--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Thu Aug 03 23:03:44 2017 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Thu Aug 03 23:06:36 2017 +0200
@@ -7,6 +7,24 @@
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
begin
+lemma finite_product_dependent: (*FIXME DELETE*)
+ assumes "finite s"
+ and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
+ shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
+ using assms
+proof induct
+ case (insert x s)
+ have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
+ (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
+ show ?case
+ unfolding *
+ apply (rule finite_UnI)
+ using insert
+ apply auto
+ done
+qed auto
+
+
lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
by (auto intro: order_trans)
@@ -1633,11 +1651,16 @@
then show "\<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
by force
qed
- from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
-
+ then obtain k where k: "\<And>x. 0 < k x"
+ "\<And>i x. \<lbrakk>i \<in> d; x \<notin> i\<rbrakk> \<Longrightarrow> ball x (k x) \<inter> i = {}"
+ by metis
have "e/2 > 0"
using e by auto
- from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
+ from henstock_lemma[OF assms(1) this]
+ obtain g where g: "gauge g"
+ "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; g fine p\<rbrakk>
+ \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
+ by (metis (no_types, lifting))
let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
show ?case
apply (rule_tac x="?g" in exI)
@@ -1716,15 +1739,12 @@
assume y: "y \<in> cbox a b"
then have "\<exists>x l. (x, l) \<in> p \<and> y\<in>l"
unfolding p'(6)[symmetric] by auto
- then guess x l by (elim exE) note xl=conjunctD2[OF this]
+ then obtain x l where xl: "(x, l) \<in> p" "y \<in> l" by metis
then have "\<exists>k. k \<in> d \<and> y \<in> k"
using y unfolding d'(6)[symmetric] by auto
- then guess i .. note i = conjunctD2[OF this]
+ then obtain i where i: "i \<in> d" "y \<in> i" by metis
have "x \<in> i"
- using fineD[OF p(3) xl(1)]
- using k(2)[OF i(1), of x]
- using i(2) xl(2)
- by auto
+ using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
unfolding p'_def Union_iff
apply (rule_tac x="i \<inter> l" in bexI)
@@ -1735,12 +1755,7 @@
qed
then have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
- apply -
- apply (rule g(2)[rule_format])
- unfolding tagged_division_of_def
- apply safe
- apply (rule gp')
- done
+ using g(2) gp' tagged_division_of_def by blast
then have **: "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
unfolding split_def
using p''
--- a/src/HOL/Analysis/Tagged_Division.thy Thu Aug 03 23:03:44 2017 +0200
+++ b/src/HOL/Analysis/Tagged_Division.thy Thu Aug 03 23:06:36 2017 +0200
@@ -10,23 +10,6 @@
Topology_Euclidean_Space
begin
-lemma finite_product_dependent: (*FIXME DELETE*)
- assumes "finite s"
- and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
- shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
- using assms
-proof induct
- case (insert x s)
- have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
- (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
- show ?case
- unfolding *
- apply (rule finite_UnI)
- using insert
- apply auto
- done
-qed auto
-
