author | huffman |
Sun, 09 May 2010 22:51:11 -0700 | |
changeset 36778 | 739a9379e29b |
parent 36725 | 34c36a5cb808 |
child 36844 | 5f9385ecc1a7 |
permissions | -rw-r--r-- |
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header {* Kurzweil-Henstock gauge integration in many dimensions. *} |
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(* Author: John Harrison |
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Translation from HOL light: Robert Himmelmann, TU Muenchen *) |
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theory Integration |
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imports Derivative SMT |
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begin |
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declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]] |
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declare [[smt_fixed=true]] |
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declare [[z3_proofs=true]] |
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subsection {* Sundries *} |
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lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto |
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lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto |
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lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto |
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lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto |
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declare smult_conv_scaleR[simp] |
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lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast |
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lemma linear_simps: assumes "bounded_linear f" |
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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)" |
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apply(rule_tac[!] additive.add additive.minus additive.diff additive.zero bounded_linear.scaleR) |
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using assms unfolding bounded_linear_def additive_def by auto |
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lemma bounded_linearI:assumes "\<And>x y. f (x + y) = f x + f y" |
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"\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K" |
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shows "bounded_linear f" |
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unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto |
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lemma real_le_inf_subset: |
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assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s" shows "Inf s <= Inf (t::real set)" |
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apply(rule isGlb_le_isLb) apply(rule Inf[OF assms(1)]) |
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using assms apply-apply(erule exE) apply(rule_tac x=b in exI) |
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unfolding isLb_def setge_def by auto |
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lemma real_ge_sup_subset: |
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assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b" shows "Sup s >= Sup (t::real set)" |
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apply(rule isLub_le_isUb) apply(rule Sup[OF assms(1)]) |
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using assms apply-apply(erule exE) apply(rule_tac x=b in exI) |
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unfolding isUb_def setle_def by auto |
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lemma dist_trans[simp]:"dist (vec1 x) (vec1 y) = dist x (y::real)" |
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unfolding dist_real_def dist_vec1 .. |
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lemma Lim_trans[simp]: fixes f::"'a \<Rightarrow> real" |
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shows "((\<lambda>x. vec1 (f x)) ---> vec1 l) net \<longleftrightarrow> (f ---> l) net" |
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using assms unfolding Lim dist_trans .. |
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lemma bounded_linear_component[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)" |
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apply(rule bounded_linearI[where K=1]) |
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using component_le_norm[of _ k] unfolding real_norm_def by auto |
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lemma bounded_vec1[intro]: "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))" |
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unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI) by auto |
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lemma transitive_stepwise_lt_eq: |
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assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)" |
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shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r") |
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proof(safe) assume ?r fix n m::nat assume "m < n" thus "R m n" apply- |
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proof(induct n arbitrary: m) case (Suc n) show ?case |
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proof(cases "m < n") case True |
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show ?thesis apply(rule assms[OF Suc(1)[OF True]]) using `?r` by auto |
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next case False hence "m = n" using Suc(2) by auto |
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thus ?thesis using `?r` by auto |
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qed qed auto qed auto |
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lemma transitive_stepwise_gt: |
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assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) " |
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shows "\<forall>n>m. R m n" |
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proof- have "\<forall>m. \<forall>n>m. R m n" apply(subst transitive_stepwise_lt_eq) |
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apply(rule assms) apply(assumption,assumption) using assms(2) by auto |
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thus ?thesis by auto qed |
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lemma transitive_stepwise_le_eq: |
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assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" |
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shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r") |
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proof safe assume ?r fix m n::nat assume "m\<le>n" thus "R m n" apply- |
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proof(induct n arbitrary: m) case (Suc n) show ?case |
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proof(cases "m \<le> n") case True show ?thesis apply(rule assms(2)) |
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apply(rule Suc(1)[OF True]) using `?r` by auto |
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next case False hence "m = Suc n" using Suc(2) by auto |
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thus ?thesis using assms(1) by auto |
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qed qed(insert assms(1), auto) qed auto |
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lemma transitive_stepwise_le: |
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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) " |
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shows "\<forall>n\<ge>m. R m n" |
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proof- have "\<forall>m. \<forall>n\<ge>m. R m n" apply(subst transitive_stepwise_le_eq) |
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apply(rule assms) apply(rule assms,assumption,assumption) using assms(3) by auto |
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thus ?thesis by auto qed |
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subsection {* Some useful lemmas about intervals. *} |
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100 |
lemma empty_as_interval: "{} = {1..0::real^'n}" |
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apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval |
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using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto |
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lemma interior_subset_union_intervals: |
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assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}" |
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shows "interior i \<subseteq> interior s" proof- |
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have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5) |
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unfolding assms(1,2) interior_closed_interval by auto |
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moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed) |
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using assms(4) unfolding assms(1,2) by auto |
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ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior) |
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unfolding assms(1,2) interior_closed_interval by auto qed |
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114 |
lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set" |
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assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}" |
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shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1 |
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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) |
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unfolding open_subset_interior[OF open_ball] using interior_subset by auto |
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have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto |
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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 |
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thus ?case proof(induct rule:finite_induct) |
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case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next |
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123 |
case (insert i f) guess x using insert(5) .. note x = this |
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124 |
then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this |
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125 |
guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this |
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126 |
show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto |
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127 |
then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] .. |
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128 |
hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto |
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129 |
hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 by auto hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto |
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130 |
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 |
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131 |
case True show ?thesis proof(cases "x\<in>{a<..<b}") |
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132 |
case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] .. |
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133 |
thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI) |
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134 |
unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next |
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135 |
case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) |
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136 |
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 |
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137 |
hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE) |
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138 |
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) |
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139 |
fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto |
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36587 | 140 |
hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto |
35172 | 141 |
hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps) |
142 |
hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed |
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moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof |
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144 |
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)" |
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145 |
apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"]) |
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146 |
unfolding norm_scaleR norm_basis by auto |
|
36587 | 147 |
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) |
148 |
finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed |
|
35172 | 149 |
ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto |
150 |
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) |
|
151 |
fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto |
|
36587 | 152 |
hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto |
35172 | 153 |
hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps) |
154 |
hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed |
|
155 |
moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof |
|
156 |
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)" |
|
157 |
apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"]) |
|
158 |
unfolding norm_scaleR norm_basis by auto |
|
36587 | 159 |
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) |
160 |
finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed |
|
35172 | 161 |
ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed |
162 |
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 |
|
163 |
thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this |
|
164 |
guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e .. |
|
165 |
hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto |
|
166 |
thus False using `t\<in>f` assms(4) by auto qed |
|
167 |
subsection {* Bounds on intervals where they exist. *} |
|
168 |
||
169 |
definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})" |
|
170 |
||
171 |
definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})" |
|
172 |
||
173 |
lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b" |
|
174 |
using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE) |
|
175 |
apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer |
|
176 |
apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI) |
|
177 |
unfolding mem_interval using assms by auto |
|
178 |
||
179 |
lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a" |
|
180 |
using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE) |
|
181 |
apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer |
|
182 |
apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI) |
|
183 |
unfolding mem_interval using assms by auto |
|
184 |
||
185 |
lemmas interval_bounds = interval_upperbound interval_lowerbound |
|
186 |
||
187 |
lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a" |
|
188 |
using assms unfolding interval_ne_empty by auto |
|
189 |
||
190 |
lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)" |
|
191 |
apply(rule interval_upperbound) by auto |
|
192 |
||
193 |
lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)" |
|
194 |
apply(rule interval_lowerbound) by auto |
|
195 |
||
196 |
lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1 |
|
197 |
||
198 |
subsection {* Content (length, area, volume...) of an interval. *} |
|
199 |
||
200 |
definition "content (s::(real^'n) set) = |
|
201 |
(if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))" |
|
202 |
||
203 |
lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}" |
|
204 |
unfolding interval_eq_empty unfolding not_ex not_less by assumption |
|
205 |
||
206 |
lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i" |
|
207 |
shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)" |
|
208 |
using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto |
|
209 |
||
210 |
lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)" |
|
211 |
apply(rule content_closed_interval) using assms unfolding interval_ne_empty . |
|
212 |
||
213 |
lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a" |
|
214 |
using content_closed_interval[of a b] by auto |
|
215 |
||
216 |
lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto |
|
217 |
||
218 |
lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof- |
|
219 |
have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto |
|
220 |
have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto |
|
221 |
thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed |
|
222 |
||
223 |
lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}") |
|
224 |
case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption |
|
225 |
have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0" |
|
226 |
apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto |
|
227 |
thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto) |
|
228 |
||
229 |
lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}" |
|
230 |
proof- have help_lemma1: "\<forall>i. a$i < b$i \<Longrightarrow> \<forall>i. a$i \<le> ((b$i)::real)" apply(rule,erule_tac x=i in allE) by auto |
|
231 |
show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos) |
|
232 |
using assms apply(erule_tac x=x in allE) by auto qed |
|
233 |
||
234 |
lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})" |
|
235 |
apply(rule content_pos_lt) by auto |
|
236 |
||
237 |
lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}") |
|
238 |
case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply- |
|
239 |
apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next |
|
240 |
guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less] |
|
241 |
show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV] |
|
242 |
apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer |
|
243 |
apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed |
|
244 |
||
245 |
lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto |
|
246 |
||
247 |
lemma content_closed_interval_cases: |
|
248 |
"content {a..b} = (if \<forall>i. a$i \<le> b$i then setprod (\<lambda>i. b$i - a$i) UNIV else 0)" apply(rule cond_cases) |
|
249 |
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 |
|
250 |
||
251 |
lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}" |
|
252 |
unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto |
|
253 |
||
254 |
lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a" |
|
255 |
unfolding content_eq_0 by auto |
|
256 |
||
257 |
lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)" |
|
258 |
apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}" |
|
259 |
hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i. a$i < b$i" unfolding content_eq_0 not_ex not_le by auto qed |
|
260 |
||
261 |
lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto |
|
262 |
||
263 |
lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}") |
|
264 |
case True thus ?thesis using content_pos_le[of c d] by auto next |
|
265 |
case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto |
|
266 |
hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto |
|
267 |
have "{c..d} \<noteq> {}" using assms False by auto |
|
268 |
hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto |
|
269 |
show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne] |
|
270 |
unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n |
|
271 |
show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto |
|
272 |
show "b $ i - a $ i \<le> d $ i - c $ i" |
|
273 |
using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i] |
|
274 |
using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed |
|
275 |
||
276 |
lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0" |
|
277 |
unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto |
|
278 |
||
279 |
subsection {* The notion of a gauge --- simply an open set containing the point. *} |
|
280 |
||
281 |
definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))" |
|
282 |
||
283 |
lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g" |
|
284 |
using assms unfolding gauge_def by auto |
|
285 |
||
286 |
lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto |
|
287 |
||
288 |
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))" |
|
289 |
unfolding gauge_def by auto |
|
290 |
||
35751 | 291 |
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto |
35172 | 292 |
|
293 |
lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto |
|
294 |
||
35751 | 295 |
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))" |
35172 | 296 |
unfolding gauge_def by auto |
297 |
||
298 |
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- |
|
299 |
have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis |
|
300 |
unfolding gauge_def unfolding * |
|
301 |
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed |
|
302 |
||
303 |
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) |
|
304 |
||
305 |
subsection {* Divisions. *} |
|
306 |
||
307 |
definition division_of (infixl "division'_of" 40) where |
|
308 |
"s division_of i \<equiv> |
|
309 |
finite s \<and> |
|
310 |
(\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and> |
|
311 |
(\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and> |
|
312 |
(\<Union>s = i)" |
|
313 |
||
314 |
lemma division_ofD[dest]: assumes "s division_of i" |
|
315 |
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})" |
|
316 |
"\<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 |
|
317 |
||
318 |
lemma division_ofI: |
|
319 |
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})" |
|
320 |
"\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" |
|
321 |
shows "s division_of i" using assms unfolding division_of_def by auto |
|
322 |
||
323 |
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s" |
|
324 |
unfolding division_of_def by auto |
|
325 |
||
326 |
lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}" |
|
327 |
unfolding division_of_def by auto |
|
328 |
||
329 |
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto |
|
330 |
||
331 |
lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof |
|
332 |
assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" |
|
333 |
ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto } |
|
334 |
ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next |
|
335 |
assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]] |
|
336 |
{ fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto } |
|
337 |
moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed |
|
338 |
||
339 |
lemma elementary_empty: obtains p where "p division_of {}" |
|
340 |
unfolding division_of_trivial by auto |
|
341 |
||
342 |
lemma elementary_interval: obtains p where "p division_of {a..b}" |
|
343 |
by(metis division_of_trivial division_of_self) |
|
344 |
||
345 |
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k" |
|
346 |
unfolding division_of_def by auto |
|
347 |
||
348 |
lemma forall_in_division: |
|
349 |
"d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))" |
|
350 |
unfolding division_of_def by fastsimp |
|
351 |
||
352 |
lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)" |
|
353 |
apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)] |
|
354 |
show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto |
|
355 |
{ 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 |
|
356 |
show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto } |
|
357 |
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 |
|
358 |
show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto |
|
359 |
||
360 |
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto |
|
361 |
||
362 |
lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0" |
|
363 |
unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)]) |
|
364 |
apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed |
|
365 |
||
366 |
lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)" |
|
367 |
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- |
|
368 |
let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto |
|
369 |
show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto |
|
370 |
moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto |
|
371 |
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 |
|
372 |
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto |
|
373 |
{ 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 |
|
374 |
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 |
|
375 |
guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this |
|
376 |
guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this |
|
377 |
show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2 |
|
378 |
assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto |
|
379 |
assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto |
|
380 |
assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto |
|
381 |
have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow> |
|
382 |
interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow> |
|
383 |
interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2) |
|
384 |
\<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto |
|
385 |
show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior) |
|
386 |
using division_ofD(5)[OF assms(1) k1(2) k2(2)] |
|
387 |
using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed |
|
388 |
||
389 |
lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i" |
|
390 |
shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}") |
|
391 |
case True show ?thesis unfolding True and division_of_trivial by auto next |
|
392 |
have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto |
|
393 |
case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed |
|
394 |
||
395 |
lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)" |
|
396 |
shows "\<exists>p. p division_of (s \<inter> t)" |
|
397 |
by(rule,rule division_inter[OF assms]) |
|
398 |
||
399 |
lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)" |
|
400 |
shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct) |
|
401 |
case (insert x f) show ?case proof(cases "f={}") |
|
402 |
case True thus ?thesis unfolding True using insert by auto next |
|
403 |
case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] .. |
|
404 |
moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately |
|
405 |
show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto |
|
406 |
||
407 |
lemma division_disjoint_union: |
|
408 |
assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}" |
|
409 |
shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) |
|
410 |
note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)] |
|
411 |
show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto |
|
412 |
show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto |
|
413 |
{ 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 = {}" |
|
414 |
{ 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)]] |
|
415 |
using assms(3) by blast } moreover |
|
416 |
{ 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)]] |
|
417 |
using assms(3) by blast} ultimately |
|
418 |
show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto } |
|
419 |
fix k assume k:"k \<in> p1 \<union> p2" show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto |
|
420 |
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 |
|
421 |
||
422 |
lemma partial_division_extend_1: |
|
423 |
assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}" |
|
424 |
obtains p where "p division_of {a..b}" "{c..d} \<in> p" |
|
425 |
proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto |
|
426 |
guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this |
|
427 |
def \<pi>' \<equiv> "inv_into {1..n} \<pi>" |
|
428 |
have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto |
|
429 |
hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto |
|
430 |
have \<pi>\<pi>'[simp]:"\<And>i. \<pi> (\<pi>' i) = i" unfolding \<pi>'_def apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto |
|
431 |
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 |
|
432 |
have "{c..d} \<noteq> {}" using assms by auto |
|
433 |
let ?p1 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else if \<pi>' i = l then c$\<pi> l else b$i)}" |
|
434 |
let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else if \<pi>' i = l then d$\<pi> l else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else b$i)}" |
|
435 |
let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}" |
|
436 |
have abcd:"\<And>i. 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 simp add:not_le not_less) |
|
437 |
show ?thesis apply(rule that[of ?p]) apply(rule division_ofI) |
|
438 |
proof- have "\<And>i. \<pi>' i < Suc n" |
|
439 |
proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i" |
|
440 |
hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto |
|
441 |
qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)" |
|
442 |
"d = (\<chi> i. if \<pi>' i < Suc n then d $ i else if \<pi>' i = n + 1 then c $ \<pi> (n + 1) else b $ i)" |
|
443 |
unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto |
|
444 |
thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto |
|
445 |
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}" |
|
446 |
unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr) |
|
447 |
proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}" |
|
448 |
then guess i unfolding mem_interval not_all .. note i=this |
|
449 |
show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE) |
|
450 |
apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto |
|
451 |
qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p" |
|
452 |
proof- fix x assume x:"x\<in>{a..b}" |
|
453 |
{ presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast } |
|
454 |
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)}" |
|
455 |
assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all .. |
|
456 |
hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI) |
|
457 |
hence M:"finite ?M" "?M \<noteq> {}" by auto |
|
458 |
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]] |
|
459 |
Min_gr_iff[OF M,unfolded l_def[symmetric]] |
|
460 |
have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le |
|
461 |
apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2) |
|
462 |
proof- assume as:"x $ \<pi> l < c $ \<pi> l" |
|
463 |
show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta |
|
464 |
proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto |
|
465 |
thus ?case using as x[unfolded mem_interval,rule_format,of i] |
|
466 |
apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"]) |
|
467 |
qed |
|
468 |
next assume as:"x $ \<pi> l > d $ \<pi> l" |
|
469 |
show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta |
|
470 |
proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto |
|
471 |
thus ?case using as x[unfolded mem_interval,rule_format,of i] |
|
472 |
apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"]) |
|
473 |
qed qed |
|
474 |
thus "x \<in> \<Union>?p" using l(2) by blast |
|
475 |
qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast) |
|
476 |
||
477 |
show "finite ?p" by auto |
|
478 |
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 |
|
479 |
show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) |
|
480 |
proof- fix i::'n and x assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto |
|
481 |
ultimately show "a$i \<le> x$i" "x$i \<le> b$i" using abcd[of i] using l by(auto elim:disjE elim!:allE[where x=i] simp add:vector_le_def) |
|
482 |
qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI) |
|
483 |
proof- case goal1 thus ?case using abcd[of x] by auto |
|
484 |
next case goal2 thus ?case using abcd[of x] by auto |
|
485 |
qed thus "k \<noteq> {}" using k by auto |
|
486 |
show "\<exists>a b. k = {a..b}" using k by auto |
|
487 |
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 |
|
488 |
{ fix k k' l l' |
|
489 |
assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" |
|
490 |
assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" |
|
491 |
assume "l \<le> l'" fix x |
|
492 |
have "x \<notin> interior k \<inter> interior k'" |
|
493 |
proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'" |
|
494 |
case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le) |
|
495 |
hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse) |
|
496 |
have ln:"l < n + 1" |
|
497 |
proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto |
|
498 |
hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le) |
|
499 |
hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse) |
|
500 |
thus False using `k\<noteq>k'` k' by auto |
|
501 |
qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto |
|
502 |
have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply- |
|
503 |
proof(erule disjE) |
|
504 |
assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
|
505 |
show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto |
|
506 |
next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
|
507 |
show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto |
|
508 |
qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval |
|
509 |
by(auto elim!:allE[where x="\<pi> l"]) |
|
510 |
next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto |
|
511 |
hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto |
|
512 |
note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)] |
|
513 |
assume x:"x \<in> interior k \<inter> interior k'" |
|
514 |
show False using l(1) l'(1) apply- |
|
515 |
proof(erule_tac[!] disjE)+ |
|
516 |
assume as:"k = ?p1 l" "k' = ?p1 l'" |
|
517 |
note * = x[unfolded as Int_iff interior_closed_interval mem_interval] |
|
518 |
have "l \<noteq> l'" using k'(2)[unfolded as] by auto |
|
519 |
thus False using * by(smt Cart_lambda_beta \<pi>l) |
|
520 |
next assume as:"k = ?p2 l" "k' = ?p2 l'" |
|
521 |
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
|
522 |
have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto |
|
523 |
thus False using *[of "\<pi> l"] *[of "\<pi> l'"] |
|
524 |
unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto |
|
525 |
next assume as:"k = ?p1 l" "k' = ?p2 l'" |
|
526 |
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
|
527 |
show False using *[of "\<pi> l"] *[of "\<pi> l'"] |
|
528 |
unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt |
|
529 |
next assume as:"k = ?p2 l" "k' = ?p1 l'" |
|
530 |
note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
|
531 |
show False using *[of "\<pi> l"] *[of "\<pi> l'"] |
|
532 |
unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt |
|
533 |
qed qed } |
|
534 |
from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'" |
|
535 |
apply - apply(cases "l' \<le> l") using k'(2) by auto |
|
536 |
thus "interior k \<inter> interior k' = {}" by auto |
|
537 |
qed qed |
|
538 |
||
539 |
lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}" |
|
540 |
obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}") |
|
541 |
case True guess q apply(rule elementary_interval[of a b]) . |
|
542 |
thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next |
|
543 |
case False note p = division_ofD[OF assms(1)] |
|
544 |
have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1 |
|
545 |
guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this |
|
546 |
have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto |
|
547 |
guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed |
|
548 |
guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]] |
|
549 |
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- |
|
550 |
fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI) |
|
551 |
using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed |
|
552 |
hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters) |
|
553 |
apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto |
|
554 |
then guess d .. note d = this |
|
555 |
show ?thesis apply(rule that[of "d \<union> p"]) proof- |
|
556 |
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 |
|
557 |
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" |
|
558 |
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 |
|
559 |
show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)]) |
|
560 |
apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule) |
|
561 |
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 |
|
562 |
show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"]) |
|
563 |
defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]] |
|
564 |
show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto |
|
565 |
show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto |
|
566 |
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}))" |
|
567 |
apply(rule subset_interior *)+ using k by auto qed qed qed auto qed |
|
568 |
||
569 |
lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)" |
|
570 |
unfolding division_of_def by(metis bounded_Union bounded_interval) |
|
571 |
||
572 |
lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}" |
|
573 |
by(meson elementary_bounded bounded_subset_closed_interval) |
|
574 |
||
575 |
lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}" |
|
576 |
obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}") |
|
577 |
case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next |
|
578 |
case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}") |
|
579 |
have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto |
|
580 |
case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union) |
|
581 |
using false True assms using interior_subset by auto next |
|
582 |
case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto |
|
583 |
have *:"{u..v} \<subseteq> {c..d}" using uv by auto |
|
584 |
guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)] |
|
585 |
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 |
|
586 |
show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union) |
|
587 |
apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer |
|
588 |
unfolding interior_inter[THEN sym] proof- |
|
589 |
have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto |
|
590 |
have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" |
|
591 |
apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto |
|
592 |
also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto |
|
593 |
finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed |
|
594 |
||
595 |
lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)" |
|
596 |
"\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}" |
|
597 |
shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+ |
|
598 |
apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)]) |
|
599 |
using division_ofD[OF assms(2)] by auto |
|
600 |
||
601 |
lemma elementary_union_interval: assumes "p division_of \<Union>p" |
|
602 |
obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof- |
|
603 |
note assm=division_ofD[OF assms] |
|
604 |
have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto |
|
605 |
have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto |
|
606 |
{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis" |
|
607 |
"p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis" |
|
608 |
thus thesis by auto |
|
609 |
next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) . |
|
610 |
thus thesis apply(rule_tac that[of p]) unfolding as by auto |
|
611 |
next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto |
|
612 |
next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}" |
|
613 |
show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI) |
|
614 |
unfolding finite_insert apply(rule assm(1)) unfolding Union_insert |
|
615 |
using assm(2-4) as apply- by(fastsimp dest: assm(5))+ |
|
616 |
next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}" |
|
617 |
have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1 |
|
618 |
from assm(4)[OF this] guess c .. then guess d .. |
|
619 |
thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto |
|
620 |
qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]] |
|
621 |
let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}" |
|
622 |
show thesis apply(rule that[of "?D"]) proof(rule division_ofI) |
|
623 |
have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto |
|
624 |
show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto |
|
625 |
show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym] |
|
626 |
using q(6) by auto |
|
627 |
fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto |
|
628 |
show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto |
|
629 |
fix k' assume k':"k'\<in>?D" "k\<noteq>k'" |
|
630 |
obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto |
|
631 |
obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto |
|
632 |
show "interior k \<inter> interior k' = {}" proof(cases "x=x'") |
|
633 |
case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto |
|
634 |
next case False |
|
635 |
{ presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" |
|
636 |
"k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis" |
|
637 |
thus ?thesis by auto } |
|
638 |
{ assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto } |
|
639 |
{ assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto } |
|
640 |
assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}" |
|
641 |
guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this |
|
642 |
have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto |
|
643 |
hence "interior k \<subseteq> interior x" apply- |
|
644 |
apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover |
|
645 |
guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this |
|
646 |
have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto |
|
647 |
hence "interior k' \<subseteq> interior x'" apply- |
|
648 |
apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto |
|
649 |
ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto |
|
650 |
qed qed } qed |
|
651 |
||
652 |
lemma elementary_unions_intervals: |
|
653 |
assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}" |
|
654 |
obtains p where "p division_of (\<Union>f)" proof- |
|
655 |
have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) |
|
656 |
show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto |
|
657 |
fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f" |
|
658 |
from this(3) guess p .. note p=this |
|
659 |
from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this |
|
660 |
have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto |
|
661 |
show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b] |
|
662 |
unfolding Union_insert ab * by auto |
|
663 |
qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed |
|
664 |
||
665 |
lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)" |
|
666 |
obtains p where "p division_of (s \<union> t)" |
|
667 |
proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto |
|
668 |
hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto |
|
669 |
show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"]) |
|
670 |
unfolding * prefer 3 apply(rule_tac p=p in that) |
|
671 |
using assms[unfolded division_of_def] by auto qed |
|
672 |
||
673 |
lemma partial_division_extend: fixes t::"(real^'n) set" |
|
674 |
assumes "p division_of s" "q division_of t" "s \<subseteq> t" |
|
675 |
obtains r where "p \<subseteq> r" "r division_of t" proof- |
|
676 |
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)] |
|
677 |
obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto |
|
678 |
guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]]) |
|
679 |
apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)] |
|
680 |
guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto |
|
681 |
then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" |
|
682 |
apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto |
|
683 |
{ fix x assume x:"x\<in>t" "x\<notin>s" |
|
684 |
hence "x\<in>\<Union>r1" unfolding r1 using ab by auto |
|
685 |
then guess r unfolding Union_iff .. note r=this moreover |
|
686 |
have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto |
|
687 |
thus False using x by auto qed |
|
688 |
ultimately have "x\<in>\<Union>(r1 - p)" by auto } |
|
689 |
hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto |
|
690 |
show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union) |
|
691 |
unfolding divp(6) apply(rule assms r2)+ |
|
692 |
proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}" |
|
693 |
proof(rule inter_interior_unions_intervals) |
|
694 |
show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto |
|
695 |
have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto |
|
696 |
show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule) |
|
697 |
fix m x assume as:"m\<in>r1-p" |
|
698 |
have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals) |
|
699 |
show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto |
|
700 |
show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto |
|
701 |
qed thus "interior s \<inter> interior m = {}" unfolding divp by auto |
|
702 |
qed qed |
|
703 |
thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto |
|
704 |
qed auto qed |
|
705 |
||
706 |
subsection {* Tagged (partial) divisions. *} |
|
707 |
||
708 |
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where |
|
709 |
"(s tagged_partial_division_of i) \<equiv> |
|
710 |
finite s \<and> |
|
711 |
(\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and> |
|
712 |
(\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2)) |
|
713 |
\<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))" |
|
714 |
||
715 |
lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i" |
|
716 |
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" |
|
717 |
"\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}" |
|
718 |
"\<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) = {}" |
|
719 |
using assms unfolding tagged_partial_division_of_def apply- by blast+ |
|
720 |
||
721 |
definition tagged_division_of (infixr "tagged'_division'_of" 40) where |
|
722 |
"(s tagged_division_of i) \<equiv> |
|
723 |
(s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
|
724 |
||
725 |
lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s" |
|
726 |
unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
|
727 |
||
728 |
lemma tagged_division_of: |
|
729 |
"(s tagged_division_of i) \<longleftrightarrow> |
|
730 |
finite s \<and> |
|
731 |
(\<forall>x k. (x,k) \<in> s |
|
732 |
\<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and> |
|
733 |
(\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2)) |
|
734 |
\<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and> |
|
735 |
(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
|
736 |
unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
|
737 |
||
738 |
lemma tagged_division_ofI: assumes |
|
739 |
"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}" |
|
740 |
"\<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) = {})" |
|
741 |
"(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
|
742 |
shows "s tagged_division_of i" |
|
743 |
unfolding tagged_division_of apply(rule) defer apply rule |
|
744 |
apply(rule allI impI conjI assms)+ apply assumption |
|
745 |
apply(rule, rule assms, assumption) apply(rule assms, assumption) |
|
746 |
using assms(1,5-) apply- by blast+ |
|
747 |
||
748 |
lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i" |
|
749 |
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}" |
|
750 |
"\<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) = {})" |
|
751 |
"(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+ |
|
752 |
||
753 |
lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i" |
|
754 |
proof(rule division_ofI) note assm=tagged_division_ofD[OF assms] |
|
755 |
show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto |
|
756 |
fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto |
|
757 |
thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+ |
|
758 |
fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto |
|
759 |
thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto |
|
760 |
qed |
|
761 |
||
762 |
lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i" |
|
763 |
shows "(snd ` s) division_of \<Union>(snd ` s)" |
|
764 |
proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms] |
|
765 |
show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto |
|
766 |
fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto |
|
767 |
thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto |
|
768 |
fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto |
|
769 |
thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto |
|
770 |
qed |
|
771 |
||
772 |
lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s" |
|
773 |
shows "t tagged_partial_division_of i" |
|
774 |
using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast |
|
775 |
||
776 |
lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector" |
|
777 |
assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0" |
|
778 |
shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)" |
|
779 |
proof- note assm=tagged_division_ofD[OF assms(1)] |
|
780 |
have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto |
|
781 |
show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero) |
|
782 |
