--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Thu Aug 24 12:45:46 2017 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Thu Aug 24 17:15:53 2017 +0100
@@ -4228,33 +4228,28 @@
obtains d where "0 < d"
and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
proof -
- have *: "(\<lambda>x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \<le> - a"
+ have intm: "(\<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
+ from indefinite_integral_continuous_left[OF intm \<open>e>0\<close>]
+ obtain d where "0 < d"
+ and d: "\<And>t. \<lbrakk>- c - d < t; t \<le> -c\<rbrakk>
+ \<Longrightarrow> norm (integral {- b..- c} (\<lambda>x. f (-x)) - integral {- b..t} (\<lambda>x. f (-x))) < e"
+ by metis
let ?d = "min d (b - c)"
show ?thesis
- apply (rule that[of "?d"])
- apply safe
- proof -
+ proof (intro that[of "?d"] allI impI)
show "0 < ?d"
- using d(1) assms(3) by auto
+ using \<open>0 < d\<close> \<open>c < b\<close> by auto
fix t :: real
- assume as: "c \<le> t" "t < c + ?d"
+ assume t: "c \<le> t \<and> t < c + ?d"
have *: "integral {a..c} f = integral {a..b} f - integral {c..b} f"
- "integral {a..t} f = integral {a..b} f - integral {t..b} f"
+ "integral {a..t} f = integral {a..b} f - integral {t..b} f"
apply (simp_all only: algebra_simps)
- apply (rule_tac[!] integral_combine)
- using assms as
- apply auto
- done
- have "(- c) - d < (- t) \<and> - t \<le> - c"
- using as by auto note d(2)[rule_format,OF this]
- then show "norm (integral {a..c} f - integral {a..t} f) < e"
- unfolding *
- unfolding integral_reflect
- apply (subst norm_minus_commute)
- apply (auto simp add: algebra_simps)
- done
+ using assms t by (auto simp: integral_combine)
+ have "(- c) - d < (- t)" "- t \<le> - c"
+ using t by auto
+ from d[OF this] show "norm (integral {a..c} f - integral {a..t} f) < e"
+ by (auto simp add: algebra_simps norm_minus_commute *)
qed
qed
@@ -4484,8 +4479,8 @@
lemma has_integral_restrict_open_subinterval:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
- assumes "(f has_integral i) (cbox c d)"
- and "cbox c d \<subseteq> cbox a b"
+ assumes intf: "(f has_integral i) (cbox c d)"
+ and cb: "cbox c d \<subseteq> cbox a b"
shows "((\<lambda>x. if x \<in> box c d then f x else 0) has_integral i) (cbox a b)"
proof -
define g where [abs_def]: "g x = (if x \<in>box c d then f x else 0)" for x
@@ -4507,7 +4502,8 @@
qed
}
assume "cbox c d \<noteq> {}"
- from partial_division_extend_1 [OF assms(2) this] guess p . note p=this
+ then obtain p where p: "p division_of cbox a b" "cbox c d \<in> p"
+ using cb partial_division_extend_1 by blast
interpret operative "lift_option plus" "Some (0 :: 'b)"
"\<lambda>i. if g integrable_on i then Some (integral i g) else None"
by (fact operative_integralI)
@@ -4536,16 +4532,13 @@
then have "x \<in> p"
by auto
note div = division_ofD(2-5)[OF p(1) this]
- from div(3) guess u v by (elim exE) note uv=this
+ then obtain u v where uv: "x = cbox u v" by blast
have "interior x \<inter> interior (cbox c d) = {}"
using div(4)[OF p(2)] x by auto
then have "(g has_integral 0) x"
unfolding uv
- apply -
- apply (rule has_integral_spike_interior[where f="\<lambda>x. 0"])
- unfolding g_def interior_cbox
- apply auto
- done
+ using has_integral_spike_interior[where f="\<lambda>x. 0"]
+ by (metis (no_types, lifting) disjoint_iff_not_equal g_def has_integral_0_eq interior_cbox)
then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
by auto
qed
@@ -4650,11 +4643,13 @@
done
}
assume "\<exists>a b. s = cbox a b"
- then guess a b by (elim exE) note s=this
- from bounded_cbox[of a b, unfolded bounded_pos] guess B ..
