Rewrite the Probability theory.
Introduced pinfreal as real numbers with infinity.
Use pinfreal as value for measures.
Introduces Lebesgue Measure based on the integral in Multivariate Analysis.
Proved Radon Nikodym for arbitrary sigma finite measure spaces.
header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
(* Author: John Harrison
Translation from HOL light: Robert Himmelmann, TU Muenchen *)
theory Integration
imports Derivative "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Indicator_Function
begin
declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.certs"]]
declare [[smt_fixed=false]]
declare [[z3_proofs=true]]
setup {* Arith_Data.add_tactic "Ferrante-Rackoff" (K FerranteRackoff.dlo_tac) *}
(*declare not_less[simp] not_le[simp]*)
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one
lemma real_arch_invD:
"0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
by(subst(asm) real_arch_inv)
subsection {* Sundries *}
(*declare basis_component[simp]*)
lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
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
lemma linear_simps: assumes "bounded_linear f"
shows "f (a + b) = f a + f b" "f (a - b) = f a - f b" "f 0 = 0" "f (- a) = - f a" "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
apply(rule_tac[!] additive.add additive.minus additive.diff additive.zero bounded_linear.scaleR)
using assms unfolding bounded_linear_def additive_def by auto
lemma bounded_linearI:assumes "\<And>x y. f (x + y) = f x + f y"
"\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<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
lemma real_le_inf_subset:
assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s" shows "Inf s <= Inf (t::real set)"
apply(rule isGlb_le_isLb) apply(rule Inf[OF assms(1)])
using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
unfolding isLb_def setge_def by auto
lemma real_ge_sup_subset:
assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b" shows "Sup s >= Sup (t::real set)"
apply(rule isLub_le_isUb) apply(rule Sup[OF assms(1)])
using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
unfolding isUb_def setle_def by auto
lemma bounded_linear_component[intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x $$ k)"
apply(rule bounded_linearI[where K=1])
using component_le_norm[of _ k] unfolding real_norm_def by auto
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) assume ?r fix n m::nat assume "m < n" thus "R m n" apply-
proof(induct n arbitrary: m) case (Suc n) show ?case
proof(cases "m < n") case True
show ?thesis apply(rule assms[OF Suc(1)[OF True]]) using `?r` by auto
next case False hence "m = n" using Suc(2) by auto
thus ?thesis using `?r` by auto
qed qed auto 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) "
shows "\<forall>n>m. R m n"
proof- have "\<forall>m. \<forall>n>m. R m n" apply(subst transitive_stepwise_lt_eq)
apply(rule assms) apply(assumption,assumption) using assms(2) by auto
thus ?thesis by auto qed
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")
proof safe assume ?r fix m n::nat assume "m\<le>n" thus "R m n" apply-
proof(induct n arbitrary: m) case (Suc n) show ?case
proof(cases "m \<le> n") case True show ?thesis apply(rule assms(2))
apply(rule Suc(1)[OF True]) using `?r` by auto
next case False hence "m = Suc n" using Suc(2) by auto
thus ?thesis using assms(1) by auto
qed qed(insert assms(1), auto) 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) "
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) by auto
thus ?thesis by auto qed
subsection {* Some useful lemmas about intervals. *}
abbreviation One where "One \<equiv> ((\<chi>\<chi> i. 1)::_::ordered_euclidean_space)"
lemma empty_as_interval: "{} = {One..0}"
apply(rule set_ext,rule) defer unfolding mem_interval
using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
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) = {}"
shows "interior i \<subseteq> interior s" proof-
have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
unfolding assms(1,2) interior_closed_interval by auto
moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
using assms(4) unfolding assms(1,2) by auto
ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
unfolding assms(1,2) interior_closed_interval by auto qed
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]) 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 defer apply(rule_tac Int_greatest)
unfolding open_subset_interior[OF open_ball] using interior_subset by auto
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)" proof- case goal1
thus ?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 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
show ?case 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)" 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" apply-apply(rule insert(3)) using insert(4) by auto thus ?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 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 by fastsimp+ next
case False then obtain k where "x$$k \<le> a$$k \<or> x$$k \<ge> b$$k" and k:"k<DIM('a)" unfolding mem_interval by(auto simp add:not_less)
hence "x$$k = a$$k \<or> x$$k = b$$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$$k = a$$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
hence "\<bar>(?z - y) $$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
hence "y$$k < a$$k" using e[THEN conjunct1] k by(auto simp add:field_simps basis_component as)
hence "y \<notin> i" unfolding ab mem_interval not_all apply(rule_tac x=k in exI) using k by auto thus 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]]) proof
fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"
apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
unfolding norm_scaleR norm_basis by auto
also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as 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 by auto
next let ?z = "x + (e/2) *\<^sub>R basis k" assume as:"x$$k = b$$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
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) $$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
hence "y$$k > b$$k" using e[THEN conjunct1] k by(auto simp add:field_simps as)
hence "y \<notin> i" unfolding ab mem_interval not_all using k by(rule_tac x=k in exI,auto) thus False using yi by auto qed
moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"
apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
unfolding norm_scaleR by auto
also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed
ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
then guess x .. hence "x \<in> s \<inter> interior (\<Union>f)" unfolding lem1[where U="\<Union>f",THEN sym] using centre_in_ball e[THEN conjunct1] by auto
thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e ..
hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
thus False using `t\<in>f` assms(4) by auto qed
subsection {* Bounds on intervals where they exist. *}
definition "interval_upperbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Sup {a. \<exists>x\<in>s. x$$i = a})::'a)"
definition "interval_lowerbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Inf {a. \<exists>x\<in>s. x$$i = a})::'a)"
lemma interval_upperbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i" shows "interval_upperbound {a..b} = b"
using assms unfolding interval_upperbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
apply(rule,rule) apply(rule_tac x="b$$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
unfolding mem_interval using assms by auto
lemma interval_lowerbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i" shows "interval_lowerbound {a..b} = a"
using assms unfolding interval_lowerbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
apply(rule,rule) apply(rule_tac x="a$$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
unfolding mem_interval using assms by auto
lemmas interval_bounds = interval_upperbound interval_lowerbound
lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
using assms unfolding interval_ne_empty by auto
subsection {* Content (length, area, volume...) of an interval. *}
definition "content (s::('a::ordered_euclidean_space) set) =
(if s = {} then 0 else (\<Prod>i<DIM('a). (interval_upperbound s)$$i - (interval_lowerbound s)$$i))"
lemma interval_not_empty:"\<forall>i<DIM('a). a$$i \<le> b$$i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
unfolding interval_eq_empty unfolding not_ex not_less by auto
lemma content_closed_interval: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a$$i \<le> b$$i"
shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
lemma content_closed_interval': fixes a::"'a::ordered_euclidean_space" assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
lemma content_real:assumes "a\<le>b" shows "content {a..b} = b-a"
proof- have *:"{..<Suc 0} = {0}" by auto
show ?thesis unfolding content_def using assms by(auto simp: *)
qed
lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1" proof-
have *:"\<forall>i<DIM('a). (0::'a)$$i \<le> (One::'a)$$i" by auto
have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
lemma content_pos_le[intro]: fixes a::"'a::ordered_euclidean_space" shows "0 \<le> content {a..b}" proof(cases "{a..b}={}")
case False hence *:"\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty by assumption
have "(\<Prod>i<DIM('a). interval_upperbound {a..b} $$ i - interval_lowerbound {a..b} $$ i) \<ge> 0"
apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
lemma content_pos_lt: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a$$i < b$$i" shows "0 < content {a..b}"
proof- have help_lemma1: "\<forall>i<DIM('a). a$$i < b$$i \<Longrightarrow> \<forall>i<DIM('a). a$$i \<le> ((b$$i)::real)" apply(rule,erule_tac x=i in allE) by auto
show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
using assms apply(erule_tac x=x in allE) by auto qed
lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i)" proof(cases "{a..b} = {}")
case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
case False note this[unfolded interval_eq_empty not_ex not_less]
hence as:"\<forall>i<DIM('a). b $$ i \<ge> a $$ i" by fastsimp
show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_lessThan]
apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
lemma content_closed_interval_cases:
"content {a..b::'a::ordered_euclidean_space} = (if \<forall>i<DIM('a). a$$i \<le> b$$i then setprod (\<lambda>i. b$$i - a$$i) {..<DIM('a)} else 0)" apply(rule cond_cases)
apply(rule content_closed_interval) unfolding content_eq_0 not_all not_le defer apply(erule exE,rule_tac x=x in exI) by auto
lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
(*lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
unfolding content_eq_0 by auto*)
lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i<DIM('a). a$$i < b$$i" unfolding content_eq_0 not_ex not_le by fastsimp qed
lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}" proof(cases "{a..b}={}")
case True thus ?thesis using content_pos_le[of c d] by auto next
case False hence ab_ne:"\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty by auto
hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
have "{c..d} \<noteq> {}" using assms False by auto
hence cd_ne:"\<forall>i<DIM('a). c $$ i \<le> d $$ i" using assms unfolding interval_ne_empty by auto
show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof
fix i assume i:"i\<in>{..<DIM('a)}"
show "0 \<le> b $$ i - a $$ i" using ab_ne[THEN spec[where x=i]] i by auto
show "b $$ i - a $$ i \<le> d $$ i - c $$ 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] using i by auto qed qed
lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastsimp
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"
using assms unfolding gauge_def by auto
lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "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
lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) 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})" proof-
have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
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)
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)"
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" 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
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
unfolding division_of_def by auto
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_sing[simp]: "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
assume ?r moreover { assume "s = {{a}}" 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 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 qed
lemma elementary_empty: obtains p where "p division_of {}"
unfolding division_of_trivial by auto
lemma elementary_interval: obtains p where "p division_of {a..b}"
by(metis division_of_trivial division_of_self)
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
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}))"
unfolding division_of_def by fastsimp
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)]
show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
{ 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 }
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 qed auto
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?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)"
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 * 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 ultimately show "finite ?A" by auto
have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto show "\<Union>?A = s1 \<inter> s2" apply(rule set_ext) unfolding * and Union_image_eq UN_iff
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] 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 thus "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
show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 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 \<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
show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
using division_ofD(5)[OF assms(1) k1(2) k2(2)]
using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto 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}" proof(cases "{a..b} = {}")
case True show ?thesis unfolding True and division_of_trivial by auto next
have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
case False 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)"
shows "\<exists>p. p division_of (s \<inter> t)"
by(rule,rule division_inter[OF assms])
lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<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 apply-proof(induct f rule:finite_induct)
case (insert x f) show ?case proof(cases "f={}")
case True thus ?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] .. ultimately
show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
lemma division_disjoint_union:
assumes "p1 division_of s1" "p2 division_of s2" "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
{ fix k1 k2 assume as:"k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2" moreover let ?g="interior k1 \<inter> interior k2 = {}"
{ assume as:"k1\<in>p1" "k2\<in>p2" have ?g using subset_interior[OF d1(2)[OF as(1)]] subset_interior[OF d2(2)[OF as(2)]]
using assms(3) by blast } moreover
{ assume as:"k1\<in>p2" "k2\<in>p1" have ?g using subset_interior[OF d1(2)[OF as(2)]] subset_interior[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 }
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 qed
(* move *)
lemma Eucl_nth_inverse[simp]: fixes x::"'a::euclidean_space" shows "(\<chi>\<chi> i. x $$ i) = x"
apply(subst euclidean_eq) by auto
lemma partial_division_extend_1:
assumes "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}" "{c..d} \<noteq> {}"
obtains p where "p division_of {a..b}" "{c..d} \<in> p"
proof- def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def using DIM_positive[where 'a='a] by auto
guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_lessThan[of "DIM('a)"]] .. note \<pi>=this
def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
have \<pi>':"bij_betw \<pi>' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
hence \<pi>'i:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
have \<pi>i:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi> i <DIM('a)" using \<pi> unfolding bij_betw_def n_def by auto
have \<pi>\<pi>'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi> (\<pi>' i) = i" unfolding \<pi>'_def
apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq)
using \<pi> unfolding n_def bij_betw_def by auto
have "{c..d} \<noteq> {}" using assms by auto
let ?p1 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else if \<pi>' i = l then c$$\<pi> l else b$$i)}"
let ?p2 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else if \<pi>' i = l then d$$\<pi> l else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else b$$i)}"
let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
have abcd:"\<And>i. i<DIM('a) \<Longrightarrow> a $$ i \<le> c $$ i \<and> c$$i \<le> d$$i \<and> d $$ i \<le> b $$ i" using assms
unfolding subset_interval interval_eq_empty by auto
show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
proof- have "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < Suc n"
proof(rule ccontr,unfold not_less) fix i assume i:"i<DIM('a)" and "Suc n \<le> \<pi>' i"
hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' i unfolding bij_betw_def by auto
qed hence "c = (\<chi>\<chi> i. if \<pi>' i < Suc n then c $$ i else a $$ i)"
"d = (\<chi>\<chi> i. if \<pi>' i < Suc n then d $$ i else if \<pi>' i = n + 1 then c $$ \<pi> (n + 1) else b $$ i)"
unfolding euclidean_eq[where 'a='a] using \<pi>' unfolding bij_betw_def by auto
thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}" "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"
unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
then guess i unfolding mem_interval not_all not_imp .. note i=conjunctD2[OF this]
show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
proof- fix x assume x:"x\<in>{a..b}"
{ presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $$ \<pi> i \<le> x $$ \<pi> i \<and> x $$ \<pi> i \<le> d $$ \<pi> i)}"
assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all not_imp ..
hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
hence M:"finite ?M" "?M \<noteq> {}" by auto
def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]
Min_gr_iff[OF M,unfolded l_def[symmetric]]
have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
proof- assume as:"x $$ \<pi> l < c $$ \<pi> l"
show "x \<in> ?p1 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
proof- case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal1 by auto
thus ?case using as x[unfolded mem_interval,rule_format,of i]
apply auto using l(3)[of "\<pi>' i"] using goal1 by(auto elim!:ballE[where x="\<pi>' i"])
next case goal2 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal2 by auto
thus ?case using as x[unfolded mem_interval,rule_format,of i]
apply auto using l(3)[of "\<pi>' i"] using goal2 by(auto elim!:ballE[where x="\<pi>' i"])
qed
next assume as:"x $$ \<pi> l > d $$ \<pi> l"
show "x \<in> ?p2 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
proof- fix i assume i:"i<DIM('a)"
have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using i by auto
thus "(if \<pi>' i < l then c $$ i else if \<pi>' i = l then d $$ \<pi> l else a $$ i) \<le> x $$ i"
"x $$ i \<le> (if \<pi>' i < l then d $$ i else b $$ i)"
using as x[unfolded mem_interval,rule_format,of i]
apply auto using l(3)[of "\<pi>' i"] i by(auto elim!:ballE[where x="\<pi>' i"])
qed qed
thus "x \<in> \<Union>?p" using l(2) by blast
qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
show "finite ?p" by auto
fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto
show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
proof fix i x assume i:"i<DIM('a)" assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
ultimately show "a$$i \<le> x$$i" "x$$i \<le> b$$i" using abcd[of i] using l using i
by(auto elim:disjE elim!:allE[where x=i] simp add:eucl_le[where 'a='a])
qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
proof- case goal1 thus ?case using abcd[of x] by auto
next case goal2 thus ?case using abcd[of x] by auto
qed thus "k \<noteq> {}" using k by auto
show "\<exists>a b. k = {a..b}" using k by auto
fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto
{ fix k k' l l'
assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
assume "l \<le> l'" fix x
have "x \<notin> interior k \<inter> interior k'"
proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
case True hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l'" using \<pi>'i using l' by(auto simp add:less_Suc_eq_le)
hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l' then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
hence k':"k' = {c..d}" using l'(1) unfolding * by auto
have ln:"l < n + 1"
proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
hence "k = {c..d}" using l(1) \<pi>'i unfolding * by(auto)
thus False using `k\<noteq>k'` k' by auto
qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
have "x $$ \<pi> l < c $$ \<pi> l \<or> d $$ \<pi> l < x $$ \<pi> l" using l(1) apply-
proof(erule disjE)
assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] by(auto simp add:** not_less)
next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] unfolding ** by auto
qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
by(auto elim!:allE[where x="\<pi> l"])
next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
assume x:"x \<in> interior k \<inter> interior k'"
show False using l(1) l'(1) apply-
proof(erule_tac[!] disjE)+
assume as:"k = ?p1 l" "k' = ?p1 l'"
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
have "l \<noteq> l'" using k'(2)[unfolded as] by auto
thus False using *[of "\<pi> l'"] *[of "\<pi> l"] ln using \<pi>i[OF ln(1)] \<pi>i[OF ln(2)] apply(cases "l<l'")
by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
next assume as:"k = ?p2 l" "k' = ?p2 l'"
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
thus False using *[of "\<pi> l"] *[of "\<pi> l'"] `l \<le> l'` ln by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
next assume as:"k = ?p1 l" "k' = ?p2 l'"
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
show False using abcd[of "\<pi> l'"] using *[of "\<pi> l"] *[of "\<pi> l'"] `l \<le> l'` ln apply(cases "l=l'")
by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
next assume as:"k = ?p2 l" "k' = ?p1 l'"
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
show False using *[of "\<pi> l"] *[of "\<pi> l'"] ln `l \<le> l'` apply(cases "l=l'") using abcd[of "\<pi> l'"]
by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
qed qed }
from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
apply - apply(cases "l' \<le> l") using k'(2) by auto
thus "interior k \<inter> interior k' = {}" by auto
qed qed
lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
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 apply- apply(rule that[of q]) unfolding True by auto 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" proof case goal1
guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
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 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) proof-
fix x 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]] 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)" apply- apply(rule elementary_inters)
apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
then guess d .. note d = this
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" apply(rule *[OF False]) proof fix i 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
show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
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
show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
defer apply(subst Int_commute) 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 "\<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 show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<subseteq> interior (\<Union>(q k - {k}))"
apply(rule subset_interior *)+ using k by auto 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}"
by(meson elementary_bounded bounded_subset_closed_interval)
lemma division_union_intervals_exists: assumes "{a..b::'a::ordered_euclidean_space} \<noteq> {}"
obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
using false True assms using interior_subset by auto next
case False 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)]
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
show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
unfolding interior_inter[THEN sym] 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}}))"
apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto 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(fastsimp 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 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)
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)"
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-
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
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)"
lemma tagged_division_of_finite[dest]: "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)"
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)"
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 fastsimp+
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
qed
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
qed
lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "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"
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
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 = {}"
unfolding tagged_partial_division_of_def by auto
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_union:
assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "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) subset_interior 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
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) = {})"
shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
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] subset_interior 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
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"
shows "p tagged_division_of (\<Union>(snd ` p))"
apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
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
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
lemma fine_inters:
"(\<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)"
unfolding fine_def by blast
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"
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))"
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"
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
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)
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"
by auto
lemma setsum_content_null:
assumes "content({a..b}) = 0" "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
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
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
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)"
unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
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<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$i"
proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
note ab=this[unfolded interval_eq_empty not_ex not_less]
{ 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<DIM('a). (c$$i = a$$i) \<and> (d$$i = (a$$i + b$$i) / 2) \<or> (c$$i = (a$$i + b$$i) / 2) \<and> (d$$i = b$$i)}"
let ?PP = "\<lambda>c d. \<forall>i<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$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)
let ?B = "(\<lambda>s.{(\<chi>\<chi> i. if i \<in> s then a$$i else (a$$i + b$$i) / 2)::'a ..