show "finite p" using assm by auto |
|
783 |
fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" |
|
784 |
obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto |
|
785 |
have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto |
|
786 |
hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto |
|
787 |
hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto |
|
788 |
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 |
|
789 |
thus "d (snd x) = 0" unfolding ab by auto qed qed |
|
790 |
||
791 |
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto |
|
792 |
||
793 |
lemma tagged_division_of_empty: "{} tagged_division_of {}" |
|
794 |
unfolding tagged_division_of by auto |
|
795 |
||
796 |
lemma tagged_partial_division_of_trivial[simp]: |
|
797 |
"p tagged_partial_division_of {} \<longleftrightarrow> p = {}" |
|
798 |
unfolding tagged_partial_division_of_def by auto |
|
799 |
||
800 |
lemma tagged_division_of_trivial[simp]: |
|
801 |
"p tagged_division_of {} \<longleftrightarrow> p = {}" |
|
802 |
unfolding tagged_division_of by auto |
|
803 |
||
804 |
lemma tagged_division_of_self: |
|
805 |
"x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}" |
|
806 |
apply(rule tagged_division_ofI) by auto |
|
807 |
||
808 |
lemma tagged_division_union: |
|
809 |
assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}" |
|
810 |
shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)" |
|
811 |
proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)] |
|
812 |
show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto |
|
813 |
show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast |
|
814 |
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 |
|
815 |
show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast |
|
816 |
fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')" |
|
817 |
have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast |
|
818 |
show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1") |
|
819 |
apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5)) |
|
820 |
using p1(3) p2(3) using xk xk' by auto qed |
|
821 |
||
822 |
lemma tagged_division_unions: |
|
823 |
assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)" |
|
824 |
"\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" |
|
825 |
shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)" |
|
826 |
proof(rule tagged_division_ofI) |
|
827 |
note assm = tagged_division_ofD[OF assms(2)[rule_format]] |
|
828 |
show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto |
|
829 |
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 |
|
830 |
also have "\<dots> = \<Union>iset" using assm(6) by auto |
|
831 |
finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . |
|
832 |
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 |
|
833 |
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 |
|
834 |
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 |
|
835 |
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) |
|
836 |
using assms(3)[rule_format] subset_interior by blast |
|
837 |
show "interior k \<inter> interior k' = {}" apply(cases "i=i'") |
|
838 |
using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto |
|
839 |
qed |
|
840 |
||
841 |
lemma tagged_partial_division_of_union_self: |
|
842 |
assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))" |
|
843 |
apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto |
|
844 |
||
845 |
lemma tagged_division_of_union_self: assumes "p tagged_division_of s" |
|
846 |
shows "p tagged_division_of (\<Union>(snd ` p))" |
|
847 |
apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto |
|
848 |
||
849 |
subsection {* Fine-ness of a partition w.r.t. a gauge. *} |
|
850 |
||
851 |
definition fine (infixr "fine" 46) where |
|
852 |
"d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))" |
|
853 |
||
854 |
lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" |
|
855 |
shows "d fine s" using assms unfolding fine_def by auto |
|
856 |
||
857 |
lemma fineD[dest]: assumes "d fine s" |
|
858 |
shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto |
|
859 |
||
860 |
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p" |
|
861 |
unfolding fine_def by auto |
|
862 |
||
863 |
lemma fine_inters: |
|
864 |
"(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)" |
|
865 |
unfolding fine_def by blast |
|
866 |
||
867 |
lemma fine_union: |
|
868 |
"d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)" |
|
869 |
unfolding fine_def by blast |
|
870 |
||
871 |
lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)" |
|
872 |
unfolding fine_def by auto |
|
873 |
||
874 |
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p" |
|
875 |
unfolding fine_def by blast |
|
876 |
||
877 |
subsection {* Gauge integral. Define on compact intervals first, then use a limit. *} |
|
878 |
||
879 |
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where |
|
880 |
"(f has_integral_compact_interval y) i \<equiv> |
|
881 |
(\<forall>e>0. \<exists>d. gauge d \<and> |
|
882 |
(\<forall>p. p tagged_division_of i \<and> d fine p |
|
883 |
\<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))" |
|
884 |
||
885 |
definition has_integral (infixr "has'_integral" 46) where |
|
886 |
"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv> |
|
887 |
if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i |
|
888 |
else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} |
|
889 |
\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and> |
|
890 |
norm(z - y) < e))" |
|
891 |
||
892 |
lemma has_integral: |
|
893 |
"(f has_integral y) ({a..b}) \<longleftrightarrow> |
|
894 |
(\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p |
|
895 |
\<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))" |
|
896 |
unfolding has_integral_def has_integral_compact_interval_def by auto |
|
897 |
||
898 |
lemma has_integralD[dest]: assumes |
|
899 |
"(f has_integral y) ({a..b})" "e>0" |
|
900 |
obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p |
|
901 |
\<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e" |
|
902 |
using assms unfolding has_integral by auto |
|
903 |
||
904 |
lemma has_integral_alt: |
|
905 |
"(f has_integral y) i \<longleftrightarrow> |
|
906 |
(if (\<exists>a b. i = {a..b}) then (f has_integral y) i |
|
907 |
else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} |
|
908 |
\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) |
|
909 |
has_integral z) ({a..b}) \<and> |
|
910 |
norm(z - y) < e)))" |
|
911 |
unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto |
|
912 |
||
913 |
lemma has_integral_altD: |
|
914 |
assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0" |
|
915 |
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)" |
|
916 |
using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto |
|
917 |
||
918 |
definition integrable_on (infixr "integrable'_on" 46) where |
|
919 |
"(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i" |
|
920 |
||
921 |
definition "integral i f \<equiv> SOME y. (f has_integral y) i" |
|
922 |
||
923 |
lemma integrable_integral[dest]: |
|
924 |
"f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i" |
|
925 |
unfolding integrable_on_def integral_def by(rule someI_ex) |
|
926 |
||
927 |
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s" |
|
928 |
unfolding integrable_on_def by auto |
|
929 |
||
930 |
lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s" |
|
931 |
by auto |
|
932 |
||
35291
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
933 |
lemma has_integral_vec1: assumes "(f has_integral k) {a..b}" |
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
934 |
shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}" |
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
935 |
proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)" |
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
936 |
unfolding vec_sub Cart_eq by(auto simp add: split_beta) |
35291
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
937 |
show ?thesis using assms unfolding has_integral apply safe |
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
938 |
apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe) |
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
939 |
apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed |
ead7bfc30b26
Support for one-dimensional integration in Multivariate-Analysis
himmelma
parents:
35173
diff
changeset
|
940 |
|
35172 | 941 |
lemma setsum_content_null: |
942 |
assumes "content({a..b}) = 0" "p tagged_division_of {a..b}" |
|
943 |
shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)" |
|
944 |
proof(rule setsum_0',rule) fix y assume y:"y\<in>p" |
|
945 |
obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast |
|
946 |
note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]] |
|
947 |
from this(2) guess c .. then guess d .. note c_d=this |
|
948 |
have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto |
|
949 |
also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d] |
|
950 |
unfolding assms(1) c_d by auto |
|
951 |
finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" . |
|
952 |
qed |
|
953 |
||
954 |
subsection {* Some basic combining lemmas. *} |
|
955 |
||
956 |
lemma tagged_division_unions_exists: |
|
957 |
assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p" |
|
958 |
"\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)" |
|
959 |
obtains p where "p tagged_division_of i" "d fine p" |
|
960 |
proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]] |
|
961 |
show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym] |
|
962 |
apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer |
|
963 |
apply(rule fine_unions) using pfn by auto |
|
964 |
qed |
|
965 |
||
966 |
subsection {* The set we're concerned with must be closed. *} |
|
967 |
||
968 |
lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)" |
|
969 |
unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval) |
|
970 |
||
971 |
subsection {* General bisection principle for intervals; might be useful elsewhere. *} |
|
972 |
||
973 |
lemma interval_bisection_step: |
|
974 |
assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::real^'n})" |
|
975 |
obtains c d where "~(P{c..d})" |
|
976 |
"\<forall>i. 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" |
|
977 |
proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto |
|
978 |
note ab=this[unfolded interval_eq_empty not_ex not_less] |
|
979 |
{ fix f have "finite f \<Longrightarrow> |
|
980 |
(\<forall>s\<in>f. P s) \<Longrightarrow> |
|
981 |
(\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow> |
|
982 |
(\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)" |
|
983 |
proof(induct f rule:finite_induct) |
|
984 |
case empty show ?case using assms(1) by auto |
|
985 |
next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format]) |
|
986 |
apply rule defer apply rule defer apply(rule inter_interior_unions_intervals) |
|
987 |
using insert by auto |
|
988 |
qed } note * = this |
|
989 |
let ?A = "{{c..d} | c d. \<forall>i. (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)}" |
|
990 |
let ?PP = "\<lambda>c d. \<forall>i. 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" |
|
991 |
{ presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False" |
|
992 |
thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto } |
|
993 |
assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}" |
|
994 |
have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) |
|
995 |
let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) .. |
|
996 |
(\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}" |
|
997 |
have "?A \<subseteq> ?B" proof case goal1 |
|
998 |
then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format] |
|
999 |
have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto |
|
1000 |
show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI) |
|
1001 |
unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta |
|
1002 |
proof(rule_tac[1-2] allI) fix i show "c $ i = (if i \<in> {i. c $ i = a $ i} then a $ i else (a $ i + b $ i) / 2)" |
|
1003 |
"d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)" |
|
1004 |
using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps) |
|
1005 |
qed auto qed |
|
1006 |
thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto |
|
1007 |
fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) |
|
1008 |
note c_d=this[rule_format] |
|
1009 |
show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case |
|
1010 |
using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed |
|
1011 |
show "\<exists>a b. s = {a..b}" unfolding c_d by auto |
|
1012 |
fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+) |
|
1013 |
note e_f=this[rule_format] |
|
1014 |
assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto |
|
1015 |
then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto |
|
1016 |
hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE) |
|
1017 |
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 |
|
1018 |
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 |
|
1019 |
qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto |
|
1020 |
show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *) |
|
1021 |
fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}" |
|
1022 |
hence x:"c$i < d$i" "e$i < f$i" "c$i < f$i" "e$i < d$i" unfolding mem_interval apply-apply(erule_tac[!] x=i in allE)+ by auto |
|
1023 |
show False using c_d(2)[of i] apply- apply(erule_tac disjE) |
|
1024 |
proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2" |
|
1025 |
show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps) |
|
1026 |
next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i" |
|
1027 |
show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps) |
|
1028 |
qed qed qed |
|
1029 |
also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule) |
|
1030 |
fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff .. |
|
1031 |
from this(1) guess c unfolding mem_Collect_eq .. then guess d .. |
|
1032 |
note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]] |
|
1033 |
show "x\<in>{a..b}" unfolding mem_interval proof |
|
1034 |
fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" |
|
1035 |
using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed |
|
1036 |
next fix x assume x:"x\<in>{a..b}" |
|
1037 |
have "\<forall>i. \<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" |
|
1038 |
(is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i |
|
1039 |
have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)" |
|
1040 |
using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast |
|
1041 |
qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq |
|
1042 |
apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto |
|
1043 |
qed finally show False using assms by auto qed |
|
1044 |
||
1045 |
lemma interval_bisection: |
|
1046 |
assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}" |
|
1047 |
obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})" |
|
1048 |
proof- |
|
1049 |
have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and> |
|
1050 |
2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof- |
|
1051 |
presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis" |
|
1052 |
thus ?thesis apply(cases "P {fst x..snd x}") by auto |
|
1053 |
next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . |
|
1054 |
thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto |
|
1055 |
qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this |
|
1056 |
def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def |
|
1057 |
have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and> |
|
1058 |
(\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> |
|
1059 |
2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n") |
|
1060 |
proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto |
|
1061 |
case goal3 note S = ab_def funpow.simps o_def id_apply show ?case |
|
1062 |
proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto |
|
1063 |
next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto |
|
1064 |
qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format] |
|
1065 |
||
1066 |
have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e" |
|
1067 |
proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this |
|
1068 |
show ?case apply(rule_tac x=n in exI) proof(rule,rule) |
|
1069 |
fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}" |
|
36587 | 1070 |
have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding dist_norm by(rule norm_le_l1) |
35172 | 1071 |
also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV" |
1072 |
proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i" |
|
1073 |
using xy[unfolded mem_interval,THEN spec[where x=i]] |
|
1074 |
unfolding vector_minus_component by auto qed |
|
1075 |
also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib |
|
1076 |
proof(rule setsum_mono) case goal1 thus ?case |
|
1077 |
proof(induct n) case 0 thus ?case unfolding AB by auto |
|
1078 |
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 n i] by auto |
|
1079 |
also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case . |
|
1080 |
qed qed |
|
1081 |
also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" . |
|
1082 |
qed qed |
|
1083 |
{ fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this |
|
1084 |
have "{A n..B n} \<subseteq> {A m..B m}" unfolding d |
|
1085 |
proof(induct d) case 0 thus ?case by auto |
|
1086 |
next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc]) |
|
1087 |
apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE) |
|
1088 |
proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps) |
|
1089 |
qed qed } note ABsubset = this |
|
1090 |
have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv]) |
|
1091 |
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 |
|
1092 |
then guess x0 .. note x0=this[rule_format] |
|
1093 |
show thesis proof(rule that[rule_format,of x0]) |
|
1094 |
show "x0\<in>{a..b}" using x0[of 0] unfolding AB . |
|
1095 |
fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this |
|
1096 |
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}" |
|
1097 |
apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer |
|
1098 |
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 |
|
1099 |
show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto |
|
1100 |
show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto |
|
1101 |
qed qed qed |
|
1102 |
||
1103 |
subsection {* Cousin's lemma. *} |
|
1104 |
||
1105 |
lemma fine_division_exists: assumes "gauge g" |
|
1106 |
obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p" |
|
1107 |
proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False" |
|
1108 |
then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto |
|
1109 |
next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)" |
|
1110 |
guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as]) |
|
1111 |
apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+ |
|
1112 |
proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto |
|
1113 |
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 = {}" |
|
1114 |
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 |
|
1115 |
apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto |
|
1116 |
qed note x=this |
|
1117 |
obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto |
|
1118 |
from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this |
|
1119 |
have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto |
|
1120 |
thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed |
|
1121 |
||
1122 |
subsection {* Basic theorems about integrals. *} |
|
1123 |
||
1124 |
lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1125 |
assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2" |
|
1126 |
proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto |
|
1127 |
have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> a b k1 k2. |
|
1128 |
(f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False" |
|
1129 |
proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto |
|
1130 |
guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this |
|
1131 |
guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this |
|
1132 |
guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this |
|
1133 |
let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)" |
|
36350 | 1134 |
using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute) |
35172 | 1135 |
also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" |
1136 |
apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto |
|
1137 |
finally show False by auto |
|
1138 |
qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False" |
|
1139 |
thus False apply-apply(cases "\<exists>a b. i = {a..b}") |
|
1140 |
using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) } |
|
1141 |
assume as:"\<not> (\<exists>a b. i = {a..b})" |
|
1142 |
guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format] |
|
1143 |
guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format] |
|
1144 |
have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval) |
|
1145 |
using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+ |
|
1146 |
note ab=conjunctD2[OF this[unfolded Un_subset_iff]] |
|
1147 |
guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this] |
|
1148 |
guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this] |
|
1149 |
have "z = w" using lem[OF w(1) z(1)] by auto |
|
1150 |
hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)" |
|
1151 |
using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) |
|
1152 |
also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2)) |
|
1153 |
finally show False by auto qed |
|
1154 |
||
1155 |
lemma integral_unique[intro]: |
|
1156 |
"(f has_integral y) k \<Longrightarrow> integral k f = y" |
|
1157 |
unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) |
|
1158 |
||
1159 |
lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1160 |
assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s" |
|
1161 |
proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a. |
|
1162 |
(\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral |
|
1163 |
proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a" |
|
1164 |
assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)" |
|
1165 |
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)" |
|
1166 |
apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball) |
|
1167 |
proof(rule,rule,erule conjE) case goal1 |
|
1168 |
have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule) |
|
1169 |
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 |
|
1170 |
thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto |
|
1171 |
qed thus ?case using as by auto |
|
1172 |
qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1173 |
thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") |
|
1174 |
using assms by(auto simp add:has_integral intro:lem) } |
|
1175 |
have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto |
|
1176 |
assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P * |
|
1177 |
apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule) |
|
1178 |
proof- fix e::real and a b assume "e>0" |
|
1179 |
thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e" |
|
1180 |
apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto |
|
1181 |
qed auto qed |
|
1182 |
||
1183 |
lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s" |
|
1184 |
apply(rule has_integral_is_0) by auto |
|
1185 |
||
1186 |
lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0" |
|
1187 |
using has_integral_unique[OF has_integral_0] by auto |
|
1188 |
||
1189 |
lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1190 |
assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s" |
|
1191 |
proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format] |
|
1192 |
have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b. |
|
1193 |
(f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})" |
|
1194 |
proof(subst has_integral,rule,rule) case goal1 |
|
1195 |
from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format] |
|
1196 |
have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto |
|
1197 |
guess g using has_integralD[OF goal1(1) *] . note g=this |
|
1198 |
show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1)) |
|
1199 |
proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" |
|
1200 |
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 |
|
1201 |
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" |
|
1202 |
unfolding o_def unfolding scaleR[THEN sym] * by simp |
|
1203 |
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 |
|
1204 |
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)" . |
|
1205 |
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym] |
|
1206 |
apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps) |
|
1207 |
qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1208 |
thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) } |
|
1209 |
assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P |
|
1210 |
proof(rule,rule) fix e::real assume e:"0<e" |
|
1211 |
have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1)) |
|
1212 |
guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this |
|
1213 |
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)" |
|
1214 |
apply(rule_tac x=M in exI) apply(rule,rule M(1)) |
|
1215 |
proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this] |
|
1216 |
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)" |
|
1217 |
unfolding o_def apply(rule ext) using zero by auto |
|
1218 |
show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym] |
|
1219 |
apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps) |
|
1220 |
qed qed qed |
|
1221 |
||
1222 |
lemma has_integral_cmul: |
|
1223 |
shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s" |
|
1224 |
unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption) |
|
1225 |
by(rule scaleR.bounded_linear_right) |
|
1226 |
||
1227 |
lemma has_integral_neg: |
|
1228 |
shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s" |
|
1229 |
apply(drule_tac c="-1" in has_integral_cmul) by auto |
|
1230 |
||
1231 |
lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1232 |
assumes "(f has_integral k) s" "(g has_integral l) s" |
|
1233 |
shows "((\<lambda>x. f x + g x) has_integral (k + l)) s" |
|
1234 |
proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l. |
|
1235 |
(f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow> |
|
1236 |
((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1 |
|
1237 |
show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto |
|
1238 |
guess d1 using has_integralD[OF goal1(1) *] . note d1=this |
|
1239 |
guess d2 using has_integralD[OF goal1(2) *] . note d2=this |
|
1240 |
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)" |
|
1241 |
apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)]) |
|
1242 |
proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p" |
|
1243 |
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)" |
|
1244 |
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] |
|
1245 |
by(rule setsum_cong2,auto) |
|
1246 |
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))" |
|
36350 | 1247 |
unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>" |
35172 | 1248 |
from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto |
1249 |
have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq]) |
|
1250 |
apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto |
|
1251 |
finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto |
|
1252 |
qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1253 |
thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) } |
|
1254 |
assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P |
|
1255 |
proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto |
|
1256 |
from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format] |
|
1257 |
from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format] |
|
1258 |
show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1) |
|
1259 |
proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}" |
|
1260 |
hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto |
|
1261 |
guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this] |
|
1262 |
guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this] |
|
1263 |
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 |
|
1264 |
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" |
|
1265 |
apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]]) |
|
1266 |
using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps) |
|
1267 |
qed qed qed |
|
1268 |
||
1269 |
lemma has_integral_sub: |
|
1270 |
shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s" |
|
36350 | 1271 |
using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto |
35172 | 1272 |
|
1273 |
lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0" |
|
1274 |
by(rule integral_unique has_integral_0)+ |
|
1275 |
||
1276 |
lemma integral_add: |
|
1277 |
shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> |
|
1278 |
integral s (\<lambda>x. f x + g x) = integral s f + integral s g" |
|
1279 |
apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1280 |
apply(rule has_integral_add) by assumption+ |
|
1281 |
||
1282 |
lemma integral_cmul: |
|
1283 |
shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f" |
|
1284 |
apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1285 |
apply(rule has_integral_cmul) by assumption+ |
|
1286 |
||
1287 |
lemma integral_neg: |
|
1288 |
shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f" |
|
1289 |
apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1290 |
apply(rule has_integral_neg) by assumption+ |
|
1291 |
||
1292 |
lemma integral_sub: |
|
1293 |
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" |
|
1294 |
apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1295 |
apply(rule has_integral_sub) by assumption+ |
|
1296 |
||
1297 |
lemma integrable_0: "(\<lambda>x. 0) integrable_on s" |
|
1298 |
unfolding integrable_on_def using has_integral_0 by auto |
|
1299 |
||
1300 |
lemma integrable_add: |
|
1301 |
shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s" |
|
1302 |
unfolding integrable_on_def by(auto intro: has_integral_add) |
|
1303 |
||
1304 |
lemma integrable_cmul: |
|
1305 |
shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s" |
|
1306 |
unfolding integrable_on_def by(auto intro: has_integral_cmul) |
|
1307 |
||
1308 |
lemma integrable_neg: |
|
1309 |
shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s" |
|
1310 |
unfolding integrable_on_def by(auto intro: has_integral_neg) |
|
1311 |
||
1312 |
lemma integrable_sub: |
|
1313 |
shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s" |
|
1314 |
unfolding integrable_on_def by(auto intro: has_integral_sub) |
|
1315 |
||
1316 |
lemma integrable_linear: |
|
1317 |
shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s" |
|
1318 |
unfolding integrable_on_def by(auto intro: has_integral_linear) |
|
1319 |
||
1320 |
lemma integral_linear: |
|
1321 |
shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)" |
|
1322 |
apply(rule has_integral_unique) defer unfolding has_integral_integral |
|
1323 |
apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym] |
|
1324 |
apply(rule integrable_linear) by assumption+ |
|
1325 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
1326 |
lemma integral_component_eq[simp]: fixes f::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
1327 |
assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
1328 |
using integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
1329 |
|
35172 | 1330 |
lemma has_integral_setsum: |
1331 |
assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s" |
|
1332 |
shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s" |
|
1333 |
proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct) |
|
1334 |
case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)] |
|
1335 |
apply(rule has_integral_add) using insert assms by auto |
|
1336 |
qed auto |
|
1337 |
||
1338 |
lemma integral_setsum: |
|
1339 |
shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow> |
|
1340 |
integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t" |
|
1341 |
apply(rule integral_unique) apply(rule has_integral_setsum) |
|
1342 |
using integrable_integral by auto |
|
1343 |
||
1344 |
lemma integrable_setsum: |
|
1345 |
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" |
|
1346 |
unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto |
|
1347 |
||
1348 |
lemma has_integral_eq: |
|
1349 |
assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s" |
|
1350 |
using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0] |
|
1351 |
using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto |
|
1352 |
||
1353 |
lemma integrable_eq: |
|
1354 |
shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s" |
|
1355 |
unfolding integrable_on_def using has_integral_eq[of s f g] by auto |
|
1356 |
||
1357 |
lemma has_integral_eq_eq: |
|
1358 |
shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)" |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1359 |
using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto |
35172 | 1360 |
|
1361 |
lemma has_integral_null[dest]: |
|
1362 |
assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})" |
|
1363 |
unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer |
|
1364 |
proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto |
|
1365 |
fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*) |
|
1366 |
have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right |
|
1367 |
using setsum_content_null[OF assms(1) p, of f] . |
|
1368 |
thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed |
|
1369 |
||
1370 |
lemma has_integral_null_eq[simp]: |
|
1371 |
shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)" |
|
1372 |
apply rule apply(rule has_integral_unique,assumption) |
|
1373 |
apply(drule has_integral_null,assumption) |
|
1374 |
apply(drule has_integral_null) by auto |
|
1375 |
||
1376 |
lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0" |
|
1377 |
by(rule integral_unique,drule has_integral_null) |
|
1378 |
||
1379 |
lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}" |
|
1380 |
unfolding integrable_on_def apply(drule has_integral_null) by auto |
|
1381 |
||
1382 |
lemma has_integral_empty[intro]: shows "(f has_integral 0) {}" |
|
1383 |
unfolding empty_as_interval apply(rule has_integral_null) |
|
1384 |
using content_empty unfolding empty_as_interval . |
|
1385 |
||
1386 |
lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0" |
|
1387 |
apply(rule,rule has_integral_unique,assumption) by auto |
|
1388 |
||
1389 |
lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto |
|
1390 |
||
1391 |
lemma integral_empty[simp]: shows "integral {} f = 0" |
|
1392 |
apply(rule integral_unique) using has_integral_empty . |
|
1393 |
||
35540 | 1394 |
lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" |
1395 |
proof- have *:"{a} = {a..a}" apply(rule set_ext) unfolding mem_interval singleton_iff Cart_eq |
|
1396 |
apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps) |
|
1397 |
show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding * |
|
1398 |
apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior |
|
1399 |
unfolding interior_closed_interval using interval_sing by auto qed |
|
35172 | 1400 |
|
1401 |
lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto |
|
1402 |
||
1403 |
lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto |
|
1404 |
||
1405 |
subsection {* Cauchy-type criterion for integrability. *} |
|
1406 |
||
1407 |
lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" |
|
1408 |
shows "f integrable_on {a..b} \<longleftrightarrow> |
|
1409 |
(\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and> |
|
1410 |
p2 tagged_division_of {a..b} \<and> d fine p2 |
|
1411 |
\<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 - |
|
1412 |
setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)") |
|
1413 |
proof assume ?l |
|
1414 |
then guess y unfolding integrable_on_def has_integral .. note y=this |
|
1415 |
show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto |
|
1416 |
then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format] |
|
1417 |
show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+) |
|
1418 |
proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2" |
|
1419 |
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" |
|
36587 | 1420 |
apply(rule dist_triangle_half_l[where y=y,unfolded dist_norm]) |
35172 | 1421 |
using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] . |
1422 |
qed qed |
|
1423 |
next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto |
|
1424 |
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format] |
|
1425 |
have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto |
|
1426 |
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- |
|
1427 |
proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed |
|
1428 |
from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]] |
|
1429 |
have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto |
|
1430 |
have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))" |
|
1431 |
proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this |
|
1432 |
show ?case apply(rule_tac x=N in exI) |
|
1433 |
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 |
|
1434 |
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" |
|
1435 |
apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2)) |
|
1436 |
using dp p(1) using mn by auto |
|
1437 |
qed qed |
|
1438 |
then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format] |
|
1439 |
show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI) |
|
1440 |
proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto |
|
1441 |
then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto |
|
1442 |
guess N2 using y[OF *] .. note N2=this |
|
1443 |
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)" |
|
1444 |
apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer |
|
1445 |
proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto |
|
1446 |
fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q" |
|
1447 |
have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto |
|
1448 |
show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r) |
|
1449 |
apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer |
|
36587 | 1450 |
using N2[rule_format,unfolded dist_norm,of "N1+N2"] |
35172 | 1451 |
using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed |
1452 |
||
1453 |
subsection {* Additivity of integral on abutting intervals. *} |
|
1454 |
||
1455 |
lemma interval_split: |
|
1456 |
"{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}" |
|
1457 |
"{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}" |
|
1458 |
apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq |
|
1459 |
unfolding Cart_lambda_beta by auto |
|
1460 |
||
1461 |
lemma content_split: |
|
1462 |
"content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})" |
|
1463 |
proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def |
|
1464 |
{ presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto } |
|
1465 |
have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto |
|
1466 |
have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))" |
|
1467 |
"(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" |
|
1468 |
apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto |
|
1469 |
assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c |
|
1470 |
\<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)" |
|
1471 |
by (auto simp add:field_simps) |
|
1472 |
moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False" |
|
1473 |
unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto |
|
1474 |
ultimately show ?thesis |
|
1475 |
unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto) |
|
1476 |
qed |
|
1477 |
||
1478 |
lemma division_split_left_inj: |
|
1479 |
assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2" |
|
1480 |
"k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}" |
|
1481 |
shows "content(k1 \<inter> {x. x$k \<le> c}) = 0" |
|
1482 |
proof- note d=division_ofD[OF assms(1)] |
|
1483 |
have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})" |
|
1484 |
unfolding interval_split content_eq_0_interior by auto |
|
1485 |
guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this |
|
1486 |
guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this |
|
1487 |
have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto |
|
1488 |
show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]]) |
|
1489 |
defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed |
|
1490 |
||
1491 |
lemma division_split_right_inj: |
|
1492 |
assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2" |
|
1493 |
"k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" |
|
1494 |
shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0" |
|
1495 |
proof- note d=division_ofD[OF assms(1)] |
|
1496 |
have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k >= c}) = {})" |
|
1497 |
unfolding interval_split content_eq_0_interior by auto |
|
1498 |
guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this |
|
1499 |
guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this |
|
1500 |
have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto |
|
1501 |
show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]]) |
|
1502 |
defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed |
|
1503 |
||
1504 |
lemma tagged_division_split_left_inj: |
|
1505 |
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}" |
|
1506 |
shows "content(k1 \<inter> {x. x$k \<le> c}) = 0" |
|
1507 |
proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto |
|
1508 |
show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]]) |
|
1509 |
apply(rule_tac[1-2] *) using assms(2-) by auto qed |
|
1510 |
||
1511 |
lemma tagged_division_split_right_inj: |
|
1512 |
assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" |
|
1513 |
shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0" |
|
1514 |
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 |
|
1515 |
show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]]) |
|
1516 |
apply(rule_tac[1-2] *) using assms(2-) by auto qed |
|
1517 |
||
1518 |
lemma division_split: |
|
1519 |
assumes "p division_of {a..b::real^'n}" |
|
1520 |
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$k \<le> c})" (is "?p1 division_of ?I1") and |
|
1521 |
"{l \<inter> {x. x$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") |
|
1522 |
proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms] |
|
1523 |
show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto |
|
1524 |
{ fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this |
|
1525 |
guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this |
|
1526 |
show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l |
|
1527 |
using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto |
|
1528 |
fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this |
|
1529 |
assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto } |
|
1530 |
{ fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this |
|
1531 |
guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this |
|
1532 |
show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l |
|
1533 |
using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto |
|
1534 |
fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this |
|
1535 |
assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto } |
|
1536 |
qed |
|
1537 |
||
1538 |
lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1539 |
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})" |
|
1540 |
shows "(f has_integral (i + j)) ({a..b})" |
|
1541 |
proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto |
|
1542 |
guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]] |
|
1543 |
guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]] |
|
1544 |
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" |
|
1545 |
show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+) |
|
1546 |
proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto |
|
1547 |
fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)] |
|
1548 |
have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c" |
|
1549 |
"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c" |
|
1550 |
proof- fix x kk assume as:"(x,kk)\<in>p" |
|
1551 |
show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c" |
|
1552 |
proof(rule ccontr) case goal1 |
|
1553 |
from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>" |
|
1554 |
using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto |
|
1555 |
hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast |
|
1556 |
then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans) |
|
36587 | 1557 |
using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:dist_norm) |
35172 | 1558 |
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps) |
1559 |
qed |
|
1560 |
show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c" |
|
1561 |
proof(rule ccontr) case goal1 |
|
1562 |
from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>" |
|
1563 |
using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto |
|
1564 |
hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast |
|
1565 |
then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans) |
|
36587 | 1566 |
using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:dist_norm) |
35172 | 1567 |
thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps) |
1568 |
qed |
|
1569 |
qed |
|
1570 |
||
1571 |
have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto |
|
1572 |
have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}" |
|
1573 |
proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed |
|
1574 |
have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow> |
|
1575 |
setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})} |
|
1576 |
= setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)" |
|
1577 |
apply(rule setsum_mono_zero_left) prefer 3 |
|
1578 |
proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)" |
|
1579 |
assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" |
|
1580 |
then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto |
|
1581 |
have "content (g k) = 0" using xk using content_empty by auto |
|
1582 |
thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto |
|
1583 |
qed auto |
|
1584 |
have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto |
|
1585 |
||
1586 |
let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}" |
|
1587 |
have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI) |
|
1588 |
apply(rule lem2 p(3))+ prefer 6 apply(rule fineI) |
|
1589 |
proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto |
|
1590 |
fix x l assume xl:"(x,l)\<in>?M1" |
|
1591 |
then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this |
|
1592 |
have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto |
|
1593 |
thus "l \<subseteq> d1 x" unfolding xl' by auto |
|
1594 |
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) |
|
1595 |
using lem0(1)[OF xl'(3-4)] by auto |
|
1596 |
show "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k]) |
|
1597 |
fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1" |
|
1598 |
then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this |
|
1599 |
assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}" |
|
1600 |
proof(cases "l' = r' \<longrightarrow> x' = y'") |
|
1601 |
case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1602 |
next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto |
|
1603 |
thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1604 |
qed qed moreover |
|
1605 |
||
1606 |
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> {}}" |
|
1607 |
have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI) |
|
1608 |
apply(rule lem2 p(3))+ prefer 6 apply(rule fineI) |
|
1609 |
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 |
|
1610 |
fix x l assume xl:"(x,l)\<in>?M2" |
|
1611 |
then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this |
|
1612 |
have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto |
|
1613 |
thus "l \<subseteq> d2 x" unfolding xl' by auto |
|
1614 |
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) |
|
1615 |
using lem0(2)[OF xl'(3-4)] by auto |
|
1616 |
show "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k]) |
|
1617 |
fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2" |
|
1618 |
then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this |
|
1619 |
assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}" |
|
1620 |
proof(cases "l' = r' \<longrightarrow> x' = y'") |
|
1621 |
case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1622 |
next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto |
|
1623 |
thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1624 |
qed qed ultimately |
|
1625 |
||
1626 |
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" |
|
1627 |
apply- apply(rule norm_triangle_lt) by auto |
|
1628 |
also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto |
|
1629 |
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) |
|
1630 |
= (\<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 |
|
1631 |
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)" |
|
1632 |
unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)]) |
|
1633 |
defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *) |
|
1634 |
proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto |
|
1635 |
next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto |
|
1636 |
qed also note setsum_addf[THEN sym] |
|
1637 |
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 |
|
1638 |
= (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv |
|
1639 |
proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this |
|
1640 |
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" |
|
1641 |
unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto |
|
1642 |
qed note setsum_cong2[OF this] |
|
1643 |
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 + |
