--- a/src/HOL/Multivariate_Analysis/Integration.thy Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Fri Sep 06 10:57:27 2013 +0200
@@ -1,6 +1,8 @@
+(* Author: John Harrison
+ Author: Robert Himmelmann, TU Muenchen (Translation from HOL light)
+*)
+
header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
-(* Author: John Harrison
- Translation from HOL light: Robert Himmelmann, TU Muenchen *)
theory Integration
imports
@@ -11,62 +13,76 @@
lemma cSup_abs_le: (* TODO: is this really needed? *)
fixes S :: "real set"
shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
-by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2)
+ by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2)
lemma cInf_abs_ge: (* TODO: is this really needed? *)
fixes S :: "real set"
shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
-by (simp add: Inf_real_def) (rule cSup_abs_le, auto)
+ by (simp add: Inf_real_def) (rule cSup_abs_le, auto)
lemma cSup_asclose: (* TODO: is this really needed? *)
fixes S :: "real set"
- assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Sup S - l\<bar> \<le> e"
-proof-
- have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
- thus ?thesis using S b cSup_bounds[of S "l - e" "l+e"] unfolding th
- by (auto simp add: setge_def setle_def)
+ assumes S: "S \<noteq> {}"
+ and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
+ shows "\<bar>Sup S - l\<bar> \<le> e"
+proof -
+ have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
+ by arith
+ then show ?thesis
+ using S b cSup_bounds[of S "l - e" "l+e"]
+ unfolding th
+ by (auto simp add: setge_def setle_def)
qed
lemma cInf_asclose: (* TODO: is this really needed? *)
fixes S :: "real set"
- assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>Inf S - l\<bar> \<le> e"
+ assumes S: "S \<noteq> {}"
+ and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
+ shows "\<bar>Inf S - l\<bar> \<le> e"
proof -
have "\<bar>- Sup (uminus ` S) - l\<bar> = \<bar>Sup (uminus ` S) - (-l)\<bar>"
by auto
- also have "... \<le> e"
- apply (rule cSup_asclose)
+ also have "\<dots> \<le> e"
+ apply (rule cSup_asclose)
apply (auto simp add: S)
apply (metis abs_minus_add_cancel b add_commute diff_minus)
done
finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
- thus ?thesis
+ then show ?thesis
by (simp add: Inf_real_def)
qed
-lemma cSup_finite_ge_iff:
- fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
+lemma cSup_finite_ge_iff:
+ fixes S :: "real set"
+ shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
by (metis cSup_eq_Max Max_ge_iff)
-lemma cSup_finite_le_iff:
- fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
+lemma cSup_finite_le_iff:
+ fixes S :: "real set"
+ shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
by (metis cSup_eq_Max Max_le_iff)
-lemma cInf_finite_ge_iff:
- fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
+lemma cInf_finite_ge_iff:
+ fixes S :: "real set"
+ shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
by (metis cInf_eq_Min Min_ge_iff)
-lemma cInf_finite_le_iff:
- fixes S :: "real set" shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
+lemma cInf_finite_le_iff:
+ fixes S :: "real set"
+ shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
by (metis cInf_eq_Min Min_le_iff)
lemma Inf: (* rename *)
fixes S :: "real set"
- shows "S \<noteq> {} ==> (\<exists>b. b <=* S) ==> isGlb UNIV S (Inf S)"
-by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def intro: cInf_lower cInf_greatest)
-
+ shows "S \<noteq> {} \<Longrightarrow> (\<exists>b. b <=* S) \<Longrightarrow> isGlb UNIV S (Inf S)"
+ by (auto simp add: isLb_def setle_def setge_def isGlb_def greatestP_def
+ intro: cInf_lower cInf_greatest)
+
lemma real_le_inf_subset:
- assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s"
- shows "Inf s <= Inf (t::real set)"
+ assumes "t \<noteq> {}"
+ and "t \<subseteq> s"
+ and "\<exists>b. b <=* s"
+ shows "Inf s \<le> Inf (t::real set)"
apply (rule isGlb_le_isLb)
apply (rule Inf[OF assms(1)])
apply (insert assms)
@@ -76,8 +92,11 @@
done
lemma real_ge_sup_subset:
- assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b"
- shows "Sup s >= Sup (t::real set)"
+ fixes t :: "real set"
+ assumes "t \<noteq> {}"
+ and "t \<subseteq> s"
+ and "\<exists>b. s *<= b"
+ shows "Sup s \<ge> Sup t"
apply (rule isLub_le_isUb)
apply (rule isLub_cSup[OF assms(1)])
apply (insert assms)
@@ -104,9 +123,10 @@
lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
-declare norm_triangle_ineq4[intro]
-
-lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast
+declare norm_triangle_ineq4[intro]
+
+lemma simple_image: "{f x |x . x \<in> s} = f ` s"
+ by blast
lemma linear_simps:
assumes "bounded_linear f"
@@ -123,24 +143,30 @@
lemma bounded_linearI:
assumes "\<And>x y. f (x + y) = f x + f y"
- and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K"
+ and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
+ and "\<And>x. norm (f x) \<le> norm x * K"
shows "bounded_linear f"
- unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto
+ unfolding bounded_linear_def additive_def bounded_linear_axioms_def
+ using assms by auto
lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \<bullet> k)"
by (rule bounded_linear_inner_left)
lemma transitive_stepwise_lt_eq:
assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
- shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r")
-proof (safe)
+ shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))"
+ (is "?l = ?r")
+proof safe
assume ?r
fix n m :: nat
assume "m < n"
then show "R m n"
proof (induct n arbitrary: m)
+ case 0
+ then show ?case by auto
+ next
case (Suc n)
- show ?case
+ show ?case
proof (cases "m < n")
case True
show ?thesis
@@ -150,14 +176,16 @@
done
next
case False
- then have "m = n" using Suc(2) by auto
- then show ?thesis using `?r` by auto
+ then have "m = n"
+ using Suc(2) by auto
+ then show ?thesis
+ using `?r` by auto
qed
- qed auto
+ qed
qed auto
lemma transitive_stepwise_gt:
- assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
+ assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n)"
shows "\<forall>n>m. R m n"
proof -
have "\<forall>m. \<forall>n>m. R m n"
@@ -172,12 +200,13 @@
lemma transitive_stepwise_le_eq:
assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
- shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r")
+ shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))"
+ (is "?l = ?r")
proof safe
assume ?r
fix m n :: nat
assume "m \<le> n"
- thus "R m n"
+ then show "R m n"
proof (induct n arbitrary: m)
case 0
with assms show ?case by auto
@@ -193,21 +222,25 @@
done
next
case False
- hence "m = Suc n" using Suc(2) by auto
- thus ?thesis using assms(1) by auto
+ then have "m = Suc n"
+ using Suc(2) by auto
+ then show ?thesis
+ using assms(1) by auto
qed
qed
qed auto
lemma transitive_stepwise_le:
- assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
+ assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
+ and "\<And>n. R n (Suc n)"
shows "\<forall>n\<ge>m. R m n"
proof -
have "\<forall>m. \<forall>n\<ge>m. R m n"
apply (subst transitive_stepwise_le_eq)
apply (rule assms)
apply (rule assms,assumption,assumption)
- using assms(3) apply auto
+ using assms(3)
+ apply auto
done
then show ?thesis by auto
qed
@@ -215,14 +248,18 @@
subsection {* Some useful lemmas about intervals. *}
-abbreviation One where "One \<equiv> ((\<Sum>Basis)::_::euclidean_space)"
+abbreviation One :: "'a::euclidean_space"
+ where "One \<equiv> \<Sum>Basis"
lemma empty_as_interval: "{} = {One..(0::'a::ordered_euclidean_space)}"
by (auto simp: set_eq_iff eucl_le[where 'a='a] intro!: bexI[OF _ SOME_Basis])
-lemma interior_subset_union_intervals:
- assumes "i = {a..b::'a::ordered_euclidean_space}" "j = {c..d}"
- "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
+lemma interior_subset_union_intervals:
+ assumes "i = {a..b::'a::ordered_euclidean_space}"
+ and "j = {c..d}"
+ and "interior j \<noteq> {}"
+ and "i \<subseteq> j \<union> s"
+ and "interior i \<inter> interior j = {}"
shows "interior i \<subseteq> interior s"
proof -
have "{a<..<b} \<inter> {c..d} = {}"
@@ -247,9 +284,12 @@
lemma inter_interior_unions_intervals:
fixes f::"('a::ordered_euclidean_space) set set"
- assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
- shows "s \<inter> interior(\<Union>f) = {}"
-proof (rule ccontr, unfold ex_in_conv[THEN sym])
+ assumes "finite f"
+ and "open s"
+ and "\<forall>t\<in>f. \<exists>a b. t = {a..b}"
+ and "\<forall>t\<in>f. s \<inter> (interior t) = {}"
+ shows "s \<inter> interior (\<Union>f) = {}"
+proof (rule ccontr, unfold ex_in_conv[symmetric])
case goal1
have lem1: "\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U"
apply rule
@@ -260,42 +300,53 @@
apply auto
done
have lem2: "\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
- have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow>
- (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)"
+ have "\<And>f. finite f \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = {a..b} \<Longrightarrow>
+ \<exists>x. x \<in> s \<inter> interior (\<Union>f) \<Longrightarrow> \<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t"
proof -
case goal1
then show ?case
proof (induct rule: finite_induct)
- case empty from this(2) guess x ..
- hence False unfolding Union_empty interior_empty by auto
- thus ?case by auto
+ case empty
+ obtain x where "x \<in> s \<inter> interior (\<Union>{})"
+ using empty(2) ..
+ then have False
+ unfolding Union_empty interior_empty by auto
+ then show ?case by auto
next
- case (insert i f) guess x using insert(5) .. note x = this
- then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
- guess a using insert(4)[rule_format,OF insertI1] ..
- then guess b .. note ab = this
+ case (insert i f)
+ obtain x where x: "x \<in> s \<inter> interior (\<Union>insert i f)"
+ using insert(5) ..
+ then obtain e where e: "0 < e \<and> ball x e \<subseteq> s \<inter> interior (\<Union>insert i f)"
+ unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
+ obtain a where "\<exists>b. i = {a..b}"
+ using insert(4)[rule_format,OF insertI1] ..
+ then obtain b where ab: "i = {a..b}" ..
show ?case
- proof (cases "x\<in>i")
+ proof (cases "x \<in> i")
case False
- hence "x \<in> UNIV - {a..b}" unfolding ab by auto
- then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
- hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" unfolding ab ball_min_Int by auto
- hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
+ then have "x \<in> UNIV - {a..b}"
+ unfolding ab by auto
+ then obtain d where "0 < d \<and> ball x d \<subseteq> UNIV - {a..b}"
+ unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
+ then have "0 < d" "ball x (min d e) \<subseteq> UNIV - i"
+ unfolding ab ball_min_Int by auto
+ then have "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
using e unfolding lem1 unfolding ball_min_Int by auto
- hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
- hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t"
+ then have "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
+ then have "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t"
apply -
apply (rule insert(3))
using insert(4)
apply auto
done
- thus ?thesis by auto
+ then show ?thesis by auto
next
case True show ?thesis
proof (cases "x\<in>{a<..<b}")
case True
- then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
- thus ?thesis
+ then obtain d where "0 < d \<and> ball x d \<subseteq> {a<..<b}"
+ unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
+ then show ?thesis
apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
unfolding ab
using interval_open_subset_closed[of a b] and e
@@ -303,38 +354,40 @@
done
next
case False
- then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k:"k\<in>Basis"
+ then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k: "k \<in> Basis"
unfolding mem_interval by (auto simp add: not_less)
- hence "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
+ then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
using True unfolding ab and mem_interval
apply (erule_tac x = k in ballE)
apply auto
done
- hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
- proof (erule_tac disjE)
+ then have "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
+ proof (rule disjE)
let ?z = "x - (e/2) *\<^sub>R k"
assume as: "x\<bullet>k = a\<bullet>k"
have "ball ?z (e / 2) \<inter> i = {}"
apply (rule ccontr)
- unfolding ex_in_conv[THEN sym]
- proof (erule exE)
+ unfolding ex_in_conv[symmetric]
+ apply (erule exE)
+ proof -
fix y
assume "y \<in> ball ?z (e / 2) \<inter> i"
- hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
- hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+ then have "dist ?z y < e/2" and yi:"y\<in>i" by auto
+ then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
- hence "y\<bullet>k < a\<bullet>k"
- using e[THEN conjunct1] k by (auto simp add: field_simps as inner_Basis inner_simps)
- hence "y \<notin> i"
+ then have "y\<bullet>k < a\<bullet>k"
+ using e[THEN conjunct1] k
+ by (auto simp add: field_simps as inner_Basis inner_simps)
+ then have "y \<notin> i"
unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
- thus False using yi by auto
+ then show False using yi by auto
qed
moreover
have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
- apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
+ apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
proof
fix y
- assume as: "y\<in> ball ?z (e/2)"
+ assume as: "y \<in> ball ?z (e/2)"
have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
apply -
apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
@@ -348,7 +401,7 @@
using e
apply (auto simp add: field_simps)
done
- finally show "y\<in>ball x e"
+ finally show "y \<in> ball x e"
unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
@@ -361,18 +414,22 @@
assume as: "x\<bullet>k = b\<bullet>k"
have "ball ?z (e / 2) \<inter> i = {}"
apply (rule ccontr)
- unfolding ex_in_conv[THEN sym]
- proof(erule exE)
+ unfolding ex_in_conv[symmetric]
+ apply (erule exE)
+ proof -
fix y
assume "y \<in> ball ?z (e / 2) \<inter> i"
- hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
- hence "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
- using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
- hence "y\<bullet>k > b\<bullet>k"
- using e[THEN conjunct1] k by(auto simp add:field_simps inner_simps inner_Basis as)
- hence "y \<notin> i"
+ then have "dist ?z y < e/2" and yi: "y \<in> i"
+ by auto
+ then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
+ using Basis_le_norm[OF k, of "?z - y"]
+ unfolding dist_norm by auto
+ then have "y\<bullet>k > b\<bullet>k"
+ using e[THEN conjunct1] k
+ by (auto simp add:field_simps inner_simps inner_Basis as)
+ then have "y \<notin> i"
unfolding ab mem_interval by (auto intro!: bexI[OF _ k])
- thus False using yi by auto
+ then show False using yi by auto
qed
moreover
have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
@@ -382,7 +439,7 @@
assume as: "y\<in> ball ?z (e/2)"
have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
apply -
- apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
+ apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
unfolding norm_scaleR
apply (auto simp: k)
done
@@ -391,79 +448,81 @@
using as unfolding mem_ball dist_norm
using e apply (auto simp add: field_simps)
done
- finally show "y\<in>ball x e"
- unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
+ finally show "y \<in> ball x e"
+ unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
apply (rule_tac x="?z" in exI)
unfolding Union_insert
apply auto
done
- qed
- then guess x ..
- hence "x \<in> s \<inter> interior (\<Union>f)"
- unfolding lem1[where U="\<Union>f",THEN sym]
+ qed
+ then obtain x where "ball x (e / 2) \<subseteq> s \<inter> \<Union>f" ..
+ then have "x \<in> s \<inter> interior (\<Union>f)"
+ unfolding lem1[where U="\<Union>f", symmetric]
using centre_in_ball e[THEN conjunct1] by auto
- thus ?thesis
+ then show ?thesis
apply -
apply (rule lem2, rule insert(3))
- using insert(4) apply auto
+ using insert(4)
+ apply auto
done
qed
qed
qed
qed
- note * = this
- guess t using *[OF assms(1,3) goal1] ..
- from this(2) guess x ..
- then guess e ..
- hence "x \<in> s" "x\<in>interior t"
- defer
- using open_subset_interior[OF open_ball, of x e t] apply auto
- done
- thus False using `t\<in>f` assms(4) by auto
+ from this[OF assms(1,3) goal1]
+ obtain t x e where "t \<in> f" "0 < e" "ball x e \<subseteq> s \<inter> t"
+ by blast
+ then have "x \<in> s" "x \<in> interior t"
+ using open_subset_interior[OF open_ball, of x e t]
+ by auto
+ then show False
+ using `t \<in> f` assms(4) by auto
qed
subsection {* Bounds on intervals where they exist. *}
-definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
- "interval_upperbound s = (\<Sum>i\<in>Basis. Sup {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
-
-definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a" where
- "interval_lowerbound s = (\<Sum>i\<in>Basis. Inf {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
+definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
+ where "interval_upperbound s = (\<Sum>i\<in>Basis. Sup {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
+
+definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
+ where "interval_lowerbound s = (\<Sum>i\<in>Basis. Inf {a. \<exists>x\<in>s. x\<bullet>i = a} *\<^sub>R i)"
lemma interval_upperbound[simp]:
"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
interval_upperbound {a..b} = (b::'a::ordered_euclidean_space)"
unfolding interval_upperbound_def euclidean_representation_setsum
by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setle_def
- intro!: cSup_unique)
+ intro!: cSup_unique)
lemma interval_lowerbound[simp]:
"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
interval_lowerbound {a..b} = (a::'a::ordered_euclidean_space)"
unfolding interval_lowerbound_def euclidean_representation_setsum
by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] setge_def
- intro!: cInf_unique)
+ intro!: cInf_unique)
lemmas interval_bounds = interval_upperbound interval_lowerbound
lemma interval_bounds'[simp]:
- assumes "{a..b}\<noteq>{}"
- shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
+ assumes "{a..b} \<noteq> {}"
+ shows "interval_upperbound {a..b} = b"
+ and "interval_lowerbound {a..b} = a"
using assms unfolding interval_ne_empty by auto
+
subsection {* Content (length, area, volume...) of an interval. *}
definition "content (s::('a::ordered_euclidean_space) set) =
(if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
-lemma interval_not_empty:"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
+lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
unfolding interval_eq_empty unfolding not_ex not_less by auto
lemma content_closed_interval:
- fixes a::"'a::ordered_euclidean_space"
+ fixes a :: "'a::ordered_euclidean_space"
assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
using interval_not_empty[OF assms]
@@ -471,8 +530,8 @@
by auto
lemma content_closed_interval':
- fixes a::"'a::ordered_euclidean_space"
- assumes "{a..b}\<noteq>{}"
+ fixes a :: "'a::ordered_euclidean_space"
+ assumes "{a..b} \<noteq> {}"
shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
apply (rule content_closed_interval)
using assms
@@ -480,13 +539,8 @@
apply assumption
done
-lemma content_real:
- assumes "a\<le>b"
- shows "content {a..b} = b-a"
-proof -
- have *: "{..<Suc 0} = {0}" by auto
- show ?thesis unfolding content_def using assms by (auto simp: *)
-qed
+lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
+ unfolding content_def by auto
lemma content_singleton[simp]: "content {a} = 0"
proof -
@@ -497,9 +551,12 @@
lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1"
proof -
- have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i" by auto
- have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
- thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto
+ have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
+ by auto
+ have "0 \<in> {0..One::'a}"
+ unfolding mem_interval by auto
+ then show ?thesis
+ unfolding content_def interval_bounds[OF *] using setprod_1 by auto
qed
lemma content_pos_le[intro]:
@@ -507,7 +564,8 @@
shows "0 \<le> content {a..b}"
proof (cases "{a..b} = {}")
case False
- hence *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty .
+ then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
+ unfolding interval_ne_empty .
have "(\<Prod>i\<in>Basis. interval_upperbound {a..b} \<bullet> i - interval_lowerbound {a..b} \<bullet> i) \<ge> 0"
apply (rule setprod_nonneg)
unfolding interval_bounds[OF *]
@@ -515,29 +573,38 @@
apply (erule_tac x=x in ballE)
apply auto
done
- thus ?thesis unfolding content_def by (auto simp del:interval_bounds')
-qed (unfold content_def, auto)
+ then show ?thesis
+ unfolding content_def by (auto simp del:interval_bounds')
+next
+ case True
+ then show ?thesis
+ unfolding content_def by auto
+qed
lemma content_pos_lt:
- fixes a::"'a::ordered_euclidean_space"
+ fixes a :: "'a::ordered_euclidean_space"
assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
shows "0 < content {a..b}"
proof -
have help_lemma1: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> \<forall>i\<in>Basis. a\<bullet>i \<le> ((b\<bullet>i)::real)"
- apply (rule, erule_tac x=i in ballE)
+ apply rule
+ apply (erule_tac x=i in ballE)
apply auto
done
- show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]]
- apply(rule setprod_pos)
- using assms apply (erule_tac x=x in ballE)
+ show ?thesis
+ unfolding content_closed_interval[OF help_lemma1[OF assms]]
+ apply (rule setprod_pos)
+ using assms
+ apply (erule_tac x=x in ballE)
apply auto
done
qed
-lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
+lemma content_eq_0:
+ "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
proof (cases "{a..b} = {}")
case True
- thus ?thesis
+ then show ?thesis
unfolding content_def if_P[OF True]
unfolding interval_eq_empty
apply -
@@ -547,15 +614,16 @@
done
next
case False
- from this[unfolded interval_eq_empty not_ex not_less]
- have as: "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i" by fastforce
- show ?thesis
+ then have "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i"
+ unfolding interval_eq_empty not_ex not_less
+ by fastforce
+ then show ?thesis
unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_Basis]
- using as
by (auto intro!: bexI)
qed
-lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
+lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
+ by auto
lemma content_closed_interval_cases:
"content {a..b::'a::ordered_euclidean_space} =
@@ -563,42 +631,51 @@
by (auto simp: not_le content_eq_0 intro: less_imp_le content_closed_interval)
lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
- unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
-
-lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
+ unfolding content_eq_0 interior_closed_interval interval_eq_empty
+ by auto
+
+lemma content_pos_lt_eq:
+ "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
apply rule
defer
apply (rule content_pos_lt, assumption)
proof -
assume "0 < content {a..b}"
- hence "content {a..b} \<noteq> 0" by auto
- thus "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
+ then have "content {a..b} \<noteq> 0" by auto
+ then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
unfolding content_eq_0 not_ex not_le by fastforce
qed
-lemma content_empty [simp]: "content {} = 0" unfolding content_def by auto
+lemma content_empty [simp]: "content {} = 0"
+ unfolding content_def by auto
lemma content_subset:
assumes "{a..b} \<subseteq> {c..d}"
shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}"
proof (cases "{a..b} = {}")
case True
- thus ?thesis using content_pos_le[of c d] by auto
+ then show ?thesis
+ using content_pos_le[of c d] by auto
next
case False
- hence ab_ne:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" unfolding interval_ne_empty by auto
- hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
+ then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
+ unfolding interval_ne_empty by auto
+ then have ab_ab: "a\<in>{a..b}" "b\<in>{a..b}"
+ unfolding mem_interval by auto
have "{c..d} \<noteq> {}" using assms False by auto
- hence cd_ne:"\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i" using assms unfolding interval_ne_empty by auto
+ then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
+ using assms unfolding interval_ne_empty by auto
show ?thesis
unfolding content_def
unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`]
- apply (rule setprod_mono, rule)
+ apply (rule setprod_mono)
+ apply rule
proof
fix i :: 'a
- assume i: "i\<in>Basis"
- show "0 \<le> b \<bullet> i - a \<bullet> i" using ab_ne[THEN bspec, OF i] i by auto
+ assume i: "i \<in> Basis"
+ show "0 \<le> b \<bullet> i - a \<bullet> i"
+ using ab_ne[THEN bspec, OF i] i by auto
show "b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i]
@@ -612,58 +689,78 @@
subsection {* The notion of a gauge --- simply an open set containing the point. *}
-definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
-
-lemma gaugeI: assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
+definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
+
+lemma gaugeI:
+ assumes "\<And>x. x \<in> g x"
+ and "\<And>x. open (g x)"
+ shows "gauge g"
using assms unfolding gauge_def by auto
-lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)"
+lemma gaugeD[dest]:
+ assumes "gauge d"
+ shows "x \<in> d x"
+ and "open (d 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))"
- unfolding gauge_def by auto
-
-lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
+ unfolding gauge_def by auto
+
+lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
+ unfolding gauge_def by auto
lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)"
by (rule gauge_ball) auto
-lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
- unfolding gauge_def by auto
+lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
+ unfolding gauge_def by auto
lemma gauge_inters:
- assumes "finite s" "\<forall>d\<in>s. gauge (f d)"
- shows "gauge(\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
+ assumes "finite s"
+ and "\<forall>d\<in>s. gauge (f d)"
+ shows "gauge (\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
proof -
- have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto
+ have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
+ by auto
show ?thesis
- unfolding gauge_def unfolding *
+ unfolding gauge_def unfolding *
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
qed
-lemma gauge_existence_lemma: "(\<forall>x. \<exists>d::real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
- by(meson zero_less_one)
+lemma gauge_existence_lemma:
+ "(\<forall>x. \<exists>d :: real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
+ by (metis zero_less_one)
subsection {* Divisions. *}
-definition division_of (infixl "division'_of" 40) where
- "s division_of i \<equiv>
- finite s \<and>
- (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
- (\<Union>s = i)"
+definition division_of (infixl "division'_of" 40)
+where
+ "s division_of i \<longleftrightarrow>
+ finite s \<and>
+ (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
+ (\<Union>s = i)"
lemma division_ofD[dest]:
assumes "s division_of i"
- shows "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow> k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
- "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
+ shows "finite s"
+ and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
+ and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
+ and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+ and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
+ and "\<Union>s = i"
using assms unfolding division_of_def by auto
lemma division_ofI:
- assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow> k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
- "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
- shows "s division_of i" using assms unfolding division_of_def by auto
+ assumes "finite s"
+ and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
+ and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
+ and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+ and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+ and "\<Union>s = i"
+ shows "s division_of i"
+ using assms unfolding division_of_def by auto
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
unfolding division_of_def by auto
@@ -671,28 +768,38 @@
lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
unfolding division_of_def by auto
-lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
+lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
+ unfolding division_of_def by auto
lemma division_of_sing[simp]:
- "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r")
+ "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}"
+ (is "?l = ?r")
proof
assume ?r
- moreover {
+ moreover
+ {
assume "s = {{a}}"
- moreover fix k assume "k\<in>s"
+ moreover fix k assume "k\<in>s"
ultimately have"\<exists>x y. k = {x..y}"
apply (rule_tac x=a in exI)+
unfolding interval_sing
apply auto
done
}
- ultimately show ?l unfolding division_of_def interval_sing by auto
+ ultimately show ?l
+ unfolding division_of_def interval_sing by auto
next
assume ?l
- note as=conjunctD4[OF this[unfolded division_of_def interval_sing]]
- { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
- moreover have "s \<noteq> {}" using as(4) by auto
- ultimately show ?r unfolding interval_sing by auto
+ note * = conjunctD4[OF this[unfolded division_of_def interval_sing]]
+ {
+ fix x
+ assume x: "x \<in> s" have "x = {a}"
+ using *(2)[rule_format,OF x] by auto
+ }
+ moreover have "s \<noteq> {}"
+ using *(4) by auto
+ ultimately show ?r
+ unfolding interval_sing by auto
qed
lemma elementary_empty: obtains p where "p division_of {}"
@@ -705,27 +812,38 @@
unfolding division_of_def by auto
lemma forall_in_division:
- "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
+ "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b})"
unfolding division_of_def by fastforce
-lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
- apply (rule division_ofI)
-proof -
- note as=division_ofD[OF assms(1)]
+lemma division_of_subset:
+ assumes "p division_of (\<Union>p)"
+ and "q \<subseteq> p"
+ shows "q division_of (\<Union>q)"
+proof (rule division_ofI)
+ note * = division_ofD[OF assms(1)]
show "finite q"
apply (rule finite_subset)
- using as(1) assms(2) apply auto
+ using *(1) assms(2)
+ apply auto
done
- { fix k
+ {
+ fix k
assume "k \<in> q"
- hence kp:"k\<in>p" using assms(2) by auto
- show "k\<subseteq>\<Union>q" using `k \<in> q` by auto
- show "\<exists>a b. k = {a..b}" using as(4)[OF kp]
- by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
+ then have kp: "k \<in> p"
+ using assms(2) by auto
+ show "k \<subseteq> \<Union>q"
+ using `k \<in> q` by auto
+ show "\<exists>a b. k = {a..b}"
+ using *(4)[OF kp] by auto
+ show "k \<noteq> {}"
+ using *(3)[OF kp] by auto
+ }
fix k1 k2
assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
- hence *: "k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto
- show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto
+ then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
+ using assms(2) by auto
+ show "interior k1 \<inter> interior k2 = {}"
+ using *(5)[OF **] by auto
qed auto
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
@@ -740,52 +858,65 @@
apply (drule content_subset) unfolding assms(1)
proof -
case goal1
- thus ?case using content_pos_le[of a b] by auto
+ then show ?case using content_pos_le[of a b] by auto
qed
lemma division_inter:
- assumes "p1 division_of s1" "p2 division_of (s2::('a::ordered_euclidean_space) set)"
+ fixes s1 s2 :: "'a::ordered_euclidean_space set"
+ assumes "p1 division_of s1"
+ and "p2 division_of s2"
shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
(is "?A' division_of _")
proof -
let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
- have *:"?A' = ?A" by auto
- show ?thesis unfolding *
+ have *: "?A' = ?A" by auto
+ show ?thesis
+ unfolding *
proof (rule division_ofI)
- have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
- moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto
+ have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
+ by auto
+ moreover have "finite (p1 \<times> p2)"
+ using assms unfolding division_of_def by auto
ultimately show "finite ?A" by auto
- have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto
+ 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 Union_image_eq UN_iff
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
apply auto
done
- { fix k
- assume "k\<in>?A"
- then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto
- thus "k \<noteq> {}" by auto
+ {
+ fix k
+ assume "k \<in> ?A"
+ then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
+ by auto
+ then show "k \<noteq> {}"
+ by auto
show "k \<subseteq> s1 \<inter> s2"
using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
unfolding k by auto
- guess a1 using division_ofD(4)[OF assms(1) k(2)] ..
- then guess b1 .. note ab1=this
- guess a2 using division_ofD(4)[OF assms(2) k(3)] ..
- then guess b2 .. note ab2=this
+ obtain a1 b1 where k1: "k1 = {a1..b1}"
+ using division_ofD(4)[OF assms(1) k(2)] by blast
+ obtain a2 b2 where k2: "k2 = {a2..b2}"
+ using division_ofD(4)[OF assms(2) k(3)] by blast
show "\<exists>a b. k = {a..b}"
- unfolding k ab1 ab2 unfolding inter_interval by auto }
+ unfolding k k1 k2 unfolding inter_interval by auto
+ }
fix k1 k2
- assume "k1\<in>?A"
- then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
- assume "k2\<in>?A"
- then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
+ assume "k1 \<in> ?A"
+ then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
+ by auto
+ assume "k2 \<in> ?A"
+ then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
+ by auto
assume "k1 \<noteq> k2"
- hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
- have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
- interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
- interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
- \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
+ then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
+ unfolding k1 k2 by auto
+ have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
+ interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
+ interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
+ interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
show "interior k1 \<inter> interior k2 = {}"
unfolding k1 k2
apply (rule *)
@@ -793,31 +924,41 @@
apply (rule_tac[1-4] interior_mono)
using division_ofD(5)[OF assms(1) k1(2) k2(2)]
using division_ofD(5)[OF assms(2) k1(3) k2(3)]
- using th apply auto done
+ using th
+ apply auto
+ done
qed
qed
lemma division_inter_1:
- assumes "d division_of i" "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
- shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}"
+ assumes "d division_of i"
+ and "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
+ shows "{{a..b} \<inter> k | k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {}} division_of {a..b}"
proof (cases "{a..b} = {}")
case True
- show ?thesis unfolding True and division_of_trivial by auto
+ show ?thesis
+ unfolding True and division_of_trivial by auto
next
case False
have *: "{a..b} \<inter> i = {a..b}" using assms(2) by auto
- show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto
+ show ?thesis
+ using division_inter[OF division_of_self[OF False] assms(1)]
+ unfolding * by auto
qed
lemma elementary_inter:
- assumes "p1 division_of s" "p2 division_of (t::('a::ordered_euclidean_space) set)"
+ fixes s t :: "'a::ordered_euclidean_space set"
+ assumes "p1 division_of s"
+ and "p2 division_of t"
shows "\<exists>p. p division_of (s \<inter> t)"
apply rule
apply (rule division_inter[OF assms])
done
lemma elementary_inters:
- assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
+ assumes "finite f"
+ and "f \<noteq> {}"
+ and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
shows "\<exists>p. p division_of (\<Inter> f)"
using assms
proof (induct f rule: finite_induct)
@@ -825,14 +966,18 @@
show ?case
proof (cases "f = {}")
case True
- thus ?thesis unfolding True using insert by auto
+ then show ?thesis
+ unfolding True using insert by auto
next
case False
- guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
- moreover guess px using insert(5)[rule_format,OF insertI1] ..
+ obtain p where "p division_of \<Inter>f"
+ using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
+ moreover obtain px where "px division_of x"
+ using insert(5)[rule_format,OF insertI1] ..
ultimately show ?thesis
+ apply -
unfolding Inter_insert
- apply (rule_tac elementary_inter)
+ apply (rule elementary_inter)
apply assumption
apply assumption
done
@@ -840,12 +985,17 @@
qed auto
lemma division_disjoint_union:
- assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
+ assumes "p1 division_of s1"
+ and "p2 division_of s2"
+ and "interior s1 \<inter> interior s2 = {}"
shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
proof (rule division_ofI)
- note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
- show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
- show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
+ note d1 = division_ofD[OF assms(1)]
+ note d2 = division_ofD[OF assms(2)]
+ show "finite (p1 \<union> p2)"
+ using d1(1) d2(1) by auto
+ show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
+ using d1(6) d2(6) by auto
{
fix k1 k2
assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
@@ -864,28 +1014,33 @@
using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
using assms(3) by blast
}
- ultimately show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
+ ultimately show ?g
+ using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
}
fix k
assume k: "k \<in> p1 \<union> p2"
- show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
- show "k \<noteq> {}" using k d1(3) d2(3) by auto
- show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto
+ show "k \<subseteq> s1 \<union> s2"
+ using k d1(2) d2(2) by auto
+ show "k \<noteq> {}"
+ using k d1(3) d2(3) by auto
+ show "\<exists>a b. k = {a..b}"
+ using k d1(4) d2(4) by auto
qed
lemma partial_division_extend_1:
- assumes incl: "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}"
+ fixes a b c d :: "'a::ordered_euclidean_space"
+ assumes incl: "{c..d} \<subseteq> {a..b}"
and nonempty: "{c..d} \<noteq> {}"
obtains p where "p division_of {a..b}" "{c..d} \<in> p"
proof
- let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a. {(\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)}"
+ let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
+ {(\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)}"
def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
show "{c .. d} \<in> p"
unfolding p_def
by (auto simp add: interval_eq_empty eucl_le[where 'a='a]
- intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
-
+ intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
{
fix i :: 'a
assume "i \<in> Basis"
@@ -896,13 +1051,15 @@
show "p division_of {a..b}"
proof (rule division_ofI)
- show "finite p" unfolding p_def by (auto intro!: finite_PiE)
+ show "finite p"
+ unfolding p_def by (auto intro!: finite_PiE)
{
fix k
assume "k \<in> p"
then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
by (auto simp: p_def)
- then show "\<exists>a b. k = {a..b}" by auto
+ then show "\<exists>a b. k = {a..b}"
+ by auto
have "k \<subseteq> {a..b} \<and> k \<noteq> {}"
proof (simp add: k interval_eq_empty subset_interval not_less, safe)
fix i :: 'a
@@ -913,50 +1070,55 @@
show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
by (auto simp: subset_iff eucl_le[where 'a='a])
qed
- then show "k \<noteq> {}" "k \<subseteq> {a .. b}" by auto
+ then show "k \<noteq> {}" "k \<subseteq> {a .. b}"
+ by auto
{
- fix l assume "l \<in> p"
+ fix l
+ assume "l \<in> p"
then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
by (auto simp: p_def)
assume "l \<noteq> k"
have "\<exists>i\<in>Basis. f i \<noteq> g i"
proof (rule ccontr)
- assume "\<not> (\<exists>i\<in>Basis. f i \<noteq> g i)"
+ assume "\<not> ?thesis"
with f g have "f = g"
by (auto simp: PiE_iff extensional_def intro!: ext)
with `l \<noteq> k` show False
by (simp add: l k)
qed
- then guess i .. note * = this
- moreover from * have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
+ then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
+ then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
"g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
using f g by (auto simp: PiE_iff)
- moreover note ord[of i]
- ultimately show "interior l \<inter> interior k = {}"
+ with * ord[of i] show "interior l \<inter> interior k = {}"
by (auto simp add: l k interior_closed_interval disjoint_interval intro!: bexI[of _ i])
}
- note `k \<subseteq> { a.. b}`
+ note `k \<subseteq> {a.. b}`
}
moreover
{
fix x assume x: "x \<in> {a .. b}"
have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
proof
- fix i :: 'a assume "i \<in> Basis"
- with x ord[of i]
+ fix i :: 'a
+ assume "i \<in> Basis"
+ with x ord[of i]
have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
(d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
by (auto simp: eucl_le[where 'a='a])
then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
by auto
qed
- then guess f unfolding bchoice_iff .. note f = this
+ then obtain f where
+ f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}"
+ unfolding bchoice_iff ..
moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
by auto
moreover from f have "x \<in> ?B (restrict f Basis)"
by (auto simp: mem_interval eucl_le[where 'a='a])
ultimately have "\<exists>k\<in>p. x \<in> k"
- unfolding p_def by blast }
+ unfolding p_def by blast
+ }
ultimately show "\<Union>p = {a..b}"
by auto
qed
@@ -967,8 +1129,9 @@
obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}"
proof (cases "p = {}")
case True
- guess q apply (rule elementary_interval[of a b]) .