(\<chi>\<chi> i. if i \<in> s then (a$$i + b$$i) / 2 else b$$i)}) ` {s. s \<subseteq> {..<DIM('a)}}"
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<DIM('a) \<and> c$$i = a$$i}" in bexI)
unfolding c_d apply(rule * ) unfolding euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta' mem_Collect_eq
proof- fix i assume "i<DIM('a)" thus " c $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then a $$ i else (a $$ i + b $$ i) / 2)"
"d $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then (a $$ i + b $$ i) / 2 else b $$ i)"
using c_d(2)[of i] ab[THEN spec[where x=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[THEN spec[where x=i]] 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$$i \<noteq> e$$i \<or> d$$i \<noteq> f$$i" and i':"i<DIM('a)" unfolding de_Morgan_conj euclidean_eq[where 'a='a] by auto
hence i:"c$$i \<noteq> e$$i" "d$$i \<noteq> f$$i" apply- apply(erule_tac[!] disjE)
proof- assume "c$$i \<noteq> e$$i" thus "d$$i \<noteq> f$$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
next assume "d$$i \<noteq> f$$i" thus "c$$i \<noteq> e$$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
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$$i < d$$i" "e$$i < f$$i" "c$$i < f$$i" "e$$i < d$$i" unfolding mem_interval using i'
apply-apply(erule_tac[!] x=i in allE)+ by auto
show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
proof(erule_tac[!] conjE) assume as:"c $$ i = a $$ i" "d $$ i = (a $$ i + b $$ i) / 2"
show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
next assume as:"c $$ i = (a $$ i + b $$ i) / 2" "d $$ i = b $$ i"
show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
qed qed qed
also have "\<Union> ?A = {a..b}" proof(rule set_ext,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 assume "i<DIM('a)" thus "a $$ i \<le> x $$ i" "x $$ i \<le> b $$ i"
using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
next fix x assume x:"x\<in>{a..b}"
have "\<forall>i<DIM('a). \<exists>c d. (c = a$$i \<and> d = (a$$i + b$$i) / 2 \<or> c = (a$$i + b$$i) / 2 \<and> d = b$$i) \<and> c\<le>x$$i \<and> x$$i \<le> d"
(is "\<forall>i<DIM('a). \<exists>c d. ?P i c d") unfolding mem_interval proof(rule,rule) fix i
have "?P i (a$$i) ((a $$ i + b $$ i) / 2) \<or> ?P i ((a $$ i + b $$ i) / 2) (b$$i)"
using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
qed thus "x\<in>\<Union>?A" unfolding Union_iff unfolding lambda_skolem' unfolding Bex_def mem_Collect_eq
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-
have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
(\<forall>i<DIM('a). fst x$$i \<le> fst y$$i \<and> fst y$$i \<le> snd y$$i \<and> snd y$$i \<le> snd x$$i \<and>
2 * (snd y$$i - fst y$$i) \<le> snd x$$i - fst x$$i))" proof case goal1 thus ?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
have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
(\<forall>i<DIM('a). A(n)$$i \<le> A(Suc n)$$i \<and> A(Suc n)$$i \<le> B(Suc n)$$i \<and> B(Suc n)$$i \<le> B(n)$$i \<and>
2 * (B(Suc n)$$i - A(Suc n)$$i) \<le> B(n)$$i - A(n)$$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$$i - a$$i) {..<DIM('a)}) / 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)$$i)) {..<DIM('a)}" unfolding dist_norm by(rule norm_le_l1)
also have "\<dots> \<le> setsum (\<lambda>i. B n$$i - A n$$i) {..<DIM('a)}"
proof(rule setsum_mono) fix i show "\<bar>(x - y) $$ i\<bar> \<le> B n $$ i - A n $$ i"
using xy[unfolded mem_interval,THEN spec[where x=i]] by auto qed
also have "\<dots> \<le> setsum (\<lambda>i. b$$i - a$$i) {..<DIM('a)} / 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) $$ i - A (Suc n) $$ i \<le> (B n $$ i - A n $$ i) / 2"
using AB(4)[of i n] using goal1 by auto
also have "\<dots> \<le> (b $$ i - a $$ 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 guess d unfolding le_Suc_ex_iff .. note d=this
have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
proof(induct d) case 0 thus ?case by auto
next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
proof- case goal1 thus ?case using AB(4)[of i "m + d"] by(auto simp add:field_simps)
qed qed } 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
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
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
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.
(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)
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
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
lemma has_integral_0[simp]: "((\<lambda>x::'n::ordered_euclidean_space. 0) has_integral 0) s"
apply(rule has_integral_is_0) by 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
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 scaleR.bounded_linear_right)
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"
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
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 "\<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
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+
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"
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"
unfolding integrable_on_def by(auto intro: has_integral_cmul)
lemma integrable_neg:
shows "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"
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"
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 $$ k) = integral s f $$ 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"
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>
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
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
lemma has_integral_eq:
assumes "\<forall>x\<in>s. f x = g x" "(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
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_ext) unfolding mem_interval singleton_iff euclidean_eq[where 'a='a]
apply safe prefer 3 apply(erule_tac x=i in allE) 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
subsection {* Cauchy-type criterion for integrability. *}
(* XXXXXXX *)
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
\<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
proof assume ?l
then guess y unfolding integrable_on_def has_integral .. note y=this
show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
apply(rule dist_triangle_half_l[where y=y,unfolded dist_norm])
using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
qed qed
next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
hence "\<forall>n. \<exists>p. p tagged_division_of {a..b} \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p" apply-
proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
show ?case apply(rule_tac x=N in exI)
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
qed qed
then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
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
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
show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
using N2[rule_format,unfolded dist_norm,of "N1+N2"]
using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
subsection {* Additivity of integral on abutting intervals. *}
lemma interval_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
"{a..b} \<inter> {x. x$$k \<le> c} = {a .. (\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)}"
"{a..b} \<inter> {x. x$$k \<ge> c} = {(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i) .. b}"
apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq using assms by auto
lemma content_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
"content {a..b} = content({a..b} \<inter> {x. x$$k \<le> c}) + content({a..b} \<inter> {x. x$$k >= c})"
proof- note simps = interval_split[OF assms] content_closed_interval_cases eucl_le[where 'a='a]
{ presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps using assms by auto }
have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" "\<And>x. finite ({..<DIM('a)}-{x})" "\<And>x. x\<notin>{..<DIM('a)}-{x}"
using assms by auto
have *:"\<And>X Y Z. (\<Prod>i\<in>{..<DIM('a)}. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>{..<DIM('a)}-{k}. Z i (Y i))"
"(\<Prod>i\<in>{..<DIM('a)}. b$$i - a$$i) = (\<Prod>i\<in>{..<DIM('a)}-{k}. b$$i - a$$i) * (b$$k - a$$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 $$ k) c = max (a $$ k) c
\<Longrightarrow> x* (b$$k - a$$k) = x*(max (a $$ k) c - a $$ k) + x*(b $$ k - max (a $$ k) c)"
by (auto simp add:field_simps)
moreover have **:"(\<Prod>i<DIM('a). ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i - a $$ i) =
(\<Prod>i<DIM('a). (if i = k then min (b $$ k) c else b $$ i) - a $$ i)"
"(\<Prod>i<DIM('a). b $$ i - ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i) =
(\<Prod>i<DIM('a). b $$ i - (if i = k then max (a $$ k) c else a $$ i))"
apply(rule_tac[!] setprod.cong) by auto
have "\<not> a $$ k \<le> c \<Longrightarrow> \<not> c \<le> b $$ k \<Longrightarrow> False"
unfolding not_le using as[unfolded eucl_le[where 'a='a],rule_format,of k] assms by auto
ultimately show ?thesis using assms unfolding simps **
unfolding *(1)[of "\<lambda>i x. b$$i - x"] *(1)[of "\<lambda>i x. x - a$$i"] unfolding *(2)
apply(subst(2) euclidean_lambda_beta''[where 'a='a])
apply(subst(3) euclidean_lambda_beta''[where 'a='a]) by auto
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"
"k1 \<inter> {x::'a. x$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}"and k:"k<DIM('a)"
shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
proof- note d=division_ofD[OF assms(1)]
have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k \<le> c}) = {})"
unfolding interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
"k1 \<inter> {x::'a. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}" and k:"k<DIM('a)"
shows "content(k1 \<inter> {x. x$$k \<ge> c}) = 0"
proof- note d=division_ofD[OF assms(1)]
have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k >= c}) = {})"
unfolding interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
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 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$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}"
and k:"k<DIM('a)"
shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
apply(rule_tac[1-2] *) using assms(2-) by auto qed
lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}"
and k:"k<DIM('a)"
shows "content(k1 \<inter> {x. x$$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
show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
apply(rule_tac[1-2] *) using assms(2-) by auto qed
lemma division_split: fixes a::"'a::ordered_euclidean_space"
assumes "p division_of {a..b}" and k:"k<DIM('a)"
shows "{l \<inter> {x. x$$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<le> c} = {})} division_of({a..b} \<inter> {x. x$$k \<le> c})" (is "?p1 division_of ?I1") and
"{l \<inter> {x. x$$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$$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
{ 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
using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
{ fix k assume "k\<in>?p2" 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>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
qed
lemma has_integral_split: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "(f has_integral i) ({a..b} \<inter> {x. x$$k \<le> c})" "(f has_integral j) ({a..b} \<inter> {x. x$$k \<ge> c})" and k:"k<DIM('a)"
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]]
let ?d = "\<lambda>x. if x$$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$$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
fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
proof- fix x kk assume as:"(x,kk)\<in>p"
show "~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
proof(rule ccontr) case goal1
from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ k - c\<bar>"
using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
hence "\<exists>y. y \<in> ball x \<bar>x $$ k - c\<bar> \<inter> {x. x $$ k \<le> c}" using goal1(1) by blast
then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<le> c" apply-apply(rule le_less_trans)
using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
qed
show "~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
proof(rule ccontr) case goal1
from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ 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 $$ k - c\<bar> \<inter> {x. x $$ k \<ge> c}" using goal1(1) by blast
then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<ge> c" apply-apply(rule le_less_trans)
using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
qed
qed
have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
have lem3: "\<And>g::('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool. finite p \<Longrightarrow>
setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
= setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
apply(rule setsum_mono_zero_left) prefer 3
proof fix g::"('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" and i::"('a) \<times> (('a) set)"
assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
have "content (g k) = 0" using xk using content_empty by auto
thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
qed auto
have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
let ?M1 = "{(x,kk \<inter> {x. x$$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$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$$k \<le> c}" unfolding p(8)[THEN sym] by auto
fix x l assume xl:"(x,l)\<in>?M1"
then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
thus "l \<subseteq> d1 x" unfolding xl' by auto
show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
using lem0(1)[OF xl'(3-4)] by auto
show "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k,where c=c])
fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
proof(cases "l' = r' \<longrightarrow> x' = y'")
case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
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$$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$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$$k \<ge> c}" unfolding p(8)[THEN sym] by auto
fix x l assume xl:"(x,l)\<in>?M2"
then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
thus "l \<subseteq> d2 x" unfolding xl' by auto
show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
using lem0(2)[OF xl'(3-4)] by auto
show "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k, where c=c])
fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
proof(cases "l' = r' \<longrightarrow> x' = y'")
case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
qed qed ultimately
have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"
apply- apply(rule norm_triangle_lt) by auto
also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0" using scaleR_zero_left by auto
have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)
= (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto
also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) +
(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $$ 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[!] *)
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]
also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $$ 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 $$ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $$ 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
qed note setsum_cong2[OF this]
finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $$ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $$ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $$ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $$ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
(*lemma has_integral_split_cart: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})" "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"
shows "(f has_integral (i + j)) ({a..b})" *)
subsection {* A sort of converse, integrability on subintervals. *}
lemma tagged_division_union_interval: fixes a::"'a::ordered_euclidean_space"
assumes "p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> (c::real)})" "p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c})"
and k:"k<DIM('a)"
shows "(p1 \<union> p2) tagged_division_of ({a..b})"
proof- have *:"{a..b} = ({a..b} \<inter> {x. x$$k \<le> c}) \<union> ({a..b} \<inter> {x. x$$k \<ge> c})" by auto
show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms(1-2)])
unfolding interval_split[OF k] interior_closed_interval using k
by(auto simp add: eucl_less[where 'a='a] elim!:allE[where x=k]) qed
lemma has_integral_separate_sides: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "(f has_integral i) ({a..b})" "e>0" and k:"k<DIM('a)"
obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> c}) \<and> d fine p1 \<and>
p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c}) \<and> d fine p2
\<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
proof- guess d using has_integralD[OF assms(1-2)] . note d=this
show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $$ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
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$$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
moreover have "interior {x::'a. x $$ k = c} = {}"
proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x$$k = c}" by auto
then guess e unfolding mem_interior .. note e=this
have x:"x$$k = c" using x interior_subset by fastsimp
have *:"\<And>i. i<DIM('a) \<Longrightarrow> \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>
= (if i = k then e/2 else 0)" using e by auto
have "(\<Sum>i<DIM('a). \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>) =
(\<Sum>i<DIM('a). (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
finally have "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball dist_norm
by(rule le_less_trans[OF norm_le_l1])
hence "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> {x. x$$k = c}" using e by auto
thus False unfolding mem_Collect_eq using e x k by auto
qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
thus "content b *\<^sub>R f a = 0" by auto
qed auto
also have "\<dots> < e" by(rule k d(2) p12 fine_union p1 p2)+
finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
lemma integrable_split[intro]: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
assumes "f integrable_on {a..b}" and k:"k<DIM('a)"
shows "f integrable_on ({a..b} \<inter> {x. x$$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$$k \<ge> c})" (is ?t2)
proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
def b' \<equiv> "(\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)::'a"
and a' \<equiv> "(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i)::'a"
show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
\<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
show "?P {x. x $$ k \<le> 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 $$ k \<le> c} \<and> d fine p1
\<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c} \<and> d fine p2"
show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
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
show "?P {x. x $$ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p1
\<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p2"
show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
using p using assms by(auto simp add:algebra_simps) qed qed qed qed
subsection {* Generalized notion of additivity. *}
definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
"operative opp f \<equiv>
(\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
(\<forall>a b c. \<forall>k<DIM('b). f({a..b}) =
opp (f({a..b} \<inter> {x. x$$k \<le> c}))
(f({a..b} \<inter> {x. x$$k \<ge> c})))"
lemma operativeD[dest]: fixes type::"'a::ordered_euclidean_space" assumes "operative opp f"
shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b::'a} = neutral(opp)"
"\<And>a b c k. k<DIM('a) \<Longrightarrow> f({a..b}) = opp (f({a..b} \<inter> {x. x$$k \<le> c})) (f({a..b} \<inter> {x. x$$k \<ge> c}))"
using assms unfolding operative_def by auto
lemma operative_trivial:
"operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
unfolding operative_def by auto
lemma property_empty_interval:
"(\<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"
unfolding operative_def apply(rule property_empty_interval) by auto
subsection {* Using additivity of lifted function to encode definedness. *}
lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
by (metis option.nchotomy)
lemma exists_option:
"(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
by (metis option.nchotomy)
fun lifted where
"lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
"lifted opp None _ = (None::'b option)" |
"lifted opp _ None = None"
lemma lifted_simp_1[simp]: "lifted opp v None = None"
apply(induct v) by auto
definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and>
(\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
(\<forall>x. opp (neutral opp) x = x)"
lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
"\<And>x y z. opp x (opp y z) = opp (opp x y) z"
"\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
unfolding monoidal_def using assms by fastsimp
lemma monoidal_ac: assumes "monoidal opp"
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
"opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)"
using assms unfolding monoidal_def apply- by metis+
lemma monoidal_simps[simp]: assumes "monoidal opp"
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
using monoidal_ac[OF assms] by auto
lemma neutral_lifted[cong]: assumes "monoidal opp"
shows "neutral (lifted opp) = Some(neutral opp)"
apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
thus "x = Some (neutral opp)" apply(induct x) defer
apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
qed(auto simp add:monoidal_ac[OF assms])
lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
lemma support_clauses:
"\<And>f g s. support opp f {} = {}"
"\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
"\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
"\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
"\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
"\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
"\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
unfolding support_def by auto
lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
unfolding support_def by auto
lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
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))"
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
note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
show ?thesis proof(cases "f x = neutral opp")
case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
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 fun_left_comm.fold_insert[OF * finite_support])
using `finite s` unfolding support_def using False x by auto qed qed
lemma iterate_some:
assumes "monoidal opp" "finite s"
shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
proof(induct s) case empty thus ?case using assms by auto
next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
subsection {* Two key instances of additivity. *}
lemma neutral_add[simp]:
"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 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)
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)"
unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
apply(rule,rule,rule,rule) defer apply(rule allI impI)+
proof- fix a b c k assume k:"k<DIM('a)" 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 $$ k \<le> c} then Some (integral ({a..b} \<inter> {x. x $$ k \<le> c}) f) else None)
(if f integrable_on {a..b} \<inter> {x. c \<le> x $$ k} then Some (integral ({a..b} \<inter> {x. c \<le> x $$ k}) f) else None)"
proof(cases "f integrable_on {a..b}")
case True show ?thesis unfolding if_P[OF True] using k apply-
unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k])
apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x $$ k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x $$ k}))"
proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
apply(rule_tac x="integral ({a..b} \<inter> {x. x $$ k \<le> c}) f + integral ({a..b} \<inter> {x. x $$ k \<ge> c}) f" in exI)
apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
thus False using False by auto
qed thus ?thesis using False by auto
qed next
fix a b assume as:"content {a..b::'a} = 0"
thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
subsection {* Points of division of a partition. *}
definition "division_points (k::('a::ordered_euclidean_space) set) d =
{(j,x). j<DIM('a) \<and> (interval_lowerbound k)$$j < x \<and> x < (interval_upperbound k)$$j \<and>
(\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
lemma division_points_finite: fixes i::"('a::ordered_euclidean_space) set"
assumes "d division_of i" shows "finite (division_points i d)"
proof- note assm = division_ofD[OF assms]
let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$$j < x \<and> x < (interval_upperbound i)$$j \<and>
(\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
have *:"division_points i d = \<Union>(?M ` {..<DIM('a)})"
unfolding division_points_def by auto
show ?thesis unfolding * using assm by auto qed
lemma division_points_subset: fixes a::"'a::ordered_euclidean_space"
assumes "d division_of {a..b}" "\<forall>i<DIM('a). a$$i < b$$i" "a$$k < c" "c < b$$k" and k:"k<DIM('a)"
shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<le> c} = {})}
\<subseteq> division_points ({a..b}) d" (is ?t1) and
"division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})}
\<subseteq> division_points ({a..b}) d" (is ?t2)
proof- note assm = division_ofD[OF assms(1)]
have *:"\<forall>i<DIM('a). a$$i \<le> b$$i" "\<forall>i<DIM('a). a$$i \<le> ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i"
"\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i \<le> b$$i" "min (b $$ k) c = c" "max (a $$ k) c = c"
using assms using less_imp_le by auto
show ?t1 unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta'
proof- fix i l x assume as:"a $$ fst x < snd x" "snd x < (if fst x = k then c else b $$ fst x)"
"interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x"
"i = l \<inter> {x. x $$ k \<le> c}" "l \<in> d" "l \<inter> {x. x $$ k \<le> c} \<noteq> {}" and fstx:"fst x <DIM('a)"
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *:"\<forall>i<DIM('a). u $$ i \<le> ((\<chi>\<chi> i. if i = k then min (v $$ k) c else v $$ i)::'a) $$ i"
using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
show "fst x <DIM('a) \<and> a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x
\<or> interval_upperbound i $$ fst x = snd x)" apply(rule,rule fstx)
using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
apply(case_tac[!] "fst x = k") using assms fstx apply- unfolding euclidean_lambda_beta by auto
qed
show ?t2 unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta' apply(rule,assumption)
proof- fix i l x assume as:"(if fst x = k then c else a $$ fst x) < snd x" "snd x < b $$ fst x"
"interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x"
"i = l \<inter> {x. c \<le> x $$ k}" "l \<in> d" "l \<inter> {x. c \<le> x $$ k} \<noteq> {}" and fstx:"fst x < DIM('a)"
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *:"\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (u $$ k) c else u $$ i)::'a) $$ i \<le> v $$ i"
using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
show "a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x \<or>
interval_upperbound i $$ fst x = snd x)"
using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
apply(case_tac[!] "fst x = k") using assms fstx apply- by(auto simp add:euclidean_lambda_beta'[OF k]) qed qed
lemma division_points_psubset: fixes a::"'a::ordered_euclidean_space"
assumes "d division_of {a..b}" "\<forall>i<DIM('a). a$$i < b$$i" "a$$k < c" "c < b$$k"
"l \<in> d" "interval_lowerbound l$$k = c \<or> interval_upperbound l$$k = c" and k:"k<DIM('a)"
shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}
\<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
"division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}
\<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D")
proof- have ab:"\<forall>i<DIM('a). a$$i \<le> b$$i" using assms(2) by(auto intro!:less_imp_le)
guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
have uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "\<forall>i<DIM('a). a$$i \<le> u$$i \<and> v$$i \<le> b$$i"
using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
have *:"interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
"interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
have *:"interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
"interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
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. *}
lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
unfolding support_def by auto
lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
unfolding iterate_def support_support by auto
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
lemma iterate_image: assumes "monoidal opp" "inj_on f s"
shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
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 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
apply(cases "finite (support opp g (f ` s))")
apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
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)
apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
(* This lemma about iterations comes up in a few places. *)
lemma iterate_nonzero_image_lemma:
assumes "monoidal opp" "finite s" "g(a) = neutral opp"
"\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
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_image[OF assms(1)]) defer
apply(subst(2) iterate_support[THEN sym]) 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])
unfolding * using assms(1) by auto qed
lemma iterate_op: assumes "monoidal opp" "finite s"
shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
shows "iterate opp s f = iterate opp s g"
proof- have *:"support opp g s = support opp f s"
unfolding support_def using assms(2) by auto
show ?thesis
proof(cases "finite (support opp f s)")
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
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)]
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
lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
lemma operative_division: fixes f::"('a::ordered_euclidean_space) set \<Rightarrow> 'b"
assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
shows "iterate opp d f = f {a..b}"
proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
proof(induct C arbitrary:a b d rule:full_nat_induct)
case goal1
{ presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
thus ?case apply-apply(cases) defer apply assumption
proof- assume as:"content {a..b} = 0"
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)]
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<DIM('a). a$$i \<le> b$$i" by (auto intro!: less_imp_le) show ?case
proof(cases "division_points {a..b} d = {}")
case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
(\<forall>j<DIM('a). u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j)"
unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule,rule)
proof- fix u v j assume j:"j<DIM('a)" assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
hence uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "u$$j \<le> v$$j" using j unfolding interval_ne_empty by auto
have *:"\<And>p r Q. \<not> j<DIM('a) \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as j by auto
have "(j, u$$j) \<notin> division_points {a..b} d"
"(j, v$$j) \<notin> division_points {a..b} d" using True by auto
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
moreover have "a$$j \<le> u$$j" "v$$j \<le> b$$j" using division_ofD(2,2,3)[OF goal1(4) as]
unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
unfolding interval_ne_empty mem_interval using j by auto
ultimately show "u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j"
unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by auto
qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
have "{a..b} \<in> d"
proof- { presume "i = {a..b}" thus ?thesis using i by auto }
{ presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
show "u = a" "v = b" unfolding euclidean_eq[where 'a='a]
proof(safe) fix j assume j:"j<DIM('a)" note i(2)[unfolded uv mem_interval,rule_format,of j]
thus "u $$ j = a $$ j" "v $$ j = b $$ j" using uv(2)[rule_format,of j] j by auto
qed qed
hence *:"d = insert {a..b} (d - {{a..b}})" by auto
have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
then guess u v apply-by(erule exE conjE)+ note uv=this
have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
then obtain j where "u$$j \<noteq> a$$j \<or> v$$j \<noteq> b$$j" and j:"j<DIM('a)" unfolding euclidean_eq[where 'a='a] by auto
hence "u$$j = v$$j" using uv(2)[rule_format,OF j] by auto
hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) using j by auto
thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
by(erule exE conjE)+ note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
from this(3) guess j .. note j=this
def d1 \<equiv> "{l \<inter> {x. x$$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}"
def d2 \<equiv> "{l \<inter> {x. x$$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}"
def cb \<equiv> "(\<chi>\<chi> i. if i = k then c else b$$i)::'a" and ca \<equiv> "(\<chi>\<chi> i. if i = k then c else a$$i)::'a"
note division_points_psubset[OF goal1(4) ab kc(1-2) j]
note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x$$k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x$$k \<ge> c})"
apply- unfolding interval_split[OF kc(4)] apply(rule_tac[!] goal1(1)[rule_format])
using division_split[OF goal1(4), where k=k and c=c]
unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto
also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$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 $$ k \<le> c} = y \<inter> {x. x $$ 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 $$ k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$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 $$ k} = y \<inter> {x. c \<le> x $$ 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 $$ k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x $$ k \<le> c})) (f (x \<inter> {x. c \<le> x $$ k}))"
unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto
have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x $$ k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x $$ 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
finally show ?thesis by auto
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"
shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
case goal1 show ?case using assms(1) by auto
next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
apply(rule finite_imageI goal2)+
apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
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
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)+
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}"
using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
finally show ?thesis . qed
subsection {* Additivity of content. *}
lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
proof- have *:"setsum f s = setsum f (support op + f s)"
apply(rule setsum_mono_zero_right)
unfolding support_def neutral_monoid using assms by auto
thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
unfolding neutral_monoid . qed
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]
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]
apply(subst setsum_iterate) using assms by auto
subsection {* Finally, the integral of a constant *}
lemma has_integral_const[intro]:
"((\<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])
defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
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 setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
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)
apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
thus "0 \<le> content x" using content_pos_le by auto
qed(insert assms,auto)
lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
proof(cases "{a..b} = {}") case True
show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
next case False show ?thesis
apply(rule order_trans,rule setsum_norm) defer 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)
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
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]
from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
show "norm (f (fst xk)) \<le> e" unfolding xk fst_conv using tagged_division_ofD(2,3)[OF assms(1) xk(2)] assms(2) by auto
qed(insert assms,auto) qed
lemma rsum_diff_bound:
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
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"
shows "norm i \<le> B * content {a..b}"
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
show ?thesis unfolding * ** using assms(1) by auto
qed auto } assume ab:?P
{ presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
from fine_division_exists[OF this(1), of a b] guess p . note p=this
have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
proof- case goal1 thus ?case unfolding not_less
using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
subsection {* Similar theorems about relationship among components. *}
lemma rsum_component_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$$i \<le> (g x)$$i"
shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$$i"
unfolding euclidean_component.setsum apply(rule setsum_mono) apply safe
proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
from this(3) guess u v apply-by(erule exE)+ note b=this
show "(content b *\<^sub>R f a) $$ i \<le> (content b *\<^sub>R g a) $$ i" unfolding b
unfolding euclidean_simps real_scaleR_def apply(rule mult_left_mono)
defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
lemma has_integral_component_le: fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
shows "i$$k \<le> j$$k"
proof- have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
(g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$$k \<le> (g x)$$k \<Longrightarrow> i$$k \<le> j$$k"
proof(rule ccontr) case goal1 hence *:"0 < (i$$k - j$$k) / 3" by auto
guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k] term g
note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
thus False unfolding euclidean_simps using rsum_component_le[OF p(1) goal1(3)] apply simp by smt
qed let ?P = "\<exists>a b. s = {a..b}"
{ presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
case True then guess a b apply-by(erule exE)+ note s=this
show ?thesis apply(rule lem) using assms[unfolded s] by auto
qed auto } assume as:"\<not> ?P"
{ presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
assume "\<not> i$$k \<le> j$$k" hence ij:"(i$$k - j$$k) / 3 > 0" by auto
note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
have "bounded (ball 0 B1 \<union> ball (0::'a) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" by smt
note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
have "w1$$k \<le> w2$$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
show False unfolding euclidean_simps by(rule *) qed
lemma integral_component_le: fixes g f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
shows "(integral s f)$$k \<le> (integral s g)$$k"
apply(rule has_integral_component_le) using integrable_integral assms by auto
(*lemma has_integral_dest_vec1_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> real^1"
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
using assms(3) unfolding vector_le_def by auto
lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto*)
lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> i$$k"
using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2-) by auto
lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> (integral s f)$$k"
apply(rule has_integral_component_nonneg) using assms by auto
(*lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
using has_integral_component_nonneg[OF assms(1), of 1]
using assms(2) unfolding vector_le_def by auto
lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
apply(rule has_integral_dest_vec1_nonneg) using assms by auto*)
lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$$k \<le> 0"shows "i$$k \<le> 0"
using has_integral_component_le[OF assms(1) has_integral_0] assms(2-) by auto
(*lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto*)
lemma has_integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)" shows "B * content {a..b} \<le> i$$k"
using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi>\<chi> i. B)::'b" k] assms(2-)
unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] by(auto simp add:field_simps)
lemma has_integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$$k \<le> B" "k<DIM('b)"
shows "i$$k \<le> B * content({a..b})"
using has_integral_component_le[OF assms(1) has_integral_const, of k "\<chi>\<chi> i. B"]
unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] using assms(2) by(auto simp add:field_simps)
lemma integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)"
shows "B * content({a..b}) \<le> (integral({a..b}) f)$$k"
apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$$k \<le> B" "k<DIM('b)"
shows "(integral({a..b}) f)$$k \<le> B * content({a..b})"
apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
subsection {* Uniform limit of integrable functions is integrable. *}
lemma integrable_uniform_limit: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
shows "f integrable_on {a..b}"
proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
show ?thesis apply cases apply(rule *,assumption)
unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
assume as:"content {a..b} > 0"
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)
then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e" apply(rule_tac x=M in exI,rule,rule,rule,rule)
proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
have lem2:"\<And>s1 s2 i1 i2. norm(s2 - s1) \<le> e/2 \<Longrightarrow> norm(s1 - i1) < e / 4 \<Longrightarrow> norm(s2 - i2) < e / 4 \<Longrightarrow>norm(i1 - i2) < e"
proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps)
also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
finally show ?case .
qed
show ?case unfolding dist_norm apply(rule lem2) defer
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
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)"
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
also have "\<dots> = 2 / real M" unfolding divide_inverse by auto
finally show "norm (g n x - g m x) \<le> 2 / real M"
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
show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
proof(rule,rule)
case goal1 hence *:"e/3 > 0" by auto
from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
have lem:"\<And>sf sg i. norm(sf - sg) \<le> e / 3 \<Longrightarrow> norm(i - s) < e / 3 \<Longrightarrow> norm(sg - i) < e / 3 \<Longrightarrow> norm(sf - s) < e"
proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
using norm_triangle_ineq[of "sf - sg" "sg - s"]
using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:algebra_simps)
also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
finally show ?case .
qed
show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
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
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,unfolded dist_norm],auto)
qed qed qed qed
subsection {* Negligible sets. *}
definition "negligible (s::('a::ordered_euclidean_space) set) \<equiv> (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) {a..b})"
subsection {* Negligibility of hyperplane. *}
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
lemma interval_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)"
shows "{a..b} \<inter> {x . abs(x$$k - c) \<le> (e::real)} =
{(\<chi>\<chi> i. if i = k then max (a$$k) (c - e) else a$$i) .. (\<chi>\<chi> i. if i = k then min (b$$k) (c + e) else b$$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
show ?thesis unfolding * ** interval_split[OF assms] by(rule refl) qed
lemma division_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "p division_of {a..b}" and k:"k<DIM('a)"
shows "{l \<inter> {x. abs(x$$k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x$$k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x$$k - c) \<le> e})"
proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
note division_split(2)[OF this, where c="c-e" and k=k,OF k]
thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $$ k}" in exI) apply rule defer apply rule
apply(rule_tac x=l in exI) by blast+ qed
lemma content_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "0 < e" and k:"k<DIM('a)"
obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$$k - c) \<le> d}) < e"
proof(cases "content {a..b} = 0")
case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
unfolding interval_doublesplit[THEN sym,OF k] using assms by auto
next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})"
note False[unfolded content_eq_0 not_ex not_le, rule_format]
hence "\<And>x. x<DIM('a) \<Longrightarrow> b$$x > a$$x" by(auto simp add:not_le)
hence prod0:"0 < setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
proof(rule that[of d]) have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" using k by auto
have **:"{a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
(\<Prod>i\<in>{..<DIM('a)} - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i
- interval_lowerbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i)
= (\<Prod>i\<in>{..<DIM('a)} - {k}. b$$i - a$$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 $$ k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
unfolding interval_doublesplit[OF k] interval_eq_empty not_ex not_less prefer 3
apply(subst interval_bounds) defer apply(subst interval_bounds) unfolding euclidean_lambda_beta'[OF k] if_P[OF refl]
proof- have "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) \<le> 2 * d" by auto
also have "... < e / (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
finally show "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) * (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i) < e"
unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
lemma negligible_standard_hyperplane[intro]: fixes type::"'a::ordered_euclidean_space" assumes k:"k<DIM('a)"
shows "negligible {x::'a. x$$k = (c::real)}"
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
let ?i = "indicator {x::'a. x$$k = c} :: 'a\<Rightarrow>real"
show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$$k - c) \<le> d}) *\<^sub>R ?i x)"
apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
apply(cases,rule disjI1,assumption,rule disjI2)
proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
show "content l = content (l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
thus "\<bar>y $$ k - c\<bar> \<le> d" unfolding euclidean_simps xk by auto
qed auto qed
note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e" unfolding diff_0_right * unfolding real_scaleR_def real_norm_def
apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
prefer 2 apply(subst(asm) eq_commute) apply assumption
apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}))"
apply(rule setsum_mono) unfolding split_paired_all split_conv
apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k] intro!:content_pos_le)
also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $$ 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
thus ?case unfolding goal1 unfolding interval_doublesplit[OF k] using content_pos_le by smt
next have *:"setsum content {l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $$ 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
proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})) < e"
apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
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 $$ k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}"
have "({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
hence "interior ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
thus "content ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
qed qed
finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) < e" .