lemma sum_Sigma_product:
assumes "finite S"
and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)"
@@ -210,18 +193,18 @@
subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
-definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
+definition "gauge \<gamma> \<longleftrightarrow> (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))"
lemma gaugeI:
- assumes "\<And>x. x \<in> g x"
- and "\<And>x. open (g x)"
- shows "gauge g"
+ assumes "\<And>x. x \<in> \<gamma> x"
+ and "\<And>x. open (\<gamma> x)"
+ shows "gauge \<gamma>"
using assms unfolding gauge_def by auto
lemma gaugeD[dest]:
- assumes "gauge d"
- shows "x \<in> d x"
- and "open (d x)"
+ assumes "gauge \<gamma>"
+ shows "x \<in> \<gamma> x"
+ and "open (\<gamma> x)"
using assms unfolding gauge_def by auto
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
@@ -233,7 +216,7 @@
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
by (rule gauge_ball) auto
-lemma gauge_Int[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
+lemma gauge_Int[intro]: "gauge \<gamma>1 \<Longrightarrow> gauge \<gamma>2 \<Longrightarrow> gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
unfolding gauge_def by auto
lemma gauge_reflect:
@@ -244,10 +227,10 @@
lemma gauge_Inter:
assumes "finite S"
- and "\<And>d. d\<in>S \<Longrightarrow> gauge (f d)"
- shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> S})"
+ and "\<And>u. u\<in>S \<Longrightarrow> gauge (f u)"
+ shows "gauge (\<lambda>x. \<Inter>{f u x | u. u \<in> S})"
proof -
- have *: "\<And>x. {f d x |d. d \<in> S} = (\<lambda>d. f d x) ` S"
+ have *: "\<And>x. {f u x |u. u \<in> S} = (\<lambda>u. f u x) ` S"
by auto
show ?thesis
unfolding gauge_def unfolding *
@@ -314,18 +297,15 @@
assume "s = {{a}}" "K\<in>s"
then have "\<exists>x y. K = cbox x y"
apply (rule_tac x=a in exI)+
- apply (force simp: cbox_sing)
+ apply force
done
}
ultimately show ?l
unfolding division_of_def cbox_sing by auto
next
assume ?l
- {
- fix x
- assume x: "x \<in> s" have "x = {a}"
- by (metis \<open>s division_of cbox a a\<close> cbox_sing division_ofD(2) division_ofD(3) subset_singletonD x)
- }
+ have "x = {a}" if "x \<in> s" for x
+ by (metis \<open>s division_of cbox a a\<close> cbox_sing division_ofD(2) division_ofD(3) subset_singletonD that)
moreover have "s \<noteq> {}"
using \<open>s division_of cbox a a\<close> by auto
ultimately show ?r
@@ -401,11 +381,8 @@
have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
by auto
show "\<Union>?A = s1 \<inter> s2"
- apply (rule set_eqI)
- unfolding * and UN_iff
- using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
- apply auto
- done
+ unfolding *
+ using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
{
fix k
assume "k \<in> ?A"
@@ -461,14 +438,14 @@
unfolding * by auto
qed
-lemma elementary_inter:
+lemma elementary_Int:
fixes s t :: "'a::euclidean_space set"
assumes "p1 division_of s"
and "p2 division_of t"
shows "\<exists>p. p division_of (s \<inter> t)"
using assms division_inter by blast
-lemma elementary_inters:
+lemma elementary_Inter:
assumes "finite f"
and "f \<noteq> {}"
and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
@@ -488,7 +465,7 @@
moreover obtain px where "px division_of x"
using insert(5)[rule_format,OF insertI1] ..
