+ with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
+ by blast
+ then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
+ using Basis_le_norm[OF k, of "x - y"]
+ by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
+ with y show False
+ using ** by (auto simp add: field_simps)
qed
- have xk_ge_c: "\<And>x kk. (x, kk) \<in> p \<Longrightarrow> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {} \<Longrightarrow> x\<bullet>k \<ge> c"
- proof -
- fix x kk
- assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}"
- show "x\<bullet>k \<ge> c"
- proof (rule ccontr)
- assume **: "\<not> ?thesis"
- from this[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
- using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
- with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
- by blast
- then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
- using Basis_le_norm[OF k, of "x - y"]
- by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
- with y show False
- using ** by (auto simp add: field_simps)
- qed
+ have xk_ge_c: "x\<bullet>k \<ge> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}" for x K
+ proof (rule ccontr)
+ assume **: "\<not> x \<bullet> k \<ge> c"
+ then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
+ using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
+ with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
+ by blast
+ then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
+ using Basis_le_norm[OF k, of "x - y"]
+ by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
+ with y show False
+ using ** by (auto simp add: field_simps)
qed
-
- have lem1: "\<And>f P Q. (\<forall>x k. (x, k) \<in> {(x, f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow>
- (\<forall>x k. P x k \<longrightarrow> Q x (f k))"
- by auto
- have fin_finite: "finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
+ have fin_finite: "finite {(x,f K) | x K. (x,K) \<in> s \<and> P x K}"
if "finite s" for s and f :: "'a set \<Rightarrow> 'a set" and P :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"
proof -
- from that have "finite ((\<lambda>(x, k). (x, f k)) ` s)"
+ from that have "finite ((\<lambda>(x,K). (x, f K)) ` s)"
by auto
then show ?thesis
by (rule rev_finite_subset) auto
qed
- { fix g :: "'a set \<Rightarrow> 'a set"
+ { fix \<G> :: "'a set \<Rightarrow> 'a set"
fix i :: "'a \<times> 'a set"
- assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
- then obtain x k where xk:
- "i = (x, g k)" "(x, k) \<in> p"
- "(x, g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
- by auto
- have "content (g k) = 0"
+ assume "i \<in> (\<lambda>(x, k). (x, \<G> k)) ` p - {(x, \<G> k) |x k. (x, k) \<in> p \<and> \<G> k \<noteq> {}}"
+ then obtain x K where xk: "i = (x, \<G> K)" "(x,K) \<in> p"
+ "(x, \<G> K) \<notin> {(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}"
+ by auto
+ have "content (\<G> K) = 0"
using xk using content_empty by auto
- then have "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0"
+ then have "(\<lambda>(x,K). content K *\<^sub>R f x) i = 0"
unfolding xk split_conv by auto
} note [simp] = this
- have lem3: "\<And>g :: 'a set \<Rightarrow> 'a set. finite p \<Longrightarrow>
- sum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
- sum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
- by (rule sum.mono_neutral_left) auto
- let ?M1 = "{(x, kk \<inter> {x. x\<bullet>k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
- have d1_fine: "d1 fine ?M1"
- by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
+ have "finite p"
+ using p by blast
+ let ?M1 = "{(x, K \<inter> {x. x\<bullet>k \<le> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
+ have \<gamma>1_fine: "\<gamma>1 fine ?M1"
+ using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
- proof (rule d1norm [OF tagged_division_ofI d1_fine])
+ proof (rule \<gamma>1norm [OF tagged_division_ofI \<gamma>1_fine])
show "finite ?M1"
- by (rule fin_finite p(3))+
+ by (rule fin_finite) (use p in blast)
show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
- unfolding p(8)[symmetric] by auto
- fix x l
- assume xl: "(x, l) \<in> ?M1"
- then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
- show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
- unfolding xl'
- using p(4-6)[OF xl'(3)] using xl'(4)
- using xk_le_c[OF xl'(3-4)] by auto
- show "\<exists>a b. l = cbox a b"
- unfolding xl'
- using p(6)[OF xl'(3)]
- by (fastforce simp add: interval_split[OF k,where c=c])
- fix y r
- let ?goal = "interior l \<inter> interior r = {}"
- assume yr: "(y, r) \<in> ?M1"
- then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
- assume as: "(x, l) \<noteq> (y, r)"
- show "interior l \<inter> interior r = {}"
- proof (cases "l' = r' \<longrightarrow> x' = y'")
+ by (auto simp: ab_eqp)