qed qed qed
subsection {* A technical lemma about "refinement" of division. *}
lemma tagged_division_finer: fixes p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set"
assumes "p tagged_division_of {a..b}" "gauge d"
obtains q where "q tagged_division_of {a..b}" "d fine q" "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
proof-
let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
(\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
(\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
{ have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
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)
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]
from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
have *:"\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}" unfolding xk by auto
note p = tagged_partial_division_ofD[OF insert(4)]
from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
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)
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,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
next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
qed qed qed
subsection {* Hence the main theorem about negligible sets. *}
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)
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
lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s = setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}" using assms
proof(induct) case (insert a s)
have *:"{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} = (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
prefer 4 apply(subst insert(3)) unfolding add_right_cancel
proof- show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in>Pair a ` t a. x xa y)" apply(subst setsum_reindex) unfolding inj_on_def by auto
show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
qed(insert insert, auto) qed auto
lemma has_integral_negligible: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
shows "(f has_integral 0) t"
proof- presume P:"\<And>f::'b::ordered_euclidean_space \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
next show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) {a..b} \<and> norm (z - 0) < e)"
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"
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"
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"]
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)
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"
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 ..
hence N:"\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N" apply(subst(asm) Sup_finite_le_iff) using as as' by auto
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)
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
apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
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 setsum_norm) defer 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)
using tagged_division_ofD(4)[OF q(1) as''] by auto
next fix i::nat show "finite (q i)" using q by auto
next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
proof(cases "x\<in>s") case False thus ?thesis using assm by auto
next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
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])
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]
apply(subst sumr_geometric) using goal1 by auto
finally show "?goal" by auto qed qed qed
lemma has_integral_spike: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
shows "(g has_integral y) t"
proof- { fix a b::"'b" and f g ::"'b \<Rightarrow> 'a" and y::'a
assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
apply(rule has_integral_negligible[OF assms(1)]) using as by auto
hence "(g has_integral y) {a..b}" by auto } note * = this
show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
apply(erule_tac x=e in allE,safe,rule_tac x=B in exI,safe) apply(erule_tac x=a in allE,erule_tac x=b in allE,safe)
apply(rule_tac x=z in exI,safe) apply(rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"]) by auto qed
lemma has_integral_spike_eq:
assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
shows "g integrable_on t"
using assms unfolding integrable_on_def apply-apply(erule exE)
apply(rule,rule has_integral_spike) by fastsimp+
lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
shows "integral t f = integral t g"
unfolding integral_def using has_integral_spike_eq[OF assms] by auto
subsection {* Some other trivialities about negligible sets. *}
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
lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
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
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
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 0 "a$$0"] by auto
lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
apply(subst insert_is_Un) unfolding negligible_union_eq by auto
lemma negligible_empty[intro]: "negligible {}" by auto
lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
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)
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)
proof- fix t::"'a set" assume as:"negligible s"
have *:"(\<lambda>x. if x \<in> s \<inter> t then 1 else 0) = (\<lambda>x. if x\<in>t then if x\<in>s then 1 else 0 else 0)" by(rule ext,auto)
show "((indicator s::'a\<Rightarrow>real) has_integral 0) t" apply(subst has_integral_alt)
apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def_raw unfolding * by auto qed auto
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"
"(f has_integral y) t" shows "(g has_integral y) t"
apply(rule has_integral_spike) using assms by auto
lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
lemma integrable_spike_finite:
assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t"
using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
apply(rule has_integral_spike_finite) by auto
subsection {* In particular, the boundary of an interval is negligible. *}
lemma negligible_frontier_interval: "negligible({a::'a::ordered_euclidean_space..b} - {a<..<b})"
proof- let ?A = "\<Union>((\<lambda>k. {x. x$$k = a$$k} \<union> {x::'a. x$$k = b$$k}) ` {..<DIM('a)})"
have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
apply(erule conjE exE)+ apply(rule_tac X="{x. x $$ xa = a $$ xa} \<union> {x. x $$ xa = b $$ xa}" in UnionI)
apply(erule_tac[!] x=xa in allE) by auto
thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
lemma has_integral_spike_interior:
assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
lemma has_integral_spike_interior_eq:
assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
lemma integrable_spike_interior: assumes "\<forall>x\<in>{a<..<b}. g x = f x" "f integrable_on {a..b}" shows "g integrable_on {a..b}"
using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
subsection {* Integrability of continuous functions. *}
lemma neutral_and[simp]: "neutral op \<and> = True"
unfolding neutral_def apply(rule some_equality) by auto
lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
apply induct unfolding iterate_insert[OF monoidal_and] by auto
lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
lemma operative_approximable: assumes "0 \<le> e" fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
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}"
apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
{ fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k<DIM('b)"
show "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
"\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2) k] by auto }
fix c k g1 g2 assume as:"\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
"\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g2 x) \<le> e" "g2 integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
assume k:"k<DIM('b)"
let ?g = "\<lambda>x. if x$$k = c then f x else if x$$k \<le> c then g1 x else g2 x"
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$$k=c", case_tac "x$$k < c") using as assms by auto
next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
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 $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
lemma approximable_on_division: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "0 \<le> e" "d division_of {a..b}" "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
lemma integrable_continuous: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
note d=conjunctD2[OF this,rule_format]
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"
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)
fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
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 fastsimp 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
subsection {* Specialization of additivity to one dimension. *}
lemma operative_1_lt: assumes "monoidal opp"
shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
(\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
unfolding operative_def content_eq_0 DIM_real less_one dnf_simps(39,41) Eucl_real_simps
(* The dnf_simps simplify "\<exists> x. x= _ \<and> _" and "\<forall>k. k = _ \<longrightarrow> _" *)
proof safe fix a b c::"real" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c}))
(f ({a..b} \<inter> {x. c \<le> x}))" "a < c" "c < b"
from this(2-) have "{a..b} \<inter> {x. x \<le> c} = {a..c}" "{a..b} \<inter> {x. x \<ge> c} = {c..b}" by auto
thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c"] by auto
next fix a b c::real
assume as:"\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}"
show "f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c})) (f ({a..b} \<inter> {x. c \<le> x}))"
proof(cases "c \<in> {a .. b}")
case False hence "c<a \<or> c>b" by auto
thus ?thesis apply-apply(erule disjE)
proof- assume "c<a" hence *:"{a..b} \<inter> {x. x \<le> c} = {1..0}" "{a..b} \<inter> {x. c \<le> x} = {a..b}" by auto
show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
next assume "b<c" hence *:"{a..b} \<inter> {x. x \<le> c} = {a..b}" "{a..b} \<inter> {x. c \<le> x} = {1..0}" by auto
show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
qed
next case True hence *:"min (b) c = c" "max a c = c" by auto
have **:"0 < DIM(real)" by auto
have ***:"\<And>P Q. (\<chi>\<chi> i. if i = 0 then P i else Q i) = (P 0::real)" apply(subst euclidean_eq)
apply safe unfolding euclidean_lambda_beta' by auto
show ?thesis unfolding interval_split[OF **,unfolded Eucl_real_simps(1,3)] unfolding *** *
proof(cases "c = a \<or> c = b")
case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
apply-apply(subst as(2)[rule_format]) using True by auto
next case True thus "f {a..b} = opp (f {a..c}) (f {c..b})" apply-
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 qed
lemma operative_1_le: assumes "monoidal opp"
shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
(\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
unfolding operative_1_lt[OF assms]
proof safe fix a b c::"real" assume as:"\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a < c" "c < b"
show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) by auto
next fix a b c ::"real" assume "\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp"
"\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a \<le> c" "c \<le> b"
note as = this[rule_format]
show "opp (f {a..c}) (f {c..b}) = f {a..b}"
proof(cases "c = a \<or> c = b")
case False thus ?thesis apply-apply(subst as(2)) using as(3-) by(auto)
next case True thus ?thesis apply-
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
subsection {* Special case of additivity we need for the FCT. *}
lemma interval_bound_sing[simp]: "interval_upperbound {a} = a" "interval_lowerbound {a} = a"
unfolding interval_upperbound_def interval_lowerbound_def by auto
lemma additive_tagged_division_1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
assumes "a \<le> b" "p tagged_division_of {a..b}"
shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
proof- let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
have ***:"\<forall>i<DIM(real). a $$ i \<le> b $$ 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
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. *}
lemma has_integral_factor_content:
"(f has_integral i) {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 - 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(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]]
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"
{ assume "\<forall>e>0. ?P e op <" thus "?P (e * content {a..b}) op \<le>" apply(erule_tac x="e * content {a..b}" in allE)
apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
using F e by(auto simp add:field_simps intro:mult_pos_pos) }
{ assume "\<forall>e>0. ?P (e * content {a..b}) op \<le>" thus "?P e op <" apply(erule_tac x="e / 2 / content {a..b}" in allE)
apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
subsection {* Fundamental theorem of calculus. *}
lemma interval_bounds_real: assumes "a\<le>(b::real)"
shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
apply(rule_tac[!] interval_bounds) using assms by auto
lemma fundamental_theorem_of_calculus: fixes f::"real \<Rightarrow> 'a::banach"
assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_vector_derivative f' x) (at x within {a..b})"
shows "(f' has_integral (f b - f a)) ({a..b})"
unfolding has_integral_factor_content
proof safe fix e::real assume e:"e>0"
note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
have *:"\<And>P Q. \<forall>x\<in>{a..b}. P x \<and> (\<forall>e>0. \<exists>d>0. Q x e d) \<Longrightarrow> \<forall>x. \<exists>(d::real)>0. x\<in>{a..b} \<longrightarrow> Q x e d" using e by blast
note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
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) - (f b - f a)) \<le> e * content {a..b})"
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]
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
have ball:"\<forall>xa\<in>k. xa \<in> ball x (d x)" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,
unfolded split_conv subset_eq] .
have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
unfolding scaleR.diff_left by(auto simp add:algebra_simps)
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)
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>
e * (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bounds_real[OF *] content_real[OF *] .
qed(insert as, auto) qed qed
subsection {* Attempt a systematic general set of "offset" results for components. *}
lemma gauge_modify:
assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
shows "gauge (\<lambda>x y. d (f x) (f y))"
using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
subsection {* Only need trivial subintervals if the interval itself is trivial. *}
lemma division_of_nontrivial: fixes s::"('a::ordered_euclidean_space) set set"
assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
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}"
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 }
assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
then obtain k where k:"k\<in>s" "content k = 0" by auto
from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
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 ..
hence i:"c$$i = d$$i" "i<DIM('a)" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
hence xi:"x$$i = d$$i" using as unfolding k mem_interval by smt
def y \<equiv> "(\<chi>\<chi> j. if j = i then if c$$i \<le> (a$$i + b$$i) / 2 then c$$i +
min e (b$$i - c$$i) / 2 else c$$i - min e (c$$i - a$$i) / 2 else x$$j)::'a"
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(fastsimp 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 spec[where x=i]]
hence xyi:"y$$i \<noteq> x$$i" unfolding y_def unfolding i xi euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2)
using assms(2)[unfolded content_eq_0] using i(2) by smt+
thus "y \<noteq> x" unfolding euclidean_eq[where 'a='a] using i by auto
have *:"{..<DIM('a)} = insert i ({..<DIM('a)} - {i})" using i by auto
have "norm (y - x) < e + setsum (\<lambda>i. 0) {..<DIM('a)}" apply(rule le_less_trans[OF norm_le_l1])
apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
proof- show "\<bar>(y - x) $$ i\<bar> < e" unfolding y_def euclidean_simps euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
show "(\<Sum>i\<in>{..<DIM('a)} - {i}. \<bar>(y - x) $$ i\<bar>) \<le> (\<Sum>i\<in>{..<DIM('a)}. 0)" unfolding y_def by auto
qed auto thus "dist y x < e" unfolding dist_norm by auto
have "y\<notin>k" unfolding k mem_interval apply rule apply(erule_tac x=i in allE) using xyi unfolding k i xi by auto
moreover have "y \<in> \<Union>s" unfolding s mem_interval
proof(rule,rule) note simps = y_def euclidean_lambda_beta' if_not_P
fix j assume j:"j<DIM('a)" show "a $$ j \<le> y $$ j \<and> y $$ j \<le> b $$ j"
proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
thus ?thesis using j apply- unfolding simps if_not_P[OF False] unfolding mem_interval by auto
next case True note T = this show ?thesis
proof(cases "c $$ i \<le> (a $$ i + b $$ i) / 2")
case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
qed qed qed
ultimately show "y \<in> \<Union>(s - {k})" by auto
qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] 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
subsection {* Integrabibility on subintervals. *}
lemma operative_integrable: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
"operative op \<and> (\<lambda>i. f integrable_on i)"
unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
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}"
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
subsection {* Combining adjacent intervals in 1 dimension. *}
lemma has_integral_combine: assumes "(a::real) \<le> c" "c \<le> b"
"(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
shows "(f has_integral (i + j)) {a..b}"
proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
apply(subst(asm) if_P) using assms(3-) by auto
with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
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_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
lemma integrable_combine: fixes f::"real \<Rightarrow> 'a::banach"
assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
subsection {* Reduce integrability to "local" integrability. *}
lemma integrable_on_little_subintervals: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "\<forall>x\<in>{a..b}. \<exists>d>0. \<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}"
shows "f integrable_on {a..b}"
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]
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
subsection {* Second FCT or existence of antiderivative. *}
lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
unfolding integrable_on_def by(rule,rule has_integral_const)
lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
assumes "continuous_on {a..b} f" "x \<in> {a..b}"
shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
unfolding has_vector_derivative_def has_derivative_within_alt
apply safe apply(rule scaleR.bounded_linear_left)
proof- fix e::real assume e:"e>0"
note compact_uniformly_continuous[OF assms(1) compact_interval,unfolded uniformly_continuous_on_def]
from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
let ?I = "\<lambda>a b. integral {a..b} f"
show "\<exists>d>0. \<forall>y\<in>{a..b}. norm (y - x) < d \<longrightarrow> norm (?I a y - ?I a x - (y - x) *\<^sub>R f x) \<le> e * norm (y - x)"
proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
case False have "f integrable_on {a..y}" apply(rule integrable_subinterval,rule integrable_continuous)
apply(rule assms) unfolding not_less using assms(2) goal1 by auto
hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
using False unfolding not_less using assms(2) goal1 by auto
have **:"norm (y - x) = content {x..y}" apply(subst content_real) using False unfolding not_less by auto
show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
defer apply(rule has_integral_sub) apply(rule integrable_integral)
apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
proof- show "{x..y} \<subseteq> {a..b}" using goal1 assms(2) by auto
have *:"y - x = norm(y - x)" using False by auto
show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {x.. y}" apply(subst *) unfolding ** by auto
show "\<forall>xa\<in>{x..y}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
qed(insert e,auto)
next case True have "f integrable_on {a..x}" apply(rule integrable_subinterval,rule integrable_continuous)
apply(rule assms)+ unfolding not_less using assms(2) goal1 by auto
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
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
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
show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {y..x}" apply(subst *) unfolding ** by auto
show "\<forall>xa\<in>{y..x}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
qed(insert e,auto) qed qed qed
lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
subsection {* Combined fundamental theorem of calculus. *}
lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u..v}"
proof- from antiderivative_continuous[OF assms] guess g . note g=this
show ?thesis apply(rule that[of g])
proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g" "f"] by auto qed qed
subsection {* General "twiddling" for interval-to-interval function image. *}
lemma has_integral_twiddle:
assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
"\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
"\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
"\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
"(f has_integral i) {a..b}"
shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
show ?thesis apply cases defer apply(rule *,assumption)
proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
def d' \<equiv> "\<lambda>x y. d (g x) (g y)" have d':"\<And>x. d' x = {y. g y \<in> (d (g x))}" unfolding d'_def by(auto simp add:mem_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
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
show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
{ fix y assume "y \<in> k" thus "g y \<in> {a..b}" "g y \<in> {a..b}" using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
unfolding image_Int[OF inj(1)] by auto thus False using as by blast
qed thus "g x = g x'" by auto
{ fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto }
{ fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto }
next fix x assume "x \<in> {a..b}" hence "h x \<in> \<Union>{k. \<exists>x. (x, k) \<in> p}" using p(6) by auto
then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
using X(2) assms(3)[rule_format,of x] by auto
qed note ** = d(2)[OF this] have *:"inj_on (\<lambda>(x, k). (g x, g ` k)) p" using inj(1) unfolding inj_on_def by fastsimp
have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding algebra_simps add_left_cancel
unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
also have "... = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r") unfolding scaleR.diff_right scaleR.scaleR_left[THEN sym]
unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e" using ** unfolding * unfolding norm_scaleR
using assms(1) by(auto simp add:field_simps) qed qed qed
subsection {* Special case of a basic affine transformation. *}
lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space) + c) ` {a..b} = {u..v}"
unfolding image_affinity_interval by auto
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:
"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 }
have *:"DIM('a) = card {..<DIM('a)}" by auto
assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
apply(rule setprod_cong2) using True as unfolding interval_ne_empty euclidean_simps not_le
by(auto simp add:field_simps intro:mult_left_mono)
next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
apply(rule setprod_cong2) using False as unfolding interval_ne_empty euclidean_simps not_le
by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
lemma has_integral_affinity: fixes a::"'a::ordered_euclidean_space" assumes "(f has_integral i) {a..b}" "m \<noteq> 0"
shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
apply(rule has_integral_twiddle,safe) apply(rule zero_less_power) unfolding euclidean_eq[where 'a='a]
unfolding scaleR_right_distrib euclidean_simps scaleR.scaleR_left[THEN sym]
defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` {a..b})"
using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
subsection {* Special case of stretching coordinate axes separately. *}
lemma image_stretch_interval:
"(\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} =
(if {a..b} = {} then {} else {(\<chi>\<chi> k. min (m(k) * a$$k) (m(k) * b$$k))::'a .. (\<chi>\<chi> k. max (m(k) * a$$k) (m(k) * b$$k))})"
(is "?l = ?r")
proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
next have *:"\<And>P Q. (\<forall>i<DIM('a). P i) \<and> (\<forall>i<DIM('a). Q i) \<longleftrightarrow> (\<forall>i<DIM('a). P i \<and> Q i)" by auto
case False note ab = this[unfolded interval_ne_empty]
show ?thesis apply-apply(rule set_ext)
proof- fix x::"'a" have **:"\<And>P Q. (\<forall>i<DIM('a). P i = Q i) \<Longrightarrow> (\<forall>i<DIM('a). P i) = (\<forall>i<DIM('a). Q i)" by auto
show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
unfolding image_iff mem_interval Bex_def euclidean_simps euclidean_eq[where 'a='a] *
unfolding imp_conjR[THEN sym] apply(subst euclidean_lambda_beta'') apply(subst lambda_skolem'[THEN sym])
apply(rule **,rule,rule) unfolding euclidean_lambda_beta'
proof- fix i assume i:"i<DIM('a)" show "(\<exists>xa. (a $$ i \<le> xa \<and> xa \<le> b $$ i) \<and> x $$ i = m i * xa) =
(min (m i * a $$ i) (m i * b $$ i) \<le> x $$ i \<and> x $$ i \<le> max (m i * a $$ i) (m i * b $$ i))"
proof(cases "m i = 0") case True thus ?thesis using ab i by auto
next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
"max (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab i unfolding min_def max_def by auto
show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
using as by(auto simp add:field_simps)