- thus ?thesis
+ obtain q where "q division_of {a..b}"
+ by (rule elementary_interval)
+ then show ?thesis
apply -
apply (rule that[of q])
unfolding True
@@ -977,31 +1140,36 @@
next
case False
note p = division_ofD[OF assms(1)]
- have *: "\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q"
+ have *: "\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k \<in> q"
proof
case goal1
- guess c using p(4)[OF goal1] ..
- then guess d .. note "cd" = this
+ obtain c d where k: "k = {c..d}"
+ using p(4)[OF goal1] by blast
have *: "{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}"
- using p(2,3)[OF goal1, unfolded "cd"] using assms(2) by auto
- guess q apply(rule partial_division_extend_1[OF *]) .
- thus ?case unfolding "cd" by auto
+ using p(2,3)[OF goal1, unfolded k] using assms(2) by auto
+ obtain q where "q division_of {a..b}" "{c..d} \<in> q"
+ by (rule partial_division_extend_1[OF *])
+ then show ?case
+ unfolding k by auto
qed
- guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
- have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
- apply (rule, rule_tac p="q x" in division_of_subset)
+ obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of {a..b}" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
+ using bchoice[OF *] by blast
+ have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
+ apply rule
+ apply (rule_tac p="q x" in division_of_subset)
proof -
fix x
- assume x: "x\<in>p"
+ assume x: "x \<in> p"
show "q x division_of \<Union>q x"
apply -
apply (rule division_ofI)
using division_ofD[OF q(1)[OF x]]
apply auto
done
- show "q x - {x} \<subseteq> q x" by auto
+ show "q x - {x} \<subseteq> q x"
+ by auto
qed
- hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
+ then have "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
apply -
apply (rule elementary_inters)
apply (rule finite_imageI[OF p(1)])
@@ -1009,16 +1177,16 @@
apply (rule False)
apply auto
done
- then guess d .. note d = this
+ then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
show ?thesis
apply (rule that[of "d \<union> p"])
proof -
- have *: "\<And>s f t. s \<noteq> {} \<Longrightarrow> (\<forall>i\<in>s. f i \<union> i = t) \<Longrightarrow> t = \<Inter> (f ` s) \<union> (\<Union>s)" by auto
- have *: "{a..b} = \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
+ have *: "\<And>s f t. s \<noteq> {} \<Longrightarrow> \<forall>i\<in>s. f i \<union> i = t \<Longrightarrow> t = \<Inter>(f ` s) \<union> \<Union>s" by auto
+ have *: "{a..b} = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
apply (rule *[OF False])
proof
fix i
- assume i: "i\<in>p"
+ assume i: "i \<in> p"
show "\<Union>(q i - {i}) \<union> i = {a..b}"
using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
qed
@@ -1027,10 +1195,12 @@
apply (rule division_disjoint_union[OF d assms(1)])
apply (rule inter_interior_unions_intervals)
apply (rule p open_interior ballI)+
- proof (assumption, rule)
+ apply assumption
+ proof
fix k
- assume k: "k\<in>p"
- have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto
+ assume k: "k \<in> p"
+ have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}"
+ by auto
show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<inter> interior k = {}"
apply (rule *[of _ "interior (\<Union>(q k - {k}))"])
defer
@@ -1038,27 +1208,34 @@
apply (rule inter_interior_unions_intervals)
proof -
note qk=division_ofD[OF q(1)[OF k]]
- show "finite (q k - {k})" "open (interior k)"
- "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
+ show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}"
+ using qk by auto
show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
using qk(5) using q(2)[OF k] by auto
- have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto
+ have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x"
+ by auto
show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<subseteq> interior (\<Union>(q k - {k}))"
apply (rule interior_mono *)+
- using k by auto
+ using k
+ apply auto
+ done
qed
qed
qed auto
qed
-lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::('a::ordered_euclidean_space) set)"
- unfolding division_of_def by(metis bounded_Union bounded_interval)
-
-lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
+lemma elementary_bounded[dest]:
+ fixes s :: "'a::ordered_euclidean_space set"
+ shows "p division_of s \<Longrightarrow> bounded s"
+ unfolding division_of_def by (metis bounded_Union bounded_interval)
+
+lemma elementary_subset_interval:
+ "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
by (meson elementary_bounded bounded_subset_closed_interval)
lemma division_union_intervals_exists:
- assumes "{a..b::'a::ordered_euclidean_space} \<noteq> {}"
+ fixes a b :: "'a::ordered_euclidean_space"
+ assumes "{a..b} \<noteq> {}"
obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})"
proof (cases "{c..d} = {}")
case True
@@ -1070,16 +1247,15 @@
done
next
case False
- note false=this
show ?thesis
proof (cases "{a..b} \<inter> {c..d} = {}")
case True
- have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
+ have *: "\<And>a b. {a, b} = {a} \<union> {b}" by auto
show ?thesis
apply (rule that[of "{{c..d}}"])
unfolding *
apply (rule division_disjoint_union)
- using false True assms
+ using `{c..d} \<noteq> {}` True assms
using interior_subset
apply auto
done
@@ -1088,10 +1264,11 @@
obtain u v where uv: "{a..b} \<inter> {c..d} = {u..v}"
unfolding inter_interval by auto
have *: "{u..v} \<subseteq> {c..d}" using uv by auto
- guess p apply (rule partial_division_extend_1[OF * False[unfolded uv]]) .
- note p=this division_ofD[OF this(1)]
+ obtain p where "p division_of {c..d}" "{u..v} \<in> p"
+ by (rule partial_division_extend_1[OF * False[unfolded uv]])
+ note p = this division_ofD[OF this(1)]
have *: "{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s"
- using p(8) unfolding uv[THEN sym] by auto
+ using p(8) unfolding uv[symmetric] by auto
show ?thesis
apply (rule that[of "p - {{u..v}}"])
unfolding *(1)
@@ -1101,10 +1278,10 @@
apply (rule division_of_subset[of p])
apply (rule division_of_union_self[OF p(1)])
defer
- unfolding interior_inter[THEN sym]
+ unfolding interior_inter[symmetric]
proof -
have *: "\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
- have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
+ have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
apply (rule arg_cong[of _ _ interior])
apply (rule *[OF _ uv])
using p(8)
@@ -1121,270 +1298,611 @@
qed
qed
-lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
- "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
- shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
- apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
- using division_ofD[OF assms(2)] by auto
-
-lemma elementary_union_interval: assumes "p division_of \<Union>p"
- obtains q where "q division_of ({a..b::'a::ordered_euclidean_space} \<union> \<Union>p)" proof-
- note assm=division_ofD[OF assms]
- 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" by auto
-{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
- "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
- thus thesis by auto
-next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
- thus thesis apply(rule_tac that[of p]) unfolding as by auto
-next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
-next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
- show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
- unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
- using assm(2-4) as apply- by(fastforce dest: assm(5))+
-next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
- have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
- from assm(4)[OF this] guess c .. then guess d ..
- thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
- qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
- let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
- show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
- have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
- show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
- show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
- using q(6) by auto
- fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
- show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
- fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
- obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto
- obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
- show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
- case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
- next case False
- { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
- "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
- thus ?thesis by auto }
- { assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
- { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto }
- assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
- guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
- have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto
- hence "interior k \<subseteq> interior x" apply-
- apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
- guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
- have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
- hence "interior k' \<subseteq> interior x'" apply-
- apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
- ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
- qed qed } qed
+lemma division_of_unions:
+ assumes "finite f"
+ and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
+ and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+ shows "\<Union>f division_of \<Union>\<Union>f"
+ apply (rule division_ofI)
+ prefer 5
+ apply (rule assms(3)|assumption)+
+ apply (rule finite_Union assms(1))+
+ prefer 3
+ apply (erule UnionE)
+ apply (rule_tac s=X in division_ofD(3)[OF assms(2)])
+ using division_ofD[OF assms(2)]
+ apply auto
+ done
+
+lemma elementary_union_interval:
+ fixes a b :: "'a::ordered_euclidean_space"
+ assumes "p division_of \<Union>p"
+ obtains q where "q division_of ({a..b} \<union> \<Union>p)"
+proof -
+ note assm = division_ofD[OF assms]
+ 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"
+ by auto
+ {
+ presume "p = {} \<Longrightarrow> thesis"
+ "{a..b} = {} \<Longrightarrow> thesis"
+ "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
+ "p \<noteq> {} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
+ then show thesis by auto
+ next
+ assume as: "p = {}"
+ obtain p where "p division_of {a..b}"
+ by (rule elementary_interval)
+ then show thesis
+ apply -
+ apply (rule that[of p])
+ unfolding as
+ apply auto
+ done
+ next
+ assume as: "{a..b} = {}"
+ show thesis
+ apply (rule that)
+ unfolding as
+ using assms
+ apply auto
+ done
+ next
+ assume as: "interior {a..b} = {}" "{a..b} \<noteq> {}"
+ show thesis
+ apply (rule that[of "insert {a..b} p"],rule division_ofI)
+ unfolding finite_insert
+ apply (rule assm(1)) unfolding Union_insert
+ using assm(2-4) as
+ apply -
+ apply (fastforce dest: assm(5))+
+ done
+ next
+ assume as: "p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b} \<noteq> {}"
+ have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)"
+ proof
+ case goal1
+ from assm(4)[OF this] obtain c d where "k = {c..d}" by blast
+ then show ?case
+ apply -
+ apply (rule division_union_intervals_exists[OF as(3), of c d])
+ apply auto
+ done
+ qed
+ from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert {a..b} (q x) division_of {a..b} \<union> x" ..
+ note q = division_ofD[OF this[rule_format]]
+ let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
+ show thesis
+ apply (rule that[of "?D"])
+ apply (rule division_ofI)
+ proof -
+ have *: "{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p"
+ by auto
+ show "finite ?D"
+ apply (rule finite_Union)
+ unfolding *
+ apply (rule finite_imageI)
+ using assm(1) q(1)
+ apply auto
+ done
+ show "\<Union>?D = {a..b} \<union> \<Union>p"
+ unfolding * lem1
+ unfolding lem2[OF as(1), of "{a..b}", symmetric]
+ using q(6)
+ by auto
+ fix k
+ assume k: "k \<in> ?D"
+ then show "k \<subseteq> {a..b} \<union> \<Union>p"
+ using q(2) by auto
+ show "k \<noteq> {}"
+ using q(3) k by auto
+ show "\<exists>a b. k = {a..b}"
+ using q(4) k by auto
+ fix k'
+ assume k': "k' \<in> ?D" "k \<noteq> k'"
+ obtain x where x: "k \<in> insert {a..b} (q x)" "x\<in>p"
+ using k by auto
+ obtain x' where x': "k'\<in>insert {a..b} (q x')" "x'\<in>p"
+ using k' by auto
+ show "interior k \<inter> interior k' = {}"
+ proof (cases "x = x'")
+ case True
+ show ?thesis
+ apply(rule q(5))
+ using x x' k'
+ unfolding True
+ apply auto
+ done
+ next
+ case False
+ {
+ presume "k = {a..b} \<Longrightarrow> ?thesis"
+ and "k' = {a..b} \<Longrightarrow> ?thesis"
+ and "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
+ then show ?thesis by auto
+ next
+ assume as': "k = {a..b}"
+ show ?thesis
+ apply (rule q(5))
+ using x' k'(2)
+ unfolding as'
+ apply auto
+ done
+ next
+ assume as': "k' = {a..b}"
+ show ?thesis
+ apply (rule q(5))
+ using x k'(2)
+ unfolding as'
+ apply auto
+ done
+ }
+ assume as': "k \<noteq> {a..b}" "k' \<noteq> {a..b}"
+ obtain c d where k: "k = {c..d}"
+ using q(4)[OF x(2,1)] by blast
+ have "interior k \<inter> interior {a..b} = {}"
+ apply (rule q(5))
+ using x k'(2)
+ using as'
+ apply auto
+ done
+ then have "interior k \<subseteq> interior x"
+ apply -
+ apply (rule interior_subset_union_intervals[OF k _ as(2) q(2)[OF x(2,1)]])
+ apply auto
+ done
+ moreover
+ obtain c d where c_d: "k' = {c..d}"
+ using q(4)[OF x'(2,1)] by blast
+ have "interior k' \<inter> interior {a..b} = {}"
+ apply (rule q(5))
+ using x' k'(2)
+ using as'
+ apply auto
+ done
+ then have "interior k' \<subseteq> interior x'"
+ apply -
+ apply (rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]])
+ apply auto
+ done
+ ultimately show ?thesis
+ using assm(5)[OF x(2) x'(2) False] by auto
+ qed
+ qed
+ }
+qed
lemma elementary_unions_intervals:
- assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
- obtains p where "p division_of (\<Union>f)" proof-
- have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
+ assumes fin: "finite f"
+ and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
+ obtains p where "p division_of (\<Union>f)"
+proof -
+ have "\<exists>p. p division_of (\<Union>f)"
+ proof (induct_tac f rule:finite_subset_induct)
show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
- fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
- from this(3) guess p .. note p=this
- from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
- have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
- show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
- unfolding Union_insert ab * by auto
- qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
-
-lemma elementary_union: assumes "ps division_of s" "pt division_of (t::('a::ordered_euclidean_space) set)"
+ next
+ fix x F
+ assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
+ from this(3) obtain p where p: "p division_of \<Union>F" ..
+ from assms(2)[OF as(4)] obtain a b where x: "x = {a..b}" by blast
+ have *: "\<Union>F = \<Union>p"
+ using division_ofD[OF p] by auto
+ show "\<exists>p. p division_of \<Union>insert x F"
+ using elementary_union_interval[OF p[unfolded *], of a b]
+ unfolding Union_insert x * by auto
+ qed (insert assms, auto)
+ then show ?thesis
+ apply -
+ apply (erule exE)
+ apply (rule that)
+ apply auto
+ done
+qed
+
+lemma elementary_union:
+ fixes s t :: "'a::ordered_euclidean_space set"
+ assumes "ps division_of s"
+ and "pt division_of t"
obtains p where "p division_of (s \<union> t)"
-proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
- hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
- show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
- unfolding * prefer 3 apply(rule_tac p=p in that)
- using assms[unfolded division_of_def] by auto qed
-
-lemma partial_division_extend: fixes t::"('a::ordered_euclidean_space) set"
- assumes "p division_of s" "q division_of t" "s \<subseteq> t"
- obtains r where "p \<subseteq> r" "r division_of t" proof-
+proof -
+ have "s \<union> t = \<Union>ps \<union> \<Union>pt"
+ using assms unfolding division_of_def by auto
+ then have *: "\<Union>(ps \<union> pt) = s \<union> t" by auto
+ show ?thesis
+ apply -
+ apply (rule elementary_unions_intervals[of "ps \<union> pt"])
+ unfolding *
+ prefer 3
+ apply (rule_tac p=p in that)
+ using assms[unfolded division_of_def]
+ apply auto
+ done
+qed
+
+lemma partial_division_extend:
+ fixes t :: "'a::ordered_euclidean_space set"
+ assumes "p division_of s"
+ and "q division_of t"
+ and "s \<subseteq> t"
+ obtains r where "p \<subseteq> r" and "r division_of t"
+proof -
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
- obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
- guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
- apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)]
- guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
- then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
- apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
- { fix x assume x:"x\<in>t" "x\<notin>s"
- hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
- then guess r unfolding Union_iff .. note r=this moreover
- have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
- thus False using x by auto qed
- ultimately have "x\<in>\<Union>(r1 - p)" by auto }
- hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
- show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
- unfolding divp(6) apply(rule assms r2)+
- proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
- proof(rule inter_interior_unions_intervals)
- show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
- have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
- show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
- fix m x assume as:"m\<in>r1-p"
- have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
- show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
- show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
- qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
- qed qed
- thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
- qed auto qed
+ obtain a b where ab: "t \<subseteq> {a..b}"
+ using elementary_subset_interval[OF assms(2)] by auto
+ obtain r1 where "p \<subseteq> r1" "r1 division_of {a..b}"
+ apply (rule partial_division_extend_interval)
+ apply (rule assms(1)[unfolded divp(6)[symmetric]])
+ apply (rule subset_trans)
+ apply (rule ab assms[unfolded divp(6)[symmetric]])+
+ apply assumption
+ done
+ note r1 = this division_ofD[OF this(2)]
+ obtain p' where "p' division_of \<Union>(r1 - p)"
+ apply (rule elementary_unions_intervals[of "r1 - p"])
+ using r1(3,6)
+ apply auto
+ done
+ then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
+ apply -
+ apply (drule elementary_inter[OF _ assms(2)[unfolded divq(6)[symmetric]]])
+ apply auto
+ done
+ {
+ fix x
+ assume x: "x \<in> t" "x \<notin> s"
+ then have "x\<in>\<Union>r1"
+ unfolding r1 using ab by auto
+ then obtain r where r: "r \<in> r1" "x \<in> r"
+ unfolding Union_iff ..
+ moreover
+ have "r \<notin> p"
+ proof
+ assume "r \<in> p"
+ then have "x \<in> s" using divp(2) r by auto
+ then show False using x by auto
+ qed
+ ultimately have "x\<in>\<Union>(r1 - p)" by auto
+ }
+ then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
+ unfolding divp divq using assms(3) by auto
+ show ?thesis
+ apply (rule that[of "p \<union> r2"])
+ unfolding *
+ defer
+ apply (rule division_disjoint_union)
+ unfolding divp(6)
+ apply(rule assms r2)+
+ proof -
+ have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
+ proof (rule inter_interior_unions_intervals)
+ show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}"
+ using r1 by auto
+ have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
+ by auto
+ show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
+ proof
+ fix m x
+ assume as: "m \<in> r1 - p"
+ have "interior m \<inter> interior (\<Union>p) = {}"
+ proof (rule inter_interior_unions_intervals)
+ show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = {a..b}"
+ using divp by auto
+ show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
+ apply (rule, rule r1(7))
+ using as
+ using r1
+ apply auto
+ done
+ qed
+ then show "interior s \<inter> interior m = {}"
+ unfolding divp by auto
+ qed
+ qed
+ then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
+ using interior_subset by auto
+ qed auto
+qed
+
subsection {* Tagged (partial) divisions. *}
-definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
- "(s tagged_partial_division_of i) \<equiv>
- finite s \<and>
- (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
- \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
-
-lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
- shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
- "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
- "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> (x1,k1) \<noteq> (x2,k2) \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
- using assms unfolding tagged_partial_division_of_def apply- by blast+
-
-definition tagged_division_of (infixr "tagged'_division'_of" 40) where
- "(s tagged_division_of i) \<equiv>
- (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
+ where "s tagged_partial_division_of i \<longleftrightarrow>
+ finite s \<and>
+ (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
+ interior k1 \<inter> interior k2 = {})"
+
+lemma tagged_partial_division_ofD[dest]:
+ assumes "s tagged_partial_division_of i"
+ shows "finite s"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+ and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
+ (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
+ using assms unfolding tagged_partial_division_of_def by blast+
+
+definition tagged_division_of (infixr "tagged'_division'_of" 40)
+ where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto
lemma tagged_division_of:
- "(s tagged_division_of i) \<longleftrightarrow>
- finite s \<and>
- (\<forall>x k. (x,k) \<in> s
- \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
- (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
- \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
- (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+ "s tagged_division_of i \<longleftrightarrow>
+ finite s \<and>
+ (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
+ (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
+ interior k1 \<inter> interior k2 = {}) \<and>
+ (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto
-lemma tagged_division_ofI: assumes
- "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i" "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
- "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
- "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+lemma tagged_division_ofI:
+ assumes "finite s"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+ and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
+ interior k1 \<inter> interior k2 = {}"
+ and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
shows "s tagged_division_of i"
- unfolding tagged_division_of apply(rule) defer apply rule
- apply(rule allI impI conjI assms)+ apply assumption
- apply(rule, rule assms, assumption) apply(rule assms, assumption)
- using assms(1,5-) apply- by blast+
-
-lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
- shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i" "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
- "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
- "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
-
-lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i"
-proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
- show "\<Union>(snd ` s) = i" "finite (snd ` s)" using assm by auto
- fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
- thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastforce+
- fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
- thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+ unfolding tagged_division_of
+ apply rule
+ defer
+ apply rule
+ apply (rule allI impI conjI assms)+
+ apply assumption
+ apply rule
+ apply (rule assms)
+ apply assumption
+ apply (rule assms)
+ apply assumption
+ using assms(1,5-)
+ apply blast+
+ done
+
+lemma tagged_division_ofD[dest]:
+ assumes "s tagged_division_of i"
+ shows "finite s"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
+ and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
+ and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
+ interior k1 \<inter> interior k2 = {}"
+ and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
+ using assms unfolding tagged_division_of by blast+
+
+lemma division_of_tagged_division:
+ assumes "s tagged_division_of i"
+ shows "(snd ` s) division_of i"
+proof (rule division_ofI)
+ note assm = tagged_division_ofD[OF assms]
+ show "\<Union>(snd ` s) = i" "finite (snd ` s)"
+ using assm by auto
+ fix k
+ assume k: "k \<in> snd ` s"
+ then obtain xk where xk: "(xk, k) \<in> s"
+ by auto
+ then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}"
+ using assm by fastforce+
+ fix k'
+ assume k': "k' \<in> snd ` s" "k \<noteq> k'"
+ from this(1) obtain xk' where xk': "(xk', k') \<in> s"
+ by auto
+ then show "interior k \<inter> interior k' = {}"
+ apply -
+ apply (rule assm(5))
+ apply (rule xk xk')+
+ using k'
+ apply auto
+ done
qed
-lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
+lemma partial_division_of_tagged_division:
+ assumes "s tagged_partial_division_of i"
shows "(snd ` s) division_of \<Union>(snd ` s)"
-proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
- show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)" using assm by auto
- fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
- thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>(snd ` s)" using assm by auto
- fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
- thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
+proof (rule division_ofI)
+ note assm = tagged_partial_division_ofD[OF assms]
+ show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
+ using assm by auto
+ fix k
+ assume k: "k \<in> snd ` s"
+ then obtain xk where xk: "(xk, k) \<in> s"
+ by auto
+ then show "k \<noteq> {}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>(snd ` s)"
+ using assm by auto
+ fix k'
+ assume k': "k' \<in> snd ` s" "k \<noteq> k'"
+ from this(1) obtain xk' where xk': "(xk', k') \<in> s"
+ by auto
+ then show "interior k \<inter> interior k' = {}"
+ apply -
+ apply (rule assm(5))
+ apply(rule xk xk')+
+ using k'
+ apply auto
+ done
qed
-lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
+lemma tagged_partial_division_subset:
+ assumes "s tagged_partial_division_of i"
+ and "t \<subseteq> s"
shows "t tagged_partial_division_of i"
- using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
-
-lemma setsum_over_tagged_division_lemma: fixes d::"('m::ordered_euclidean_space) set \<Rightarrow> 'a::real_normed_vector"
- assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
+ using assms
+ unfolding tagged_partial_division_of_def
+ using finite_subset[OF assms(2)]
+ by blast
+
+lemma setsum_over_tagged_division_lemma:
+ fixes d :: "'m::ordered_euclidean_space set \<Rightarrow> 'a::real_normed_vector"
+ assumes "p tagged_division_of i"
+ and "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
-proof- note assm=tagged_division_ofD[OF assms(1)]
- have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
- show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
- show "finite p" using assm by auto
- fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
- obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
- have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
- hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
- hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
- hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto
- thus "d (snd x) = 0" unfolding ab by auto qed qed
-
-lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
+proof -
+ note assm = tagged_division_ofD[OF assms(1)]
+ have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
+ unfolding o_def by (rule ext) auto
+ show ?thesis
+ unfolding *
+ apply (subst eq_commute)
+ proof (rule setsum_reindex_nonzero)
+ show "finite p"
+ using assm by auto
+ fix x y
+ assume as: "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
+ obtain a b where ab: "snd x = {a..b}"
+ using assm(4)[of "fst x" "snd x"] as(1) by auto
+ have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
+ unfolding as(4)[symmetric] using as(1-3) by auto
+ then have "interior (snd x) \<inter> interior (snd y) = {}"
+ apply -
+ apply (rule assm(5)[of "fst x" _ "fst y"])
+ using as
+ apply auto
+ done
+ then have "content {a..b} = 0"
+ unfolding as(4)[symmetric] ab content_eq_0_interior by auto
+ then have "d {a..b} = 0"
+ apply -
+ apply (rule assms(2))
+ using assm(2)[of "fst x" "snd x"] as(1)
+ unfolding ab[symmetric]
+ apply auto
+ done
+ then show "d (snd x) = 0"
+ unfolding ab by auto
+ qed
+qed
+
+lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
+ by auto
lemma tagged_division_of_empty: "{} tagged_division_of {}"
unfolding tagged_division_of by auto
-lemma tagged_partial_division_of_trivial[simp]:
- "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
+lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
unfolding tagged_partial_division_of_def by auto
-lemma tagged_division_of_trivial[simp]:
- "p tagged_division_of {} \<longleftrightarrow> p = {}"
+lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
unfolding tagged_division_of by auto
-lemma tagged_division_of_self:
- "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
- apply(rule tagged_division_ofI) by auto
+lemma tagged_division_of_self: "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
+ by (rule tagged_division_ofI) auto
lemma tagged_division_union:
- assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
+ assumes "p1 tagged_division_of s1"
+ and "p2 tagged_division_of s2"
+ and "interior s1 \<inter> interior s2 = {}"
shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
-proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
- show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
- show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
- fix x k assume xk:"(x,k)\<in>p1\<union>p2" show "x\<in>k" "\<exists>a b. k = {a..b}" using xk p1(2,4) p2(2,4) by auto
- show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
- fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
- have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) interior_mono by blast
- show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
- apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
- using p1(3) p2(3) using xk xk' by auto qed
+proof (rule tagged_division_ofI)
+ note p1 = tagged_division_ofD[OF assms(1)]
+ note p2 = tagged_division_ofD[OF assms(2)]
+ show "finite (p1 \<union> p2)"
+ using p1(1) p2(1) by auto
+ show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
+ using p1(6) p2(6) by blast
+ fix x k
+ assume xk: "(x, k) \<in> p1 \<union> p2"
+ show "x \<in> k" "\<exists>a b. k = {a..b}"
+ using xk p1(2,4) p2(2,4) by auto
+ show "k \<subseteq> s1 \<union> s2"
+ using xk p1(3) p2(3) by blast
+ fix x' k'
+ assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
+ have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
+ using assms(3) interior_mono by blast
+ show "interior k \<inter> interior k' = {}"
+ apply (cases "(x, k) \<in> p1")
+ apply (case_tac[!] "(x',k') \<in> p1")
+ apply (rule p1(5))
+ prefer 4
+ apply (rule *)
+ prefer 6
+ apply (subst Int_commute)
+ apply (rule *)
+ prefer 8
+ apply (rule p2(5))
+ using p1(3) p2(3)
+ using xk xk'
+ apply auto
+ done
+qed
lemma tagged_division_unions:
- assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
- "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
+ 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)"
-proof(rule tagged_division_ofI)
+proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF assms(2)[rule_format]]
- show "finite (\<Union>(pfn ` iset))" apply(rule finite_Union) 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)" by blast
- also have "\<dots> = \<Union>iset" using assm(6) by auto
- finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
- fix x k assume xk:"(x,k)\<in>\<Union>(pfn ` iset)" then obtain i where i:"i \<in> iset" "(x, k) \<in> pfn i" by auto
- show "x\<in>k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset" 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'" 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)
- using assms(3)[rule_format] interior_mono by blast
- show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
- using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
+ show "finite (\<Union>(pfn ` iset))"
+ apply (rule finite_Union)
+ using assms
+ apply auto
+ done
+ have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
+ by blast
+ also have "\<dots> = \<Union>iset"
+ using assm(6) by auto
+ finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
+ fix x k
+ assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
+ then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
+ by auto
+ show "x \<in> k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset"
+ 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'"
+ 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)
+ using assms(3)[rule_format] interior_mono
+ by blast
+ show "interior k \<inter> interior k' = {}"
+ apply (cases "i = i'")
+ using assm(5)[OF i _ xk'(2)] i'(2)
+ using assm(3)[OF i] assm(3)[OF i']
+ defer
+ apply -
+ apply (rule *)
+ apply auto
+ done
qed
lemma tagged_partial_division_of_union_self:
- assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
- apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
-
-lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
+ assumes "p tagged_partial_division_of s"
shows "p tagged_division_of (\<Union>(snd ` p))"
- apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
+ apply (rule tagged_division_ofI)
+ using tagged_partial_division_ofD[OF assms]
+ apply auto
+ done
+
+lemma tagged_division_of_union_self:
+ assumes "p tagged_division_of s"
+ shows "p tagged_division_of (\<Union>(snd ` p))"
+ apply (rule tagged_division_ofI)
+ using tagged_division_ofD[OF assms]
+ apply auto
+ done
+
subsection {* Fine-ness of a partition w.r.t. a gauge. *}
-definition fine (infixr "fine" 46) where
- "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
-
-lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
- shows "d fine s" using assms unfolding fine_def by auto
-
-lemma fineD[dest]: assumes "d fine s"
- shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
+definition fine (infixr "fine" 46)
+ where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
+
+lemma fineI:
+ assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
+ shows "d fine s"
+ using assms unfolding fine_def by auto
+
+lemma fineD[dest]:
+ assumes "d fine s"
+ shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
+ using assms unfolding fine_def by auto
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
unfolding fine_def by auto
@@ -1393,570 +1911,1222 @@
"(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
unfolding fine_def by blast
-lemma fine_union:
- "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
+lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
unfolding fine_def by blast
-lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
+lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
unfolding fine_def by auto
-lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
+lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
unfolding fine_def by blast
+
subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
-definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
- "(f has_integral_compact_interval y) i \<equiv>
- (\<forall>e>0. \<exists>d. gauge d \<and>
- (\<forall>p. p tagged_division_of i \<and> d fine p
- \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
-
-definition has_integral (infixr "has'_integral" 46) where
-"((f::('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
- if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
- else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
- \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
- norm(z - y) < e))"
+definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
+ where "(f has_integral_compact_interval y) i \<longleftrightarrow>
+ (\<forall>e>0. \<exists>d. gauge d \<and>
+ (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
+ norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
+
+definition has_integral ::
+ "('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
+ (infixr "has'_integral" 46)
+ where "(f has_integral y) i \<longleftrightarrow>
+ (if \<exists>a b. i = {a..b}
+ then (f has_integral_compact_interval y) i
+ else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+ (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
+ norm (z - y) < e)))"
lemma has_integral:
- "(f has_integral y) ({a..b}) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
- \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
- unfolding has_integral_def has_integral_compact_interval_def by auto
-
-lemma has_integralD[dest]: assumes
- "(f has_integral y) ({a..b})" "e>0"
- obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
- \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
+ "(f has_integral y) {a..b} \<longleftrightarrow>
+ (\<forall>e>0. \<exists>d. gauge d \<and>
+ (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
+ norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
+ unfolding has_integral_def has_integral_compact_interval_def
+ by auto
+
+lemma has_integralD[dest]:
+ assumes "(f has_integral y) ({a..b})"
+ and "e > 0"
+ obtains d where "gauge d"
+ and "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p \<Longrightarrow>
+ norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
using assms unfolding has_integral by auto
lemma has_integral_alt:
- "(f has_integral y) i \<longleftrightarrow>
- (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
- else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
- \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
- has_integral z) ({a..b}) \<and>
- norm(z - y) < e)))"
- unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
+ "(f has_integral y) i \<longleftrightarrow>
+ (if \<exists>a b. i = {a..b}
+ then (f has_integral y) i
+ else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+ (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm (z - y) < e)))"
+ unfolding has_integral
+ unfolding has_integral_compact_interval_def has_integral_def
+ by auto
lemma has_integral_altD:
- assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
- obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
- using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
-
-definition integrable_on (infixr "integrable'_on" 46) where
- "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
-
-definition "integral i f \<equiv> SOME y. (f has_integral y) i"
-
-lemma integrable_integral[dest]:
- "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
- unfolding integrable_on_def integral_def by(rule someI_ex)
+ assumes "(f has_integral y) i"
+ and "\<not> (\<exists>a b. i = {a..b})"
+ and "e>0"
+ obtains B where "B > 0"
+ and "\<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+ (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
+ using assms
+ unfolding has_integral
+ unfolding has_integral_compact_interval_def has_integral_def
+ by auto
+
+definition integrable_on (infixr "integrable'_on" 46)
+ where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
+
+definition "integral i f = (SOME y. (f has_integral y) i)"
+
+lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
+ unfolding integrable_on_def integral_def by (rule someI_ex)
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
unfolding integrable_on_def by auto
-lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
+lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
by auto
lemma setsum_content_null:
- assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
+ assumes "content {a..b} = 0"
+ and "p tagged_division_of {a..b}"
shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
-proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
- obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
+proof (rule setsum_0', rule)
+ fix y
+ assume y: "y \<in> p"
+ obtain x k where xk: "y = (x, k)"
+ using surj_pair[of y] by blast
note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
- from this(2) guess c .. then guess d .. note c_d=this
- have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
- also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
- unfolding assms(1) c_d by auto
+ from this(2) obtain c d where k: "k = {c..d}" by blast
+ have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
+ unfolding xk by auto
+ also have "\<dots> = 0"
+ using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
+ unfolding assms(1) k
+ by auto
finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
qed
+
subsection {* Some basic combining lemmas. *}
lemma tagged_division_unions_exists:
- assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
- "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
- obtains p where "p tagged_division_of i" "d fine p"
-proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
- show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
- apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
- apply(rule fine_unions) using pfn by auto
+ 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"
+ 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"
+ using bchoice[OF assms(2)] by auto
+ show thesis
+ apply (rule_tac p="\<Union>(pfn ` iset)" in that)
+ unfolding assms(4)[symmetric]
+ apply (rule tagged_division_unions[OF assms(1) _ assms(3)])
+ defer
+ apply (rule fine_unions)
+ using pfn
+ apply auto
+ done
qed
+
subsection {* The set we're concerned with must be closed. *}
-lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::('n::ordered_euclidean_space) set)"
+lemma division_of_closed:
+ fixes i :: "'n::ordered_euclidean_space set"
+ shows "s division_of i \<Longrightarrow> closed i"
unfolding division_of_def by fastforce
subsection {* General bisection principle for intervals; might be useful elsewhere. *}
-lemma interval_bisection_step: fixes type::"'a::ordered_euclidean_space"
- assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::'a})"
- obtains c d where "~(P{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"
-proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
- then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" by (auto simp: interval_eq_empty not_le)
- { fix f have "finite f \<Longrightarrow>
- (\<forall>s\<in>f. P s) \<Longrightarrow>
- (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
- (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<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])
- apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
- using insert by auto
- qed } note * = this
- let ?A = "{{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)}"
+lemma interval_bisection_step:
+ fixes type :: "'a::ordered_euclidean_space"
+ assumes "P {}"
+ and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
+ and "\<not> P {a..b::'a}"
+ obtains c d where "\<not> P{c..d}"
+ and "\<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"
+proof -
+ have "{a..b} \<noteq> {}"
+ using assms(1,3) by auto
+ then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
+ by (auto simp: interval_eq_empty not_le)
+ {
+ fix f
+ have "finite f \<Longrightarrow>
+ \<forall>s\<in>f. P s \<Longrightarrow>
+ \<forall>s\<in>f. \<exists>a b. s = {a..b} \<Longrightarrow>
+ \<forall>s\<in>f.\<forall>t\<in>f. s \<noteq> t \<longrightarrow> interior s \<inter> interior t = {} \<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])
+ apply rule
+ defer
+ apply rule
+ defer
+ apply (rule inter_interior_unions_intervals)
+ using insert
+ apply auto
+ done
+ qed
+ } note * = this
+ let ?A = "{{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"
- { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
- thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
- assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
- have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
+ {
+ presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
+ then show thesis
+ unfolding atomize_not not_all
+ apply -
+ apply (erule exE)+
+ apply (rule_tac c=x and d=xa in that)
+ apply auto
+ done
+ }
+ assume as: "\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
+ have "P (\<Union> ?A)"
+ apply (rule *)
+ apply (rule_tac[2-] ballI)
+ apply (rule_tac[4] ballI)
+ apply (rule_tac[4] impI)
+ proof -
let ?B = "(\<lambda>s.{(\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i)::'a ..