+
+ fix x L
+ assume xL: "(x, L) \<in> ?M1"
+ then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<le> c}"
+ "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
+ by blast
+ then obtain a' b' where ab': "L' = cbox a' b'"
+ using p by blast
+ show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
+ using p xk_le_c xL' by auto
+ show "\<exists>a b. L = cbox a b"
+ using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
+
+ fix y R
+ assume yR: "(y, R) \<in> ?M1"
+ then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<le> c}"
+ "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
+ by blast
+ assume as: "(x, L) \<noteq> (y, R)"
+ show "interior L \<inter> interior R = {}"
+ proof (cases "L' = R' \<longrightarrow> x' = y'")
case False
+ have "interior R' = {}"
+ by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
then show ?thesis
- using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ using yR' by simp
next
case True
- then have "l' \<noteq> r'"
- using as unfolding xl' yr' by auto
+ then have "L' \<noteq> R'"
+ using as unfolding xL' yR' by auto
+ have "interior L' \<inter> interior R' = {}"
+ by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
then show ?thesis
- using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ using xL'(2) yR'(2) by auto
qed
qed
moreover
- let ?M2 = "{(x,kk \<inter> {x. x\<bullet>k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
- have d2_fine: "d2 fine ?M2"
- by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
+ let ?M2 = "{(x,K \<inter> {x. x\<bullet>k \<ge> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
+ have \<gamma>2_fine: "\<gamma>2 fine ?M2"
+ using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
- proof (rule d2norm [OF tagged_division_ofI d2_fine])
+ proof (rule \<gamma>2norm [OF tagged_division_ofI \<gamma>2_fine])
show "finite ?M2"
- by (rule fin_finite p(3))+
+ by (rule fin_finite) (use p in blast)
show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
- unfolding p(8)[symmetric] by auto
- fix x l
- assume xl: "(x, l) \<in> ?M2"
- then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
- show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
- unfolding xl'
- using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
- by auto
- show "\<exists>a b. l = cbox a b"
- unfolding xl'
- using p(6)[OF xl'(3)]
- by (fastforce simp add: interval_split[OF k, where c=c])
- fix y r
- let ?goal = "interior l \<inter> interior r = {}"
- assume yr: "(y, r) \<in> ?M2"
- then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
- assume as: "(x, l) \<noteq> (y, r)"
- show "interior l \<inter> interior r = {}"
- proof (cases "l' = r' \<longrightarrow> x' = y'")
+ by (auto simp: ab_eqp)
+
+ fix x L
+ assume xL: "(x, L) \<in> ?M2"
+ then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<ge> c}"
+ "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
+ by blast
+ then obtain a' b' where ab': "L' = cbox a' b'"
+ using p by blast
+ show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
+ using p xk_ge_c xL' by auto
+ show "\<exists>a b. L = cbox a b"
+ using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
+
+ fix y R
+ assume yR: "(y, R) \<in> ?M2"
+ then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<ge> c}"
+ "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
+ by blast
+ assume as: "(x, L) \<noteq> (y, R)"
+ show "interior L \<inter> interior R = {}"
+ proof (cases "L' = R' \<longrightarrow> x' = y'")
case False
+ have "interior R' = {}"
+ by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
then show ?thesis
- using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ using yR' by simp
next
case True
- then have "l' \<noteq> r'"
- using as unfolding xl' yr' by auto
+ then have "L' \<noteq> R'"
+ using as unfolding xL' yR' by auto
+ have "interior L' \<inter> interior R' = {}"
+ by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
then show ?thesis
- using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
+ using xL'(2) yR'(2) by auto
qed
qed
ultimately
- have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"
+ have "norm (((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) + ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j)) < e/2 + e/2"
using norm_add_less by blast
- also {
+ moreover have "((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) +
+ ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j) =
+ (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
+ proof -
have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
- using scaleR_zero_left by auto
+ by auto
have cont_eq: "\<And>g. (\<lambda>(x,l). content l *\<^sub>R f x) \<circ> (\<lambda>(x,l). (x,g l)) = (\<lambda>(x,l). content (g l) *\<^sub>R f x)"
by auto
+ have *: "\<And>\<G> :: 'a set \<Rightarrow> 'a set.
