1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy Wed Feb 17 18:33:45 2010 +0100
1.3 @@ -0,0 +1,3465 @@
1.4 +
1.5 +header {* Kurzweil-Henstock gauge integration in many dimensions. *}
1.6 +(* Author: John Harrison
1.7 + Translation from HOL light: Robert Himmelmann, TU Muenchen *)
1.8 +
1.9 +theory Integration_Aleph
1.10 + imports Derivative SMT
1.11 +begin
1.12 +
1.13 +declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
1.14 +declare [[smt_record=true]]
1.15 +declare [[z3_proofs=true]]
1.16 +
1.17 +lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
1.18 +lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
1.19 +lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
1.20 +lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
1.21 +
1.22 +declare smult_conv_scaleR[simp]
1.23 +
1.24 +subsection {* Some useful lemmas about intervals. *}
1.25 +
1.26 +lemma empty_as_interval: "{} = {1..0::real^'n}"
1.27 + apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
1.28 + using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
1.29 +
1.30 +lemma interior_subset_union_intervals:
1.31 + assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
1.32 + shows "interior i \<subseteq> interior s" proof-
1.33 + have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
1.34 + unfolding assms(1,2) interior_closed_interval by auto
1.35 + moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
1.36 + using assms(4) unfolding assms(1,2) by auto
1.37 + ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
1.38 + unfolding assms(1,2) interior_closed_interval by auto qed
1.39 +
1.40 +lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
1.41 + assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
1.42 + shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
1.43 + 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)
1.44 + unfolding open_subset_interior[OF open_ball] using interior_subset by auto
1.45 + have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
1.46 + 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
1.47 + thus ?case proof(induct rule:finite_induct)
1.48 + case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
1.49 + case (insert i f) guess x using insert(5) .. note x = this
1.50 + then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
1.51 + guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
1.52 + show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
1.53 + then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
1.54 + hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
1.55 + 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
1.56 + 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
1.57 + case True show ?thesis proof(cases "x\<in>{a<..<b}")
1.58 + case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
1.59 + thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
1.60 + unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
1.61 + 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)
1.62 + 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
1.63 + hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
1.64 + 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)
1.65 + fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
1.66 + hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
1.67 + hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
1.68 + 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
1.69 + moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
1.70 + 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)"
1.71 + apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
1.72 + unfolding norm_scaleR norm_basis by auto
1.73 + also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
1.74 + finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
1.75 + ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
1.76 + 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)
1.77 + fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
1.78 + hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
1.79 + hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
1.80 + 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
1.81 + moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
1.82 + 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)"
1.83 + apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
1.84 + unfolding norm_scaleR norm_basis by auto
1.85 + also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
1.86 + finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
1.87 + ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
1.88 + 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
1.89 + thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
1.90 + guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e ..
1.91 + hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
1.92 + thus False using `t\<in>f` assms(4) by auto qed
1.93 +subsection {* Bounds on intervals where they exist. *}
1.94 +
1.95 +definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
1.96 +
1.97 +definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
1.98 +
1.99 +lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
1.100 + using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
1.101 + apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
1.102 + apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
1.103 + unfolding mem_interval using assms by auto
1.104 +
1.105 +lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
1.106 + using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
1.107 + apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
1.108 + apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
1.109 + unfolding mem_interval using assms by auto
1.110 +
1.111 +lemmas interval_bounds = interval_upperbound interval_lowerbound
1.112 +
1.113 +lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
1.114 + using assms unfolding interval_ne_empty by auto
1.115 +
1.116 +lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
1.117 + apply(rule interval_upperbound) by auto
1.118 +
1.119 +lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
1.120 + apply(rule interval_lowerbound) by auto
1.121 +
1.122 +lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
1.123 +
1.124 +subsection {* Content (length, area, volume...) of an interval. *}
1.125 +
1.126 +definition "content (s::(real^'n) set) =
1.127 + (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
1.128 +
1.129 +lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
1.130 + unfolding interval_eq_empty unfolding not_ex not_less by assumption
1.131 +
1.132 +lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
1.133 + shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
1.134 + using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
1.135 +
1.136 +lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
1.137 + apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
1.138 +
1.139 +lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
1.140 + using content_closed_interval[of a b] by auto
1.141 +
1.142 +lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
1.143 +
1.144 +lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
1.145 + have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
1.146 + have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
1.147 + thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
1.148 +
1.149 +lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
1.150 + case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
1.151 + have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
1.152 + apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
1.153 + thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
1.154 +
1.155 +lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
1.156 +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
1.157 + show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
1.158 + using assms apply(erule_tac x=x in allE) by auto qed
1.159 +
1.160 +lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
1.161 + apply(rule content_pos_lt) by auto
1.162 +
1.163 +lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
1.164 + case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
1.165 + apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
1.166 + guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