next assume as:"0 > m i" hence *:"max (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
"min (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab as i unfolding min_def max_def
by(auto simp add:field_simps mult_le_cancel_left_neg intro: order_antisym)
show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
using as by(auto simp add:field_simps) qed qed qed qed qed
lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
unfolding image_stretch_interval by auto
lemma content_image_stretch_interval:
"content((\<lambda>x::'a::ordered_euclidean_space. (\<chi>\<chi> k. m k * x$$k)::'a) ` {a..b}) = abs(setprod m {..<DIM('a)}) * content({a..b})"
proof(cases "{a..b} = {}") case True thus ?thesis
unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
next case False hence "(\<lambda>x. (\<chi>\<chi> k. m k * x $$ 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 euclidean_lambda_beta'
proof- fix i assume i:"i<DIM('a)" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
thus "max (m i * a $$ i) (m i * b $$ i) - min (m i * a $$ i) (m i * b $$ i) = \<bar>m i\<bar> * (b $$ i - a $$ 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"
assumes "(f has_integral i) {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
shows "((\<lambda>x. f(\<chi>\<chi> k. m k * x$$k)) has_integral
((1/(abs(setprod m {..<DIM('a)}))) *\<^sub>R i)) ((\<lambda>x. (\<chi>\<chi> k. 1/(m k) * x$$k)::'a) ` {a..b})"
apply(rule has_integral_twiddle[where f=f]) unfolding zero_less_abs_iff content_image_stretch_interval
unfolding image_stretch_interval empty_as_interval euclidean_eq[where 'a='a] using assms
proof- show "\<forall>y::'a. continuous (at y) (\<lambda>x. (\<chi>\<chi> k. m k * x $$ k)::'a)"
apply(rule,rule linear_continuous_at) unfolding linear_linear
unfolding linear_def euclidean_simps euclidean_eq[where 'a='a] by(auto simp add:field_simps) qed auto
lemma integrable_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
assumes "f integrable_on {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
shows "(\<lambda>x::'a. f(\<chi>\<chi> k. m k * x$$k)) integrable_on ((\<lambda>x. \<chi>\<chi> k. 1/(m k) * x$$k) ` {a..b})"
using assms unfolding integrable_on_def apply-apply(erule exE)
apply(drule has_integral_stretch,assumption) by auto
subsection {* even more special cases. *}
lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::'a::ordered_euclidean_space}"
apply(rule set_ext,rule) defer unfolding image_iff
apply(rule_tac x="-x" in bexI) by(auto simp add:minus_le_iff le_minus_iff eucl_le[where 'a='a])
lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
using has_integral_affinity[OF assms, of "-1" 0] by auto
lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
unfolding integrable_on_def by auto
lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
unfolding integral_def by auto
subsection {* Stronger form of FCT; quite a tedious proof. *}
lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)" by(meson zero_less_one)
lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
assumes "a \<le> b" "p tagged_division_of {a..b}"
shows "setsum (\<lambda>(x,k). f (interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
using additive_tagged_division_1[OF _ assms(2), of f] using assms(1) by auto
lemma split_minus[simp]:"(\<lambda>(x, k). f x k) x - (\<lambda>(x, k). g x k) x = (\<lambda>(x, k). f x k - g x k) x"
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(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"
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)
show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0 using * `a=b` by auto
qed } assume ab:"a < b"
let ?P = "\<lambda>e. \<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) - (f b - f a)) \<le> e * content {a..b})"
{ presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
fix e::real assume e:"e>0"
note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
note conjunctD2[OF this] note bounded=this(1) and this(2)
from this(2) have "\<forall>x\<in>{a<..<b}. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
apply-apply safe apply(erule_tac x=x in ballE,erule_tac x="e/2" in allE) using e by auto note this[unfolded bgauge_existence_lemma]
from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_interval assms by auto
from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
\<longrightarrow> norm(content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
proof- have "a\<in>{a..b}" using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
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)
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)
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)"
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"
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
apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
qed finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow>
norm(content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
proof- have "b\<in>{a..b}" using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
using e ab by(auto simp add:field_simps)
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
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)"
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"
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)
apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
qed finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
show "?P e" apply(rule_tac x="?d" in exI)
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]
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
unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
proof(rule norm_triangle_le,rule **)
case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
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
< norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k)))"
from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
hence "u \<le> v" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto
note result = as(2)[unfolded k interval_bounds_real[OF this(1)] content_real[OF this(1)]]
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
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)
also have "... \<le> e / 2 * norm (v - u)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
finally have "e * (v - u) / 2 < e * (v - u) / 2"
apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
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]
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
with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
thus "0 \<le> e * ((interval_upperbound k) - (interval_lowerbound k))"
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"
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}"
hence xk:"(x,k)\<in>p" "content k = 0" by auto from p(4)[OF xk(1)] guess u v apply-by(erule exE)+ note uv=this
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} =
{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"
proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
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 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"
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 \<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"
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 "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:)
qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
qed
show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<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:"( a, k) \<in> p" "( a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (min (v) (v'))"
have "{ a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
moreover have " ((a + ?v)/2) \<in> { a <..< ?v}" using k(3-)
unfolding v v' content_eq_0 not_le by(auto simp add:not_le)
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
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"
guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (max (v) (v'))"
have "{?v <..< b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
moreover have " ((b + ?v)/2) \<in> {?v <..< b}" using k(3-) unfolding v v' content_eq_0 not_le by auto
ultimately have " ((b + ?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
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
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)]
apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x=" x" in ballE)
by(auto simp add:subset_eq dist_real_def v) ultimately
show ?case unfolding v interval_bounds_real[OF v(2)] apply- apply(rule da(2)[of "v"])
using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
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
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
moreover have "{v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe
apply(erule_tac x=" x" in ballE) using ab
by(auto simp add:subset_eq v dist_real_def) ultimately
show ?case unfolding v unfolding interval_bounds_real[OF v(2)] apply- apply(rule db(2)[of "v"])
using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
qed
qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
subsection {* Stronger form with finite number of exceptional points. *}
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-
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
apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
show ?case proof(cases "c\<in>{a<..<b}")
case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
case True hence "a \<le> c" "c \<le> b" by auto
thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
qed auto qed qed
lemma fundamental_theorem_of_calculus_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}"
apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
using assms(4) by auto
lemma indefinite_integral_continuous_left: fixes f::"real \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e"
obtains d where "0 < d" "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e"
proof- have "\<exists>w>0. \<forall>t. c - w < t \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3"
proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)"
apply-apply(rule divide_pos_pos) using `e>0` by auto
thus ?thesis apply-apply(rule,rule,assumption,safe)
proof- fix t assume as:"t < c" and "c - e / 3 / norm (f c) < t"
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
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 ..
note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x"
have "gauge d" unfolding d_def using w(1) d1 by auto
note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this]
from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
let ?d = "min k (c - a)/2" show ?thesis apply(rule that[of ?d])
proof safe show "?d > 0" using k(1) using assms(2) by auto
fix t assume as:"c - ?d < t" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
{ presume *:"t < c \<Longrightarrow> ?thesis"
show ?thesis apply(cases "t = c") defer apply(rule *)
apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
note d2 = conjunctD2[OF this,rule_format]
def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x"
have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto
from fine_division_exists[OF this, of a t] guess p . note p=this
note p'=tagged_division_ofD[OF this(1)]
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 $$0 \<le> t} = {a..t}" "{a..c} \<inter> {x. x$$0 \<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
apply(rule tagged_division_union_interval[of _ _ _ 0 "t"]) unfolding * apply(rule p)
apply(rule tagged_division_of_self) unfolding fine_def
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)
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"
"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
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)
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
lemma indefinite_integral_continuous_right: fixes f::"real \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e"
proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a" using assms by auto
from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b - c)"
show ?thesis apply(rule that[of "?d"])
proof safe show "0 < ?d" using d(1) assms(3) by auto
fix t::"real" assume as:"c \<le> t" "t < c + ?d"
have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
"integral{a..t} f = integral{a..b} f - integral{t..b} f" 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 *
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)"
let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e"
{ presume *:"a<b \<Longrightarrow> ?thesis"
show ?thesis apply(cases,rule *,assumption)
proof- case goal1 hence "{a..b} = {x}" using as(1) apply-apply(rule set_ext)
unfolding atLeastAtMost_iff by(auto simp only:field_simps not_less DIM_real)
thus ?case using `e>0` by auto
qed } assume "a<b"
have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add:)
thus ?thesis apply-apply(erule disjE)+
proof- assume "x=a" have "a \<le> a" by auto
from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
unfolding `x=a` dist_norm apply(rule d(2)[rule_format]) by auto
next assume "x=b" have "b \<le> b" by auto
from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
unfolding `x=b` dist_norm apply(rule d(2)[rule_format]) by auto
next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add: )
from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
show ?thesis apply(rule_tac x="min d1 d2" in exI)
proof safe show "0 < min d1 d2" using d1 d2 by auto
fix y assume "y\<in>{a..b}" "dist y x < min d1 d2"
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
subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach"
assumes "finite k" "continuous_on {a..b} f" "f a = y"
"\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
shows "f x = y"
proof- have ab:"a\<le>b" using assms by auto
have *:"a\<le>x" using assms(5) by auto
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 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]
thus ?thesis unfolding assms by auto qed
subsection {* Generalize a bit to any convex set. *}
lemma has_derivative_zero_unique_strong_convex: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
"\<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"
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"
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)
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
have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
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])
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)
unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
apply(rule has_derivative_vmul_within,rule has_derivative_id)+
apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
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
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"
assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
"\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
shows "f x = y"
proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
apply(rule continuous_closed_in_preimage[OF assms(4) closed_sing])
apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
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
show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball])
apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+
apply(subst centre_in_ball,rule e,rule) apply safe
apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format])
using y e by auto qed qed
thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed
subsection {* Integrating characteristic function of an interval. *}
lemma has_integral_restrict_open_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
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- 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
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]
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
thus ?thesis using integrable_integral unfolding g_def by auto }
note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
note * = this[unfolded neutral_monoid]
have iterate:"iterate (lifted op +) (p - {{c..d}})
(\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0"
proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this]
from div(3) guess u v apply-by(erule exE)+ note uv=this
have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto
hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"])
unfolding g_def interior_closed_interval by auto thus ?case by auto
qed
have *:"p = insert {c..d} (p - {{c..d}})" using p by auto
have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f])
unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto
moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)])
apply(rule has_integral_spike_interior[where f=g]) defer
apply(rule integrable_integral[OF **]) unfolding g_def by auto
ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+
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}"
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}"
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])
proof assumption assume ?l hence "?g integrable_on {c..d}"
apply-apply(rule integrable_subinterval[OF _ assms]) by auto
hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto
hence "i = integral {c..d} f" apply-apply(rule has_integral_unique)
apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto
thus ?r using * by auto qed qed auto
subsection {* Hence we can apply the limit process uniformly to all integrals. *}
lemma has_integral': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
"(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<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 - i) < e))" (is "?l \<longleftrightarrow> (\<forall>e>0. ?r e)")
proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis"
show ?thesis apply(cases,rule *,assumption)
apply(subst has_integral_alt) by auto }
assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this
from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B ..
note B = conjunctD2[OF this,rule_format] show ?thesis apply safe
proof- fix e assume ?l "e>(0::real)"
show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI)
proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::'n::ordered_euclidean_space}"
thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s
apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s])
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"
from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'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 component_le_norm[of x i] unfolding c_def d_def
by(auto simp add:field_simps) qed
have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto 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
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> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'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 component_le_norm[of x i] unfolding c_def d_def
by(auto simp add:field_simps) qed
have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto 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
(*lemma has_integral_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
"((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s"
unfolding has_integral'[unfolded has_integral]
proof case goal1 thus ?case apply safe
apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
apply(erule_tac x=a in allE, erule_tac x=b in allE,safe)
apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe)
apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
apply(subst(asm)(2) norm_vector_1) unfolding split_def
unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
apply(subst(asm)(2) norm_vector_1) by auto
next case goal2 thus ?case apply safe
apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
apply(erule_tac x=a in allE, erule_tac x=b in allE,safe)
apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe)
apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
apply(subst norm_vector_1) unfolding split_def
unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
apply(subst norm_vector_1) by auto qed
lemma integral_trans[simp]: assumes "(f::'n::ordered_euclidean_space \<Rightarrow> real) integrable_on s"
shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)"
apply(rule integral_unique) using assms by auto
lemma integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
"(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)"
unfolding integrable_on_def
apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans
apply safe defer apply(rule_tac x="vec1 y" in exI) by auto *)
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)"
shows "i \<le> j" using has_integral_component_le[OF assms(1-2), of 0] using assms(3) by auto
lemma integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
shows "integral s f \<le> integral s g"
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"
using has_integral_component_nonneg[of "f" "i" s 0]
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"
using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
subsection {* Hence a general restriction property. *}
lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows
"((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
proof- have *:"\<And>x. (if x \<in> t then if x \<in> s then f x else 0 else 0) = (if x\<in>s then f x else 0)" using assms by auto
show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed
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"
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"
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"
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
lemma integrable_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
"(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
unfolding integrable_on_def by auto
lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r")
proof assume ?r show ?l unfolding negligible_def
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"
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 (safe, 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"
shows "(f has_integral y) t"
using assms has_integral_spike_set_eq by auto
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
unfolding has_integral_spike_set_eq[OF assms(1)] .
lemma integrable_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "negligible((s - t) \<union> (t - s))"
shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)"
apply(rule,rule_tac[!] integrable_spike_set) using assms by auto
(*lemma integral_spike_set:
"\<forall>f:real^M->real^N g s t.
negligible(s DIFF t \<union> t DIFF s)
\<longrightarrow> integral s f = integral t f"
qed REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
ASM_MESON_TAC[]);;
lemma has_integral_interior:
"\<forall>f:real^M->real^N y s.
negligible(frontier s)
\<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
qed REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
NEGLIGIBLE_SUBSET)) THEN
REWRITE_TAC[frontier] THEN
MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
SET_TAC[]);;
lemma has_integral_closure:
"\<forall>f:real^M->real^N y s.
negligible(frontier s)
\<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
qed REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
NEGLIGIBLE_SUBSET)) THEN
REWRITE_TAC[frontier] THEN
MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
SET_TAC[]);;*)
subsection {* More lemmas that are useful later. *}
lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)$$k"
shows "i$$k \<le> j$$k"
proof- note has_integral_restrict_univ[THEN sym, of f]
note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
show ?thesis apply(rule *) using assms(1,4) by auto qed
lemma has_integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "f" "i" "j" 0] using assms by auto
lemma integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)$$k"
shows "(integral s f)$$k \<le> (integral t f)$$k"
apply(rule has_integral_subset_component_le) using assms by auto
lemma integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)"
shows "(integral s f) \<le> (integral t f)"
apply(rule has_integral_subset_le) using assms by auto
lemma has_integral_alt': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
(\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e)" (is "?l = ?r")
proof assume ?r
show ?l apply- apply(subst has_integral')
proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
show ?case apply(rule,rule,rule B,safe)
apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI)
apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto
qed next
assume ?l note as = this[unfolded has_integral'[of f],rule_format]
let ?f = "\<lambda>x. if x \<in> s then f x else 0"
show ?r proof safe fix a b::"'n::ordered_euclidean_space"
from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
let ?a = "(\<chi>\<chi> i. min (a$$i) (-B))::'n::ordered_euclidean_space" and ?b = "(\<chi>\<chi> i. max (b$$i) B)::'n::ordered_euclidean_space"
show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval dist_norm
proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
from B(2)[OF this] guess z .. note conjunct1[OF this]
thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {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
subsection {* Continuity of the integral (for a 1-dimensional interval). *}
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}
\<longrightarrow> 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)" (is "?l = ?r")
proof assume ?l then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
note y=conjunctD2[OF this,rule_format] show ?r apply safe apply(rule y)
proof- case goal1 hence "e/2 > 0" by auto from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
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]
have "Cauchy (\<lambda>n. integral ({(\<chi>\<chi> i. - real n)::'n .. (\<chi>\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))"
proof(unfold Cauchy_def,safe) case goal1
from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
from real_arch_simple[of B] guess N .. note N = this
{ fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {(\<chi>\<chi> i. - real n)::'n..\<chi>\<chi> i. real n}" apply safe
unfolding mem_ball mem_interval dist_norm
proof case goal1 thus ?case using component_le_norm[of x i]
using n N by(auto simp add:field_simps) 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 ..