ultimately show ?thesis
- by (simp add: elementary_inter Inter_insert)
+ by (simp add: elementary_Int Inter_insert)
qed
qed auto
@@ -548,13 +525,9 @@
show "cbox c d \<in> p"
unfolding p_def
by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
- {
- fix i :: 'a
- assume "i \<in> Basis"
- with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
- unfolding box_eq_empty subset_box by (auto simp: not_le)
- }
- note ord = this
+ have ord: "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i" if "i \<in> Basis" for i
+ using incl nonempty that
+ unfolding box_eq_empty subset_box by (auto simp: not_le)
show "p division_of (cbox a b)"
proof (rule division_ofI)
@@ -665,8 +638,8 @@
then have di: "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
by (meson Diff_subset division_of_subset)
have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
- apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
- apply (auto dest: di simp: False elementary_inters [OF finite_imageI[OF p(1)]])
+ apply (rule elementary_Inter [OF finite_imageI[OF p(1)]])
+ apply (auto dest: di simp: False elementary_Inter [OF finite_imageI[OF p(1)]])
done
then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
have "d \<union> p division_of cbox a b"
@@ -719,12 +692,7 @@
obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
proof (cases "cbox c d = {}")
case True
- show ?thesis
- apply (rule that[of "{}"])
- unfolding True
- using assms
- apply auto
- done
+ with assms that show ?thesis by force
next
case False
show ?thesis
@@ -812,14 +780,13 @@
by auto
show "finite ?D"
using "*" pdiv(1) q(1) by auto
- have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
- by auto
- have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
+ have "\<Union>?D = (\<Union>x\<in>p. \<Union>insert (cbox a b) (q x))"
by auto
- show "\<Union>?D = cbox a b \<union> \<Union>p"
- unfolding * lem1
- unfolding lem2[OF \<open>p \<noteq> {}\<close>, of "cbox a b", symmetric]
- using q(6) by auto
+ also have "... = \<Union>{cbox a b \<union> t |t. t \<in> p}"
+ using q(6) by auto
+ also have "... = cbox a b \<union> \<Union>p"
+ using \<open>p \<noteq> {}\<close> by auto
+ finally show "\<Union>?D = cbox a b \<union> \<Union>p" .
show "K \<subseteq> cbox a b \<union> \<Union>p" "K \<noteq> {}" if "K \<in> ?D" for K
using q that by blast+
show "\<exists>a b. K = cbox a b" if "K \<in> ?D" for K
@@ -920,7 +887,7 @@
apply auto
done
then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
- by (metis assms(2) divq(6) elementary_inter)
+ by (metis assms(2) divq(6) elementary_Int)
{
fix x
assume x: "x \<in> T" "x \<notin> S"
@@ -1130,9 +1097,8 @@
assumes "s tagged_partial_division_of i"
and "t \<subseteq> s"
shows "t tagged_partial_division_of i"
- using assms
+ using assms finite_subset[OF assms(2)]
unfolding tagged_partial_division_of_def
- using finite_subset[OF assms(2)]
by blast
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
@@ -1183,28 +1149,28 @@
qed
lemma tagged_division_unions:
- assumes "finite iset"
- and "\<forall>i\<in>iset. pfn i tagged_division_of i"
- and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
- shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
+ assumes "finite I"
+ and "\<forall>i\<in>I. pfn i tagged_division_of i"
+ and "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
+ shows "\<Union>(pfn ` I) tagged_division_of (\<Union>I)"
proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF assms(2)[rule_format]]
- show "finite (\<Union>(pfn ` iset))"
+ show "finite (\<Union>(pfn ` I))"
using assms by auto
- have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
+ have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` I)"
by blast
- also have "\<dots> = \<Union>iset"
+ also have "\<dots> = \<Union>I"
using assm(6) by auto
- finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
+ finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>I" .