(\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)}) ` {s. s \<subseteq> Basis}"
- have "?A \<subseteq> ?B" proof case goal1
- then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
- have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
- show "x\<in>?B" unfolding image_iff
- apply(rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
- unfolding c_d
- apply(rule *)
+ have "?A \<subseteq> ?B"
+ proof
+ case goal1
+ then obtain c d where x: "x = {c..d}"
+ "\<And>i. i \<in> Basis \<Longrightarrow>
+ 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" by blast
+ have *: "\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}"
+ by auto
+ show "x \<in> ?B"
+ unfolding image_iff
+ apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
+ unfolding x
+ apply (rule *)
apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
- cong: ball_cong)
+ cong: ball_cong)
apply safe
- proof-
- fix i :: 'a assume i: "i\<in>Basis"
- thus " c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
- "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
- using c_d(2)[of i] ab[OF i] by(auto simp add:field_simps)
- qed qed
- thus "finite ?A" apply(rule finite_subset) by auto
- fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
- note c_d=this[rule_format]
- show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 thus ?case
- using c_d(2)[of i] using ab[OF `i \<in> Basis`] by auto qed
- show "\<exists>a b. s = {a..b}" unfolding c_d by auto
- fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
- note e_f=this[rule_format]
- assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
- then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i':"i\<in>Basis"
+ proof -
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then show "c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
+ and "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
+ using x(2)[of i] ab[OF i] by (auto simp add:field_simps)
+ qed
+ qed
+ then show "finite ?A"
+ by (rule finite_subset) auto
+ fix s
+ assume "s \<in> ?A"
+ then obtain c d where s:
+ "s = {c..d}"
+ "\<And>i. i \<in> Basis \<Longrightarrow>
+ 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"
+ by blast
+ show "P s"
+ unfolding s
+ apply (rule as[rule_format])
+ proof -
+ case goal1
+ then show ?case
+ using s(2)[of i] using ab[OF `i \<in> Basis`] by auto
+ qed
+ show "\<exists>a b. s = {a..b}"
+ unfolding s by auto
+ fix t
+ assume "t \<in> ?A"
+ then obtain e f where t:
+ "t = {e..f}"
+ "\<And>i. i \<in> Basis \<Longrightarrow>
+ e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+ e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
+ by blast
+ assume "s \<noteq> t"
+ then have "\<not> (c = e \<and> d = f)"
+ unfolding s t by auto
+ then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
unfolding euclidean_eq_iff[where 'a='a] by auto
- hence i:"c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i" apply- apply(erule_tac[!] disjE)
- proof- assume "c\<bullet>i \<noteq> e\<bullet>i" thus "d\<bullet>i \<noteq> f\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
- next assume "d\<bullet>i \<noteq> f\<bullet>i" thus "c\<bullet>i \<noteq> e\<bullet>i" using c_d(2)[OF i'] e_f(2)[OF i'] by fastforce
- qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
- show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
- fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
- hence x:"c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i" unfolding mem_interval using i'
- apply-apply(erule_tac[!] x=i in ballE)+ by auto
- show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
- proof(erule_tac[!] conjE) assume as:"c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
- show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
- next assume as:"c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
- show False using e_f(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
- qed qed qed
- also have "\<Union> ?A = {a..b}" proof(rule set_eqI,rule)
- fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
- from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
- note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
- show "x\<in>{a..b}" unfolding mem_interval proof safe
- fix i :: 'a assume i: "i\<in>Basis" thus "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
- using c_d(1)[OF i] c_d(2)[unfolded mem_interval,THEN bspec, OF i] by auto qed
- next fix x assume x:"x\<in>{a..b}"
- have "\<forall>i\<in>Basis. \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
- (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d") unfolding mem_interval
+ then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
+ apply -
+ apply(erule_tac[!] disjE)
+ proof -
+ assume "c\<bullet>i \<noteq> e\<bullet>i"
+ then show "d\<bullet>i \<noteq> f\<bullet>i"
+ using s(2)[OF i'] t(2)[OF i'] by fastforce
+ next
+ assume "d\<bullet>i \<noteq> f\<bullet>i"
+ then show "c\<bullet>i \<noteq> e\<bullet>i"
+ using s(2)[OF i'] t(2)[OF i'] by fastforce
+ qed
+ have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
+ by auto
+ show "interior s \<inter> interior t = {}"
+ unfolding s t interior_closed_interval
+ proof (rule *)
+ fix x
+ assume "x \<in> {c<..<d}" "x \<in> {e<..<f}"
+ then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
+ unfolding mem_interval using i'
+ apply -
+ apply (erule_tac[!] x=i in ballE)+
+ apply auto
+ done
+ show False
+ using s(2)[OF i']
+ apply -
+ apply (erule_tac disjE)
+ apply (erule_tac[!] conjE)
+ proof -
+ assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
+ show False
+ using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
+ next
+ assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
+ show False
+ using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
+ qed
+ qed
+ qed
+ also have "\<Union> ?A = {a..b}"
+ proof (rule set_eqI,rule)
+ fix x
+ assume "x \<in> \<Union>?A"
+ then obtain c d where x:
+ "x \<in> {c..d}"
+ "\<And>i. i \<in> Basis \<Longrightarrow>
+ 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" by blast
+ show "x\<in>{a..b}"
+ unfolding mem_interval
+ proof safe
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
+ using x(2)[OF i] x(1)[unfolded mem_interval,THEN bspec, OF i] by auto
+ qed
+ next
+ fix x
+ assume x: "x \<in> {a..b}"
+ have "\<forall>i\<in>Basis.
+ \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
+ (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
+ unfolding mem_interval
proof
- fix i :: 'a assume i: "i \<in> Basis"
+ fix i :: 'a
+ assume i: "i \<in> Basis"
have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
- using x[unfolded mem_interval,THEN bspec, OF i] by auto thus "\<exists>c d. ?P i c d" by blast
+ using x[unfolded mem_interval,THEN bspec, OF i] by auto
+ then show "\<exists>c d. ?P i c d"
+ by blast
qed
- thus "x\<in>\<Union>?A"
+ then show "x\<in>\<Union>?A"
unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
- apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
- qed finally show False using assms by auto qed
-
-lemma interval_bisection: fixes type::"'a::ordered_euclidean_space"
- assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::'a}"
- obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
-proof-
+ apply -
+ apply (erule exE)+
+ apply (rule_tac x="{xa..xaa}" in exI)
+ unfolding mem_interval
+ apply auto
+ done
+ qed
+ finally show False
+ using assms by auto
+qed
+
+lemma interval_bisection:
+ fixes type :: "'a::ordered_euclidean_space"
+ assumes "P {}"
+ and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
+ and "\<not> P {a..b::'a}"
+ obtains x where "x \<in> {a..b}"
+ and "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
+proof -
have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
(\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
- 2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" proof case goal1 thus ?case proof-
+ 2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
+ proof
+ case goal1
+ then show ?case
+ proof -
presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
- thus ?thesis apply(cases "P {fst x..snd x}") by auto
- next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
- thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
- qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
- def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
+ then show ?thesis by (cases "P {fst x..snd x}") auto
+ next
+ assume as: "\<not> P {fst x..snd x}"
+ obtain c d where "\<not> P {c..d}"
+ "\<forall>i\<in>Basis.
+ fst x \<bullet> i \<le> c \<bullet> i \<and>
+ c \<bullet> i \<le> d \<bullet> i \<and>
+ d \<bullet> i \<le> snd x \<bullet> i \<and>
+ 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
+ qed
+ qed
+ then guess f
+ apply -
+ apply (drule choice)
+ apply (erule exE)
+ done
+ note f = this
+ def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
+ def A \<equiv> "\<lambda>n. fst(AB n)"
+ def B \<equiv> "\<lambda>n. snd(AB n)"
+ note ab_def = A_def B_def AB_def
have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {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>
+ (\<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>
2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
- proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
- case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
- proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
- next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
- qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
-
- have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
- proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] .. note n=this
- show ?case apply(rule_tac x=n in exI) proof(rule,rule)
- fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
- have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis" unfolding dist_norm by(rule norm_le_l1)
+ proof -
+ show "A 0 = a" "B 0 = b"
+ unfolding ab_def by auto
+ case goal3
+ note S = ab_def funpow.simps o_def id_apply
+ show ?case
+ proof (induct n)
+ case 0
+ then show ?case
+ unfolding S
+ apply (rule f[rule_format]) using assms(3)
+ apply auto
+ done
+ next
+ case (Suc n)
+ show ?case
+ unfolding S
+ apply (rule f[rule_format])
+ using Suc
+ unfolding S
+ apply auto
+ done
+ qed
+ qed
+ note AB = this(1-2) conjunctD2[OF this(3),rule_format]
+
+ have interv: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
+ proof -
+ case goal1
+ obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
+ using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
+ show ?case
+ apply (rule_tac x=n in exI)
+ apply rule
+ apply rule
+ proof -
+ fix x y
+ assume xy: "x\<in>{A n..B n}" "y\<in>{A n..B n}"
+ have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
+ unfolding dist_norm by(rule norm_le_l1)
also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
- proof(rule setsum_mono)
- fix i :: 'a assume i: "i \<in> Basis" show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
- using xy[unfolded mem_interval,THEN bspec, OF i] by (auto simp: inner_diff_left) qed
- also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n" unfolding setsum_divide_distrib
- proof(rule setsum_mono) case goal1 thus ?case
- proof(induct n) case 0 thus ?case unfolding AB by auto
- 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"
+ proof (rule setsum_mono)
+ fix i :: 'a
+ assume i: "i \<in> Basis"
+ show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
+ using xy[unfolded mem_interval,THEN bspec, OF i]
+ by (auto simp: inner_diff_left)
+ qed
+ also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
+ unfolding setsum_divide_distrib
+ proof (rule setsum_mono)
+ case goal1
+ then show ?case
+ proof (induct n)
+ case 0
+ then show ?case
+ unfolding AB by auto
+ 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(4)[of i n] using goal1 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 .
- qed qed
- also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
- qed qed
- { fix n m :: nat assume "m \<le> n" then have "{A n..B n} \<subseteq> {A m..B m}"
- proof(induct rule: inc_induct)
- case (step i) show ?case
+ also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
+ using Suc by (auto simp add:field_simps)
+ finally show ?case .
+ qed
+ qed
+ also have "\<dots> < e"
+ using n using goal1 by (auto simp add:field_simps)
+ finally show "dist x y < e" .
+ qed
+ qed
+ {
+ fix n m :: nat
+ assume "m \<le> n"
+ then have "{A n..B n} \<subseteq> {A m..B m}"
+ proof (induct rule: inc_induct)
+ case (step i)
+ show ?case
using AB(4) by (intro order_trans[OF step(2)] subset_interval_imp) auto
- qed simp } note ABsubset = this
- have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
- proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
- then guess x0 .. note x0=this[rule_format]
- show thesis proof(rule that[rule_format,of x0])
- show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
- fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
+ qed simp
+ } note ABsubset = this
+ have "\<exists>a. \<forall>n. a\<in>{A n..B n}"
+ apply (rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
+ proof -
+ fix n
+ show "{A n..B n} \<noteq> {}"
+ apply (cases "0 < n")
+ using AB(3)[of "n - 1"] assms(1,3) AB(1-2)
+ apply auto
+ done
+ qed auto
+ then obtain x0 where x0: "\<And>n. x0 \<in> {A n..B n}"
+ by blast
+ show thesis
+ proof (rule that[rule_format, of x0])
+ show "x0\<in>{a..b}"
+ using x0[of 0] unfolding AB .
+ fix e :: real
+ assume "e > 0"
+ from interv[OF this] obtain n
+ where n: "\<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e" ..
show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
- apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
- proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto
- show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
- show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
- qed qed qed
+ apply (rule_tac x="A n" in exI)
+ apply (rule_tac x="B n" in exI)
+ apply rule
+ apply (rule x0)
+ apply rule
+ defer
+ apply rule
+ proof -
+ show "\<not> P {A n..B n}"
+ apply (cases "0 < n")
+ using AB(3)[of "n - 1"] assms(3) AB(1-2)
+ apply auto
+ done
+ show "{A n..B n} \<subseteq> ball x0 e"
+ using n using x0[of n] by auto
+ show "{A n..B n} \<subseteq> {a..b}"
+ unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
+ qed
+ qed
+qed
+
subsection {* Cousin's lemma. *}
-lemma fine_division_exists: assumes "gauge g"
- obtains p where "p tagged_division_of {a..b::'a::ordered_euclidean_space}" "g fine p"
-proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
- then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
-next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
- guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
- apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
- proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
- fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"
- thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule
- apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
- qed note x=this
- obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
- from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
- have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
- thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
+lemma fine_division_exists:
+ fixes a b :: "'a::ordered_euclidean_space"
+ assumes "gauge g"
+ obtains p where "p tagged_division_of {a..b}" "g fine p"
+proof -
+ presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
+ then obtain p where "p tagged_division_of {a..b}" "g fine p"
+ by blast
+ then show thesis ..
+next
+ assume as: "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
+ guess x
+ apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
+ apply (rule_tac x="{}" in exI)
+ defer
+ apply (erule conjE exE)+
+ proof -
+ show "{} tagged_division_of {} \<and> g fine {}"
+ unfolding fine_def by auto
+ fix s t p p'
+ assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
+ "interior s \<inter> interior t = {}"
+ then show "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p"
+ apply -
+ apply (rule_tac x="p \<union> p'" in exI)
+ apply rule
+ apply (rule tagged_division_union)
+ prefer 4
+ apply (rule fine_union)
+ apply auto
+ done
+ qed note x = this
+ obtain e where e: "e > 0" "ball x e \<subseteq> g x"
+ using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
+ from x(2)[OF e(1)] obtain c d where c_d:
+ "x \<in> {c..d}"
+ "{c..d} \<subseteq> ball x e"
+ "{c..d} \<subseteq> {a..b}"
+ "\<not> (\<exists>p. p tagged_division_of {c..d} \<and> g fine p)"
+ by blast
+ have "g fine {(x, {c..d})}"
+ unfolding fine_def using e using c_d(2) by auto
+ then show False
+ using tagged_division_of_self[OF c_d(1)] using c_d by auto
+qed
+
subsection {* Basic theorems about integrals. *}
-lemma has_integral_unique: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
-proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
- have lem:"\<And>f::'n \<Rightarrow> 'a. \<And> a b k1 k2.
+lemma has_integral_unique:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral k1) i"
+ and "(f has_integral k2) i"
+ shows "k1 = k2"
+proof (rule ccontr)
+ let ?e = "norm(k1 - k2) / 2"
+ assume as:"k1 \<noteq> k2"
+ then have e: "?e > 0"
+ by auto
+ have lem: "\<And>f::'n \<Rightarrow> 'a. \<And>a b k1 k2.
(f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
- proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
- guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
- guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
- guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
- let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
- using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
+ proof -
+ case goal1
+ let ?e = "norm (k1 - k2) / 2"
+ from goal1(3) have e: "?e > 0" by auto
+ guess d1 by (rule has_integralD[OF goal1(1) e]) note d1=this
+ guess d2 by (rule has_integralD[OF goal1(2) e]) note d2=this
+ guess p by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
+ let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
+ have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
+ using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
+ by (auto simp add:algebra_simps norm_minus_commute)
also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
- apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
+ apply (rule add_strict_mono)
+ apply (rule_tac[!] d2(2) d1(2))
+ using p unfolding fine_def
+ apply auto
+ done
finally show False by auto
- qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
- thus False apply-apply(cases "\<exists>a b. i = {a..b}")
- using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
- assume as:"\<not> (\<exists>a b. i = {a..b})"
- guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
- guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
- have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
- using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
- note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
- guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
- guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
- have "z = w" using lem[OF w(1) z(1)] by auto
- hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
- using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
- also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
- finally show False by auto qed
-
-lemma integral_unique[intro]:
- "(f has_integral y) k \<Longrightarrow> integral k f = y"
- unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
-
-lemma has_integral_is_0: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
- assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
-proof- have lem:"\<And>a b. \<And>f::'n \<Rightarrow> 'a.
- (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
- proof(rule,rule) fix a b e and f::"'n \<Rightarrow> 'a"
- assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
- show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
- apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
- proof(rule,rule,erule conjE) case goal1
- have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
- fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
- thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
- qed thus ?case using as by auto
- qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
- using assms by(auto simp add:has_integral intro:lem) }
- have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
- assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
- apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
- proof- fix e::real and a b assume "e>0"
- thus "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
- apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
- qed auto qed
+ qed
+ {
+ presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
+ then show False
+ apply -
+ apply (cases "\<exists>a b. i = {a..b}")
+ using assms
+ apply (auto simp add:has_integral intro:lem[OF _ _ as])
+ done
+ }
+ assume as: "\<not> (\<exists>a b. i = {a..b})"
+ guess B1 by (rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
+ guess B2 by (rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
+ have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}"
+ apply (rule bounded_subset_closed_interval)
+ using bounded_Un bounded_ball
+ apply auto
+ done
+ then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> {a..b}" "ball 0 B2 \<subseteq> {a..b}"
+ by blast
+ obtain w where w:
+ "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) {a..b}"
+ "norm (w - k1) < norm (k1 - k2) / 2"
+ using B1(2)[OF ab(1)] by blast
+ obtain z where z:
+ "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) {a..b}"
+ "norm (z - k2) < norm (k1 - k2) / 2"
+ using B2(2)[OF ab(2)] by blast
+ have "z = w"
+ using lem[OF w(1) z(1)] by auto
+ then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
+ using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
+ by (auto simp add: norm_minus_commute)
+ also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
+ apply (rule add_strict_mono)
+ apply (rule_tac[!] z(2) w(2))
+ done
+ finally show False by auto
+qed
+
+lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
+ unfolding integral_def
+ by (rule some_equality) (auto intro: has_integral_unique)
+
+lemma has_integral_is_0:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+ assumes "\<forall>x\<in>s. f x = 0"
+ shows "(f has_integral 0) s"
+proof -
+ have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
+ (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
+ unfolding has_integral
+ apply rule
+ apply rule
+ proof -
+ fix a b e
+ fix f :: "'n \<Rightarrow> 'a"
+ assume as: "\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
+ show "\<exists>d. gauge d \<and>
+ (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
+ apply (rule_tac x="\<lambda>x. ball x 1" in exI)
+ apply rule
+ apply (rule gaugeI)
+ unfolding centre_in_ball
+ defer
+ apply (rule open_ball)
+ apply rule
+ apply rule
+ apply (erule conjE)
+ proof -
+ case goal1
+ have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
+ proof (rule setsum_0', rule)
+ fix x
+ assume x: "x \<in> p"
+ have "f (fst x) = 0"
+ using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
+ then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
+ apply (subst surjective_pairing[of x])
+ unfolding split_conv
+ apply auto
+ done
+ qed
+ then show ?case
+ using as by auto
+ qed auto
+ qed
+ {
+ presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ then show ?thesis
+ apply -
+ apply (cases "\<exists>a b. s = {a..b}")
+ using assms
+ apply (auto simp add:has_integral intro: lem)
+ done
+ }
+ have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
+ apply (rule ext)
+ using assms
+ apply auto
+ done
+ assume "\<not> (\<exists>a b. s = {a..b})"
+ then show ?thesis
+ apply (subst has_integral_alt)
+ unfolding if_not_P *
+ apply rule
+ apply rule
+ apply (rule_tac x=1 in exI)
+ apply rule
+ defer
+ apply rule
+ apply rule
+ apply rule
+ proof -
+ fix e :: real
+ fix a b
+ assume "e > 0"
+ then show "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
+ apply (rule_tac x=0 in exI)
+ apply(rule,rule lem)
+ apply auto
+ done
+ qed auto
+qed
lemma has_integral_0[simp]: "((\<lambda>x::'n::ordered_euclidean_space. 0) has_integral 0) s"
- apply(rule has_integral_is_0) by auto
+ by (rule has_integral_is_0) auto
lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
using has_integral_unique[OF has_integral_0] by auto
-lemma has_integral_linear: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
-proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
- have lem:"\<And>f::'n \<Rightarrow> 'a. \<And> y a b.
- (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
- proof(subst has_integral,rule,rule) case goal1
- from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
- have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
- guess g using has_integralD[OF goal1(1) *] . note g=this
- show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
- proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
- have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto
+lemma has_integral_linear:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral y) s"
+ and "bounded_linear h"
+ shows "((h o f) has_integral ((h y))) s"
+proof -
+ interpret bounded_linear h
+ using assms(2) .
+ from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
+ by blast
+ have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
+ (f has_integral y) {a..b} \<Longrightarrow> ((h o f) has_integral h y) {a..b}"
+ apply (subst has_integral)
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ from pos_bounded
+ obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
+ by blast
+ have *: "e / B > 0"
+ apply (rule divide_pos_pos)
+ using goal1(2) B
+ apply auto
+ done
+ thm has_integralD[OF goal1(1) *]
+ obtain g where g:
+ "gauge g"
+ "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> g fine p \<Longrightarrow>
+ norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
+ by (rule has_integralD[OF goal1(1) *]) blast
+ show ?case
+ apply (rule_tac x=g in exI)
+ apply rule
+ apply (rule g(1))
+ apply rule
+ apply rule
+ apply (erule conjE)
+ proof -
+ fix p
+ assume as: "p tagged_division_of {a..b}" "g fine p"
+ have *: "\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x"
+ by auto
have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
- unfolding o_def unfolding scaleR[THEN sym] * by simp
- also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
- finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
- apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
- qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
- assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
- proof(rule,rule) fix e::real assume e:"0<e"
- have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
- guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
- show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
- apply(rule_tac x=M in exI) apply(rule,rule M(1))
- proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
- have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
- unfolding o_def apply(rule ext) using zero by auto
- show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
- apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
- qed qed qed
-
-lemma has_integral_cmul:
- shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
- unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
- by(rule bounded_linear_scaleR_right)
+ unfolding o_def unfolding scaleR[symmetric] * by simp
+ also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
+ using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
+ finally have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
+ unfolding * diff[symmetric]
+ apply (rule le_less_trans[OF B(2)])
+ using g(2)[OF as] B(1)
+ apply (auto simp add: field_simps)
+ done
+ qed
+ qed
+ {
+ presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ then show ?thesis
+ apply -
+ apply (cases "\<exists>a b. s = {a..b}")
+ using assms
+ apply (auto simp add:has_integral intro!:lem)
+ done
+ }
+ assume as: "\<not> (\<exists>a b. s = {a..b})"
+ then show ?thesis
+ apply (subst has_integral_alt)
+ unfolding if_not_P
+ apply rule
+ apply rule
+ proof -
+ fix e :: real
+ assume e: "e > 0"
+ have *: "0 < e/B"
+ by (rule divide_pos_pos,rule e,rule B(1))
+ obtain M where M:
+ "M > 0"
+ "\<And>a b. ball 0 M \<subseteq> {a..b} \<Longrightarrow>
+ \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) {a..b} \<and> norm (z - y) < e / B"
+ using has_integral_altD[OF assms(1) as *] by blast
+ show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
+ (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
+ apply (rule_tac x=M in exI)
+ apply rule
+ apply (rule M(1))
+ apply rule
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ obtain z where z:
+ "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) {a..b}"
+ "norm (z - y) < e / B"
+ using M(2)[OF goal1(1)] by blast
+ have *: "(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
+ unfolding o_def
+ apply (rule ext)
+ using zero
+ apply auto
+ done
+ show ?case
+ apply (rule_tac x="h z" in exI)
+ apply rule
+ unfolding *
+ apply (rule lem[OF z(1)])
+ unfolding diff[symmetric]
+ apply (rule le_less_trans[OF B(2)])
+ using B(1) z(2)
+ apply (auto simp add: field_simps)
+ done
+ qed
+ qed
+qed
+
+lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
+ unfolding o_def[symmetric]
+ apply (rule has_integral_linear,assumption)
+ apply (rule bounded_linear_scaleR_right)
+ done
lemma has_integral_cmult_real:
fixes c :: real
assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
shows "((\<lambda>x. c * f x) has_integral c * x) A"
-proof cases
- assume "c \<noteq> 0"
+proof (cases "c = 0")
+ case True
+ then show ?thesis by simp
+next
+ case False
from has_integral_cmul[OF assms[OF this], of c] show ?thesis
unfolding real_scaleR_def .
-qed simp
-
-lemma has_integral_neg:
- shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
- apply(drule_tac c="-1" in has_integral_cmul) by auto
-
-lemma has_integral_add: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
- assumes "(f has_integral k) s" "(g has_integral l) s"
+qed
+
+lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
+ apply (drule_tac c="-1" in has_integral_cmul)
+ apply auto
+ done
+
+lemma has_integral_add:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
+ assumes "(f has_integral k) s"
+ and "(g has_integral l) s"
shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
-proof- have lem:"\<And>f g::'n \<Rightarrow> 'a. \<And>a b k l.
- (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
- ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
- show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
- guess d1 using has_integralD[OF goal1(1) *] . note d1=this
- guess d2 using has_integralD[OF goal1(2) *] . note d2=this
- show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
- apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
- proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
- have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
- unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
- by(rule setsum_cong2,auto)
- have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
- unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
- from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
- have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
- apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
- finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
- qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
- thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
- assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
- proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
+proof -
+ have lem:"\<And>(f:: 'n \<Rightarrow> 'a) g a b k l.
+ (f has_integral k) {a..b} \<Longrightarrow>
+ (g has_integral l) {a..b} \<Longrightarrow>
+ ((\<lambda>x. f x + g x) has_integral (k + l)) {a..b}"
+ proof -
+ case goal1
+ show ?case
+ unfolding has_integral
+ apply rule
+ apply rule
+ proof -
+ fix e :: real
+ assume e: "e > 0"
+ then have *: "e/2 > 0"
+ by auto
+ obtain d1 where d1:
+ "gauge d1"
+ "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d1 fine p \<Longrightarrow>
+ norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) < e / 2"
+ using has_integralD[OF goal1(1) *] by blast
+ obtain d2 where d2:
+ "gauge d2"
+ "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d2 fine p \<Longrightarrow>
+ norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l) < e / 2"
+ using has_integralD[OF goal1(2) *] by blast
+ show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
+ norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
+ apply (rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI)
+ apply rule
+ apply (rule gauge_inter[OF d1(1) d2(1)])
+ apply (rule,rule,erule conjE)
+ proof -
+ fix p
+ assume as: "p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
+ have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) =
+ (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
+ unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,symmetric]
+ by (rule setsum_cong2) auto
+ have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) =
+ norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
+ unfolding * by (auto simp add: algebra_simps)
+ also
+ let ?res = "\<dots>"
+ from as have *: "d1 fine p" "d2 fine p"
+ unfolding fine_inter by auto
+ have "?res < e/2 + e/2"
+ apply (rule le_less_trans[OF norm_triangle_ineq])
+ apply (rule add_strict_mono)
+ using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)]
+ apply auto
+ done
+ finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e"
+ by auto
+ qed
+ qed
+ qed
+ {
+ presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
+ then show ?thesis
+ apply -
+ apply (cases "\<exists>a b. s = {a..b}")
+ using assms
+ apply (auto simp add:has_integral intro!:lem)
+ done
+ }
+ assume as: "\<not> (\<exists>a b. s = {a..b})"
+ then show ?thesis
+ apply (subst has_integral_alt)
+ unfolding if_not_P
+ apply rule
+ apply rule
+ proof -
+ case goal1
+ then have *: "e/2 > 0"
+ by auto
from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
- show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
- proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
- hence *:"ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}" by auto
- guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
- guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
- have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto
+ show ?case
+ apply (rule_tac x="max B1 B2" in exI)
+ apply rule
+ apply (rule min_max.less_supI1)
+ apply (rule B1)
+ apply rule
+ apply rule
+ apply rule
+ proof -
+ fix a b
+ assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
+ then have *: "ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}"
+ by auto
+ obtain w where w:
+ "((\<lambda>x. if x \<in> s then f x else 0) has_integral w) {a..b}"
+ "norm (w - k) < e / 2"
+ using B1(2)[OF *(1)] by blast
+ obtain z where z:
+ "((\<lambda>x. if x \<in> s then g x else 0) has_integral z) {a..b}"
+ "norm (z - l) < e / 2"
+ using B2(2)[OF *(2)] by blast
+ have *: "\<And>x. (if x \<in> s then f x + g x else 0) =
+ (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)"
+ by auto
show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"
- apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
- using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
- qed qed qed
+ apply (rule_tac x="w + z" in exI)
+ apply rule
+ apply (rule lem[OF w(1) z(1), unfolded *[symmetric]])
+ using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
+ apply (auto simp add: field_simps)
+ done
+ qed
+ qed
+qed
lemma has_integral_sub:
- shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
- using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
-
-lemma integral_0: "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
- by(rule integral_unique has_integral_0)+
-
-lemma integral_add:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
- integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_add) by assumption+
-
-lemma integral_cmul:
- shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_cmul) by assumption+
-
-lemma integral_neg:
- shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_neg) by assumption+
-
-lemma integral_sub:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
- apply(rule integral_unique) apply(drule integrable_integral)+
- apply(rule has_integral_sub) by assumption+
+ "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow>
+ ((\<lambda>x. f x - g x) has_integral (k - l)) s"
+ using has_integral_add[OF _ has_integral_neg, of f k s g l]
+ unfolding algebra_simps
+ by auto
+
+lemma integral_0:
+ "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
+ by (rule integral_unique has_integral_0)+
+
+lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
+ integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
+ apply (rule integral_unique)
+ apply (drule integrable_integral)+
+ apply (rule has_integral_add)
+ apply assumption+
+ done
+
+lemma integral_cmul: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
+ apply (rule integral_unique)
+ apply (drule integrable_integral)+
+ apply (rule has_integral_cmul)
+ apply assumption+
+ done
+
+lemma integral_neg: "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
+ apply (rule integral_unique)
+ apply (drule integrable_integral)+
+ apply (rule has_integral_neg)
+ apply assumption+
+ done
+
+lemma integral_sub: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
+ integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
+ apply (rule integral_unique)
+ apply (drule integrable_integral)+
+ apply (rule has_integral_sub)
+ apply assumption+
+ done
lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
unfolding integrable_on_def using has_integral_0 by auto
-lemma integrable_add:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
+lemma integrable_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_add)
-lemma integrable_cmul:
- shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
+lemma integrable_cmul: "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_cmul)
lemma integrable_on_cmult_iff:
- fixes c :: real assumes "c \<noteq> 0"
+ fixes c :: real
+ assumes "c \<noteq> 0"
shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] `c \<noteq> 0`
by auto
-lemma integrable_neg:
- shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
+lemma integrable_neg: "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_neg)
lemma integrable_sub:
- shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
+ "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_sub)
lemma integrable_linear:
- shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
+ "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_linear)
lemma integral_linear:
- shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
- apply(rule has_integral_unique) defer unfolding has_integral_integral
- apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
- apply(rule integrable_linear) by assumption+
-
-lemma integral_component_eq[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
- assumes "f integrable_on s" shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
+ "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
+ apply (rule has_integral_unique)
+ defer
+ unfolding has_integral_integral
+ apply (drule (2) has_integral_linear)
+ unfolding has_integral_integral[symmetric]
+ apply (rule integrable_linear)
+ apply assumption+
+ done
+
+lemma integral_component_eq[simp]:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
+ assumes "f integrable_on s"
+ shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
lemma has_integral_setsum:
- assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
+ assumes "finite t"
+ and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
-proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
- case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
- apply(rule has_integral_add) using insert assms by auto
-qed auto
-
-lemma integral_setsum:
- shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
+ using assms(1) subset_refl[of t]
+proof (induct rule: finite_subset_induct)
+ case empty
+ then show ?case by auto
+next
+ case (insert x F)
+ show ?case
+ unfolding setsum_insert[OF insert(1,3)]
+ apply (rule has_integral_add)
+ using insert assms
+ apply auto
+ done
+qed
+
+lemma integral_setsum: "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
- apply(rule integral_unique) apply(rule has_integral_setsum)
- using integrable_integral by auto
+ apply (rule integral_unique)
+ apply (rule has_integral_setsum)
+ using integrable_integral
+ apply auto
+ done
lemma integrable_setsum:
- shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
- unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
+ "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
+ unfolding integrable_on_def
+ apply (drule bchoice)
+ using has_integral_setsum[of t]
+ apply auto
+ done
lemma has_integral_eq:
- assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
+ assumes "\<forall>x\<in>s. f x = g x"
+ and "(f has_integral k) s"
+ shows "(g has_integral k) s"
using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
- using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
-
-lemma integrable_eq:
- shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
- unfolding integrable_on_def using has_integral_eq[of s f g] by auto
-
-lemma has_integral_eq_eq:
- shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
- using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
+ using has_integral_is_0[of s "\<lambda>x. f x - g x"]
+ using assms(1)
+ by auto
+
+lemma integrable_eq: "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
+ unfolding integrable_on_def
+ using has_integral_eq[of s f g]
+ by auto
+
+lemma has_integral_eq_eq: "\<forall>x\<in>s. f x = g x \<Longrightarrow> (f has_integral k) s \<longleftrightarrow> (g has_integral k) s"
+ using has_integral_eq[of s f g] has_integral_eq[of s g f]
+ by auto
lemma has_integral_null[dest]:
- assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})"
- unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
-proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
- fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
- have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
- using setsum_content_null[OF assms(1) p, of f] .
- thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
-
-lemma has_integral_null_eq[simp]:
- shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
- apply rule apply(rule has_integral_unique,assumption)
- apply(drule has_integral_null,assumption)
- apply(drule has_integral_null) by auto
-
-lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
- by(rule integral_unique,drule has_integral_null)
-
-lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
- unfolding integrable_on_def apply(drule has_integral_null) by auto
-
-lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
- unfolding empty_as_interval apply(rule has_integral_null)
- using content_empty unfolding empty_as_interval .
-
-lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
- apply(rule,rule has_integral_unique,assumption) by auto
-
-lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
-
-lemma integral_empty[simp]: shows "integral {} f = 0"
- apply(rule integral_unique) using has_integral_empty .
-
-lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a::'a::ordered_euclidean_space}"
-proof-
- have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq_iff[where 'a='a]
- apply safe prefer 3 apply(erule_tac x=b in ballE) by(auto simp add: field_simps)
- show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
- apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
- unfolding interior_closed_interval using interval_sing by auto qed
-
-lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
-
-lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
+ assumes "content({a..b}) = 0"
+ shows "(f has_integral 0) ({a..b})"
+ unfolding has_integral
+ apply rule
+ apply rule
+ apply (rule_tac x="\<lambda>x. ball x 1" in exI)
+ apply rule
+ defer
+ apply rule
+ apply rule
+ apply (erule conjE)
+proof -
+ fix e :: real
+ assume e: "e > 0"
+ then show "gauge (\<lambda>x. ball x 1)"
+ by auto
+ fix p
+ assume p: "p tagged_division_of {a..b}"
+ have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0"
+ unfolding norm_eq_zero diff_0_right
+ using setsum_content_null[OF assms(1) p, of f] .