|
1644 |
((\<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) = |
|
1645 |
(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto } |
|
1646 |
finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed |
|
1647 |
||
1648 |
subsection {* A sort of converse, integrability on subintervals. *} |
|
1649 |
||
1650 |
lemma tagged_division_union_interval: |
|
1651 |
assumes "p1 tagged_division_of ({a..b} \<inter> {x::real^'n. x$k \<le> (c::real)})" "p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c})" |
|
1652 |
shows "(p1 \<union> p2) tagged_division_of ({a..b})" |
|
1653 |
proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto |
|
1654 |
show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms]) |
|
1655 |
unfolding interval_split interior_closed_interval |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1656 |
by(auto simp add: vector_less_def elim!:allE[where x=k]) qed |
35172 | 1657 |
|
1658 |
lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector" |
|
1659 |
assumes "(f has_integral i) ({a..b})" "e>0" |
|
1660 |
obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and> |
|
1661 |
p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2 |
|
1662 |
\<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + |
|
1663 |
setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)" |
|
1664 |
proof- guess d using has_integralD[OF assms] . note d=this |
|
1665 |
show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+) |
|
1666 |
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 |
|
1667 |
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 |
|
1668 |
note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this |
|
1669 |
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)" |
|
1670 |
apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv |
|
1671 |
proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2" |
|
1672 |
have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this |
|
1673 |
have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp |
|
1674 |
moreover have "interior {x. x $ k = c} = {}" |
|
1675 |
proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto |
|
1676 |
then guess e unfolding mem_interior .. note e=this |
|
1677 |
have x:"x$k = c" using x interior_subset by fastsimp |
|
1678 |
have *:"\<And>i. \<bar>(x - (x + (\<chi> i. if i = k then e / 2 else 0))) $ i\<bar> = (if i = k then e/2 else 0)" using e by auto |
|
36587 | 1679 |
have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball dist_norm |
35172 | 1680 |
apply(rule le_less_trans[OF norm_le_l1]) unfolding * |
1681 |
unfolding setsum_delta[OF finite_UNIV] using e by auto |
|
1682 |
hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto |
|
1683 |
thus False unfolding mem_Collect_eq using e x by auto |
|
1684 |
qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto |
|
1685 |
thus "content b *\<^sub>R f a = 0" by auto |
|
1686 |
qed auto |
|
1687 |
also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+ |
|
1688 |
finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed |
|
1689 |
||
1690 |
lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}" |
|
1691 |
shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) |
|
1692 |
proof- guess y using assms unfolding integrable_on_def .. note y=this |
|
1693 |
def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n" |
|
1694 |
and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n" |
|
1695 |
show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym] |
|
1696 |
proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto |
|
1697 |
from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format] |
|
1698 |
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> |
|
1699 |
norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)" |
|
1700 |
show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule) |
|
1701 |
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" |
|
1702 |
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" |
|
1703 |
proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this |
|
1704 |
show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]] |
|
1705 |
using as unfolding interval_split b'_def[symmetric] a'_def[symmetric] |
|
36350 | 1706 |
using p using assms by(auto simp add:algebra_simps) |
35172 | 1707 |
qed qed |
1708 |
show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule) |
|
1709 |
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" |
|
1710 |
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" |
|
1711 |
proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this |
|
1712 |
show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]] |
|
1713 |
using as unfolding interval_split b'_def[symmetric] a'_def[symmetric] |
|
36350 | 1714 |
using p using assms by(auto simp add:algebra_simps) qed qed qed qed |
35172 | 1715 |
|
1716 |
subsection {* Generalized notion of additivity. *} |
|
1717 |
||
1718 |
definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)" |
|
1719 |
||
1720 |
definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where |
|
1721 |
"operative opp f \<equiv> |
|
1722 |
(\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and> |
|
1723 |
(\<forall>a b c k. f({a..b}) = |
|
1724 |
opp (f({a..b} \<inter> {x. x$k \<le> c})) |
|
1725 |
(f({a..b} \<inter> {x. x$k \<ge> c})))" |
|
1726 |
||
1727 |
lemma operativeD[dest]: assumes "operative opp f" |
|
1728 |
shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)" |
|
1729 |
"\<And>a b c k. f({a..b}) = opp (f({a..b} \<inter> {x. x$k \<le> c})) (f({a..b} \<inter> {x. x$k \<ge> c}))" |
|
1730 |
using assms unfolding operative_def by auto |
|
1731 |
||
1732 |
lemma operative_trivial: |
|
1733 |
"operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp" |
|
1734 |
unfolding operative_def by auto |
|
1735 |
||
1736 |
lemma property_empty_interval: |
|
1737 |
"(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" |
|
1738 |
using content_empty unfolding empty_as_interval by auto |
|
1739 |
||
1740 |
lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp" |
|
1741 |
unfolding operative_def apply(rule property_empty_interval) by auto |
|
1742 |
||
1743 |
subsection {* Using additivity of lifted function to encode definedness. *} |
|
1744 |
||
1745 |
lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))" |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1746 |
by (metis option.nchotomy) |
35172 | 1747 |
|
1748 |
lemma exists_option: |
|
1749 |
"(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1750 |
by (metis option.nchotomy) |
35172 | 1751 |
|
1752 |
fun lifted where |
|
1753 |
"lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" | |
|
1754 |
"lifted opp None _ = (None::'b option)" | |
|
1755 |
"lifted opp _ None = None" |
|
1756 |
||
1757 |
lemma lifted_simp_1[simp]: "lifted opp v None = None" |
|
1758 |
apply(induct v) by auto |
|
1759 |
||
1760 |
definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and> |
|
1761 |
(\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and> |
|
1762 |
(\<forall>x. opp (neutral opp) x = x)" |
|
1763 |
||
1764 |
lemma monoidalI: assumes "\<And>x y. opp x y = opp y x" |
|
1765 |
"\<And>x y z. opp x (opp y z) = opp (opp x y) z" |
|
1766 |
"\<And>x. opp (neutral opp) x = x" shows "monoidal opp" |
|
1767 |
unfolding monoidal_def using assms by fastsimp |
|
1768 |
||
1769 |
lemma monoidal_ac: assumes "monoidal opp" |
|
1770 |
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a" |
|
1771 |
"opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)" |
|
1772 |
using assms unfolding monoidal_def apply- by metis+ |
|
1773 |
||
1774 |
lemma monoidal_simps[simp]: assumes "monoidal opp" |
|
1775 |
shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" |
|
1776 |
using monoidal_ac[OF assms] by auto |
|
1777 |
||
1778 |
lemma neutral_lifted[cong]: assumes "monoidal opp" |
|
1779 |
shows "neutral (lifted opp) = Some(neutral opp)" |
|
1780 |
apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3 |
|
1781 |
proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y" |
|
1782 |
thus "x = Some (neutral opp)" apply(induct x) defer |
|
1783 |
apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality) |
|
1784 |
apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto |
|
1785 |
qed(auto simp add:monoidal_ac[OF assms]) |
|
1786 |
||
1787 |
lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)" |
|
1788 |
unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto |
|
1789 |
||
1790 |
definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}" |
|
1791 |
definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)" |
|
1792 |
definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)" |
|
1793 |
||
1794 |
lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto |
|
1795 |
lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto |
|
1796 |
||
1797 |
lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp" |
|
1798 |
unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto |
|
1799 |
||
1800 |
lemma support_clauses: |
|
1801 |
"\<And>f g s. support opp f {} = {}" |
|
1802 |
"\<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))" |
|
1803 |
"\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}" |
|
1804 |
"\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)" |
|
1805 |
"\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)" |
|
1806 |
"\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)" |
|
1807 |
"\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)" |
|
1808 |
unfolding support_def by auto |
|
1809 |
||
1810 |
lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)" |
|
1811 |
unfolding support_def by auto |
|
1812 |
||
1813 |
lemma iterate_empty[simp]:"iterate opp {} f = neutral opp" |
|
1814 |
unfolding iterate_def fold'_def by auto |
|
1815 |
||
1816 |
lemma iterate_insert[simp]: assumes "monoidal opp" "finite s" |
|
1817 |
shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" |
|
1818 |
proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto |
|
1819 |
show ?thesis unfolding iterate_def if_P[OF True] * by auto |
|
1820 |
next case False note x=this |
|
1821 |
note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]] |
|
1822 |
show ?thesis proof(cases "f x = neutral opp") |
|
1823 |
case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True] |
|
1824 |
unfolding True monoidal_simps[OF assms(1)] by auto |
|
1825 |
next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False] |
|
1826 |
apply(subst fun_left_comm.fold_insert[OF * finite_support]) |
|
1827 |
using `finite s` unfolding support_def using False x by auto qed qed |
|
1828 |
||
1829 |
lemma iterate_some: |
|
1830 |
assumes "monoidal opp" "finite s" |
|
1831 |
shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2) |
|
1832 |
proof(induct s) case empty thus ?case using assms by auto |
|
1833 |
next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P) |
|
1834 |
defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed |
|
1835 |
||
1836 |
subsection {* Two key instances of additivity. *} |
|
1837 |
||
1838 |
lemma neutral_add[simp]: |
|
1839 |
"neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def |
|
1840 |
apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto |
|
1841 |
||
1842 |
lemma operative_content[intro]: "operative (op +) content" |
|
1843 |
unfolding operative_def content_split[THEN sym] neutral_add by auto |
|
1844 |
||
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1845 |
lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0" |
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1846 |
by (rule neutral_add) (* FIXME: duplicate *) |
35172 | 1847 |
|
1848 |
lemma monoidal_monoid[intro]: |
|
1849 |
shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)" |
|
36350 | 1850 |
unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps) |
35172 | 1851 |
|
1852 |
lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
1853 |
shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)" |
|
1854 |
unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add |
|
1855 |
apply(rule,rule,rule,rule) defer apply(rule allI)+ |
|
1856 |
proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = |
|
1857 |
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) |
|
1858 |
(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)" |
|
1859 |
proof(cases "f integrable_on {a..b}") |
|
1860 |
case True show ?thesis unfolding if_P[OF True] |
|
1861 |
unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]] |
|
1862 |
unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) |
|
1863 |
apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+ |
|
1864 |
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}))" |
|
1865 |
proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def |
|
1866 |
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) |
|
1867 |
apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto |
|
1868 |
thus False using False by auto |
|
1869 |
qed thus ?thesis using False by auto |
|
1870 |
qed next |
|
1871 |
fix a b assume as:"content {a..b::real^'n} = 0" |
|
1872 |
thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0" |
|
1873 |
unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed |
|
1874 |
||
1875 |
subsection {* Points of division of a partition. *} |
|
1876 |
||
1877 |
definition "division_points (k::(real^'n) set) d = |
|
1878 |
{(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and> |
|
1879 |
(\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}" |
|
1880 |
||
1881 |
lemma division_points_finite: assumes "d division_of i" |
|
1882 |
shows "finite (division_points i d)" |
|
1883 |
proof- note assm = division_ofD[OF assms] |
|
1884 |
let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and> |
|
1885 |
(\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}" |
|
1886 |
have *:"division_points i d = \<Union>(?M ` UNIV)" |
|
1887 |
unfolding division_points_def by auto |
|
1888 |
show ?thesis unfolding * using assm by auto qed |
|
1889 |
||
1890 |
lemma division_points_subset: |
|
1891 |
assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k" |
|
1892 |
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$k \<le> c} = {})} |
|
1893 |
\<subseteq> division_points ({a..b}) d" (is ?t1) and |
|
1894 |
"division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})} |
|
1895 |
\<subseteq> division_points ({a..b}) d" (is ?t2) |
|
1896 |
proof- note assm = division_ofD[OF assms(1)] |
|
1897 |
have *:"\<forall>i. a$i \<le> b$i" "\<forall>i. a$i \<le> (\<chi> i. if i = k then min (b $ k) c else b $ i) $ i" |
|
1898 |
"\<forall>i. (\<chi> i. if i = k then max (a $ k) c else a $ i) $ i \<le> b$i" "min (b $ k) c = c" "max (a $ k) c = c" |
|
1899 |
using assms using less_imp_le by auto |
|
1900 |
show ?t1 unfolding division_points_def interval_split[of a b] |
|
1901 |
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding * |
|
1902 |
unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+ |
|
1903 |
proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)" |
|
1904 |
"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> {}" |
|
1905 |
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this |
|
1906 |
have *:"\<forall>i. u $ i \<le> (\<chi> i. if i = k then min (v $ k) c else v $ i) $ i" using as(6) unfolding l interval_split interval_ne_empty as . |
|
1907 |
have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto |
|
1908 |
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)" |
|
1909 |
using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply- |
|
1910 |
apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **] |
|
1911 |
apply(case_tac[!] "fst x = k") using assms by auto |
|
1912 |
qed |
|
1913 |
show ?t2 unfolding division_points_def interval_split[of a b] |
|
1914 |
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding * |
|
1915 |
unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+ |
|
1916 |
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" |
|
1917 |
"i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}" |
|
1918 |
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this |
|
1919 |
have *:"\<forall>i. (\<chi> i. if i = k then max (u $ k) c else u $ i) $ i \<le> v $ i" using as(6) unfolding l interval_split interval_ne_empty as . |
|
1920 |
have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto |
|
1921 |
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)" |
|
1922 |
using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply- |
|
1923 |
apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **] |
|
1924 |
apply(case_tac[!] "fst x = k") using assms by auto qed qed |
|
1925 |
||
1926 |
lemma division_points_psubset: |
|
1927 |
assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k" |
|
1928 |
"l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c" |
|
1929 |
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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") |
|
1930 |
"division_points ({a..b} \<inter> {x. x$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") |
|
1931 |
proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le) |
|
1932 |
guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this |
|
1933 |
have uv:"\<forall>i. u$i \<le> v$i" "\<forall>i. 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 |
|
1934 |
unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto |
|
1935 |
have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k" |
|
1936 |
"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k" |
|
1937 |
unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds) |
|
1938 |
unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto |
|
1939 |
have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE) |
|
1940 |
apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer |
|
1941 |
apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI) |
|
1942 |
unfolding division_points_def unfolding interval_bounds[OF ab] |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1943 |
apply auto unfolding * by auto |
35172 | 1944 |
thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto |
1945 |
||
1946 |
have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k" |
|
1947 |
"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k" |
|
1948 |
unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds) |
|
1949 |
unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto |
|
1950 |
have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE) |
|
1951 |
apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer |
|
1952 |
apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI) |
|
1953 |
unfolding division_points_def unfolding interval_bounds[OF ab] |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
1954 |
apply auto unfolding * by auto |
35172 | 1955 |
thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed |
1956 |
||
1957 |
subsection {* Preservation by divisions and tagged divisions. *} |
|
1958 |
||
1959 |
lemma support_support[simp]:"support opp f (support opp f s) = support opp f s" |
|
1960 |
unfolding support_def by auto |
|
1961 |
||
1962 |
lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f" |
|
1963 |
unfolding iterate_def support_support by auto |
|
1964 |
||
1965 |
lemma iterate_expand_cases: |
|
1966 |
"iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)" |
|
1967 |
apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto |
|
1968 |
||
1969 |
lemma iterate_image: assumes "monoidal opp" "inj_on f s" |
|
1970 |
shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" |
|
1971 |
proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow> |
|
1972 |
iterate opp (f ` s) g = iterate opp s (g \<circ> f)" |
|
1973 |
proof- case goal1 show ?case using goal1 |
|
1974 |
proof(induct s) case empty thus ?case using assms(1) by auto |
|
1975 |
next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] |
|
1976 |
unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym]) |
|
1977 |
unfolding image_insert defer apply(subst iterate_insert[OF assms(1)]) |
|
1978 |
apply(rule finite_imageI insert)+ apply(subst if_not_P) |
|
1979 |
unfolding image_iff o_def using insert(2,4) by auto |
|
1980 |
qed qed |
|
1981 |
show ?thesis |
|
1982 |
apply(cases "finite (support opp g (f ` s))") |
|
1983 |
apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym]) |
|
1984 |
unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric] |
|
1985 |
apply(rule subset_inj_on[OF assms(2) support_subset])+ |
|
1986 |
apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False) |
|
1987 |
apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed |
|
1988 |
||
1989 |
||
1990 |
(* This lemma about iterations comes up in a few places. *) |
|
1991 |
lemma iterate_nonzero_image_lemma: |
|
1992 |
assumes "monoidal opp" "finite s" "g(a) = neutral opp" |
|
1993 |
"\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp" |
|
1994 |
shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)" |
|
1995 |
proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto |
|
1996 |
have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s" |
|
1997 |
unfolding support_def using assms(3) by auto |
|
1998 |
show ?thesis unfolding * |
|
1999 |
apply(subst iterate_support[THEN sym]) unfolding support_clauses |
|
2000 |
apply(subst iterate_image[OF assms(1)]) defer |
|
2001 |
apply(subst(2) iterate_support[THEN sym]) apply(subst **) |
|
2002 |
unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed |
|
2003 |
||
2004 |
lemma iterate_eq_neutral: |
|
2005 |
assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)" |
|
2006 |
shows "(iterate opp s f = neutral opp)" |
|
2007 |
proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto |
|
2008 |
show ?thesis apply(subst iterate_support[THEN sym]) |
|
2009 |
unfolding * using assms(1) by auto qed |
|
2010 |
||
2011 |
lemma iterate_op: assumes "monoidal opp" "finite s" |
|
2012 |
shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2) |
|
2013 |
proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto |
|
2014 |
next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3) |
|
2015 |
unfolding monoidal_ac[OF assms(1)] by(rule refl) qed |
|
2016 |
||
2017 |
lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x" |
|
2018 |
shows "iterate opp s f = iterate opp s g" |
|
2019 |
proof- have *:"support opp g s = support opp f s" |
|
2020 |
unfolding support_def using assms(2) by auto |
|
2021 |
show ?thesis |
|
2022 |
proof(cases "finite (support opp f s)") |
|
2023 |
case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases) |
|
2024 |
unfolding * by auto |
|
2025 |
next def su \<equiv> "support opp f s" |
|
2026 |
case True note support_subset[of opp f s] |
|
2027 |
thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True |
|
2028 |
unfolding su_def[symmetric] |
|
2029 |
proof(induct su) case empty show ?case by auto |
|
2030 |
next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] |
|
2031 |
unfolding if_not_P[OF insert(2)] apply(subst insert(3)) |
|
2032 |
defer apply(subst assms(2)[of x]) using insert by auto qed qed qed |
|
2033 |
||
2034 |
lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto |
|
2035 |
||
2036 |
lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a" |
|
2037 |
assumes "monoidal opp" "operative opp f" "d division_of {a..b}" |
|
2038 |
shows "iterate opp d f = f {a..b}" |
|
2039 |
proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms |
|
2040 |
proof(induct C arbitrary:a b d rule:full_nat_induct) |
|
2041 |
case goal1 |
|
2042 |
{ presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case" |
|
2043 |
thus ?case apply-apply(cases) defer apply assumption |
|
2044 |
proof- assume as:"content {a..b} = 0" |
|
2045 |
show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)]) |
|
2046 |
proof fix x assume x:"x\<in>d" |
|
2047 |
then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+ |
|
2048 |
thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] |
|
2049 |
using operativeD(1)[OF assms(2)] x by auto |
|
2050 |
qed qed } |
|
2051 |
assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq] |
|
2052 |
hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case |
|
2053 |
proof(cases "division_points {a..b} d = {}") |
|
2054 |
case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and> |
|
2055 |
(\<forall>j. 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)" |
|
2056 |
unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule) |
|
2057 |
apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule) |
|
2058 |
proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this] |
|
2059 |
hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto |
|
2060 |
have *:"\<And>p r Q. p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as by auto |
|
2061 |
have "(j, u$j) \<notin> division_points {a..b} d" |
|
2062 |
"(j, v$j) \<notin> division_points {a..b} d" using True by auto |
|
2063 |
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps] |
|
2064 |
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]] |
|
2065 |
moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] |
|
2066 |
unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) |
|
2067 |
unfolding interval_ne_empty mem_interval by auto |
|
2068 |
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" |
|
2069 |
unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto |
|
2070 |
qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le) |
|
2071 |
note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff] |
|
2072 |
then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this |
|
2073 |
have "{a..b} \<in> d" |
|
2074 |
proof- { presume "i = {a..b}" thus ?thesis using i by auto } |
|
2075 |
{ presume "u = a" "v = b" thus "i = {a..b}" using uv by auto } |
|
2076 |
show "u = a" "v = b" unfolding Cart_eq |
|
2077 |
proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j] |
|
2078 |
thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto |
|
2079 |
qed qed |
|
2080 |
hence *:"d = insert {a..b} (d - {{a..b}})" by auto |
|
2081 |
have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)]) |
|
2082 |
proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this] |
|
2083 |
then guess u v apply-by(erule exE conjE)+ note uv=this |
|
2084 |
have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto |
|
2085 |
then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto |
|
2086 |
hence "u$j = v$j" using uv(2)[rule_format,of j] by auto |
|
2087 |
hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) by auto |
|
2088 |
thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)]) |
|
2089 |
qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) |
|
2090 |
apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto |
|
2091 |
next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto |
|
2092 |
then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv |
|
2093 |
by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']] |
|
2094 |
from this(3) guess j .. note j=this |
|
2095 |
def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}" |
|
2096 |
def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}" |
|
2097 |
def cb \<equiv> "(\<chi> i. if i = k then c else b$i)" and ca \<equiv> "(\<chi> i. if i = k then c else a$i)" |
|
2098 |
note division_points_psubset[OF goal1(4) ab kc(1-2) j] |
|
2099 |
note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)] |
|
2100 |
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})" |
|
2101 |
apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format]) |
|
2102 |
using division_split[OF goal1(4), where k=k and c=c] |
|
2103 |
unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono |
|
2104 |
using goal1(2-3) using division_points_finite[OF goal1(4)] by auto |
|
2105 |
have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev") |
|
2106 |
unfolding * apply(rule operativeD(2)) using goal1(3) . |
|
2107 |
also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))" |
|
2108 |
unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def]) |
|
2109 |
unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+ |
|
2110 |
unfolding empty_as_interval[THEN sym] apply(rule content_empty) |
|
2111 |
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" |
|
2112 |
from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this |
|
2113 |
show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split |
|
2114 |
apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj) |
|
2115 |
apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+ |
|
2116 |
qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))" |
|
2117 |
unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def]) |
|
2118 |
unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+ |
|
2119 |
unfolding empty_as_interval[THEN sym] apply(rule content_empty) |
|
2120 |
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" |
|
2121 |
from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this |
|
2122 |
show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split |
|
2123 |
apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj) |
|
2124 |
apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+ |
|
2125 |
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}))" |
|
2126 |
unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) . |
|
2127 |
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}))) |
|
2128 |
= iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3 |
|
2129 |
apply(rule iterate_op[THEN sym]) using goal1 by auto |
|
2130 |
finally show ?thesis by auto |
|
2131 |
qed qed qed |
|
2132 |
||
2133 |
lemma iterate_image_nonzero: assumes "monoidal opp" |
|
2134 |
"finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp" |
|
2135 |
shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms |
|
2136 |
proof(induct rule:finite_subset_induct[OF assms(2) subset_refl]) |
|
2137 |
case goal1 show ?case using assms(1) by auto |
|
2138 |
next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto |
|
2139 |
show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)]) |
|
2140 |
apply(rule finite_imageI goal2)+ |
|
2141 |
apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer |
|
2142 |
apply(subst iterate_insert[OF assms(1) goal2(1)]) defer |
|
2143 |
apply(subst iterate_insert[OF assms(1) goal2(1)]) |
|
2144 |
unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE) |
|
2145 |
apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format]) |
|
2146 |
using goal2 unfolding o_def by auto qed |
|
2147 |
||
2148 |
lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}" |
|
2149 |
shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}" |
|
2150 |
proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)] |
|
2151 |
have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding * |
|
2152 |
apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ |
|
2153 |
unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE) |
|
2154 |
proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba" |
|
2155 |
guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this |
|
2156 |
show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)]) |
|
2157 |
unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)] |
|
2158 |
unfolding as(4)[THEN sym] uv by auto |
|
2159 |
qed also have "\<dots> = f {a..b}" |
|
2160 |
using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] . |
|
2161 |
finally show ?thesis . qed |
|
2162 |
||
2163 |
subsection {* Additivity of content. *} |
|
2164 |
||
2165 |
lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f" |
|
2166 |
proof- have *:"setsum f s = setsum f (support op + f s)" |
|
2167 |
apply(rule setsum_mono_zero_right) |
|
2168 |
unfolding support_def neutral_monoid using assms by auto |
|
2169 |
thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def |
|
2170 |
unfolding neutral_monoid . qed |
|
2171 |
||
2172 |
lemma additive_content_division: assumes "d division_of {a..b}" |
|
2173 |
shows "setsum content d = content({a..b})" |
|
2174 |
unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym] |
|
2175 |
apply(subst setsum_iterate) using assms by auto |
|
2176 |
||
2177 |
lemma additive_content_tagged_division: |
|
2178 |
assumes "d tagged_division_of {a..b}" |
|
2179 |
shows "setsum (\<lambda>(x,l). content l) d = content({a..b})" |
|
2180 |
unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym] |
|
2181 |
apply(subst setsum_iterate) using assms by auto |
|
2182 |
||
36334 | 2183 |
subsection {* Finally, the integral of a constant *} |
35172 | 2184 |
|
2185 |
lemma has_integral_const[intro]: |
|
2186 |
"((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})" |
|
2187 |
unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI) |
|
2188 |
apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE) |
|
2189 |
unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def]) |
|
2190 |
defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto |
|
2191 |
||
2192 |
subsection {* Bounds on the norm of Riemann sums and the integral itself. *} |
|
2193 |
||
2194 |
lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e" |
|
2195 |
shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r") |
|
2196 |
apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym] |
|
2197 |
apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero) |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
2198 |
apply(subst mult_commute) apply(rule mult_left_mono) |
35172 | 2199 |
apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2) |
2200 |
apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)] |
|
2201 |
proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" . |
|
2202 |
fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+ |
|
2203 |
thus "0 \<le> content x" using content_pos_le by auto |
|
2204 |
qed(insert assms,auto) |
|
2205 |
||
2206 |
lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e" |
|
2207 |
shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})" |
|
2208 |
proof(cases "{a..b} = {}") case True |
|
2209 |
show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto |
|
2210 |
next case False show ?thesis |
|
2211 |
apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR |
|
2212 |
apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
2213 |
unfolding setsum_left_distrib[THEN sym] apply(subst mult_commute) apply(rule mult_left_mono) |
35172 | 2214 |
apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2) |
2215 |
apply(subst o_def, rule abs_of_nonneg) |
|
2216 |
proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl) |
|
2217 |
unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto |
|
2218 |
guess w using nonempty_witness[OF False] . |
|
2219 |
thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto |
|
2220 |
fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this] |
|
2221 |
from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this |
|
2222 |
show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le) |
|
2223 |
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 |
|
2224 |
qed(insert assms,auto) qed |
|
2225 |
||
2226 |
lemma rsum_diff_bound: |
|
2227 |
assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e" |
|
2228 |
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})" |
|
2229 |
apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm]) |
|
2230 |
unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto |
|
2231 |
||
2232 |
lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2233 |
assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B" |
|
2234 |
shows "norm i \<le> B * content {a..b}" |
|
2235 |
proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis" |
|
2236 |
thus ?thesis proof(cases ?P) case False |
|
2237 |
hence *:"content {a..b} = 0" using content_lt_nz by auto |
|
2238 |
hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto |
|
2239 |
show ?thesis unfolding * ** using assms(1) by auto |
|
2240 |
qed auto } assume ab:?P |
|
2241 |
{ presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto } |
|
2242 |
assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto |
|
2243 |
from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format] |
|
2244 |
from fine_division_exists[OF this(1), of a b] guess p . note p=this |
|
2245 |
have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)" |
|
2246 |
proof- case goal1 thus ?case unfolding not_less |
|
2247 |
using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto |
|
2248 |
qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed |
|
2249 |
||
2250 |
subsection {* Similar theorems about relationship among components. *} |
|
2251 |
||
2252 |
lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2253 |
assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i" |
|
2254 |
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" |
|
2255 |
unfolding setsum_component apply(rule setsum_mono) |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
2256 |
apply(rule mp) defer apply assumption unfolding split_paired_all apply rule unfolding split_conv |
35172 | 2257 |
proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab] |
2258 |
from this(3) guess u v apply-by(erule exE)+ note b=this |
|
2259 |
show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b |
|
2260 |
unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono) |
|
2261 |
defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed |
|
2262 |
||
2263 |
lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2264 |
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k" |
|
2265 |
shows "i$k \<le> j$k" |
|
2266 |
proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> |
|
2267 |
(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" |
|
2268 |
proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto |
|
2269 |
guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format] |
|
2270 |
guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format] |
|
2271 |
guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter . |
|
2272 |
note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k] |
|
2273 |
note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]] |
|
2274 |
thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt |
|
2275 |
qed let ?P = "\<exists>a b. s = {a..b}" |
|
2276 |
{ presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P) |
|
2277 |
case True then guess a b apply-by(erule exE)+ note s=this |
|
2278 |
show ?thesis apply(rule lem) using assms[unfolded s] by auto |
|
2279 |
qed auto } assume as:"\<not> ?P" |
|
2280 |
{ presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto } |
|
2281 |
assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto |
|
2282 |
note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format] |
|
2283 |
have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+ |
|
2284 |
from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+ |
|
2285 |
note ab = conjunctD2[OF this[unfolded Un_subset_iff]] |
|
2286 |
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this] |
|
2287 |
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this] |
|
2288 |
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(*SMTSMT*) |
|
2289 |
note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover |
|
2290 |
have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately |
|
2291 |
show False unfolding Cart_nth.diff by(rule *) qed |
|
2292 |
||
2293 |
lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2294 |
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k" |
|
2295 |
shows "(integral s f)$k \<le> (integral s g)$k" |
|
2296 |
apply(rule has_integral_component_le) using integrable_integral assms by auto |
|
2297 |
||
2298 |
lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1" |
|
2299 |
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x" |
|
2300 |
shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)]) |
|
2301 |
using assms(3) unfolding vector_le_def by auto |
|
2302 |
||
2303 |
lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1" |
|
2304 |
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x" |
|
2305 |
shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)" |
|
2306 |
apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto |
|
2307 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2308 |
lemma has_integral_component_nonneg: fixes f::"real^'n \<Rightarrow> real^'m" |
35172 | 2309 |
assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k" |
2310 |
using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto |
|
2311 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2312 |
lemma integral_component_nonneg: fixes f::"real^'n \<Rightarrow> real^'m" |
35172 | 2313 |
assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k" |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2314 |
apply(rule has_integral_component_nonneg) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2315 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2316 |
lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1" |
35172 | 2317 |
assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2318 |
using has_integral_component_nonneg[OF assms(1), of 1] |
35172 | 2319 |
using assms(2) unfolding vector_le_def by auto |
2320 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2321 |
lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1" |
35172 | 2322 |
assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
2323 |
apply(rule has_integral_dest_vec1_nonneg) using assms by auto |
35172 | 2324 |
|
2325 |
lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2326 |
assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0" |
|
2327 |
using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto |
|
2328 |
||
2329 |
lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1" |
|
2330 |
assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0" |
|
2331 |
using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto |
|
2332 |
||
2333 |
lemma has_integral_component_lbound: |
|
2334 |
assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k" shows "B * content {a..b} \<le> i$k" |
|
2335 |
using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2) |
|
2336 |
unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps) |
|
2337 |
||
2338 |
lemma has_integral_component_ubound: |
|
2339 |
assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B" |
|
2340 |
shows "i$k \<le> B * content({a..b})" |
|
2341 |
using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"] |
|
2342 |
unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps) |
|
2343 |
||
2344 |
lemma integral_component_lbound: |
|
2345 |
assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k" |
|
2346 |
shows "B * content({a..b}) \<le> (integral({a..b}) f)$k" |
|
2347 |
apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto |
|
2348 |
||
2349 |
lemma integral_component_ubound: |
|
2350 |
assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B" |
|
2351 |
shows "(integral({a..b}) f)$k \<le> B * content({a..b})" |
|
2352 |
apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto |
|
2353 |
||
2354 |
subsection {* Uniform limit of integrable functions is integrable. *} |
|
2355 |
||
2356 |
lemma real_arch_invD: |
|
2357 |
"0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)" |
|
2358 |
by(subst(asm) real_arch_inv) |
|
2359 |
||
2360 |
lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2361 |
assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}" |
|
2362 |
shows "f integrable_on {a..b}" |
|
2363 |
proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis" |
|
2364 |
show ?thesis apply cases apply(rule *,assumption) |
|
2365 |
unfolding content_lt_nz integrable_on_def using has_integral_null by auto } |
|
2366 |
assume as:"content {a..b} > 0" |
|
2367 |
have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto |
|
2368 |
from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format] |
|
2369 |
from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format] |
|
2370 |
||
2371 |
have "Cauchy i" unfolding Cauchy_def |
|
2372 |
proof(rule,rule) fix e::real assume "e>0" |
|
2373 |
hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps) |
|
2374 |
then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this |
|
2375 |
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) |
|
2376 |
proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this] |
|
2377 |
from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format] |
|
2378 |
from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format] |
|
2379 |
from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this |
|
2380 |
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" |
|
2381 |
proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)" |
|
2382 |
using norm_triangle_ineq[of "i1 - s1" "s1 - i2"] |
|
36350 | 2383 |
using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps) |
2384 |
also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps) |
|
35172 | 2385 |
finally show ?case . |
2386 |
qed |
|
36587 | 2387 |
show ?case unfolding dist_norm apply(rule lem2) defer |
35172 | 2388 |
apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]]) |
2389 |
using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans) |
|
2390 |
apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"]) |
|
2391 |
proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse |
|
2392 |
using M as by(auto simp add:field_simps) |
|
2393 |
fix x assume x:"x \<in> {a..b}" |
|
2394 |
have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" |
|
2395 |
using g(1)[OF x, of n] g(1)[OF x, of m] by auto |
|
2396 |
also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono) |
|
2397 |
apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
2398 |
also have "\<dots> = 2 / real M" unfolding divide_inverse by auto |
35172 | 2399 |
finally show "norm (g n x - g m x) \<le> 2 / real M" |
2400 |
using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"] |
|
36350 | 2401 |
by(auto simp add:algebra_simps simp add:norm_minus_commute) |
35172 | 2402 |
qed qed qed |
2403 |
from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this |
|
2404 |
||
2405 |
show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral |
|
2406 |
proof(rule,rule) |
|
2407 |
case goal1 hence *:"e/3 > 0" by auto |
|
2408 |
from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this |
|
2409 |
from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps) |
|
2410 |
from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this |
|
2411 |
from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format] |
|
2412 |
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" |
|
2413 |
proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)" |
|
2414 |
using norm_triangle_ineq[of "sf - sg" "sg - s"] |
|
36350 | 2415 |
using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:algebra_simps) |
2416 |
also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps) |
|
35172 | 2417 |
finally show ?case . |
2418 |
qed |
|
2419 |
show ?case apply(rule_tac x=g' in exI) apply(rule,rule g') |
|
2420 |
proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this] |
|
2421 |
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *]) |
|
2422 |
apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption) |
|
2423 |
proof- have "content {a..b} < e / 3 * (real N2)" |
|
2424 |
using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps) |
|
2425 |
hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)" |
|
2426 |
apply-apply(rule less_le_trans,assumption) using `e>0` by auto |
|
2427 |
thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3" |
|
2428 |
unfolding inverse_eq_divide by(auto simp add:field_simps) |
|
36587 | 2429 |
show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded dist_norm],auto) |
35172 | 2430 |
qed qed qed qed |
2431 |
||
2432 |
subsection {* Negligible sets. *} |
|
2433 |
||
2434 |
definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))" |
|
2435 |
||
2436 |
lemma dest_vec1_indicator: |
|
2437 |
"indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto |
|
2438 |
||
2439 |
lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto |
|
2440 |
||
2441 |
lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto |
|
2442 |
||
2443 |
lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1" |
|
2444 |
unfolding indicator_def by auto |
|
2445 |
||
2446 |
definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})" |
|
2447 |
||
2448 |
lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0" |
|
2449 |
unfolding indicator_def by auto |
|
2450 |
||
2451 |
subsection {* Negligibility of hyperplane. *} |
|
2452 |
||
2453 |
lemma vsum_nonzero_image_lemma: |
|
2454 |
assumes "finite s" "g(a) = 0" |
|
2455 |
"\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0" |
|
2456 |
shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s" |
|
2457 |
unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer |
|
2458 |
apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+ |
|
2459 |
unfolding assms using neutral_add unfolding neutral_add using assms by auto |
|
2460 |
||
2461 |
lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} = |
|
2462 |
{(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}" |
|
2463 |
proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto |
|
2464 |
have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast |
|
2465 |
show ?thesis unfolding * ** interval_split by(rule refl) qed |
|
2466 |
||
2467 |
lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" |
|
2468 |
shows "{l \<inter> {x. abs(x$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})" |
|
2469 |
proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto |
|
2470 |
have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto |
|
2471 |
note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split] |
|
2472 |
note division_split(2)[OF this, where c="c-e" and k=k] |
|
2473 |
thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit |
|
2474 |
apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer |
|
2475 |
apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule |
|
2476 |
apply(rule_tac x=l in exI) by blast+ qed |
|
2477 |
||
2478 |
lemma content_doublesplit: assumes "0 < e" |
|
2479 |
obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e" |
|
2480 |
proof(cases "content {a..b} = 0") |
|
2481 |
case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit |
|
2482 |
apply(rule le_less_trans[OF content_subset]) defer apply(subst True) |
|
2483 |