note i = this[unfolded Lim_sequentially, rule_format]
show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
apply safe apply(rule as(1)[unfolded integrable_on_def])
proof- case goal1 hence *:"e/2 > 0" by auto
from i[OF this] guess N .. note N =this[rule_format]
from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format] let ?B = "max (real N) B"
show ?case apply(rule_tac x="?B" in exI)
proof safe show "0 < ?B" using B(1) by auto
fix a b assume ab:"ball 0 ?B \<subseteq> {a..b::'n::ordered_euclidean_space}"
from real_arch_simple[of ?B] guess n .. note n=this
show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
apply(rule norm_triangle_half_l) apply(rule B(2)) defer apply(subst norm_minus_commute)
apply(rule N[unfolded dist_norm, 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
show "x\<in>{\<chi>\<chi> i. - real n..\<chi>\<chi> i. real n}" using x unfolding mem_interval mem_ball dist_norm apply-
proof case goal1 thus ?case using component_le_norm[of x i]
using n by(auto simp add:field_simps) qed qed qed qed qed
lemma integrable_altD: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on s"
shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
"\<And>e. e>0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
\<longrightarrow> 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"
using assms[unfolded integrable_alt[of f]] by auto
lemma integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on s" "{a..b} \<subseteq> s" shows "f integrable_on {a..b}"
apply(rule integrable_eq) defer apply(rule integrable_altD(1)[OF assms(1)])
using assms(2) by auto
subsection {* A straddling criterion for integrability. *}
lemma integrable_straddle_interval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) ({a..b}) \<and> (h has_integral j) ({a..b}) \<and>
norm(i - j) < e \<and> (\<forall>x\<in>{a..b}. (g x) \<le> (f x) \<and> (f x) \<le>(h x))"
shows "f integrable_on {a..b}"
proof(subst integrable_cauchy,safe)
case goal1 hence e:"e/3 > 0" by auto note assms[rule_format,OF this]
then guess g h i j apply- by(erule exE conjE)+ note obt = this
from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
show ?case apply(rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI) apply(rule conjI gauge_inter d1 d2)+ unfolding fine_inter
proof safe 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(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)"
"(\<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 mult.diff_right[THEN sym]
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
thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
unfolding real_norm_def[THEN sym] 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
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))"
shows "f integrable_on s"
proof- have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
proof(rule integrable_straddle_interval,safe) case goal1 hence *:"e/4 > 0" by auto
from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
def c \<equiv> "(\<chi>\<chi> i. min (a$$i) (- (max B1 B2)))::'n" and d \<equiv> "(\<chi>\<chi> i. max (b$$i) (max B1 B2))::'n"
have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval dist_norm
proof(rule_tac[!] allI)
case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next
case goal2 thus ?case using component_le_norm[of x i] 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
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)
apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then h x else 0)" in exI)
apply safe apply(rule_tac[1-2] integrable_integral,rule g,rule h)
apply(rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
proof- have *:"\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
(if x \<in> s then f x - g x else (0::real))" by auto
note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_sub, OF h g]]
show " norm (integral {a..b} (\<lambda>x. if x \<in> s then h x else 0) -
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
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 allE) by auto
qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this
show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv)
proof- case goal1 hence *:"e/3 > 0" by auto
from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
show ?case apply(rule_tac x="max B1 B2" in exI) apply safe apply(rule min_max.less_supI1,rule B1)
proof- fix a b c d::"'n::ordered_euclidean_space" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}"
have **:"ball 0 B1 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" by auto
have *:"\<And>ga gc ha hc fa fc::real. abs(ga - i) < e / 3 \<and> abs(gc - i) < e / 3 \<and> abs(ha - j) < e / 3 \<and>
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 smt
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))
apply(rule obt) apply(rule_tac[!] integral_le) using obt
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]
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]
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
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
proof(cases "x\<in>\<Union>t") case True then guess s unfolding Union_iff .. note s=this
hence *:"\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s" using goal1(3) by blast
show ?thesis unfolding if_P[OF True] apply(rule trans) defer
apply(rule setsum_cong2) apply(subst *, assumption) apply(rule refl)
unfolding setsum_delta[OF assms(1)] using s by auto qed auto qed qed
subsection {* In particular adding integrals over a division, maybe not of an interval. *}
lemma has_integral_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
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)
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
from d(5)[OF goal1] show ?case unfolding obt interior_closed_interval
apply-apply(rule negligible_subset[of "({a..b}-{a<..<b}) \<union> ({c..d}-{c<..<d})"])
apply(rule negligible_union negligible_frontier_interval)+ by auto qed qed
lemma integral_combine_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "d division_of s" "\<forall>k\<in>d. f integrable_on k"
shows "integral s f = setsum (\<lambda>i. integral i f) d"
apply(rule integral_unique) apply(rule has_integral_combine_division[OF assms(1)])
using assms(2) unfolding has_integral_integral .
lemma has_integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
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]
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
lemma integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on s" "d division_of s"
shows "integral s f = setsum (\<lambda>i. integral i f) d"
apply(rule integral_unique,rule has_integral_combine_division_topdown) using assms by auto
lemma integrable_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "d division_of s" "\<forall>i\<in>d. f integrable_on i"
shows "f integrable_on s"
using assms(2) unfolding integrable_on_def
by(metis has_integral_combine_division[OF assms(1)])
lemma integrable_on_subdivision: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "d division_of i" "f integrable_on s" "i \<subseteq> s"
shows "f integrable_on i"
apply(rule integrable_combine_division assms)+
proof safe case goal1 note division_ofD(2,4)[OF assms(1) this]
thus ?case apply safe apply(rule integrable_on_subinterval[OF assms(2)])
using assms(3) by auto qed
subsection {* Also tagged divisions. *}
lemma has_integral_combine_tagged_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "p tagged_division_of s" "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
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)
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)
apply(rule setsum_cong2) using assms(2) by auto qed
lemma integral_combine_tagged_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "p tagged_division_of {a..b}" "\<forall>(x,k)\<in>p. f integrable_on k"
shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
apply(rule integral_unique) apply(rule has_integral_combine_tagged_division[OF assms(1)])
using assms(2) by auto
lemma has_integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) {a..b}"
apply(rule has_integral_combine_tagged_division[OF assms(2)])
proof safe case goal1 note tagged_division_ofD(3-4)[OF assms(2) this]
thus ?case using integrable_subinterval[OF assms(1)] by auto qed
lemma integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
apply(rule integral_unique,rule has_integral_combine_tagged_division_topdown) using assms by auto
subsection {* Henstock's lemma. *}
lemma henstock_lemma_part1: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on {a..b}" "0 < e" "gauge d"
"(\<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)"
and p:"p tagged_partial_division_of {a..b}" "d fine p"
shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e" (is "?x \<le> e")
proof- { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" thus ?thesis by arith }
fix k::real assume k:"k>0" note p' = tagged_partial_division_ofD[OF p(1)]
have "\<Union>snd ` p \<subseteq> {a..b}" using p'(3) by fastsimp
note partial_division_of_tagged_division[OF p(1)] this
from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
def r \<equiv> "q - snd ` p" have "snd ` p \<inter> r = {}" unfolding r_def by auto
have r:"finite r" using q' unfolding r_def by auto
have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
norm(setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
proof safe case goal1 hence i:"i \<in> q" unfolding r_def by auto
from q'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
have *:"k / (real (card r) + 1) > 0" apply(rule divide_pos_pos,rule k) by auto
have "f integrable_on {u..v}" apply(rule integrable_subinterval[OF assms(1)])
using q'(2)[OF i] unfolding uv by auto
note integrable_integral[OF this, unfolded has_integral[of f]]
from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
note gauge_inter[OF `gauge d` dd(1)] from fine_division_exists[OF this,of u v] guess qq .
thus ?case apply(rule_tac x=qq in exI) using dd(2)[of qq] unfolding fine_inter uv by auto qed
from bchoice[OF this] guess qq .. note qq=this[rule_format]
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
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 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
ultimately show "?p tagged_division_of {a..b}" by fastsimp 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
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)]
from this(2) guess u v apply-by(erule exE)+ note uv=this
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 subset_interior[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
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)
prefer 4 apply assumption apply(rule finite_imageI,fact)
unfolding split_paired_all split_conv image_iff defer apply(erule bexE)+
proof- fix x m k l T1 T2 assume "(x,m)\<in>T1" "(x,m)\<in>T2" "T1\<noteq>T2" "k\<in>r" "l\<in>r" "T1 = qq k" "T2 = qq l"
note as = this(1-5)[unfolded this(6-)] note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
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]
apply(rule subset_interior) 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
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)]
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 "?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"])
proof- case goal2 have *:"(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
apply(subst setsum_reindex_nonzero) apply fact
unfolding split_paired_all snd_conv split_def o_def
proof- fix x l y m assume as:"(x,l)\<in>p" "(y,m)\<in>p" "(x,l)\<noteq>(y,m)" "l = m"
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
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
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,fact)
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)
qed finally show "?x \<le> e + k" . qed
lemma henstock_lemma_part2: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
assumes "f integrable_on {a..b}" "0 < e" "gauge d"
"\<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
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"
"\<forall>p. p tagged_partial_division_of {a..b} \<and> d fine p
\<longrightarrow> setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
proof- have *:"e / (2 * (real DIM('n) + 1)) > 0" apply(rule divide_pos_pos) using assms(2) by auto
from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
guess d .. note d = conjunctD2[OF this] show thesis apply(rule that,rule d)
proof safe case goal1 note * = henstock_lemma_part2[OF assms(1) * d this]
show ?case apply(rule le_less_trans[OF *]) using `e>0` by(auto simp add:field_simps) qed qed
subsection {* monotone convergence (bounded interval first). *}
lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>k. (f k) integrable_on {a..b}"
"\<forall>k. \<forall>x\<in>{a..b}.(f k x) \<le> (f (Suc k) x)"
"\<forall>x\<in>{a..b}. ((\<lambda>k. f k x) ---> g x) sequentially"
"bounded {integral {a..b} (f k) | k . k \<in> UNIV}"
shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially"
proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using Lim_const by auto
next assume ab:"content {a..b} \<noteq> 0"
have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) $$ 0 \<le> (g x) $$ 0"
proof safe case goal1 note assms(3)[rule_format,OF this]
note * = Lim_component_ge[OF this trivial_limit_sequentially]
show ?case apply(rule *) unfolding eventually_sequentially
apply(rule_tac x=k in exI) apply- apply(rule transitive_stepwise_le)
using assms(2)[rule_format,OF goal1] by auto qed
have "\<exists>i. ((\<lambda>k. integral ({a..b}) (f k)) ---> i) sequentially"
apply(rule bounded_increasing_convergent) defer
apply rule apply(rule integral_le) apply safe
apply(rule assms(1-2)[rule_format])+ using assms(4) by auto
then guess i .. note i=this
have i':"\<And>k. (integral({a..b}) (f k)) \<le> i$$0"
apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially)
unfolding eventually_sequentially apply(rule_tac x=k in exI)
apply(rule transitive_stepwise_le) prefer 3 unfolding Eucl_real_simps apply(rule integral_le)
apply(rule assms(1-2)[rule_format])+ using assms(2) by auto
have "(g has_integral i) {a..b}" unfolding has_integral
proof safe case goal1 note e=this
hence "\<forall>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral {a..b} (f k)) < e / 2 ^ (k + 2)))"
apply-apply(rule,rule assms(1)[unfolded has_integral_integral has_integral,rule_format])
apply(rule divide_pos_pos) by auto
from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i$$0 - (integral {a..b} (f k)) \<and> i$$0 - (integral {a..b} (f k)) < e / 4"
proof- case goal1 have "e/4 > 0" using e by auto
from i[unfolded Lim_sequentially,rule_format,OF this] guess r ..
thus ?case apply(rule_tac x=r in exI) apply rule
apply(erule_tac x=k in allE)
proof- case goal1 thus ?case using i'[of k] unfolding dist_real_def by auto qed qed
then guess r .. note r=conjunctD2[OF this[rule_format]]
have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)$$0 - (f k x)$$0 \<and>
(g x)$$0 - (f k x)$$0 < e / (4 * content({a..b}))"
proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact)
using ab content_pos_le[of a b] by auto
from assms(3)[rule_format,OF goal1,unfolded Lim_sequentially,rule_format,OF this]
guess n .. note n=this
thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE)
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"
show ?case apply(rule_tac x=d in exI)
proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
next fix p assume p:"p tagged_division_of {a..b}" "d fine p"
note p'=tagged_division_ofD[OF p(1)]
have "\<exists>a. \<forall>x\<in>p. m (fst x) \<le> a" by(rule upper_bound_finite_set,fact)
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"
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
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where
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 setsum_norm[OF p'(1)])
apply(rule setsum_mono) unfolding split_paired_all split_conv
unfolding split_def setsum_left_distrib[THEN sym] scaleR.diff_right[THEN sym]
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]
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
apply(subst split_def)+ unfolding setsum_subtractf apply rule
proof- show "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))) < e / 2"
apply(rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"])
apply(rule setsum_norm_le[OF finite_atLeastAtMost])
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 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}"
show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x -
integral k (f (m x))) \<le> e / 2 ^ (t + 2)"apply(rule order_trans[of _
"norm(setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
apply(rule eq_refl) apply(rule arg_cong[where f=norm]) apply(rule setsum_cong2) defer
apply(rule henstock_lemma_part1) apply(rule assms(1)[rule_format])
apply(rule divide_pos_pos,rule e) defer apply safe apply(rule c)+
apply rule apply assumption+ apply(rule tagged_partial_division_subset[of p])
apply(rule p(1)[unfolded tagged_division_of_def,THEN conjunct1]) defer
unfolding fine_def apply safe apply(drule p(2)[unfolded fine_def,rule_format])
unfolding d_def by auto qed
qed(insert s, auto)
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$$0 - kr$$0
\<and> i$$0 - kr$$0 < 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 Eucl_real_simps])
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$$0 - (integral {a..b} (f r))$$0" using r(1) by auto
show "i$$0 - (integral {a..b} (f r))$$0 < 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"
unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
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
have "integral {a..b} g = i" apply(rule integral_unique) using * .
thus ?thesis using i * by auto qed
lemma monotone_convergence_increasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x)"
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
proof- have lem:"\<And>f::nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real. \<And> g s. \<forall>k.\<forall>x\<in>s. 0 \<le> (f k x) \<Longrightarrow> \<forall>k. (f k) integrable_on s \<Longrightarrow>
\<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x) \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially \<Longrightarrow>
bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
proof- case goal1 note assms=this[rule_format]
have "\<forall>x\<in>s. \<forall>k. (f k x)$$0 \<le> (g x)$$0" apply safe apply(rule Lim_component_ge)
apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially)
unfolding eventually_sequentially apply(rule_tac x=k in exI)
apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format]
have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially" apply(rule bounded_increasing_convergent)
apply(rule goal1(5)) apply(rule,rule integral_le) apply(rule goal1(2)[rule_format])+
using goal1(3) by auto then guess i .. note i=this
have "\<And>k. \<forall>x\<in>s. \<forall>n\<ge>k. f k x \<le> f n x" apply(rule,rule transitive_stepwise_le) using goal1(3) by auto
hence i':"\<forall>k. (integral s (f k))$$0 \<le> i$$0" apply-apply(rule,rule Lim_component_ge)
apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
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 "\<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"
proof(rule monotone_convergence_interval,safe)
case goal1 show ?case using int .