- note B = conjunctD2[OF this,rule_format] show ?thesis
- apply safe
- proof -
+ then obtain a b where s: "s = cbox a b"
+ by blast
+ from bounded_cbox[of a b, unfolded bounded_pos]
+ obtain B where " 0 < B" and B: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm x \<le> B"
+ by blast
+ show ?thesis
+ proof safe
fix e :: real
assume ?l and "e > 0"
show "?r e"
@@ -4671,12 +4666,12 @@
apply (rule has_integral_restrict_closed_subinterval)
apply (rule \<open>?l\<close>[unfolded s])
apply safe
- apply (drule B(2)[rule_format])
+ apply (drule B[rule_format])
unfolding subset_eq
apply (erule_tac x=x in ballE)
apply (auto simp add: dist_norm)
done
- qed (insert B \<open>e>0\<close>, auto)
+ qed (insert \<open>B>0\<close> \<open>e>0\<close>, auto)
next
assume as: "\<forall>e>0. ?r e"
from this[rule_format,OF zero_less_one] guess C..note C=conjunctD2[OF this,rule_format]
@@ -4684,7 +4679,7 @@
define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
have c_d: "cbox a b \<subseteq> cbox c d"
apply safe
- apply (drule B(2))
+ apply (drule B)
unfolding mem_box
proof
fix x i
@@ -4720,7 +4715,7 @@
define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
have c_d: "cbox a b \<subseteq> cbox c d"
apply safe
- apply (drule B(2))
+ apply (drule B)
unfolding mem_box
proof
fix x i :: 'n
@@ -5029,75 +5024,55 @@
lemma has_integral_alt':
fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
- shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
- (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
- norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e)"
+ shows "(f has_integral i) s \<longleftrightarrow>
+ (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
+ (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
+ norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e)"
(is "?l = ?r")
proof
- assume ?r
+ assume rhs: ?r
show ?l
- apply (subst has_integral')
- apply safe
- proof goal_cases
- case (1 e)
- from \<open>?r\<close>[THEN conjunct2,rule_format,OF this] guess B..note B=conjunctD2[OF this]
- show ?case
- apply rule
- apply rule
- apply (rule B)
- apply safe
- apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0)" in exI)
- apply (drule B(2)[rule_format])
- using integrable_integral[OF \<open>?r\<close>[THEN conjunct1,rule_format]]
- apply auto
- done
+ proof (subst has_integral', intro allI impI)
+ fix e::real
+ assume "e > 0"
+ from rhs[THEN conjunct2,rule_format,OF this]
+ show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
+ (\<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z)
+ (cbox a b) \<and> norm (z - i) < e)"
+ apply (rule ex_forward)
+ using rhs by blast
qed
next
- assume ?l note as = this[unfolded has_integral'[of f],rule_format]
+ let ?\<Phi> = "\<lambda>e a b. \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - i) < e"
+ assume ?l
+ then have lhs: "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> ?\<Phi> e a b" if "e > 0" for e
+ using that has_integral'[of f] by auto
let ?f = "\<lambda>x. if x \<in> s then f x else 0"
show ?r
- proof safe
+ proof (intro conjI allI impI)
fix a b :: 'n
- from as[OF zero_less_one] guess B..note B=conjunctD2[OF this,rule_format]
+ from lhs[OF zero_less_one]
+ obtain B where "0 < B" and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> ?\<Phi> 1 a b"
+ by blast
let ?a = "\<Sum>i\<in>Basis. min (a\<bullet>i) (-B) *\<^sub>R i::'n"
let ?b = "\<Sum>i\<in>Basis. max (b\<bullet>i) B *\<^sub>R i::'n"
show "?f integrable_on cbox a b"
proof (rule integrable_subinterval[of _ ?a ?b])
- have "ball 0 B \<subseteq> cbox ?a ?b"
- apply (rule subsetI)