+ (\<Sum>(x,K)\<in>{(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}. content K *\<^sub>R f x) =
+ (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x, \<G> K)) ` p. content K *\<^sub>R f x)"
+ by (rule sum.mono_neutral_left) (auto simp: \<open>finite p\<close>)
have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) =
(\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
by auto
- also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
- (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
- unfolding lem3[OF p(3)]
- by (subst (1 2) sum.reindex_nontrivial[OF p(3)])
- (auto intro!: k eq0 tagged_division_split_left_inj_content[OF p(1)] tagged_division_split_right_inj_content[OF p(1)]
- simp: cont_eq)+
- also note sum.distrib[symmetric]
- also have "\<And>x. x \<in> p \<Longrightarrow>
- (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
- (\<lambda>(x,ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x =
- (\<lambda>(x,ka). content ka *\<^sub>R f x) x"
+ moreover have "\<dots> = (\<Sum>(x,K) \<in> p. content (K \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
+ (\<Sum>(x,K) \<in> p. content (K \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
+ unfolding *
+ apply (subst (1 2) sum.reindex_nontrivial)
+ apply (auto intro!: k p eq0 tagged_division_split_left_inj_content tagged_division_split_right_inj_content
+ simp: cont_eq \<open>finite p\<close>)
+ done
+ moreover have "\<And>x. x \<in> p \<Longrightarrow> (\<lambda>(a,B). content (B \<inter> {a. a \<bullet> k \<le> c}) *\<^sub>R f a) x +
+ (\<lambda>(a,B). content (B \<inter> {a. c \<le> a \<bullet> k}) *\<^sub>R f a) x =
+ (\<lambda>(a,B). content B *\<^sub>R f a) x"
proof clarify
- fix a b
- assume "(a, b) \<in> p"
- from p(6)[OF this] guess u v by (elim exE) note uv=this
- then show "content (b \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a =
- content b *\<^sub>R f a"
- unfolding scaleR_left_distrib[symmetric]
- unfolding uv content_split[OF k,of u v c]
- by auto
+ fix a B
+ assume "(a, B) \<in> p"
+ with p obtain u v where uv: "B = cbox u v" by blast
+ then show "content (B \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (B \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a = content B *\<^sub>R f a"
+ by (auto simp: scaleR_left_distrib uv content_split[OF k,of u v c])
qed
- note sum.cong [OF _ this]
- finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x \<bullet> k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
- ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x \<bullet> k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
- (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
- by auto
- }
- finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
+ ultimately show ?thesis
+ by (auto simp: sum.distrib[symmetric])
+ qed
+ ultimately show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
by auto
qed
qed
@@ -1303,7 +1306,7 @@
norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
by (subst sum.union_inter_neutral) (auto simp: p1 p2)
also have "\<dots> < e"
- by (rule k d(2) p12 fine_union p1 p2)+
+ by (rule k d(2) p12 fine_Un p1 p2)+
finally have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" .
}
then show ?thesis
@@ -1352,7 +1355,7 @@
qed
qed
with f show ?thesis1
- by (simp add: interval_split[OF k] integrable_cauchy)
+ 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>
@@ -1384,7 +1387,7 @@
qed
qed
with f show ?thesis2
- by (simp add: interval_split[OF k] integrable_cauchy)
+ by (simp add: interval_split[OF k] integrable_Cauchy)
qed
lemma operative_integral:
@@ -1509,21 +1512,25 @@
lemma has_integral_bound:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "0 \<le> B"
- and *: "(f has_integral i) (cbox a b)"
- and "\<forall>x\<in>cbox a b. norm (f x) \<le> B"
+ and f: "(f has_integral i) (cbox a b)"
+ and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
shows "norm i \<le> B * content (cbox a b)"
proof (rule ccontr)
assume "\<not> ?thesis"
- then have *: "norm i - B * content (cbox a b) > 0"
+ then have "norm i - B * content (cbox a b) > 0"
by auto
- from assms(2)[unfolded has_integral,rule_format,OF *]
- guess d by (elim exE conjE) note d=this[rule_format]
- from fine_division_exists[OF this(1), of a b] guess p . note p=this
- have *: "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
+ with f[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) - i) < norm i - B * content (cbox a b)"
+ by metis
+ then obtain p where p: "p tagged_division_of cbox a b" and "\<gamma> fine p"
+ using fine_division_exists by blast
+ have "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
unfolding not_less
- by (metis norm_triangle_sub[of i] add.commute le_less_trans less_diff_eq linorder_not_le norm_minus_commute)
- show False
- using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
+ by (metis diff_left_mono dist_commute dist_norm norm_triangle_ineq2 order_trans)
+ then show False
+ using \<gamma> [OF p \<open>\<gamma> fine p\<close>] rsum_bound[OF p] assms by metis
qed
corollary has_integral_bound_real:
@@ -1547,20 +1554,17 @@
lemma rsum_component_le:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "p tagged_division_of (cbox a b)"
- and "\<forall>x\<in>cbox a b. (f x)\<bullet>i \<le> (g x)\<bullet>i"
- shows "(sum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (sum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
+ assumes p: "p tagged_division_of (cbox a b)"
+ and "\<And>x. x \<in> cbox a b \<Longrightarrow> (f x)\<bullet>i \<le> (g x)\<bullet>i"
+ shows "(\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) \<bullet> i \<le> (\<Sum>(x, K)\<in>p. content K *\<^sub>R g x) \<bullet> i"
unfolding inner_sum_left
proof (rule sum_mono, clarify)
- fix a b
- assume ab: "(a, b) \<in> p"
- note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
- from this(3) guess u v by (elim exE) note b=this
- show "(content b *\<^sub>R f a) \<bullet> i \<le> (content b *\<^sub>R g a) \<bullet> i"
- unfolding b inner_simps real_scaleR_def
- apply (rule mult_left_mono)
- using assms(2) tagged
- by (auto simp add: content_pos_le)
+ fix x K
+ assume ab: "(x, K) \<in> p"
+ with p obtain u v where K: "K = cbox u v"
+ by blast
+ then show "(content K *\<^sub>R f x) \<bullet> i \<le> (content K *\<^sub>R g x) \<bullet> i"
+ by (metis ab assms inner_scaleR_left measure_nonneg mult_left_mono tag_in_interval)
qed
lemma has_integral_component_le:
@@ -1582,7 +1586,7 @@
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_Int[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_Int
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"
@@ -1591,8 +1595,8 @@
by blast+
then show False
unfolding inner_simps
- using rsum_component_le[OF p(1) le]
- by (simp add: abs_real_def split: if_split_asm)
+ using rsum_component_le[OF p(1)] le
+ by (force simp add: abs_real_def split: if_split_asm)
qed
show ?thesis
proof (cases "\<exists>a b. s = cbox a b")
@@ -1611,7 +1615,8 @@
have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
unfolding bounded_Un by(rule conjI bounded_ball)+
from bounded_subset_cbox[OF this] guess a b by (elim exE)
- note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
+ then have ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
+ by blast+
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
have *: "\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False"
@@ -1787,7 +1792,7 @@
finally have "norm (i1 - i2) < e" .