unfolding interval_doublesplit[THEN sym] using assms by auto |
|
2484 |
next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" |
|
2485 |
note False[unfolded content_eq_0 not_ex not_le, rule_format] |
|
2486 |
hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt |
|
2487 |
hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis |
|
2488 |
proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto |
|
2489 |
have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> |
|
2490 |
(\<Prod>i\<in>UNIV - {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) |
|
2491 |
= (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl) |
|
2492 |
unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto |
|
2493 |
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) |
|
2494 |
unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding ** |
|
2495 |
unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl] |
|
2496 |
proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto |
|
2497 |
also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps) |
|
2498 |
finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e" |
|
2499 |
unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed |
|
2500 |
||
2501 |
lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" |
|
2502 |
unfolding negligible_def has_integral apply(rule,rule,rule,rule) |
|
2503 |
proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}" |
|
2504 |
show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d) |
|
2505 |
proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p" |
|
2506 |
have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)" |
|
2507 |
apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv |
|
2508 |
apply(cases,rule disjI1,assumption,rule disjI2) |
|
2509 |
proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto) |
|
2510 |
show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content]) |
|
2511 |
apply(rule set_ext,rule,rule) unfolding mem_Collect_eq |
|
2512 |
proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv] |
|
36587 | 2513 |
note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this] |
35172 | 2514 |
thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto |
2515 |
qed auto qed |
|
2516 |
note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]] |
|
2517 |
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 |
|
2518 |
apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv |
|
2519 |
apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst) |
|
2520 |
prefer 2 apply(subst(asm) eq_commute) apply assumption |
|
2521 |
apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le) |
|
2522 |
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}))" |
|
2523 |
apply(rule setsum_mono) unfolding split_paired_all split_conv |
|
2524 |
apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le) |
|
2525 |
also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]]) |
|
2526 |
proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}" |
|
2527 |
unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto |
|
2528 |
thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt |
|
2529 |
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" |
|
2530 |
apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all |
|
2531 |
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)" |
|
2532 |
guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this |
|
2533 |
show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le) |
|
2534 |
qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit] |
|
2535 |
note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]] |
|
36725 | 2536 |
note this[unfolded real_scaleR_def real_norm_def normalizing.semiring_rules, of k c d] note le_less_trans[OF this d(2)] |
35172 | 2537 |
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" |
2538 |
apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym]) |
|
2539 |
apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p''] |
|
2540 |
proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v |
|
2541 |
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}" |
|
2542 |
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 |
|
2543 |
note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]] |
|
2544 |
hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto |
|
2545 |
thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] . |
|
2546 |
qed qed |
|
2547 |
finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" . |
|
2548 |
qed qed qed |
|
2549 |
||
2550 |
subsection {* A technical lemma about "refinement" of division. *} |
|
2551 |
||
2552 |
lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set" |
|
2553 |
assumes "p tagged_division_of {a..b}" "gauge d" |
|
2554 |
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" |
|
2555 |
proof- |
|
2556 |
let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow> |
|
2557 |
(\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and> |
|
2558 |
(\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))" |
|
2559 |
{ have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto |
|
2560 |
presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this |
|
2561 |
thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto |
|
2562 |
} fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p" |
|
2563 |
show "?P p" apply(rule,rule) using as proof(induct p) |
|
2564 |
case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto |
|
2565 |
next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this |
|
2566 |
note tagged_partial_division_subset[OF insert(4) subset_insertI] |
|
2567 |
from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this] |
|
2568 |
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 |
|
2569 |
note p = tagged_partial_division_ofD[OF insert(4)] |
|
2570 |
from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this |
|
2571 |
||
2572 |
have "finite {k. \<exists>x. (x, k) \<in> p}" |
|
2573 |
apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq |
|
2574 |
apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto |
|
2575 |
hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}" |
|
2576 |
apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI) |
|
2577 |
unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption) |
|
2578 |
apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption |
|
2579 |
using insert(2) unfolding uv xk by auto |
|
2580 |
||
2581 |
show ?case proof(cases "{u..v} \<subseteq> d x") |
|
2582 |
case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule |
|
2583 |
unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self) |
|
2584 |
apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int) |
|
2585 |
apply(rule,rule fine_union,subst fine_def) defer apply(rule q1) |
|
2586 |
unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule) |
|
2587 |
apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto |
|
2588 |
next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this |
|
2589 |
show ?thesis apply(rule_tac x="q2 \<union> q1" in exI) |
|
2590 |
apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+ |
|
2591 |
unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union) |
|
2592 |
apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE) |
|
2593 |
apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto |
|
2594 |
qed qed qed |
|
2595 |
||
2596 |
subsection {* Hence the main theorem about negligible sets. *} |
|
2597 |
||
2598 |
lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)" |
|
2599 |
shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms |
|
2600 |
proof(induct) case (insert x s) |
|
2601 |
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 |
|
2602 |
show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto |
|
2603 |
||
2604 |
lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)" |
|
2605 |
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 |
|
2606 |
proof(induct) case (insert a s) |
|
2607 |
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 |
|
2608 |
show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)] |
|
2609 |
prefer 4 apply(subst insert(3)) unfolding add_right_cancel |
|
2610 |
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 |
|
2611 |
show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto |
|
2612 |
qed(insert insert, auto) qed auto |
|
2613 |
||
2614 |
lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2615 |
assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0" |
|
2616 |
shows "(f has_integral 0) t" |
|
2617 |
proof- presume P:"\<And>f::real^'n \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" |
|
2618 |
let ?f = "(\<lambda>x. if x \<in> t then f x else 0)" |
|
2619 |
show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl) |
|
2620 |
apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P |
|
2621 |
proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this |
|
2622 |
show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto |
|
2623 |
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)" |
|
2624 |
apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI) |
|
2625 |
apply(rule,rule P) using assms(2) by auto |
|
2626 |
qed |
|
2627 |
next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" |
|
2628 |
show "(f has_integral 0) {a..b}" unfolding has_integral |
|
2629 |
proof(safe) case goal1 |
|
2630 |
hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" |
|
2631 |
apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps) |
|
2632 |
note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] |
|
2633 |
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]] |
|
2634 |
show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) |
|
2635 |
proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto |
|
2636 |
fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" |
|
2637 |
let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" |
|
2638 |
{ presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto } |
|
2639 |
assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N .. |
|
2640 |
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 |
|
2641 |
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)" |
|
2642 |
apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto |
|
2643 |
from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]] |
|
2644 |
have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) |
|
2645 |
unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto |
|
2646 |
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" |
|
2647 |
proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4 |
|
2648 |
apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed |
|
2649 |
have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) * |
|
2650 |
norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}" |
|
2651 |
unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right |
|
2652 |
apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 |
|
2653 |
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 |
|
2654 |
fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)" |
|
2655 |
unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg) |
|
2656 |
using tagged_division_ofD(4)[OF q(1) as''] by auto |
|
2657 |
next fix i::nat show "finite (q i)" using q by auto |
|
2658 |
next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>" |
|
2659 |
have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto |
|
2660 |
have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto |
|
2661 |
hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto |
|
2662 |
moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv] |
|
2663 |
note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]] |
|
2664 |
moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)" |
|
2665 |
proof(cases "x\<in>s") case False thus ?thesis using assm by auto |
|
2666 |
next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto |
|
2667 |
moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto |
|
2668 |
ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps) |
|
2669 |
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)" |
|
2670 |
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 |
|
2671 |
qed(insert as, auto) |
|
2672 |
also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) |
|
2673 |
proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym]) |
|
2674 |
using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps) |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
2675 |
qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[THEN sym] |
35172 | 2676 |
apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym] |
2677 |
apply(subst sumr_geometric) using goal1 by auto |
|
2678 |
finally show "?goal" by auto qed qed qed |
|
2679 |
||
2680 |
lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2681 |
assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t" |
|
2682 |
shows "(g has_integral y) t" |
|
2683 |
proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a |
|
2684 |
assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}" |
|
2685 |
have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)]) |
|
2686 |
apply(rule has_integral_negligible[OF assms(1)]) using as by auto |
|
2687 |
hence "(g has_integral y) {a..b}" by auto } note * = this |
|
2688 |
show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe) |
|
2689 |
apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P |
|
2690 |
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) |
|
2691 |
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 |
|
2692 |
||
2693 |
lemma has_integral_spike_eq: |
|
2694 |
assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" |
|
2695 |
shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)" |
|
2696 |
apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto |
|
2697 |
||
2698 |
lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t" |
|
2699 |
shows "g integrable_on t" |
|
2700 |
using assms unfolding integrable_on_def apply-apply(erule exE) |
|
2701 |
apply(rule,rule has_integral_spike) by fastsimp+ |
|
2702 |
||
2703 |
lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" |
|
2704 |
shows "integral t f = integral t g" |
|
2705 |
unfolding integral_def using has_integral_spike_eq[OF assms] by auto |
|
2706 |
||
2707 |
subsection {* Some other trivialities about negligible sets. *} |
|
2708 |
||
2709 |
lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def |
|
2710 |
proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b] |
|
2711 |
apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption |
|
2712 |
using assms(2) unfolding indicator_def by auto qed |
|
2713 |
||
2714 |
lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto |
|
2715 |
||
2716 |
lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto |
|
2717 |
||
2718 |
lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def |
|
2719 |
proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b] |
|
2720 |
thus ?case apply(subst has_integral_spike_eq[OF assms(2)]) |
|
2721 |
defer apply assumption unfolding indicator_def by auto qed |
|
2722 |
||
2723 |
lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)" |
|
2724 |
using negligible_union by auto |
|
2725 |
||
2726 |
lemma negligible_sing[intro]: "negligible {a::real^'n}" |
|
2727 |
proof- guess x using UNIV_witness[where 'a='n] .. |
|
2728 |
show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed |
|
2729 |
||
2730 |
lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s" |
|
2731 |
apply(subst insert_is_Un) unfolding negligible_union_eq by auto |
|
2732 |
||
2733 |
lemma negligible_empty[intro]: "negligible {}" by auto |
|
2734 |
||
2735 |
lemma negligible_finite[intro]: assumes "finite s" shows "negligible s" |
|
2736 |
using assms apply(induct s) by auto |
|
2737 |
||
2738 |
lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)" |
|
2739 |
using assms by(induct,auto) |
|
2740 |
||
2741 |
lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)" |
|
2742 |
apply safe defer apply(subst negligible_def) |
|
2743 |
proof- fix t::"(real^'n) set" assume as:"negligible s" |
|
2744 |
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) |
|
2745 |
show "(indicator s has_integral 0) t" apply(subst has_integral_alt) |
|
2746 |
apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format]) |
|
2747 |
apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI) |
|
2748 |
using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto |
|
2749 |
||
2750 |
subsection {* Finite case of the spike theorem is quite commonly needed. *} |
|
2751 |
||
2752 |
lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" |
|
2753 |
"(f has_integral y) t" shows "(g has_integral y) t" |
|
2754 |
apply(rule has_integral_spike) using assms by auto |
|
2755 |
||
2756 |
lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" |
|
2757 |
shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)" |
|
2758 |
apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto |
|
2759 |
||
2760 |
lemma integrable_spike_finite: |
|
2761 |
assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t" |
|
2762 |
using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI) |
|
2763 |
apply(rule has_integral_spike_finite) by auto |
|
2764 |
||
2765 |
subsection {* In particular, the boundary of an interval is negligible. *} |
|
2766 |
||
2767 |
lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})" |
|
2768 |
proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)" |
|
2769 |
have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all |
|
2770 |
apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI) |
|
2771 |
apply(erule_tac[!] x=xa in allE) by auto |
|
2772 |
thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed |
|
2773 |
||
2774 |
lemma has_integral_spike_interior: |
|
2775 |
assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})" |
|
2776 |
apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto |
|
2777 |
||
2778 |
lemma has_integral_spike_interior_eq: |
|
2779 |
assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))" |
|
2780 |
apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto |
|
2781 |
||
2782 |
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}" |
|
2783 |
using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto |
|
2784 |
||
2785 |
subsection {* Integrability of continuous functions. *} |
|
2786 |
||
2787 |
lemma neutral_and[simp]: "neutral op \<and> = True" |
|
2788 |
unfolding neutral_def apply(rule some_equality) by auto |
|
2789 |
||
2790 |
lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto |
|
2791 |
||
2792 |
lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms |
|
2793 |
apply induct unfolding iterate_insert[OF monoidal_and] by auto |
|
2794 |
||
2795 |
lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}" |
|
2796 |
shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}" |
|
2797 |
using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto |
|
2798 |
||
2799 |
lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2800 |
shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::real^'n)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and |
|
2801 |
proof safe fix a b::"real^'n" { assume "content {a..b} = 0" |
|
2802 |
thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" |
|
2803 |
apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) } |
|
2804 |
{ fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" |
|
2805 |
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}" |
|
2806 |
"\<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}" |
|
2807 |
apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto } |
|
2808 |
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}" |
|
2809 |
"\<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}" |
|
2810 |
let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x" |
|
2811 |
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) |
|
2812 |
proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto |
|
2813 |
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}" |
|
2814 |
then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this] |
|
2815 |
show ?case unfolding integrable_on_def by auto |
|
2816 |
next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}" |
|
2817 |
apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed |
|
2818 |
||
2819 |
lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2820 |
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" |
|
2821 |
obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" |
|
2822 |
proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)] |
|
2823 |
note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]] |
|
2824 |
guess g .. thus thesis apply-apply(rule that[of g]) by auto qed |
|
2825 |
||
2826 |
lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2827 |
assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}" |
|
2828 |
proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e" |
|
2829 |
from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d .. |
|
2830 |
note d=conjunctD2[OF this,rule_format] |
|
2831 |
from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this |
|
2832 |
note p' = tagged_division_ofD[OF p(1)] |
|
2833 |
have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i" |
|
2834 |
proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" |
|
2835 |
from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this |
|
2836 |
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) |
|
2837 |
proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const) |
|
2838 |
fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this] |
|
2839 |
note d(2)[OF _ _ this[unfolded mem_ball]] |
|
36587 | 2840 |
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 |
35172 | 2841 |
from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g . |
2842 |
thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed |
|
2843 |
||
2844 |
subsection {* Specialization of additivity to one dimension. *} |
|
2845 |
||
2846 |
lemma operative_1_lt: assumes "monoidal opp" |
|
2847 |
shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and> |
|
2848 |
(\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))" |
|
2849 |
unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def |
|
2850 |
proof safe fix a b c::"real^1" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))" "a $ 1 < c $ 1" "c $ 1 < b $ 1" |
|
2851 |
from this(2-) have "{a..b} \<inter> {x. x $ 1 \<le> c $ 1} = {a..c}" "{a..b} \<inter> {x. x $ 1 \<ge> c $ 1} = {c..b}" by auto |
|
2852 |
thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto |
|
2853 |
next fix a b::"real^1" and c::real |
|
2854 |
assume as:"\<forall>a b. b $ 1 \<le> a $ 1 \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a $ 1 < c $ 1 \<and> c $ 1 < b $ 1 \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" |
|
2855 |
show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))" |
|
2856 |
proof(cases "c \<in> {a$1 .. b$1}") |
|
2857 |
case False hence "c<a$1 \<or> c>b$1" by auto |
|
2858 |
thus ?thesis apply-apply(erule disjE) |
|
2859 |
proof- assume "c<a$1" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {1..0}" "{a..b} \<inter> {x. c \<le> x $ 1} = {a..b}" by auto |
|
2860 |
show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto |
|
2861 |
next assume "b$1<c" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {a..b}" "{a..b} \<inter> {x. c \<le> x $ 1} = {1..0}" by auto |
|
2862 |
show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto |
|
2863 |
qed |
|
2864 |
next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto |
|
2865 |
show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym] |
|
2866 |
proof(cases "c = a$1 \<or> c = b$1") |
|
2867 |
case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" |
|
2868 |
apply-apply(subst as(2)[rule_format]) using True by auto |
|
2869 |
next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply- |
|
2870 |
proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto |
|
2871 |
hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2872 |
thus ?thesis using assms unfolding * by auto |
|
2873 |
next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto |
|
2874 |
hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2875 |
thus ?thesis using assms unfolding * by auto qed qed qed qed |
|
2876 |
||
2877 |
lemma operative_1_le: assumes "monoidal opp" |
|
2878 |
shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and> |
|
2879 |
(\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))" |
|
2880 |
unfolding operative_1_lt[OF assms] |
|
2881 |
proof safe fix a b c::"real^1" 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" |
|
2882 |
show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) unfolding vector_le_def vector_less_def by auto |
|
2883 |
next fix a b c ::"real^1" |
|
2884 |
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" |
|
2885 |
note as = this[rule_format] |
|
2886 |
show "opp (f {a..c}) (f {c..b}) = f {a..b}" |
|
2887 |
proof(cases "c = a \<or> c = b") |
|
2888 |
case False thus ?thesis apply-apply(subst as(2)) using as(3-) unfolding vector_le_def vector_less_def Cart_eq by(auto simp del:dest_vec1_eq) |
|
2889 |
next case True thus ?thesis apply- |
|
2890 |
proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2891 |
thus ?thesis using assms unfolding * by auto |
|
2892 |
next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2893 |
thus ?thesis using assms unfolding * by auto qed qed qed |
|
2894 |
||
2895 |
subsection {* Special case of additivity we need for the FCT. *} |
|
2896 |
||
35540 | 2897 |
lemma interval_bound_sing[simp]: "interval_upperbound {a} = a" "interval_lowerbound {a} = a" |
2898 |
unfolding interval_upperbound_def interval_lowerbound_def unfolding Cart_eq by auto |
|
2899 |
||
35172 | 2900 |
lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector" |
2901 |
assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}" |
|
2902 |
shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a" |
|
2903 |
proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))" |
|
2904 |
have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1 |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
2905 |
by(auto simp add:not_less vector_less_def) |
35172 | 2906 |
have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)] |
2907 |
note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ] |
|
2908 |
show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer |
|
2909 |
apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed |
|
2910 |
||
2911 |
subsection {* A useful lemma allowing us to factor out the content size. *} |
|
2912 |
||
2913 |
lemma has_integral_factor_content: |
|
2914 |
"(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 |
|
2915 |
\<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))" |
|
2916 |
proof(cases "content {a..b} = 0") |
|
2917 |
case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe |
|
2918 |
apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer |
|
2919 |
apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption) |
|
2920 |
apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto |
|
2921 |
next case False note F = this[unfolded content_lt_nz[THEN sym]] |
|
2922 |
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)" |
|
2923 |
show ?thesis apply(subst has_integral) |
|
2924 |
proof safe fix e::real assume e:"e>0" |
|
2925 |
{ 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) |
|
2926 |
apply(erule impE) defer apply(erule exE,rule_tac x=d in exI) |
|
2927 |
using F e by(auto simp add:field_simps intro:mult_pos_pos) } |
|
2928 |
{ 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) |
|
2929 |
apply(erule impE) defer apply(erule exE,rule_tac x=d in exI) |
|
2930 |
using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed |
|
2931 |
||
2932 |
subsection {* Fundamental theorem of calculus. *} |
|
2933 |
||
2934 |
lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
2935 |
assumes "a \<le> b" "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})" |
|
2936 |
shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})" |
|
2937 |
unfolding has_integral_factor_content |
|
2938 |
proof safe fix e::real assume e:"e>0" have ab:"dest_vec1 (vec1 a) \<le> dest_vec1 (vec1 b)" using assms(1) by auto |
|
2939 |
note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt] |
|
2940 |
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 |
|
2941 |
note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]] |
|
2942 |
guess d .. note d=conjunctD2[OF this[rule_format],rule_format] |
|
2943 |
show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow> |
|
2944 |
norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b})" |
|
2945 |
apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe) |
|
2946 |
apply(rule gauge_ball_dependent,rule,rule d(1)) |
|
2947 |
proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p" |
|
2948 |
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b}" |
|
2949 |
unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym] |
|
2950 |
unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym] |
|
2951 |
apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] |
|
2952 |
proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p" |
|
2953 |
note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this |
|
2954 |
have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto |
|
2955 |
have ball:"\<forall>xa\<in>k. xa \<in> ball x (d (dest_vec1 x))" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,unfolded split_conv subset_eq] . |
|
2956 |
have "norm ((v$1 - u$1) *\<^sub>R f' x - (f v - f u)) \<le> norm (f u - f x - (u$1 - x$1) *\<^sub>R f' x) + norm (f v - f x - (v$1 - x$1) *\<^sub>R f' x)" |
|
2957 |
apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm]) |
|
36350 | 2958 |
unfolding scaleR.diff_left by(auto simp add:algebra_simps) |
35172 | 2959 |
also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)" |
2960 |
apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4 |
|
2961 |
apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1]) |
|
2962 |
using ball[rule_format,of u] ball[rule_format,of v] |
|
36587 | 2963 |
using xk(1-2) unfolding k subset_eq by(auto simp add:dist_norm norm_real) |
35172 | 2964 |
also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" |
36587 | 2965 |
unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:dist_norm norm_real field_simps) |
35172 | 2966 |
finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le> |
2967 |
e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] . |
|
2968 |
qed(insert as, auto) qed qed |
|
2969 |
||
2970 |
subsection {* Attempt a systematic general set of "offset" results for components. *} |
|
2971 |
||
2972 |
lemma gauge_modify: |
|
2973 |
assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d" |
|
2974 |
shows "gauge (\<lambda>x y. d (f x) (f y))" |
|
2975 |
using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE) |
|
2976 |
apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto |
|
2977 |
||
2978 |
subsection {* Only need trivial subintervals if the interval itself is trivial. *} |
|
2979 |
||
2980 |
lemma division_of_nontrivial: fixes s::"(real^'n) set set" |
|
2981 |
assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0" |
|
2982 |
shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply- |
|
2983 |
proof(induct "card s" arbitrary:s rule:nat_less_induct) |
|
2984 |
fix s::"(real^'n) set set" assume assm:"s division_of {a..b}" |
|
2985 |
"\<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}" |
|
2986 |
note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}" |
|
2987 |
{ presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis" |
|
2988 |
show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto } |
|
2989 |
assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s" |
|
2990 |
then obtain k where k:"k\<in>s" "content k = 0" by auto |
|
2991 |
from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this |
|
2992 |
from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto |
|
2993 |
hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto |
|
2994 |
have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4)) |
|
2995 |
apply safe apply(rule closed_interval) using assm(1) by auto |
|
2996 |
have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable |
|
2997 |
proof safe fix x and e::real assume as:"x\<in>k" "e>0" |
|
2998 |
from k(2)[unfolded k content_eq_0] guess i .. |
|
2999 |
hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt |
|
3000 |
hence xi:"x$i = d$i" using as unfolding k mem_interval by smt |
|
3001 |
def y \<equiv> "(\<chi> j. if j = i then if c$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$j)" |
|
3002 |
show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) |
|
3003 |
proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less) |
|
3004 |
hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]] |
|
3005 |
hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl] |
|
3006 |
apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2) using assms(2)[unfolded content_eq_0] by smt+ |
|
3007 |
thus "y \<noteq> x" unfolding Cart_eq by auto |
|
3008 |
have *:"UNIV = insert i (UNIV - {i})" by auto |
|
3009 |
have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1]) |
|
3010 |
apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono) |
|
3011 |
proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl] |
|
3012 |
apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto |
|
3013 |
show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto |
|
36587 | 3014 |
qed auto thus "dist y x < e" unfolding dist_norm by auto |
35172 | 3015 |
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 |
3016 |
moreover have "y \<in> \<Union>s" unfolding s mem_interval |
|
3017 |
proof note simps = y_def Cart_lambda_beta if_not_P |
|
3018 |
fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" |
|
3019 |
proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto |
|
3020 |
thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto |
|
3021 |
next case True note T = this show ?thesis |
|
3022 |
proof(cases "c $ i \<le> (a $ i + b $ i) / 2") |
|
3023 |
case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i |
|
3024 |
using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) |
|
3025 |
next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i |
|
3026 |
using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) |
|
3027 |
qed qed qed |
|
3028 |
ultimately show "y \<in> \<Union>(s - {k})" by auto |
|
3029 |
qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto |
|
3030 |
hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl]) |
|
3031 |
apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto |
|
3032 |
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 |
|
3033 |
||
3034 |
subsection {* Integrabibility on subintervals. *} |
|
3035 |
||
3036 |
lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
|
3037 |
"operative op \<and> (\<lambda>i. f integrable_on i)" |
|
3038 |
unfolding operative_def neutral_and apply safe apply(subst integrable_on_def) |
|
3039 |
unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+ |
|
3040 |
unfolding integrable_on_def by(auto intro: has_integral_split) |
|
3041 |
||
3042 |
lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3043 |
assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" |
|
3044 |
apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption) |
|
3045 |
using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto |
|
3046 |
||
3047 |
subsection {* Combining adjacent intervals in 1 dimension. *} |
|
3048 |
||
3049 |
lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b" |
|
3050 |
"(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}" |
|
3051 |
shows "(f has_integral (i + j)) {a..b}" |
|
3052 |
proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]] |
|
3053 |
note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]] |
|
3054 |
hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer |
|
3055 |
apply(subst(asm) if_P) using assms(3-) by auto |
|
3056 |
with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P) |
|
3057 |
unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed |
|
3058 |
||
3059 |
lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3060 |
assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})" |
|
3061 |
shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f" |
|
3062 |
apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)]) |
|
3063 |
apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto |
|
3064 |
||
3065 |
lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3066 |
assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}" |
|
3067 |
shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine) |
|
3068 |
||
3069 |
subsection {* Reduce integrability to "local" integrability. *} |
|
3070 |
||
3071 |
lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3072 |
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}" |
|
3073 |
shows "f integrable_on {a..b}" |
|
3074 |
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})" |
|
3075 |
using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format] |
|
3076 |
guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2) |
|
3077 |
note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f] |
|
3078 |
show ?thesis unfolding * apply safe unfolding snd_conv |
|
3079 |
proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this] |
|
3080 |
thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed |
|
3081 |
||
3082 |
subsection {* Second FCT or existence of antiderivative. *} |
|
3083 |
||
3084 |
lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}" |
|
3085 |
unfolding integrable_on_def by(rule,rule has_integral_const) |
|
3086 |
||
3087 |
lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach" |
|
3088 |
assumes "continuous_on {a..b} f" "x \<in> {a..b}" |
|
3089 |
shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})" |
|
3090 |
unfolding has_vector_derivative_def has_derivative_within_alt |
|
3091 |
apply safe apply(rule scaleR.bounded_linear_left) |
|
3092 |
proof- fix e::real assume e:"e>0" |
|
3093 |
note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def] |
|
3094 |
from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format] |
|
3095 |
let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)" |
|
3096 |
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)" |
|
3097 |
proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x") |
|
3098 |
case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous) |
|
3099 |
apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto |
|
36350 | 3100 |
hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine) |
35172 | 3101 |
using False unfolding not_less using assms(2) goal1 by auto |
3102 |
have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto |
|
3103 |
show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def |
|
3104 |
defer apply(rule has_integral_sub) apply(rule integrable_integral) |
|
3105 |
apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+ |
|
3106 |
proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto |
|
3107 |
have *:"y - x = norm(y - x)" using False by auto |
|
3108 |
show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto |
|
3109 |
show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le) |
|
36587 | 3110 |
apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto |
35172 | 3111 |
qed(insert e,auto) |
3112 |
next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous) |
|
3113 |
apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto |
|
36350 | 3114 |
hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine) |
35172 | 3115 |
using True using assms(2) goal1 by auto |
3116 |
have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto |
|
3117 |
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 |
|
3118 |
show ?thesis apply(subst ***) unfolding norm_minus_cancel ** |
|
3119 |
apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def |
|
3120 |
defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus |
|
3121 |
apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) |
|
3122 |
apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+ |
|
3123 |
proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto |
|
3124 |
have *:"x - y = norm(y - x)" using True by auto |
|
3125 |
show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto |
|
3126 |
show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le) |
|
36587 | 3127 |
apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto |
35172 | 3128 |
qed(insert e,auto) qed qed qed |
3129 |
||
3130 |
lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3131 |
assumes "continuous_on {a..b} f" "x \<in> {a..b}" |
|
3132 |
shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})" |
|
3133 |
using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"] |
|
3134 |
unfolding o_def vec1_dest_vec1 using assms(2) by auto |
|
3135 |
||
3136 |
lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f" |
|
3137 |
obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})" |
|
3138 |
apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto |
|
3139 |
||
3140 |
subsection {* Combined fundamental theorem of calculus. *} |
|
3141 |
||
3142 |
lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f" |
|
3143 |
obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> ((f o dest_vec1) has_integral (g v - g u)) {vec u..vec v}" |
|
3144 |
proof- from antiderivative_continuous[OF assms] guess g . note g=this |
|
3145 |
show ?thesis apply(rule that[of g]) |
|
3146 |
proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})" |
|
3147 |
apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto |
|
3148 |
thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"] |
|
3149 |
unfolding o_def vec1_dest_vec1 by auto qed qed |
|
3150 |
||
3151 |
subsection {* General "twiddling" for interval-to-interval function image. *} |
|
3152 |
||
3153 |
lemma has_integral_twiddle: |
|
3154 |
assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g" |
|
3155 |
"\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}" |
|
3156 |
"\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}" |
|
3157 |
"\<forall>u v. content(g ` {u..v}) = r * content {u..v}" |
|
3158 |
"(f has_integral i) {a..b}" |
|
3159 |
shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})" |
|
3160 |
proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis" |
|
3161 |
show ?thesis apply cases defer apply(rule *,assumption) |
|
3162 |
proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed } |
|
3163 |
assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this |
|
3164 |
have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr) |
|
3165 |
using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer |
|
3166 |
using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto |
|
3167 |
show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz) |
|
3168 |
proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos) |
|
3169 |
from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format] |
|
3170 |
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) |
|
3171 |
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)" |
|
3172 |
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 |
|
3173 |
fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] |
|
3174 |
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 |
|
3175 |
proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto |
|
3176 |
show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto |
|
3177 |
fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto |
|
3178 |
show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto |
|
3179 |
{ 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)"] |
|
3180 |
using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto } |
|
3181 |
fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')" |
|
3182 |
hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto |
|
3183 |
have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk']) |
|
3184 |
proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z . |
|
3185 |
hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)] |
|
3186 |
unfolding image_Int[OF inj(1)] by auto thus False using as by blast |
|
3187 |
qed thus "g x = g x'" by auto |
|
3188 |
{ fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto } |
|
3189 |
{ fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto } |
|
3190 |
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 |
|
3191 |
then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq .. |
|
3192 |
thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply- |
|
3193 |
apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI) |
|
3194 |
using X(2) assms(3)[rule_format,of x] by auto |
|
3195 |
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 |
|
36350 | 3196 |
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 |
35172 | 3197 |
unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv |
3198 |
apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto |
|
3199 |
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] |
|
3200 |
unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" . |
|
3201 |
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 |
|
3202 |
using assms(1) by(auto simp add:field_simps) qed qed qed |
|
3203 |
||
3204 |
subsection {* Special case of a basic affine transformation. *} |
|
3205 |
||
3206 |
lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}" |
|
3207 |
unfolding image_affinity_interval by auto |
|
3208 |
||
3209 |
lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta |
|
3210 |
Cart_eq vector_le_def vector_less_def |
|
3211 |
||
3212 |
lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A" |
|
3213 |
apply(rule setprod_cong) using assms by auto |
|
3214 |
||
3215 |
lemma content_image_affinity_interval: |
|
3216 |
"content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r") |
|
3217 |
proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption) |
|
3218 |
unfolding not_not using content_empty by auto } |
|
3219 |
assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0") |
|
3220 |
case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True] |
|
3221 |
unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') |
|
3222 |
defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym] |
|
3223 |
apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le |
|
3224 |
by(auto simp add:field_simps intro:mult_left_mono) |
|
3225 |
next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False] |
|
3226 |
unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') |
|
3227 |
defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym] |
|
3228 |
apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le |
|
3229 |
by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed |
|
3230 |
||
3231 |
lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0" |
|
3232 |
shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ CARD('n::finite))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})" |
|
3233 |
apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power) |
|
3234 |
defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps) |
|
3235 |
apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto |
|
3236 |
||
3237 |
lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0" |
|
3238 |
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})" |
|
3239 |
using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto |
|
3240 |
||
3241 |
subsection {* Special case of stretching coordinate axes separately. *} |
|
3242 |
||
3243 |
lemma image_stretch_interval: |
|
3244 |
"(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = |
|
3245 |
(if {a..b} = {} then {} else {(\<chi> k. min (m(k) * a$k) (m(k) * b$k)) .. (\<chi> k. max (m(k) * a$k) (m(k) * b$k))})" (is "?l = ?r") |
|
3246 |
proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto |
|
3247 |
next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto |
|
3248 |
case False note ab = this[unfolded interval_ne_empty] |
|
3249 |
show ?thesis apply-apply(rule set_ext) |
|
3250 |
proof- fix x::"real^'n" have **:"\<And>P Q. (\<forall>i. P i = Q i) \<Longrightarrow> (\<forall>i. P i) = (\<forall>i. Q i)" by auto |
|
3251 |
show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] |
|
3252 |
unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq * |
|
3253 |
unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"] |
|
3254 |
proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) = |
|
3255 |
(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))" |
|
3256 |
proof(cases "m i = 0") case True thus ?thesis using ab by auto |
|
3257 |
next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply- |
|
3258 |
proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i" |
|
3259 |
"max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto |
|
3260 |
show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI) |
|
3261 |
using as by(auto simp add:field_simps) |
|
3262 |
next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i" |
|
3263 |
"min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
3264 |
by(auto simp add:field_simps mult_le_cancel_left_neg intro: order_antisym) |
35172 | 3265 |
show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI) |
3266 |
using as by(auto simp add:field_simps) qed qed qed qed qed |
|
3267 |
||
3268 |
lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}" |
|
3269 |
unfolding image_stretch_interval by auto |
|
3270 |
||
3271 |
lemma content_image_stretch_interval: |
|
3272 |
"content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})" |
|
3273 |
proof(cases "{a..b} = {}") case True thus ?thesis |
|
3274 |
unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto |
|
3275 |
next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto |
|
3276 |
thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P |
|
3277 |
unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta |
|
3278 |
proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto |
|
3279 |
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)" |
|
3280 |
apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] |
|
3281 |
by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed |
|
3282 |
||
3283 |
lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)" |
|
3284 |
shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral |
|
3285 |
((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})" |
|
3286 |
apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval |
|
3287 |
unfolding image_stretch_interval empty_as_interval Cart_eq using assms |
|
3288 |
proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)" |
|
3289 |
apply(rule,rule linear_continuous_at) unfolding linear_linear |
|
3290 |
unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto |
|
3291 |
||
3292 |
lemma integrable_stretch: |
|
3293 |
assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)" |
|
3294 |
shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})" |
|
3295 |
using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto |
|
3296 |
||
3297 |
subsection {* even more special cases. *} |
|
3298 |
||
3299 |
lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}" |
|
3300 |
apply(rule set_ext,rule) defer unfolding image_iff |
|
3301 |
apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff) |
|
3302 |
||
3303 |
lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}" |
|
3304 |
shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}" |
|
3305 |
using has_integral_affinity[OF assms, of "-1" 0] by auto |
|
3306 |
||
3307 |
lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})" |
|
3308 |
apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto |
|
3309 |
||
3310 |
lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}" |
|
3311 |
unfolding integrable_on_def by auto |
|
3312 |
||
3313 |
lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f" |
|
3314 |
unfolding integral_def by auto |
|
3315 |
||
3316 |
subsection {* Stronger form of FCT; quite a tedious proof. *} |
|
3317 |
||
3318 |
(** move this **) |
|
3319 |
declare norm_triangle_ineq4[intro] |
|
3320 |
||
3321 |
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) |
|
3322 |
||
3323 |
lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector" |
|
3324 |
assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}" |
|
3325 |
shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a" |
|
3326 |
using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"] |
|
3327 |
unfolding o_def vec1_dest_vec1 using assms(1) by auto |
|
3328 |
||
36318 | 3329 |
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" |
35172 | 3330 |
unfolding split_def by(rule refl) |
3331 |
||
3332 |
lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e" |
|
3333 |
apply(subst(asm)(2) norm_minus_cancel[THEN sym]) |
|
36350 | 3334 |
apply(drule norm_triangle_le) by(auto simp add:algebra_simps) |
35172 | 3335 |
|
3336 |
lemma fundamental_theorem_of_calculus_interior: |
|
3337 |
assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)" |
|
3338 |
shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" |
|
3339 |
proof- { presume *:"a < b \<Longrightarrow> ?thesis" |
|
3340 |
show ?thesis proof(cases,rule *,assumption) |
|
3341 |
assume "\<not> a < b" hence "a = b" using assms(1) by auto |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
3342 |
hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" by(auto simp add: Cart_eq vector_le_def order_antisym) |
35172 | 3343 |
show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto |
3344 |
qed } assume ab:"a < b" |
|
3345 |
let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow> |
|
3346 |
norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f' \<circ> dest_vec1) x) - (f b - f a)) \<le> e * content {vec1 a..vec1 b})" |
|
3347 |
{ presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto } |
|
3348 |
fix e::real assume e:"e>0" |
|
3349 |
note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib] |
|
3350 |
note conjunctD2[OF this] note bounded=this(1) and this(2) |
|
3351 |
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)" |
|
3352 |
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] |
|
3353 |
from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format] |
|
3354 |
have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto |
|
3355 |
from this[unfolded bounded_pos] guess B .. note B = this[rule_format] |
|
3356 |
||
3357 |
have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da |
|
3358 |
\<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)" |
|
3359 |
proof- have "a\<in>{a..b}" using ab by auto |
|
3360 |
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this] |
|
3361 |
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps) |
|
3362 |
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format] |
|
3363 |
have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8" |
|
3364 |
proof(cases "f' a = 0") case True |
|
3365 |
thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) |
|
3366 |
next case False thus ?thesis |
|
3367 |
apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) |
|
3368 |
using ab e by(auto simp add:field_simps) |
|
3369 |
qed then guess l .. note l = conjunctD2[OF this] |
|
3370 |
show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+) |
|
3371 |
proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" |
|
3372 |
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval] |
|
3373 |
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) |
|
3374 |
also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" |
|
3375 |
proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto |
|
3376 |
thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto |
|
3377 |
next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer |
|
36587 | 3378 |
apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps) |
35172 | 3379 |
qed finally show "norm (content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto |
3380 |
qed qed then guess da .. note da=conjunctD2[OF this,rule_format] |
|
3381 |
||
3382 |
have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow> norm(content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4" |
|
3383 |
proof- have "b\<in>{a..b}" using ab by auto |
|
3384 |
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this] |
|
3385 |
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps) |
|
3386 |
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format] |
|
3387 |
have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8" |
|
3388 |
proof(cases "f' b = 0") case True |
|
3389 |
thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) |
|
3390 |
next case False thus ?thesis |
|
3391 |
apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI) |
|
3392 |
using ab e by(auto simp add:field_simps) |
|
3393 |
qed then guess l .. note l = conjunctD2[OF this] |
|
3394 |
show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+) |
|
3395 |
proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" |
|
3396 |
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval] |
|
3397 |
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) |
|
3398 |
also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" |
|
3399 |
proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto |
|
3400 |
thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto |
|
3401 |
next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute) |
|
36587 | 3402 |
apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps) |
35172 | 3403 |
qed finally show "norm (content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto |
3404 |
qed qed then guess db .. note db=conjunctD2[OF this,rule_format] |
|
3405 |
||
3406 |
let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))" |
|
3407 |
show "?P e" apply(rule_tac x="?d" in exI) |
|
3408 |
proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto |
|
3409 |
next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)] |
|
3410 |
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 |
|
3411 |
note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym] |
|
3412 |
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 |
|
3413 |
show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus |
|
3414 |
unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)] |
|
3415 |
proof(rule norm_triangle_le,rule **) |
|
3416 |
case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum) |
|
3417 |
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" |
|
3418 |
"e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2 |
|
3419 |
< norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))" |
|
3420 |
from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this |
|
3421 |
hence "\<forall>i. u$i \<le> v$i" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto note this(1) this(1)[unfolded forall_1] |
|
3422 |
note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]] |
|
3423 |
||
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
3424 |
assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add: Cart_eq) note * = d(2)[OF this] |
35172 | 3425 |
have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) = |
3426 |
norm ((f (u$1) - f (x$1) - (u$1 - x$1) *\<^sub>R f' (x$1)) - (f (v$1) - f (x$1) - (v$1 - x$1) *\<^sub>R f' (x$1)))" |
|
3427 |
apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto |
|
3428 |
also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub) |
|
3429 |
apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq |
|
3430 |
apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real) |
|
3431 |
also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps) |
|
3432 |
finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2" |
|
3433 |
apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed |
|
3434 |
||
3435 |
next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto |
|
3436 |
case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv) |
|
3437 |
defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] |
|
3438 |
apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms) |
|
3439 |
proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this] |
|
3440 |
from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this |
|
3441 |
with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto |
|
3442 |
thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)" |
|
3443 |
unfolding uv using e by(auto simp add:field_simps) |
|
3444 |
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 |
|
3445 |
show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x - |
|
3446 |
(f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" |
|
3447 |
apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"]) |
|
3448 |
apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def |
|
3449 |
proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}} - p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content (snd t) \<noteq> 0}" |
|
3450 |
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 |
|
3451 |
have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk unfolding uv content_eq_0_1 interval_eq_empty by auto |
|
3452 |
thus "content k *\<^sub>R (f' (x$1)) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1)) = 0" using xk unfolding uv by auto |
|
3453 |
next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = |
|
3454 |
{t. t\<in>p \<and> fst t = vec1 a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = vec1 b \<and> content(snd t) \<noteq> 0}" by blast |
|
3455 |
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" |
|
3456 |
proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto |
|
3457 |
thus ?case using `x\<in>s` goal2(2) by auto |
|
3458 |
qed auto |
|
3459 |
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"]) |
|
3460 |
apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **) |
|
3461 |
proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}" |
|
3462 |
have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" |
|
3463 |
proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this |
|
3464 |
have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto |
|
3465 |
have u:"u = vec1 a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto |
|
3466 |
have "u \<ge> vec1 a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>vec1 a" ultimately |
|
3467 |
have "u > vec1 a" unfolding Cart_simps by auto |
|
3468 |
thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps) |
|
3469 |
qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto |
|
3470 |
qed |
|
3471 |
have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" |
|
3472 |
proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this |
|
3473 |
have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto |
|
3474 |
have u:"v = vec1 b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto |
|
3475 |
have "v \<le> vec1 b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq>vec1 b" ultimately |
|
3476 |
have "v < vec1 b" unfolding Cart_simps by auto |
|
3477 |
thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps) |
|
3478 |
qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto |
|
3479 |
qed |
|
3480 |
||
3481 |
show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all) |
|
3482 |
unfolding mem_Collect_eq fst_conv snd_conv apply safe |
|
3483 |
proof- fix x k k' assume k:"(vec1 a, k) \<in> p" "(vec1 a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0" |
|
3484 |
guess v using pa[OF k(1)] .. note v = conjunctD2[OF this] |
|
3485 |
guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))" |
|
3486 |
have "{vec1 a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter] |
|
3487 |
moreover have "vec1 ((a + ?v$1)/2) \<in> {vec1 a <..< ?v}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps) |
|
3488 |
ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto |
|
3489 |
hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto |
|
3490 |
{ assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . } |
|
3491 |
qed |
|
3492 |
show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all) |
|
3493 |
unfolding mem_Collect_eq fst_conv snd_conv apply safe |
|
3494 |
proof- fix x k k' assume k:"(vec1 b, k) \<in> p" "(vec1 b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0" |
|
3495 |
guess v using pb[OF k(1)] .. note v = conjunctD2[OF this] |
|
3496 |
guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))" |
|
3497 |
have "{?v <..< vec1 b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter] |
|
3498 |
moreover have "vec1 ((b + ?v$1)/2) \<in> {?v <..< vec1 b}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps) |
|
3499 |
ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto |
|
3500 |
hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto |
|
3501 |
{ assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . } |
|
3502 |
qed |
|
3503 |
||
3504 |
let ?a = a and ?b = b (* a is something else while proofing the next theorem. *) |
|
3505 |
show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x) |
|
3506 |
\<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1 |
|
3507 |
proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this] |
|
3508 |
have "vec1 ?a\<in>{vec1 ?a..v}" using v(2) by auto hence "dest_vec1 v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto |
|
3509 |
moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)] |
|
3510 |
apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) |
|
3511 |
by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately |
|
3512 |
show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply- |
|
3513 |
apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1]) |
|
3514 |
using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto |
|
3515 |
qed |
|
3516 |
show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x) |
|
3517 |
\<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1 |
|
3518 |
proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this] |
|
3519 |
have "vec1 ?b\<in>{v..vec1 ?b}" using v(2) by auto hence "dest_vec1 v \<ge> ?a" using p(3)[OF goal1(1)] unfolding subset_eq v by auto |
|
3520 |
moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)] |
|
3521 |
apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab |
|
3522 |
by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately |
|
3523 |
show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply- |
|
3524 |
apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1]) |
|
3525 |
using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto |
|
3526 |
qed |
|
3527 |
qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed |
|
3528 |
||
3529 |
subsection {* Stronger form with finite number of exceptional points. *} |
|
3530 |
||
3531 |
lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach" |
|
3532 |
assumes"finite s" "a \<le> b" "continuous_on {a..b} f" |
|
3533 |
"\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)" |
|
3534 |
shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply- |
|
3535 |
proof(induct "card s" arbitrary:s a b) |
|
3536 |
case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto |
|
3537 |
next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq |
|
3538 |
apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format] |
|
3539 |
show ?case proof(cases "c\<in>{a<..<b}") |
|
3540 |
case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer |
|
3541 |
apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto |
|
3542 |
next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto |
|
3543 |
case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto |
|
3544 |
thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+ |
|
3545 |
apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs |
|
3546 |
proof- show "continuous_on {a..c} f" "continuous_on {c..b} f" |
|
3547 |
apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto |
|
3548 |
let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)" |
|
3549 |
show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto |
|
3550 |
qed auto qed qed |
|
3551 |
||
3552 |
lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach" |
|
3553 |
assumes "finite s" "a \<le> b" "continuous_on {a..b} f" |
|
3554 |
"\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)" |
|
3555 |
shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}" |
|
3556 |
apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f']) |
|
3557 |
using assms(4) by auto |
|
3558 |
||
35751 | 3559 |
lemma indefinite_integral_continuous_left: fixes f::"real^1 \<Rightarrow> 'a::banach" |
3560 |
assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e" |
|
3561 |
obtains d where "0 < d" "\<forall>t. c$1 - d < t$1 \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e" |
|
3562 |
proof- have "\<exists>w>0. \<forall>t. c$1 - w < t$1 \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3" |
|
3563 |
proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)" |
|
3564 |
apply-apply(rule divide_pos_pos) using `e>0` by auto |
|
3565 |
thus ?thesis apply-apply(rule,rule,assumption,safe) |
|
3566 |
proof- fix t assume as:"t < c" and "c$1 - e / 3 / norm (f c) < t$(1::1)" |
|
3567 |
hence "c$1 - t$1 < e / 3 / norm (f c)" by auto |
|
3568 |
hence "norm (c - t) < e / 3 / norm (f c)" using as unfolding norm_vector_1 vector_less_def by auto |
|
3569 |
thus "norm (f c) * norm (c - t) < e / 3" using False apply- |
|
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
3570 |
apply(subst mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto |
35751 | 3571 |
qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto |
3572 |
qed then guess w .. note w = conjunctD2[OF this,rule_format] |
|
3573 |
||
3574 |
have *:"e / 3 > 0" using assms by auto |
|
3575 |
have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto |
|
3576 |
from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 .. |
|
3577 |
note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x" |
|
3578 |
have "gauge d" unfolding d_def using w(1) d1 by auto |
|
3579 |
note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this] |
|
3580 |
from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this] |
|
3581 |
||
3582 |
let ?d = "min k (c$1 - a$1)/2" show ?thesis apply(rule that[of ?d]) |
|
3583 |
proof safe show "?d > 0" using k(1) using assms(2) unfolding vector_less_def by auto |
|
3584 |
fix t assume as:"c$1 - ?d < t$1" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e" |
|
3585 |
{ presume *:"t < c \<Longrightarrow> ?thesis" |
|
3586 |
show ?thesis apply(cases "t = c") defer apply(rule *) |
|
3587 |
unfolding vector_less_def apply(subst less_le) using `e>0` as(2) by auto } assume "t < c" |
|
3588 |
||
3589 |
have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto |
|
3590 |
from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 .. |
|
3591 |
note d2 = conjunctD2[OF this,rule_format] |
|
3592 |
def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x" |
|
3593 |
have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto |
|
3594 |
from fine_division_exists[OF this, of a t] guess p . note p=this |
|
3595 |
note p'=tagged_division_ofD[OF this(1)] |
|
3596 |
have pt:"\<forall>(x,k)\<in>p. x$1 \<le> t$1" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed |
|
3597 |
with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto |
|
3598 |
note d2_fin = d2(2)[OF conjI[OF p(1) this]] |
|
3599 |
||
3600 |
have *:"{a..c} \<inter> {x. x$1 \<le> t$1} = {a..t}" "{a..c} \<inter> {x. x$1 \<ge> t$1} = {t..c}" |
|
3601 |
using assms(2-3) as by(auto simp add:field_simps) |
|
3602 |
have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule |
|
3603 |
apply(rule tagged_division_union_interval[of _ _ _ 1 "t$1"]) unfolding * apply(rule p) |
|
3604 |
apply(rule tagged_division_of_self) unfolding fine_def |
|
3605 |
proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x" |
|
3606 |
using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto |
|
3607 |
next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real |
|
3608 |
using as(1) by(auto simp add:field_simps) |
|
3609 |
thus "x \<in> d1 c" using k(2) unfolding d_def by auto |
|
3610 |
qed(insert as(2), auto) note d1_fin = d1(2)[OF this] |
|
3611 |
||
3612 |
have *:"integral{a..c} f - integral {a..t} f = -(((c$1 - t$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) - |
|
3613 |
integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c$1 - t$1) *\<^sub>R f c" |
|
3614 |
"e = (e/3 + e/3) + e/3" by auto |
|
3615 |
have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c$1 - t$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" |
|
3616 |
proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto |
|
3617 |
have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this] |
|
3618 |
have "c \<in> {a..t}" by auto thus False using `t<c` unfolding vector_less_def by auto |
|
3619 |
qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p') |
|
3620 |
unfolding split_conv defer apply(subst content_1) using as(2) by auto qed |
|
3621 |
||
3622 |
have ***:"c$1 - w < t$1 \<and> t < c" |
|
3623 |
proof- have "c$1 - k < t$1" using `k>0` as(1) by(auto simp add:field_simps) |
|
3624 |
moreover have "k \<le> w" apply(rule ccontr) using k(2) |
|
3625 |
unfolding subset_eq apply(erule_tac x="c + vec ((k + w)/2)" in ballE) |
|
3626 |
unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real) |
|
3627 |
ultimately show ?thesis using `t<c` by(auto simp add:field_simps) qed |
|
3628 |
||
3629 |
show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+ |
|
3630 |
unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin) |
|
3631 |
using w(2)[OF ***] unfolding norm_scaleR norm_real by(auto simp add:field_simps) qed qed |
|
3632 |
||
3633 |
lemma indefinite_integral_continuous_right: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3634 |
assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e" |
|
3635 |
obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t$1 < c$1 + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e" |
|
3636 |
proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a" |
|
3637 |
using assms unfolding Cart_simps by auto |
|
3638 |
from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b$1 - c$1)" |
|
3639 |
show ?thesis apply(rule that[of "?d"]) |
|
3640 |
proof safe show "0 < ?d" using d(1) assms(3) unfolding Cart_simps by auto |
|
3641 |
fix t::"_^1" assume as:"c \<le> t" "t$1 < c$1 + ?d" |
|
3642 |
have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f" |
|
36350 | 3643 |
"integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps |
35751 | 3644 |
apply(rule_tac[!] integral_combine) using assms as unfolding Cart_simps by auto |
3645 |
have "(- c)$1 - d < (- t)$1 \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this] |
|
3646 |
thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding * |
|
36350 | 3647 |
unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed |
35751 | 3648 |
|
3649 |
declare dest_vec1_eq[simp del] not_less[simp] not_le[simp] |
|
3650 |
||
3651 |
lemma indefinite_integral_continuous: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3652 |
assumes "f integrable_on {a..b}" shows "continuous_on {a..b} (\<lambda>x. integral {a..x} f)" |
|
36359 | 3653 |
proof(unfold continuous_on_iff, safe) fix x e assume as:"x\<in>{a..b}" "0<(e::real)" |
35751 | 3654 |
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" |
3655 |
{ presume *:"a<b \<Longrightarrow> ?thesis" |
|
3656 |
show ?thesis apply(cases,rule *,assumption) |
|
3657 |
proof- case goal1 hence "{a..b} = {x}" using as(1) unfolding Cart_simps |
|
3658 |
by(auto simp only:field_simps not_less Cart_eq forall_1 mem_interval) |
|
3659 |
thus ?case using `e>0` by auto |
|
3660 |
qed } assume "a<b" |
|
3661 |
have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add: Cart_simps) |
|
3662 |
thus ?thesis apply-apply(erule disjE)+ |
|
3663 |
proof- assume "x=a" have "a \<le> a" by auto |
|
3664 |
from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this |
|
3665 |
show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute) |
|
36587 | 3666 |
unfolding `x=a` dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto |
35751 | 3667 |
next assume "x=b" have "b \<le> b" by auto |
3668 |
from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this |
|
3669 |
show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute) |
|
36587 | 3670 |
unfolding `x=b` dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto |
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
3671 |
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: vector_less_def) |
35751 | 3672 |
from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this |
3673 |
from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this |
|
3674 |
show ?thesis apply(rule_tac x="min d1 d2" in exI) |
|
3675 |
proof safe show "0 < min d1 d2" using d1 d2 by auto |
|
3676 |
fix y assume "y\<in>{a..b}" "dist y x < min d1 d2" |
|
3677 |
thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute) |
|
36587 | 3678 |
apply(cases "y < x") unfolding dist_norm apply(rule d1(2)[rule_format]) defer |
35751 | 3679 |
apply(rule d2(2)[rule_format]) unfolding Cart_simps not_less norm_real by(auto simp add:field_simps) |
3680 |
qed qed qed |
|
3681 |
||
3682 |
subsection {* This doesn't directly involve integration, but that gives an easy proof. *} |
|
3683 |
||
3684 |
lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach" |
|
3685 |
assumes "finite k" "continuous_on {a..b} f" "f a = y" |
|
3686 |
"\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}" |
|
3687 |
shows "f x = y" |
|
3688 |
proof- have ab:"a\<le>b" using assms by auto |
|
3689 |
have *:"(\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 = (\<lambda>x. 0)" unfolding o_def by auto have **:"a \<le> x" using assms by auto |
|
3690 |
have "((\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 has_integral f x - f a) {vec1 a..vec1 x}" |
|
3691 |
apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) ** ]) |
|
3692 |
apply(rule continuous_on_subset[OF assms(2)]) defer |
|
3693 |
apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym]) |
|
3694 |
apply assumption apply(rule open_interval_real) apply(rule has_derivative_within_subset[where s="{a..b}"]) |
|
3695 |
using assms(4) assms(5) by auto note this[unfolded *] |
|
3696 |
note has_integral_unique[OF has_integral_0 this] |
|
3697 |
thus ?thesis unfolding assms by auto qed |
|
3698 |
||
3699 |
subsection {* Generalize a bit to any convex set. *} |
|
3700 |
||
3701 |
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib |
|
3702 |
scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff |
|
3703 |
scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one |
|
3704 |
||
3705 |
lemma has_derivative_zero_unique_strong_convex: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3706 |
assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y" |
|
3707 |
"\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s" |
|
3708 |
shows "f x = y" |
|
3709 |
proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption) |
|
3710 |
unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c" |
|
3711 |
note conv = assms(1)[unfolded convex_alt,rule_format] |
|
3712 |
have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))" |
|
3713 |
apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)]) |
|
3714 |
apply safe apply(rule conv) using assms(4,7) by auto |
|
3715 |
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" |
|
3716 |
proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c" |
|
36350 | 3717 |
unfolding scaleR_simps by(auto simp add:algebra_simps) |
35751 | 3718 |
thus ?case using `x\<noteq>c` by auto qed |
3719 |
have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2) |
|
3720 |
apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"]) |
|
3721 |
apply safe unfolding image_iff apply rule defer apply assumption |
|
3722 |
apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto |
|
3723 |
have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y" |
|
3724 |
apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ]) |
|
3725 |
unfolding o_def using assms(5) defer apply-apply(rule) |
|
3726 |
proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}" |
|
3727 |
have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps]) |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
3728 |
using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps) |
35751 | 3729 |
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})" |
3730 |
apply(rule diff_chain_within) apply(rule has_derivative_add) |
|
3731 |
unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const) |
|
3732 |
apply(rule has_derivative_vmul_within,rule has_derivative_id)+ |
|
3733 |
apply(rule has_derivative_within_subset,rule assms(6)[rule_format]) |
|
3734 |
apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto |
|
3735 |
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 . |
|
3736 |
qed auto thus ?thesis by auto qed |
|
3737 |
||
3738 |
subsection {* Also to any open connected set with finite set of exceptions. Could |
|
3739 |
generalize to locally convex set with limpt-free set of exceptions. *} |
|
3740 |
||
3741 |
lemma has_derivative_zero_unique_strong_connected: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3742 |
assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y" |
|
3743 |
"\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s" |
|
3744 |
shows "f x = y" |
|
3745 |
proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s" |
|
3746 |
apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer |
|
3747 |
apply(rule continuous_closed_in_preimage[OF assms(4) closed_sing]) |
|
3748 |
apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball |
|
3749 |
proof safe fix x assume "x\<in>s" |
|
3750 |
from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this] |
|
3751 |
show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e) |
|
3752 |
proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto |
|
3753 |
show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball]) |
|
3754 |
apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+ |
|
3755 |
apply(subst centre_in_ball,rule e,rule) apply safe |
|
3756 |
apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format]) |
|
3757 |
using y e by auto qed qed |
|
3758 |
thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed |
|
3759 |
||
3760 |
subsection {* Integrating characteristic function of an interval. *} |
|
3761 |
||
3762 |
lemma has_integral_restrict_open_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3763 |
assumes "(f has_integral i) {c..d}" "{c..d} \<subseteq> {a..b}" |
|
3764 |
shows "((\<lambda>x. if x \<in> {c<..<d} then f x else 0) has_integral i) {a..b}" |
|
3765 |
proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0" |
|
3766 |
{ presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis" |
|
3767 |
show ?thesis apply(cases,rule *,assumption) |
|
3768 |
proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto |
|
3769 |
show ?thesis using assms(1) unfolding * using goal1 by auto |
|
3770 |
qed } assume "{c..d}\<noteq>{}" |
|
3771 |
from partial_division_extend_1[OF assms(2) this] guess p . note p=this |
|
3772 |
note mon = monoidal_lifted[OF monoidal_monoid] |
|
3773 |
note operat = operative_division[OF this operative_integral p(1), THEN sym] |
|
3774 |
let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i" |
|
3775 |
{ presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i" |
|
3776 |
apply- apply(cases,subst(asm) if_P,assumption) by auto |
|
3777 |
thus ?thesis using integrable_integral unfolding g_def by auto } |
|
3778 |
||
3779 |
note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]] |
|
3780 |
note * = this[unfolded neutral_monoid] |
|
3781 |
have iterate:"iterate (lifted op +) (p - {{c..d}}) |
|
3782 |
(\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0" |
|
3783 |
proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this] |
|
3784 |
from div(3) guess u v apply-by(erule exE)+ note uv=this |
|
3785 |
have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto |
|
3786 |
hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"]) |
|
3787 |
unfolding g_def interior_closed_interval by auto thus ?case by auto |
|
3788 |
qed |
|
3789 |
||
3790 |
have *:"p = insert {c..d} (p - {{c..d}})" using p by auto |
|
3791 |
have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f]) |
|
3792 |
unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto |
|
3793 |
moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)]) |
|
3794 |
apply(rule has_integral_spike_interior[where f=g]) defer |
|
3795 |
apply(rule integrable_integral[OF **]) unfolding g_def by auto |
|
3796 |
ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+ |
|
3797 |
unfolding iterate defer apply(subst if_not_P) defer using p by auto qed |
|
3798 |
||
3799 |
lemma has_integral_restrict_closed_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3800 |
assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}" |
|
3801 |
shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}" |
|
3802 |
proof- note has_integral_restrict_open_subinterval[OF assms] |
|
3803 |
note * = has_integral_spike[OF negligible_frontier_interval _ this] |
|
3804 |
show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed |
|
3805 |
||
3806 |
lemma has_integral_restrict_closed_subintervals_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" assumes "{c..d} \<subseteq> {a..b}" |
|
3807 |
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") |
|
3808 |
proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0" |
|
3809 |
show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) |
|
3810 |
proof assumption assume ?l hence "?g integrable_on {c..d}" |
|
3811 |
apply-apply(rule integrable_subinterval[OF _ assms]) by auto |
|
3812 |
hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto |
|
3813 |
hence "i = integral {c..d} f" apply-apply(rule has_integral_unique) |
|
3814 |
apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto |
|
3815 |
thus ?r using * by auto qed qed auto |
|
3816 |
||
3817 |
subsection {* Hence we can apply the limit process uniformly to all integrals. *} |
|
3818 |
||
3819 |
lemma has_integral': fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
|
3820 |
"(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} |
|
3821 |
\<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)") |
|
3822 |
proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis" |
|
3823 |
show ?thesis apply(cases,rule *,assumption) |
|
3824 |
apply(subst has_integral_alt) by auto } |
|
3825 |
assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this |
|
3826 |
from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B .. |
|
3827 |
note B = conjunctD2[OF this,rule_format] show ?thesis apply safe |
|
3828 |
proof- fix e assume ?l "e>(0::real)" |
|
3829 |
show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI) |
|
3830 |
proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::real^'n}" |
|
3831 |
thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s |
|
3832 |
apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s]) |
|
3833 |
apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE) |
|
36587 | 3834 |
by(auto simp add:dist_norm) |
35751 | 3835 |
qed(insert B `e>0`, auto) |
3836 |
next assume as:"\<forall>e>0. ?r e" |
|
3837 |
from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format] |
|
3838 |
def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n" |
|
3839 |
have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval |
|
3840 |
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def |
|
3841 |
by(auto simp add:field_simps) qed |
|
36587 | 3842 |
have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm |
35751 | 3843 |
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed |
3844 |
from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}" |
|
3845 |
unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto |
|
3846 |
then guess y .. note y=this |
|
3847 |
||
3848 |
have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto |
|
3849 |
from as[rule_format,OF this] guess C .. note C=conjunctD2[OF this,rule_format] |
|
3850 |
def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n" |
|
3851 |
have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval |
|
3852 |
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def |
|
3853 |
by(auto simp add:field_simps) qed |
|
36587 | 3854 |
have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm |
35751 | 3855 |
proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed |
3856 |
note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s] |
|
3857 |
note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]] |
|
3858 |
hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) . |
|
3859 |
thus False by auto qed |
|
3860 |
thus ?l using y unfolding s by auto qed qed |
|
3861 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3862 |
lemma has_integral_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3863 |
"((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3864 |
unfolding has_integral'[unfolded has_integral] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3865 |
proof case goal1 thus ?case apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3866 |
apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3867 |
apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3868 |
apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3869 |
apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3870 |
apply(subst(asm)(2) norm_vector_1) unfolding split_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3871 |
unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3872 |
Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3873 |
apply(subst(asm)(2) norm_vector_1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3874 |
next case goal2 thus ?case apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3875 |
apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3876 |
apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3877 |
apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3878 |
apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3879 |
apply(subst norm_vector_1) unfolding split_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3880 |
unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3881 |
Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3882 |
apply(subst norm_vector_1) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3883 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3884 |
lemma integral_trans[simp]: assumes "(f::real^'n \<Rightarrow> real) integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3885 |
shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3886 |
apply(rule integral_unique) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3887 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3888 |
lemma integrable_on_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3889 |
"(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3890 |
unfolding integrable_on_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3891 |
apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3892 |
apply safe defer apply(rule_tac x="vec1 y" in exI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3893 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3894 |
lemma has_integral_le: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3895 |
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x) \<le> (g x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3896 |
shows "i \<le> j" using has_integral_component_le[of "vec1 o f" "vec1 i" s "vec1 o g" "vec1 j" 1] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3897 |
unfolding o_def using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3898 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3899 |
lemma integral_le: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3900 |
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3901 |
shows "integral s f \<le> integral s g" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3902 |
using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3903 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3904 |
lemma has_integral_nonneg: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3905 |
assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3906 |
using has_integral_component_nonneg[of "vec1 o f" "vec1 i" s 1] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3907 |
unfolding o_def using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3908 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3909 |
lemma integral_nonneg: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3910 |
assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3911 |
using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
3912 |
|
35751 | 3913 |
subsection {* Hence a general restriction property. *} |
3914 |
||
3915 |
lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows |
|
3916 |
"((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s" |
|
3917 |
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 |
|
3918 |
show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed |
|
3919 |
||
3920 |
lemma has_integral_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
|
3921 |
"((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto |
|
3922 |
||
3923 |
lemma has_integral_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3924 |
assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s" |
|
3925 |
shows "(f has_integral i) t" |
|
3926 |
proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)" |
|
3927 |
apply(rule) using assms(1-2) by auto |
|
3928 |
thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym]) |
|
3929 |
apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed |
|
3930 |
||
3931 |
lemma integrable_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3932 |
assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s" |
|
3933 |
shows "f integrable_on t" |
|
3934 |
using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset) |
|
3935 |
||
3936 |
lemma integral_restrict_univ[intro]: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3937 |
shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f" |
|
3938 |
apply(rule integral_unique) unfolding has_integral_restrict_univ by auto |
|
3939 |
||
3940 |
lemma integrable_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
|
3941 |
"(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s" |
|
3942 |
unfolding integrable_on_def by auto |
|
3943 |
||
3944 |
lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r") |
|
3945 |
proof assume ?r show ?l unfolding negligible_def |
|
3946 |
proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]]) |
|
3947 |
unfolding indicator_def by auto qed qed auto |
|
3948 |
||
3949 |
lemma has_integral_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3950 |
assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)" |
|
36362
06475a1547cb
fix lots of looping simp calls and other warnings
huffman
parents:
36359
diff
changeset
|
3951 |
unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (safe, auto split: split_if_asm) |
35751 | 3952 |
|
3953 |
lemma has_integral_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3954 |
assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s" |
|
3955 |
shows "(f has_integral y) t" |
|
3956 |
using assms has_integral_spike_set_eq by auto |
|
3957 |
||
3958 |
lemma integrable_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3959 |
assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s" |
|
3960 |
shows "f integrable_on t" using assms(2) unfolding integrable_on_def |
|
3961 |
unfolding has_integral_spike_set_eq[OF assms(1)] . |
|
3962 |
||
3963 |
lemma integrable_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
3964 |
assumes "negligible((s - t) \<union> (t - s))" |
|
3965 |
shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)" |
|
3966 |
apply(rule,rule_tac[!] integrable_spike_set) using assms by auto |
|
3967 |
||
3968 |
(*lemma integral_spike_set: |
|
3969 |
"\<forall>f:real^M->real^N g s t. |
|
3970 |
negligible(s DIFF t \<union> t DIFF s) |
|
3971 |
\<longrightarrow> integral s f = integral t f" |
|
3972 |
qed REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN |
|
3973 |
AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN |
|
3974 |
ASM_MESON_TAC[]);; |
|
3975 |
||
3976 |
lemma has_integral_interior: |
|
3977 |
"\<forall>f:real^M->real^N y s. |
|
3978 |
negligible(frontier s) |
|
3979 |
\<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)" |
|
3980 |
qed REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN |
|
3981 |
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] |
|
3982 |
NEGLIGIBLE_SUBSET)) THEN |
|
3983 |
REWRITE_TAC[frontier] THEN |
|
3984 |
MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN |
|
3985 |
MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN |
|
3986 |
SET_TAC[]);; |
|
3987 |
||
3988 |
lemma has_integral_closure: |
|
3989 |
"\<forall>f:real^M->real^N y s. |
|
3990 |
negligible(frontier s) |
|
3991 |
\<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)" |
|
3992 |
qed REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN |
|
3993 |
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ] |
|
3994 |
NEGLIGIBLE_SUBSET)) THEN |
|
3995 |
REWRITE_TAC[frontier] THEN |
|
3996 |
MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN |
|
3997 |
MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN |
|
3998 |
SET_TAC[]);;*) |
|
3999 |
||
4000 |
subsection {* More lemmas that are useful later. *} |
|
4001 |
||
4002 |
lemma has_integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
4003 |
assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)$k" |
|
4004 |
shows "i$k \<le> j$k" |
|
4005 |
proof- note has_integral_restrict_univ[THEN sym, of f] |
|
4006 |
note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this] |
|
4007 |
show ?thesis apply(rule *) using assms(1,4) by auto qed |
|
4008 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4009 |
lemma has_integral_subset_le: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4010 |
assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4011 |
shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "vec1 o f" "vec1 i" "vec1 j" 1] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4012 |
unfolding o_def using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4013 |
|
35751 | 4014 |
lemma integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
4015 |
assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)$k" |
|
4016 |
shows "(integral s f)$k \<le> (integral t f)$k" |
|
4017 |
apply(rule has_integral_subset_component_le) using assms by auto |
|
4018 |
||
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4019 |
lemma integral_subset_le: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4020 |
assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4021 |
shows "(integral s f) \<le> (integral t f)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4022 |
apply(rule has_integral_subset_le) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4023 |
|
35751 | 4024 |
lemma has_integral_alt': fixes f::"real^'n \<Rightarrow> 'a::banach" |
4025 |
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> |
|
4026 |
(\<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") |
|
4027 |
proof assume ?r |
|
4028 |
show ?l apply- apply(subst has_integral') |
|
4029 |
proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this] |
|
4030 |
show ?case apply(rule,rule,rule B,safe) |
|
4031 |
apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI) |
|
4032 |
apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto |
|
4033 |
qed next |
|
4034 |
assume ?l note as = this[unfolded has_integral'[of f],rule_format] |
|
4035 |
let ?f = "\<lambda>x. if x \<in> s then f x else 0" |
|
4036 |
show ?r proof safe fix a b::"real^'n" |
|
4037 |
from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format] |
|
4038 |
let ?a = "(\<chi> i. min (a$i) (-B))::real^'n" and ?b = "(\<chi> i. max (b$i) B)::real^'n" |
|
4039 |
show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b]) |
|
36587 | 4040 |
proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval dist_norm |
35751 | 4041 |
proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed |
4042 |
from B(2)[OF this] guess z .. note conjunct1[OF this] |
|
4043 |
thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto |
|
4044 |
show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed |
|
4045 |
||
4046 |
fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format] |
|
4047 |
show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> |
|
4048 |
norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e" |
|
4049 |
proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this] |
|
4050 |
from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed |
|
4051 |
||
35752 | 4052 |
|
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4053 |
subsection {* Continuity of the integral (for a 1-dimensional interval). *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4054 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4055 |
lemma integrable_alt: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4056 |
"f integrable_on s \<longleftrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4057 |
(\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4058 |
(\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4059 |
\<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4060 |
integral {c..d} (\<lambda>x. if x \<in> s then f x else 0)) < e)" (is "?l = ?r") |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4061 |
proof assume ?l then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4062 |
note y=conjunctD2[OF this,rule_format] show ?r apply safe apply(rule y) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4063 |
proof- case goal1 hence "e/2 > 0" by auto from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4064 |
show ?case apply(rule,rule,rule B) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4065 |
proof safe case goal1 show ?case apply(rule norm_triangle_half_l) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4066 |
using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4067 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4068 |
next assume ?r note as = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4069 |
have "Cauchy (\<lambda>n. integral ({(\<chi> i. - real n) .. (\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4070 |
proof(unfold Cauchy_def,safe) case goal1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4071 |
from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4072 |
from real_arch_simple[of B] guess N .. note N = this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4073 |
{ fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {\<chi> i. - real n..\<chi> i. real n}" apply safe |
36587 | 4074 |
unfolding mem_ball mem_interval dist_norm |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4075 |
proof case goal1 thus ?case using component_le_norm[of x i] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4076 |
using n N by(auto simp add:field_simps) qed } |
36587 | 4077 |
thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4078 |
qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4079 |
note i = this[unfolded Lim_sequentially, rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4080 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4081 |
show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4082 |
apply safe apply(rule as(1)[unfolded integrable_on_def]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4083 |
proof- case goal1 hence *:"e/2 > 0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4084 |
from i[OF this] guess N .. note N =this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4085 |
from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format] let ?B = "max (real N) B" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4086 |
show ?case apply(rule_tac x="?B" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4087 |
proof safe show "0 < ?B" using B(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4088 |
fix a b assume ab:"ball 0 ?B \<subseteq> {a..b::real^'n}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4089 |
from real_arch_simple[of ?B] guess n .. note n=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4090 |
show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4091 |
apply(rule norm_triangle_half_l) apply(rule B(2)) defer apply(subst norm_minus_commute) |
36587 | 4092 |
apply(rule N[unfolded dist_norm, of n]) |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4093 |
proof safe show "N \<le> n" using n by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4094 |
fix x::"real^'n" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4095 |
thus "x\<in>{a..b}" using ab by blast |
36587 | 4096 |
show "x\<in>{\<chi> i. - real n..\<chi> i. real n}" using x unfolding mem_interval mem_ball dist_norm apply- |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4097 |
proof case goal1 thus ?case using component_le_norm[of x i] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4098 |
using n by(auto simp add:field_simps) qed qed qed qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4099 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4100 |
lemma integrable_altD: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4101 |
assumes "f integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4102 |
shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4103 |
"\<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} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4104 |
\<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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4105 |
using assms[unfolded integrable_alt[of f]] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4106 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4107 |
lemma integrable_on_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4108 |
assumes "f integrable_on s" "{a..b} \<subseteq> s" shows "f integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4109 |
apply(rule integrable_eq) defer apply(rule integrable_altD(1)[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4110 |
using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4111 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4112 |
subsection {* A straddling criterion for integrability. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4113 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4114 |
lemma integrable_straddle_interval: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4115 |
assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) ({a..b}) \<and> (h has_integral j) ({a..b}) \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4116 |
norm(i - j) < e \<and> (\<forall>x\<in>{a..b}. (g x) \<le> (f x) \<and> (f x) \<le>(h x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4117 |
shows "f integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4118 |
proof(subst integrable_cauchy,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4119 |
case goal1 hence e:"e/3 > 0" by auto note assms[rule_format,OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4120 |
then guess g h i j apply- by(erule exE conjE)+ note obt = this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4121 |
from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4122 |
from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4123 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4124 |
proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4125 |
abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow> abs(g1 - i) < e / 3 \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4126 |
abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4127 |
case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4128 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4129 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4130 |
"0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4131 |
"(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4132 |
"0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4133 |
unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4134 |
apply safe unfolding real_scaleR_def mult.diff_right[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4135 |
apply(rule_tac[!] mult_nonneg_nonneg) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4136 |
proof- fix a b assume ab:"(a,b) \<in> p1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4137 |
show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4138 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4139 |
next fix a b assume ab:"(a,b) \<in> p2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4140 |
show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4141 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4142 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4143 |
thus ?case apply- unfolding real_norm_def apply(rule **) defer defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4144 |
unfolding real_norm_def[THEN sym] apply(rule obt(3)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4145 |
apply(rule d1(2)[OF conjI[OF goal1(4,5)]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4146 |
apply(rule d1(2)[OF conjI[OF goal1(1,2)]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4147 |
apply(rule d2(2)[OF conjI[OF goal1(4,6)]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4148 |
apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4149 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4150 |
lemma integrable_straddle: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4151 |
assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4152 |
norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4153 |
shows "f integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4154 |
proof- have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4155 |
proof(rule integrable_straddle_interval,safe) case goal1 hence *:"e/4 > 0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4156 |
from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4157 |
note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4158 |
note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4159 |
note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4160 |
note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4161 |
def c \<equiv> "\<chi> i. min (a$i) (- (max B1 B2))" and d \<equiv> "\<chi> i. max (b$i) (max B1 B2)" |
36587 | 4162 |
have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval dist_norm |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4163 |
proof(rule_tac[!] allI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4164 |
case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4165 |
case goal2 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4166 |
have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4167 |
norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4168 |
using obt(3) unfolding real_norm_def by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4169 |
show ?case apply(rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4170 |
apply(rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4171 |
apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4172 |
apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then h x else 0)" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4173 |
apply safe apply(rule_tac[1-2] integrable_integral,rule g,rule h) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4174 |
apply(rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4175 |
proof- have *:"\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) = |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4176 |
(if x \<in> s then f x - g x else (0::real))" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4177 |
note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_sub, OF h g]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4178 |
show " norm (integral {a..b} (\<lambda>x. if x \<in> s then h x else 0) - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4179 |
integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4180 |
\<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4181 |
integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4182 |
unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4183 |
apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4184 |
proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4185 |
apply - apply rule apply(erule_tac x=i in allE) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4186 |
qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4187 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4188 |
show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4189 |
proof- case goal1 hence *:"e/3 > 0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4190 |
from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4191 |
note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4192 |
note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4193 |
note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4194 |
note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4195 |
show ?case apply(rule_tac x="max B1 B2" in exI) apply safe apply(rule min_max.less_supI1,rule B1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4196 |
proof- fix a b c d::"real^'n" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4197 |
have **:"ball 0 B1 \<subseteq> ball (0::real^'n) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::real^'n) (max B1 B2)" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4198 |
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> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4199 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4200 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4201 |
unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4202 |
apply(rule B1(2),rule order_trans,rule **,rule as(1)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4203 |
apply(rule B1(2),rule order_trans,rule **,rule as(2)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4204 |
apply(rule B2(2),rule order_trans,rule **,rule as(1)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4205 |
apply(rule B2(2),rule order_trans,rule **,rule as(2)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4206 |
apply(rule obt) apply(rule_tac[!] integral_le) using obt |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4207 |
by(auto intro!: h g interv) qed qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4208 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4209 |
subsection {* Adding integrals over several sets. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4210 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4211 |
lemma has_integral_union: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4212 |
assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4213 |
shows "(f has_integral (i + j)) (s \<union> t)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4214 |
proof- note * = has_integral_restrict_univ[THEN sym, of f] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4215 |
show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4216 |
defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4217 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4218 |
lemma has_integral_unions: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4219 |
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')" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4220 |
shows "(f has_integral (setsum i t)) (\<Union>t)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4221 |
proof- note * = has_integral_restrict_univ[THEN sym, of f] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4222 |
have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4223 |
apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4224 |
apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4225 |
note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4226 |
thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4227 |
proof safe case goal1 thus ?case |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4228 |
proof(cases "x\<in>\<Union>t") case True then guess s unfolding Union_iff .. note s=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4229 |
hence *:"\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s" using goal1(3) by blast |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4230 |
show ?thesis unfolding if_P[OF True] apply(rule trans) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4231 |
apply(rule setsum_cong2) apply(subst *, assumption) apply(rule refl) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4232 |
unfolding setsum_delta[OF assms(1)] using s by auto qed auto qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4233 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4234 |
subsection {* In particular adding integrals over a division, maybe not of an interval. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4235 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4236 |
lemma has_integral_combine_division: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4237 |
assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4238 |
shows "(f has_integral (setsum i d)) s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4239 |
proof- note d = division_ofD[OF assms(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4240 |
show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4241 |
apply(rule d assms)+ apply(rule,rule,rule) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4242 |
proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4243 |
guess a c b d apply-by(erule exE)+ note obt=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4244 |
from d(5)[OF goal1] show ?case unfolding obt interior_closed_interval |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4245 |
apply-apply(rule negligible_subset[of "({a..b}-{a<..<b}) \<union> ({c..d}-{c<..<d})"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4246 |
apply(rule negligible_union negligible_frontier_interval)+ by auto qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4247 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4248 |
lemma integral_combine_division_bottomup: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4249 |
assumes "d division_of s" "\<forall>k\<in>d. f integrable_on k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4250 |
shows "integral s f = setsum (\<lambda>i. integral i f) d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4251 |
apply(rule integral_unique) apply(rule has_integral_combine_division[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4252 |
using assms(2) unfolding has_integral_integral . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4253 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4254 |
lemma has_integral_combine_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4255 |
assumes "f integrable_on s" "d division_of k" "k \<subseteq> s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4256 |
shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4257 |
apply(rule has_integral_combine_division[OF assms(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4258 |
apply safe unfolding has_integral_integral[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4259 |
proof- case goal1 from division_ofD(2,4)[OF assms(2) this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4260 |
show ?case apply safe apply(rule integrable_on_subinterval) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4261 |
apply(rule assms) using assms(3) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4262 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4263 |
lemma integral_combine_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4264 |
assumes "f integrable_on s" "d division_of s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4265 |
shows "integral s f = setsum (\<lambda>i. integral i f) d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4266 |
apply(rule integral_unique,rule has_integral_combine_division_topdown) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4267 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4268 |
lemma integrable_combine_division: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4269 |
assumes "d division_of s" "\<forall>i\<in>d. f integrable_on i" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4270 |
shows "f integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4271 |
using assms(2) unfolding integrable_on_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4272 |
by(metis has_integral_combine_division[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4273 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4274 |
lemma integrable_on_subdivision: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4275 |
assumes "d division_of i" "f integrable_on s" "i \<subseteq> s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4276 |
shows "f integrable_on i" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4277 |
apply(rule integrable_combine_division assms)+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4278 |
proof safe case goal1 note division_ofD(2,4)[OF assms(1) this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4279 |
thus ?case apply safe apply(rule integrable_on_subinterval[OF assms(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4280 |
using assms(3) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4281 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4282 |
subsection {* Also tagged divisions. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4283 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4284 |
lemma has_integral_combine_tagged_division: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4285 |
assumes "p tagged_division_of s" "\<forall>(x,k) \<in> p. (f has_integral (i k)) k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4286 |
shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4287 |
proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4288 |
apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4289 |
using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4290 |
thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4291 |
apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4292 |
apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4293 |
apply(rule setsum_cong2) using assms(2) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4294 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4295 |
lemma integral_combine_tagged_division_bottomup: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4296 |
assumes "p tagged_division_of {a..b}" "\<forall>(x,k)\<in>p. f integrable_on k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4297 |
shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4298 |
apply(rule integral_unique) apply(rule has_integral_combine_tagged_division[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4299 |
using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4300 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4301 |
lemma has_integral_combine_tagged_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4302 |
assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4303 |
shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4304 |
apply(rule has_integral_combine_tagged_division[OF assms(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4305 |
proof safe case goal1 note tagged_division_ofD(3-4)[OF assms(2) this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4306 |
thus ?case using integrable_subinterval[OF assms(1)] by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4307 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4308 |
lemma integral_combine_tagged_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4309 |
assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4310 |
shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4311 |
apply(rule integral_unique,rule has_integral_combine_tagged_division_topdown) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4312 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4313 |
subsection {* Henstock's lemma. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4314 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4315 |
lemma henstock_lemma_part1: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4316 |
assumes "f integrable_on {a..b}" "0 < e" "gauge d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4317 |
"(\<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)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4318 |
and p:"p tagged_partial_division_of {a..b}" "d fine p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4319 |
shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e" (is "?x \<le> e") |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4320 |
proof- { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" thus ?thesis by arith } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4321 |
fix k::real assume k:"k>0" note p' = tagged_partial_division_ofD[OF p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4322 |
have "\<Union>snd ` p \<subseteq> {a..b}" using p'(3) by fastsimp |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4323 |
note partial_division_of_tagged_division[OF p(1)] this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4324 |
from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4325 |
def r \<equiv> "q - snd ` p" have "snd ` p \<inter> r = {}" unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4326 |
have r:"finite r" using q' unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4327 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4328 |
have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4329 |
norm(setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4330 |
proof safe case goal1 hence i:"i \<in> q" unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4331 |
from q'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4332 |
have *:"k / (real (card r) + 1) > 0" apply(rule divide_pos_pos,rule k) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4333 |
have "f integrable_on {u..v}" apply(rule integrable_subinterval[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4334 |
using q'(2)[OF i] unfolding uv by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4335 |
note integrable_integral[OF this, unfolded has_integral[of f]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4336 |
from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4337 |
note gauge_inter[OF `gauge d` dd(1)] from fine_division_exists[OF this,of u v] guess qq . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4338 |
thus ?case apply(rule_tac x=qq in exI) using dd(2)[of qq] unfolding fine_inter uv by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4339 |
from bchoice[OF this] guess qq .. note qq=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4340 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4341 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4342 |
apply(rule assms(4)[rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4343 |
proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4344 |
note * = tagged_partial_division_of_union_self[OF p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4345 |
have "p \<union> \<Union>qq ` r tagged_division_of \<Union>snd ` p \<union> \<Union>r" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4346 |
proof(rule tagged_division_union[OF * tagged_division_unions]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4347 |
show "finite r" by fact case goal2 thus ?case using qq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4348 |
next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4349 |
next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4350 |
apply(rule,rule q') defer apply(rule,subst Int_commute) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4351 |
apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4352 |
apply(rule,rule q') using q(1) p' unfolding r_def by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4353 |
moreover have "\<Union>snd ` p \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4354 |
unfolding Union_Un_distrib[THEN sym] r_def using q by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4355 |
ultimately show "?p tagged_division_of {a..b}" by fastsimp qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4356 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4357 |
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) - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4358 |
integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4359 |
apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4360 |
proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4361 |
note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4362 |
from this(2) guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4363 |
have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4364 |
hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4365 |
note q'(5)[OF this] hence "interior l = {}" using subset_interior[OF `l \<subseteq> k`] by blast |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4366 |
thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4367 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4368 |
hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4369 |
(qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4370 |
prefer 4 apply assumption apply(rule finite_imageI,fact) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4371 |
unfolding split_paired_all split_conv image_iff defer apply(erule bexE)+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4372 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4373 |
note as = this(1-5)[unfolded this(6-)] note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4374 |
from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4375 |
have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q') |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4376 |
using as unfolding r_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4377 |
have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4378 |
apply(rule subset_interior) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4379 |
thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4380 |
qed(insert qq, auto) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4381 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4382 |
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 - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4383 |
integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4384 |
apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4385 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4386 |
note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4387 |
show "content m *\<^sub>R f x = 0" using as(3) unfolding as by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4388 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4389 |
have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4390 |
ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4391 |
proof- case goal1 thus ?case using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"] |
36350 | 4392 |
unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4393 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4394 |
have "?x = norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4395 |
unfolding split_def setsum_subtractf .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4396 |
also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4397 |
proof- case goal2 have *:"(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4398 |
apply(subst setsum_reindex_nonzero) apply fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4399 |
unfolding split_paired_all snd_conv split_def o_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4400 |
proof- fix x l y m assume as:"(x,l)\<in>p" "(y,m)\<in>p" "(x,l)\<noteq>(y,m)" "l = m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4401 |
from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4402 |
show "integral l f = 0" unfolding uv apply(rule integral_unique) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4403 |
apply(rule has_integral_null) unfolding content_eq_0_interior |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4404 |
using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4405 |
qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4406 |
show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4407 |
apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4408 |
next case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4409 |
show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4410 |
unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le,fact) |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
4411 |
apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[THEN sym] |
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
4412 |
unfolding divide_inverse[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat) |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4413 |
qed finally show "?x \<le> e + k" . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4414 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4415 |
lemma henstock_lemma_part2: fixes f::"real^'m \<Rightarrow> real^'n" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4416 |
assumes "f integrable_on {a..b}" "0 < e" "gauge d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4417 |
"\<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 - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4418 |
integral({a..b}) f) < e" "p tagged_partial_division_of {a..b}" "d fine p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4419 |
shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (CARD('n)) * e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4420 |
unfolding split_def apply(rule vsum_norm_allsubsets_bound) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4421 |
apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4422 |
apply safe apply(rule assms[rule_format,unfolded split_def]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4423 |
apply(rule tagged_partial_division_subset,rule assms,assumption) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4424 |
apply(rule fine_subset,assumption,rule assms) using assms(5) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4425 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4426 |
lemma henstock_lemma: fixes f::"real^'m \<Rightarrow> real^'n" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4427 |
assumes "f integrable_on {a..b}" "e>0" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4428 |
obtains d where "gauge d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4429 |
"\<forall>p. p tagged_partial_division_of {a..b} \<and> d fine p |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4430 |
\<longrightarrow> setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4431 |
proof- have *:"e / (2 * (real CARD('n) + 1)) > 0" apply(rule divide_pos_pos) using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4432 |
from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4433 |
guess d .. note d = conjunctD2[OF this] show thesis apply(rule that,rule d) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4434 |
proof safe case goal1 note * = henstock_lemma_part2[OF assms(1) * d this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4435 |
show ?case apply(rule le_less_trans[OF *]) using `e>0` by(auto simp add:field_simps) qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4436 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4437 |
subsection {* monotone convergence (bounded interval first). *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4438 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4439 |
lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4440 |
assumes "\<forall>k. (f k) integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4441 |
"\<forall>k. \<forall>x\<in>{a..b}. dest_vec1(f k x) \<le> dest_vec1(f (Suc k) x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4442 |
"\<forall>x\<in>{a..b}. ((\<lambda>k. f k x) ---> g x) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4443 |
"bounded {integral {a..b} (f k) | k . k \<in> UNIV}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4444 |
shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4445 |
proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4446 |
show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using Lim_const by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4447 |
next assume ab:"content {a..b} \<noteq> 0" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4448 |
have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x)$1 \<le> (g x)$1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4449 |
proof safe case goal1 note assms(3)[rule_format,OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4450 |
note * = Lim_component_ge[OF this trivial_limit_sequentially] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4451 |
show ?case apply(rule *) unfolding eventually_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4452 |
apply(rule_tac x=k in exI) apply- apply(rule transitive_stepwise_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4453 |
using assms(2)[rule_format,OF goal1] by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4454 |
have "\<exists>i. ((\<lambda>k. integral ({a..b}) (f k)) ---> i) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4455 |
apply(rule bounded_increasing_convergent) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4456 |
apply rule apply(rule integral_component_le) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4457 |
apply(rule assms(1-2)[rule_format])+ using assms(4) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4458 |
then guess i .. note i=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4459 |
have i':"\<And>k. dest_vec1(integral({a..b}) (f k)) \<le> dest_vec1 i" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4460 |
apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4461 |
unfolding eventually_sequentially apply(rule_tac x=k in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4462 |
apply(rule transitive_stepwise_le) prefer 3 apply(rule integral_dest_vec1_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4463 |
apply(rule assms(1-2)[rule_format])+ using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4464 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4465 |
have "(g has_integral i) {a..b}" unfolding has_integral |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4466 |
proof safe case goal1 note e=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4467 |
hence "\<forall>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4468 |
norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral {a..b} (f k)) < e / 2 ^ (k + 2)))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4469 |
apply-apply(rule,rule assms(1)[unfolded has_integral_integral has_integral,rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4470 |
apply(rule divide_pos_pos) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4471 |
from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4472 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4473 |
have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i$1 - dest_vec1(integral {a..b} (f k)) \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4474 |
i$1 - dest_vec1(integral {a..b} (f k)) < e / 4" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4475 |
proof- case goal1 have "e/4 > 0" using e by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4476 |
from i[unfolded Lim_sequentially,rule_format,OF this] guess r .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4477 |
thus ?case apply(rule_tac x=r in exI) apply rule |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4478 |
apply(erule_tac x=k in allE) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4479 |
proof- case goal1 thus ?case using i'[of k] unfolding dist_real by auto qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4480 |
then guess r .. note r=conjunctD2[OF this[rule_format]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4481 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4482 |
have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)$1 - (f k x)$1 \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4483 |
(g x)$1 - (f k x)$1 < e / (4 * content({a..b}))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4484 |
proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4485 |
using ab content_pos_le[of a b] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4486 |
from assms(3)[rule_format,OF goal1,unfolded Lim_sequentially,rule_format,OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4487 |
guess n .. note n=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4488 |
thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4489 |
unfolding dist_real using fg[rule_format,OF goal1] by(auto simp add:field_simps) qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4490 |
from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4491 |
def d \<equiv> "\<lambda>x. c (m x) x" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4492 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4493 |
show ?case apply(rule_tac x=d in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4494 |
proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4495 |
next fix p assume p:"p tagged_division_of {a..b}" "d fine p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4496 |
note p'=tagged_division_ofD[OF p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4497 |
have "\<exists>a. \<forall>x\<in>p. m (fst x) \<le> a" by(rule upper_bound_finite_set,fact) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4498 |
then guess s .. note s=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4499 |
have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4500 |
norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4501 |
proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4502 |
norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel |
36350 | 4503 |
by(auto simp add:algebra_simps) qed |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4504 |
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4505 |
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))"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4506 |
proof safe case goal1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4507 |
show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4508 |
unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule setsum_norm[OF p'(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4509 |
apply(rule setsum_mono) unfolding split_paired_all split_conv |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4510 |
unfolding split_def setsum_left_distrib[THEN sym] scaleR.diff_right[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4511 |
unfolding additive_content_tagged_division[OF p(1), unfolded split_def] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4512 |
proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4513 |
from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4514 |
show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4515 |
unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4516 |
apply(rule mult_left_mono) unfolding norm_real using m(2)[OF x,of "m x"] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4517 |
qed(insert ab,auto) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4518 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4519 |
next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4520 |
\<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4521 |
apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4522 |
apply(subst split_def)+ unfolding setsum_subtractf apply rule |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4523 |
proof- show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4524 |
m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e / 2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4525 |
apply(rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4526 |
apply(rule setsum_norm_le[OF finite_atLeastAtMost]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4527 |
proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2" |
36778
739a9379e29b
avoid using real-specific versions of generic lemmas
huffman
parents:
36725
diff
changeset
|
4528 |
unfolding power_add divide_inverse inverse_mult_distrib |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4529 |
unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4530 |
unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4531 |
unfolding power2_eq_square by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4532 |
fix t assume "t\<in>{0..s}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4533 |
show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4534 |
integral k (f (m x))) \<le> e / 2 ^ (t + 2)"apply(rule order_trans[of _ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4535 |
"norm(setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4536 |
apply(rule eq_refl) apply(rule arg_cong[where f=norm]) apply(rule setsum_cong2) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4537 |
apply(rule henstock_lemma_part1) apply(rule assms(1)[rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4538 |
apply(rule divide_pos_pos,rule e) defer apply safe apply(rule c)+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4539 |
apply rule apply assumption+ apply(rule tagged_partial_division_subset[of p]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4540 |
apply(rule p(1)[unfolded tagged_division_of_def,THEN conjunct1]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4541 |
unfolding fine_def apply safe apply(drule p(2)[unfolded fine_def,rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4542 |
unfolding d_def by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4543 |
qed(insert s, auto) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4544 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4545 |
next case goal3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4546 |
note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4547 |
have *:"\<And>sr sx ss ks kr::real^1. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i$1 - kr$1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4548 |
\<and> i$1 - kr$1 < e/4 \<longrightarrow> abs(sx$1 - i$1) < e/4" unfolding Cart_eq forall_1 vector_le_def by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4549 |
show ?case unfolding norm_real Cart_nth.diff apply(rule *[rule_format],safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4550 |
apply(rule comb[of r],rule comb[of s]) unfolding vector_le_def forall_1 setsum_component |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4551 |
apply(rule i') apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4552 |
apply(rule_tac[1-2] integral_component_le[OF ]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4553 |
proof safe show "0 \<le> i$1 - (integral {a..b} (f r))$1" using r(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4554 |
show "i$1 - (integral {a..b} (f r))$1 < e / 4" using r(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4555 |
fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4556 |
show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4557 |
unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4558 |
using p'(3)[OF xk] unfolding uv by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4559 |
fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4560 |
hence *:"\<And>m. \<forall>n\<ge>m. (f m y)$1 \<le> (f n y)$1" apply-apply(rule transitive_stepwise_le) using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4561 |
show "(f r y)$1 \<le> (f (m x) y)$1" "(f (m x) y)$1 \<le> (f s y)$1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4562 |
apply(rule_tac[!] *[rule_format]) using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4563 |
qed qed qed qed note * = this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4564 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4565 |
have "integral {a..b} g = i" apply(rule integral_unique) using * . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4566 |
thus ?thesis using i * by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4567 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4568 |
lemma monotone_convergence_increasing: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4569 |
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f k x)$1 \<le> (f (Suc k) x)$1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4570 |
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4571 |
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4572 |
proof- have lem:"\<And>f::nat \<Rightarrow> real^'n \<Rightarrow> real^1. \<And> g s. \<forall>k.\<forall>x\<in>s. 0\<le>dest_vec1 (f k x) \<Longrightarrow> \<forall>k. (f k) integrable_on s \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4573 |
\<forall>k. \<forall>x\<in>s. (f k x)$1 \<le> (f (Suc k) x)$1 \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4574 |
bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4575 |
proof- case goal1 note assms=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4576 |
have "\<forall>x\<in>s. \<forall>k. dest_vec1(f k x) \<le> dest_vec1(g x)" apply safe apply(rule Lim_component_ge) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4577 |
apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4578 |
unfolding eventually_sequentially apply(rule_tac x=k in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4579 |
apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4580 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4581 |
have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially" apply(rule bounded_increasing_convergent) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4582 |
apply(rule goal1(5)) apply(rule,rule integral_component_le) apply(rule goal1(2)[rule_format])+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4583 |
using goal1(3) by auto then guess i .. note i=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4584 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4585 |
hence i':"\<forall>k. (integral s (f k))$1 \<le> i$1" apply-apply(rule,rule Lim_component_ge) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4586 |
apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4587 |
apply(rule_tac x=k in exI,safe) apply(rule integral_dest_vec1_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4588 |
apply(rule goal1(2)[rule_format])+ by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4589 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4590 |
note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4591 |