- unfolding mem_ball mem_box dist_norm
- proof (rule, goal_cases)
- case (1 x i)
- then show ?case using Basis_le_norm[of i x]
- by (auto simp add:field_simps)
- qed
- from B(2)[OF this] guess z..note conjunct1[OF this]
+ have "?a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> ?b \<bullet> i" if "norm (0 - x) < B" "i \<in> Basis" for x i
+ using Basis_le_norm[of i x] that by (auto simp add:field_simps)
+ then have "ball 0 B \<subseteq> cbox ?a ?b"
+ by (auto simp: mem_box dist_norm)
then show "?f integrable_on cbox ?a ?b"
- unfolding integrable_on_def by auto
+ unfolding integrable_on_def using B by blast
show "cbox a b \<subseteq> cbox ?a ?b"
- apply safe
- unfolding mem_box
- apply rule
- apply (erule_tac x=i in ballE)
- apply auto
- done
+ by (force simp: mem_box)
qed
-
+
fix e :: real
assume "e > 0"
- from as[OF this] guess B..note B=conjunctD2[OF this,rule_format]
- show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
+ with lhs show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
- apply rule
- apply rule
- apply (rule B)
- apply safe
- proof goal_cases
- case 1
- from B(2)[OF this] guess z..note z=conjunctD2[OF this]
- from integral_unique[OF this(1)] show ?case
- using z(2) by auto
- qed
+ by (metis (no_types, lifting) has_integral_integrable_integral)
qed
qed
@@ -5695,8 +5670,7 @@
and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
shows "norm (sum (\<lambda>(x,K). content K *\<^sub>R f x - integral K f) p) \<le> e"
(is "?x \<le> e")
-proof -
- { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" then show ?thesis by (blast intro: field_le_epsilon) }
+proof (rule field_le_epsilon)
fix k :: real
assume k: "k > 0"
note p' = tagged_partial_division_ofD[OF p(1)]
@@ -6365,7 +6339,7 @@
and le_g: "\<forall>x\<in>S. norm (f x) \<le> g x"
shows "norm (integral S f) \<le> integral S g"
proof -
- have norm: "norm \<eta> < y + e"
+ have norm: "norm \<eta> \<le> y + e"
if "norm \<zeta> \<le> x" and "\<bar>x - y\<bar> < e/2" and "norm (\<zeta> - \<eta>) < e/2"
for e x y and \<zeta> \<eta> :: 'a
proof -
@@ -6374,14 +6348,14 @@
moreover have "x \<le> y + e/2"
using that(2) by linarith
ultimately show ?thesis
- using that(1) le_less_trans[OF norm_triangle_sub[of \<eta> \<zeta>]] by auto
+ using that(1) le_less_trans[OF norm_triangle_sub[of \<eta> \<zeta>]] by (auto simp: less_imp_le)
qed
have lem: "norm (integral(cbox a b) f) \<le> integral (cbox a b) g"
if f: "f integrable_on cbox a b"
and g: "g integrable_on cbox a b"
and nle: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x"
for f :: "'n \<Rightarrow> 'a" and g a b
- proof (rule eps_leI)
+ proof (rule field_le_epsilon)
fix e :: real
assume "e > 0"
then have e: "e/2 > 0"
@@ -6404,7 +6378,7 @@
by metis
have "\<gamma> fine \<D>" "\<delta> fine \<D>"
using fine_Int p(2) by blast+
- show "norm (integral (cbox a b) f) < integral (cbox a b) g + e"
+ show "norm (integral (cbox a b) f) \<le> integral (cbox a b) g + e"
proof (rule norm)
have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if "(x, K) \<in> \<D>" for x K
proof-
@@ -6426,7 +6400,7 @@
qed
qed
show ?thesis
- proof (rule eps_leI)
+ proof (rule field_le_epsilon)
fix e :: real
assume "e > 0"
then have e: "e/2 > 0"
@@ -6453,7 +6427,7 @@
using ab by auto
with Bg obtain w where int_gw: "(?g has_integral w) (cbox a b)" and w: "norm (w - integral S g) < e/2"
by meson
- show "norm (integral S f) < integral S g + e"
+ show "norm (integral S f) \<le> integral S g + e"
proof (rule norm)
show "norm (integral (cbox a b) ?f) \<le> integral (cbox a b) ?g"
by (simp add: le_g lem[OF f g, of a b])