} note triangle3 = this
have finep: "gm fine p" "gn fine p"
- using fine_inter p by auto
+ using fine_Int p by auto
{ fix x
assume x: "x \<in> cbox a b"
have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
@@ -4444,7 +4449,7 @@
have *: "(\<lambda>x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \<le> - a"
using assms by auto
from indefinite_integral_continuous_left[OF * \<open>e>0\<close>] guess d . note d = this
- let ?d = "min d (b - c)"
+ let ?d = "min d (b - c)"
show ?thesis
apply (rule that[of "?d"])
apply safe
@@ -4471,92 +4476,62 @@
qed
qed
-lemma indefinite_integral_continuous:
+lemma indefinite_integral_continuous_1:
fixes f :: "real \<Rightarrow> 'a::banach"
assumes "f integrable_on {a .. b}"
shows "continuous_on {a .. b} (\<lambda>x. integral {a .. x} f)"
-proof (unfold continuous_on_iff, safe)
- fix x e :: real
- assume as: "x \<in> {a .. b}" "e > 0"
- let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a .. b}. dist x' x < d \<longrightarrow> dist (integral {a .. x'} f) (integral {a .. x} f) < e"
- {
- presume *: "a < b \<Longrightarrow> ?thesis"
- show ?thesis
- apply cases
- apply (rule *)
- apply assumption
- proof goal_cases
- case 1
- then have "cbox a b = {x}"
- using as(1)
- apply -
- apply (rule set_eqI)
- apply auto
+proof -
+ have "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e"
+ if x: "x \<in> {a..b}" and "e > 0" for x e :: real
+ proof (cases "a = b")
+ case True
+ with that show ?thesis by force
+ next
+ case False
+ with x have "a < b" by force
+ with x consider "x = a" | "x = b" | "a < x" "x < b"
+ by force
+ then show ?thesis
+ proof cases
+ case 1 show ?thesis
+ apply (rule indefinite_integral_continuous_right [OF assms _ \<open>a < b\<close> \<open>e > 0\<close>], force)
+ using \<open>x = a\<close> apply (force simp: dist_norm algebra_simps)
+ done
+ next
+ case 2 show ?thesis
+ apply (rule indefinite_integral_continuous_left [OF assms \<open>a < b\<close> _ \<open>e > 0\<close>], force)
+ using \<open>x = b\<close> apply (force simp: dist_norm norm_minus_commute algebra_simps)
done
- then show ?case using \<open>e > 0\<close> by auto
- qed
- }
- assume "a < b"
- have "(x = a \<or> x = b) \<or> (a < x \<and> x < b)"
- using as(1) by auto
- then show ?thesis
- apply (elim disjE)
- proof -
- assume "x = a"
- have "a \<le> a" ..
- from indefinite_integral_continuous_right[OF assms(1) this \<open>a<b\<close> \<open>e>0\<close>] guess d . note d=this
- show ?thesis
- apply rule
- apply rule
- apply (rule d)
- apply safe
- apply (subst dist_commute)
- unfolding \<open>x = a\<close> dist_norm
- apply (rule d(2)[rule_format])
- apply auto
- done
- next
- assume "x = b"
- have "b \<le> b" ..