1.167 + show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
1.168 + apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
1.169 + apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
1.170 +
1.171 +lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
1.172 +
1.173 +lemma content_closed_interval_cases:
1.174 + "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)
1.175 + 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
1.176 +
1.177 +lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
1.178 + unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
1.179 +
1.180 +lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
1.181 + unfolding content_eq_0 by auto
1.182 +
1.183 +lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
1.184 + apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
1.185 + 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
1.186 +
1.187 +lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
1.188 +
1.189 +lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
1.190 + case True thus ?thesis using content_pos_le[of c d] by auto next
1.191 + case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
1.192 + hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
1.193 + have "{c..d} \<noteq> {}" using assms False by auto
1.194 + hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
1.195 + show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
1.196 + unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
1.197 + show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
1.198 + show "b $ i - a $ i \<le> d $ i - c $ i"
1.199 + using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
1.200 + using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
1.201 +
1.202 +lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
1.203 + unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
1.204 +
1.205 +subsection {* The notion of a gauge --- simply an open set containing the point. *}
1.206 +
1.207 +definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
1.208 +
1.209 +lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
1.210 + using assms unfolding gauge_def by auto
1.211 +
1.212 +lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
1.213 +
1.214 +lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
1.215 + unfolding gauge_def by auto
1.216 +
1.217 +lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
1.218 +
1.219 +lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
1.220 +
1.221 +lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
1.222 + unfolding gauge_def by auto
1.223 +
1.224 +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-
1.225 + have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
1.226 + unfolding gauge_def unfolding *
1.227 + using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
1.228 +
1.229 +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)
1.230 +
1.231 +subsection {* Divisions. *}
1.232 +
1.233 +definition division_of (infixl "division'_of" 40) where
1.234 + "s division_of i \<equiv>
1.235 + finite s \<and>
1.236 + (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
1.237 + (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
1.238 + (\<Union>s = i)"
1.239 +
1.240 +lemma division_ofD[dest]: assumes "s division_of i"
1.241 + 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})"
1.242 + "\<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
1.243 +
1.244 +lemma division_ofI:
1.245 + 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})"
1.246 + "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
1.247 + shows "s division_of i" using assms unfolding division_of_def by auto
1.248 +
1.249 +lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
1.250 + unfolding division_of_def by auto
1.251 +
1.252 +lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
1.253 + unfolding division_of_def by auto
1.254 +
1.255 +lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
1.256 +
1.257 +lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
1.258 + assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s"
1.259 + ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
1.260 + ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
1.261 + assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
1.262 + { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
1.263 + moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
1.264 +
1.265 +lemma elementary_empty: obtains p where "p division_of {}"
1.266 + unfolding division_of_trivial by auto
1.267 +
1.268 +lemma elementary_interval: obtains p where "p division_of {a..b}"
1.269 + by(metis division_of_trivial division_of_self)
1.270 +
1.271 +lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
1.272 + unfolding division_of_def by auto
1.273 +
1.274 +lemma forall_in_division:
1.275 + "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
1.276 + unfolding division_of_def by fastsimp
1.277 +
1.278 +lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
1.279 + apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
1.280 + show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
1.281 + { 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
1.282 + show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
1.283 + 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
1.284 + show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
1.285 +
1.286 +lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
1.287 +
1.288 +lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
1.289 + unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
1.290 + apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
1.291 +
1.292 +lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
1.293 + 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-
1.294 +let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
1.295 +show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
1.296 + moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
1.297 + 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
1.298 + using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
1.299 + { 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
1.300 + 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
1.301 + guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
1.302 + guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
1.303 + show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
1.304 + assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
1.305 + assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
1.306 + assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
1.307 + have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
1.308 + interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
1.309 + interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
1.310 + \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
1.311 + show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
1.312 + using division_ofD(5)[OF assms(1) k1(2) k2(2)]
1.313 + using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
1.314 +
1.315 +lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
1.316 + shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
1.317 + case True show ?thesis unfolding True and division_of_trivial by auto next
1.318 + have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
1.319 + case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
1.320 +
1.321 +lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
1.322 + shows "\<exists>p. p division_of (s \<inter> t)"
1.323 + by(rule,rule division_inter[OF assms])
1.324 +
1.325 +lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
1.326 + shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
1.327 +case (insert x f) show ?case proof(cases "f={}")
1.328 + case True thus ?thesis unfolding True using insert by auto next
1.329 + case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
1.330 + moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
1.331 + show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
1.332 +
1.333 +lemma division_disjoint_union:
1.334 + assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
1.335 + shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI)
1.336 + note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
1.337 + show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