fix x k
- assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
- then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
+ assume xk: "(x, k) \<in> \<Union>(pfn ` I)"
+ then obtain i where i: "i \<in> I" "(x, k) \<in> pfn i"
by auto
- show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
+ show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>I"
using assm(2-4)[OF i] using i(1) by auto
fix x' k'
- assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
- then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
+ assume xk': "(x', k') \<in> \<Union>(pfn ` I)" "(x, k) \<noteq> (x', k')"
+ then obtain i' where i': "i' \<in> I" "(x', k') \<in> pfn i'"
by auto
have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
using i(1) i'(1)
@@ -1379,8 +1345,8 @@
lemma division_points_psubset:
fixes a :: "'a::euclidean_space"
- assumes "d division_of (cbox a b)"
- and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k"
+ assumes d: "d division_of (cbox a b)"
+ and altb: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k"
and "l \<in> d"
and disj: "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c"
and k: "k \<in> Basis"
@@ -1390,14 +1356,12 @@
division_points (cbox a b) d" (is "?D2 \<subset> ?D")
proof -
have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
- using assms(2) by (auto intro!:less_imp_le)
- guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
+ using altb by (auto intro!:less_imp_le)
+ obtain u v where l: "l = cbox u v"
+ using d \<open>l \<in> d\<close> by blast
have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i"
- using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
- using subset_box(1)
- apply auto
- apply blast+
- done
+ apply (metis assms(5) box_ne_empty(1) cbox_division_memE d l)
+ by (metis assms(5) box_ne_empty(1) cbox_division_memE d l subset_box(1))
have *: "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
"interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
@@ -1406,11 +1370,7 @@
have "\<exists>x. x \<in> ?D - ?D1"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
- apply -
- apply (erule disjE)
- apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
- apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
- done
+ by (force simp add: *)
moreover have "?D1 \<subseteq> ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D1 \<subset> ?D"
@@ -1423,11 +1383,7 @@
have "\<exists>x. x \<in> ?D - ?D2"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
- apply -
- apply (erule disjE)
- apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
- apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
- done
+ by (force simp add: *)
moreover have "?D2 \<subseteq> ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D2 \<subset> ?D"
@@ -1769,39 +1725,28 @@
done
next
fix a b c :: real
- assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
- and "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
+ assume eq1: "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
+ and eqg: "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
and "a \<le> c"
and "c \<le> b"
- note as = this[rule_format]
show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
proof (cases "c = a \<or> c = b")
case False
then show ?thesis
- apply -
- apply (subst as(2))
- using as(3-)
- apply auto
- done
+ using eqg \<open>a \<le> c\<close> \<open>c \<le> b\<close> by auto
next
case True
then show ?thesis
proof
assume *: "c = a"
then have "g {a .. c} = \<^bold>1"
- apply -
- apply (rule as(1)[rule_format])
- apply auto
- done
+ using eq1 by blast
then show ?thesis
unfolding * by auto
next
assume *: "c = b"
then have "g {c .. b} = \<^bold>1"
- apply -
- apply (rule as(1)[rule_format])
- apply auto
- done
+ using eq1 by blast
then show ?thesis
unfolding * by auto
qed
@@ -1934,18 +1879,18 @@
subsection \<open>Some basic combining lemmas.\<close>
lemma tagged_division_Union_exists:
- assumes "finite iset"
- and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
- and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
- and "\<Union>iset = i"
+ assumes "finite I"
+ and "\<forall>i\<in>I. \<exists>p. p tagged_division_of i \<and> d fine p"
+ and "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
+ and "\<Union>I = i"
obtains p where "p tagged_division_of i" and "d fine p"
proof -
obtain pfn where pfn:
- "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
- "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
+ "\<And>x. x \<in> I \<Longrightarrow> pfn x tagged_division_of x"
+ "\<And>x. x \<in> I \<Longrightarrow> d fine pfn x"
using bchoice[OF assms(2)] by auto
show thesis
- apply (rule_tac p="\<Union>(pfn ` iset)" in that)
+ apply (rule_tac p="\<Union>(pfn ` I)" in that)
using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
by (metis (mono_tags, lifting) fine_Union imageE pfn(2))
qed
@@ -1972,25 +1917,22 @@
using assms(1,3) by metis
then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
by (force simp: mem_box)
- { fix f
- have "\<lbrakk>finite f;
+ have UN_cases: "\<lbrakk>finite f;
\<And>s. s\<in>f \<Longrightarrow> P s;
\<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
- \<And>s t. s\<in>f \<Longrightarrow> t\<in>f \<Longrightarrow> s \<noteq> t \<Longrightarrow> interior s \<inter> interior t = {}\<rbrakk> \<Longrightarrow> P (\<Union>f)"
- proof (induct f rule: finite_induct)
- case empty
- show ?case
- using assms(1) by auto
- next
- case (insert x f)
- show ?case
- unfolding Union_insert
- apply (rule assms(2)[rule_format])
- using Int_interior_Union_intervals [of f "interior x"]
- apply (auto simp: insert)
- by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
- qed
- } note UN_cases = this
+ \<And>s t. s\<in>f \<Longrightarrow> t\<in>f \<Longrightarrow> s \<noteq> t \<Longrightarrow> interior s \<inter> interior t = {}\<rbrakk> \<Longrightarrow> P (\<Union>f)" for f
+ proof (induct f rule: finite_induct)
+ case empty
+ show ?case
+ using assms(1) by auto
+ next
+ case (insert x f)
+ show ?case
+ unfolding Union_insert
+ apply (rule assms(2)[rule_format])
+ using Int_interior_Union_intervals [of f "interior x"]
+ by (metis (no_types, lifting) insert insert_iff open_interior)
+ qed
let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
(c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
@@ -2110,13 +2052,12 @@
then show "\<exists>c d. ?P i c d"
by blast
qed
+ then obtain \<alpha> \<beta> where
+ "\<forall>i\<in>Basis. (\<alpha> \<bullet> i = a \<bullet> i \<and> \<beta> \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+ \<alpha> \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> \<beta> \<bullet> i = b \<bullet> i) \<and> \<alpha> \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> \<beta> \<bullet> i"
+ by (auto simp: choice_Basis_iff)
then show "x\<in>\<Union>?A"
- unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
- apply auto
- apply (rule_tac x="cbox xa xaa" in exI)
- unfolding mem_box
- apply auto
- done
+ by (force simp add: mem_box)
qed
finally show False
using assms by auto
@@ -2148,10 +2089,7 @@
2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
by (rule interval_bisection_step[of P, OF assms(1-2) as])
then show ?thesis
- apply -
- apply (rule_tac x="(c,d)" in exI)
- apply auto
- done
+ by (rule_tac x="(c,d)" in exI) auto
qed
qed
then obtain f where f:
@@ -2162,11 +2100,7 @@
fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
- 2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
- apply -
- apply (drule choice)
- apply blast
- done
+ 2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)" by metis
define AB A B where ab_def: "AB n = (f ^^ n) (a,b)" "A n = fst(AB n)" "B n = snd(AB n)" for n
have [simp]: "A 0 = a" "B 0 = b" and ABRAW: "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
(\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and>
@@ -2179,10 +2113,7 @@
proof (induct n)
case 0
then show ?case
- unfolding S
- apply (rule f[rule_format]) using assms(3)
- apply auto
- done
+ unfolding S using \<open>\<not> P (cbox a b)\<close> f by auto
next
case (Suc n)
show ?case
@@ -2230,8 +2161,7 @@
next
case (Suc n)
have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
- using AB(3) that
- using AB(4)[of i n] using i by auto
+ using AB(3) that AB(4)[of i n] using i by auto
also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
using Suc by (auto simp add: field_simps)
finally show ?case .
@@ -2295,13 +2225,13 @@
fixes a b :: "'a::euclidean_space"
assumes "gauge g"
obtains p where "p tagged_division_of (cbox a b)" "g fine p"
-proof -
- presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
- then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
- by blast
- then show thesis ..