- from indefinite_integral_continuous_left[OF assms(1) \<open>a<b\<close> this \<open>e>0\<close>] guess d . note d=this
- show ?thesis
- apply rule
- apply rule
- apply (rule d)
- apply safe
- apply (subst dist_commute)
- unfolding \<open>x = b\<close> dist_norm
- apply (rule d(2)[rule_format])
- apply auto
- done
- next
- assume "a < x \<and> x < b"
- then have xl: "a < x" "x \<le> b" and xr: "a \<le> x" "x < b"
- by auto
- from indefinite_integral_continuous_left [OF assms(1) xl \<open>e>0\<close>] guess d1 . note d1=this
- from indefinite_integral_continuous_right[OF assms(1) xr \<open>e>0\<close>] guess d2 . note d2=this
- show ?thesis
- apply (rule_tac x="min d1 d2" in exI)
- proof safe
- show "0 < min d1 d2"
- using d1 d2 by auto
- fix y
- assume "y \<in> {a .. b}" and "dist y x < min d1 d2"
- then show "dist (integral {a .. y} f) (integral {a .. x} f) < e"
- apply (subst dist_commute)
- apply (cases "y < x")
- unfolding dist_norm
- apply (rule d1(2)[rule_format])
- defer
- apply (rule d2(2)[rule_format])
- unfolding not_less
- apply (auto simp add: field_simps)
- done
+ next
+ case 3
+ obtain d1 where "0 < d1"
+ and d1: "\<And>t. \<lbrakk>x - d1 < t; t \<le> x\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
+ using 3 by (auto intro: indefinite_integral_continuous_left [OF assms \<open>a < x\<close> _ \<open>e > 0\<close>])
+ obtain d2 where "0 < d2"
+ and d2: "\<And>t. \<lbrakk>x \<le> t; t < x + d2\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
+ using 3 by (auto intro: indefinite_integral_continuous_right [OF assms _ \<open>x < b\<close> \<open>e > 0\<close>])
+ show ?thesis
+ proof (intro exI ballI conjI impI)
+ show "0 < min d1 d2"
+ using \<open>0 < d1\<close> \<open>0 < d2\<close> by simp
+ show "dist (integral {a..y} f) (integral {a..x} f) < e"
+ if "y \<in> {a..b}" "dist y x < min d1 d2" for y
+ proof (cases "y < x")
+ case True
+ with that d1 show ?thesis by (auto simp: dist_commute dist_norm)
+ next
+ case False
+ with that d2 show ?thesis
+ by (auto simp: dist_commute dist_norm)
+ qed
+ qed
qed
qed
+ then show ?thesis
+ by (auto simp: continuous_on_iff)
qed
-lemma indefinite_integral_continuous':
+lemma indefinite_integral_continuous_1':
fixes f::"real \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}"
shows "continuous_on {a..b} (\<lambda>x. integral {x..b} f)"
@@ -4565,7 +4540,7 @@
using integral_combine[OF _ _ assms, of x] that
by (auto simp: algebra_simps)
with _ show ?thesis
- by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous assms)
+ by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous_1 assms)
qed
@@ -5248,7 +5223,7 @@
assumes "negligible ((S - T) \<union> (T - S))"
shows "f integrable_on S \<longleftrightarrow> f integrable_on T"
by (blast intro: integrable_spike_set assms negligible_subset)
-
+
lemma has_integral_interior:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (interior S) \<longleftrightarrow> (f has_integral y) S"
@@ -5256,7 +5231,7 @@
apply (auto simp: frontier_def Un_Diff closure_def)
apply (metis Diff_eq_empty_iff interior_subset negligible_empty)
done
-
+
lemma has_integral_closure:
fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (closure S) \<longleftrightarrow> (f has_integral y) S"
@@ -5593,10 +5568,10 @@
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)
+ (auto simp: fine_Int intro: \<open>gauge d1\<close> \<open>gauge d2\<close> d1 d2)
qed
then show ?thesis
- by (simp add: integrable_cauchy)
+ by (simp add: integrable_Cauchy)
qed
lemma integrable_straddle:
@@ -5605,142 +5580,123 @@
\<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, goal_cases)
- case (1 a b e)
+ let ?fs = "(\<lambda>x. if x \<in> s then f x else 0)"
+ have "?fs integrable_on cbox a b" for a b
+ proof (rule integrable_straddle_interval)
+ fix e::real
+ assume "e > 0"
then have *: "e/4 > 0"
by auto
- from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
- note obt(1)[unfolded has_integral_alt'[of g]]
- note conjunctD2[OF this, rule_format]
- note g = this(1) and this(2)[OF *]
- from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
- note obt(2)[unfolded has_integral_alt'[of h]]
- note conjunctD2[OF this, rule_format]
- note h = this(1) and this(2)[OF *]
- from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
- define c :: 'n where "c = (\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max B1 B2)) *\<^sub>R i)"
- define d :: 'n where "d = (\<Sum>i\<in>Basis. max (b\<bullet>i) (max B1 B2) *\<^sub>R i)"
- have *: "ball 0 B1 \<subseteq> cbox c d" "ball 0 B2 \<subseteq> cbox c d"
- apply safe