+ then show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
+ using e by auto
+qed
+
+lemma has_integral_null_eq[simp]: "content {a..b} = 0 \<Longrightarrow> (f has_integral i) {a..b} \<longleftrightarrow> i = 0"
+ apply rule
+ apply (rule has_integral_unique)
+ apply assumption
+ apply (drule (1) has_integral_null)
+ apply (drule has_integral_null)
+ apply auto
+ done
+
+lemma integral_null[dest]: "content {a..b} = 0 \<Longrightarrow> integral {a..b} f = 0"
+ apply (rule integral_unique)
+ apply (drule has_integral_null)
+ apply assumption
+ done
+
+lemma integrable_on_null[dest]: "content {a..b} = 0 \<Longrightarrow> f integrable_on {a..b}"
+ unfolding integrable_on_def
+ apply (drule has_integral_null)
+ apply auto
+ done
+
+lemma has_integral_empty[intro]: "(f has_integral 0) {}"
+ unfolding empty_as_interval
+ apply (rule has_integral_null)
+ using content_empty
+ unfolding empty_as_interval
+ apply assumption
+ done
+
+lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
+ apply rule
+ apply (rule has_integral_unique)
+ apply assumption
+ apply auto
+ done
+
+lemma integrable_on_empty[intro]: "f integrable_on {}"
+ unfolding integrable_on_def by auto
+
+lemma integral_empty[simp]: "integral {} f = 0"
+ by (rule integral_unique) (rule has_integral_empty)
+
+lemma has_integral_refl[intro]:
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "(f has_integral 0) {a..a}"
+ and "(f has_integral 0) {a}"
+proof -
+ have *: "{a} = {a..a}"
+ apply (rule set_eqI)
+ unfolding mem_interval singleton_iff euclidean_eq_iff[where 'a='a]
+ apply safe
+ prefer 3
+ apply (erule_tac x=b in ballE)
+ apply (auto simp add: field_simps)
+ done
+ show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}"
+ unfolding *
+ apply (rule_tac[!] has_integral_null)
+ unfolding content_eq_0_interior
+ unfolding interior_closed_interval
+ using interval_sing
+ apply auto
+ done
+qed
+
+lemma integrable_on_refl[intro]: "f integrable_on {a..a}"
+ unfolding integrable_on_def by auto
+
+lemma integral_refl: "integral {a..a} f = 0"
+ by (rule integral_unique) auto
+
subsection {* Cauchy-type criterion for integrability. *}
(* XXXXXXX *)
-lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
+lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
shows "f integrable_on {a..b} \<longleftrightarrow>
(\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
p2 tagged_division_of {a..b} \<and> d fine p2
@@ -1985,15 +3155,15 @@
proof(rule,rule,rule,rule) 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 *) apply(rule d(2))
- using dp p(1) using mn by auto
+ using dp p(1) using mn by auto
qed qed
- then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[THEN LIMSEQ_D]
+ then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
guess N2 using y[OF *] .. note N2=this
show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
- apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
+ apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
@@ -2019,12 +3189,12 @@
have *:"Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
using assms by auto
have *:"\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
- "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
+ "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
assume as:"a\<le>b" moreover have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c
\<Longrightarrow> x* (b\<bullet>k - a\<bullet>k) = x*(max (a \<bullet> k) c - a \<bullet> k) + x*(b \<bullet> k - max (a \<bullet> k) c)"
by (auto simp add:field_simps)
- moreover have **:"(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
+ moreover have **:"(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
(\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
"(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
(\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
@@ -2041,7 +3211,7 @@
qed
lemma division_split_left_inj: fixes type::"'a::ordered_euclidean_space"
- assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
+ assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
"k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"and k:"k\<in>Basis"
shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
proof- note d=division_ofD[OF assms(1)]
@@ -2052,7 +3222,7 @@
have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
-
+
lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
"k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" and k:"k\<in>Basis"
@@ -2067,7 +3237,7 @@
defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
lemma tagged_division_split_left_inj: fixes x1::"'a::ordered_euclidean_space"
- assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
+ assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
and k:"k\<in>Basis"
shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
@@ -2075,7 +3245,7 @@
apply(rule_tac[1-2] *) using assms(2-) by auto qed
lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
- assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
+ assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
and k:"k\<in>Basis"
shows "content(k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
@@ -2084,10 +3254,10 @@
lemma division_split: fixes a::"'a::ordered_euclidean_space"
assumes "p division_of {a..b}" and k:"k\<in>Basis"
- shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})} division_of({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is "?p1 division_of ?I1") and
+ shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<le> c} = {})} division_of({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is "?p1 division_of ?I1") and
"{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is "?p2 division_of ?I2")
proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms(1)]
- show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
+ show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[symmetric] by auto
{ fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
@@ -2106,8 +3276,8 @@
assumes "(f has_integral i) ({a..b} \<inter> {x. x\<bullet>k \<le> c})" "(f has_integral j) ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" and k:"k\<in>Basis"
shows "(f has_integral (i + j)) ({a..b})"
proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
- guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[THEN sym,OF k]]
- guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[THEN sym,OF k]]
+ guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[symmetric,OF k]]
+ guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[symmetric,OF k]]
let ?d = "\<lambda>x. if x\<bullet>k = c then (d1 x \<inter> d2 x) else ball x (abs(x\<bullet>k - c)) \<inter> d1 x \<inter> d2 x"
show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
@@ -2119,7 +3289,7 @@
proof(rule ccontr) case goal1
from this(2)[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
- hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<le> c}" using goal1(1) by blast
+ hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<le> c}" using goal1(1) by blast
then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c" apply-apply(rule le_less_trans)
using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
@@ -2128,7 +3298,7 @@
proof(rule ccontr) case goal1
from this(2)[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
- hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<ge> c}" using goal1(1) by blast
+ hence "\<exists>y. y \<in> ball x \<bar>x \<bullet> k - c\<bar> \<inter> {x. x \<bullet> k \<ge> c}" using goal1(1) by blast
then guess y .. hence "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c" apply-apply(rule le_less_trans)
using Basis_le_norm[OF k, of "x - y"] by (auto simp add: dist_norm inner_diff_left)
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
@@ -2153,7 +3323,7 @@
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 "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
- proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x\<bullet>k \<le> c}" unfolding p(8)[THEN sym] by auto
+ proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {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 apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
@@ -2170,10 +3340,10 @@
thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' 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> {}}"
+ 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 "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
- proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x\<bullet>k \<ge> c}" unfolding p(8)[THEN sym] by auto
+ proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {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 apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
@@ -2198,15 +3368,15 @@
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)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
- defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
+ defer unfolding lem4[symmetric] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) using k by auto
next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) using k by auto
- qed also note setsum_addf[THEN sym]
+ qed also note setsum_addf[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" unfolding split_paired_all split_conv
proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
thus "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[THEN sym] unfolding uv content_split[OF k,of u v c] by auto
+ unfolding scaleR_left_distrib[symmetric] unfolding uv content_split[OF k,of u v c] by auto
qed note setsum_cong2[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) =
@@ -2240,7 +3410,7 @@
proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
have "b \<subseteq> {x. x\<bullet>k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastforce
- moreover have "interior {x::'a. x \<bullet> k = c} = {}"
+ moreover have "interior {x::'a. x \<bullet> k = c} = {}"
proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x\<bullet>k = c}" by auto
then guess e unfolding mem_interior .. note e=this
have x:"x\<bullet>k = c" using x interior_subset by fastforce
@@ -2248,7 +3418,7 @@
= (if i = k then e/2 else 0)" using e k by (auto simp: inner_simps inner_not_same_Basis)
have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
(\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
- also have "... < e" apply(subst setsum_delta) using e by auto
+ also have "... < e" apply(subst setsum_delta) using e by auto
finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
hence "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}" using e by auto
@@ -2262,11 +3432,11 @@
lemma integrable_split[intro]:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
assumes "f integrable_on {a..b}" and k:"k\<in>Basis"
- shows "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
+ shows "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
def b' \<equiv> "\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i::'a"
def a' \<equiv> "\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i::'a"
- show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
+ show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[symmetric,OF k]
proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
@@ -2280,7 +3450,7 @@
show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
using p using assms by(auto simp add:algebra_simps)
- qed qed
+ qed qed
show "?P {x. x \<bullet> k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1
\<and> p2 tagged_division_of {a..b} \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p2"
@@ -2295,7 +3465,7 @@
definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
- "operative opp f \<equiv>
+ "operative opp f \<equiv>
(\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
(\<forall>a b c. \<forall>k\<in>Basis. f({a..b}) =
opp (f({a..b} \<inter> {x. x\<bullet>k \<le> c}))
@@ -2311,7 +3481,7 @@
unfolding operative_def by auto
lemma property_empty_interval:
- "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
+ "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
using content_empty unfolding empty_as_interval by auto
lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
@@ -2395,10 +3565,10 @@
unfolding support_def by auto
lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
- unfolding iterate_def fold'_def by auto
+ unfolding iterate_def fold'_def by auto
lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
- shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
+ shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
show ?thesis unfolding iterate_def if_P[OF True] * by auto
next case False note x=this
@@ -2408,7 +3578,7 @@
unfolding True monoidal_simps[OF assms(1)] by auto
next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False]
apply(subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
- using `finite s` unfolding support_def using False x by auto qed qed
+ using `finite s` unfolding support_def using False x by auto qed qed
lemma iterate_some:
assumes "monoidal opp" "finite s"
@@ -2419,19 +3589,19 @@
subsection {* Two key instances of additivity. *}
lemma neutral_add[simp]:
- "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
+ "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
-lemma operative_content[intro]: "operative (op +) content"
- unfolding operative_def neutral_add apply safe
- unfolding content_split[THEN sym] ..
+lemma operative_content[intro]: "operative (op +) content"
+ unfolding operative_def neutral_add apply safe
+ unfolding content_split[symmetric] ..
lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
by (rule neutral_add) (* FIXME: duplicate *)
lemma monoidal_monoid[intro]:
shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
- unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps)
+ unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps)
lemma operative_integral: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
@@ -2442,25 +3612,25 @@
show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
lifted op + (if f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c} then Some (integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f) else None)
(if f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k} then Some (integral ({a..b} \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
- proof(cases "f integrable_on {a..b}")
+ proof(cases "f integrable_on {a..b}")
case True show ?thesis unfolding if_P[OF True] using k apply-
unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
- unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k])
+ unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k])
apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x \<bullet> k}))"
proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
apply(rule_tac x="integral ({a..b} \<inter> {x. x \<bullet> k \<le> c}) f + integral ({a..b} \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
thus False using False by auto
- qed thus ?thesis using False by auto
- qed next
+ qed thus ?thesis using False by auto
+ qed next
fix a b assume as:"content {a..b::'a} = 0"
thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
subsection {* Points of division of a partition. *}
-definition "division_points (k::('a::ordered_euclidean_space) set) d =
+definition "division_points (k::('a::ordered_euclidean_space) set) d =
{(j,x). j\<in>Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
(\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
@@ -2502,7 +3672,7 @@
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *:"\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i"
using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
- have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+ have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[symmetric] by auto
show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
apply (rule bexI[OF _ `l \<in> d`])
using as(1-3,5) fstx
@@ -2520,12 +3690,12 @@
apply(erule exE conjE)+
proof
fix i l x assume as:"(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
- "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+ "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
"i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}" and fstx:"fst x \<in> Basis"
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *:"\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i"
using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
- have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
+ have **:"\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" using l using as(6) unfolding interval_ne_empty[symmetric] by auto
show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
apply (rule bexI[OF _ `l \<in> d`])
using as(1-3,5) fstx
@@ -2540,9 +3710,9 @@
assumes "d division_of {a..b}" "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k"
"l \<in> d" "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c" and k:"k\<in>Basis"
shows "division_points ({a..b} \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}
- \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
+ \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
"division_points ({a..b} \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}
- \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D")
+ \<subset> division_points ({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)] apply-by(erule exE)+ note l=this
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"
@@ -2555,7 +3725,7 @@
have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
- unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
+ unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
have *:"interval_lowerbound ({a..b} \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
@@ -2565,7 +3735,7 @@
have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
apply(rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI)
- unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
+ unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4) k]) by auto qed
subsection {* Preservation by divisions and tagged divisions. *}
@@ -2578,7 +3748,7 @@
lemma iterate_expand_cases:
"iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
- apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
+ apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
lemma iterate_image: assumes "monoidal opp" "inj_on f s"
shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
@@ -2587,14 +3757,14 @@
proof- case goal1 show ?case using goal1
proof(induct s) case empty thus ?case using assms(1) by auto
next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
- unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
+ unfolding if_not_P[OF insert(2)] apply(subst insert(3)[symmetric])
unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
apply(rule finite_imageI insert)+ apply(subst if_not_P)
unfolding image_iff o_def using insert(2,4) by auto
qed qed
- show ?thesis
+ show ?thesis
apply(cases "finite (support opp g (f ` s))")
- apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
+ apply(subst (1) iterate_support[symmetric],subst (2) iterate_support[symmetric])
unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
apply(rule subset_inj_on[OF assms(2) support_subset])+
apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
@@ -2610,16 +3780,16 @@
have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
unfolding support_def using assms(3) by auto
show ?thesis unfolding *
- apply(subst iterate_support[THEN sym]) unfolding support_clauses
+ apply(subst iterate_support[symmetric]) unfolding support_clauses
apply(subst iterate_image[OF assms(1)]) defer
- apply(subst(2) iterate_support[THEN sym]) apply(subst **)
+ apply(subst(2) iterate_support[symmetric]) apply(subst **)
unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
lemma iterate_eq_neutral:
assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)"
shows "(iterate opp s f = neutral opp)"
proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
- show ?thesis apply(subst iterate_support[THEN sym])
+ show ?thesis apply(subst iterate_support[symmetric])
unfolding * using assms(1) by auto qed
lemma iterate_op: assumes "monoidal opp" "finite s"
@@ -2637,11 +3807,11 @@
case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
unfolding * by auto
next def su \<equiv> "support opp f s"
- case True note support_subset[of opp f s]
- thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
+ case True note support_subset[of opp f s]
+ thus ?thesis apply- apply(subst iterate_support[symmetric],subst(2) iterate_support[symmetric]) unfolding * using True
unfolding su_def[symmetric]
proof(induct su) case empty show ?case by auto
- next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
+ next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
unfolding if_not_P[OF insert(2)] apply(subst insert(3))
defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
@@ -2659,11 +3829,11 @@
show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
proof fix x assume x:"x\<in>d"
then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
- thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
+ thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
using operativeD(1)[OF assms(2)] x by auto
qed qed }
- assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
- hence ab':"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" by (auto intro!: less_imp_le) show ?case
+ assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
+ hence ab':"\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i" by (auto intro!: less_imp_le) show ?case
proof(cases "division_points {a..b} d = {}")
case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
(\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)"
@@ -2677,7 +3847,7 @@
"(j, v\<bullet>j) \<notin> division_points {a..b} d" using True by auto
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
- moreover have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" using division_ofD(2,2,3)[OF goal1(4) as]
+ moreover have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j" using division_ofD(2,2,3)[OF goal1(4) as]
unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
unfolding interval_ne_empty mem_interval using j by auto
ultimately show "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j"
@@ -2685,7 +3855,7 @@
qed
have "(1/2) *\<^sub>R (a+b) \<in> {a..b}"
unfolding mem_interval using ab by(auto intro!: less_imp_le simp: inner_simps)
- note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
+ note this[unfolded division_ofD(6)[OF goal1(4),symmetric] Union_iff]
then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
have "{a..b} \<in> d"
proof- { presume "i = {a..b}" thus ?thesis using i by auto }
@@ -2700,12 +3870,12 @@
have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
then guess u v apply-by(erule exE conjE)+ note uv=this
- have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
+ have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j:"j\<in>Basis" unfolding euclidean_eq_iff[where 'a='a] by auto
hence "u\<bullet>j = v\<bullet>j" using uv(2)[rule_format,OF j] by auto
hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in bexI) using j by auto
thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
- qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
+ qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
@@ -2723,32 +3893,32 @@
unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
- unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto
+ unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto
also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<le> c}))"
unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
- unfolding empty_as_interval[THEN sym] apply(rule content_empty)
- proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
+ unfolding empty_as_interval[symmetric] apply(rule content_empty)
+ proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
- show "f (l \<inter> {x. x \<bullet> k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
- apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
- apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
+ show "f (l \<inter> {x. x \<bullet> k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
+ apply(rule operativeD(1) goal1)+ unfolding interval_split[symmetric,OF kc(4)] apply(rule division_split_left_inj)
+ apply(rule goal1) unfolding l[symmetric] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\<bullet>k \<ge> c}))"
unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
- unfolding empty_as_interval[THEN sym] apply(rule content_empty)
- proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
+ unfolding empty_as_interval[symmetric] apply(rule content_empty)
+ proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
- show "f (l \<inter> {x. x \<bullet> k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
- apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
- apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
+ show "f (l \<inter> {x. x \<bullet> k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
+ apply(rule operativeD(1) goal1)+ unfolding interval_split[symmetric,OF kc(4)] apply(rule division_split_right_inj)
+ apply(rule goal1) unfolding l[symmetric] apply(rule as(1),rule as(2)) by(rule as kc(4))+
qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x \<bullet> k \<le> c})) (f (x \<inter> {x. c \<le> x \<bullet> k}))"
- unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto
+ unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto
have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x \<bullet> k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x \<bullet> k})))
= iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
- apply(rule iterate_op[THEN sym]) using goal1 by auto
+ apply(rule iterate_op[symmetric]) using goal1 by auto
finally show ?thesis by auto
- qed qed qed
+ qed qed qed
lemma iterate_image_nonzero: assumes "monoidal opp"
"finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
@@ -2763,20 +3933,20 @@
apply(subst iterate_insert[OF assms(1) goal2(1)])
unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
- using goal2 unfolding o_def by auto qed
+ using goal2 unfolding o_def by auto qed
lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
- apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
+ apply(rule iterate_image_nonzero[symmetric,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
- unfolding as(4)[THEN sym] uv by auto
- qed also have "\<dots> = f {a..b}"
+ unfolding as(4)[symmetric] uv by auto
+ qed also have "\<dots> = f {a..b}"
using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
finally show ?thesis . qed
@@ -2794,13 +3964,13 @@
lemma additive_content_division: assumes "d division_of {a..b}"
shows "setsum content d = content({a..b})"
- unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
+ unfolding operative_division[OF monoidal_monoid operative_content assms,symmetric]
apply(subst setsum_iterate) using assms by auto
lemma additive_content_tagged_division:
assumes "d tagged_division_of {a..b}"
shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
- unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
+ unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,symmetric]
apply(subst setsum_iterate) using assms by auto
subsection {* Finally, the integral of a constant *}
@@ -2809,7 +3979,7 @@
"((\<lambda>x. c) has_integral (content({a..b::'a::ordered_euclidean_space}) *\<^sub>R c)) ({a..b})"
unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
- unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
+ unfolding split_def apply(subst scaleR_left.setsum[symmetric, unfolded o_def])
defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
lemma integral_const[simp]:
@@ -2821,7 +3991,7 @@
lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
- apply(rule order_trans,rule norm_setsum) unfolding norm_scaleR setsum_left_distrib[THEN sym]
+ apply(rule order_trans,rule norm_setsum) unfolding norm_scaleR setsum_left_distrib[symmetric]
apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
apply(subst mult_commute) apply(rule mult_left_mono)
apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
@@ -2838,11 +4008,11 @@
next case False show ?thesis
apply(rule order_trans,rule norm_setsum) unfolding split_def norm_scaleR
apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
- unfolding setsum_left_distrib[THEN sym] apply(subst mult_commute) apply(rule mult_left_mono)
+ unfolding setsum_left_distrib[symmetric] apply(subst mult_commute) apply(rule mult_left_mono)
apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
apply(subst o_def, rule abs_of_nonneg)
proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
- unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
+ unfolding additive_content_tagged_division[OF assms(1),symmetric] split_def by auto
guess w using nonempty_witness[OF False] .
thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
@@ -2855,7 +4025,7 @@
assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le> e * content({a..b})"
apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
- unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR_diff_right by auto
+ unfolding setsum_subtractf[symmetric] apply(rule setsum_cong2) unfolding scaleR_diff_right by auto
lemma has_integral_bound: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
@@ -2863,7 +4033,7 @@
proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
thus ?thesis proof(cases ?P) case False
hence *:"content {a..b} = 0" using content_lt_nz by auto
- hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
+ hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[symmetric]) by auto
show ?thesis unfolding * ** using assms(1) by auto
qed auto } assume ab:?P
{ presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
@@ -2893,7 +4063,7 @@
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
shows "i\<bullet>k \<le> j\<bullet>k"
proof -
- have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
+ have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
(g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)\<bullet>k \<le> (g x)\<bullet>k \<Longrightarrow> i\<bullet>k \<le> j\<bullet>k"
proof (rule ccontr)
case goal1
@@ -2935,7 +4105,7 @@
apply(rule has_integral_component_le) using integrable_integral assms by auto
lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> i\<bullet>k"
+ assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k" shows "0 \<le> i\<bullet>k"
using has_integral_component_le[OF assms(1) has_integral_0 assms(2)] using assms(3-) by auto
lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
@@ -2943,7 +4113,7 @@
apply(rule has_integral_component_nonneg) using assms by auto
lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
- assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"shows "i\<bullet>k \<le> 0"
+ assumes "k\<in>Basis" "(f has_integral i) s" "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"shows "i\<bullet>k \<le> 0"
using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-) by auto
lemma has_integral_component_lbound:
@@ -2966,7 +4136,7 @@
apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
- assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\<bullet>k \<le> B" "k\<in>Basis"
+ assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\<bullet>k \<le> B" "k\<in>Basis"
shows "(integral({a..b}) f)\<bullet>k \<le> B * content({a..b})"
apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
@@ -2982,7 +4152,7 @@
have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
-
+
have "Cauchy i" unfolding Cauchy_def
proof(rule,rule) fix e::real assume "e>0"
hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
@@ -3003,10 +4173,10 @@
apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
- proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
+ proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
using M as by(auto simp add:field_simps)
fix x assume x:"x \<in> {a..b}"
- have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
+ have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
using g(1)[OF x, of n] g(1)[OF x, of m] by auto
also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
@@ -3015,10 +4185,10 @@
using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
by(auto simp add:algebra_simps simp add:norm_minus_commute)
qed qed qed
- from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
+ from this[unfolded convergent_eq_cauchy[symmetric]] guess s .. note s=this
show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
- proof(rule,rule)
+ proof(rule,rule)
case goal1 hence *:"e/3 > 0" by auto
from LIMSEQ_D [OF s this] guess N1 .. note N1=this
from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
@@ -3038,7 +4208,7 @@
proof- have "content {a..b} < e / 3 * (real N2)"
using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
- apply-apply(rule less_le_trans,assumption) using `e>0` by auto
+ apply-apply(rule less_le_trans,assumption) using `e>0` by auto
thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
unfolding inverse_eq_divide by(auto simp add:field_simps)
show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format],auto)
@@ -3050,17 +4220,17 @@
subsection {* Negligibility of hyperplane. *}
-lemma vsum_nonzero_image_lemma:
+lemma vsum_nonzero_image_lemma:
assumes "finite s" "g(a) = 0"
"\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
- unfolding assms using neutral_add unfolding neutral_add using assms by auto
+ unfolding assms using neutral_add unfolding neutral_add using assms by auto
lemma interval_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "k\<in>Basis"
- shows "{a..b} \<inter> {x . abs(x\<bullet>k - c) \<le> (e::real)} =
- {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) ..
+ shows "{a..b} \<inter> {x . abs(x\<bullet>k - c) \<le> (e::real)} =
+ {(\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i) ..
(\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)}"
proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
@@ -3071,7 +4241,7 @@
proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
- note division_split(2)[OF this, where c="c-e" and k=k,OF k]
+ note division_split(2)[OF this, where c="c-e" and k=k,OF k]
thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
apply(rule set_eqI) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI) apply rule defer apply rule
@@ -3082,17 +4252,17 @@
proof(cases "content {a..b} = 0")
case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
- unfolding interval_doublesplit[THEN sym,OF k] using assms by auto
+ unfolding interval_doublesplit[symmetric,OF k] using assms by auto
next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})"
note False[unfolded content_eq_0 not_ex not_le, rule_format]
hence "\<And>x. x\<in>Basis \<Longrightarrow> b\<bullet>x > a\<bullet>x" by(auto simp add:not_le)
hence prod0:"0 < setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
proof(rule that[of d]) have *:"Basis = insert k (Basis - {k})" using k by auto
- have **:"{a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
+ have **:"{a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
(\<Prod>i\<in>Basis - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i
- interval_lowerbound ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i)
- = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)" apply(rule setprod_cong,rule refl)
+ = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)" apply(rule setprod_cong,rule refl)
unfolding interval_doublesplit[OF k] apply(subst interval_bounds) defer apply(subst interval_bounds)
unfolding interval_eq_empty not_ex not_less by auto
show "content ({a..b} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
@@ -3109,10 +4279,10 @@
qed
qed
-lemma negligible_standard_hyperplane[intro]:
+lemma negligible_standard_hyperplane[intro]:
fixes k :: "'a::ordered_euclidean_space"
assumes k: "k \<in> Basis"
- shows "negligible {x. x\<bullet>k = c}"
+ shows "negligible {x. x\<bullet>k = c}"
unfolding negligible_def has_integral apply(rule,rule,rule,rule)
proof-
case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
@@ -3136,30 +4306,30 @@
prefer 2 apply(subst(asm) eq_commute) apply assumption
apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
- apply(rule setsum_mono) unfolding split_paired_all split_conv
+ apply(rule setsum_mono) unfolding split_paired_all split_conv
apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k])
also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content {u..v}"
- unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
+ unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[symmetric,OF k] by auto
thus ?case unfolding goal1 unfolding interval_doublesplit[OF k]
by (blast intro: antisym)
next have *:"setsum content {l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
- apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
+ apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
- note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
+ note dsum_bound[OF this **,unfolded interval_doublesplit[symmetric,OF k]]
note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e"
- apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
+ apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,symmetric])
apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}" "{m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
have "({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
note interior_mono[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
hence "interior ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
- thus "content ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
+ thus "content ({m..n} \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[symmetric] .
qed qed
finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
qed qed qed
@@ -3177,7 +4347,7 @@
presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
} fix p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set" assume as:"finite p"
- show "?P p" apply(rule,rule) using as proof(induct p)
+ show "?P p" apply(rule,rule) using as proof(induct p)
case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
note tagged_partial_division_subset[OF insert(4) subset_insertI]
@@ -3186,19 +4356,19 @@
note p = tagged_partial_division_ofD[OF insert(4)]
from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
- have "finite {k. \<exists>x. (x, k) \<in> p}"
+ have "finite {k. \<exists>x. (x, k) \<in> p}"
apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
- unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
+ unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
using insert(2) unfolding uv xk by auto
show ?case proof(cases "{u..v} \<subseteq> d x")
case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
- apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int)
+ apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int)
apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
@@ -3214,7 +4384,7 @@
lemma finite_product_dependent: assumes "finite s" "\<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)
+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 by auto qed auto
@@ -3241,16 +4411,16 @@
apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
apply(rule,rule P) using assms(2) by auto
qed
-next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
+next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
show "(f has_integral 0) {a..b}" unfolding has_integral
proof(safe) case goal1
- hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
+ hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
- note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
+ note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
- show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
+ show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
+ fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
{ presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto }
assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
@@ -3258,7 +4428,7 @@
have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
- have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe)
+ have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe)
unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
have **:"\<And>f g s t. finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow> (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t"
proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
@@ -3266,7 +4436,7 @@
have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {0..N+1}"
unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
- apply(rule order_trans,rule norm_setsum) apply(subst sum_sum_product) prefer 3
+ apply(rule order_trans,rule norm_setsum) apply(subst sum_sum_product) prefer 3
proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto
fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
@@ -3286,11 +4456,11 @@
qed ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le> (real y + 1) * (content k *\<^sub>R indicator s x)"
apply(rule_tac x=n in exI,safe) apply(rule_tac x=n in exI,rule_tac x="(x,k)" in exI,safe) by auto
qed(insert as, auto)
- also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
- proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
+ also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
+ proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[symmetric])
using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
- qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[THEN sym]
- apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
+ qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[symmetric]
+ apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[symmetric]
apply(subst sumr_geometric) using goal1 by auto
finally show "?goal" by auto qed qed qed
@@ -3323,7 +4493,7 @@
subsection {* Some other trivialities about negligible sets. *}
-lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
+lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
using assms(2) unfolding indicator_def by auto qed
@@ -3332,7 +4502,7 @@
lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
-lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
+lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
defer apply assumption unfolding indicator_def by auto qed
@@ -3340,8 +4510,8 @@
lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
using negligible_union by auto
-lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}"
- using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto
+lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}"
+ using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto
lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
apply(subst insert_is_Un) unfolding negligible_union_eq by auto
@@ -3352,7 +4522,7 @@
using assms apply(induct s) by auto
lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
- using assms by(induct,auto)
+ using assms by(induct,auto)
lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
apply safe defer apply(subst negligible_def)
@@ -3377,7 +4547,7 @@
subsection {* Finite case of the spike theorem is quite commonly needed. *}
-lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
+lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
"(f has_integral y) t" shows "(g has_integral y) t"
apply(rule has_integral_spike) using assms by auto
@@ -3438,7 +4608,7 @@
proof safe
fix a b::"'b"
{ assume "content {a..b} = 0"
- thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
+ thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
{ fix c g and k :: 'b
assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k\<in>Basis"
@@ -3452,7 +4622,7 @@
show "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" apply(rule_tac x="?g" in exI)
proof safe case goal1 thus ?case apply- apply(cases "x\<bullet>k=c", case_tac "x\<bullet>k < c") using as assms by auto
next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
- then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k]
+ then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k]
show ?case unfolding integrable_on_def by auto
next show "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \<bullet> k \<ge> c}"
apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
@@ -3472,7 +4642,7 @@
from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
note p' = tagged_division_ofD[OF p(1)]
have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
- proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
+ proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l" apply(rule_tac x="\<lambda>y. f x" in exI)
proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
@@ -3480,11 +4650,11 @@
note d(2)[OF _ _ this[unfolded mem_ball]]
thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce qed qed
from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
- thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
+ thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
subsection {* Specialization of additivity to one dimension. *}
-lemma
+lemma
shows real_inner_1_left: "inner 1 x = x"
and real_inner_1_right: "inner x 1 = x"
by simp_all
@@ -3510,9 +4680,9 @@
qed
next case True hence *:"min (b) c = c" "max a c = c" by auto
have **: "(1::real) \<in> Basis" by simp
- have ***:"\<And>P Q. (\<Sum>i\<in>Basis. (if i = 1 then P i else Q i) *\<^sub>R i) = (P 1::real)"
+ have ***:"\<And>P Q. (\<Sum>i\<in>Basis. (if i = 1 then P i else Q i) *\<^sub>R i) = (P 1::real)"
by simp
- show ?thesis
+ show ?thesis
unfolding interval_split[OF **, unfolded real_inner_1_right] unfolding *** *
proof(cases "c = a \<or> c = b")
case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
@@ -3540,7 +4710,7 @@
proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
thus ?thesis using assms unfolding * by auto
next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
- thus ?thesis using assms unfolding * by auto qed qed qed
+ thus ?thesis using assms unfolding * by auto qed qed qed
subsection {* Special case of additivity we need for the FCT. *}
@@ -3554,8 +4724,8 @@
have ***:"\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" using assms by auto
have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty by auto
have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
- note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],THEN sym]
- show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
+ note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
+ show ?thesis unfolding * apply(subst setsum_iterate[symmetric]) defer
apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
subsection {* A useful lemma allowing us to factor out the content size. *}
@@ -3565,10 +4735,10 @@
\<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
proof(cases "content {a..b} = 0")
case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
- apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
+ apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
-next case False note F = this[unfolded content_lt_nz[THEN sym]]
+next case False note F = this[unfolded content_lt_nz[symmetric]]
let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
show ?thesis apply(subst has_integral)
proof safe fix e::real assume e:"e>0"
@@ -3599,10 +4769,10 @@
apply(rule_tac x="\<lambda>x. ball x (d x)" in exI,safe)
apply(rule gauge_ball_dependent,rule,rule d(1))
proof- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. ball x (d x)) fine p"
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
- unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,THEN sym]
- unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",THEN sym]
- unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
+ unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,symmetric]
+ unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",symmetric]
+ unfolding setsum_right_distrib defer unfolding setsum_subtractf[symmetric]
proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
have *:"u \<le> v" using xk unfolding k by auto
@@ -3615,8 +4785,8 @@
also have "... \<le> e * norm (u - x) + e * norm (v - x)"
apply(rule add_mono) apply(rule d(2)[of "x" "u",unfolded o_def]) prefer 4
apply(rule d(2)[of "x" "v",unfolded o_def])
- using ball[rule_format,of u] ball[rule_format,of v]
- using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def)
+ using ball[rule_format,of u] ball[rule_format,of v]
+ using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def)
also have "... \<le> e * (interval_upperbound k - interval_lowerbound k)"
unfolding k interval_bounds_real[OF *] using xk(1) unfolding k by(auto simp add:dist_real_def field_simps)
finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
@@ -3638,7 +4808,7 @@
shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
proof(induct "card s" arbitrary:s rule:nat_less_induct)
fix s::"'a set set" assume assm:"s division_of {a..b}"
- "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}"
+ "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}"
note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
{ presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
@@ -3651,12 +4821,12 @@
apply safe apply(rule closed_interval) using assm(1) by auto
have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
proof safe fix x and e::real assume as:"x\<in>k" "e>0"
- from k(2)[unfolded k content_eq_0] guess i ..
+ from k(2)[unfolded k content_eq_0] guess i ..
hence i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
hence xi:"x\<bullet>i = d\<bullet>i" using as unfolding k mem_interval by (metis antisym)
def y \<equiv> "(\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i) / 2 then c\<bullet>i +
min e (b\<bullet>i - c\<bullet>i) / 2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i) / 2 else x\<bullet>j) *\<^sub>R j)::'a"
- show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
+ show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastforce simp add: not_less)
hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN bspec[where x=i]]
hence xyi:"y\<bullet>i \<noteq> x\<bullet>i"
@@ -3677,7 +4847,7 @@
using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i unfolding s mem_interval y_def
by (auto simp: field_simps elim!: ballE[of _ _ i])
ultimately show "y \<in> \<Union>(s - {k})" by auto
- qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
+ qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[symmetric] by auto
hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
moreover have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}" using k by auto ultimately show ?thesis by auto qed
@@ -3690,10 +4860,10 @@
unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption,assumption)+
unfolding integrable_on_def by(auto intro!: has_integral_split)
-lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
- assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
+lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
+ assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
- using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
+ using operative_division_and[OF operative_integrable,symmetric,of _ _ _ f] assms(1) by auto
subsection {* Combining adjacent intervals in 1 dimension. *}
@@ -3710,7 +4880,7 @@
lemma integral_combine: fixes f::"real \<Rightarrow> 'a::banach"
assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
- apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
+ apply(rule integral_unique[symmetric]) apply(rule has_integral_combine[OF assms(1-2)])
apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
lemma integrable_combine: fixes f::"real \<Rightarrow> 'a::banach"
@@ -3725,7 +4895,7 @@
proof- have "\<forall>x. \<exists>d. x\<in>{a..b} \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v})"
using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
- note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
+ note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,symmetric,of f]
show ?thesis unfolding * apply safe unfolding snd_conv
proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
@@ -3765,10 +4935,10 @@
hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
using True using assms(2) goal1 by auto
have **:"norm (y - x) = content {y..x}" apply(subst content_real) using True unfolding not_less by auto
- have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto
+ have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto
show ?thesis apply(subst ***) unfolding norm_minus_cancel **
apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
- defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
+ defer apply(rule has_integral_sub) apply(subst minus_minus[symmetric]) unfolding minus_minus
apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
proof- show "{y..x} \<subseteq> {a..b}" using goal1 assms(2) by auto
have *:"x - y = norm(y - x)" using True by auto
@@ -3813,8 +4983,8 @@
def d' \<equiv> "\<lambda>x. {y. g y \<in> d (g x)}" have d':"\<And>x. d' x = {y. g y \<in> (d (g x))}" unfolding d'_def ..
show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
proof(rule_tac x=d' in exI,safe) show "gauge d'" using d(1) unfolding gauge_def d' using continuous_open_preimage_univ[OF assms(4)] by auto
- fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
- have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of
+ fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
+ have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of
proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
@@ -3852,12 +5022,12 @@
lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
apply(rule setprod_cong) using assms by auto
-lemma content_image_affinity_interval:
+lemma content_image_affinity_interval:
"content((\<lambda>x::'a::ordered_euclidean_space. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ DIM('a) * content {a..b}" (is "?l = ?r")
proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
unfolding not_not using content_empty by auto }
- assume as: "{a..b}\<noteq>{}"
- show ?thesis
+ assume as: "{a..b}\<noteq>{}"
+ show ?thesis
proof (cases "m \<ge> 0")
case True
with as have "{m *\<^sub>R a + c..m *\<^sub>R b + c} \<noteq> {}"
@@ -3903,10 +5073,10 @@
lemma image_stretch_interval:
"(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k)) *\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} =
(if {a..b} = {} then {} else
- {(\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)::'a ..
+ {(\<Sum>k\<in>Basis. (min (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)::'a ..