+proof (cases "\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p")
+ case True
+ then show ?thesis
+ using that by auto
next
- assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
+ case False
+ assume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
obtain x where x:
"x \<in> (cbox a b)"
"\<And>e. 0 < e \<Longrightarrow>
@@ -2310,10 +2240,10 @@
cbox c d \<subseteq> ball x e \<and>
cbox c d \<subseteq> (cbox a b) \<and>
\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
- apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
+ apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ False])
apply (simp add: fine_def)
apply (metis tagged_division_union fine_Un)
- apply (auto simp: )
+ apply auto
done
obtain e where e: "e > 0" "ball x e \<subseteq> g x"
using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
@@ -2325,7 +2255,7 @@
by blast
have "g fine {(x, cbox c d)}"
unfolding fine_def using e using c_d(2) by auto
- then show False
+ then show ?thesis
using tagged_division_of_self[OF c_d(1)] using c_d by auto
qed
@@ -2369,70 +2299,52 @@
proof (induct p)
case empty
show ?case
- apply (rule_tac x="{}" in exI)
- unfolding fine_def
- apply auto
- done
+ by (force simp add: fine_def)
next
case (insert xk p)
- guess x k using surj_pair[of xk] by (elim exE) note xk=this
- note tagged_partial_division_subset[OF insert(4) subset_insertI]
- from insert(3)[OF this insert(5)]
+ obtain x k where xk: "xk = (x, k)"
+ using surj_pair[of xk] by metis
obtain q1 where q1: "q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> p}"
and "d fine q1"
and q1I: "\<And>x k. \<lbrakk>(x, k)\<in>p; k \<subseteq> d x\<rbrakk> \<Longrightarrow> (x, k) \<in> q1"
- by (force simp add: )
+ using case_prodD tagged_partial_division_subset[OF insert(4) subset_insertI]
+ by (metis (mono_tags, lifting) insert.hyps(3) insert.prems(2))
have *: "\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}"
unfolding xk by auto
note p = tagged_partial_division_ofD[OF insert(4)]
- from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
-
+ obtain u v where uv: "k = cbox u v"
+ using p(4)[unfolded xk, OF insertI1] by blast
have "finite {k. \<exists>x. (x, k) \<in> p}"
apply (rule finite_subset[of _ "snd ` p"])
- using p
- apply safe
- apply (metis image_iff snd_conv)
- apply auto
- done
+ using image_iff apply fastforce
+ using insert.hyps(1) by blast
then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
- apply (rule Int_interior_Union_intervals)
- apply (rule open_interior)
- unfolding mem_Collect_eq
- apply (erule_tac[!] exE)
- apply (drule p(4)[OF insertI2])
- apply assumption
- apply (rule p(5))
- unfolding uv xk
- apply (rule insertI1)
- apply (rule insertI2)
- apply assumption
- using insert(2)
- unfolding uv xk
- apply auto
- done
+ proof (rule Int_interior_Union_intervals)
+ show "open (interior (cbox u v))"
+ by auto
+ show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> \<exists>a b. T = cbox a b"
+ using p(4) by auto
+ show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> interior (cbox u v) \<inter> interior T = {}"
+ by clarify (metis insert.hyps(2) insert_iff interior_cbox p(5) uv xk)
+ qed
show ?case
proof (cases "cbox u v \<subseteq> d x")
case True
- then show ?thesis
+ have "{(x, cbox u v)} tagged_division_of cbox u v"
+ by (simp add: p(2) uv xk tagged_division_of_self)
+ then have "{(x, cbox u v)} \<union> q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> insert xk p}"
+ unfolding * uv by (metis (no_types, lifting) int q1 tagged_division_union)
+ with True show ?thesis
apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
- apply rule
- unfolding * uv
- apply (rule tagged_division_union)
- apply (rule tagged_division_of_self)
- apply (rule p[unfolded xk uv] insertI1)+
- apply (rule q1)
- apply (rule int)
- apply rule
- apply (rule fine_Un)
- apply (subst fine_def)
- apply (auto simp add: \<open>d fine q1\<close> q1I uv xk)
+ using \<open>d fine q1\<close> fine_def q1I uv xk apply fastforce
done
next
case False
- from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
+ obtain q2 where q2: "q2 tagged_division_of cbox u v" "d fine q2"
+ using fine_division_exists[OF assms(2)] by blast
show ?thesis
apply (rule_tac x="q2 \<union> q1" in exI)
- apply rule
+ apply (intro conjI)
unfolding * uv
apply (rule tagged_division_union q2 q1 int fine_Un)+
apply (auto intro: q1 q2 fine_Un \<open>d fine q1\<close> simp add: False q1I uv xk)