- unfolding mem_ball mem_box dist_norm
- apply (rule_tac[!] ballI)
- proof goal_cases
- case (1 x i)
- then show ?case using Basis_le_norm[of i x]
- unfolding c_def d_def by auto
- next
- case (2 x i)
- then show ?case using Basis_le_norm[of i x]
- 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> abs (ag - ah) < e"
- using obt(3)
- unfolding real_norm_def
- by arith
- show ?case
- apply (rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
- apply (rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
- apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then g x else 0)" in exI)
- 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 **[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) =
- (if x \<in> s then f x - g x else (0::real))"
+ with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
+ and ij: "\<bar>i - j\<bar> < e/4"
+ and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
+ by metis
+ let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
+ let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
+ obtain Bg where Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/4"
+ and int_g: "\<And>a b. ?gs integrable_on cbox a b"
+ using g * unfolding has_integral_alt' real_norm_def by meson
+ obtain Bh where
+ Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/4"
+ and int_h: "\<And>a b. ?hs integrable_on cbox a b"
+ using h * unfolding has_integral_alt' real_norm_def by meson
+ define c where "c = (\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max Bg Bh)) *\<^sub>R i)"
+ define d where "d = (\<Sum>i\<in>Basis. max (b\<bullet>i) (max Bg Bh) *\<^sub>R i)"
+ have "\<lbrakk>norm (0 - x) < Bg; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
+ using Basis_le_norm[of i x] unfolding c_def d_def by auto
+ then have ballBg: "ball 0 Bg \<subseteq> cbox c d"
+ by (auto simp: mem_box dist_norm)
+ have "\<lbrakk>norm (0 - x) < Bh; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
+ using Basis_le_norm[of i x] unfolding c_def d_def by auto
+ then have ballBh: "ball 0 Bh \<subseteq> cbox c d"
+ by (auto simp: mem_box dist_norm)
+ have ab_cd: "cbox a b \<subseteq> cbox c d"
+ by (auto simp: c_def d_def subset_box_imp)
+ have **: "\<And>ch cg ag ah::real. \<lbrakk>\<bar>ah - ag\<bar> \<le> \<bar>ch - cg\<bar>; \<bar>cg - i\<bar> < e/4; \<bar>ch - j\<bar> < e/4\<rbrakk>
+ \<Longrightarrow> \<bar>ag - ah\<bar> < e"
+ using ij by arith
+ show "\<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> (if x \<in> s then f x else 0) \<and>
+ (if x \<in> s then f x else 0) \<le> h x)"
+ proof (intro exI ballI conjI)
+ have eq: "\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
+ (if x \<in> s then f x - g x else (0::real))"
by auto
- note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF h g]]
- show "norm (integral (cbox a b) (\<lambda>x. if x \<in> s then h x else 0) -
- integral (cbox a b) (\<lambda>x. if x \<in> s then g x else 0)) \<le>
- norm (integral (cbox c d) (\<lambda>x. if x \<in> s then h x else 0) -
- integral (cbox c d) (\<lambda>x. if x \<in> s then g x else 0))"
- unfolding integral_diff[OF h g,symmetric] real_norm_def
- apply (subst **)
- defer
- apply (subst **)
- defer
- apply (rule has_integral_subset_le)
- defer
- apply (rule integrable_integral integrable_diff h g)+
- proof safe
- fix x
- assume "x \<in> cbox a b"
- then show "x \<in> cbox c d"
- unfolding mem_box c_def d_def
- apply -
- apply rule
- apply (erule_tac x=i in ballE)
- apply auto
- done
- qed (insert obt(4), auto)
- qed (insert obt(4), auto)
- qed
- note interv = this
-
- show ?thesis
- unfolding integrable_alt[of f]
- apply safe
- apply (rule interv)
- proof goal_cases
- case (1 e)
- then have *: "e/3 > 0"
- by auto
- from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
- note obt(1)[unfolded has_integral_alt'[of g]]
- note conjunctD2[OF this, rule_format]
- note g = this(1) and this(2)[OF *]
- from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
- note obt(2)[unfolded has_integral_alt'[of h]]
- note conjunctD2[OF this, rule_format]
- note h = this(1) and this(2)[OF *]
- from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
- show ?case
- apply (rule_tac x="max B1 B2" in exI)
- apply safe
- apply (rule max.strict_coboundedI1)
- apply (rule B1)
- proof -
- fix a b c d :: 'n
- assume as: "ball 0 (max B1 B2) \<subseteq> cbox a b" "ball 0 (max B1 B2) \<subseteq> cbox c d"
- have **: "ball 0 B1 \<subseteq> ball (0::'n) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n) (max B1 B2)"
- by auto
- have *: "\<And>ga gc ha hc fa fc::real.