have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) = |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4592 |
(\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4593 |
have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4594 |
apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4595 |
have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4596 |
((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) ---> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4597 |
integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4598 |
proof(rule monotone_convergence_interval,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4599 |
case goal1 show ?case using int . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4600 |
next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4601 |
next case goal3 thus ?case apply-apply(cases "x\<in>s") using assms(4) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4602 |
next case goal4 note * = integral_dest_vec1_nonneg[unfolded vector_le_def forall_1 zero_index] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4603 |
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))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4604 |
unfolding norm_real apply(subst abs_of_nonneg) apply(rule *[OF int]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4605 |
apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4606 |
apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4607 |
apply(subst integral_restrict_univ[THEN sym,OF int]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4608 |
unfolding ifif unfolding integral_restrict_univ[OF int'] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4609 |
apply(rule integral_subset_component_le[OF _ int' assms(2)]) using assms(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4610 |
thus ?case using assms(5) unfolding bounded_iff apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4611 |
apply(rule_tac x=aa in exI,safe) apply(erule_tac x="integral s (f k)" in ballE) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4612 |
apply(rule order_trans) apply assumption by auto qed note g = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4613 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4614 |
have "(g has_integral i) s" unfolding has_integral_alt' apply safe apply(rule g(1)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4615 |
proof- case goal1 hence "e/4>0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4616 |
from i[unfolded Lim_sequentially,rule_format,OF this] guess N .. note N=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4617 |
note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4618 |
from this[THEN conjunct2,rule_format,OF `e/4>0`] guess B .. note B=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4619 |
show ?case apply(rule,rule,rule B,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4620 |
proof- fix a b::"real^'n" assume ab:"ball 0 B \<subseteq> {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4621 |
from `e>0` have "e/2>0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4622 |
from g(2)[unfolded Lim_sequentially,of a b,rule_format,OF this] guess M .. note M=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4623 |
have **:"norm (integral {a..b} (\<lambda>x. if x \<in> s then f N x else 0) - i) < e/2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4624 |
apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N] |
36587 | 4625 |
unfolding dist_norm apply-defer apply(subst norm_minus_commute) by auto |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4626 |
have *:"\<And>f1 f2 g. abs(f1 - i$1) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i$1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4627 |
\<longrightarrow> abs(g - i$1) < e" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4628 |
show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4629 |
unfolding norm_real Cart_simps apply(rule *[rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4630 |
apply(rule **[unfolded norm_real Cart_simps]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4631 |
apply(rule M[rule_format,of "M + N",unfolded dist_real]) apply(rule le_add1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4632 |
apply(rule integral_component_le[OF int int]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4633 |
apply(rule order_trans[OF _ i'[rule_format,of "M + N"]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4634 |
proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)$1 \<le> (f n x)$1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4635 |
apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4636 |
next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4637 |
unfolding ifif integral_restrict_univ[OF int'] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4638 |
apply(rule integral_subset_component_le[OF _ int']) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4639 |
qed qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4640 |
thus ?case apply safe defer apply(drule integral_unique) using i by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4641 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4642 |
have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4643 |
apply(subst integral_sub) apply(rule assms(1)[rule_format])+ by rule |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4644 |
have "\<And>x m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. dest_vec1 (f m x) \<le> dest_vec1 (f n x)" apply(rule transitive_stepwise_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4645 |
using assms(2) by auto note * = this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4646 |
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)) ---> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4647 |
integral s (\<lambda>x. g x - f 0 x)) sequentially" apply(rule lem,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4648 |
proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4649 |
next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4650 |
next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4651 |
next case goal4 thus ?case apply-apply(rule Lim_sub) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4652 |
using seq_offset[OF assms(3)[rule_format],of x 1] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4653 |
next case goal5 thus ?case using assms(4) unfolding bounded_iff |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4654 |
apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4655 |
apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4656 |
apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4657 |
note conjunctD2[OF this] note Lim_add[OF this(2) Lim_const[of "integral s (f 0)"]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4658 |
integrable_add[OF this(1) assms(1)[rule_format,of 0]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4659 |
thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4660 |
using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4661 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4662 |
lemma monotone_convergence_decreasing: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4663 |
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f (Suc k) x)$1 \<le> (f k x)$1" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4664 |
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4665 |
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4666 |
proof- note assm = assms[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4667 |
have *:"{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R -1 ` {integral s (f k)| k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4668 |
apply safe unfolding image_iff apply(rule_tac x="integral s (f k)" in bexI) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4669 |
apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4670 |
have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4671 |
---> integral s (\<lambda>x. - g x)) sequentially" apply(rule monotone_convergence_increasing) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4672 |
apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule Lim_neg) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4673 |
apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4674 |
note * = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4675 |
show ?thesis apply rule using integrable_neg[OF *(1)] defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4676 |
using Lim_neg[OF *(2)] apply- unfolding integral_neg[OF assm(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4677 |
unfolding integral_neg[OF *(1),THEN sym] by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4678 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4679 |
lemma monotone_convergence_increasing_real: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4680 |
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<ge> (f k x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4681 |
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4682 |
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4683 |
proof- have *:"{integral s (\<lambda>x. vec1 (f k x)) |k. True} = vec1 ` {integral s (f k) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4684 |
unfolding integral_trans[OF assms(1)[rule_format]] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4685 |
have "vec1 \<circ> g integrable_on s \<and> ((\<lambda>k. integral s (vec1 \<circ> f k)) ---> integral s (vec1 \<circ> g)) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4686 |
apply(rule monotone_convergence_increasing) unfolding o_def integrable_on_trans |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4687 |
unfolding vec1_dest_vec1 apply(rule assms)+ unfolding Lim_trans unfolding * using assms(3,4) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4688 |
thus ?thesis unfolding o_def unfolding integral_trans[OF assms(1)[rule_format]] by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4689 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4690 |
lemma monotone_convergence_decreasing_real: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4691 |
assumes "\<forall>k. (f k) integrable_on s" "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<le> (f k x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4692 |
"\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4693 |
shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4694 |
proof- have *:"{integral s (\<lambda>x. vec1 (f k x)) |k. True} = vec1 ` {integral s (f k) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4695 |
unfolding integral_trans[OF assms(1)[rule_format]] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4696 |
have "vec1 \<circ> g integrable_on s \<and> ((\<lambda>k. integral s (vec1 \<circ> f k)) ---> integral s (vec1 \<circ> g)) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4697 |
apply(rule monotone_convergence_decreasing) unfolding o_def integrable_on_trans |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4698 |
unfolding vec1_dest_vec1 apply(rule assms)+ unfolding Lim_trans unfolding * using assms(3,4) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4699 |
thus ?thesis unfolding o_def unfolding integral_trans[OF assms(1)[rule_format]] by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4700 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4701 |
subsection {* absolute integrability (this is the same as Lebesgue integrability). *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4702 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4703 |
definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46) where |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4704 |
"f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. (norm(f x))) integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4705 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4706 |
lemma absolutely_integrable_onI[intro?]: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4707 |
"f integrable_on s \<Longrightarrow> (\<lambda>x. (norm(f x))) integrable_on s \<Longrightarrow> f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4708 |
unfolding absolutely_integrable_on_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4709 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4710 |
lemma absolutely_integrable_onD[dest]: assumes "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4711 |
shows "f integrable_on s" "(\<lambda>x. (norm(f x))) integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4712 |
using assms unfolding absolutely_integrable_on_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4713 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4714 |
lemma absolutely_integrable_on_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4715 |
"(vec1 o f) absolutely_integrable_on s \<longleftrightarrow> f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4716 |
unfolding absolutely_integrable_on_def o_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4717 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4718 |
lemma integral_norm_bound_integral: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4719 |
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> g x" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4720 |
shows "norm(integral s f) \<le> (integral s g)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4721 |
proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4722 |
apply(erule_tac x="x - y" in allE) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4723 |
have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4724 |
\<longrightarrow> norm(ig) < dia + e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4725 |
proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]]) |
36725 | 4726 |
apply(subst real_sum_of_halves[of e,THEN sym]) unfolding normalizing.add_a |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4727 |
apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4728 |
apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4729 |
qed note norm=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4730 |
have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> g a b. f integrable_on {a..b} \<Longrightarrow> g integrable_on {a..b} \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4731 |
\<forall>x\<in>{a..b}. norm(f x) \<le> (g x) \<Longrightarrow> norm(integral({a..b}) f) \<le> (integral({a..b}) g)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4732 |
proof(rule *[rule_format]) case goal1 hence *:"e/2>0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4733 |
from integrable_integral[OF goal1(1),unfolded has_integral[of f],rule_format,OF *] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4734 |
guess d1 .. note d1 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4735 |
from integrable_integral[OF goal1(2),unfolded has_integral[of g],rule_format,OF *] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4736 |
guess d2 .. note d2 = conjunctD2[OF this,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4737 |
note gauge_inter[OF d1(1) d2(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4738 |
from fine_division_exists[OF this, of a b] guess p . note p=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4739 |
show ?case apply(rule norm) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4740 |
apply(rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4741 |
apply(rule d1(2)[OF conjI[OF p(1)]]) defer apply(rule setsum_norm_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4742 |
proof safe fix x k assume "(x,k)\<in>p" note as = tagged_division_ofD(2-4)[OF p(1) this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4743 |
from this(3) guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4744 |
show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x" unfolding uv norm_scaleR |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4745 |
unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4746 |
apply(rule mult_left_mono) using goal1(3) as by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4747 |
qed(insert p[unfolded fine_inter],auto) qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4748 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4749 |
{ presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4750 |
thus ?thesis apply-apply(rule *[rule_format]) by auto } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4751 |
fix e::real assume "e>0" hence e:"e/2 > 0" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4752 |
note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4753 |
note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4754 |
from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4755 |
guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4756 |
from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4757 |
guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4758 |
from bounded_subset_closed_interval[OF bounded_ball, of "0::real^'n" "max B1 B2"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4759 |
guess a b apply-by(erule exE)+ note ab=this[unfolded ball_max_Un] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4760 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4761 |
have "ball 0 B1 \<subseteq> {a..b}" using ab by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4762 |
from B1(2)[OF this] guess z .. note z=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4763 |
have "ball 0 B2 \<subseteq> {a..b}" using ab by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4764 |
from B2(2)[OF this] guess w .. note w=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4765 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4766 |
show "norm (integral s f) < integral s g + e" apply(rule norm) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4767 |
apply(rule lem[OF f g, of a b]) unfolding integral_unique[OF z(1)] integral_unique[OF w(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4768 |
defer apply(rule w(2)[unfolded real_norm_def],rule z(2)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4769 |
apply safe apply(case_tac "x\<in>s") unfolding if_P apply(rule assms(3)[rule_format]) by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4770 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4771 |
lemma integral_norm_bound_integral_component: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4772 |
assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4773 |
shows "norm(integral s f) \<le> (integral s g)$k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4774 |
proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x $ k) o g)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4775 |
apply(rule integral_norm_bound_integral[OF assms(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4776 |
apply(rule integrable_linear[OF assms(2)],rule) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4777 |
unfolding o_def by(rule assms) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4778 |
thus ?thesis unfolding o_def integral_component_eq[OF assms(2)] . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4779 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4780 |
lemma has_integral_norm_bound_integral_component: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4781 |
assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4782 |
shows "norm(i) \<le> j$k" using integral_norm_bound_integral_component[of f s g k] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4783 |
unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4784 |
using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4785 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4786 |
lemma absolutely_integrable_le: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4787 |
assumes "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4788 |
shows "norm(integral s f) \<le> integral s (\<lambda>x. norm(f x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4789 |
apply(rule integral_norm_bound_integral) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4790 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4791 |
lemma absolutely_integrable_0[intro]: "(\<lambda>x. 0) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4792 |
unfolding absolutely_integrable_on_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4793 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4794 |
lemma absolutely_integrable_cmul[intro]: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4795 |
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4796 |
unfolding absolutely_integrable_on_def using integrable_cmul[of f s c] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4797 |
using integrable_cmul[of "\<lambda>x. norm (f x)" s "abs c"] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4798 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4799 |
lemma absolutely_integrable_neg[intro]: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4800 |
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4801 |
apply(drule absolutely_integrable_cmul[where c="-1"]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4802 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4803 |
lemma absolutely_integrable_norm[intro]: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4804 |
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. norm(f x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4805 |
unfolding absolutely_integrable_on_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4806 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4807 |
lemma absolutely_integrable_abs[intro]: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4808 |
"f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. abs(f x::real)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4809 |
apply(drule absolutely_integrable_norm) unfolding real_norm_def . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4810 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4811 |
lemma absolutely_integrable_on_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4812 |
"f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4813 |
unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4814 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4815 |
lemma absolutely_integrable_bounded_variation: fixes f::"real^'n \<Rightarrow> 'a::banach" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4816 |
assumes "f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4817 |
obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4818 |
apply(rule that[of "integral UNIV (\<lambda>x. norm (f x))"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4819 |
proof safe case goal1 note d = division_ofD[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4820 |
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. integral i (\<lambda>x. norm (f x)))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4821 |
apply(rule setsum_mono,rule absolutely_integrable_le) apply(drule d(4),safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4822 |
apply(rule absolutely_integrable_on_subinterval[OF assms]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4823 |
also have "... \<le> integral (\<Union>d) (\<lambda>x. norm (f x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4824 |
apply(subst integral_combine_division_topdown[OF _ goal1(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4825 |
using integrable_on_subdivision[OF goal1(2)] using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4826 |
also have "... \<le> integral UNIV (\<lambda>x. norm (f x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4827 |
apply(rule integral_subset_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4828 |
using integrable_on_subdivision[OF goal1(2)] using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4829 |
finally show ?case . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4830 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4831 |
lemma helplemma: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4832 |
assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4833 |
shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4834 |
unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4835 |
apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4836 |
using norm_triangle_ineq3 . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4837 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4838 |
lemma bounded_variation_absolutely_integrable_interval: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4839 |
fixes f::"real^'n \<Rightarrow> real^'m" assumes "f integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4840 |
"\<forall>d. d division_of {a..b} \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4841 |
shows "f absolutely_integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4842 |
proof- let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of {a..b} }" def i \<equiv> "Sup ?S" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4843 |
have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4844 |
apply(rule elementary_interval) defer apply(rule_tac x=B in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4845 |
apply(rule setleI) using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4846 |
show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4847 |
proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) ` |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4848 |
{d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4849 |
unfolding setge_def ubs_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4850 |
hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4851 |
unfolding mem_Collect_eq isUb_def setle_def by simp then guess d .. note d=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4852 |
note d' = division_ofD[OF this(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4853 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4854 |
have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4855 |
proof case goal1 have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4856 |
apply(rule separate_point_closed) apply(rule closed_Union) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4857 |
apply(rule finite_subset[OF _ d'(1)]) apply safe apply(drule d'(4)) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4858 |
thus ?case apply safe apply(rule_tac x=da in exI,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4859 |
apply(erule_tac x=xa in ballE) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4860 |
qed from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4861 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4862 |
have "e/2 > 0" using goal1 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4863 |
from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4864 |
let ?g = "\<lambda>x. g x \<inter> ball x (k x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4865 |
show ?case apply(rule_tac x="?g" in exI) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4866 |
proof- show "gauge ?g" using g(1) unfolding gauge_def using k(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4867 |
fix p assume "p tagged_division_of {a..b}" "?g fine p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4868 |
note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4869 |
note p' = tagged_division_ofD[OF p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4870 |
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}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4871 |
have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4872 |
have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4873 |
proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4874 |
` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4875 |
defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4876 |
apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4877 |
fix x k assume "(x,k)\<in>p'" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4878 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4879 |
then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4880 |
show "x\<in>k" "k\<subseteq>{a..b}" using p'(2-3)[OF il(3)] il by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4881 |
show "\<exists>a b. k = {a..b}" unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4882 |
apply safe unfolding inter_interval by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4883 |
next fix x1 k1 assume "(x1,k1)\<in>p'" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4884 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4885 |
then guess i1 l1 apply-by(erule exE)+ note il1=conjunctD4[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4886 |
fix x2 k2 assume "(x2,k2)\<in>p'" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4887 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4888 |
then guess i2 l2 apply-by(erule exE)+ note il2=conjunctD4[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4889 |
assume "(x1, k1) \<noteq> (x2, k2)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4890 |
hence "interior(i1) \<inter> interior(i2) = {} \<or> interior(l1) \<inter> interior(l2) = {}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4891 |
using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] unfolding il1 il2 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4892 |
thus "interior k1 \<inter> interior k2 = {}" unfolding il1 il2 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4893 |
next have *:"\<forall>(x, X) \<in> p'. X \<subseteq> {a..b}" unfolding p'_def using d' by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4894 |
show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4895 |
unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4896 |
proof- fix y assume y:"y\<in>{a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4897 |
hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4898 |
then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4899 |
hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4900 |
then guess i .. note i = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4901 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4902 |
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) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4903 |
defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4904 |
apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4905 |
qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4906 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4907 |
hence "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4908 |
apply-apply(rule g(2)[rule_format]) unfolding tagged_division_of_def apply safe using gp' . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4909 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4910 |
unfolding split_def apply(rule helplemma) using p'' by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4911 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4912 |
have p'alt:"p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> ~(i \<inter> l = {})}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4913 |
proof safe case goal2 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4914 |
have "x\<in>i" using fineD[OF p(3) goal2(1)] k(2)[OF goal2(2), of x] goal2(4-) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4915 |
hence "(x, i \<inter> l) \<in> p'" unfolding p'_def apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4916 |
apply(rule_tac x=x in exI,rule_tac x="i\<inter>l" in exI) apply safe using goal2 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4917 |
thus ?case using goal2(3) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4918 |
next fix x k assume "(x,k)\<in>p'" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4919 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4920 |
then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4921 |
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> {}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4922 |
apply(rule_tac x=x in exI,rule_tac x=i in exI,rule_tac x=l in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4923 |
using p'(2)[OF il(3)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4924 |
qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4925 |
have sum_p':"(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4926 |
apply(subst setsum_over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4927 |
unfolding norm_eq_zero apply(rule integral_null,assumption) .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4928 |
note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4929 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4930 |
have *:"\<And>sni sni' sf sf'. abs(sf' - sni') < e / 2 \<longrightarrow> i - e / 2 < sni \<and> sni' \<le> i \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4931 |
sni \<le> sni' \<and> sf' = sf \<longrightarrow> abs(sf - i) < e" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4932 |
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4933 |
unfolding real_norm_def apply(rule *[rule_format,OF **],safe) apply(rule d(2)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4934 |
proof- case goal1 show ?case unfolding sum_p' |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4935 |
apply(rule isLubD2[OF i]) using division_of_tagged_division[OF p''] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4936 |
next case goal2 have *:"{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4937 |
(\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4938 |
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))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4939 |
proof(rule setsum_mono) case goal1 note k=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4940 |
from d'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4941 |
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]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4942 |
have "d' division_of {u..v}" apply(subst d'_def) apply(rule division_inter_1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4943 |
apply(rule division_of_tagged_division[OF p(1)]) using uvab . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4944 |
hence "norm (integral k f) \<le> setsum (\<lambda>k. norm (integral k f)) d'" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4945 |
unfolding uv apply(subst integral_combine_division_topdown[of _ _ d']) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4946 |
apply(rule integrable_on_subinterval[OF assms(1) uvab]) apply assumption |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4947 |
apply(rule setsum_norm_le) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4948 |
also have "... = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4949 |
apply(rule setsum_mono_zero_left) apply(subst simple_image) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4950 |
apply(rule finite_imageI)+ apply fact unfolding d'_def uv apply blast |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4951 |
proof case goal1 hence "i \<in> {{u..v} \<inter> l |l. l \<in> snd ` p}" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4952 |
from this[unfolded mem_Collect_eq] guess l .. note l=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4953 |
hence "{u..v} \<inter> l = {}" using goal1 by auto thus ?case using l by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4954 |
qed also have "... = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))" unfolding simple_image |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4955 |
apply(rule setsum_reindex_nonzero[unfolded o_def])apply(rule finite_imageI,rule p') |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4956 |
proof- case goal1 have "interior (k \<inter> l) \<subseteq> interior (l \<inter> y)" apply(subst(2) interior_inter) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4957 |
apply(rule Int_greatest) defer apply(subst goal1(4)) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4958 |
hence *:"interior (k \<inter> l) = {}" using snd_p(5)[OF goal1(1-3)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4959 |
from d'(4)[OF k] snd_p(4)[OF goal1(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4960 |
show ?case using * unfolding uv inter_interval content_eq_0_interior[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4961 |
qed finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4962 |
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))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4963 |
apply(subst sum_sum_product[THEN sym],fact) using p'(1) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4964 |
also have "... = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (split op \<inter> x) f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4965 |
unfolding split_def .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4966 |
also have "... = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4967 |
unfolding * apply(rule setsum_reindex_nonzero[THEN sym,unfolded o_def]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4968 |
apply(rule finite_product_dependent) apply(fact,rule finite_imageI,rule p') |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4969 |
unfolding split_paired_all mem_Collect_eq split_conv o_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4970 |
proof- note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4971 |
fix l1 l2 k1 k2 assume as:"(l1, k1) \<noteq> (l2, k2)" "l1 \<inter> k1 = l2 \<inter> k2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4972 |
"\<exists>i l. (l1, k1) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4973 |
"\<exists>i l. (l2, k2) = (i, l) \<and> i \<in> d \<and> l \<in> snd ` p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4974 |
hence "l1 \<in> d" "k1 \<in> snd ` p" by auto from d'(4)[OF this(1)] *(1)[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4975 |
guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4976 |
have "l1 \<noteq> l2 \<or> k1 \<noteq> k2" using as by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4977 |
hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4978 |
apply(erule disjE) apply(rule disjI2) apply(rule d'(5)) prefer 4 apply(rule disjI1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4979 |
apply(rule *) using as by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4980 |
moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" using as(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4981 |
ultimately have "interior(l1 \<inter> k1) = {}" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4982 |
thus "norm (integral (l1 \<inter> k1) f) = 0" unfolding uv inter_interval |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4983 |
unfolding content_eq_0_interior[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4984 |
qed also have "... = (\<Sum>(x, k)\<in>p'. norm (integral k f))" unfolding sum_p' |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4985 |
apply(rule setsum_mono_zero_right) apply(subst *) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4986 |
apply(rule finite_imageI[OF finite_product_dependent]) apply fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4987 |
apply(rule finite_imageI[OF p'(1)]) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4988 |
proof- case goal2 have "ia \<inter> b = {}" using goal2 unfolding p'alt image_iff Bex_def not_ex |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4989 |
apply(erule_tac x="(a,ia\<inter>b)" in allE) by auto thus ?case by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4990 |
next case goal1 thus ?case unfolding p'_def apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4991 |
apply(rule_tac x=i in exI,rule_tac x=l in exI) unfolding snd_conv image_iff |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4992 |
apply safe apply(rule_tac x="(a,l)" in bexI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4993 |
qed finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4994 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4995 |
next case goal3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4996 |
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4997 |
have Sigma_alt:"\<And>s t. s \<times> t = {(i, j) |i j. i \<in> s \<and> j \<in> t}" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4998 |
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}"**) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
4999 |
apply safe unfolding image_iff apply(rule_tac x="((x,l),i)" in bexI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5000 |
note pdfin = finite_cartesian_product[OF p'(1) d'(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5001 |
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))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5002 |
unfolding norm_scaleR apply(rule setsum_mono_zero_left) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5003 |
apply(subst *, rule finite_imageI) apply fact unfolding p'alt apply blast |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5004 |
apply safe apply(rule_tac x=x in exI,rule_tac x=i in exI,rule_tac x=l in exI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5005 |
also have "... = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))" unfolding * |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5006 |
apply(subst setsum_reindex_nonzero,fact) unfolding split_paired_all |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5007 |
unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff Pair_eq apply(erule_tac conjE)+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5008 |
proof- fix x1 l1 k1 x2 l2 k2 assume as:"(x1,l1)\<in>p" "(x2,l2)\<in>p" "k1\<in>d" "k2\<in>d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5009 |
"x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> k1 = k2)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5010 |
from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5011 |
from as have "l1 \<noteq> l2 \<or> k1 \<noteq> k2" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5012 |
hence "(interior(k1) \<inter> interior(k2) = {} \<or> interior(l1) \<inter> interior(l2) = {})" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5013 |
apply-apply(erule disjE) apply(rule disjI2) defer apply(rule disjI1) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5014 |
apply(rule d'(5)[OF as(3-4)],assumption) apply(rule p'(5)[OF as(1-2)]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5015 |
moreover have "interior(l1 \<inter> k1) = interior(l2 \<inter> k2)" unfolding as .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5016 |
ultimately have "interior (l1 \<inter> k1) = {}" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5017 |
thus "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0" unfolding uv inter_interval |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5018 |
unfolding content_eq_0_interior[THEN sym] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5019 |
qed safe also have "... = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))" unfolding Sigma_alt |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5020 |
apply(subst sum_sum_product[THEN sym]) apply(rule p', rule,rule d') |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5021 |
apply(rule setsum_cong2) unfolding split_paired_all split_conv |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5022 |
proof- fix x l assume as:"(x,l)\<in>p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5023 |
note xl = p'(2-4)[OF this] from this(3) guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5024 |
have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> {u..v}))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5025 |
apply(rule setsum_cong2) apply(drule d'(4),safe) apply(subst Int_commute) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5026 |
unfolding inter_interval uv apply(subst abs_of_nonneg) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5027 |
also have "... = setsum content {k\<inter>{u..v}| k. k\<in>d}" unfolding simple_image |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5028 |
apply(rule setsum_reindex_nonzero[unfolded o_def,THEN sym]) apply(rule d') |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5029 |
proof- case goal1 from d'(4)[OF this(1)] d'(4)[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5030 |
guess u1 v1 u2 v2 apply- by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5031 |
have "{} = interior ((k \<inter> y) \<inter> {u..v})" apply(subst interior_inter) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5032 |
using d'(5)[OF goal1(1-3)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5033 |
also have "... = interior (y \<inter> (k \<inter> {u..v}))" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5034 |
also have "... = interior (k \<inter> {u..v})" unfolding goal1(4) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5035 |
finally show ?case unfolding uv inter_interval content_eq_0_interior .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5036 |
qed also have "... = setsum content {{u..v} \<inter> k |k. k \<in> d \<and> ~({u..v} \<inter> k = {})}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5037 |
apply(rule setsum_mono_zero_right) unfolding simple_image |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5038 |
apply(rule finite_imageI,rule d') apply blast apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5039 |
apply(rule_tac x=k in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5040 |
proof- case goal1 from d'(4)[OF this(1)] guess a b apply-by(erule exE)+ note ab=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5041 |
have "interior (k \<inter> {u..v}) \<noteq> {}" using goal1(2) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5042 |
unfolding ab inter_interval content_eq_0_interior by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5043 |
thus ?case using goal1(1) using interior_subset[of "k \<inter> {u..v}"] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5044 |
qed finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5045 |
unfolding setsum_left_distrib[THEN sym] real_scaleR_def apply - |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5046 |
apply(subst(asm) additive_content_division[OF division_inter_1[OF d(1)]]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5047 |
using xl(2)[unfolded uv] unfolding uv by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5048 |
qed finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5049 |
qed qed qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5050 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5051 |
lemma bounded_variation_absolutely_integrable: fixes f::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5052 |
assumes "f integrable_on UNIV" "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5053 |
shows "f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5054 |
proof(rule absolutely_integrable_onI,fact,rule) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5055 |
let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of (\<Union>d)}" def i \<equiv> "Sup ?S" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5056 |
have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5057 |
apply(rule elementary_interval) defer apply(rule_tac x=B in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5058 |
apply(rule setleI) using assms(2) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5059 |
have f_int:"\<And>a b. f absolutely_integrable_on {a..b}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5060 |
apply(rule bounded_variation_absolutely_integrable_interval[where B=B]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5061 |
apply(rule integrable_on_subinterval[OF assms(1)]) defer apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5062 |
apply(rule assms(2)[rule_format]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5063 |
show "((\<lambda>x. norm (f x)) has_integral i) UNIV" apply(subst has_integral_alt',safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5064 |
proof- case goal1 show ?case using f_int[of a b] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5065 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5066 |
proof(rule ccontr) case goal1 hence "i \<le> i - e" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5067 |
apply(rule isLub_le_isUb[OF i]) apply(rule isUbI) unfolding setle_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5068 |
thus False using goal2 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5069 |
qed then guess K .. note * = this[unfolded image_iff not_le] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5070 |
from this(1) guess d .. note this[unfolded mem_Collect_eq] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5071 |
note d = this(1) *(2)[unfolded this(2)] note d'=division_ofD[OF this(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5072 |
have "bounded (\<Union>d)" by(rule elementary_bounded,fact) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5073 |
from this[unfolded bounded_pos] guess K .. note K=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5074 |
show ?case apply(rule_tac x="K + 1" in exI,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5075 |
proof- fix a b assume ab:"ball 0 (K + 1) \<subseteq> {a..b::real^'n}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5076 |
have *:"\<forall>s s1. i - e < s1 \<and> s1 \<le> s \<and> s < i + e \<longrightarrow> abs(s - i) < (e::real)" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5077 |
show "norm (integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) - i) < e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5078 |
unfolding real_norm_def apply(rule *[rule_format],safe) apply(rule d(2)) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5079 |
proof- case goal1 have "(\<Sum>k\<in>d. norm (integral k f)) \<le> setsum (\<lambda>k. integral k (\<lambda>x. norm (f x))) d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5080 |
apply(rule setsum_mono) apply(rule absolutely_integrable_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5081 |
apply(drule d'(4),safe) by(rule f_int) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5082 |
also have "... = integral (\<Union>d) (\<lambda>x. norm(f x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5083 |