(\<Sum>k\<in>Basis. (max (m k * (a\<bullet>k)) (m k * (b\<bullet>k))) *\<^sub>R k)})"
proof cases
- assume *: "{a..b} \<noteq> {}"
+ assume *: "{a..b} \<noteq> {}"
show ?thesis
unfolding interval_ne_empty if_not_P[OF *]
apply (simp add: interval image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
@@ -3929,14 +5099,14 @@
"max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) = (if 0 < m i then m i * (b \<bullet> i) else m i * (a \<bullet> i))"
using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
with False show ?thesis using a_le_b
- unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps)
+ unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps)
qed
qed
qed simp
-lemma interval_image_stretch_interval:
+lemma interval_image_stretch_interval:
"\<exists>u v. (\<lambda>x. \<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
- unfolding image_stretch_interval by auto
+ unfolding image_stretch_interval by auto
lemma content_image_stretch_interval:
"content((\<lambda>x::'a::ordered_euclidean_space. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b}) = abs(setprod m Basis) * content({a..b})"
@@ -3944,12 +5114,12 @@
unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
next case False hence "(\<lambda>x. (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)::'a) ` {a..b} \<noteq> {}" by auto
thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
- unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff
+ unfolding abs_setprod setprod_timesf[symmetric] apply(rule setprod_cong2) unfolding lessThan_iff
proof (simp only: inner_setsum_left_Basis)
fix i :: 'a assume i:"i\<in>Basis" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
- thus "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) - min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) =
+ thus "max (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) - min (m i * (a \<bullet> i)) (m i * (b \<bullet> i)) =
\<bar>m i\<bar> * (b \<bullet> i - a \<bullet> i)"
- apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i
+ apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i
by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
lemma has_integral_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
@@ -3966,7 +5136,7 @@
lemma integrable_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
assumes "f integrable_on {a..b}" "\<forall>k\<in>Basis. ~(m k = 0)"
shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` {a..b})"
- using assms unfolding integrable_on_def apply-apply(erule exE)
+ using assms unfolding integrable_on_def apply-apply(erule exE)
apply(drule has_integral_stretch,assumption) by auto
subsection {* even more special cases. *}
@@ -4001,13 +5171,13 @@
unfolding split_def by(rule refl)
lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
- apply(subst(asm)(2) norm_minus_cancel[THEN sym])
+ apply(subst(asm)(2) norm_minus_cancel[symmetric])
apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
lemma fundamental_theorem_of_calculus_interior: fixes f::"real => 'a::real_normed_vector"
assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a..b}"
-proof- { presume *:"a < b \<Longrightarrow> ?thesis"
+proof- { presume *:"a < b \<Longrightarrow> ?thesis"
show ?thesis proof(cases,rule *,assumption)
assume "\<not> a < b" hence "a = b" using assms(1) by auto
hence *:"{a .. b} = {b}" "f b - f a = 0" by(auto simp add: order_antisym)
@@ -4034,15 +5204,15 @@
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
proof(cases "f' a = 0") case True
- thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
+ thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
next case False thus ?thesis
- apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps)
+ apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps)
qed then guess l .. note l = conjunctD2[OF this]
show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
- proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
+ proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)" by(rule norm_triangle_ineq4)
- also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
+ also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
@@ -4060,16 +5230,16 @@
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
proof(cases "f' b = 0") case True
- thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
- next case False thus ?thesis
+ thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
+ next case False thus ?thesis
apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
using ab e by(auto simp add:field_simps)
qed then guess l .. note l = conjunctD2[OF this]
show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
- proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
+ proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)" by(rule norm_triangle_ineq4)
- also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
+ also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
@@ -4083,11 +5253,11 @@
proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
next case goal2 note as=this let ?A = "{t. fst t \<in> {a, b}}" note p = tagged_division_ofD[OF goal2(1)]
have pA:"p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}" using goal2 by auto
- note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
+ note * = additive_tagged_division_1'[OF assms(1) goal2(1), symmetric]
have **:"\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2" by arith
- show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
+ show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[symmetric] split_minus
unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
- proof(rule norm_triangle_le,rule **)
+ proof(rule norm_triangle_le,rule **)
case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) defer apply(subst setsum_divide_distrib)
proof(rule order_refl,safe,unfold not_le o_def split_conv fst_conv,rule ccontr) fix x k assume as:"(x,k) \<in> p"
"e * (interval_upperbound k - interval_lowerbound k) / 2
@@ -4099,8 +5269,8 @@
assume as':"x \<noteq> a" "x \<noteq> b" hence "x \<in> {a<..<b}" using p(2-3)[OF as(1)] by auto
note * = d(2)[OF this]
have "norm ((v - u) *\<^sub>R f' (x) - (f (v) - f (u))) =
- norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))"
- apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
+ norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))"
+ apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
also have "... \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)" apply(rule norm_triangle_le_sub)
apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp:dist_real_def)
@@ -4110,7 +5280,7 @@
next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
- defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
+ defer unfolding setsum_Un_disjoint[OF pA(2-),symmetric] pA(1)[symmetric] unfolding setsum_right_distrib[symmetric]
apply(subst additive_tagged_division_1[OF _ as(1)]) apply(rule assms)
proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}}" note xk=IntD1[OF this]
from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
@@ -4119,7 +5289,7 @@
unfolding uv using e by(auto simp add:field_simps)
next have *:"\<And>s f t e. setsum f s = setsum f t \<Longrightarrow> norm(setsum f t) \<le> e \<Longrightarrow> norm(setsum f s) \<le> e" by auto
show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
- (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2"
+ (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2"
apply(rule *[where t="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
@@ -4127,7 +5297,7 @@
have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk
unfolding uv content_eq_0 interval_eq_empty by auto
thus "content k *\<^sub>R (f' (x)) - (f ((interval_upperbound k)) - f ((interval_lowerbound k))) = 0" using xk unfolding uv by auto
- next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
+ next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
{t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}" by blast
have **:"\<And>s f. \<And>e::real. (\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y) \<Longrightarrow> (\<forall>x. x \<in> s \<longrightarrow> norm(f x) \<le> e)
\<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
@@ -4135,22 +5305,22 @@
thus ?case using `x\<in>s` goal2(2) by auto
qed auto
case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4
- apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
+ apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
proof- let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
- have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v"
+ have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v"
proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
- have u:"u = a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
+ have u:"u = a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
have "u \<ge> a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>a" ultimately
have "u > a" by auto
thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
qed
- have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v"
+ have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v"
proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
- have u:"v = b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
+ have u:"v = b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
have "v \<le> b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq> b" ultimately
have "v < b" by auto
thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
@@ -4168,7 +5338,7 @@
ultimately have " ((a + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
{ assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
- qed
+ qed
show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
unfolding mem_Collect_eq fst_conv snd_conv apply safe
proof- fix x k k' assume k:"( b, k) \<in> p" "( b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
@@ -4184,7 +5354,7 @@
let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) - (f ((interval_upperbound k)) -
f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4" apply(rule,rule) unfolding mem_Collect_eq
- unfolding split_paired_all fst_conv snd_conv
+ unfolding split_paired_all fst_conv snd_conv
proof safe case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
have " ?a\<in>{ ?a..v}" using v(2) by auto hence "v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
moreover have "{?a..v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
@@ -4195,7 +5365,7 @@
qed
show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) -
(f ((interval_upperbound k)) - f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4"
- apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv
+ apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv
proof safe case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
have " ?b\<in>{v.. ?b}" using v(2) by auto hence "v \<ge> ?a" using p(3)[OF goal1(1)]
unfolding subset_eq v by auto
@@ -4213,7 +5383,7 @@
lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
"\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
- shows "(f' has_integral (f b - f a)) {a..b}" using assms apply-
+ shows "(f' has_integral (f b - f a)) {a..b}" using assms apply-
proof(induct "card s" arbitrary:s a b)
case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
@@ -4249,10 +5419,10 @@
hence "c - t < e / 3 / norm (f c)" by auto
hence "norm (c - t) < e / 3 / norm (f c)" using as by auto
thus "norm (f c) * norm (c - t) < e / 3" using False apply-
- apply(subst mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
+ apply(subst mult_commute) apply(subst pos_less_divide_eq[symmetric]) by auto
qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
qed then guess w .. note w = conjunctD2[OF this,rule_format]
-
+
have *:"e / 3 > 0" using assms by auto
have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
@@ -4281,7 +5451,7 @@
have pt:"\<forall>(x,k)\<in>p. x \<le> t" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
note d2_fin = d2(2)[OF conjI[OF p(1) this]]
-
+
have *:"{a..c} \<inter> {x. x \<bullet> 1 \<le> t} = {a..t}" "{a..c} \<inter> {x. x \<bullet> 1 \<ge> t} = {t..c}"
using assms(2-3) as by(auto simp add:field_simps)
have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
@@ -4290,30 +5460,30 @@
proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real_def
- using as(1) by(auto simp add:field_simps)
+ using as(1) by(auto simp add:field_simps)
thus "x \<in> d1 c" using k(2) unfolding d_def by auto
qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
have *:"integral{a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
- integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c"
+ integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c"
"e = (e/3 + e/3) + e/3" by auto
have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
have "c \<in> {a..t}" by auto thus False using `t<c` by auto
qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
- unfolding split_conv defer apply(subst content_real) using as(2) by auto qed
+ unfolding split_conv defer apply(subst content_real) using as(2) by auto qed
have ***:"c - w < t \<and> t < c"
proof- have "c - k < t" using `k>0` as(1) by(auto simp add:field_simps)
- moreover have "k \<le> w" apply(rule ccontr) using k(2)
+ moreover have "k \<le> w" apply(rule ccontr) using k(2)
unfolding subset_eq apply(erule_tac x="c + ((k + w)/2)" in ballE)
unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real_def)
ultimately show ?thesis using `t<c` by(auto simp add:field_simps) qed
show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
- using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed
+ using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed
lemma indefinite_integral_continuous_right: fixes f::"real \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
@@ -4327,9 +5497,9 @@
"integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
apply(rule_tac[!] integral_combine) using assms as by auto
have "(- c) - d < (- t) \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
- thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
+ thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
-
+
lemma indefinite_integral_continuous: 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 assume as:"x\<in>{a..b}" "0<(e::real)"
@@ -4359,7 +5529,7 @@
thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-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 by(auto simp add:field_simps)
- qed qed qed
+ qed qed qed
subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
@@ -4372,7 +5542,7 @@
have "((\<lambda>x. 0\<Colon>'a) has_integral f x - f a) {a..x}"
apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
apply(rule continuous_on_subset[OF assms(2)]) defer
- apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
+ apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[symmetric])
apply assumption apply(rule open_interval) apply(rule has_derivative_within_subset[where s="{a..b}"])
using assms(4) assms(5) by auto note this[unfolded *]
note has_integral_unique[OF has_integral_0 this]
@@ -4385,16 +5555,16 @@
"\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
shows "f x = y"
proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
- unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
+ unfolding assms(5)[symmetric] by auto } assume "x\<noteq>c"
note conv = assms(1)[unfolded convex_alt,rule_format]
have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
apply safe apply(rule conv) using assms(4,7) by auto
have *:"\<And>t xa. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x \<Longrightarrow> t = xa"
- proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
+ proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
unfolding scaleR_simps by(auto simp add:algebra_simps)
thus ?case using `x\<noteq>c` by auto qed
- have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
+ have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
apply safe unfolding image_iff apply rule defer apply assumption
apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
@@ -4402,7 +5572,7 @@
apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
unfolding o_def using assms(5) defer apply-apply(rule)
proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
- have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
+ have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
apply(rule diff_chain_within) apply(rule has_derivative_add)
@@ -4414,7 +5584,7 @@
thus "((\<lambda>xa. f ((1 - xa) *\<^sub>R c + xa *\<^sub>R x)) has_derivative (\<lambda>h. 0)) (at t within {0..1})" unfolding o_def .
qed auto thus ?thesis by auto qed
-subsection {* Also to any open connected set with finite set of exceptions. Could
+subsection {* Also to any open connected set with finite set of exceptions. Could
generalize to locally convex set with limpt-free set of exceptions. *}
lemma has_derivative_zero_unique_strong_connected: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
@@ -4425,7 +5595,7 @@
apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
apply(rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
- proof safe fix x assume "x\<in>s"
+ proof safe fix x assume "x\<in>s"
from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
@@ -4444,12 +5614,12 @@
proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
{ presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
show ?thesis apply(cases,rule *,assumption)
- proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto
+ proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto
show ?thesis using assms(1) unfolding * using goal1 by auto
qed } assume "{c..d}\<noteq>{}"
from partial_division_extend_1[OF assms(2) this] guess p . note p=this
- note mon = monoidal_lifted[OF monoidal_monoid]
- note operat = operative_division[OF this operative_integral p(1), THEN sym]
+ note mon = monoidal_lifted[OF monoidal_monoid]
+ note operat = operative_division[OF this operative_integral p(1), symmetric]
let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
{ presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
apply- apply(cases,subst(asm) if_P,assumption) by auto
@@ -4476,13 +5646,13 @@
unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
lemma has_integral_restrict_closed_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
- assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}"
+ assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}"
shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
proof- note has_integral_restrict_open_subinterval[OF assms]
note * = has_integral_spike[OF negligible_frontier_interval _ this]
show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
-lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}"
+lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}"
shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b} \<longleftrightarrow> (f has_integral i) {c..d}" (is "?l = ?r")
proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
@@ -4512,38 +5682,38 @@
apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
by(auto simp add:dist_norm)
qed(insert B `e>0`, auto)
- next assume as:"\<forall>e>0. ?r e"
+ 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]
- def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
+ def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
proof
case goal1 thus ?case using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def
by(auto simp add:field_simps setsum_negf)
qed
- have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
+ have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
proof
case goal1 thus ?case
using Basis_le_norm[OF `i\<in>Basis`, of x] unfolding c_def d_def by (auto simp: setsum_negf)
qed
from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
- unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
+ unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,symmetric] unfolding s by auto
then guess y .. note y=this
have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
from as[rule_format,OF this] guess C .. note C=conjunctD2[OF this,rule_format]
- def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
+ def c \<equiv> "(\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)::'n::ordered_euclidean_space"
def d \<equiv> "(\<Sum>i\<in>Basis. max B C *\<^sub>R i)::'n::ordered_euclidean_space"
have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def
by(auto simp add:field_simps setsum_negf) qed
- have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
+ have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
proof case goal1 thus ?case using Basis_le_norm[of i x] unfolding c_def d_def by (auto simp: setsum_negf) qed
note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
thus False by auto qed
- thus ?l using y unfolding s by auto qed qed
+ thus ?l using y unfolding s by auto qed qed
lemma has_integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x) \<le> (g x)"
@@ -4556,12 +5726,12 @@
using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] .
lemma has_integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
- assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
+ assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
using has_integral_component_nonneg[of 1 f i s]
unfolding o_def using assms by auto
lemma integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
- assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
+ assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
subsection {* Hence a general restriction property. *}
@@ -4574,20 +5744,20 @@
lemma has_integral_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
"((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
-lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
+lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
shows "(f has_integral i) t"
proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
apply(rule) using assms(1-2) by auto
- thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
- apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
-
-lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
+ thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[symmetric])
+ apply- apply(subst(asm) has_integral_restrict_univ[symmetric]) by auto qed
+
+lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
shows "f integrable_on t"
using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
-lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
+lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
@@ -4600,9 +5770,9 @@
proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
unfolding indicator_def by auto qed qed auto
-lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
+lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
- unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (auto split: split_if_asm)
+ unfolding has_integral_restrict_univ[symmetric,of f] apply(rule has_integral_spike_eq[OF assms]) by (auto split: split_if_asm)
lemma has_integral_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
@@ -4611,7 +5781,7 @@
lemma integrable_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
- shows "f integrable_on t" using assms(2) unfolding integrable_on_def
+ shows "f integrable_on t" using assms(2) unfolding integrable_on_def
unfolding has_integral_spike_set_eq[OF assms(1)] .
lemma integrable_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
@@ -4656,7 +5826,7 @@
lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes k: "k\<in>Basis" and as: "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\<bullet>k"
shows "i\<bullet>k \<le> j\<bullet>k"
-proof- note has_integral_restrict_univ[THEN sym, of f]
+proof- note has_integral_restrict_univ[symmetric, of f]
note as(2-3)[unfolded this] note * = has_integral_component_le[OF k this]
show ?thesis apply(rule *) using as(1,4) by auto qed
@@ -4701,12 +5871,12 @@
show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
proof(rule,rule,rule B,safe) case goal1 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 qed qed
+ from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed
subsection {* Continuity of the integral (for a 1-dimensional interval). *}
-lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
+lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
"f integrable_on s \<longleftrightarrow>
(\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
(\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
@@ -4718,7 +5888,7 @@
show ?case apply(rule,rule,rule B)
proof safe case goal1 show ?case apply(rule norm_triangle_half_l)
using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
-
+
next assume ?r note as = conjunctD2[OF this,rule_format]
let ?cube = "\<lambda>n. {(\<Sum>i\<in>Basis. - real n *\<^sub>R i)::'n .. (\<Sum>i\<in>Basis. real n *\<^sub>R i)} :: 'n set"
have "Cauchy (\<lambda>n. integral (?cube n) (\<lambda>x. if x \<in> s then f x else 0))"
@@ -4730,7 +5900,7 @@
proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
using n N by(auto simp add:field_simps setsum_negf) qed }
thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto
- qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
+ qed from this[unfolded convergent_eq_cauchy[symmetric]] guess i ..
note i = this[THEN LIMSEQ_D]
show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
@@ -4747,7 +5917,7 @@
apply(rule N[of n])
proof safe show "N \<le> n" using n by auto
fix x::"'n::ordered_euclidean_space" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
- thus "x\<in>{a..b}" using ab by blast
+ thus "x\<in>{a..b}" using ab by blast
show "x\<in>?cube n" using x unfolding mem_interval mem_ball dist_norm apply-
proof case goal1 thus ?case using Basis_le_norm[of i x] `i\<in>Basis`
using n by(auto simp add:field_simps setsum_negf) qed qed qed qed qed
@@ -4777,31 +5947,31 @@
from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
show ?case apply(rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI) apply(rule conjI gauge_inter d1 d2)+ unfolding fine_inter
proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
- abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow> abs(g1 - i) < e / 3 \<Longrightarrow>
+ abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow> abs(g1 - i) < e / 3 \<Longrightarrow>
abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith
case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)]
have "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
- "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
+ "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
"(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
- "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
- unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg)
- apply safe unfolding real_scaleR_def right_diff_distrib[THEN sym]
+ "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
+ unfolding setsum_subtractf[symmetric] apply- apply(rule_tac[!] setsum_nonneg)
+ apply safe unfolding real_scaleR_def right_diff_distrib[symmetric]
apply(rule_tac[!] mult_nonneg_nonneg)
proof- fix a b assume ab:"(a,b) \<in> p1"
show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(1-2)[OF ab] using obt(4)[rule_format,of a] by auto
next fix a b assume ab:"(a,b) \<in> p2"
show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
- show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto qed
+ show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto qed
thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
- unfolding real_norm_def[THEN sym] apply(rule obt(3))
+ unfolding real_norm_def[symmetric] apply(rule obt(3))
apply(rule d1(2)[OF conjI[OF goal1(4,5)]])
apply(rule d1(2)[OF conjI[OF goal1(1,2)]])
apply(rule d2(2)[OF conjI[OF goal1(4,6)]])
- apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed
-
+ apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed
+
lemma integrable_straddle: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))"
@@ -4822,7 +5992,7 @@
case goal2 thus ?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> norm(ag - ah) < e"
- using obt(3) unfolding real_norm_def by arith
+ 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 {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI)
@@ -4836,7 +6006,7 @@
integral {a..b} (\<lambda>x. if x \<in> s then g x else 0))
\<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) -
integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))"
- unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
+ unfolding integral_sub[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_sub h g)+
proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
apply - apply rule apply(erule_tac x=i in ballE) by auto
@@ -4856,30 +6026,30 @@
abs(hc - j) < e / 3 \<and> abs(i - j) < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc\<Longrightarrow> abs(fa - fc) < e"
by (simp add: abs_real_def split: split_if_asm)
show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - integral {c..d} (\<lambda>x. if x \<in> s then f x else 0)) < e"
- unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
- apply(rule B1(2),rule order_trans,rule **,rule as(1))
- apply(rule B1(2),rule order_trans,rule **,rule as(2))
- apply(rule B2(2),rule order_trans,rule **,rule as(1))
- apply(rule B2(2),rule order_trans,rule **,rule as(2))
+ unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[symmetric]
+ apply(rule B1(2),rule order_trans,rule **,rule as(1))
+ apply(rule B1(2),rule order_trans,rule **,rule as(2))
+ apply(rule B2(2),rule order_trans,rule **,rule as(1))
+ apply(rule B2(2),rule order_trans,rule **,rule as(2))
apply(rule obt) apply(rule_tac[!] integral_le) using obt
- by(auto intro!: h g interv) qed qed qed
+ by(auto intro!: h g interv) qed qed qed
subsection {* Adding integrals over several sets. *}
lemma has_integral_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)"
shows "(f has_integral (i + j)) (s \<union> t)"
-proof- note * = has_integral_restrict_univ[THEN sym, of f]
+proof- note * = has_integral_restrict_univ[symmetric, of f]
show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)])
defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed
lemma has_integral_unions: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "finite t" "\<forall>s\<in>t. (f has_integral (i s)) s" "\<forall>s\<in>t. \<forall>s'\<in>t. ~(s = s') \<longrightarrow> negligible(s \<inter> s')"
shows "(f has_integral (setsum i t)) (\<Union>t)"
-proof- note * = has_integral_restrict_univ[THEN sym, of f]
+proof- note * = has_integral_restrict_univ[symmetric, of f]
have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))"
- apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer
- apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto
+ apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer
+ apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto
note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this]
thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption
proof safe case goal1 thus ?case
@@ -4895,7 +6065,7 @@
assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k"
shows "(f has_integral (setsum i d)) s"
proof- note d = division_ofD[OF assms(1)]
- show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions)
+ show ?thesis unfolding d(6)[symmetric] apply(rule has_integral_unions)
apply(rule d assms)+ apply(rule,rule,rule)
proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)]
guess a c b d apply-by(erule exE)+ note obt=this
@@ -4913,7 +6083,7 @@
assumes "f integrable_on s" "d division_of k" "k \<subseteq> s"
shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
apply(rule has_integral_combine_division[OF assms(2)])
- apply safe unfolding has_integral_integral[THEN sym]
+ apply safe unfolding has_integral_integral[symmetric]
proof- case goal1 from division_ofD(2,4)[OF assms(2) this]
show ?case apply safe apply(rule integrable_on_subinterval)
apply(rule assms) using assms(3) by auto qed
@@ -4944,7 +6114,7 @@
shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)])
- using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto)
+ using assms(2) unfolding has_integral_integral[symmetric] by(safe,auto)
thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption
apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute)
apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption)
@@ -4998,22 +6168,22 @@
let ?p = "p \<union> \<Union>(qq ` r)" have "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - integral {a..b} f) < e"
apply(rule assms(4)[rule_format])
- proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto
+ proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto
note * = tagged_partial_division_of_union_self[OF p(1)]
have "p \<union> \<Union>(qq ` r) tagged_division_of \<Union>(snd ` p) \<union> \<Union>r"
proof(rule tagged_division_union[OF * tagged_division_unions])
show "finite r" by fact case goal2 thus ?case using qq by auto
next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto
next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule)
- apply(rule,rule q') defer apply(rule,subst Int_commute)
+ apply(rule,rule q') defer apply(rule,subst Int_commute)
apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer
apply(rule,rule q') using q(1) p' unfolding r_def by auto qed
moreover have "\<Union>(snd ` p) \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r"
- unfolding Union_Un_distrib[THEN sym] r_def using q by auto
+ unfolding Union_Un_distrib[symmetric] r_def using q by auto
ultimately show "?p tagged_division_of {a..b}" by fastforce qed
hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>\<Union>(qq ` r). content k *\<^sub>R f x) -
- integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3
+ integral {a..b} f) < e" apply(subst setsum_Un_zero[symmetric]) apply(rule p') prefer 3
apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq)
proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r"
note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
@@ -5021,7 +6191,7 @@
have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto
note q'(5)[OF this] hence "interior l = {}" using interior_mono[OF `l \<subseteq> k`] by blast
- thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto
+ thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto qed auto
hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
(qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero)
@@ -5032,23 +6202,23 @@
from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this
have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q')
using as unfolding r_def by auto
- have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym]
+ have "interior m = {}" unfolding subset_empty[symmetric] unfolding *[symmetric]
apply(rule interior_mono) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto
- thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto
+ thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[symmetric] by auto
qed(insert qq, auto)
hence **:"norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x) \<circ> qq) r -
integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact
apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption
proof- fix k l x m assume as:"k\<in>r" "l\<in>r" "k\<noteq>l" "qq k = qq l" "(x,m)\<in>qq k"
- note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)]
+ note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)]
show "content m *\<^sub>R f x = 0" using as(3) unfolding as by auto qed
-
- have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow>
- ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k"
- proof- case goal1 thus ?case using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
- unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
-
+
+ have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow>
+ ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k"
+ proof- case goal1 thus ?case using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
+ unfolding goal1(3)[symmetric] norm_minus_cancel by(auto simp add:algebra_simps) qed
+
have "?x = norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
unfolding split_def setsum_subtractf ..
also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"])
@@ -5059,15 +6229,15 @@
from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this
show "integral l f = 0" unfolding uv apply(rule integral_unique)
apply(rule has_integral_null) unfolding content_eq_0_interior
- using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto
- qed auto
+ using p'(5)[OF as(1-3)] unfolding uv as(4)[symmetric] by auto
+ qed auto
show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
- apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto
+ apply(rule setsum_Un_disjoint'[symmetric]) using q(1) q'(1) p'(1) by auto
next case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps)
show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
- unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le)
- apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[THEN sym]
- unfolding divide_inverse[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat)
+ unfolding setsum_subtractf[symmetric] apply(rule setsum_norm_le)
+ apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[symmetric]
+ unfolding divide_inverse[symmetric] using * by(auto simp add:field_simps real_eq_of_nat)
qed finally show "?x \<le> e + k" . qed
lemma henstock_lemma_part2: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
@@ -5075,12 +6245,12 @@
"\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p -
integral({a..b}) f) < e" "p tagged_partial_division_of {a..b}" "d fine p"
shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
- unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer
+ unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer
apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
apply safe apply(rule assms[rule_format,unfolded split_def]) defer
apply(rule tagged_partial_division_subset,rule assms,assumption)
apply(rule fine_subset,assumption,rule assms) using assms(5) by auto
-
+
lemma henstock_lemma: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
assumes "f integrable_on {a..b}" "e>0"
obtains d where "gauge d"
@@ -5201,7 +6371,7 @@
unfolding dist_real_def using fg[rule_format,OF goal1]
by (auto simp add:field_simps) qed
from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
- def d \<equiv> "\<lambda>x. c (m x) x"
+ def d \<equiv> "\<lambda>x. c (m x) x"
show ?case apply(rule_tac x=d in exI)
proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
@@ -5211,7 +6381,7 @@
by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
then guess s .. note s=this
have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and>
- norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e"
+ norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e"
proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
by(auto simp add:algebra_simps) qed
@@ -5219,17 +6389,17 @@
b="\<Sum>(x, k)\<in>p. content k *\<^sub>R f (m x) x" and c="\<Sum>(x, k)\<in>p. integral k (f (m x))"])
proof safe case goal1
show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"])
- unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule norm_setsum)
+ unfolding setsum_subtractf[symmetric] apply(rule order_trans,rule norm_setsum)
apply(rule setsum_mono) unfolding split_paired_all split_conv
- unfolding split_def setsum_left_distrib[THEN sym] scaleR_diff_right[THEN sym]
+ unfolding split_def setsum_left_distrib[symmetric] scaleR_diff_right[symmetric]
unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto
from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this
show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))"
- unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le]
+ unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le]
apply(rule mult_left_mono) using m(2)[OF x,of "m x"] by auto
qed(insert ab,auto)
-
+
next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
\<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer
@@ -5240,7 +6410,7 @@
apply(rule setsum_norm_le)
proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
unfolding power_add divide_inverse inverse_mult_distrib
- unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym]
+ unfolding setsum_right_distrib[symmetric] setsum_left_distrib[symmetric]
unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e])
unfolding power2_eq_square by auto
fix t assume "t\<in>{0..s}"
@@ -5259,22 +6429,22 @@
next case goal3
note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i\<bullet>1 - kr\<bullet>1
- \<and> i\<bullet>1 - kr\<bullet>1 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto
+ \<and> i\<bullet>1 - kr\<bullet>1 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto
show ?case unfolding real_norm_def apply(rule *[rule_format],safe)
- apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded real_inner_1_right])
+ apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded real_inner_1_right])
apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
apply(rule_tac[1-2] integral_le[OF ])
proof safe show "0 \<le> i\<bullet>1 - (integral {a..b} (f r))\<bullet>1" using r(1) by auto
show "i\<bullet>1 - (integral {a..b} (f r))\<bullet>1 < e / 4" using r(2) by auto
fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
- show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k"
+ show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k"
unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
- using p'(3)[OF xk] unfolding uv by auto
+ using p'(3)[OF xk] unfolding uv by auto
fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto
hence *:"\<And>m. \<forall>n\<ge>m. (f m y) \<le> (f n y)" apply-apply(rule transitive_stepwise_le) using assms(2) by auto
show "(f r y) \<le> (f (m x) y)" "(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] by auto
- qed qed qed qed note * = this
+ qed qed qed qed note * = this
have "integral {a..b} g = i" apply(rule integral_unique) using * .
thus ?thesis using i * by auto qed
@@ -5300,13 +6470,13 @@
apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
apply simp
- apply(rule goal1(2)[rule_format])+ by auto
+ apply(rule goal1(2)[rule_format])+ by auto
note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) =
(\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto
- have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym])
- apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int .
+ have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[symmetric])
+ apply(subst ifif[symmetric]) apply(subst integrable_restrict_univ) using int .
have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and>
((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) --->
integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
@@ -5320,7 +6490,7 @@
unfolding real_norm_def apply(subst abs_of_nonneg) apply(rule *[OF int])
apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3
apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]])
- apply(subst integral_restrict_univ[THEN sym,OF int])
+ apply(subst integral_restrict_univ[symmetric,OF int])
unfolding ifif unfolding integral_restrict_univ[OF int']
apply(rule integral_subset_le[OF _ int' assms(2)]) using assms(1) by auto
thus ?case using assms(5) unfolding bounded_iff apply safe
@@ -5341,7 +6511,7 @@
apply-defer apply(subst norm_minus_commute) by auto
have *:"\<And>f1 f2 g. abs(f1 - i) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i
\<longrightarrow> abs(g - i) < e" unfolding real_inner_1_right by arith
- show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e"
+ show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e"
unfolding real_norm_def apply(rule *[rule_format])
apply(rule **[unfolded real_norm_def])
apply(rule M[rule_format,of "M + N",unfolded real_norm_def]) apply(rule le_add1)
@@ -5349,10 +6519,10 @@
apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded real_inner_1_right]])
proof safe case goal2 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) by auto thus ?case by auto
- next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int])
+ next case goal1 show ?case apply(subst integral_restrict_univ[symmetric,OF int])
unfolding ifif integral_restrict_univ[OF int']
apply(rule integral_subset_le[OF _ int']) using assms by auto
- qed qed qed
+ qed qed qed
thus ?case apply safe defer apply(drule integral_unique) using i by auto qed
have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
@@ -5364,7 +6534,7 @@
proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
- next case goal4 thus ?case apply-apply(rule tendsto_diff)
+ next case goal4 thus ?case apply-apply(rule tendsto_diff)
using seq_offset[OF assms(3)[rule_format],of x 1] by auto
next case goal5 thus ?case using assms(4) unfolding bounded_iff
apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
@@ -5390,7 +6560,7 @@
note * = conjunctD2[OF this]
show ?thesis apply rule using integrable_neg[OF *(1)] defer
using tendsto_minus[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
- unfolding integral_neg[OF *(1),THEN sym] by auto qed
+ unfolding integral_neg[OF *(1),symmetric] by auto qed
subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
@@ -5415,9 +6585,9 @@
proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr)
apply(erule_tac x="x - y" in allE) by auto
have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
- \<longrightarrow> norm(ig) < dia + e"
+ \<longrightarrow> norm(ig) < dia + e"
proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
- apply(subst real_sum_of_halves[of e,THEN sym]) unfolding add_assoc[symmetric]
+ apply(subst real_sum_of_halves[of e,symmetric]) unfolding add_assoc[symmetric]
apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
qed note norm=this[rule_format]
@@ -5440,7 +6610,7 @@
apply(rule mult_left_mono) using goal1(3) as by auto
qed(insert p[unfolded fine_inter],auto) qed
- { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
+ { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
thus ?thesis apply-apply(rule *[rule_format]) by auto }
fix e::real assume "e>0" hence e:"e/2 > 0" by auto
note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
@@ -5505,7 +6675,7 @@
apply(drule absolutely_integrable_norm) unfolding real_norm_def .
lemma absolutely_integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
- "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}"
+ "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}"
unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
lemma absolutely_integrable_bounded_variation: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
@@ -5520,14 +6690,14 @@
apply(subst integral_combine_division_topdown[OF _ goal1(2)])
using integrable_on_subdivision[OF goal1(2)] using assms by auto
also have "... \<le> integral UNIV (\<lambda>x. norm (f x))"
- apply(rule integral_subset_le)
+ apply(rule integral_subset_le)
using integrable_on_subdivision[OF goal1(2)] using assms by auto
finally show ?case . qed
lemma helplemma:
assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s"
shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e"
- unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs])
+ unfolding setsum_subtractf[symmetric] apply(rule le_less_trans[OF setsum_abs])
apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
using norm_triangle_ineq3 .
@@ -5542,7 +6712,7 @@
show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i])
proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) `
{d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format]
- unfolding setge_def ubs_def by auto
+ unfolding setge_def ubs_def by auto
hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))"
unfolding mem_Collect_eq isUb_def setle_def by(simp add:not_le) then guess d .. note d=conjunctD2[OF this]
note d' = division_ofD[OF this(1)]
@@ -5567,7 +6737,7 @@
have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto
have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI)
proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l))
- ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def
+ ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def
defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto
fix x k assume "(x,k)\<in>p'"
@@ -5590,15 +6760,15 @@
show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least)
unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe
proof- fix y assume y:"y\<in>{a..b}"
- hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto
+ hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[symmetric] by auto
then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this]
- hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto
+ hence "\<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]
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
thus "y\<in>\<Union>{k. \<exists>x. (x, k) \<in> p'}" unfolding p'_def Union_iff apply(rule_tac x="i \<inter> l" in bexI)
defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI)
- apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto
- qed qed
+ apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto
+ qed qed
hence "(\<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 using gp' .
@@ -5625,7 +6795,7 @@
have *:"\<And>sni sni' sf sf'. abs(sf' - sni') < e / 2 \<longrightarrow> i - e / 2 < sni \<and> sni' \<le> i \<and>
sni \<le> sni' \<and> sf' = sf \<longrightarrow> abs(sf - i) < e" by arith
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e"
+ show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e"
unfolding real_norm_def apply(rule *[rule_format,OF **],safe) apply(rule d(2))
proof- case goal1 show ?case unfolding sum_p'
apply(rule isLubD2[OF i]) using division_of_tagged_division[OF p''] by auto
@@ -5635,7 +6805,7 @@
proof(rule setsum_mono) case goal1 note k=this
from d'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
def d' \<equiv> "{{u..v} \<inter> l |l. l \<in> snd ` p \<and> ~({u..v} \<inter> l = {})}" note uvab = d'(2)[OF k[unfolded uv]]
- have "d' division_of {u..v}" apply(subst d'_def) apply(rule division_inter_1)
+ have "d' division_of {u..v}" apply(subst d'_def) apply(rule division_inter_1)
apply(rule division_of_tagged_division[OF p(1)]) using uvab .
hence "norm (integral k f) \<le> setsum (\<lambda>k. norm (integral k f)) d'"
unfolding uv apply(subst integral_combine_division_topdown[of _ _ d'])
@@ -5653,18 +6823,18 @@
apply(rule Int_greatest) defer apply(subst goal1(4)) by auto
hence *:"interior (k \<inter> l) = {}" using snd_p(5)[OF goal1(1-3)] by auto
from d'(4)[OF k] snd_p(4)[OF goal1(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this
- show ?case using * unfolding uv inter_interval content_eq_0_interior[THEN sym] by auto
+ show ?case using * unfolding uv inter_interval content_eq_0_interior[symmetric] by auto
qed finally show ?case .
qed also have "... = (\<Sum>(i,l)\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (i\<inter>l) f))"
- apply(subst sum_sum_product[THEN sym],fact) using p'(1) by auto
+ apply(subst sum_sum_product[symmetric],fact) using p'(1) by auto
also have "... = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (split op \<inter> x) f))"
unfolding split_def ..
also have "... = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
- unfolding * apply(rule setsum_reindex_nonzero[THEN sym,unfolded o_def])
+ unfolding * apply(rule setsum_reindex_nonzero[symmetric,unfolded o_def])
apply(rule finite_product_dependent) apply(fact,rule finite_imageI,rule p')
unfolding split_paired_all mem_Collect_eq split_conv o_def
proof- note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)]
- fix l1 l2 k1 k2 assume as:"(l1, k1) \<noteq> (l2, k2)" "l1 \<inter> k1 = l2 \<inter> k2"
+ fix l1 l2 k1 k2 assume as:"(l1, k1) \<noteq> (l2, k2)" "l1 \<inter> k1 = l2 \<inter> k2"
"\<exists>i l. (l1, k1) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
"\<exists>i l. (l2, k2) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p"
hence "l1 \<in> d" "k1 \<in> snd ` p" by auto from d'(4)[OF this(1)] *(1)[OF this(2)]
@@ -5676,7 +6846,7 @@
moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" using as(2) by auto
ultimately have "interior(l1 \<inter> k1) = {}" by auto
thus "norm (integral (l1 \<inter> k1) f) = 0" unfolding uv inter_interval
- unfolding content_eq_0_interior[THEN sym] by auto
+ unfolding content_eq_0_interior[symmetric] by auto
qed also have "... = (\<Sum>(x, k)\<in>p'. norm (integral k f))" unfolding sum_p'
apply(rule setsum_mono_zero_right) apply(subst *)
apply(rule finite_imageI[OF finite_product_dependent]) apply fact
@@ -5684,7 +6854,7 @@
proof- case goal2 have "ia \<inter> b = {}" using goal2 unfolding p'alt image_iff Bex_def not_ex
apply(erule_tac x="(a,ia\<inter>b)" in allE) by auto thus ?case by auto
next case goal1 thus ?case unfolding p'_def apply safe
- apply(rule_tac x=i in exI,rule_tac x=l in exI) unfolding snd_conv image_iff
+ apply(rule_tac x=i in exI,rule_tac x=l in exI) unfolding snd_conv image_iff
apply safe apply(rule_tac x="(a,l)" in bexI) by auto
qed finally show ?case .
@@ -5705,15 +6875,15 @@
"x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> k1 = k2)"
from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this
from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2" by auto
- hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})"
+ hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})"
apply-apply(erule disjE) apply(rule disjI2) defer apply(rule disjI1)
apply(rule d'(5)[OF as(3-4)],assumption) apply(rule p'(5)[OF as(1-2)]) by auto
moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" unfolding as ..
ultimately have "interior (l1 \<inter> k1) = {}" by auto
thus "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0" unfolding uv inter_interval
- unfolding content_eq_0_interior[THEN sym] by auto
+ unfolding content_eq_0_interior[symmetric] by auto
qed safe also have "... = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))" unfolding Sigma_alt
- apply(subst sum_sum_product[THEN sym]) apply(rule p', rule,rule d')
+ apply(subst sum_sum_product[symmetric]) apply(rule p', rule,rule d')
apply(rule setsum_cong2) unfolding split_paired_all split_conv
proof- fix x l assume as:"(x,l)\<in>p"
note xl = p'(2-4)[OF this] from this(3) guess u v apply-by(erule exE)+ note uv=this
@@ -5721,7 +6891,7 @@
apply(rule setsum_cong2) apply(drule d'(4),safe) apply(subst Int_commute)
unfolding inter_interval uv apply(subst abs_of_nonneg) by auto
also have "... = setsum content {k\<inter>{u..v}| k. k\<in>d}" unfolding simple_image
- apply(rule setsum_reindex_nonzero[unfolded o_def,THEN sym]) apply(rule d')
+ apply(rule setsum_reindex_nonzero[unfolded o_def,symmetric]) apply(rule d')
proof- case goal1 from d'(4)[OF this(1)] d'(4)[OF this(2)]
guess u1 v1 u2 v2 apply- by(erule exE)+ note uv=this
have "{} = interior ((k \<inter> y) \<inter> {u..v})" apply(subst interior_inter)
@@ -5738,11 +6908,11 @@
unfolding ab inter_interval content_eq_0_interior by auto
thus ?case using goal1(1) using interior_subset[of "k \<inter> {u..v}"] by auto
qed finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
- unfolding setsum_left_distrib[THEN sym] real_scaleR_def apply -
+ unfolding setsum_left_distrib[symmetric] real_scaleR_def apply -
apply(subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
using xl(2)[unfolded uv] unfolding uv by auto
- qed finally show ?case .
- qed qed qed qed
+ qed finally show ?case .