- \<bar>ga - i\<bar> < e / 3 \<and> \<bar>gc - i\<bar> < e / 3 \<and> \<bar>ha - j\<bar> < e / 3 \<and>
- \<bar>hc - j\<bar> < e / 3 \<and> \<bar>i - j\<bar> < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc \<Longrightarrow>
- \<bar>fa - fc\<bar> < e"
- by (simp add: abs_real_def split: if_split_asm)
- show "norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)
- (\<lambda>x. if x \<in> s then f x else 0)) < e"
- unfolding real_norm_def
- apply (rule *)
- apply safe
- unfolding real_norm_def[symmetric]
- apply (rule B1(2))
- apply (rule order_trans)
+ have int_hg: "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox a b"
+ "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox c d"
+ by (metis (no_types) integrable_diff g h has_integral_integrable integrable_altD(1))+
+ show "(?gs has_integral integral (cbox a b) ?gs) (cbox a b)"
+ "(?hs has_integral integral (cbox a b) ?hs) (cbox a b)"
+ by (intro integrable_integral int_g int_h)+
+ then have "integral (cbox a b) ?gs \<le> integral (cbox a b) ?hs"
+ apply (rule has_integral_le)
+ using fgh by force
+ then have "0 \<le> integral (cbox a b) ?hs - integral (cbox a b) ?gs"
+ by simp
+ then have "\<bar>integral (cbox a b) ?hs - integral (cbox a b) ?gs\<bar>
+ \<le> \<bar>integral (cbox c d) ?hs - integral (cbox c d) ?gs\<bar>"
+ apply (simp add: integral_diff [symmetric] int_g int_h)
+ apply (subst abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF int_h int_g]])
+ using fgh apply (force simp: eq intro!: integral_subset_le [OF ab_cd int_hg])+
+ done
+ then show "\<bar>integral (cbox a b) ?gs - integral (cbox a b) ?hs\<bar> < e"
apply (rule **)
- apply (rule as(1))
- apply (rule B1(2))
- apply (rule order_trans)
- apply (rule **)
- apply (rule as(2))
- apply (rule B2(2))
- apply (rule order_trans)
- apply (rule **)
- apply (rule as(1))
- apply (rule B2(2))
- apply (rule order_trans)
- apply (rule **)
- apply (rule as(2))
- using obt(3) apply auto[1]
- apply (rule_tac[!] integral_le)
- using obt
- apply (auto intro!: h g interv)
+ apply (rule Bg ballBg Bh ballBh)+
done
+ show "\<And>x. x \<in> cbox a b \<Longrightarrow> ?gs x \<le> ?fs x" "\<And>x. x \<in> cbox a b \<Longrightarrow> ?fs x \<le> ?hs x"
+ using fgh by auto
qed
qed
+ then have int_f: "?fs integrable_on cbox a b" for a b
+ by simp
+ have "\<exists>B>0. \<forall>a b c d.
+ ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
+ abs (integral (cbox a b) ?fs - integral (cbox c d) ?fs) < e"
+ if "0 < e" for e
+ proof -
+ have *: "e/3 > 0"
+ using that by auto
+ with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
+ and ij: "\<bar>i - j\<bar> < e/3"
+ and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
+ by metis
+ let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
+ let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
+ obtain Bg where "Bg > 0"
+ and Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
+ and int_g: "\<And>a b. ?gs integrable_on cbox a b"
+ using g * unfolding has_integral_alt' real_norm_def by meson
+ obtain Bh where "Bh > 0"
+ and Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
+ and int_h: "\<And>a b. ?hs integrable_on cbox a b"
+ using h * unfolding has_integral_alt' real_norm_def by meson
+ { fix a b c d :: 'n
+ assume as: "ball 0 (max Bg Bh) \<subseteq> cbox a b" "ball 0 (max Bg Bh) \<subseteq> cbox c d"
+ have **: "ball 0 Bg \<subseteq> ball (0::'n) (max Bg Bh)" "ball 0 Bh \<subseteq> ball (0::'n) (max Bg Bh)"
+ by auto
+ have *: "\<And>ga gc ha hc fa fc. \<lbrakk>\<bar>ga - i\<bar> < e/3; \<bar>gc - i\<bar> < e/3; \<bar>ha - j\<bar> < e/3;
+ \<bar>hc - j\<bar> < e/3; ga \<le> fa; fa \<le> ha; gc \<le> fc; fc \<le> hc\<rbrakk> \<Longrightarrow>
+ \<bar>fa - fc\<bar> < e"
+ using ij by arith
+ have "abs (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)
+ (\<lambda>x. if x \<in> s then f x else 0)) < e"
+ proof (rule *)
+ show "\<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
+ using "**" Bg as by blast