next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto
next case goal3 thus ?case apply-apply(cases "x\<in>s") using assms(4) by auto
next case goal4 note * = integral_nonneg
have "\<And>k. norm (integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) \<le> norm (integral s (f k))"
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])
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
apply(rule_tac x=aa in exI,safe) apply(erule_tac x="integral s (f k)" in ballE)
apply(rule order_trans) apply assumption by auto qed note g = conjunctD2[OF this]
have "(g has_integral i) s" unfolding has_integral_alt' apply safe apply(rule g(1))
proof- case goal1 hence "e/4>0" by auto
from i[unfolded Lim_sequentially,rule_format,OF this] guess N .. note N=this
note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]]
from this[THEN conjunct2,rule_format,OF `e/4>0`] guess B .. note B=conjunctD2[OF this]
show ?case apply(rule,rule,rule B,safe)
proof- fix a b::"'n::ordered_euclidean_space" assume ab:"ball 0 B \<subseteq> {a..b}"
from `e>0` have "e/2>0" by auto
from g(2)[unfolded Lim_sequentially,of a b,rule_format,OF this] guess M .. note M=this
have **:"norm (integral {a..b} (\<lambda>x. if x \<in> s then f N x else 0) - i) < e/2"
apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N]
unfolding dist_norm 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 Eucl_real_simps by arith
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 dist_real_def]) apply(rule le_add1)
apply(rule integral_le[OF int int]) defer
apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded Eucl_real_simps]])
proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)$$0 \<le> (f n x)$$0"
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])
unfolding ifif integral_restrict_univ[OF int']
apply(rule integral_subset_le[OF _ int']) using assms by auto
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)"
apply(subst integral_sub) apply(rule assms(1)[rule_format])+ by rule
have "\<And>x m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x) \<le> (f n x)" apply(rule transitive_stepwise_le)
using assms(2) by auto note * = this[rule_format]
have "(\<lambda>x. g x - f 0 x) integrable_on s \<and>((\<lambda>k. integral s (\<lambda>x. f (Suc k) x - f 0 x)) --->
integral s (\<lambda>x. g x - f 0 x)) sequentially" apply(rule lem,safe)
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 Lim_sub)
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)
apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub
apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed
note conjunctD2[OF this] note Lim_add[OF this(2) Lim_const[of "integral s (f 0)"]]
integrable_add[OF this(1) assms(1)[rule_format,of 0]]
thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub)
using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed
lemma monotone_convergence_decreasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<le> (f k x)"
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
proof- note assm = assms[rule_format]
have *:"{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R -1 ` {integral s (f k)| k. True}"
apply safe unfolding image_iff apply(rule_tac x="integral s (f k)" in bexI) prefer 3
apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto
have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x))
---> integral s (\<lambda>x. - g x)) sequentially" apply(rule monotone_convergence_increasing)
apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule Lim_neg)
apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto
note * = conjunctD2[OF this]
show ?thesis apply rule using integrable_neg[OF *(1)] defer
using Lim_neg[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
unfolding integral_neg[OF *(1),THEN sym] by auto qed
subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46) where
"f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. (norm(f x))) integrable_on s"
lemma absolutely_integrable_onI[intro?]:
"f integrable_on s \<Longrightarrow> (\<lambda>x. (norm(f x))) integrable_on s \<Longrightarrow> f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def by auto
lemma absolutely_integrable_onD[dest]: assumes "f absolutely_integrable_on s"
shows "f integrable_on s" "(\<lambda>x. (norm(f x))) integrable_on s"
using assms unfolding absolutely_integrable_on_def by auto
(*lemma absolutely_integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
"(vec1 o f) absolutely_integrable_on s \<longleftrightarrow> f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def o_def by auto*)
lemma integral_norm_bound_integral: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> g x"
shows "norm(integral s f) \<le> (integral s g)"
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"
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(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]
have lem:"\<And>f::'n::ordered_euclidean_space \<Rightarrow> 'a. \<And> g a b. f integrable_on {a..b} \<Longrightarrow> g integrable_on {a..b} \<Longrightarrow>
\<forall>x\<in>{a..b}. norm(f x) \<le> (g x) \<Longrightarrow> norm(integral({a..b}) f) \<le> (integral({a..b}) g)"
proof(rule *[rule_format]) case goal1 hence *:"e/2>0" by auto
from integrable_integral[OF goal1(1),unfolded has_integral[of f],rule_format,OF *]
guess d1 .. note d1 = conjunctD2[OF this,rule_format]
from integrable_integral[OF goal1(2),unfolded has_integral[of g],rule_format,OF *]
guess d2 .. note d2 = conjunctD2[OF this,rule_format]
note gauge_inter[OF d1(1) d2(1)]
from fine_division_exists[OF this, of a b] guess p . note p=this
show ?case apply(rule norm) defer
apply(rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def]) defer
apply(rule d1(2)[OF conjI[OF p(1)]]) defer apply(rule setsum_norm_le)
proof safe fix x k assume "(x,k)\<in>p" note as = tagged_division_ofD(2-4)[OF p(1) this]
from this(3) guess u v apply-by(erule exE)+ note uv=this
show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x" unfolding uv norm_scaleR
unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
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"
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]
note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format]
from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e]
guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format]
from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e]
guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format]
from bounded_subset_closed_interval[OF bounded_ball, of "0::'n::ordered_euclidean_space" "max B1 B2"]
guess a b apply-by(erule exE)+ note ab=this[unfolded ball_max_Un]
have "ball 0 B1 \<subseteq> {a..b}" using ab by auto
from B1(2)[OF this] guess z .. note z=conjunctD2[OF this]
have "ball 0 B2 \<subseteq> {a..b}" using ab by auto
from B2(2)[OF this] guess w .. note w=conjunctD2[OF this]
show "norm (integral s f) < integral s g + e" apply(rule norm)
apply(rule lem[OF f g, of a b]) unfolding integral_unique[OF z(1)] integral_unique[OF w(1)]
defer apply(rule w(2)[unfolded real_norm_def],rule z(2))
apply safe apply(case_tac "x\<in>s") unfolding if_P apply(rule assms(3)[rule_format]) by auto qed
lemma integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
fixes g::"'n => 'b::ordered_euclidean_space"
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
shows "norm(integral s f) \<le> (integral s g)$$k"
proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x $$ k) o g)"
apply(rule integral_norm_bound_integral[OF assms(1)])
apply(rule integrable_linear[OF assms(2)],rule)
unfolding o_def by(rule assms)
thus ?thesis unfolding o_def integral_component_eq[OF assms(2)] . qed
lemma has_integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
fixes g::"'n => 'b::ordered_euclidean_space"
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
shows "norm(i) \<le> j$$k" using integral_norm_bound_integral_component[of f s g k]
unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
using assms by auto
lemma absolutely_integrable_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f absolutely_integrable_on s"
shows "norm(integral s f) \<le> integral s (\<lambda>x. norm(f x))"
apply(rule integral_norm_bound_integral) using assms by auto
lemma absolutely_integrable_0[intro]: "(\<lambda>x. 0) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def by auto
lemma absolutely_integrable_cmul[intro]:
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def using integrable_cmul[of f s c]
using integrable_cmul[of "\<lambda>x. norm (f x)" s "abs c"] by auto
lemma absolutely_integrable_neg[intro]:
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) absolutely_integrable_on s"
apply(drule absolutely_integrable_cmul[where c="-1"]) by auto
lemma absolutely_integrable_norm[intro]:
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. norm(f x)) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def by auto
lemma absolutely_integrable_abs[intro]:
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. abs(f x::real)) absolutely_integrable_on s"
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}"
unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
lemma absolutely_integrable_bounded_variation: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
assumes "f absolutely_integrable_on UNIV"
obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
apply(rule that[of "integral UNIV (\<lambda>x. norm (f x))"])
proof safe case goal1 note d = division_ofD[OF this(2)]
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. integral i (\<lambda>x. norm (f x)))"
apply(rule setsum_mono,rule absolutely_integrable_le) apply(drule d(4),safe)
apply(rule absolutely_integrable_on_subinterval[OF assms]) by auto
also have "... \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
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)
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])
apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
using norm_triangle_ineq3 .
lemma bounded_variation_absolutely_integrable_interval:
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assumes "f integrable_on {a..b}"
"\<forall>d. d division_of {a..b} \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
shows "f absolutely_integrable_on {a..b}"
proof- let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of {a..b} }" def i \<equiv> "Sup ?S"
have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
apply(rule elementary_interval) defer apply(rule_tac x=B in exI)
apply(rule setleI) using assms(2) by auto
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
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)]
have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
proof case goal1 have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
apply(rule separate_point_closed) apply(rule closed_Union)
apply(rule finite_subset[OF _ d'(1)]) apply safe apply(drule d'(4)) by auto
thus ?case apply safe apply(rule_tac x=da in exI,safe)
apply(erule_tac x=xa in ballE) by auto
qed from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
have "e/2 > 0" using goal1 by auto
from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
show ?case apply(rule_tac x="?g" in exI) apply safe
proof- show "gauge ?g" using g(1) unfolding gauge_def using k(1) by auto
fix p assume "p tagged_division_of {a..b}" "?g fine p"
note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
note p' = tagged_division_ofD[OF p(1)]
def p' \<equiv> "{(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
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
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'"
hence "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l" unfolding p'_def by auto
then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
show "x\<in>k" "k\<subseteq>{a..b}" using p'(2-3)[OF il(3)] il by auto
show "\<exists>a b. k = {a..b}" unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
apply safe unfolding inter_interval by auto
next fix x1 k1 assume "(x1,k1)\<in>p'"
hence "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> k1 = i \<inter> l" unfolding p'_def by auto
then guess i1 l1 apply-by(erule exE)+ note il1=conjunctD4[OF this]
fix x2 k2 assume "(x2,k2)\<in>p'"
hence "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> k2 = i \<inter> l" unfolding p'_def by auto
then guess i2 l2 apply-by(erule exE)+ note il2=conjunctD4[OF this]
assume "(x1, k1) \<noteq> (x2, k2)"
hence "interior(i1) \<inter> interior(i2) = {} \<or> interior(l1) \<inter> interior(l2) = {}"
using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] unfolding il1 il2 by auto
thus "interior k1 \<inter> interior k2 = {}" unfolding il1 il2 by auto
next have *:"\<forall>(x, X) \<in> p'. X \<subseteq> {a..b}" unfolding p'_def using d' by auto
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
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
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
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' .
hence **:" \<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 p'' by auto
have p'alt:"p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> ~(i \<inter> l = {})}"
proof safe case goal2
have "x\<in>i" using fineD[OF p(3) goal2(1)] k(2)[OF goal2(2), of x] goal2(4-) by auto
hence "(x, i \<inter> l) \<in> p'" unfolding p'_def apply safe
apply(rule_tac x=x in exI,rule_tac x="i\<inter>l" in exI) apply safe using goal2 by auto
thus ?case using goal2(3) by auto
next fix x k assume "(x,k)\<in>p'"
hence "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l" unfolding p'_def by auto
then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
thus "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
apply(rule_tac x=x in exI,rule_tac x=i in exI,rule_tac x=l in exI)
using p'(2)[OF il(3)] by auto
qed
have sum_p':"(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
apply(subst setsum_over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
unfolding norm_eq_zero apply(rule integral_null,assumption) ..
note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
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"
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
next case goal2 have *:"{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
(\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}" by auto
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
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)
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'])
apply(rule integrable_on_subinterval[OF assms(1) uvab]) apply assumption
apply(rule setsum_norm_le) by auto
also have "... = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))"
apply(rule setsum_mono_zero_left) apply(subst simple_image)
apply(rule finite_imageI)+ apply fact unfolding d'_def uv apply blast
proof case goal1 hence "i \<in> {{u..v} \<inter> l |l. l \<in> snd ` p}" by auto
from this[unfolded mem_Collect_eq] guess l .. note l=this
hence "{u..v} \<inter> l = {}" using goal1 by auto thus ?case using l by auto
qed also have "... = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))" unfolding simple_image
apply(rule setsum_reindex_nonzero[unfolded o_def])apply(rule finite_imageI,rule p')
proof- case goal1 have "interior (k \<inter> l) \<subseteq> interior (l \<inter> y)" apply(subst(2) interior_inter)
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
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
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])
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"
"\<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)]
guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this
have "l1 \<noteq> l2 \<or> k1 \<noteq> k2" using as by auto
hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})" apply-
apply(erule disjE) apply(rule disjI2) apply(rule d'(5)) prefer 4 apply(rule disjI1)
apply(rule *) using as by auto
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
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
apply(rule finite_imageI[OF p'(1)]) apply safe
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 safe apply(rule_tac x="(a,l)" in bexI) by auto
qed finally show ?case .
next case goal3
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
have Sigma_alt:"\<And>s t. s \<times> t = {(i, j) |i j. i \<in> s \<and> j \<in> t}" by auto
have *:"?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)" (*{(xl,i)|xl i. xl\<in>p \<and> i\<in>d}"**)
apply safe unfolding image_iff apply(rule_tac x="((x,l),i)" in bexI) by auto
note pdfin = finite_cartesian_product[OF p'(1) d'(1)]
have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
unfolding norm_scaleR apply(rule setsum_mono_zero_left)
apply(subst *, rule finite_imageI) apply fact unfolding p'alt apply blast
apply safe apply(rule_tac x=x in exI,rule_tac x=i in exI,rule_tac x=l in exI) by auto
also have "... = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))" unfolding *
apply(subst setsum_reindex_nonzero,fact) unfolding split_paired_all
unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff Pair_eq apply(erule_tac conjE)+
proof- fix x1 l1 k1 x2 l2 k2 assume as:"(x1,l1)\<in>p" "(x2,l2)\<in>p" "k1\<in>d" "k2\<in>d"
"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) = {})"
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
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(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
have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> {u..v}))"
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')
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)
using d'(5)[OF goal1(1-3)] by auto
also have "... = interior (y \<inter> (k \<inter> {u..v}))" by auto
also have "... = interior (k \<inter> {u..v})" unfolding goal1(4) by auto
finally show ?case unfolding uv inter_interval content_eq_0_interior ..
qed also have "... = setsum content {{u..v} \<inter> k |k. k \<in> d \<and> ~({u..v} \<inter> k = {})}"
apply(rule setsum_mono_zero_right) unfolding simple_image
apply(rule finite_imageI,rule d') apply blast apply safe
apply(rule_tac x=k in exI)
proof- case goal1 from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this
have "interior (k \<inter> {u..v}) \<noteq> {}" using goal1(2)
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 -
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
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"
shows "f absolutely_integrable_on UNIV"
proof(rule absolutely_integrable_onI,fact,rule)
let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of (\<Union>d)}" def i \<equiv> "Sup ?S"
have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
apply(rule elementary_interval) defer apply(rule_tac x=B in exI)
apply(rule setleI) using assms(2) by auto
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
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"
proof(rule ccontr) case goal1 hence "i \<le> i - e" apply-
apply(rule isLub_le_isUb[OF i]) apply(rule isUbI) unfolding setle_def by auto
thus False using goal2 by auto
qed then guess K .. note * = this[unfolded image_iff not_le]
from this(1) guess d .. note this[unfolded mem_Collect_eq]
note d = this(1) *(2)[unfolded this(2)] note d'=division_ofD[OF this(1)]
have "bounded (\<Union>d)" by(rule elementary_bounded,fact)
from this[unfolded bounded_pos] guess K .. note K=conjunctD2[OF this]
show ?case apply(rule_tac x="K + 1" in exI,safe)
proof- fix a b assume ab:"ball 0 (K + 1) \<subseteq> {a..b::'n::ordered_euclidean_space}"
have *:"\<forall>s s1. i - e < s1 \<and> s1 \<le> s \<and> s < i + e \<longrightarrow> abs(s - i) < (e::real)" by arith
show "norm (integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) - i) < e"
unfolding real_norm_def apply(rule *[rule_format],safe) apply(rule d(2))
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])
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])
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}"])
apply(rule d) using f_int[of a b] by auto
qed finally show ?case .
next note f = absolutely_integrable_onD[OF f_int[of a b]]
note * = this(2)[unfolded has_integral_integral has_integral[of "\<lambda>x. norm (f x)"],rule_format]
have "e/2>0" using `e>0` by auto from *[OF this] guess d1 .. note d1=conjunctD2[OF this]
from henstock_lemma[OF f(1) `e/2>0`] guess d2 . note d2=this
from fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] guess p .
note p=this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
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])
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
show "abs ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral {a..b} (\<lambda>x. norm(f x))) < e / 2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]] unfolding real_norm_def .
show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
apply(rule setsum_cong2) unfolding split_paired_all split_conv
apply(drule tagged_division_ofD(4)[OF p(1)]) unfolding norm_scaleR
apply(subst abs_of_nonneg) by auto
show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> i"
apply(subst setsum_over_tagged_division_lemma[OF p(1)]) defer apply(rule isLubD2[OF i])
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
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"
unfolding absolutely_integrable_on_def if_distrib norm_zero integrable_restrict_univ ..
lemma absolutely_integrable_add[intro]: fixes f g::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
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]
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"
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"
apply(rule bounded_variation_absolutely_integrable[of _ "B1+B2"])
apply(rule integrable_add) prefer 3
proof safe case goal1 have "\<And>k. k \<in> d \<Longrightarrow> f integrable_on k \<and> g integrable_on k"
apply(drule division_ofD(4)[OF goal1]) apply safe
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)
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 .
qed(insert assms,auto) qed
lemma absolutely_integrable_sub[intro]: fixes f g::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
shows "(\<lambda>x. f(x) - g(x)) absolutely_integrable_on s"
using absolutely_integrable_add[OF assms(1) absolutely_integrable_neg[OF assms(2)]]
unfolding algebra_simps .
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.
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]
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"
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)])
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)
proof- case goal1 from division_ofD(4)[OF goal2 this]
guess u v apply-by(erule exE)+ note uv=this
have *:"f integrable_on k" unfolding uv apply(rule integrable_on_subinterval[of _ UNIV])
using assms by auto note this[unfolded has_integral_integral]
note has_integral_linear[OF this assms(2)] integrable_linear[OF * assms(2)]
note * = has_integral_unique[OF this(2)[unfolded has_integral_integral] this(1)]
show ?case unfolding * using b by auto
qed also have "... \<le> B * b" apply(rule mult_right_mono) using B goal2 b by auto
finally show ?case .
qed(insert assms,auto) qed
lemma absolutely_integrable_setsum: fixes f::"'a \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "finite t" "\<And>a. a \<in> t \<Longrightarrow> (f a) absolutely_integrable_on s"
shows "(\<lambda>x. setsum (\<lambda>a. f a x) t) absolutely_integrable_on s"
using assms(1,2) apply induct defer apply(subst setsum.insert) apply assumption+ by(rule,auto)
lemma absolutely_integrable_vector_abs: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
assumes "f absolutely_integrable_on s"
shows "(\<lambda>x. (\<chi>\<chi> i. abs(f x$$i))::'c::ordered_euclidean_space) absolutely_integrable_on s"
proof- have *:"\<And>x. ((\<chi>\<chi> i. abs(f x$$i))::'c::ordered_euclidean_space) = (setsum (\<lambda>i.