apply(rule integral_combine_division_bottomup[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5084 |
apply(rule d) unfolding forall_in_division[OF d(1)] using f_int by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5085 |
also have "... \<le> integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5086 |
proof- case goal1 have "\<Union>d \<subseteq> {a..b}" apply rule apply(drule K(2)[rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5087 |
apply(rule ab[unfolded subset_eq,rule_format]) by(auto simp add:dist_norm) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5088 |
thus ?case apply- apply(subst if_P,rule) apply(rule integral_subset_le) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5089 |
apply(rule integrable_on_subdivision[of _ _ _ "{a..b}"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5090 |
apply(rule d) using f_int[of a b] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5091 |
qed finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5092 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5093 |
next note f = absolutely_integrable_onD[OF f_int[of a b]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5094 |
note * = this(2)[unfolded has_integral_integral has_integral[of "\<lambda>x. norm (f x)"],rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5095 |
have "e/2>0" using `e>0` by auto from *[OF this] guess d1 .. note d1=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5096 |
from henstock_lemma[OF f(1) `e/2>0`] guess d2 . note d2=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5097 |
from fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] guess p . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5098 |
note p=this(1) conjunctD2[OF this(2)[unfolded fine_inter]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5099 |
have *:"\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> i \<longrightarrow> abs(sf - si) < e / 2 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5100 |
\<longrightarrow> abs(sf' - di) < e / 2 \<longrightarrow> di < i + e" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5101 |
show "integral {a..b} (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < i + e" apply(subst if_P,rule) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5102 |
proof(rule *[rule_format]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5103 |
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" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5104 |
unfolding split_def apply(rule helplemma) using d2(2)[rule_format,of p] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5105 |
using p(1,3) unfolding tagged_division_of_def split_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5106 |
show "abs ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral {a..b} (\<lambda>x. norm(f x))) < e / 2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5107 |
using d1(2)[rule_format,OF conjI[OF p(1,2)]] unfolding real_norm_def . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5108 |
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))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5109 |
apply(rule setsum_cong2) unfolding split_paired_all split_conv |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5110 |
apply(drule tagged_division_ofD(4)[OF p(1)]) unfolding norm_scaleR |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5111 |
apply(subst abs_of_nonneg) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5112 |
show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> i" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5113 |
apply(subst setsum_over_tagged_division_lemma[OF p(1)]) defer apply(rule isLubD2[OF i]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5114 |
unfolding image_iff apply(rule_tac x="snd ` p" in bexI) unfolding mem_Collect_eq defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5115 |
apply(rule partial_division_of_tagged_division[of _ "{a..b}"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5116 |
using p(1) unfolding tagged_division_of_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5117 |
qed qed qed(insert K,auto) qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5118 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5119 |
lemma absolutely_integrable_restrict_univ: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5120 |
"(\<lambda>x. if x \<in> s then f x else (0::'a::banach)) absolutely_integrable_on UNIV \<longleftrightarrow> f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5121 |
unfolding absolutely_integrable_on_def if_distrib norm_zero integrable_restrict_univ .. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5122 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5123 |
lemma absolutely_integrable_add[intro]: fixes f g::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5124 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5125 |
shows "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5126 |
proof- let ?P = "\<And>f g::real^'n \<Rightarrow> real^'m. f absolutely_integrable_on UNIV \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5127 |
g absolutely_integrable_on UNIV \<Longrightarrow> (\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5128 |
{ presume as:"PROP ?P" note a = absolutely_integrable_restrict_univ[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5129 |
have *:"\<And>x. (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5130 |
= (if x \<in> s then f x + g x else 0)" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5131 |
show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]] unfolding * . } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5132 |
fix f g::"real^'n \<Rightarrow> real^'m" assume assms:"f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5133 |
"g absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5134 |
note absolutely_integrable_bounded_variation |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5135 |
from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5136 |
show "(\<lambda>x. f(x) + g(x)) absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5137 |
apply(rule bounded_variation_absolutely_integrable[of _ "B1+B2"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5138 |
apply(rule integrable_add) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5139 |
proof safe case goal1 have "\<And>k. k \<in> d \<Longrightarrow> f integrable_on k \<and> g integrable_on k" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5140 |
apply(drule division_ofD(4)[OF goal1]) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5141 |
apply(rule_tac[!] integrable_on_subinterval[of _ UNIV]) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5142 |
hence "(\<Sum>k\<in>d. norm (integral k (\<lambda>x. f x + g x))) \<le> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5143 |
(\<Sum>k\<in>d. norm (integral k f)) + (\<Sum>k\<in>d. norm (integral k g))" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5144 |
unfolding setsum_addf[THEN sym] apply(rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5145 |
apply(subst integral_add) prefer 3 apply(rule norm_triangle_ineq) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5146 |
also have "... \<le> B1 + B2" using B(1)[OF goal1] B(2)[OF goal1] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5147 |
finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5148 |
qed(insert assms,auto) qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5149 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5150 |
lemma absolutely_integrable_sub[intro]: fixes f g::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5151 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5152 |
shows "(\<lambda>x. f(x) - g(x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5153 |
using absolutely_integrable_add[OF assms(1) absolutely_integrable_neg[OF assms(2)]] |
36350 | 5154 |
unfolding algebra_simps . |
36243
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5155 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5156 |
lemma absolutely_integrable_linear: fixes f::"real^'m \<Rightarrow> real^'n" and h::"real^'n \<Rightarrow> real^'p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5157 |
assumes "f absolutely_integrable_on s" "bounded_linear h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5158 |
shows "(h o f) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5159 |
proof- { presume as:"\<And>f::real^'m \<Rightarrow> real^'n. \<And>h::real^'n \<Rightarrow> real^'p. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5160 |
f absolutely_integrable_on UNIV \<Longrightarrow> bounded_linear h \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5161 |
(h o f) absolutely_integrable_on UNIV" note a = absolutely_integrable_restrict_univ[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5162 |
show ?thesis apply(subst a) using as[OF assms[unfolded a[of f] a[of g]]] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5163 |
unfolding o_def if_distrib linear_simps[OF assms(2)] . } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5164 |
fix f::"real^'m \<Rightarrow> real^'n" and h::"real^'n \<Rightarrow> real^'p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5165 |
assume assms:"f absolutely_integrable_on UNIV" "bounded_linear h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5166 |
from absolutely_integrable_bounded_variation[OF assms(1)] guess B . note B=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5167 |
from bounded_linear.pos_bounded[OF assms(2)] guess b .. note b=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5168 |
show "(h o f) absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5169 |
apply(rule bounded_variation_absolutely_integrable[of _ "B * b"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5170 |
apply(rule integrable_linear[OF _ assms(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5171 |
proof safe case goal2 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5172 |
have "(\<Sum>k\<in>d. norm (integral k (h \<circ> f))) \<le> setsum (\<lambda>k. norm(integral k f)) d * b" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5173 |
unfolding setsum_left_distrib apply(rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5174 |
proof- case goal1 from division_ofD(4)[OF goal2 this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5175 |
guess u v apply-by(erule exE)+ note uv=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5176 |
have *:"f integrable_on k" unfolding uv apply(rule integrable_on_subinterval[of _ UNIV]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5177 |
using assms by auto note this[unfolded has_integral_integral] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5178 |
note has_integral_linear[OF this assms(2)] integrable_linear[OF * assms(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5179 |
note * = has_integral_unique[OF this(2)[unfolded has_integral_integral] this(1)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5180 |
show ?case unfolding * using b by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5181 |
qed also have "... \<le> B * b" apply(rule mult_right_mono) using B goal2 b by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5182 |
finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5183 |
qed(insert assms,auto) qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5184 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5185 |
lemma absolutely_integrable_setsum: fixes f::"'a \<Rightarrow> real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5186 |
assumes "finite t" "\<And>a. a \<in> t \<Longrightarrow> (f a) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5187 |
shows "(\<lambda>x. setsum (\<lambda>a. f a x) t) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5188 |
using assms(1,2) apply induct defer apply(subst setsum.insert) apply assumption+ by(rule,auto) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5189 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5190 |
lemma absolutely_integrable_vector_abs: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5191 |
assumes "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5192 |
shows "(\<lambda>x. (\<chi> i. abs(f x$i))::real^'n) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5193 |
proof- have *:"\<And>x. ((\<chi> i. abs(f x$i))::real^'n) = (setsum (\<lambda>i. |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5194 |
(((\<lambda>y. (\<chi> j. if j = i then y$1 else 0)) o |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5195 |
(vec1 o ((\<lambda>x. (norm((\<chi> j. if j = i then x$i else 0)::real^'n))) o f))) x)) UNIV)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5196 |
unfolding Cart_eq setsum_component Cart_lambda_beta |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5197 |
proof case goal1 have *:"\<And>i xa. ((if i = xa then f x $ xa else 0) \<bullet> (if i = xa then f x $ xa else 0)) = |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5198 |
(if i = xa then (f x $ xa) * (f x $ xa) else 0)" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5199 |
have "\<bar>f x $ i\<bar> = (setsum (\<lambda>k. if k = i then abs ((f x)$i) else 0) UNIV)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5200 |
unfolding setsum_delta[OF finite_UNIV] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5201 |
also have "... = (\<Sum>xa\<in>UNIV. ((\<lambda>y. \<chi> j. if j = xa then dest_vec1 y else 0) \<circ> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5202 |
(\<lambda>x. vec1 (norm (\<chi> j. if j = xa then x $ xa else 0))) \<circ> f) x $ i)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5203 |
unfolding norm_eq_sqrt_inner inner_vector_def Cart_lambda_beta o_def * |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5204 |
apply(rule setsum_cong2) unfolding setsum_delta[OF finite_UNIV] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5205 |
finally show ?case unfolding o_def . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5206 |
show ?thesis unfolding * apply(rule absolutely_integrable_setsum) apply(rule finite_UNIV) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5207 |
apply(rule absolutely_integrable_linear) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5208 |
unfolding absolutely_integrable_on_trans unfolding o_def apply(rule absolutely_integrable_norm) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5209 |
apply(rule absolutely_integrable_linear[OF assms,unfolded o_def]) unfolding linear_linear |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5210 |
apply(rule_tac[!] linearI) unfolding Cart_eq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5211 |
qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5212 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5213 |
lemma absolutely_integrable_max: fixes f::"real^'m \<Rightarrow> real^'n" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5214 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5215 |
shows "(\<lambda>x. (\<chi> i. max (f(x)$i) (g(x)$i))::real^'n) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5216 |
proof- have *:"\<And>x. (1 / 2) *\<^sub>R ((\<chi> i. \<bar>(f x - g x) $ i\<bar>) + (f x + g x)) = (\<chi> i. max (f(x)$i) (g(x)$i))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5217 |
unfolding Cart_eq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5218 |
note absolutely_integrable_sub[OF assms] absolutely_integrable_add[OF assms] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5219 |
note absolutely_integrable_vector_abs[OF this(1)] this(2) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5220 |
note absolutely_integrable_add[OF this] note absolutely_integrable_cmul[OF this,of "1/2"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5221 |
thus ?thesis unfolding * . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5222 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5223 |
lemma absolutely_integrable_max_real: fixes f::"real^'m \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5224 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5225 |
shows "(\<lambda>x. max (f x) (g x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5226 |
proof- have *:"(\<lambda>x. \<chi> i. max ((vec1 \<circ> f) x $ i) ((vec1 \<circ> g) x $ i)) = vec1 o (\<lambda>x. max (f x) (g x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5227 |
apply rule unfolding Cart_eq by auto note absolutely_integrable_max[of "vec1 o f" s "vec1 o g"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5228 |
note this[unfolded absolutely_integrable_on_trans,OF assms] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5229 |
thus ?thesis unfolding * by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5230 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5231 |
lemma absolutely_integrable_min: fixes f::"real^'m \<Rightarrow> real^'n" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5232 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5233 |
shows "(\<lambda>x. (\<chi> i. min (f(x)$i) (g(x)$i))::real^'n) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5234 |
proof- have *:"\<And>x. (1 / 2) *\<^sub>R ((f x + g x) - (\<chi> i. \<bar>(f x - g x) $ i\<bar>)) = (\<chi> i. min (f(x)$i) (g(x)$i))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5235 |
unfolding Cart_eq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5236 |
note absolutely_integrable_add[OF assms] absolutely_integrable_sub[OF assms] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5237 |
note this(1) absolutely_integrable_vector_abs[OF this(2)] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5238 |
note absolutely_integrable_sub[OF this] note absolutely_integrable_cmul[OF this,of "1/2"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5239 |
thus ?thesis unfolding * . qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5240 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5241 |
lemma absolutely_integrable_min_real: fixes f::"real^'m \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5242 |
assumes "f absolutely_integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5243 |
shows "(\<lambda>x. min (f x) (g x)) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5244 |
proof- have *:"(\<lambda>x. \<chi> i. min ((vec1 \<circ> f) x $ i) ((vec1 \<circ> g) x $ i)) = vec1 o (\<lambda>x. min (f x) (g x))" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5245 |
apply rule unfolding Cart_eq by auto note absolutely_integrable_min[of "vec1 o f" s "vec1 o g"] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5246 |
note this[unfolded absolutely_integrable_on_trans,OF assms] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5247 |
thus ?thesis unfolding * by auto qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5248 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5249 |
lemma absolutely_integrable_abs_eq: fixes f::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5250 |
shows "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5251 |
(\<lambda>x. (\<chi> i. abs(f x$i))::real^'m) integrable_on s" (is "?l = ?r") |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5252 |
proof assume ?l thus ?r apply-apply rule defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5253 |
apply(drule absolutely_integrable_vector_abs) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5254 |
next assume ?r { presume lem:"\<And>f::real^'n \<Rightarrow> real^'m. f integrable_on UNIV \<Longrightarrow> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5255 |
(\<lambda>x. (\<chi> i. abs(f(x)$i))::real^'m) integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5256 |
have *:"\<And>x. (\<chi> i. \<bar>(if x \<in> s then f x else 0) $ i\<bar>) = (if x\<in>s then (\<chi> i. \<bar>f x $ i\<bar>) else 0)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5257 |
unfolding Cart_eq by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5258 |
show ?l apply(subst absolutely_integrable_restrict_univ[THEN sym]) apply(rule lem) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5259 |
unfolding integrable_restrict_univ * using `?r` by auto } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5260 |
fix f::"real^'n \<Rightarrow> real^'m" assume assms:"f integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5261 |
"(\<lambda>x. (\<chi> i. abs(f(x)$i))::real^'m) integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5262 |
let ?B = "setsum (\<lambda>i. integral UNIV (\<lambda>x. (\<chi> j. abs(f x$j)) ::real^'m) $ i) UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5263 |
show "f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5264 |
apply(rule bounded_variation_absolutely_integrable[OF assms(1), where B="?B"],safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5265 |
proof- case goal1 note d=this and d'=division_ofD[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5266 |
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5267 |
(\<Sum>k\<in>d. setsum (op $ (integral k (\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>))) UNIV)" apply(rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5268 |
apply(rule order_trans[OF norm_le_l1],rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5269 |
proof- fix k and i::'m assume "k\<in>d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5270 |
from d'(4)[OF this] guess a b apply-by(erule exE)+ note ab=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5271 |
show "\<bar>integral k f $ i\<bar> \<le> integral k (\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>) $ i" apply(rule abs_leI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5272 |
unfolding vector_uminus_component[THEN sym] defer apply(subst integral_neg[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5273 |
defer apply(rule_tac[1-2] integral_component_le) apply(rule integrable_neg) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5274 |
using integrable_on_subinterval[OF assms(1),of a b] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5275 |
integrable_on_subinterval[OF assms(2),of a b] unfolding ab by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5276 |
qed also have "... \<le> setsum (op $ (integral UNIV (\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>))) UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5277 |
apply(subst setsum_commute,rule setsum_mono) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5278 |
proof- case goal1 have *:"(\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>) integrable_on \<Union>d" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5279 |
using integrable_on_subdivision[OF d assms(2)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5280 |
have "(\<Sum>i\<in>d. integral i (\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>) $ j) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5281 |
= integral (\<Union>d) (\<lambda>x. (\<chi> j. abs(f x$j)) ::real^'m) $ j" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5282 |
unfolding setsum_component[THEN sym] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5283 |
apply(subst integral_combine_division_topdown[THEN sym,OF * d]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5284 |
also have "... \<le> integral UNIV (\<lambda>x. \<chi> j. \<bar>f x $ j\<bar>) $ j" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5285 |
apply(rule integral_subset_component_le) using assms * by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5286 |
finally show ?case . |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5287 |
qed finally show ?case . qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5288 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5289 |
lemma nonnegative_absolutely_integrable: fixes f::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5290 |
assumes "\<forall>x\<in>s. \<forall>i. 0 \<le> f(x)$i" "f integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5291 |
shows "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5292 |
unfolding absolutely_integrable_abs_eq apply rule defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5293 |
apply(rule integrable_eq[of _ f]) unfolding Cart_eq using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5294 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5295 |
lemma absolutely_integrable_integrable_bound: fixes f::"real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5296 |
assumes "\<forall>x\<in>s. norm(f x) \<le> g x" "f integrable_on s" "g integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5297 |
shows "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5298 |
proof- { presume *:"\<And>f::real^'n \<Rightarrow> real^'m. \<And> g. \<forall>x. norm(f x) \<le> g x \<Longrightarrow> f integrable_on UNIV |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5299 |
\<Longrightarrow> g integrable_on UNIV \<Longrightarrow> f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5300 |
show ?thesis apply(subst absolutely_integrable_restrict_univ[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5301 |
apply(rule *[of _ "\<lambda>x. if x\<in>s then g x else 0"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5302 |
using assms unfolding integrable_restrict_univ by auto } |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5303 |
fix g and f :: "real^'n \<Rightarrow> real^'m" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5304 |
assume assms:"\<forall>x. norm(f x) \<le> g x" "f integrable_on UNIV" "g integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5305 |
show "f absolutely_integrable_on UNIV" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5306 |
apply(rule bounded_variation_absolutely_integrable[OF assms(2),where B="integral UNIV g"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5307 |
proof safe case goal1 note d=this and d'=division_ofD[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5308 |
have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>k\<in>d. integral k g)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5309 |
apply(rule setsum_mono) apply(rule integral_norm_bound_integral) apply(drule_tac[!] d'(4),safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5310 |
apply(rule_tac[1-2] integrable_on_subinterval) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5311 |
also have "... = integral (\<Union>d) g" apply(rule integral_combine_division_bottomup[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5312 |
apply(rule d,safe) apply(drule d'(4),safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5313 |
apply(rule integrable_on_subinterval[OF assms(3)]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5314 |
also have "... \<le> integral UNIV g" apply(rule integral_subset_le) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5315 |
apply(rule integrable_on_subdivision[OF d,of _ UNIV]) prefer 4 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5316 |
apply(rule,rule_tac y="norm (f x)" in order_trans) using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5317 |
finally show ?case . qed qed |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5318 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5319 |
lemma absolutely_integrable_integrable_bound_real: fixes f::"real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5320 |
assumes "\<forall>x\<in>s. norm(f x) \<le> g x" "f integrable_on s" "g integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5321 |
shows "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5322 |
apply(subst absolutely_integrable_on_trans[THEN sym]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5323 |
apply(rule absolutely_integrable_integrable_bound[where g=g]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5324 |
using assms unfolding o_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5325 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5326 |
lemma absolutely_integrable_absolutely_integrable_bound: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5327 |
fixes f::"real^'n \<Rightarrow> real^'m" and g::"real^'n \<Rightarrow> real^'p" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5328 |
assumes "\<forall>x\<in>s. norm(f x) \<le> norm(g x)" "f integrable_on s" "g absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5329 |
shows "f absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5330 |
apply(rule absolutely_integrable_integrable_bound[of s f "\<lambda>x. norm (g x)"]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5331 |
using assms by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5332 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5333 |
lemma absolutely_integrable_inf_real: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5334 |
assumes "finite k" "k \<noteq> {}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5335 |
"\<forall>i\<in>k. (\<lambda>x. (fs x i)::real) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5336 |
shows "(\<lambda>x. (Inf ((fs x) ` k))) absolutely_integrable_on s" using assms |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5337 |
proof induct case (insert a k) let ?P = " (\<lambda>x. if fs x ` k = {} then fs x a |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5338 |
else min (fs x a) (Inf (fs x ` k))) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5339 |
show ?case unfolding image_insert |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5340 |
apply(subst Inf_insert_finite) apply(rule finite_imageI[OF insert(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5341 |
proof(cases "k={}") case True |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5342 |
thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5343 |
next case False thus ?P apply(subst if_not_P) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5344 |
apply(rule absolutely_integrable_min_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5345 |
defer apply(rule insert(3)[OF False]) using insert(5) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5346 |
qed qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5347 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5348 |
lemma absolutely_integrable_sup_real: |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5349 |
assumes "finite k" "k \<noteq> {}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5350 |
"\<forall>i\<in>k. (\<lambda>x. (fs x i)::real) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5351 |
shows "(\<lambda>x. (Sup ((fs x) ` k))) absolutely_integrable_on s" using assms |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5352 |
proof induct case (insert a k) let ?P = " (\<lambda>x. if fs x ` k = {} then fs x a |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5353 |
else max (fs x a) (Sup (fs x ` k))) absolutely_integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5354 |
show ?case unfolding image_insert |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5355 |
apply(subst Sup_insert_finite) apply(rule finite_imageI[OF insert(1)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5356 |
proof(cases "k={}") case True |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5357 |
thus ?P apply(subst if_P) defer apply(rule insert(5)[rule_format]) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5358 |
next case False thus ?P apply(subst if_not_P) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5359 |
apply(rule absolutely_integrable_max_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5360 |
defer apply(rule insert(3)[OF False]) using insert(5) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5361 |
qed qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5362 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5363 |
subsection {* Dominated convergence. *} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5364 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5365 |
lemma dominated_convergence: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5366 |
assumes "\<And>k. (f k) integrable_on s" "h integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5367 |
"\<And>k. \<forall>x \<in> s. norm(f k x) \<le> (h x)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5368 |
"\<forall>x \<in> s. ((\<lambda>k. f k x) ---> g x) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5369 |
shows "g integrable_on s" "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5370 |
proof- have "\<And>m. (\<lambda>x. Inf {f j x |j. m \<le> j}) integrable_on s \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5371 |
((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) ---> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5372 |
integral s (\<lambda>x. Inf {f j x |j. m \<le> j}))sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5373 |
proof(rule monotone_convergence_decreasing_real,safe) fix m::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5374 |
show "bounded {integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}}) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5375 |
unfolding bounded_iff apply(rule_tac x="integral s h" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5376 |
proof safe fix k::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5377 |
show "norm (integral s (\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}})) \<le> integral s h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5378 |
apply(rule integral_norm_bound_integral) unfolding simple_image |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5379 |
apply(rule absolutely_integrable_onD) apply(rule absolutely_integrable_inf_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5380 |
prefer 5 unfolding real_norm_def apply(rule) apply(rule Inf_abs_ge) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5381 |
prefer 5 apply rule apply(rule_tac g=h in absolutely_integrable_integrable_bound_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5382 |
using assms unfolding real_norm_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5383 |
qed fix k::nat show "(\<lambda>x. Inf {f j x |j. j \<in> {m..m + k}}) integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5384 |
unfolding simple_image apply(rule absolutely_integrable_onD) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5385 |
apply(rule absolutely_integrable_inf_real) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5386 |
using absolutely_integrable_integrable_bound_real[OF assms(3,1,2)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5387 |
fix x assume x:"x\<in>s" show "Inf {f j x |j. j \<in> {m..m + Suc k}} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5388 |
\<le> Inf {f j x |j. j \<in> {m..m + k}}" apply(rule Inf_ge) unfolding setge_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5389 |
defer apply rule apply(subst Inf_finite_le_iff) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5390 |
apply(rule_tac x=xa in bexI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5391 |
let ?S = "{f j x| j. m \<le> j}" def i \<equiv> "Inf ?S" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5392 |
show "((\<lambda>k. Inf {f j x |j. j \<in> {m..m + k}}) ---> i) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5393 |
unfolding Lim_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5394 |
proof safe case goal1 note e=this have i:"isGlb UNIV ?S i" unfolding i_def apply(rule Inf) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5395 |
defer apply(rule_tac x="- h x - 1" in exI) unfolding setge_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5396 |
proof safe case goal1 thus ?case using assms(3)[rule_format,OF x, of j] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5397 |
qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5398 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5399 |
have "\<exists>y\<in>?S. \<not> y \<ge> i + e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5400 |
proof(rule ccontr) case goal1 hence "i \<ge> i + e" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5401 |
apply(rule isGlb_le_isLb[OF i]) apply(rule isLbI) unfolding setge_def by fastsimp+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5402 |
thus False using e by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5403 |
qed then guess y .. note y=this[unfolded not_le] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5404 |
from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5405 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5406 |
show ?case apply(rule_tac x=N in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5407 |
proof safe case goal1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5408 |
have *:"\<And>y ix. y < i + e \<longrightarrow> i \<le> ix \<longrightarrow> ix \<le> y \<longrightarrow> abs(ix - i) < e" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5409 |
show ?case unfolding dist_real_def apply(rule *[rule_format,OF y(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5410 |
unfolding i_def apply(rule real_le_inf_subset) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5411 |
apply(rule,rule isGlbD1[OF i]) prefer 3 apply(subst Inf_finite_le_iff) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5412 |
prefer 3 apply(rule_tac x=y in bexI) using N goal1 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5413 |
qed qed qed note dec1 = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5414 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5415 |
have "\<And>m. (\<lambda>x. Sup {f j x |j. m \<le> j}) integrable_on s \<and> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5416 |
((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) ---> |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5417 |
integral s (\<lambda>x. Sup {f j x |j. m \<le> j})) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5418 |
proof(rule monotone_convergence_increasing_real,safe) fix m::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5419 |
show "bounded {integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}}) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5420 |
unfolding bounded_iff apply(rule_tac x="integral s h" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5421 |
proof safe fix k::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5422 |
show "norm (integral s (\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}})) \<le> integral s h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5423 |
apply(rule integral_norm_bound_integral) unfolding simple_image |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5424 |
apply(rule absolutely_integrable_onD) apply(rule absolutely_integrable_sup_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5425 |
prefer 5 unfolding real_norm_def apply(rule) apply(rule Sup_abs_le) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5426 |
prefer 5 apply rule apply(rule_tac g=h in absolutely_integrable_integrable_bound_real) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5427 |
using assms unfolding real_norm_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5428 |
qed fix k::nat show "(\<lambda>x. Sup {f j x |j. j \<in> {m..m + k}}) integrable_on s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5429 |
unfolding simple_image apply(rule absolutely_integrable_onD) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5430 |
apply(rule absolutely_integrable_sup_real) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5431 |
using absolutely_integrable_integrable_bound_real[OF assms(3,1,2)] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5432 |
fix x assume x:"x\<in>s" show "Sup {f j x |j. j \<in> {m..m + Suc k}} |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5433 |
\<ge> Sup {f j x |j. j \<in> {m..m + k}}" apply(rule Sup_le) unfolding setle_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5434 |
defer apply rule apply(subst Sup_finite_ge_iff) prefer 3 apply(rule_tac x=y in bexI) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5435 |
let ?S = "{f j x| j. m \<le> j}" def i \<equiv> "Sup ?S" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5436 |
show "((\<lambda>k. Sup {f j x |j. j \<in> {m..m + k}}) ---> i) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5437 |
unfolding Lim_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5438 |
proof safe case goal1 note e=this have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5439 |
defer apply(rule_tac x="h x" in exI) unfolding setle_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5440 |
proof safe case goal1 thus ?case using assms(3)[rule_format,OF x, of j] by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5441 |
qed auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5442 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5443 |
have "\<exists>y\<in>?S. \<not> y \<le> i - e" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5444 |
proof(rule ccontr) case goal1 hence "i \<le> i - e" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5445 |
apply(rule isLub_le_isUb[OF i]) apply(rule isUbI) unfolding setle_def by fastsimp+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5446 |
thus False using e by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5447 |
qed then guess y .. note y=this[unfolded not_le] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5448 |
from this(1)[unfolded mem_Collect_eq] guess N .. note N=conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5449 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5450 |
show ?case apply(rule_tac x=N in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5451 |
proof safe case goal1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5452 |
have *:"\<And>y ix. i - e < y \<longrightarrow> ix \<le> i \<longrightarrow> y \<le> ix \<longrightarrow> abs(ix - i) < e" by arith |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5453 |
show ?case unfolding dist_real_def apply(rule *[rule_format,OF y(2)]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5454 |
unfolding i_def apply(rule real_ge_sup_subset) prefer 3 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5455 |
apply(rule,rule isLubD1[OF i]) prefer 3 apply(subst Sup_finite_ge_iff) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5456 |
prefer 3 apply(rule_tac x=y in bexI) using N goal1 by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5457 |
qed qed qed note inc1 = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5458 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5459 |
have "g integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5460 |
apply(rule monotone_convergence_increasing_real,safe) apply fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5461 |
proof- show "bounded {integral s (\<lambda>x. Inf {f j x |j. k \<le> j}) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5462 |
unfolding bounded_iff apply(rule_tac x="integral s h" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5463 |
proof safe fix k::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5464 |
show "norm (integral s (\<lambda>x. Inf {f j x |j. k \<le> j})) \<le> integral s h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5465 |
apply(rule integral_norm_bound_integral) apply fact+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5466 |
unfolding real_norm_def apply(rule) apply(rule Inf_abs_ge) using assms(3) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5467 |
qed fix k::nat and x assume x:"x\<in>s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5468 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5469 |
have *:"\<And>x y::real. x \<ge> - y \<Longrightarrow> - x \<le> y" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5470 |
show "Inf {f j x |j. k \<le> j} \<le> Inf {f j x |j. Suc k \<le> j}" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5471 |
apply(rule real_le_inf_subset) prefer 3 unfolding setge_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5472 |
apply(rule_tac x="- h x" in exI) apply safe apply(rule *) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5473 |
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5474 |
show "((\<lambda>k. Inf {f j x |j. k \<le> j}) ---> g x) sequentially" unfolding Lim_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5475 |
proof safe case goal1 hence "0<e/2" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5476 |
from assms(4)[unfolded Lim_sequentially,rule_format,OF x this] guess N .. note N=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5477 |
show ?case apply(rule_tac x=N in exI,safe) unfolding dist_real_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5478 |
apply(rule le_less_trans[of _ "e/2"]) apply(rule Inf_asclose) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5479 |
defer apply(rule less_imp_le) using N goal1 unfolding dist_real_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5480 |
qed qed note inc2 = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5481 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5482 |
have "g integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) ---> integral s g) sequentially" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5483 |
apply(rule monotone_convergence_decreasing_real,safe) apply fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5484 |
proof- show "bounded {integral s (\<lambda>x. Sup {f j x |j. k \<le> j}) |k. True}" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5485 |
unfolding bounded_iff apply(rule_tac x="integral s h" in exI) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5486 |
proof safe fix k::nat |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5487 |
show "norm (integral s (\<lambda>x. Sup {f j x |j. k \<le> j})) \<le> integral s h" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5488 |
apply(rule integral_norm_bound_integral) apply fact+ |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5489 |
unfolding real_norm_def apply(rule) apply(rule Sup_abs_le) using assms(3) by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5490 |
qed fix k::nat and x assume x:"x\<in>s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5491 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5492 |
show "Sup {f j x |j. k \<le> j} \<ge> Sup {f j x |j. Suc k \<le> j}" apply- |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5493 |
apply(rule real_ge_sup_subset) prefer 3 unfolding setle_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5494 |
apply(rule_tac x="h x" in exI) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5495 |
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5496 |
show "((\<lambda>k. Sup {f j x |j. k \<le> j}) ---> g x) sequentially" unfolding Lim_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5497 |
proof safe case goal1 hence "0<e/2" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5498 |
from assms(4)[unfolded Lim_sequentially,rule_format,OF x this] guess N .. note N=this |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5499 |
show ?case apply(rule_tac x=N in exI,safe) unfolding dist_real_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5500 |
apply(rule le_less_trans[of _ "e/2"]) apply(rule Sup_asclose) apply safe |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5501 |
defer apply(rule less_imp_le) using N goal1 unfolding dist_real_def by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5502 |
qed qed note dec2 = conjunctD2[OF this] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5503 |
|
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5504 |
show "g integrable_on s" by fact |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5505 |
show "((\<lambda>k. integral s (f k)) ---> integral s g) sequentially" unfolding Lim_sequentially |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5506 |
proof safe case goal1 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5507 |
from inc2(2)[unfolded Lim_sequentially,rule_format,OF goal1] guess N1 .. note N1=this[unfolded dist_real_def] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5508 |
from dec2(2)[unfolded Lim_sequentially,rule_format,OF goal1] guess N2 .. note N2=this[unfolded dist_real_def] |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5509 |
show ?case apply(rule_tac x="N1+N2" in exI,safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5510 |
proof- fix n assume n:"n \<ge> N1 + N2" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5511 |
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 |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5512 |
show "dist (integral s (f n)) (integral s g) < e" unfolding dist_real_def |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5513 |
apply(rule *[rule_format,OF N1[rule_format] N2[rule_format], of n n]) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5514 |
proof- show "integral s (\<lambda>x. Inf {f j x |j. n \<le> j}) \<le> integral s (f n)" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5515 |
proof(rule integral_le[OF dec1(1) assms(1)],safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5516 |
fix x assume x:"x \<in> s" have *:"\<And>x y::real. x \<ge> - y \<Longrightarrow> - x \<le> y" by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5517 |
show "Inf {f j x |j. n \<le> j} \<le> f n x" apply(rule Inf_lower[where z="- h x"]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5518 |
apply(rule *) using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5519 |
qed show "integral s (f n) \<le> integral s (\<lambda>x. Sup {f j x |j. n \<le> j})" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5520 |
proof(rule integral_le[OF assms(1) inc1(1)],safe) |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5521 |
fix x assume x:"x \<in> s" |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5522 |
show "f n x \<le> Sup {f j x |j. n \<le> j}" apply(rule Sup_upper[where z="h x"]) defer |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5523 |
using assms(3)[rule_format,OF x] unfolding real_norm_def abs_le_iff by auto |
027ae62681be
Translated remaining theorems about integration from HOL light.
himmelma
parents:
36081
diff
changeset
|
5524 |
qed qed(insert n,auto) qed qed qed |
35752 | 5525 |
|
5526 |
declare [[smt_certificates=""]] |
|
36244
009b0ee1b838
Only use provided SMT-certificates in HOL-Multivariate_Analysis.
hoelzl
parents:
36243
diff
changeset
|
5527 |
declare [[smt_fixed=false]] |
35752 | 5528 |
|
35173
9b24bfca8044
Renamed Multivariate-Analysis/Integration to Multivariate-Analysis/Integration_MV to avoid name clash with Integration.
hoelzl
parents:
35172
diff
changeset
|
5529 |
end |