+ show "\<bar>integral (cbox c d) ?gs - i\<bar> < e/3"
+ using "**" Bg as by blast
+ show "\<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
+ using "**" Bh as by blast
+ show "\<bar>integral (cbox c d) ?hs - j\<bar> < e/3"
+ using "**" Bh as by blast
+ qed (use int_f int_g int_h fgh in \<open>simp_all add: integral_le\<close>)
+ }
+ then show ?thesis
+ apply (rule_tac x="max Bg Bh" in exI)
+ using \<open>Bg > 0\<close> by auto
+ qed
+ then show ?thesis
+ unfolding integrable_alt[of f] real_norm_def by (blast intro: int_f)
qed
@@ -6043,7 +5999,7 @@
then show ?case
apply (rule_tac x=qq in exI)
using dd(2)[of qq]
- unfolding fine_inter uv
+ unfolding fine_Int uv
apply auto
done
qed
@@ -6054,9 +6010,9 @@
apply (rule assms(4)[rule_format])
proof
show "d fine ?p"
- apply (rule fine_union)
+ apply (rule fine_Un)
apply (rule p)
- apply (rule fine_unions)
+ apply (rule fine_Union)
using qq
apply auto
done
@@ -6316,7 +6272,7 @@
apply (rule *)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
- apply -
+ apply clarify
apply (rule transitive_stepwise_le)
using assms(2)[rule_format, OF x]
apply auto
@@ -6339,7 +6295,8 @@
apply (rule trivial_limit_sequentially)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
- apply (rule transitive_stepwise_le)
+ apply clarify
+ apply (erule transitive_stepwise_le)
prefer 3
unfolding inner_simps real_inner_1_right
apply (rule integral_le)
@@ -6539,11 +6496,7 @@
then have "y \<in> cbox a b"
using p'(3)[OF xk] by auto
then have *: "\<And>m. \<forall>n\<ge>m. f m y \<le> f n y"
- apply -
- apply (rule transitive_stepwise_le)
- using assms(2)
- apply auto
- done
+ using assms(2) by (auto intro: transitive_stepwise_le)
show "f r y \<le> f (m x) y" and "f (m x) y \<le> f s y"
apply (rule_tac[!] *[rule_format])
using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk]
@@ -6585,10 +6538,7 @@
apply (rule trivial_limit_sequentially)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
- apply (rule transitive_stepwise_le)
- using that(3)
- apply auto
- done
+ using assms(3) by (force intro: transitive_stepwise_le)
note fg=this[rule_format]
have "\<exists>i. ((\<lambda>k. integral s (f k)) \<longlongrightarrow> i) sequentially"
@@ -6602,11 +6552,7 @@
done
then guess i .. note i=this
have "\<And>k. \<forall>x\<in>s. \<forall>n\<ge>k. f k x \<le> f n x"
- apply rule
- apply (rule transitive_stepwise_le)
- using that(3)
- apply auto
- done
+ using assms(3) by (force intro: transitive_stepwise_le)
then have i': "\<forall>k. (integral s (f k))\<bullet>1 \<le> i\<bullet>1"
apply -
apply rule
@@ -6731,10 +6677,7 @@
proof (safe, goal_cases)
case (2 x)
have "\<And>m. x \<in> s \<Longrightarrow> \<forall>n\<ge>m. (f m x)\<bullet>1 \<le> (f n x)\<bullet>1"
- apply (rule transitive_stepwise_le)
- using assms(3)
- apply auto
- done
+ using assms(3) by (force intro: transitive_stepwise_le)
then show ?case
by auto
next
@@ -6764,10 +6707,7 @@
apply rule
done
have "\<And>x m. x \<in> s \<Longrightarrow> \<forall>n\<ge>m. f m x \<le> f n x"
- apply (rule transitive_stepwise_le)
- using assms(2)
- apply auto
- done
+ using assms(2) by (force intro: transitive_stepwise_le)
note * = this[rule_format]
have "(\<lambda>x. g x - f 0 x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. f (Suc k) x - f 0 x)) \<longlongrightarrow>
integral s (\<lambda>x. g x - f 0 x)) sequentially"
@@ -6952,7 +6892,7 @@
using that(3) as
apply auto
done
- qed (insert p[unfolded fine_inter], auto)
+ qed (insert p[unfolded fine_Int], auto)
qed
{ presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
@@ -7400,7 +7340,7 @@
proof -
let ?F = "\<lambda>x. integral {c..g x} f"
have cont_int: "continuous_on {a..b} ?F"
- by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous
+ by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous_1
f integrable_continuous_real)+
have deriv: "(((\<lambda>x. integral {c..x} f) \<circ> g) has_vector_derivative g' x *\<^sub>R f (g x))
(at x within {a..b})" if "x \<in> {a..b} - s" for x