(((\<lambda>y. (\<chi>\<chi> j. if j = i then y else 0)) o
(((\<lambda>x. (norm((\<chi>\<chi> j. if j = i then x$$i else 0)::'c::ordered_euclidean_space))) o f))) x)) {..<DIM('c)})"
unfolding euclidean_eq[where 'a='c] euclidean_component.setsum apply safe
unfolding euclidean_lambda_beta'
proof- case goal1 have *:"\<And>i xa. ((if i = xa then f x $$ xa else 0) * (if i = xa then f x $$ xa else 0)) =
(if i = xa then (f x $$ xa) * (f x $$ xa) else 0)" by auto
have *:"\<And>xa. norm ((\<chi>\<chi> j. if j = xa then f x $$ xa else 0)::'c) = (if xa<DIM('c) then abs (f x $$ xa) else 0)"
unfolding norm_eq_sqrt_inner euclidean_inner[where 'a='c]
by(auto simp add:setsum_delta[OF finite_lessThan] *)
have "\<bar>f x $$ i\<bar> = (setsum (\<lambda>k. if k = i then abs ((f x)$$i) else 0) {..<DIM('c)})"
unfolding setsum_delta[OF finite_lessThan] using goal1 by auto
also have "... = (\<Sum>xa<DIM('c). ((\<lambda>y. (\<chi>\<chi> j. if j = xa then y else 0)::'c) \<circ>
(\<lambda>x. (norm ((\<chi>\<chi> j. if j = xa then x $$ xa else 0)::'c))) \<circ> f) x $$ i)"
unfolding o_def * apply(rule setsum_cong2)
unfolding euclidean_lambda_beta'[OF goal1 ] by auto
finally show ?case unfolding o_def . qed
show ?thesis unfolding * apply(rule absolutely_integrable_setsum) apply(rule finite_lessThan)
apply(rule absolutely_integrable_linear) unfolding o_def apply(rule absolutely_integrable_norm)
apply(rule absolutely_integrable_linear[OF assms,unfolded o_def]) unfolding linear_linear
apply(rule_tac[!] linearI) unfolding euclidean_eq[where 'a='c]
by(auto simp:euclidean_scaleR[where 'a=real,unfolded real_scaleR_def])
qed
lemma absolutely_integrable_max: fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
shows "(\<lambda>x. (\<chi>\<chi> i. max (f(x)$$i) (g(x)$$i))::'n::ordered_euclidean_space) absolutely_integrable_on s"
proof- have *:"\<And>x. (1 / 2) *\<^sub>R (((\<chi>\<chi> i. \<bar>(f x - g x) $$ i\<bar>)::'n) + (f x + g x)) = (\<chi>\<chi> i. max (f(x)$$i) (g(x)$$i))"
unfolding euclidean_eq[where 'a='n] by auto
note absolutely_integrable_sub[OF assms] absolutely_integrable_add[OF assms]
note absolutely_integrable_vector_abs[OF this(1)] this(2)
note absolutely_integrable_add[OF this] note absolutely_integrable_cmul[OF this,of "1/2"]
thus ?thesis unfolding * . qed
lemma absolutely_integrable_min: fixes f g::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s"
shows "(\<lambda>x. (\<chi>\<chi> i. min (f(x)$$i) (g(x)$$i))::'n::ordered_euclidean_space) absolutely_integrable_on s"
proof- have *:"\<And>x. (1 / 2) *\<^sub>R ((f x + g x) - ((\<chi>\<chi> i. \<bar>(f x - g x) $$ i\<bar>)::'n)) = (\<chi>\<chi> i. min (f(x)$$i) (g(x)$$i))"
unfolding euclidean_eq[where 'a='n] by auto
note absolutely_integrable_add[OF assms] absolutely_integrable_sub[OF assms]
note this(1) absolutely_integrable_vector_abs[OF this(2)]
note absolutely_integrable_sub[OF this] note absolutely_integrable_cmul[OF this,of "1/2"]
thus ?thesis unfolding * . qed
lemma absolutely_integrable_abs_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
shows "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and>
(\<lambda>x. (\<chi>\<chi> i. abs(f x$$i))::'m) integrable_on s" (is "?l = ?r")
proof assume ?l thus ?r apply-apply rule defer
apply(drule absolutely_integrable_vector_abs) by auto
next assume ?r { presume lem:"\<And>f::'n \<Rightarrow> 'm. f integrable_on UNIV \<Longrightarrow>
(\<lambda>x. (\<chi>\<chi> i. abs(f(x)$$i))::'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV"
have *:"\<And>x. (\<chi>\<chi> i. \<bar>(if x \<in> s then f x else 0) $$ i\<bar>) = (if x\<in>s then (\<chi>\<chi> i. \<bar>f x $$ i\<bar>) else (0::'m))"
unfolding euclidean_eq[where 'a='m] by auto
show ?l apply(subst absolutely_integrable_restrict_univ[THEN sym]) 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. (\<chi>\<chi> i. abs(f(x)$$i))::'m::ordered_euclidean_space) integrable_on UNIV"
let ?B = "setsum (\<lambda>i. integral UNIV (\<lambda>x. (\<chi>\<chi> j. abs(f x$$j)) ::'m::ordered_euclidean_space) $$ i) {..<DIM('m)}"
show "f absolutely_integrable_on UNIV"
apply(rule bounded_variation_absolutely_integrable[OF assms(1), where B="?B"],safe)
proof- case goal1 note d=this and d'=division_ofD[OF this]
have "(\<Sum>k\<in>d. norm (integral k f)) \<le>
(\<Sum>k\<in>d. setsum (op $$ (integral k (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m))) {..<DIM('m)})" apply(rule setsum_mono)
apply(rule order_trans[OF norm_le_l1]) apply(rule setsum_mono) unfolding lessThan_iff
proof- fix k and i assume "k\<in>d" and i:"i<DIM('m)"
from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this
show "\<bar>integral k f $$ i\<bar> \<le> integral k (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ i" apply(rule abs_leI)
unfolding euclidean_component.minus[THEN sym] defer apply(subst integral_neg[THEN sym])
defer apply(rule_tac[1-2] integral_component_le) apply(rule integrable_neg)
using integrable_on_subinterval[OF assms(1),of a b]
integrable_on_subinterval[OF assms(2),of a b] unfolding ab by auto
qed also have "... \<le> setsum (op $$ (integral UNIV (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>))::'m)) {..<DIM('m)}"
apply(subst setsum_commute,rule setsum_mono)
proof- case goal1 have *:"(\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) integrable_on \<Union>d"
using integrable_on_subdivision[OF d assms(2)] by auto
have "(\<Sum>i\<in>d. integral i (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ j)
= integral (\<Union>d) (\<lambda>x. (\<chi>\<chi> j. abs(f x$$j)) ::'m::ordered_euclidean_space) $$ j"
unfolding euclidean_component.setsum[THEN sym] integral_combine_division_topdown[OF * d] ..
also have "... \<le> integral UNIV (\<lambda>x. (\<chi>\<chi> j. \<bar>f x $$ j\<bar>)::'m) $$ j"
apply(rule integral_subset_component_le) using assms * by auto
finally show ?case .
qed finally show ?case . qed qed
lemma nonnegative_absolutely_integrable: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "\<forall>x\<in>s. \<forall>i<DIM('m). 0 \<le> f(x)$$i" "f integrable_on s"
shows "f absolutely_integrable_on s"
unfolding absolutely_integrable_abs_eq apply rule defer
apply(rule integrable_eq[of _ f]) using assms apply-apply(subst euclidean_eq) by auto
lemma absolutely_integrable_integrable_bound: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
assumes "\<forall>x\<in>s. norm(f x) \<le> g x" "f integrable_on s" "g integrable_on s"
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])
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"
assume assms:"\<forall>x. norm(f x) \<le> g x" "f integrable_on UNIV" "g integrable_on UNIV"
show "f absolutely_integrable_on UNIV"
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_tac[1-2] integrable_on_subinterval) using assms by auto
also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[THEN sym])
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
apply(rule integrable_on_subdivision[OF d,of _ UNIV]) prefer 4
apply(rule,rule_tac y="norm (f x)" in order_trans) using assms by auto
finally show ?case . qed qed
lemma absolutely_integrable_integrable_bound_real: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<forall>x\<in>s. norm(f x) \<le> g x" "f integrable_on s" "g integrable_on s"
shows "f absolutely_integrable_on s"
apply(rule absolutely_integrable_integrable_bound[where g=g])
using assms unfolding o_def by auto
lemma absolutely_integrable_absolutely_integrable_bound:
fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" and g::"'n::ordered_euclidean_space \<Rightarrow> 'p::ordered_euclidean_space"
assumes "\<forall>x\<in>s. norm(f x) \<le> norm(g x)" "f integrable_on s" "g absolutely_integrable_on s"
shows "f absolutely_integrable_on s"
apply(rule absolutely_integrable_integrable_bound[of s f "\<lambda>x. norm (g x)"])
using assms by auto
lemma absolutely_integrable_inf_real:
assumes "finite k" "k \<noteq> {}"
"\<forall>i\<in>k. (\<lambda>x. (fs x i)::real) absolutely_integrable_on s"
shows "(\<lambda>x. (Inf ((fs x) ` k))) absolutely_integrable_on s" using assms
proof induct case (insert a k) let ?P = " (\<lambda>x. if fs x ` k = {} then fs x a
else min (fs x a) (Inf (fs x ` k))) absolutely_integrable_on s"
show ?case unfolding image_insert
apply(subst Inf_insert_finite) apply(rule finite_imageI[OF insert(1)])
proof(cases "k={}") case True
thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto
next case False thus ?P apply(subst if_not_P) defer
apply(rule absolutely_integrable_min[where 'n=real,unfolded Eucl_real_simps])
defer apply(rule insert(3)[OF False]) using insert(5) by auto
qed qed auto
lemma absolutely_integrable_sup_real:
assumes "finite k" "k \<noteq> {}"
"\<forall>i\<in>k. (\<lambda>x. (fs x i)::real) absolutely_integrable_on s"
shows "(\<lambda>x. (Sup ((fs x) ` k))) absolutely_integrable_on s" using assms
proof induct case (insert a k) let ?P = " (\<lambda>x. if fs x ` k = {} then fs x a
else max (fs x a) (Sup (fs x ` k))) absolutely_integrable_on s"
show ?case unfolding image_insert
apply(subst Sup_insert_finite) apply(rule finite_imageI[OF insert(1)])
proof(cases "k={}") case True
thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto
next case False thus ?P apply(subst if_not_P) defer
apply(rule absolutely_integrable_max[where 'n=real,unfolded Eucl_real_simps])
defer apply(rule insert(3)[OF False]) using insert(5) by auto
qed qed auto
subsection {* Dominated convergence. *}
lemma dominated_convergence: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
assumes "\<And>k. (f k) integrable_on s" "h integrable_on s"
"\<And>k. \<forall>x \<in> s. norm(f k x) \<le> (h x)"
"\<forall>x \<in> s. ((\<lambda>k. f k x) ---> g x) sequentially"
shows "g integrable_on s" "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
proof- have "\<And>m. (\<lambda>x. Inf {f j x |j. m \<le> j}) integrable_on s \<and>
((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) --->
integral s (\<lambda>x. Inf {f j x |j. m \<le> j}))sequentially"
proof(rule monotone_convergence_decreasing,safe) fix m::nat
show "bounded {integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}}) |k. True}"
unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
proof safe fix k::nat
show "norm (integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) \<le> integral s h"
apply(rule integral_norm_bound_integral) unfolding simple_image
apply(rule absolutely_integrable_onD) apply(rule absolutely_integrable_inf_real)
prefer 5 unfolding real_norm_def apply(rule) apply(rule Inf_abs_ge)
prefer 5 apply rule apply(rule_tac g=h in absolutely_integrable_integrable_bound_real)
using assms unfolding real_norm_def by auto
qed fix k::nat show "(\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}}) integrable_on s"
unfolding simple_image apply(rule absolutely_integrable_onD)
apply(rule absolutely_integrable_inf_real) prefer 3
using absolutely_integrable_integrable_bound_real[OF assms(3,1,2)] by auto
fix x assume x:"x\<in>s" show "Inf {f j x |j. j \<in> {m..m + Suc k}}
\<le> Inf {f j x |j. j \<in> {m..m + k}}" apply(rule Inf_ge) unfolding setge_def
defer apply rule apply(subst Inf_finite_le_iff) prefer 3
apply(rule_tac x=xa in bexI) by auto
let ?S = "{f j x| j. m \<le> j}" def i \<equiv> "Inf ?S"
show "((\<lambda>k. Inf {f j x |j. j \<in> {m..m + k}}) ---> i) sequentially"
unfolding Lim_sequentially
proof safe case goal1 note e=this have i:"isGlb UNIV ?S i" unfolding i_def apply(rule Inf)
defer apply(rule_tac x="- h x - 1" in exI) unfolding setge_def
proof safe case goal1 thus ?case using assms(3)[rule_format,OF x, of j] by auto
qed auto
have "\<exists>y\<in>?S. \<not> y \<ge> i + e"
proof(rule ccontr) case goal1 hence "i \<ge> i + e" apply-
apply(rule isGlb_le_isLb[OF i]) apply(rule isLbI) unfolding setge_def by fastsimp+
thus False using e by auto
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 case goal1
have *:"\<And>y ix. y < i + e \<longrightarrow> i \<le> ix \<longrightarrow> ix \<le> y \<longrightarrow> abs(ix - i) < e" by arith
show ?case unfolding dist_real_def apply(rule *[rule_format,OF y(2)])
unfolding i_def apply(rule real_le_inf_subset) prefer 3
apply(rule,rule isGlbD1[OF i]) prefer 3 apply(subst Inf_finite_le_iff)
prefer 3 apply(rule_tac x=y in bexI) using N goal1 by auto
qed qed qed note dec1 = conjunctD2[OF this]
have "\<And>m. (\<lambda>x. Sup {f j x |j. m \<le> j}) integrable_on s \<and>
((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) --->
integral s (\<lambda>x. Sup {f j x |j. m \<le> j})) sequentially"
proof(rule monotone_convergence_increasing,safe) fix m::nat
show "bounded {integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}}) |k. True}"
unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
proof safe fix k::nat
show "norm (integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) \<le> integral s h"
apply(rule integral_norm_bound_integral) unfolding simple_image
apply(rule absolutely_integrable_onD) apply(rule absolutely_integrable_sup_real)
prefer 5 unfolding real_norm_def apply(rule) apply(rule Sup_abs_le)
prefer 5 apply rule apply(rule_tac g=h in absolutely_integrable_integrable_bound_real)
using assms unfolding real_norm_def by auto
qed fix k::nat show "(\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}}) integrable_on s"
unfolding simple_image apply(rule absolutely_integrable_onD)
apply(rule absolutely_integrable_sup_real) prefer 3
using absolutely_integrable_integrable_bound_real[OF assms(3,1,2)] by auto
fix x assume x:"x\<in>s" show "Sup {f j x |j. j \<in> {m..m + Suc k}}
\<ge> Sup {f j x |j. j \<in> {m..m + k}}" apply(rule Sup_le) unfolding setle_def
defer apply rule apply(subst Sup_finite_ge_iff) prefer 3 apply(rule_tac x=y in bexI) by auto
let ?S = "{f j x| j. m \<le> j}" def i \<equiv> "Sup ?S"
show "((\<lambda>k. Sup {f j x |j. j \<in> {m..m + k}}) ---> i) sequentially"
unfolding Lim_sequentially
proof safe case goal1 note e=this have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
defer apply(rule_tac x="h x" in exI) unfolding setle_def
proof safe 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 - e"
proof(rule ccontr) case goal1 hence "i \<le> i - e" apply-
apply(rule isLub_le_isUb[OF i]) apply(rule isUbI) unfolding setle_def by fastsimp+
thus False using e by auto
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 case goal1
have *:"\<And>y ix. i - e < y \<longrightarrow> ix \<le> i \<longrightarrow> y \<le> ix \<longrightarrow> abs(ix - i) < e" by arith
show ?case unfolding dist_real_def apply(rule *[rule_format,OF y(2)])
unfolding i_def apply(rule real_ge_sup_subset) prefer 3
apply(rule,rule isLubD1[OF i]) prefer 3 apply(subst Sup_finite_ge_iff)
prefer 3 apply(rule_tac x=y in bexI) using N goal1 by auto
qed qed qed note inc1 = conjunctD2[OF this]
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
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)
proof safe fix k::nat
show "norm (integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) \<le> integral s h"
apply(rule integral_norm_bound_integral) apply fact+
unfolding real_norm_def apply(rule) apply(rule Inf_abs_ge) using assms(3) by auto
qed fix k::nat and x 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. k \<le> j} \<le> Inf {f j x |j. Suc k \<le> j}" apply-
apply(rule real_le_inf_subset) prefer 3 unfolding setge_def
apply(rule_tac x="- h x" in exI) apply safe apply(rule *)
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto
show "((\<lambda>k. Inf {f j x |j. k \<le> j}) ---> g x) sequentially" unfolding Lim_sequentially
proof safe case goal1 hence "0<e/2" by auto
from assms(4)[unfolded Lim_sequentially,rule_format,OF x this] guess N .. note N=this
show ?case apply(rule_tac x=N in exI,safe) unfolding dist_real_def
apply(rule le_less_trans[of _ "e/2"]) apply(rule Inf_asclose) apply safe
defer apply(rule less_imp_le) using N goal1 unfolding dist_real_def by auto
qed qed note inc2 = conjunctD2[OF this]
have "g integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) ---> integral s g) sequentially"
apply(rule monotone_convergence_decreasing,safe) apply fact
proof- show "bounded {integral s (\<lambda>x. Sup {f j x |j. k \<le> j}) |k. True}"
unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
proof safe fix k::nat
show "norm (integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) \<le> integral s h"
apply(rule integral_norm_bound_integral) apply fact+
unfolding real_norm_def apply(rule) apply(rule Sup_abs_le) using assms(3) by auto
qed fix k::nat and x assume x:"x\<in>s"
show "Sup {f j x |j. k \<le> j} \<ge> Sup {f j x |j. Suc k \<le> j}" apply-
apply(rule real_ge_sup_subset) prefer 3 unfolding setle_def
apply(rule_tac x="h x" in exI) apply safe
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto
show "((\<lambda>k. Sup {f j x |j. k \<le> j}) ---> g x) sequentially" unfolding Lim_sequentially
proof safe case goal1 hence "0<e/2" by auto
from assms(4)[unfolded Lim_sequentially,rule_format,OF x this] guess N .. note N=this
show ?case apply(rule_tac x=N in exI,safe) unfolding dist_real_def
apply(rule le_less_trans[of _ "e/2"]) apply(rule Sup_asclose) apply safe
defer apply(rule less_imp_le) using N goal1 unfolding dist_real_def by auto
qed qed note dec2 = conjunctD2[OF this]
show "g integrable_on s" by fact
show "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" unfolding Lim_sequentially
proof safe case goal1
from inc2(2)[unfolded Lim_sequentially,rule_format,OF goal1] guess N1 .. note N1=this[unfolded dist_real_def]
from dec2(2)[unfolded Lim_sequentially,rule_format,OF goal1] guess N2 .. note N2=this[unfolded dist_real_def]
show ?case apply(rule_tac x="N1+N2" in exI,safe)
proof- fix n assume n:"n \<ge> N1 + N2"
have *:"\<And>i0 i i1 g. \<bar>i0 - g\<bar> < e \<longrightarrow> \<bar>i1 - g\<bar> < e \<longrightarrow> i0 \<le> i \<longrightarrow> i \<le> i1 \<longrightarrow> \<bar>i - g\<bar> < e" by arith
show "dist (integral s (f n)) (integral s g) < e" unfolding dist_real_def
apply(rule *[rule_format,OF N1[rule_format] N2[rule_format], of n n])
proof- 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" 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 Inf_lower[where z="- h x"]) defer
apply(rule *) using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto
qed show "integral s (f n) \<le> integral s (\<lambda>x. Sup {f j x |j. n \<le> j})"
proof(rule integral_le[OF assms(1) inc1(1)],safe)
fix x assume x:"x \<in> s"
show "f n x \<le> Sup {f j x |j. n \<le> j}" apply(rule Sup_upper[where z="h x"]) defer
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto
qed qed(insert n,auto) qed qed qed
declare [[smt_certificates=""]]
declare [[smt_fixed=false]]
end