+ qed qed qed qed
lemma bounded_variation_absolutely_integrable: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "f integrable_on UNIV" "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
@@ -5755,7 +6925,7 @@
have f_int:"\<And>a b. f absolutely_integrable_on {a..b}"
apply(rule bounded_variation_absolutely_integrable_interval[where B=B])
apply(rule integrable_on_subinterval[OF assms(1)]) defer apply safe
- apply(rule assms(2)[rule_format]) by auto
+ apply(rule assms(2)[rule_format]) by auto
show "((\<lambda>x. norm (f x)) has_integral i) UNIV" apply(subst has_integral_alt',safe)
proof- case goal1 show ?case using f_int[of a b] by auto
next case goal2 have "\<exists>y\<in>setsum (\<lambda>k. norm (integral k f)) ` {d. d division_of \<Union>d}. \<not> y \<le> i - e"
@@ -5775,11 +6945,11 @@
proof- case goal1 have "(\<Sum>k\<in>d. norm (integral k f)) \<le> setsum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d"
apply(rule setsum_mono) apply(rule absolutely_integrable_le)
apply(drule d'(4),safe) by(rule f_int)
- also have "... = integral (\<Union>d) (\<lambda>x. norm(f x))"
- apply(rule integral_combine_division_bottomup[THEN sym])
+ also have "... = integral (\<Union>d) (\<lambda>x. norm(f x))"
+ apply(rule integral_combine_division_bottomup[symmetric])
apply(rule d) unfolding forall_in_division[OF d(1)] using f_int by auto
- also have "... \<le> integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)"
- proof- case goal1 have "\<Union>d \<subseteq> {a..b}" apply rule apply(drule K(2)[rule_format])
+ also have "... \<le> integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)"
+ proof- case goal1 have "\<Union>d \<subseteq> {a..b}" apply rule apply(drule K(2)[rule_format])
apply(rule ab[unfolded subset_eq,rule_format]) by(auto simp add:dist_norm)
thus ?case apply- apply(subst if_P,rule) apply(rule integral_subset_le) defer
apply(rule integrable_on_subdivision[of _ _ _ "{a..b}"])
@@ -5795,7 +6965,7 @@
have *:"\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> i \<longrightarrow> abs(sf - si) < e / 2
\<longrightarrow> abs(sf' - di) < e / 2 \<longrightarrow> di < i + e" by arith
show "integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < i + e" apply(subst if_P,rule)
- proof(rule *[rule_format])
+ proof(rule *[rule_format])
show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e / 2"
unfolding split_def apply(rule helplemma) using d2(2)[rule_format,of p]
using p(1,3) unfolding tagged_division_of_def split_def by auto
@@ -5810,7 +6980,7 @@
unfolding image_iff apply(rule_tac x="snd ` p" in bexI) unfolding mem_Collect_eq defer
apply(rule partial_division_of_tagged_division[of _ "{a..b}"])
using p(1) unfolding tagged_division_of_def by auto
- qed qed qed(insert K,auto) qed qed
+ qed qed qed(insert K,auto) qed qed
lemma absolutely_integrable_restrict_univ:
"(\<lambda>x. if x \<in> s then f x else (0::'a::banach)) absolutely_integrable_on UNIV \<longleftrightarrow> f absolutely_integrable_on s"
@@ -5821,12 +6991,12 @@
shows "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on s"
proof- let ?P = "\<And>f g::'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space. f absolutely_integrable_on UNIV \<Longrightarrow>
g absolutely_integrable_on UNIV \<Longrightarrow> (\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV"
- { presume as:"PROP ?P" note a = absolutely_integrable_restrict_univ[THEN sym]
+ { presume as:"PROP ?P" note a = absolutely_integrable_restrict_univ[symmetric]
have *:"\<And>x. (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)" by auto
show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]] unfolding * . }
fix f g::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assume assms:"f absolutely_integrable_on UNIV"
- "g absolutely_integrable_on UNIV"
+ "g absolutely_integrable_on UNIV"
note absolutely_integrable_bounded_variation
from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
show "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV"
@@ -5837,7 +7007,7 @@
apply(rule_tac[!] integrable_on_subinterval[of _ UNIV]) using assms by auto
hence "(\<Sum>k\<in>d. norm (integral k (\<lambda>x. f x + g x))) \<le>
(\<Sum>k\<in>d. norm (integral k f)) + (\<Sum>k\<in>d. norm (integral k g))" apply-
- unfolding setsum_addf[THEN sym] apply(rule setsum_mono)
+ unfolding setsum_addf[symmetric] apply(rule setsum_mono)
apply(subst integral_add) prefer 3 apply(rule norm_triangle_ineq) by auto
also have "... \<le> B1 + B2" using B(1)[OF goal1] B(2)[OF goal1] by auto
finally show ?case .
@@ -5852,18 +7022,18 @@
lemma absolutely_integrable_linear: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space" and h::"'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space"
assumes "f absolutely_integrable_on s" "bounded_linear h"
shows "(h o f) absolutely_integrable_on s"
-proof- { presume as:"\<And>f::'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space. \<And>h::'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space.
+proof- { presume as:"\<And>f::'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space. \<And>h::'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space.
f absolutely_integrable_on UNIV \<Longrightarrow> bounded_linear h \<Longrightarrow>
- (h o f) absolutely_integrable_on UNIV" note a = absolutely_integrable_restrict_univ[THEN sym]
+ (h o f) absolutely_integrable_on UNIV" note a = absolutely_integrable_restrict_univ[symmetric]
show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]]
unfolding o_def if_distrib linear_simps[OF assms(2)] . }
fix f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space" and h::"'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space"
- assume assms:"f absolutely_integrable_on UNIV" "bounded_linear h"
+ assume assms:"f absolutely_integrable_on UNIV" "bounded_linear h"
from absolutely_integrable_bounded_variation[OF assms(1)] guess B . note B=this
from bounded_linear.pos_bounded[OF assms(2)] guess b .. note b=conjunctD2[OF this]
show "(h o f) absolutely_integrable_on UNIV"
apply(rule bounded_variation_absolutely_integrable[of _ "B * b"])
- apply(rule integrable_linear[OF _ assms(2)])
+ apply(rule integrable_linear[OF _ assms(2)])
proof safe case goal2
have "(\<Sum>k\<in>d. norm (integral k (h \<circ> f))) \<le> setsum (\<lambda>k. norm(integral k f)) d * b"
unfolding setsum_left_distrib apply(rule setsum_mono)
@@ -5953,14 +7123,14 @@
proof
assume ?l thus ?r apply-apply rule defer
apply(drule absolutely_integrable_vector_abs) by auto
-next
+next
assume ?r
{ presume lem:"\<And>f::'n \<Rightarrow> 'm. f integrable_on UNIV \<Longrightarrow>
(\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
have *:"\<And>x. (\<Sum>i\<in>Basis. \<bar>(if x \<in> s then f x else 0) \<bullet> i\<bar> *\<^sub>R i) =
(if x\<in>s then (\<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) else (0::'m))"
unfolding euclidean_eq_iff[where 'a='m] by auto
- show ?l apply(subst absolutely_integrable_restrict_univ[THEN sym]) apply(rule lem)
+ show ?l apply(subst absolutely_integrable_restrict_univ[symmetric]) apply(rule lem)
unfolding integrable_restrict_univ * using `?r` by auto }
fix f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assume assms:"f integrable_on UNIV" "(\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) integrable_on UNIV"
@@ -5976,7 +7146,7 @@
from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this
show "\<bar>integral k f \<bullet> i\<bar> \<le> integral k (\<lambda>x. (\<Sum>i\<in>Basis. \<bar>f x\<bullet>i\<bar> *\<^sub>R i)::'m) \<bullet> i"
apply (rule abs_leI)
- unfolding inner_minus_left[THEN sym] defer apply(subst integral_neg[THEN sym])
+ unfolding inner_minus_left[symmetric] defer apply(subst integral_neg[symmetric])
defer apply(rule_tac[1-2] integral_component_le[OF i]) apply(rule integrable_neg)
using integrable_on_subinterval[OF assms(1),of a b]
integrable_on_subinterval[OF assms(2),of a b] i unfolding ab by auto
@@ -6009,7 +7179,7 @@
shows "f absolutely_integrable_on s"
proof- { presume *:"\<And>f::'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space. \<And> g. \<forall>x. norm(f x) \<le> g x \<Longrightarrow> f integrable_on UNIV
\<Longrightarrow> g integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
- show ?thesis apply(subst absolutely_integrable_restrict_univ[THEN sym])
+ show ?thesis apply(subst absolutely_integrable_restrict_univ[symmetric])
apply(rule *[of _ "\<lambda>x. if x\<in>s then g x else 0"])
using assms unfolding integrable_restrict_univ by auto }
fix g and f :: "'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
@@ -6018,9 +7188,9 @@
apply(rule bounded_variation_absolutely_integrable[OF assms(2),where B="integral UNIV g"])
proof safe case goal1 note d=this and d'=division_ofD[OF this]
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>k\<in>d. integral k g)"
- apply(rule setsum_mono) apply(rule integral_norm_bound_integral) apply(drule_tac[!] d'(4),safe)
+ apply(rule setsum_mono) apply(rule integral_norm_bound_integral) apply(drule_tac[!] d'(4),safe)
apply(rule_tac[1-2] integrable_on_subinterval) using assms by auto
- also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[THEN sym])
+ also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[symmetric])
apply(rule d,safe) apply(drule d'(4),safe)
apply(rule integrable_on_subinterval[OF assms(3)]) by auto
also have "... \<le> integral UNIV g" apply(rule integral_subset_le) defer
@@ -6161,7 +7331,7 @@
qed
then guess y .. note y=this[unfolded not_le]
from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this]
-
+
show ?case
apply (rule_tac x=N in exI)
proof safe
@@ -6247,7 +7417,7 @@
case goal1
thus ?case using assms(3)[rule_format,OF x, of j] by auto
qed auto
-
+
have "\<exists>y\<in>?S. \<not> y \<le> i - r"
proof (rule ccontr)
case goal1
@@ -6262,7 +7432,7 @@
qed
then guess y .. note y=this[unfolded not_le]
from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this]
-
+
show ?case
apply (rule_tac x=N in exI)
proof safe
@@ -6291,7 +7461,7 @@
have "g integrable_on s \<and>
((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) ---> integral s g) sequentially"
apply (rule monotone_convergence_increasing,safe)
- apply fact
+ apply fact
proof -
show "bounded {integral s (\<lambda>x. Inf {f j x |j. k \<le> j}) |k. True}"
unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
@@ -6418,7 +7588,7 @@
show "integral s (\<lambda>x. Inf {f j x |j. n \<le> j}) \<le> integral s (f n)"
proof (rule integral_le[OF dec1(1) assms(1)], safe)
fix x
- assume x: "x \<in> s"
+ assume x: "x \<in> s"
have *: "\<And>x y::real. x \<ge> - y \<Longrightarrow> - x \<le> y" by auto
show "Inf {f j x |j. n \<le> j} \<le> f n x"
apply (rule cInf_lower[where z="- h x"])
--- a/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 06 10:56:40 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Fri Sep 06 10:57:27 2013 +0200
@@ -17,11 +17,15 @@
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
proof -
- have "(x + 1/2)\<^sup>2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
- then show ?thesis by (simp add: field_simps power2_eq_square)
+ have "(x + 1/2)\<^sup>2 + 3/4 > 0"
+ using zero_le_power2[of "x+1/2"] by arith
+ then show ?thesis
+ by (simp add: field_simps power2_eq_square)
qed
-lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
+lemma square_continuous:
+ fixes e :: real
+ shows "e > 0 \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. abs (y - x) < d \<longrightarrow> abs (y * y - x * x) < e)"
using isCont_power[OF isCont_ident, of x, unfolded isCont_def LIM_eq, rule_format, of e 2]
apply (auto simp add: power2_eq_square)
apply (rule_tac x="s" in exI)
@@ -30,7 +34,7 @@
apply auto
done
-lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y\<^sup>2 ==> sqrt x <= y"
+lemma real_le_lsqrt: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x \<le> y\<^sup>2 \<Longrightarrow> sqrt x \<le> y"
using real_sqrt_le_iff[of x "y\<^sup>2"] by simp
lemma real_le_rsqrt: "x\<^sup>2 \<le> y \<Longrightarrow> x \<le> sqrt y"
@@ -41,46 +45,49 @@
lemma sqrt_even_pow2:
assumes n: "even n"
- shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
+ shows "sqrt (2 ^ n) = 2 ^ (n div 2)"
proof -
- from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
- from m have "sqrt(2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
+ from n obtain m where m: "n = 2 * m"
+ unfolding even_mult_two_ex ..
+ from m have "sqrt (2 ^ n) = sqrt ((2 ^ m)\<^sup>2)"
by (simp only: power_mult[symmetric] mult_commute)
- then show ?thesis using m by simp
+ then show ?thesis
+ using m by simp
qed
-lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
- apply (cases "x = 0", simp_all)
+lemma real_div_sqrt: "0 \<le> x \<Longrightarrow> x / sqrt x = sqrt x"
+ apply (cases "x = 0")
+ apply simp_all
using sqrt_divide_self_eq[of x]
apply (simp add: inverse_eq_divide field_simps)
done
text{* Hence derive more interesting properties of the norm. *}
-lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
+lemma norm_eq_0_dot: "norm x = 0 \<longleftrightarrow> x \<bullet> x = (0::real)"
by simp (* TODO: delete *)
-lemma norm_cauchy_schwarz: "inner x y <= norm x * norm y"
+lemma norm_cauchy_schwarz: "x \<bullet> y \<le> norm x * norm y"
(* TODO: move to Inner_Product.thy *)
using Cauchy_Schwarz_ineq2[of x y] by auto
lemma norm_triangle_sub:
fixes x y :: "'a::real_normed_vector"
- shows "norm x \<le> norm y + norm (x - y)"
+ shows "norm x \<le> norm y + norm (x - y)"
using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
-lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
- by (simp add: norm_eq_sqrt_inner)
-
-lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> x \<bullet> x \<le> y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
-lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+lemma norm_lt: "norm x < norm y \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+ by (simp add: norm_eq_sqrt_inner)
+
+lemma norm_eq: "norm x = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
apply (subst order_eq_iff)
apply (auto simp: norm_le)
done
-lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+lemma norm_eq_1: "norm x = 1 \<longleftrightarrow> x \<bullet> x = 1"
by (simp add: norm_eq_sqrt_inner)
text{* Squaring equations and inequalities involving norms. *}
@@ -88,7 +95,7 @@
lemma dot_square_norm: "x \<bullet> x = (norm x)\<^sup>2"
by (simp only: power2_norm_eq_inner) (* TODO: move? *)
-lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a\<^sup>2"
+lemma norm_eq_square: "norm x = a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x = a\<^sup>2"
by (auto simp add: norm_eq_sqrt_inner)
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)\<^sup>2 \<le> y\<^sup>2"
@@ -102,13 +109,13 @@
then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
qed
-lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a\<^sup>2"
+lemma norm_le_square: "norm x \<le> a \<longleftrightarrow> 0 \<le> a \<and> x \<bullet> x \<le> a\<^sup>2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done
-lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a\<^sup>2"
+lemma norm_ge_square: "norm x \<ge> a \<longleftrightarrow> a \<le> 0 \<or> x \<bullet> x \<ge> a\<^sup>2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
@@ -116,16 +123,17 @@
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a\<^sup>2"
by (metis not_le norm_ge_square)
+
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a\<^sup>2"
by (metis norm_le_square not_less)
text{* Dot product in terms of the norm rather than conversely. *}
-lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
+lemmas inner_simps = inner_add_left inner_add_right inner_diff_right inner_diff_left
inner_scaleR_left inner_scaleR_right
lemma dot_norm: "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
- unfolding power2_norm_eq_inner inner_simps inner_commute by auto
+ unfolding power2_norm_eq_inner inner_simps inner_commute by auto
lemma dot_norm_neg: "x \<bullet> y = (((norm x)\<^sup>2 + (norm y)\<^sup>2) - (norm (x - y))\<^sup>2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute
@@ -133,32 +141,37 @@
text{* Equality of vectors in terms of @{term "op \<bullet>"} products. *}
-lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then show ?rhs by simp
next
assume ?rhs
- then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
- then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_diff inner_commute)
- then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_diff inner_commute)
- then show "x = y" by (simp)
+ then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0"
+ by simp
+ then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
+ by (simp add: inner_diff inner_commute)
+ then have "(x - y) \<bullet> (x - y) = 0"
+ by (simp add: field_simps inner_diff inner_commute)
+ then show "x = y" by simp
qed
lemma norm_triangle_half_r:
- shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
- using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
+ "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
+ using dist_triangle_half_r unfolding dist_norm[symmetric] by auto
lemma norm_triangle_half_l:
- assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
+ assumes "norm (x - y) < e / 2"
+ and "norm (x' - (y)) < e / 2"
shows "norm (x - x') < e"
- using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
- unfolding dist_norm[THEN sym] .
-
-lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
+ using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]]
+ unfolding dist_norm[symmetric] .
+
+lemma norm_triangle_le: "norm x + norm y \<le> e \<Longrightarrow> norm (x + y) \<le> e"
by (rule norm_triangle_ineq [THEN order_trans])
-lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
+lemma norm_triangle_lt: "norm x + norm y < e \<Longrightarrow> norm (x + y) < e"
by (rule norm_triangle_ineq [THEN le_less_trans])
lemma setsum_clauses:
@@ -191,7 +204,8 @@
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
assume "\<forall>x. x \<bullet> y = x \<bullet> z"
- then have "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_diff)
+ then have "\<forall>x. x \<bullet> (y - z) = 0"
+ by (simp add: inner_diff)
then have "(y - z) \<bullet> (y - z) = 0" ..
then show "y = z" by simp
qed simp
@@ -199,7 +213,8 @@
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
proof
assume "\<forall>z. x \<bullet> z = y \<bullet> z"
- then have "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_diff)
+ then have "\<forall>z. (x - y) \<bullet> z = 0"
+ by (simp add: inner_diff)
then have "(x - y) \<bullet> (x - y) = 0" ..
then show "x = y" by simp
qed simp
@@ -237,31 +252,35 @@
where "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
lemma linearI:
- assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+ assumes "\<And>x y. f (x + y) = f x + f y"
+ and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "linear f"
using assms unfolding linear_def by auto
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+lemma linear_compose_cmul: "linear f \<Longrightarrow> linear (\<lambda>x. c *\<^sub>R f x)"
by (simp add: linear_def algebra_simps)
-lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
+lemma linear_compose_neg: "linear f \<Longrightarrow> linear (\<lambda>x. - f x)"
by (simp add: linear_def)
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x + g x)"
by (simp add: linear_def algebra_simps)
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (\<lambda>x. f x - g x)"
by (simp add: linear_def algebra_simps)
-lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
+lemma linear_compose: "linear f \<Longrightarrow> linear g \<Longrightarrow> linear (g \<circ> f)"
by (simp add: linear_def)
-lemma linear_id: "linear id" by (simp add: linear_def id_def)
-
-lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
+lemma linear_id: "linear id"
+ by (simp add: linear_def id_def)
+
+lemma linear_zero: "linear (\<lambda>x. 0)"
+ by (simp add: linear_def)
lemma linear_compose_setsum:
- assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
+ assumes fS: "finite S"
+ and lS: "\<forall>a \<in> S. linear (f a)"
shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
using lS
apply (induct rule: finite_induct[OF fS])
@@ -275,88 +294,100 @@
apply simp
done
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x"
+lemma linear_cmul: "linear f \<Longrightarrow> f (c *\<^sub>R x) = c *\<^sub>R f x"
by (simp add: linear_def)
-lemma linear_neg: "linear f ==> f (-x) = - f x"
+lemma linear_neg: "linear f \<Longrightarrow> f (- x) = - f x"
using linear_cmul [where c="-1"] by simp
-lemma linear_add: "linear f ==> f(x + y) = f x + f y"
+lemma linear_add: "linear f \<Longrightarrow> f(x + y) = f x + f y"
by (metis linear_def)
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+lemma linear_sub: "linear f \<Longrightarrow> f(x - y) = f x - f y"
by (simp add: diff_minus linear_add linear_neg)
lemma linear_setsum:
- assumes lf: "linear f" and fS: "finite S"
- shows "f (setsum g S) = setsum (f o g) S"
- using fS
-proof (induct rule: finite_induct)
+ assumes lin: "linear f"
+ and fin: "finite S"
+ shows "f (setsum g S) = setsum (f \<circ> g) S"
+ using fin
+proof induct
case empty
- then show ?case by (simp add: linear_0[OF lf])
+ then show ?case
+ by (simp add: linear_0[OF lin])
next
case (insert x F)
- have "f (setsum g (insert x F)) = f (g x + setsum g F)" using insert.hyps
- by simp
- also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
- also have "\<dots> = setsum (f o g) (insert x F)" using insert.hyps by simp
+ have "f (setsum g (insert x F)) = f (g x + setsum g F)"
+ using insert.hyps by simp
+ also have "\<dots> = f (g x) + f (setsum g F)"
+ using linear_add[OF lin] by simp
+ also have "\<dots> = setsum (f \<circ> g) (insert x F)"
+ using insert.hyps by simp
finally show ?case .
qed
lemma linear_setsum_mul:
- assumes lf: "linear f" and fS: "finite S"
+ assumes lin: "linear f"
+ and fin: "finite S"
shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
- using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lf]
+ using linear_setsum[OF lin fin, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def] linear_cmul[OF lin]
by simp
lemma linear_injective_0:
- assumes lf: "linear f"
+ assumes lin: "linear f"
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
proof -
- have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
- also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
+ have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)"
+ by (simp add: inj_on_def)
+ also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)"
+ by simp
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
- by (simp add: linear_sub[OF lf])
- also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
+ by (simp add: linear_sub[OF lin])
+ also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)"
+ by auto
finally show ?thesis .
qed
subsection {* Bilinear functions. *}
-definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
-
-lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
+definition "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))"
+
+lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z"
by (simp add: bilinear_def linear_def)
-lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
+lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z"
by (simp add: bilinear_def linear_def)
-lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
+lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y"
by (simp add: bilinear_def linear_def)
-lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
+lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y"
by (simp add: bilinear_def linear_def)
-lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
+lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y"
by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
-lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
+lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y"
by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
-lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
+lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
using add_imp_eq[of x y 0] by auto
-lemma bilinear_lzero: assumes "bilinear h" shows "h 0 x = 0"
+lemma bilinear_lzero:
+ assumes "bilinear h"
+ shows "h 0 x = 0"
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps)
-lemma bilinear_rzero: assumes "bilinear h" shows "h x 0 = 0"
+lemma bilinear_rzero:
+ assumes "bilinear h"
+ shows "h x 0 = 0"
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps)
-lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
+lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z"
by (simp add: diff_minus bilinear_ladd bilinear_lneg)
-lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
+lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y"
by (simp add: diff_minus bilinear_radd bilinear_rneg)
lemma bilinear_setsum:
@@ -367,7 +398,8 @@
proof -
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
apply (rule linear_setsum[unfolded o_def])
- using bh fS apply (auto simp add: bilinear_def)
+ using bh fS
+ apply (auto simp add: bilinear_def)
done
also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
apply (rule setsum_cong, simp)
@@ -375,7 +407,8 @@
using bh fT
apply (auto simp add: bilinear_def)
done
- finally show ?thesis unfolding setsum_cartesian_product .
+ finally show ?thesis
+ unfolding setsum_cartesian_product .
qed
@@ -388,13 +421,19 @@
shows "adjoint f = g"
unfolding adjoint_def
proof (rule some_equality)
- show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
+ show "\<forall>x y. inner (f x) y = inner x (g y)"
+ by (rule assms)
next
- fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
- then have "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
- then have "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
- then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
- then have "\<forall>y. h y = g y" by simp
+ fix h
+ assume "\<forall>x y. inner (f x) y = inner x (h y)"
+ then have "\<forall>x y. inner x (g y) = inner x (h y)"
+ using assms by simp
+ then have "\<forall>x y. inner x (g y - h y) = 0"
+ by (simp add: inner_diff_right)
+ then have "\<forall>y. inner (g y - h y) (g y - h y) = 0"
+ by simp
+ then have "\<forall>y. h y = g y"
+ by simp
then show "h = g" by (simp add: ext)
qed
@@ -418,7 +457,7 @@
unfolding linear_setsum[OF lf finite_Basis]
by (simp add: linear_cmul[OF lf])
finally show "f x \<bullet> y = x \<bullet> ?w"
- by (simp add: inner_setsum_left inner_setsum_right mult_commute)
+ by (simp add: inner_setsum_left inner_setsum_right mult_commute)
qed
then show ?thesis
unfolding adjoint_def choice_iff
@@ -445,18 +484,22 @@
shows "adjoint (adjoint f) = f"
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
+
subsection {* Interlude: Some properties of real sets *}
-lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
+lemma seq_mono_lemma:
+ assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
+ and "\<forall>n \<ge> m. e n \<le> e m"
shows "\<forall>n \<ge> m. d n < e m"
- using assms apply auto
+ using assms
+ apply auto
apply (erule_tac x="n" in allE)
apply (erule_tac x="n" in allE)
apply auto
done
-
-lemma infinite_enumerate: assumes fS: "infinite S"
+lemma infinite_enumerate:
+ assumes fS: "infinite S"
shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
unfolding subseq_def
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
@@ -467,53 +510,57 @@
apply auto
done
-
lemma triangle_lemma:
- assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x\<^sup>2 <= y\<^sup>2 + z\<^sup>2"
- shows "x <= y + z"
+ fixes x y z :: real
+ assumes x: "0 \<le> x"
+ and y: "0 \<le> y"
+ and z: "0 \<le> z"
+ and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+ shows "x \<le> y + z"
proof -
- have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2*y*z + z\<^sup>2" using z y by (simp add: mult_nonneg_nonneg)
- with xy have th: "x\<^sup>2 \<le> (y+z)\<^sup>2" by (simp add: power2_eq_square field_simps)
- from y z have yz: "y + z \<ge> 0" by arith
+ have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 *y * z + z\<^sup>2"
+ using z y by (simp add: mult_nonneg_nonneg)
+ with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+ by (simp add: power2_eq_square field_simps)
+ from y z have yz: "y + z \<ge> 0"
+ by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
-definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
- where "S hull s = Inter {t. S t \<and> s \<subseteq> t}"
+definition hull :: "('a set \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75)
+ where "S hull s = \<Inter>{t. S t \<and> s \<subseteq> t}"
lemma hull_same: "S s \<Longrightarrow> S hull s = s"
unfolding hull_def by auto
-lemma hull_in: "(\<And>T. Ball T S ==> S (Inter T)) ==> S (S hull s)"
+lemma hull_in: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> S (S hull s)"
unfolding hull_def Ball_def by auto
-lemma hull_eq: "(\<And>T. Ball T S ==> S (Inter T)) ==> (S hull s) = s \<longleftrightarrow> S s"
+lemma hull_eq: "(\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)) \<Longrightarrow> (S hull s) = s \<longleftrightarrow> S s"
using hull_same[of S s] hull_in[of S s] by metis
-
lemma hull_hull: "S hull (S hull s) = S hull s"
unfolding hull_def by blast
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
unfolding hull_def by blast
-lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
+lemma hull_mono: "s \<subseteq> t \<Longrightarrow> (S hull s) \<subseteq> (S hull t)"
unfolding hull_def by blast
-lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x ==> (T hull s) \<subseteq> (S hull s)"
+lemma hull_antimono: "\<forall>x. S x \<longrightarrow> T x \<Longrightarrow> (T hull s) \<subseteq> (S hull s)"
unfolding hull_def by blast
-lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t ==> (S hull s) \<subseteq> t"
+lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (S hull s) \<subseteq> t"
unfolding hull_def by blast
-lemma subset_hull: "S t ==> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
+lemma subset_hull: "S t \<Longrightarrow> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
unfolding hull_def by blast
-lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow>
- (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
+lemma hull_unique: "s \<subseteq> t \<Longrightarrow> S t \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> S t' \<Longrightarrow> t \<subseteq> t') \<Longrightarrow> (S hull s = t)"
unfolding hull_def by auto
lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
@@ -527,7 +574,7 @@
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
lemma hull_union:
- assumes T: "\<And>T. Ball T S ==> S (Inter T)"
+ assumes T: "\<And>T. Ball T S \<Longrightarrow> S (\<Inter>T)"
shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
apply rule
apply (rule hull_mono)
@@ -541,13 +588,13 @@
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
unfolding hull_def by blast
-lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
+lemma hull_redundant: "a \<in> (S hull s) \<Longrightarrow> (S hull (insert a s) = S hull s)"
by (metis hull_redundant_eq)
subsection {* Archimedean properties and useful consequences *}
-lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
+lemma real_arch_simple: "\<exists>n. x \<le> real (n::nat)"
unfolding real_of_nat_def by (rule ex_le_of_nat)
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
@@ -558,60 +605,77 @@
apply simp
done
-lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
+lemma real_pow_lbound: "0 \<le> x \<Longrightarrow> 1 + real n * x \<le> (1 + x) ^ n"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
- then have h: "1 + real n * x \<le> (1 + x) ^ n" by simp
- from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
- from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
- also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
+ then have h: "1 + real n * x \<le> (1 + x) ^ n"
+ by simp
+ from h have p: "1 \<le> (1 + x) ^ n"
+ using Suc.prems by simp
+ from h have "1 + real n * x + x \<le> (1 + x) ^ n + x"
+ by simp
+ also have "\<dots> \<le> (1 + x) ^ Suc n"
+ apply (subst diff_le_0_iff_le[symmetric])
apply (simp add: field_simps)
- using mult_left_mono[OF p Suc.prems] apply simp
+ using mult_left_mono[OF p Suc.prems]
+ apply simp
done
- finally show ?case by (simp add: real_of_nat_Suc field_simps)
+ finally show ?case
+ by (simp add: real_of_nat_Suc field_simps)
qed
-lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
+lemma real_arch_pow:
+ fixes x :: real
+ assumes x: "1 < x"
+ shows "\<exists>n. y < x^n"
proof -
- from x have x0: "x - 1 > 0" by arith
+ from x have x0: "x - 1 > 0"
+ by arith
from reals_Archimedean3[OF x0, rule_format, of y]
- obtain n::nat where n:"y < real n * (x - 1)" by metis
+ obtain n :: nat where n: "y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 \<ge> 0" by arith
from real_pow_lbound[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
qed
-lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
+lemma real_arch_pow2:
+ fixes x :: real
+ shows "\<exists>n. x < 2^ n"
using real_arch_pow[of 2 x] by simp
lemma real_arch_pow_inv:
- assumes y: "(y::real) > 0" and x1: "x < 1"
+ fixes x y :: real
+ assumes y: "y > 0"
+ and x1: "x < 1"
shows "\<exists>n. x^n < y"
-proof -
- { assume x0: "x > 0"
- from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
- from real_arch_pow[OF ix, of "1/y"]
- obtain n where n: "1/y < (1/x)^n" by blast
- then have ?thesis using y x0
- by (auto simp add: field_simps power_divide) }
- moreover
- { assume "\<not> x > 0"
- with y x1 have ?thesis apply auto by (rule exI[where x=1], auto) }
- ultimately show ?thesis by metis
+proof (cases "x > 0")
+ case True
+ with x1 have ix: "1 < 1/x" by (simp add: field_simps)
+ from real_arch_pow[OF ix, of "1/y"]
+ obtain n where n: "1/y < (1/x)^n" by blast
+ then show ?thesis using y `x > 0`
+ by (auto simp add: field_simps power_divide)
+next
+ case False
+ with y x1 show ?thesis
+ apply auto
+ apply (rule exI[where x=1])
+ apply auto
+ done
qed
lemma forall_pos_mono:
- "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
- (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
+ "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+ (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
by (metis real_arch_inv)
lemma forall_pos_mono_1:
- "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow>
- (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
+ "(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow>
+ (\<And>n. P(inverse(real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
apply (rule forall_pos_mono)
apply auto
apply (atomize)
@@ -620,15 +684,20 @@
done
lemma real_archimedian_rdiv_eq_0:
- assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
+ assumes x0: "x \<ge> 0"
+ and c: "c \<ge> 0"
+ and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
shows "x = 0"
-proof -
- { assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
- from reals_Archimedean3[OF xp, rule_format, of c]
- obtain n::nat where n: "c < real n * x" by blast
- with xc[rule_format, of n] have "n = 0" by arith
- with n c have False by simp }
- then show ?thesis by blast
+proof (rule ccontr)
+ assume "x \<noteq> 0"
+ with x0 have xp: "x > 0" by arith
+ from reals_Archimedean3[OF xp, rule_format, of c]
+ obtain n :: nat where n: "c < real n * x"
+ by blast
+ with xc[rule_format, of n] have "n = 0"
+ by arith
+ with n c show False
+ by simp
qed
@@ -639,15 +708,17 @@
definition (in real_vector) "span S = (subspace hull S)"
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
-abbreviation (in real_vector) "independent s == ~(dependent s)"
+abbreviation (in real_vector) "independent s \<equiv> \<not> dependent s"
text {* Closure properties of subspaces. *}
-lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+lemma subspace_UNIV[simp]: "subspace UNIV"
+ by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
+ by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
by (metis subspace_def)
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
@@ -660,7 +731,8 @@
by (metis diff_minus subspace_add subspace_neg)
lemma (in real_vector) subspace_setsum:
- assumes sA: "subspace A" and fB: "finite B"
+ assumes sA: "subspace A"
+ and fB: "finite B"
and f: "\<forall>x\<in> B. f x \<in> A"
shows "setsum f B \<in> A"
using fB f sA
@@ -668,36 +740,39 @@
(simp add: subspace_def sA, auto simp add: sA subspace_add)
lemma subspace_linear_image:
- assumes lf: "linear f" and sS: "subspace S"
- shows "subspace(f ` S)"
+ assumes lf: "linear f"
+ and sS: "subspace S"
+ shows "subspace (f ` S)"
using lf sS linear_0[OF lf]
unfolding linear_def subspace_def
apply (auto simp add: image_iff)
- apply (rule_tac x="x + y" in bexI, auto)
- apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
+ apply (rule_tac x="x + y" in bexI)
+ apply auto
+ apply (rule_tac x="c *\<^sub>R x" in bexI)
+ apply auto
done
lemma subspace_linear_vimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace (f -` S)"
by (auto simp add: subspace_def linear_def linear_0[of f])
-lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
+lemma subspace_linear_preimage: "linear f \<Longrightarrow> subspace S \<Longrightarrow> subspace {x. f x \<in> S}"
by (auto simp add: subspace_def linear_def linear_0[of f])
lemma subspace_trivial: "subspace {0}"
by (simp add: subspace_def)
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
by (simp add: subspace_def)
-lemma subspace_Times: "\<lbrakk>subspace A; subspace B\<rbrakk> \<Longrightarrow> subspace (A \<times> B)"
+lemma subspace_Times: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<times> B)"
unfolding subspace_def zero_prod_def by simp
text {* Properties of span. *}
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+lemma (in real_vector) span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
by (metis span_def hull_mono)
-lemma (in real_vector) subspace_span: "subspace(span S)"
+lemma (in real_vector) subspace_span: "subspace (span S)"
unfolding span_def
apply (rule hull_in)
apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
@@ -705,12 +780,11 @@
done
lemma (in real_vector) span_clauses:
- "a \<in> S ==> a \<in> span S"
+ "a \<in> S \<Longrightarrow> a \<in> span S"
"0 \<in> span S"
- "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+ "x\<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
"x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
- by (metis span_def hull_subset subset_eq)
- (metis subspace_span subspace_def)+
+ by (metis span_def hull_subset subset_eq) (metis subspace_span subspace_def)+
lemma span_unique:
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
@@ -722,12 +796,14 @@
lemma (in real_vector) span_induct:
assumes x: "x \<in> span S"
and P: "subspace P"
- and SP: "\<And>x. x \<in> S ==> x \<in> P"
+ and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
shows "x \<in> P"
proof -
- from SP have SP': "S \<subseteq> P" by (simp add: subset_eq)
+ from SP have SP': "S \<subseteq> P"
+ by (simp add: subset_eq)
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
- show "x \<in> P" by (metis subset_eq)
+ show "x \<in> P"
+ by (metis subset_eq)
qed
lemma span_empty[simp]: "span {} = {0}"
@@ -742,7 +818,7 @@
lemma dependent_single[simp]: "dependent {x} \<longleftrightarrow> x = 0"
unfolding dependent_def by auto
-lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
+lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
apply (clarsimp simp add: dependent_def span_mono)
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
apply force
@@ -760,34 +836,46 @@
using span_induct SP P by blast
inductive_set (in real_vector) span_induct_alt_help for S:: "'a set"
- where
+where
span_induct_alt_help_0: "0 \<in> span_induct_alt_help S"
| span_induct_alt_help_S:
- "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow> (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
+ "x \<in> S \<Longrightarrow> z \<in> span_induct_alt_help S \<Longrightarrow>
+ (c *\<^sub>R x + z) \<in> span_induct_alt_help S"
lemma span_induct_alt':
- assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+ assumes h0: "h 0"
+ and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
shows "\<forall>x \<in> span S. h x"
proof -
- { fix x:: "'a" assume x: "x \<in> span_induct_alt_help S"
+ {
+ fix x :: 'a
+ assume x: "x \<in> span_induct_alt_help S"
have "h x"
apply (rule span_induct_alt_help.induct[OF x])
apply (rule h0)
- apply (rule hS, assumption, assumption)
- done }
+ apply (rule hS)
+ apply assumption
+ apply assumption
+ done
+ }
note th0 = this
- { fix x assume x: "x \<in> span S"
+ {
+ fix x
+ assume x: "x \<in> span S"
have "x \<in> span_induct_alt_help S"
proof (rule span_induct[where x=x and S=S])
- show "x \<in> span S" using x .
+ show "x \<in> span S" by (rule x)
next
- fix x assume xS : "x \<in> S"
- from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
- show "x \<in> span_induct_alt_help S" by simp
+ fix x
+ assume xS: "x \<in> S"
+ from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
+ show "x \<in> span_induct_alt_help S"
+ by simp
next
have "0 \<in> span_induct_alt_help S" by (rule span_induct_alt_help_0)
moreover
- { fix x y
+ {
+ fix x y
assume h: "x \<in> span_induct_alt_help S" "y \<in> span_induct_alt_help S"
from h have "(x + y) \<in> span_induct_alt_help S"
apply (induct rule: span_induct_alt_help.induct)
@@ -796,9 +884,11 @@
apply (rule span_induct_alt_help_S)
apply assumption
apply simp
- done }
+ done
+ }
moreover
- { fix c x
+ {
+ fix c x
assume xt: "x \<in> span_induct_alt_help S"
then have "(c *\<^sub>R x) \<in> span_induct_alt_help S"
apply (induct rule: span_induct_alt_help.induct)
@@ -808,15 +898,17 @@
apply assumption
apply simp
done }
- ultimately
- show "subspace (span_induct_alt_help S)"
+ ultimately show "subspace (span_induct_alt_help S)"
unfolding subspace_def Ball_def by blast
- qed }
+ qed
+ }
with th0 show ?thesis by blast
qed
lemma span_induct_alt:
- assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
+ assumes h0: "h 0"
+ and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)"
+ and x: "x \<in> span S"
shows "h x"
using span_induct_alt'[of h S] h0 hS x by blast
@@ -825,35 +917,43 @@
lemma span_span: "span (span A) = span A"
unfolding span_def hull_hull ..
-lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
-
-lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
+lemma (in real_vector) span_superset: "x \<in> S \<Longrightarrow> x \<in> span S"
+ by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S"
+ by (metis subspace_span subspace_0)
lemma span_inc: "S \<subseteq> span S"
by (metis subset_eq span_superset)
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
- unfolding dependent_def apply(rule_tac x=0 in bexI)
- using assms span_0 by auto
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+lemma (in real_vector) dependent_0:
+ assumes "0 \<in> A"
+ shows "dependent A"
+ unfolding dependent_def
+ apply (rule_tac x=0 in bexI)
+ using assms span_0
+ apply auto
+ done
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
by (metis subspace_add subspace_span)
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+lemma (in real_vector) span_mul: "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
by (metis subspace_span subspace_mul)
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
by (metis subspace_neg subspace_span)
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
by (metis subspace_span subspace_sub)
-lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
+lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S \<Longrightarrow> setsum f A \<in> span S"
by (rule subspace_setsum, rule subspace_span)
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
apply (auto simp only: span_add span_sub)
- apply (subgoal_tac "(x + y) - x \<in> span S", simp)
+ apply (subgoal_tac "(x + y) - x \<in> span S")
+ apply simp
apply (simp only: span_add span_sub)
done
@@ -871,7 +971,8 @@
show "subspace (f ` span S)"
using lf subspace_span by (rule subspace_linear_image)
next
- fix T assume "f ` S \<subseteq> T" and "subspace T"
+ fix T
+ assume "f ` S \<subseteq> T" and "subspace T"
then show "f ` span S \<subseteq> T"
unfolding image_subset_iff_subset_vimage
by (intro span_minimal subspace_linear_vimage lf)
@@ -904,7 +1005,10 @@
show "subspace (range (\<lambda>k. k *\<^sub>R x))"
unfolding subspace_def
by (auto intro: scaleR_add_left [symmetric])
- fix T assume "{x} \<subseteq> T" and "subspace T" then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
+next
+ fix T
+ assume "{x} \<subseteq> T" and "subspace T"
+ then show "range (\<lambda>k. k *\<^sub>R x) \<subseteq> T"
unfolding subspace_def by auto
qed
@@ -922,12 +1026,13 @@
qed
lemma span_breakdown:
- assumes bS: "b \<in> S" and aS: "a \<in> span S"
+ assumes bS: "b \<in> S"
+ and aS: "a \<in> span S"
shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})"
using assms span_insert [of b "S - {b}"]
by (simp add: insert_absorb)
-lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)"
+lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *\<^sub>R a \<in> span S)"
by (simp add: span_insert)
text {* Hence some "reversal" results. *}
@@ -939,7 +1044,9 @@
proof -
from span_breakdown[of b "insert b S" a, OF insertI1 a]
obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
- { assume k0: "k = 0"
+ show ?thesis
+ proof (cases "k = 0")
+ case True
with k have "a \<in> span S"
apply (simp)
apply (rule set_rev_mp)
@@ -947,19 +1054,17 @@
apply (rule span_mono)
apply blast
done
- with na have ?thesis by blast }
- moreover
- { assume k0: "k \<noteq> 0"
+ with na show ?thesis by blast
+ next
+ case False
have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
- from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
+ from False have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
by (simp add: algebra_simps)
from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
by (rule span_mul)
then have th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
unfolding eq' .
-
- from k
- have ?thesis
+ from k show ?thesis
apply (subst eq)
apply (rule span_sub)
apply (rule span_mul)
@@ -968,8 +1073,10 @@
apply (rule set_rev_mp)
apply (rule th)
apply (rule span_mono)
- using na by blast }
- ultimately show ?thesis by blast
+ using na
+ apply blast
+ done
+ qed
qed
lemma in_span_delete:
@@ -990,7 +1097,8 @@
unfolding span_def by (rule hull_redundant)
lemma span_trans:
- assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
+ assumes x: "x \<in> span S"
+ and y: "y \<in> span (insert x S)"
shows "y \<in> span S"
using assms by (simp only: span_redundant)
@@ -1003,7 +1111,9 @@
"span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
proof -
- { fix x assume x: "x \<in> ?E"
+ {
+ fix x
+ assume x: "x \<in> ?E"
then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
by blast
have "x \<in> span P"
@@ -1011,7 +1121,8 @@
apply (rule span_setsum[OF fS])
using span_mono[OF SP]
apply (auto intro: span_superset span_mul)
- done }
+ done
+ }
moreover
have "\<forall>x \<in> span P. x \<in> ?E"
proof (rule span_induct_alt')
@@ -1022,15 +1133,20 @@
done
next
fix c x y
- assume x: "x \<in> P" and hy: "y \<in> Collect ?h"
+ assume x: "x \<in> P"
+ assume hy: "y \<in> Collect ?h"
from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
let ?S = "insert x S"
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c) else u y"
- from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
- { assume xS: "x \<in> S"
+ from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P"
+ by blast+
+ have "?Q ?S ?u (c*\<^sub>R x + y)"
+ proof cases
+ assume xS: "x \<in> S"
have S1: "S = (S - {x}) \<union> {x}"
- and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
+ and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}"
+ using xS fS by auto
have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
using xS
by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
@@ -1042,17 +1158,18 @@
also have "\<dots> = c*\<^sub>R x + y"
by (simp add: add_commute u)
finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
- then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast }
- moreover
- { assume xS: "x \<notin> S"
+ then show ?thesis using th0 by blast
+ next
+ assume xS: "x \<notin> S"
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
unfolding u[symmetric]
apply (rule setsum_cong2)
- using xS apply auto
+ using xS
+ apply auto
done
- have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
- by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong) }
- ultimately have "?Q ?S ?u (c*\<^sub>R x + y)" by (cases "x \<in> S") simp_all
+ show ?thesis using fS xS th0
+ by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)
+ qed
then show "(c*\<^sub>R x + y) \<in> Collect ?h"
unfolding mem_Collect_eq
apply -
@@ -1068,15 +1185,18 @@
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))"
(is "?lhs = ?rhs")
proof -
- { assume dP: "dependent P"
+ {
+ assume dP: "dependent P"
then obtain a S u where aP: "a \<in> P" and fS: "finite S"
and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
unfolding dependent_def span_explicit by blast
let ?S = "insert a S"
let ?u = "\<lambda>y. if y = a then - 1 else u y"
let ?v = a
- from aP SP have aS: "a \<notin> S" by blast
- from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
+ from aP SP have aS: "a \<notin> S"
+ by blast
+ from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0"
+ by auto
have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
using fS aS
apply (simp add: setsum_clauses field_simps)
@@ -1092,18 +1212,24 @@
done
}
moreover
- { fix S u v
+ {
+ fix S u v
assume fS: "finite S"
- and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
+ and SP: "S \<subseteq> P"
+ and vS: "v \<in> S"
+ and uv: "u v \<noteq> 0"
and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
let ?a = v
let ?S = "S - {v}"
let ?u = "\<lambda>i. (- u i) / u v"
- have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" using fS SP vS by auto
- have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
+ have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"
+ using fS SP vS by auto
+ have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =
+ setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
using fS vS uv by (simp add: setsum_diff1 divide_inverse field_simps)
- also have "\<dots> = ?a" unfolding scaleR_right.setsum [symmetric] u using uv by simp
- finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
+ also have "\<dots> = ?a"
+ unfolding scaleR_right.setsum [symmetric] u using uv by simp
+ finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
with th0 have ?lhs
unfolding dependent_def span_explicit
apply -
@@ -1122,61 +1248,72 @@
shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
(is "_ = ?rhs")
proof -
- { fix y
+ {
+ fix y
assume y: "y \<in> span S"
- from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
- u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y" unfolding span_explicit by blast
+ from y obtain S' u where fS': "finite S'"
+ and SS': "S' \<subseteq> S"
+ and u: "setsum (\<lambda>v. u v *\<^sub>R v) S' = y"
+ unfolding span_explicit by blast
let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
then have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
- then have "y \<in> ?rhs" by auto }
+ then have "y \<in> ?rhs" by auto
+ }
moreover
- { fix y u
+ {
+ fix y u
assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
- then have "y \<in> span S" using fS unfolding span_explicit by auto }
+ then have "y \<in> span S" using fS unfolding span_explicit by auto
+ }
ultimately show ?thesis by blast
qed
text {* This is useful for building a basis step-by-step. *}
lemma independent_insert:
- "independent(insert a S) \<longleftrightarrow>
- (if a \<in> S then independent S
- else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
- { assume aS: "a \<in> S"
- then have ?thesis using insert_absorb[OF aS] by simp }
- moreover
- { assume aS: "a \<notin> S"
- { assume i: ?lhs
- then have ?rhs using aS
- apply simp
- apply (rule conjI)
- apply (rule independent_mono)
- apply assumption
- apply blast
- apply (simp add: dependent_def)
- done }
- moreover
- { assume i: ?rhs
- have ?lhs using i aS
- apply simp
- apply (auto simp add: dependent_def)
- apply (case_tac "aa = a", auto)
- apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
- apply simp
- apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
- apply (subgoal_tac "insert aa (S - {aa}) = S")
- apply simp
- apply blast
- apply (rule in_span_insert)
- apply assumption
- apply blast
- apply blast
- done }
- ultimately have ?thesis by blast }
- ultimately show ?thesis by blast
+ "independent (insert a S) \<longleftrightarrow>
+ (if a \<in> S then independent S else independent S \<and> a \<notin> span S)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof (cases "a \<in> S")
+ case True
+ then show ?thesis
+ using insert_absorb[OF True] by simp
+next
+ case False
+ show ?thesis
+ proof
+ assume i: ?lhs
+ then show ?rhs
+ using False
+ apply simp
+ apply (rule conjI)
+ apply (rule independent_mono)
+ apply assumption
+ apply blast
+ apply (simp add: dependent_def)
+ done
+ next
+ assume i: ?rhs
+ show ?lhs
+ using i False
+ apply simp
+ apply (auto simp add: dependent_def)
+ apply (case_tac "aa = a")
+ apply auto
+ apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
+ apply simp
+ apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
+ apply (subgoal_tac "insert aa (S - {aa}) = S")
+ apply simp
+ apply blast
+ apply (rule in_span_insert)
+ apply assumption
+ apply blast
+ apply blast
+ done
+ qed
qed
text {* The degenerate case of the Exchange Lemma. *}
@@ -1195,18 +1332,29 @@
from span_mono[OF BA] span_mono[OF AsB]
have sAB: "span A = span B" unfolding span_span by blast
- { fix x assume x: "x \<in> A"
+ {
+ fix x
+ assume x: "x \<in> A"
from iA have th0: "x \<notin> span (A - {x})"
unfolding dependent_def using x by blast
- from x have xsA: "x \<in> span A" by (blast intro: span_superset)
+ from x have xsA: "x \<in> span A"
+ by (blast intro: span_superset)
have "A - {x} \<subseteq> A" by blast
- then have th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
- { assume xB: "x \<notin> B"
- from xB BA have "B \<subseteq> A -{x}" by blast
- then have "span B \<subseteq> span (A - {x})" by (metis span_mono)
- with th1 th0 sAB have "x \<notin> span A" by blast
- with x have False by (metis span_superset) }
- then have "x \<in> B" by blast }
+ then have th1: "span (A - {x}) \<subseteq> span A"
+ by (metis span_mono)
+ {
+ assume xB: "x \<notin> B"
+ from xB BA have "B \<subseteq> A - {x}"
+ by blast
+ then have "span B \<subseteq> span (A - {x})"
+ by (metis span_mono)
+ with th1 th0 sAB have "x \<notin> span A"
+ by blast
+ with x have False
+ by (metis span_superset)
+ }
+ then have "x \<in> B" by blast
+ }
then show "A \<subseteq> B" by blast
qed
@@ -1216,75 +1364,110 @@
assumes f:"finite t"
and i: "independent s"
and sp: "s \<subseteq> span t"
- shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+ shows "\<exists>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
proof (induct "card (t - s)" arbitrary: s t rule: less_induct)
case less
note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
- let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
+ let ?P = "\<lambda>t'. card t' = card t \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
let ?ths = "\<exists>t'. ?P t'"
- { assume st: "s \<subseteq> t"
- from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
+ {
+ assume st: "s \<subseteq> t"
+ from st ft span_mono[OF st]
+ have ?ths
+ apply -
+ apply (rule exI[where x=t])
apply (auto intro: span_superset)
- done }
+ done
+ }
moreover
- { assume st: "t \<subseteq> s"
- from spanning_subset_independent[OF st s sp]
- st ft span_mono[OF st] have ?ths
- apply -
- apply (rule exI[where x=t])
- apply (auto intro: span_superset)
- done }
+ {
+ assume st: "t \<subseteq> s"
+ from spanning_subset_independent[OF st s sp] st ft span_mono[OF st]
+ have ?ths
+ apply -
+ apply (rule exI[where x=t])
+ apply (auto intro: span_superset)
+ done
+ }
moreover
- { assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
- from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
- from b have "t - {b} - s \<subset> t - s" by blast
- then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
- by (auto intro: psubset_card_mono)
- from b ft have ct0: "card t \<noteq> 0" by auto
- { assume stb: "s \<subseteq> span(t -{b})"
- from ft have ftb: "finite (t -{b})" by auto
+ {
+ assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
+ from st(2) obtain b where b: "b \<in> t" "b \<notin> s"
+ by blast
+ from b have "t - {b} - s \<subset> t - s"
+ by blast
+ then have cardlt: "card (t - {b} - s) < card (t - s)"
+ using ft by (auto intro: psubset_card_mono)
+ from b ft have ct0: "card t \<noteq> 0"
+ by auto
+ have ?ths
+ proof cases
+ assume stb: "s \<subseteq> span(t - {b})"
+ from ft have ftb: "finite (t -{b})"
+ by auto
from less(1)[OF cardlt ftb s stb]
obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u"
and fu: "finite u" by blast
let ?w = "insert b u"
- have th0: "s \<subseteq> insert b u" using u by blast
- from u(3) b have "u \<subseteq> s \<union> t" by blast
- then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
- have bu: "b \<notin> u" using b u by blast
- from u(1) ft b have "card u = (card t - 1)" by auto
+ have th0: "s \<subseteq> insert b u"
+ using u by blast
+ from u(3) b have "u \<subseteq> s \<union> t"
+ by blast
+ then have th1: "insert b u \<subseteq> s \<union> t"
+ using u b by blast
+ have bu: "b \<notin> u"
+ using b u by blast
+ from u(1) ft b have "card u = (card t - 1)"
+ by auto
then have th2: "card (insert b u) = card t"
using card_insert_disjoint[OF fu bu] ct0 by auto
from u(4) have "s \<subseteq> span u" .
- also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
+ also have "\<dots> \<subseteq> span (insert b u)"
+ by (rule span_mono) blast
finally have th3: "s \<subseteq> span (insert b u)" .
- from th0 th1 th2 th3 fu have th: "?P ?w" by blast
- from th have ?ths by blast }
- moreover
- { assume stb: "\<not> s \<subseteq> span(t -{b})"
- from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
- have ab: "a \<noteq> b" using a b by blast
- have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
+ from th0 th1 th2 th3 fu have th: "?P ?w"
+ by blast
+ from th show ?thesis by blast
+ next
+ assume stb: "\<not> s \<subseteq> span(t - {b})"
+ from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})"
+ by blast
+ have ab: "a \<noteq> b"
+ using a b by blast
+ have at: "a \<notin> t"
+ using a ab span_superset[of a "t- {b}"] by auto
have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
using cardlt ft a b by auto
- have ft': "finite (insert a (t - {b}))" using ft by auto
- { fix x assume xs: "x \<in> s"
- have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
- from b(1) have "b \<in> span t" by (simp add: span_superset)
- have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
- using a sp unfolding subset_eq apply auto done
- from xs sp have "x \<in> span t" by blast
- with span_mono[OF t]
- have x: "x \<in> span (insert b (insert a (t - {b})))" ..
- from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" . }
- then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
-
- from less(1)[OF mlt ft' s sp'] obtain u where
- u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
- "s \<subseteq> span u" by blast
- from u a b ft at ct0 have "?P u" by auto
- then have ?ths by blast }
- ultimately have ?ths by blast
+ have ft': "finite (insert a (t - {b}))"
+ using ft by auto
+ {
+ fix x
+ assume xs: "x \<in> s"
+ have t: "t \<subseteq> insert b (insert a (t - {b}))"
+ using b by auto
+ from b(1) have "b \<in> span t"
+ by (simp add: span_superset)
+ have bs: "b \<in> span (insert a (t - {b}))"
+ apply (rule in_span_delete)
+ using a sp unfolding subset_eq
+ apply auto
+ done
+ from xs sp have "x \<in> span t"
+ by blast
+ with span_mono[OF t] have x: "x \<in> span (insert b (insert a (t - {b})))" ..
+ from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .
+ }
+ then have sp': "s \<subseteq> span (insert a (t - {b}))"
+ by blast
+ from less(1)[OF mlt ft' s sp'] obtain u where u:
+ "card u = card (insert a (t -{b}))"
+ "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
+ "s \<subseteq> span u" by blast
+ from u a b ft at ct0 have "?P u"
+ by auto
+ then show ?thesis by blast
+ qed
}
ultimately show ?ths by blast
qed
@@ -1292,21 +1475,24 @@
text {* This implies corresponding size bounds. *}
lemma independent_span_bound:
- assumes f: "finite t" and i: "independent s" and sp:"s \<subseteq> span t"
+ assumes f: "finite t"
+ and i: "independent s"
+ and sp: "s \<subseteq> span t"
shows "finite s \<and> card s \<le> card t"
by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
-
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
proof -
- have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
+ have eq: "{f x |x. x\<in> UNIV} = f ` UNIV"
+ by auto
show ?thesis unfolding eq
apply (rule finite_imageI)
apply (rule finite)
done
qed
-subsection{* Euclidean Spaces as Typeclass*}
+
+subsection {* Euclidean Spaces as Typeclass *}
lemma independent_Basis: "independent Basis"
unfolding dependent_def
@@ -1345,7 +1531,8 @@
lemma setsum_norm_allsubsets_bound:
fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
- assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
+ assumes fP: "finite P"
+ and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('n) * e"
proof -
have "(\<Sum>x\<in>P. norm (f x)) \<le> (\<Sum>x\<in>P. \<Sum>b\<in>Basis. \<bar>f x \<bullet> b\<bar>)"
@@ -1354,13 +1541,14 @@
by (rule setsum_commute)
also have "\<dots> \<le> of_nat (card (Basis :: 'n set)) * (2 * e)"
proof (rule setsum_bounded)
- fix i :: 'n assume i: "i \<in> Basis"
- have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
+ fix i :: 'n
+ assume i: "i \<in> Basis"
+ have "norm (\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le>
norm ((\<Sum>x\<in>P \<inter> - {x. f x \<bullet> i < 0}. f x) \<bullet> i) + norm ((\<Sum>x\<in>P \<inter> {x. f x \<bullet> i < 0}. f x) \<bullet> i)"
by (simp add: abs_real_def setsum_cases[OF fP] setsum_negf uminus_add_conv_diff
- norm_triangle_ineq4 inner_setsum_left
- del: real_norm_def)
- also have "\<dots> \<le> e + e" unfolding real_norm_def
+ norm_triangle_ineq4 inner_setsum_left del: real_norm_def)
+ also have "\<dots> \<le> e + e"
+ unfolding real_norm_def
by (intro add_mono norm_bound_Basis_le i fPs) auto
finally show "(\<Sum>x\<in>P. \<bar>f x \<bullet> i\<bar>) \<le> 2*e" by simp
qed
@@ -1369,6 +1557,7 @@
finally show ?thesis .
qed
+
subsection {* Linearity and Bilinearity continued *}
lemma linear_bounded:
@@ -1377,25 +1566,32 @@
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof -
let ?B = "\<Sum>b\<in>Basis. norm (f b)"
- { fix x:: "'a"
+ {
+ fix x :: 'a
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b"
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))"
unfolding euclidean_representation ..
also have "\<dots> = norm (setsum ?g Basis)"
- using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf] by auto
+ using linear_setsum[OF lf finite_Basis, of "\<lambda>b. (x \<bullet> b) *\<^sub>R b", unfolded o_def] linear_cmul[OF lf]
+ by auto
finally have th0: "norm (f x) = norm (setsum ?g Basis)" .
- { fix i :: 'a assume i: "i \<in> Basis"
+ {
+ fix i :: 'a
+ assume i: "i \<in> Basis"
from Basis_le_norm[OF i, of x]
have "norm (?g i) \<le> norm (f i) * norm x"
unfolding norm_scaleR
apply (subst mult_commute)
apply (rule mult_mono)
apply (auto simp add: field_simps)
- done }
+ done
+ }
then have th: "\<forall>b\<in>Basis. norm (?g b) \<le> norm (f b) * norm x"
by metis
from setsum_norm_le[of _ ?g, OF th]
- have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
+ have "norm (f x) \<le> ?B * norm x"
+ unfolding th0 setsum_left_distrib by metis
+ }
then show ?thesis by blast
qed
@@ -1408,7 +1604,8 @@
B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
- { assume C: "B < 0"
+ {
+ assume C: "B < 0"
def One \<equiv> "\<Sum>Basis ::'a"
then have "One \<noteq> 0"
unfolding euclidean_eq_iff[where 'a='a]
@@ -1419,14 +1616,18 @@
with B[rule_format, of One] norm_ge_zero[of "f One"]
have False by simp
}
- then have Bp: "B \<ge> 0" by (metis not_leE)
- { fix x::"'a"
+ then have Bp: "B \<ge> 0"
+ by (metis not_leE)
+ {
+ fix x::"'a"
have "norm (f x) \<le> ?K * norm x"
using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
apply (auto simp add: field_simps split add: abs_split)
apply (erule order_trans, simp)
done
- } then show ?thesis using Kp by blast
+ }
+ then show ?thesis
+ using Kp by blast
qed
lemma linear_conv_bounded_linear:
@@ -1436,10 +1637,12 @@
assume "linear f"
show "bounded_linear f"
proof
- fix x y show "f (x + y) = f x + f y"
+ fix x y
+ show "f (x + y) = f x + f y"
using `linear f` unfolding linear_def by simp
next
- fix r x show "f (scaleR r x) = scaleR r (f x)"
+ fix r x
+ show "f (scaleR r x) = scaleR r (f x)"
using `linear f` unfolding linear_def by simp
next
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
@@ -1450,43 +1653,43 @@
next
assume "bounded_linear f"
then interpret f: bounded_linear f .
- show "linear f"
- by (simp add: f.add f.scaleR linear_def)
+ show "linear f" by (simp add: f.add f.scaleR linear_def)
qed
lemma bounded_linearI':
fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+ assumes "\<And>x y. f (x + y) = f x + f y"
+ and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
shows "bounded_linear f"
- unfolding linear_conv_bounded_linear[THEN sym]
+ unfolding linear_conv_bounded_linear[symmetric]
by (rule linearI[OF assms])
-
lemma bilinear_bounded:
fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"])
- fix x:: "'m" and y :: "'n"
- have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
- apply(subst euclidean_representation[where 'a='m])
- apply(subst euclidean_representation[where 'a='n])
+ fix x :: 'm
+ fix y :: 'n
+ have "norm (h x y) = norm (h (setsum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (setsum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))"
+ apply (subst euclidean_representation[where 'a='m])
+ apply (subst euclidean_representation[where 'a='n])
apply rule
done
- also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
+ also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))"
unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
finally have th: "norm (h x y) = \<dots>" .
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
- apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
- apply (rule setsum_norm_le)
- apply simp
- apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
- field_simps simp del: scaleR_scaleR)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- apply (rule mult_mono)
- apply (auto simp add: zero_le_mult_iff Basis_le_norm)
- done
+ apply (auto simp add: setsum_left_distrib th setsum_cartesian_product)
+ apply (rule setsum_norm_le)
+ apply simp
+ apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
+ field_simps simp del: scaleR_scaleR)
+ apply (rule mult_mono)
+ apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+ apply (rule mult_mono)
+ apply (auto simp add: zero_le_mult_iff Basis_le_norm)
+ done
qed
lemma bilinear_bounded_pos:
@@ -1499,15 +1702,17 @@
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
have KB: "B < ?K" by arith
- { fix x::'a and y::'b
- from KB Kp
- have "B * norm x * norm y \<le> ?K * norm x * norm y"
+ {
+ fix x :: 'a
+ fix y :: 'b
+ from KB Kp have "B * norm x * norm y \<le> ?K * norm x * norm y"
apply -
apply (rule mult_right_mono, rule mult_right_mono)
apply auto
done
then have "norm (h x y) \<le> ?K * norm x * norm y"
- using B[rule_format, of x y] by simp }
+ using B[rule_format, of x y] by simp
+ }
with Kp show ?thesis by blast
qed
@@ -1518,17 +1723,21 @@
assume "bilinear h"
show "bounded_bilinear h"
proof
- fix x y z show "h (x + y) z = h x z + h y z"
+ fix x y z
+ show "h (x + y) z = h x z + h y z"
using `bilinear h` unfolding bilinear_def linear_def by simp
next
- fix x y z show "h x (y + z) = h x y + h x z"
+ fix x y z
+ show "h x (y + z) = h x y + h x z"
using `bilinear h` unfolding bilinear_def linear_def by simp
next
- fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
+ fix r x y
+ show "h (scaleR r x) y = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by simp
next
- fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
+ fix r x y
+ show "h x (scaleR r y) = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by simp
next
@@ -1554,13 +1763,14 @@
using independent_span_bound[OF finite_Basis, of S] by auto
lemma dependent_biggerset:
- "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
+ "(finite (S::('a::euclidean_space) set) \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S"
by (metis independent_bound not_less)
text {* Hence we can create a maximal independent subset. *}
lemma maximal_independent_subset_extend:
- assumes sv: "(S::('a::euclidean_space) set) \<subseteq> V"
+ fixes S :: "'a::euclidean_space set"
+ assumes sv: "S \<subseteq> V"
and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
@@ -1570,15 +1780,22 @@
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
let ?ths = "\<exists>x. ?P x"
let ?d = "DIM('a)"
- { assume "V \<subseteq> span S"
- then have ?ths using sv i by blast }
- moreover
- { assume VS: "\<not> V \<subseteq> span S"
- from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
- from a have aS: "a \<notin> S" by (auto simp add: span_superset)
- have th0: "insert a S \<subseteq> V" using a sv by blast
+ show ?ths
+ proof (cases "V \<subseteq> span S")
+ case True
+ then show ?thesis
+ using sv i by blast
+ next
+ case False
+ then obtain a where a: "a \<in> V" "a \<notin> span S"
+ by blast
+ from a have aS: "a \<notin> S"
+ by (auto simp add: span_superset)
+ have th0: "insert a S \<subseteq> V"
+ using a sv by blast
from independent_insert[of a S] i a
- have th1: "independent (insert a S)" by auto
+ have th1: "independent (insert a S)"
+ by auto
have mlt: "?d - card (insert a S) < ?d - card S"
using aS a independent_bound[OF th1] by auto
@@ -1586,8 +1803,8 @@
obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto
- then have ?ths by blast }
- ultimately show ?ths by blast
+ then show ?thesis by blast
+ qed
qed
lemma maximal_independent_subset:
@@ -1598,7 +1815,7 @@
text {* Notion of dimension. *}
-definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
+definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> card B = n)"
lemma basis_exists:
"\<exists>B. (B :: ('a::euclidean_space) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
@@ -1608,58 +1825,76 @@
text {* Consequences of independence or spanning for cardinality. *}
-lemma independent_card_le_dim:
- assumes "(B::('a::euclidean_space) set) \<subseteq> V" and "independent B"
+lemma independent_card_le_dim:
+ fixes B :: "'a::euclidean_space set"
+ assumes "B \<subseteq> V"
+ and "independent B"
shows "card B \<le> dim V"
proof -
from basis_exists[of V] `B \<subseteq> V`
- obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
+ obtain B' where "independent B'"
+ and "B \<subseteq> span B'"
+ and "card B' = dim V"
+ by blast
with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
show ?thesis by auto
qed
lemma span_card_ge_dim:
- "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
by (metis basis_exists[of V] independent_span_bound subset_trans)
lemma basis_card_eq_dim:
- "B \<subseteq> (V:: ('a::euclidean_space) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
- independent B \<Longrightarrow> finite B \<and> card B = dim V"
+ fixes V :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
-lemma dim_unique: "(B::('a::euclidean_space) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow>
- independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
+lemma dim_unique:
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
text {* More lemmas about dimension. *}
-lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
+lemma dim_UNIV: "dim (UNIV :: 'a::euclidean_space set) = DIM('a)"
using independent_Basis
by (intro dim_unique[of Basis]) auto
lemma dim_subset:
- "(S:: ('a::euclidean_space) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
+ fixes S :: "'a::euclidean_space set"
+ shows "S \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
using basis_exists[of T] basis_exists[of S]
by (metis independent_card_le_dim subset_trans)
-lemma dim_subset_UNIV: "dim (S:: ('a::euclidean_space) set) \<le> DIM('a)"
+lemma dim_subset_UNIV:
+ fixes S :: "'a::euclidean_space set"
+ shows "dim S \<le> DIM('a)"
by (metis dim_subset subset_UNIV dim_UNIV)
text {* Converses to those. *}
lemma card_ge_dim_independent:
- assumes BV:"(B::('a::euclidean_space) set) \<subseteq> V"
- and iB:"independent B" and dVB:"dim V \<le> card B"
+ fixes B :: "'a::euclidean_space set"
+ assumes BV: "B \<subseteq> V"
+ and iB: "independent B"
+ and dVB: "dim V \<le> card B"
shows "V \<subseteq> span B"
-proof -
- { fix a assume aV: "a \<in> V"
- { assume aB: "a \<notin> span B"
- then have iaB: "independent (insert a B)" using iB aV BV by (simp add: independent_insert)
- from aV BV have th0: "insert a B \<subseteq> V" by blast
- from aB have "a \<notin>B" by (auto simp add: span_superset)
- with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
- then have "a \<in> span B" by blast }
- then show ?thesis by blast
+proof
+ fix a
+ assume aV: "a \<in> V"
+ {
+ assume aB: "a \<notin> span B"
+ then have iaB: "independent (insert a B)"
+ using iB aV BV by (simp add: independent_insert)
+ from aV BV have th0: "insert a B \<subseteq> V"
+ by blast
+ from aB have "a \<notin>B"
+ by (auto simp add: span_superset)
+ with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB]
+ have False by auto
+ }
+ then show "a \<in> span B" by blast
qed
lemma card_le_dim_spanning:
@@ -1669,54 +1904,81 @@
and dVB: "dim V \<ge> card B"
shows "independent B"
proof -
- { fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
- from a fB have c0: "card B \<noteq> 0" by auto
- from a fB have cb: "card (B -{a}) = card B - 1" by auto
- from BV a have th0: "B -{a} \<subseteq> V" by blast
- { fix x assume x: "x \<in> V"
- from a have eq: "insert a (B -{a}) = B" by blast
- from x VB have x': "x \<in> span B" by blast
+ {
+ fix a
+ assume a: "a \<in> B" "a \<in> span (B -{a})"
+ from a fB have c0: "card B \<noteq> 0"
+ by auto
+ from a fB have cb: "card (B -{a}) = card B - 1"
+ by auto
+ from BV a have th0: "B -{a} \<subseteq> V"
+ by blast
+ {
+ fix x
+ assume x: "x \<in> V"
+ from a have eq: "insert a (B -{a}) = B"
+ by blast
+ from x VB have x': "x \<in> span B"
+ by blast
from span_trans[OF a(2), unfolded eq, OF x']
- have "x \<in> span (B -{a})" . }
- then have th1: "V \<subseteq> span (B -{a})" by blast
- have th2: "finite (B -{a})" using fB by auto
+ have "x \<in> span (B -{a})" .
+ }
+ then have th1: "V \<subseteq> span (B -{a})"
+ by blast
+ have th2: "finite (B -{a})"
+ using fB by auto
from span_card_ge_dim[OF th0 th1 th2]
have c: "dim V \<le> card (B -{a})" .
- from c c0 dVB cb have False by simp }
- then show ?thesis unfolding dependent_def by blast
+ from c c0 dVB cb have False by simp
+ }
+ then show ?thesis
+ unfolding dependent_def by blast
qed
-lemma card_eq_dim: "(B:: ('a::euclidean_space) set) \<subseteq> V \<Longrightarrow>
- card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
+lemma card_eq_dim:
+ fixes B :: "'a::euclidean_space set"
+ shows "B \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
by (metis order_eq_iff card_le_dim_spanning card_ge_dim_independent)
text {* More general size bound lemmas. *}
lemma independent_bound_general:
- "independent (S:: ('a::euclidean_space) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
+ fixes S :: "'a::euclidean_space set"
+ shows "independent S \<Longrightarrow> finite S \<and> card S \<le> dim S"
by (metis independent_card_le_dim independent_bound subset_refl)
lemma dependent_biggerset_general:
- "(finite (S:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+ fixes S :: "'a::euclidean_space set"
+ shows "(finite S \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
-lemma dim_span: "dim (span (S:: ('a::euclidean_space) set)) = dim S"
+lemma dim_span:
+ fixes S :: "'a::euclidean_space set"
+ shows "dim (span S) = dim S"
proof -
have th0: "dim S \<le> dim (span S)"
by (auto simp add: subset_eq intro: dim_subset span_superset)
from basis_exists[of S]
- obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
- from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
- have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
- have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
+ obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
+ by blast
+ from B have fB: "finite B" "card B = dim S"
+ using independent_bound by blast+
+ have bSS: "B \<subseteq> span S"
+ using B(1) by (metis subset_eq span_inc)
+ have sssB: "span S \<subseteq> span B"
+ using span_mono[OF B(3)] by (simp add: span_span)
from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
using fB(2) by arith
qed
-lemma subset_le_dim: "(S:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
+lemma subset_le_dim:
+ fixes S :: "'a::euclidean_space set"
+ shows "S \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
by (metis dim_span dim_subset)
-lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
+lemma span_eq_dim:
+ fixes S:: "'a::euclidean_space set"
+ shows "span S = span T \<Longrightarrow> dim S = dim T"
by (metis dim_span)
lemma spans_image:
@@ -1732,12 +1994,15 @@
proof -
from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
- from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
+ from B have fB: "finite B" "card B = dim S"
+ using independent_bound by blast+
have "dim (f ` S) \<le> card (f ` B)"
apply (rule span_card_ge_dim)
- using lf B fB apply (auto simp add: span_linear_image spans_image subset_image_iff)
+ using lf B fB
+ apply (auto simp add: span_linear_image spans_image subset_image_iff)
done
- also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
+ also have "\<dots> \<le> dim S"
+ using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
@@ -1745,12 +2010,15 @@
lemma spanning_surjective_image:
assumes us: "UNIV \<subseteq> span S"
- and lf: "linear f" and sf: "surj f"
+ and lf: "linear f"
+ and sf: "surj f"
shows "UNIV \<subseteq> span (f ` S)"
proof -
- have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
- also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
-finally show ?thesis .
+ have "UNIV \<subseteq> f ` UNIV"
+ using sf by (auto simp add: surj_def)
+ also have " \<dots> \<subseteq> span (f ` S)"
+ using spans_image[OF lf us] .
+ finally show ?thesis .
qed
lemma independent_injective_image:
@@ -1759,23 +2027,30 @@
and fi: "inj f"
shows "independent (f ` S)"
proof -
- { fix a
+ {
+ fix a
assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
- have eq: "f ` S - {f a} = f ` (S - {a})" using fi
- by (auto simp add: inj_on_def)
+ have eq: "f ` S - {f a} = f ` (S - {a})"
+ using fi by (auto simp add: inj_on_def)
from a have "f a \<in> f ` span (S -{a})"
- unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
- then have "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
- with a(1) iS have False by (simp add: dependent_def) }
- then show ?thesis unfolding dependent_def by blast
+ unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
+ then have "a \<in> span (S -{a})"
+ using fi by (auto simp add: inj_on_def)
+ with a(1) iS have False
+ by (simp add: dependent_def)
+ }
+ then show ?thesis
+ unfolding dependent_def by blast
qed
text {* Picking an orthogonal replacement for a spanning set. *}
- (* FIXME : Move to some general theory ?*)
+(* FIXME : Move to some general theory ?*)
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
-lemma vector_sub_project_orthogonal: "(b::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
+lemma vector_sub_project_orthogonal:
+ fixes b x :: "'a::euclidean_space"
+ shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
unfolding inner_simps by auto
lemma pairwise_orthogonal_insert:
@@ -1786,14 +2061,17 @@
by (auto simp add: orthogonal_commute)
lemma basis_orthogonal:
- fixes B :: "('a::real_inner) set"
+ fixes B :: "'a::real_inner set"
assumes fB: "finite B"
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
(is " \<exists>C. ?P B C")
using fB
proof (induct rule: finite_induct)
case empty
- then show ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
+ then show ?case
+ apply (rule exI[where x="{}"])
+ apply (auto simp add: pairwise_def)
+ done
next
case (insert a B)
note fB = `finite B` and aB = `a \<notin> B`
@@ -1802,10 +2080,12 @@
"span C = span B" "pairwise orthogonal C" by blast
let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C"
let ?C = "insert ?a C"
- from C(1) have fC: "finite ?C" by simp
+ from C(1) have fC: "finite ?C"
+ by simp
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)"
by (simp add: card_insert_if)
- { fix x k
+ {
+ fix x k
have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)"
by (simp add: field_simps)
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
@@ -1817,12 +2097,17 @@
apply (rule span_mul)
apply (rule span_superset)
apply assumption
- done }
+ done
+ }
then have SC: "span ?C = span (insert a B)"
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto
- { fix y assume yC: "y \<in> C"
- then have Cy: "C = insert y (C - {y})" by blast
- have fth: "finite (C - {y})" using C by simp
+ {
+ fix y
+ assume yC: "y \<in> C"
+ then have Cy: "C = insert y (C - {y})"
+ by blast
+ have fth: "finite (C - {y})"
+ using C by simp
have "orthogonal ?a y"
unfolding orthogonal_def
unfolding inner_diff inner_setsum_left diff_eq_0_iff_eq
@@ -1831,10 +2116,12 @@
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
- using `y \<in> C` by auto }
+ using `y \<in> C` by auto
+ }
with `pairwise orthogonal C` have CPO: "pairwise orthogonal ?C"
by (rule pairwise_orthogonal_insert)
- from fC cC SC CPO have "?P (insert a B) ?C" by blast
+ from fC cC SC CPO have "?P (insert a B) ?C"
+ by blast
then show ?case by blast
qed
@@ -1843,19 +2130,29 @@
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
proof -
from basis_exists[of V] obtain B where
- B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
- from B have fB: "finite B" "card B = dim V" using independent_bound by auto
+ B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V"
+ by blast
+ from B have fB: "finite B" "card B = dim V"
+ using independent_bound by auto
from basis_orthogonal[OF fB(1)] obtain C where
- C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
- from C B have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
- from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
+ C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C"
+ by blast
+ from C B have CSV: "C \<subseteq> span V"
+ by (metis span_inc span_mono subset_trans)
+ from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C"
+ by (simp add: span_span)
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
- have iC: "independent C" by (simp add: dim_span)
- from C fB have "card C \<le> dim V" by simp
- moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
+ have iC: "independent C"
by (simp add: dim_span)
- ultimately have CdV: "card C = dim V" using C(1) by simp
- from C B CSV CdV iC show ?thesis by auto
+ from C fB have "card C \<le> dim V"
+ by simp
+ moreover have "dim V \<le> card C"
+ using span_card_ge_dim[OF CSV SVC C(1)]
+ by (simp add: dim_span)
+ ultimately have CdV: "card C = dim V"
+ using C(1) by simp
+ from C B CSV CdV iC show ?thesis
+ by auto
qed
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
@@ -1865,17 +2162,20 @@
text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
lemma span_not_univ_orthogonal:
- fixes S::"('a::euclidean_space) set"
+ fixes S :: "'a::euclidean_space set"
assumes sU: "span S \<noteq> UNIV"
shows "\<exists>(a::'a). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
proof -
- from sU obtain a where a: "a \<notin> span S" by blast
+ from sU obtain a where a: "a \<notin> span S"
+ by blast
from orthogonal_basis_exists obtain B where
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
by blast
- from B have fB: "finite B" "card B = dim S" using independent_bound by auto
+ from B have fB: "finite B" "card B = dim S"
+ using independent_bound by auto
from span_mono[OF B(2)] span_mono[OF B(3)]
- have sSB: "span S = span B" by (simp add: span_span)
+ have sSB: "span S = span B"
+ by (simp add: span_span)
let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B"
have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S"
unfolding sSB
@@ -1885,17 +2185,23 @@
apply (rule span_superset)
apply assumption
done
- with a have a0:"?a \<noteq> 0" by auto
+ with a have a0:"?a \<noteq> 0"
+ by auto
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
proof (rule span_induct')
show "subspace {x. ?a \<bullet> x = 0}"
by (auto simp add: subspace_def inner_add)
next
- { fix x assume x: "x \<in> B"
- from x have B': "B = insert x (B - {x})" by blast
- have fth: "finite (B - {x})" using fB by simp
+ {
+ fix x
+ assume x: "x \<in> B"
+ from x have B': "B = insert x (B - {x})"
+ by blast
+ have fth: "finite (B - {x})"
+ using fB by simp
have "?a \<bullet> x = 0"
- apply (subst B') using fB fth
+ apply (subst B')
+ using fB fth
unfolding setsum_clauses(2)[OF fth]
apply simp unfolding inner_simps
apply (clarsimp simp add: inner_add inner_setsum_left)
@@ -1903,27 +2209,36 @@
unfolding inner_commute
apply (auto simp add: x field_simps
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])
- done }
- then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
+ done
+ }
+ then show "\<forall>x \<in> B. ?a \<bullet> x = 0"
+ by blast
qed
- with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
+ with a0 show ?thesis
+ unfolding sSB by (auto intro: exI[where x="?a"])
qed
lemma span_not_univ_subset_hyperplane:
- assumes SU: "span S \<noteq> (UNIV ::('a::euclidean_space) set)"
+ fixes S :: "'a::euclidean_space set"
+ assumes SU: "span S \<noteq> UNIV"
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
using span_not_univ_orthogonal[OF SU] by auto
lemma lowdim_subset_hyperplane:
- fixes S::"('a::euclidean_space) set"
+ fixes S :: "'a::euclidean_space set"
assumes d: "dim S < DIM('a)"
shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
proof -
- { assume "span S = UNIV"
- then have "dim (span S) = dim (UNIV :: ('a) set)" by simp
- then have "dim S = DIM('a)" by (simp add: dim_span dim_UNIV)
- with d have False by arith }
- then have th: "span S \<noteq> UNIV" by blast
+ {
+ assume "span S = UNIV"
+ then have "dim (span S) = dim (UNIV :: ('a) set)"
+ by simp
+ then have "dim S = DIM('a)"
+ by (simp add: dim_span dim_UNIV)
+ with d have False by arith
+ }
+ then have th: "span S \<noteq> UNIV"
+ by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
@@ -1945,7 +2260,9 @@
case (2 a b x)
have fb: "finite b" using "2.prems" by simp
have th0: "f ` b \<subseteq> f ` (insert a b)"
- apply (rule image_mono) by blast
+ apply (rule image_mono)
+ apply blast
+ done
from independent_mono[ OF "2.prems"(2) th0]
have ifb: "independent (f ` b)" .
have fib: "inj_on f b"
@@ -1953,23 +2270,27 @@
apply blast
done
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
- obtain k where k: "x - k*\<^sub>R a \<in> span (b -{a})" by blast
+ obtain k where k: "x - k*\<^sub>R a \<in> span (b - {a})"
+ by blast
have "f (x - k*\<^sub>R a) \<in> span (f ` b)"
unfolding span_linear_image[OF lf]
apply (rule imageI)
- using k span_mono[of "b-{a}" b] apply blast
+ using k span_mono[of "b-{a}" b]
+ apply blast
done
then have "f x - k*\<^sub>R f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
then have th: "-k *\<^sub>R f a \<in> span (f ` b)"
using "2.prems"(5) by simp
- { assume k0: "k = 0"
- from k0 k have "x \<in> span (b -{a})" by simp
- then have "x \<in> span b" using span_mono[of "b-{a}" b]
- by blast }
- moreover
- { assume k0: "k \<noteq> 0"
- from span_mul[OF th, of "- 1/ k"] k0
+ have xsb: "x \<in> span b"
+ proof (cases "k = 0")
+ case True
+ with k have "x \<in> span (b -{a})" by simp
+ then show ?thesis using span_mono[of "b-{a}" b]
+ by blast
+ next
+ case False
+ with span_mul[OF th, of "- 1/ k"]
have th1: "f a \<in> span (f ` b)"
by auto
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
@@ -1979,20 +2300,21 @@
using "2.hyps"(2)
"2.prems"(3) by auto
with th1 have False by blast
- then have "x \<in> span b" by blast }
- ultimately have xsb: "x \<in> span b" by blast
- from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
- show "x = 0" .
+ then show ?thesis by blast
+ qed
+ from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)] show "x = 0" .
qed
text {* We can extend a linear mapping from basis. *}
lemma linear_independent_extend_lemma:
fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
- assumes fi: "finite B" and ib: "independent B"
- shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
- \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
- \<and> (\<forall>x\<in> B. g x = f x)"
+ assumes fi: "finite B"
+ and ib: "independent B"
+ shows "\<exists>g.
+ (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y) \<and>
+ (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
+ (\<forall>x\<in> B. g x = f x)"
using ib fi
proof (induct rule: finite_induct[OF fi])
case 1
@@ -2005,39 +2327,56 @@
g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
"\<forall>x\<in>span b. \<forall>c. g (c *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
- { fix z assume z: "z \<in> span (insert a b)"
+ {
+ fix z
+ assume z: "z \<in> span (insert a b)"
have th0: "z - ?h z *\<^sub>R a \<in> span b"
apply (rule someI_ex)
unfolding span_breakdown_eq[symmetric]
- using z .
- { fix k assume k: "z - k *\<^sub>R a \<in> span b"
+ apply (rule z)
+ done
+ {
+ fix k
+ assume k: "z - k *\<^sub>R a \<in> span b"
have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
by (simp add: field_simps scaleR_left_distrib [symmetric])
- from span_sub[OF th0 k]
- have khz: "(k - ?h z) *\<^sub>R a \<in> span b" by (simp add: eq)
- { assume "k \<noteq> ?h z" then have k0: "k - ?h z \<noteq> 0" by simp
+ from span_sub[OF th0 k] have khz: "(k - ?h z) *\<^sub>R a \<in> span b"
+ by (simp add: eq)
+ {
+ assume "k \<noteq> ?h z"
+ then have k0: "k - ?h z \<noteq> 0" by simp
from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> span b" by simp
with "2.prems"(1) "2.hyps"(2) have False
- by (auto simp add: dependent_def)}
- then have "k = ?h z" by blast}
- with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
+ by (auto simp add: dependent_def)
+ }
+ then have "k = ?h z" by blast
+ }
+ with th0 have "z - ?h z *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)"
+ by blast
+ }
note h = this
let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R a)"
- { fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
+ {
+ fix x y
+ assume x: "x \<in> span (insert a b)"
+ and y: "y \<in> span (insert a b)"
have tha: "\<And>(x::'a) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
by (simp add: algebra_simps)
have addh: "?h (x + y) = ?h x + ?h y"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (rule span_add[OF x y])
unfolding tha
- by (metis span_add x y conjunct1[OF h, rule_format])
+ apply (metis span_add x y conjunct1[OF h, rule_format])
+ done
have "?g (x + y) = ?g x + ?g y"
unfolding addh tha
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: scaleR_left_distrib)}
moreover
- { fix x:: "'a" and c:: real
+ {
+ fix x :: "'a"
+ fix c :: real
assume x: "x \<in> span (insert a b)"
have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
by (simp add: algebra_simps)
@@ -2048,24 +2387,29 @@
done
have "?g (c *\<^sub>R x) = c*\<^sub>R ?g x"
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
- by (simp add: algebra_simps) }
+ by (simp add: algebra_simps)
+ }
moreover
- { fix x assume x: "x \<in> (insert a b)"
- { assume xa: "x = a"
+ {
+ fix x
+ assume x: "x \<in> insert a b"
+ {
+ assume xa: "x = a"
have ha1: "1 = ?h a"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset insertI1)
using conjunct1[OF h, OF span_superset, OF insertI1]
apply (auto simp add: span_0)
done
-
from xa ha1[symmetric] have "?g x = f x"
apply simp
using g(2)[rule_format, OF span_0, of 0]
apply simp
- done }
+ done
+ }
moreover
- { assume xb: "x \<in> b"
+ {
+ assume xb: "x \<in> b"
have h0: "0 = ?h x"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset x)
@@ -2073,8 +2417,11 @@
apply (metis span_superset xb)
done
have "?g x = f x"
- by (simp add: h0[symmetric] g(3)[rule_format, OF xb]) }
- ultimately have "?g x = f x" using x by blast }
+ by (simp add: h0[symmetric] g(3)[rule_format, OF xb])
+ }
+ ultimately have "?g x = f x"
+ using x by blast
+ }
ultimately show ?case
apply -
apply (rule exI[where x="?g"])
@@ -2083,17 +2430,22 @@
qed
lemma linear_independent_extend:
- assumes iB: "independent (B:: ('a::euclidean_space) set)"
+ fixes B :: "'a::euclidean_space set"
+ assumes iB: "independent B"
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
proof -
from maximal_independent_subset_extend[of B UNIV] iB
- obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
+ obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C"
+ by auto
from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
- obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
- \<and> (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x)
- \<and> (\<forall>x\<in> C. g x = f x)" by blast
- from g show ?thesis unfolding linear_def using C
+ obtain g where g:
+ "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y) \<and>
+ (\<forall>x\<in> span C. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R g x) \<and>
+ (\<forall>x\<in> C. g x = f x)" by blast
+ from g show ?thesis
+ unfolding linear_def
+ using C
apply clarsimp
apply blast
done
@@ -2118,10 +2470,12 @@
then show ?case by simp
next
case (2 y t)
- from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
- from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
- f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
- from f "2.prems"(2) "2.hyps"(2) show ?case
+ from "2.prems"(1,2,5) "2.hyps"(1,2) have cst: "card s \<le> card t"
+ by simp
+ from "2.prems"(3) [OF "2.hyps"(1) cst]
+ obtain f where "f ` s \<subseteq> t" "inj_on f s"
+ by blast
+ with "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
apply (auto simp add: inj_on_def)
@@ -2135,54 +2489,74 @@
and c: "card A = card B"
shows "A = B"
proof -
- from fB AB have fA: "finite A" by (auto intro: finite_subset)
- from fA fB have fBA: "finite (B - A)" by auto
- have e: "A \<inter> (B - A) = {}" by blast
- have eq: "A \<union> (B - A) = B" using AB by blast
- from card_Un_disjoint[OF fA fBA e, unfolded eq c]
- have "card (B - A) = 0" by arith
- then have "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
- with AB show "A = B" by blast
+ from fB AB have fA: "finite A"
+ by (auto intro: finite_subset)
+ from fA fB have fBA: "finite (B - A)"
+ by auto
+ have e: "A \<inter> (B - A) = {}"
+ by blast
+ have eq: "A \<union> (B - A) = B"
+ using AB by blast
+ from card_Un_disjoint[OF fA fBA e, unfolded eq c] have "card (B - A) = 0"
+ by arith
+ then have "B - A = {}"
+ unfolding card_eq_0_iff using fA fB by simp
+ with AB show "A = B"
+ by blast
qed
lemma subspace_isomorphism:
- assumes s: "subspace (S:: ('a::euclidean_space) set)"
- and t: "subspace (T :: ('b::euclidean_space) set)"
+ fixes S :: "'a::euclidean_space set"
+ and T :: "'b::euclidean_space set"
+ assumes s: "subspace S"
+ and t: "subspace T"
and d: "dim S = dim T"
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
proof -
- from basis_exists[of S] independent_bound obtain B where
- B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
- from basis_exists[of T] independent_bound obtain C where
- C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
- from B(4) C(4) card_le_inj[of B C] d obtain f where
- f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
- from linear_independent_extend[OF B(2)] obtain g where
- g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
- from inj_on_iff_eq_card[OF fB, of f] f(2)
- have "card (f ` B) = card B" by simp
- with B(4) C(4) have ceq: "card (f ` B) = card C" using d
+ from basis_exists[of S] independent_bound
+ obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B"
+ by blast
+ from basis_exists[of T] independent_bound
+ obtain C where C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C"
+ by blast
+ from B(4) C(4) card_le_inj[of B C] d
+ obtain f where f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C`
+ by auto
+ from linear_independent_extend[OF B(2)]
+ obtain g where g: "linear g" "\<forall>x\<in> B. g x = f x"
+ by blast
+ from inj_on_iff_eq_card[OF fB, of f] f(2) have "card (f ` B) = card B"
by simp
- have "g ` B = f ` B" using g(2)
- by (auto simp add: image_iff)
+ with B(4) C(4) have ceq: "card (f ` B) = card C"
+ using d by simp
+ have "g ` B = f ` B"
+ using g(2) by (auto simp add: image_iff)
also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
finally have gBC: "g ` B = C" .
- have gi: "inj_on g B" using f(2) g(2)
- by (auto simp add: inj_on_def)
+ have gi: "inj_on g B"
+ using f(2) g(2) by (auto simp add: inj_on_def)
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
- { fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
- from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
- from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
- have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
- have "x=y" using g0[OF th1 th0] by simp }
+ {
+ fix x y
+ assume x: "x \<in> S" and y: "y \<in> S" and gxy: "g x = g y"
+ from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B"
+ by blast+
+ from gxy have th0: "g (x - y) = 0"
+ by (simp add: linear_sub[OF g(1)])
+ have th1: "x - y \<in> span B"
+ using x' y' by (metis span_sub)
+ have "x = y"
+ using g0[OF th1 th0] by simp
+ }
then have giS: "inj_on g S"
unfolding inj_on_def by blast
- from span_subspace[OF B(1,3) s]
- have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
+ from span_subspace[OF B(1,3) s] have "g ` S = span (g ` B)"
+ by (simp add: span_linear_image[OF g(1)])
also have "\<dots> = span C" unfolding gBC ..
also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
finally have gS: "g ` S = T" .
- from g(1) gS giS show ?thesis by blast
+ from g(1) gS giS show ?thesis
+ by blast
qed
text {* Linear functions are equal on a subspace if they are on a spanning set. *}
@@ -2232,7 +2606,8 @@
lemma bilinear_eq:
assumes bf: "bilinear f"
and bg: "bilinear g"
- and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
+ and SB: "S \<subseteq> span B"
+ and TC: "T \<subseteq> span C"
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
proof -
@@ -2252,11 +2627,12 @@
apply (auto simp add: span_0 bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def
intro: bilinear_ladd[OF bf])
done
- then show ?thesis using SB TC by auto
+ then show ?thesis
+ using SB TC by auto
qed
lemma bilinear_eq_stdbasis:
- fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
assumes bf: "bilinear f"
and bg: "bilinear g"
and fg: "\<forall>i\<in>Basis. \<forall>j\<in>Basis. f i j = g i j"
@@ -2266,50 +2642,53 @@
text {* Detailed theorems about left and right invertibility in general case. *}
lemma linear_injective_left_inverse:
- fixes f::"'a::euclidean_space => 'b::euclidean_space"
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes lf: "linear f" and fi: "inj f"
shows "\<exists>g. linear g \<and> g o f = id"
proof -
from linear_independent_extend[OF independent_injective_image, OF independent_Basis, OF lf fi]
- obtain h:: "'b => 'a" where
- h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x" by blast
+ obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x \<in> f ` Basis. h x = inv f x"
+ by blast
from h(2) have th: "\<forall>i\<in>Basis. (h \<circ> f) i = id i"
using inv_o_cancel[OF fi, unfolded fun_eq_iff id_def o_def]
by auto
-
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
have "h o f = id" .
- then show ?thesis using h(1) by blast
+ then show ?thesis
+ using h(1) by blast
qed
lemma linear_surjective_right_inverse:
- fixes f::"'a::euclidean_space => 'b::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes lf: "linear f"
+ and sf: "surj f"
shows "\<exists>g. linear g \<and> f o g = id"
proof -
from linear_independent_extend[OF independent_Basis[where 'a='b],of "inv f"]
- obtain h:: "'b \<Rightarrow> 'a" where
- h: "linear h" "\<forall>x\<in>Basis. h x = inv f x" by blast
- from h(2)
- have th: "\<forall>i\<in>Basis. (f o h) i = id i"
+ obtain h:: "'b \<Rightarrow> 'a" where h: "linear h" "\<forall>x\<in>Basis. h x = inv f x"
+ by blast
+ from h(2) have th: "\<forall>i\<in>Basis. (f o h) i = id i"
using sf by (auto simp add: surj_iff_all)
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
have "f o h = id" .
- then show ?thesis using h(1) by blast
+ then show ?thesis
+ using h(1) by blast
qed
text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
lemma linear_injective_imp_surjective:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and fi: "inj f"
+ fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and fi: "inj f"
shows "surj f"
proof -
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
by blast
- from B(4) have d: "dim ?U = card B" by simp
+ from B(4) have d: "dim ?U = card B"
+ by simp
have th: "?U \<subseteq> span (f ` B)"
apply (rule card_ge_dim_independent)
apply blast
@@ -2333,51 +2712,66 @@
and fT: "finite T"
and c: "card S = card T"
and ST: "f ` S \<subseteq> T"
- shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
- { assume h: "?lhs"
- { fix x y
- assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
- from x fS have S0: "card S \<noteq> 0" by auto
- { assume xy: "x \<noteq> y"
- have th: "card S \<le> card (f ` (S - {y}))"
- unfolding c
- apply (rule card_mono)
- apply (rule finite_imageI)
- using fS apply simp
- using h xy x y f unfolding subset_eq image_iff
- apply auto
- apply (case_tac "xa = f x")
- apply (rule bexI[where x=x])
- apply auto
- done
- also have " \<dots> \<le> card (S -{y})"
- apply (rule card_image_le)
- using fS by simp
- also have "\<dots> \<le> card S - 1" using y fS by simp
- finally have False using S0 by arith }
- then have "x = y" by blast}
- then have ?rhs unfolding inj_on_def by blast}
- moreover
- { assume h: ?rhs
- have "f ` S = T"
- apply (rule card_subset_eq[OF fT ST])
- unfolding card_image[OF h] using c .
- then have ?lhs by blast}
- ultimately show ?thesis by blast
+ shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume h: "?lhs"
+ {
+ fix x y
+ assume x: "x \<in> S"
+ assume y: "y \<in> S"
+ assume f: "f x = f y"
+ from x fS have S0: "card S \<noteq> 0"
+ by auto
+ have "x = y"
+ proof (rule ccontr)
+ assume xy: "x \<noteq> y"
+ have th: "card S \<le> card (f ` (S - {y}))"
+ unfolding c
+ apply (rule card_mono)
+ apply (rule finite_imageI)
+ using fS apply simp
+ using h xy x y f unfolding subset_eq image_iff
+ apply auto
+ apply (case_tac "xa = f x")
+ apply (rule bexI[where x=x])
+ apply auto
+ done
+ also have " \<dots> \<le> card (S -{y})"
+ apply (rule card_image_le)
+ using fS by simp
+ also have "\<dots> \<le> card S - 1" using y fS by simp
+ finally show False using S0 by arith
+ qed
+ }
+ then show ?rhs
+ unfolding inj_on_def by blast
+next
+ assume h: ?rhs
+ have "f ` S = T"
+ apply (rule card_subset_eq[OF fT ST])
+ unfolding card_image[OF h]
+ apply (rule c)
+ done
+ then show ?lhs by blast
qed
lemma linear_surjective_imp_injective:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and sf: "surj f"
shows "inj f"
proof -
let ?U = "UNIV :: 'a set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
by blast
- { fix x assume x: "x \<in> span B" and fx: "f x = 0"
- from B(2) have fB: "finite B" using independent_bound by auto
+ {
+ fix x
+ assume x: "x \<in> span B"
+ assume fx: "f x = 0"
+ from B(2) have fB: "finite B"
+ using independent_bound by auto
have fBi: "independent (f ` B)"
apply (rule card_le_dim_spanning[of "f ` B" ?U])
apply blast
@@ -2394,81 +2788,98 @@
apply blast
unfolding span_linear_image[OF lf]
apply (rule subset_trans[where B = "f ` UNIV"])
- using sf unfolding surj_def apply blast
+ using sf unfolding surj_def
+ apply blast
apply (rule image_mono)
apply (rule B(3))
apply (metis finite_imageI fB)
done
-
moreover have "card (f ` B) \<le> card B"
by (rule card_image_le, rule fB)
- ultimately have th1: "card B = card (f ` B)" unfolding d by arith
+ ultimately have th1: "card B = card (f ` B)"
+ unfolding d by arith
have fiB: "inj_on f B"
unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric]
by blast
from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
- have "x = 0" by blast}
- note th = this
- from th show ?thesis unfolding linear_injective_0[OF lf]
- using B(3) by blast
+ have "x = 0" by blast
+ }
+ then show ?thesis
+ unfolding linear_injective_0[OF lf]
+ using B(3)
+ by blast
qed
text {* Hence either is enough for isomorphism. *}
lemma left_right_inverse_eq:
- assumes fg: "f o g = id" and gh: "g o h = id"
+ assumes fg: "f \<circ> g = id"
+ and gh: "g \<circ> h = id"
shows "f = h"
proof -
- have "f = f o (g o h)" unfolding gh by simp
- also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
- finally show "f = h" unfolding fg by simp
+ have "f = f \<circ> (g \<circ> h)"
+ unfolding gh by simp
+ also have "\<dots> = (f \<circ> g) \<circ> h"
+ by (simp add: o_assoc)
+ finally show "f = h"
+ unfolding fg by simp
qed
lemma isomorphism_expand:
- "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
+ "f \<circ> g = id \<and> g \<circ> f = id \<longleftrightarrow> (\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x)"
by (simp add: fun_eq_iff o_def id_def)
lemma linear_injective_isomorphism:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and fi: "inj f"
+ fixes f::"'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and fi: "inj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]]
linear_injective_left_inverse[OF lf fi]
by (metis left_right_inverse_eq)
-lemma linear_surjective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
+lemma linear_surjective_isomorphism:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and sf: "surj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf sf]
linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
by (metis left_right_inverse_eq)
-text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
+text {* Left and right inverses are the same for
+ @{typ "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"}. *}
lemma linear_inverse_left:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and lf': "linear f'"
- shows "f o f' = id \<longleftrightarrow> f' o f = id"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and lf': "linear f'"
+ shows "f \<circ> f' = id \<longleftrightarrow> f' \<circ> f = id"
proof -
- { fix f f':: "'a => 'a"
- assume lf: "linear f" "linear f'" and f: "f o f' = id"
+ {
+ fix f f':: "'a \<Rightarrow> 'a"
+ assume lf: "linear f" "linear f'"
+ assume f: "f \<circ> f' = id"
from f have sf: "surj f"
apply (auto simp add: o_def id_def surj_def)
apply metis
done
from linear_surjective_isomorphism[OF lf(1) sf] lf f
- have "f' o f = id" unfolding fun_eq_iff o_def id_def
- by metis }
- then show ?thesis using lf lf' by metis
+ have "f' \<circ> f = id"
+ unfolding fun_eq_iff o_def id_def by metis
+ }
+ then show ?thesis
+ using lf lf' by metis
qed
text {* Moreover, a one-sided inverse is automatically linear. *}
lemma left_inverse_linear:
- fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and gf: "g o f = id"
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+ assumes lf: "linear f"
+ and gf: "g \<circ> f = id"
shows "linear g"
proof -
from gf have fi: "inj f"
@@ -2476,8 +2887,8 @@
apply metis
done
from linear_injective_isomorphism[OF lf fi]
- obtain h:: "'a \<Rightarrow> 'a" where
- h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
+ obtain h :: "'a \<Rightarrow> 'a" where h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x"
+ by blast
have "h = g"
apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def fun_eq_iff)
@@ -2495,22 +2906,26 @@
by auto
lemma infnorm_set_image:
- "{ abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
+ "{abs ((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis} = (\<lambda>i. abs(x \<bullet> i)) ` Basis"
by blast
lemma infnorm_Max: "infnorm (x::'a::euclidean_space) = Max ((\<lambda>i. abs(x \<bullet> i)) ` Basis)"
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max)
lemma infnorm_set_lemma:
- shows "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
- and "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
+ "finite {abs((x::'a::euclidean_space) \<bullet> i) |i. i \<in> Basis}"
+ "{abs(x \<bullet> i) |i. i \<in> Basis} \<noteq> {}"
unfolding infnorm_set_image
by auto
-lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
+lemma infnorm_pos_le:
+ fixes x :: "'a::euclidean_space"
+ shows "0 \<le> infnorm x"
by (simp add: infnorm_Max Max_ge_iff ex_in_conv)
-lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + y) \<le> infnorm x + infnorm y"
+lemma infnorm_triangle:
+ fixes x :: "'a::euclidean_space"
+ shows "infnorm (x + y) \<le> infnorm x + infnorm y"
proof -
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d"
by simp
@@ -2518,7 +2933,9 @@
by (auto simp: infnorm_Max inner_add_left intro!: *)
qed
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 0"
+lemma infnorm_eq_0:
+ fixes x :: "'a::euclidean_space"
+ shows "infnorm x = 0 \<longleftrightarrow> x = 0"
proof -
have "infnorm x \<le> 0 \<longleftrightarrow> x = 0"
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff)
@@ -2539,41 +2956,47 @@
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
proof -
have "y - x = - (x - y)" by simp
- then show ?thesis by (metis infnorm_neg)
+ then show ?thesis
+ by (metis infnorm_neg)
qed
-lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
+lemma real_abs_sub_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
proof -
- have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
+ have th: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny <= n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
"infnorm y \<le> infnorm (x - y) + infnorm x"
by (simp_all add: field_simps infnorm_neg)
- from th[OF ths] show ?thesis .
+ from th[OF ths] show ?thesis .
qed
-lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
+lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x"
using infnorm_pos_le[of x] by arith
lemma Basis_le_infnorm:
- "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm (x::'a::euclidean_space)"
+ fixes x :: "'a::euclidean_space"
+ shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x"
by (simp add: infnorm_Max)
lemma infnorm_mul: "infnorm(a *\<^sub>R x) = abs a * infnorm x"
unfolding infnorm_Max
proof (safe intro!: Max_eqI)
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis"
- show "\<And>b :: 'a. b \<in> Basis \<Longrightarrow> \<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
- by (simp add: abs_mult mult_left_mono)
-
- from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
- by (auto simp del: Max_in)
- then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
- by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+ {
+ fix b :: 'a
+ assume "b \<in> Basis"
+ then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B"
+ by (simp add: abs_mult mult_left_mono)
+ next
+ from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>"
+ by (auto simp del: Max_in)
+ then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis"
+ by (intro image_eqI[where x=b]) (auto simp: abs_mult)
+ }
qed simp
-lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
+lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x"
unfolding infnorm_mul ..
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
@@ -2591,7 +3014,8 @@
lemma norm_le_infnorm: "norm x \<le> sqrt DIM('a) * infnorm(x::'a::euclidean_space)"
proof -
let ?d = "DIM('a)"
- have "real ?d \<ge> 0" by simp
+ have "real ?d \<ge> 0"
+ by simp
then have d2: "(sqrt (real ?d))\<^sup>2 = real ?d"
by (auto intro: real_sqrt_pow2)
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
@@ -2608,29 +3032,37 @@
apply (auto simp: infnorm_Max)
done
from real_le_lsqrt[OF inner_ge_zero th th1]
- show ?thesis unfolding norm_eq_sqrt_inner id_def .
+ show ?thesis
+ unfolding norm_eq_sqrt_inner id_def .
qed
lemma tendsto_infnorm [tendsto_intros]:
assumes "(f ---> a) F"
shows "((\<lambda>x. infnorm (f x)) ---> infnorm a) F"
proof (rule tendsto_compose [OF LIM_I assms])
- fix r :: real assume "0 < r"
+ fix r :: real
+ assume "r > 0"
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r"
by (metis real_norm_def le_less_trans real_abs_sub_infnorm infnorm_le_norm)
qed
text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
-lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
- { assume h: "x = 0"
- then have ?thesis by simp }
+ {
+ assume h: "x = 0"
+ then have ?thesis by simp
+ }
moreover
- { assume h: "y = 0"
- then have ?thesis by simp }
+ {
+ assume h: "y = 0"
+ then have ?thesis by simp
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ {
+ assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"]
have "?rhs \<longleftrightarrow>
(norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) -
@@ -2648,49 +3080,58 @@
apply simp
apply metis
done
- finally have ?thesis by blast }
+ finally have ?thesis by blast
+ }
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_abs_eq:
"abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
- norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x" (is "?lhs \<longleftrightarrow> ?rhs")
+ norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R x"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
- have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
+ have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a"
+ by arith
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
by simp
- also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
- (-x) \<bullet> y = norm x * norm y)"
+ also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding norm_minus_cancel norm_scaleR ..
also have "\<dots> \<longleftrightarrow> ?lhs"
- unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps by auto
+ unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps
+ by auto
finally show ?thesis ..
qed
lemma norm_triangle_eq:
fixes x y :: "'a::real_inner"
- shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
+ shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
proof -
- { assume x: "x = 0 \<or> y = 0"
- then have ?thesis by (cases "x = 0") simp_all }
+ {
+ assume x: "x = 0 \<or> y = 0"
+ then have ?thesis
+ by (cases "x = 0") simp_all
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ {
+ assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
then have "norm x \<noteq> 0" "norm y \<noteq> 0"
by simp_all
then have n: "norm x > 0" "norm y > 0"
using norm_ge_zero[of x] norm_ge_zero[of y] by arith+
- have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2)"
+ have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 \<Longrightarrow> a = b + c \<longleftrightarrow> a\<^sup>2 = (b + c)\<^sup>2"
by algebra
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2"
- apply (rule th) using n norm_ge_zero[of "x + y"]
+ apply (rule th)
+ using n norm_ge_zero[of "x + y"]
apply arith
done
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding power2_norm_eq_inner inner_simps
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
- finally have ?thesis .}
+ finally have ?thesis .
+ }
ultimately show ?thesis by blast
qed
@@ -2700,7 +3141,8 @@
definition collinear :: "'a::real_vector set \<Rightarrow> bool"
where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
-lemma collinear_empty: "collinear {}" by (simp add: collinear_def)
+lemma collinear_empty: "collinear {}"
+ by (simp add: collinear_def)
lemma collinear_sing: "collinear {x}"
by (simp add: collinear_def)
@@ -2713,14 +3155,20 @@
apply (rule exI[where x="- 1"], simp)
done
-lemma collinear_lemma:
- "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
proof -
- { assume "x=0 \<or> y = 0"
- then have ?thesis by (cases "x = 0") (simp_all add: collinear_2 insert_commute) }
+ {
+ assume "x = 0 \<or> y = 0"
+ then have ?thesis
+ by (cases "x = 0") (simp_all add: collinear_2 insert_commute)
+ }
moreover
- { assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
- { assume h: "?lhs"
+ {
+ assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
+ have ?thesis
+ proof
+ assume h: "?lhs"
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u"
unfolding collinear_def by blast
from u[rule_format, of x 0] u[rule_format, of y 0]
@@ -2732,11 +3180,13 @@
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *\<^sub>R x"
by simp
- then have ?rhs using x y by blast }
- moreover
- { assume h: "?rhs"
- then obtain c where c: "y = c *\<^sub>R x" using x y by blast
- have ?lhs unfolding collinear_def c
+ then show ?rhs using x y by blast
+ next
+ assume h: "?rhs"
+ then obtain c where c: "y = c *\<^sub>R x"
+ using x y by blast
+ show ?lhs
+ unfolding collinear_def c
apply (rule exI[where x=x])
apply auto
apply (rule exI[where x="- 1"], simp)
@@ -2744,12 +3194,13 @@
apply (rule exI[where x=1], simp)
apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
- done }
- ultimately have ?thesis by blast }
+ done
+ qed
+ }
ultimately show ?thesis by blast
qed
-lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,x,y}"
+lemma norm_cauchy_schwarz_equal: "abs (x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0, x, y}"
unfolding norm_cauchy_schwarz_abs_eq
apply (cases "x=0", simp_all add: collinear_2)
apply (cases "y=0", simp_all add: collinear_2 insert_commute)
@@ -2773,9 +3224,9 @@
unfolding scaleR_scaleR
unfolding norm_scaleR
apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
- apply (case_tac "c <= 0", simp add: field_simps)
+ apply (case_tac "c \<le> 0", simp add: field_simps)
apply (simp add: field_simps)
- apply (case_tac "c <= 0", simp add: field_simps)
+ apply (case_tac "c \<le> 0", simp add: field_simps)
apply (simp add: field_simps)
apply simp
apply simp
@@ -2801,11 +3252,12 @@
fast intro: order_trans)
lemma atLeastAtMost_singleton_euclidean[simp]:
- fixes a :: "'a::ordered_euclidean_space" shows "{a .. a} = {a}"
+ fixes a :: "'a::ordered_euclidean_space"
+ shows "{a .. a} = {a}"
by (force simp: eucl_le[where 'a='a] euclidean_eq_iff[where 'a='a])
instance real :: ordered_euclidean_space
- by default (auto simp add: Basis_real_def)
+ by default auto
instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
begin