1.338 + show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
1.339 + { 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 = {}"
1.340 + { 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)]]
1.341 + using assms(3) by blast } moreover
1.342 + { 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)]]
1.343 + using assms(3) by blast} ultimately
1.344 + show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
1.345 + fix k assume k:"k \<in> p1 \<union> p2" show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
1.346 + 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
1.347 +
1.348 +lemma partial_division_extend_1:
1.349 + assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
1.350 + obtains p where "p division_of {a..b}" "{c..d} \<in> p"
1.351 +proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
1.352 + guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
1.353 + def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
1.354 + have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
1.355 + hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
1.356 + 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
1.357 + 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
1.358 + have "{c..d} \<noteq> {}" using assms by auto
1.359 + 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$i)}"
1.360 + let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}"
1.361 + let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
1.362 + 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)
1.363 + show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
1.364 + proof- have "\<And>i. \<pi>' i < Suc n"
1.365 + proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
1.366 + hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
1.367 + qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
1.368 + "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)"
1.369 + unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
1.370 + thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
1.371 + 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}"
1.372 + unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
1.373 + proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
1.374 + then guess i unfolding mem_interval not_all .. note i=this
1.375 + show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
1.376 + apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
1.377 + qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
1.378 + proof- fix x assume x:"x\<in>{a..b}"
1.379 + { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
1.380 + 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)}"
1.381 + assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
1.382 + hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
1.383 + hence M:"finite ?M" "?M \<noteq> {}" by auto
1.384 + 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]]
1.385 + Min_gr_iff[OF M,unfolded l_def[symmetric]]
1.386 + have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
1.387 + apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
1.388 + proof- assume as:"x $ \<pi> l < c $ \<pi> l"
1.389 + show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
1.390 + proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
1.391 + thus ?case using as x[unfolded mem_interval,rule_format,of i]
1.392 + apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
1.393 + qed
1.394 + next assume as:"x $ \<pi> l > d $ \<pi> l"
1.395 + show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
1.396 + proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
1.397 + thus ?case using as x[unfolded mem_interval,rule_format,of i]
1.398 + apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
1.399 + qed qed
1.400 + thus "x \<in> \<Union>?p" using l(2) by blast
1.401 + qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
1.402 +
1.403 + show "finite ?p" by auto
1.404 + 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
1.405 + show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
1.406 + 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
1.407 + 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)
1.408 + qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
1.409 + proof- case goal1 thus ?case using abcd[of x] by auto
1.410 + next case goal2 thus ?case using abcd[of x] by auto
1.411 + qed thus "k \<noteq> {}" using k by auto
1.412 + show "\<exists>a b. k = {a..b}" using k by auto
1.413 + 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
1.414 + { fix k k' l l'
1.415 + assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
1.416 + assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
1.417 + assume "l \<le> l'" fix x
1.418 + have "x \<notin> interior k \<inter> interior k'"
1.419 + proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
1.420 + case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
1.421 + hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
1.422 + have ln:"l < n + 1"
1.423 + proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
1.424 + hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
1.425 + hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
1.426 + thus False using `k\<noteq>k'` k' by auto
1.427 + qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
1.428 + have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
1.429 + proof(erule disjE)
1.430 + assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
1.431 + show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
1.432 + next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
1.433 + show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
1.434 + qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
1.435 + by(auto elim!:allE[where x="\<pi> l"])
1.436 + next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
1.437 + hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
1.438 + note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
1.439 + assume x:"x \<in> interior k \<inter> interior k'"
1.440 + show False using l(1) l'(1) apply-
1.441 + proof(erule_tac[!] disjE)+
1.442 + assume as:"k = ?p1 l" "k' = ?p1 l'"
1.443 + note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
1.444 + have "l \<noteq> l'" using k'(2)[unfolded as] by auto
1.445 + thus False using * by(smt Cart_lambda_beta \<pi>l)
1.446 + next assume as:"k = ?p2 l" "k' = ?p2 l'"
1.447 + note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
1.448 + have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
1.449 + thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
1.450 + unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
1.451 + next assume as:"k = ?p1 l" "k' = ?p2 l'"
1.452 + note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
1.453 + show False using *[of "\<pi> l"] *[of "\<pi> l'"]
1.454 + unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
1.455 + next assume as:"k = ?p2 l" "k' = ?p1 l'"
1.456 + note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
1.457 + show False using *[of "\<pi> l"] *[of "\<pi> l'"]
1.458 + unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
1.459 + qed qed }
1.460 + from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
1.461 + apply - apply(cases "l' \<le> l") using k'(2) by auto
1.462 + thus "interior k \<inter> interior k' = {}" by auto
1.463 +qed qed
1.464 +
1.465 +lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
1.466 + obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
1.467 + case True guess q apply(rule elementary_interval[of a b]) .
1.468 + thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
1.469 + case False note p = division_ofD[OF assms(1)]
1.470 + have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
1.471 + guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
1.472 + have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
1.473 + guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
1.474 + guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
1.475 + 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-
1.476 + fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
1.477 + using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
1.478 + hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
1.479 + apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
1.480 + then guess d .. note d = this
1.481 + show ?thesis apply(rule that[of "d \<union> p"]) proof-
1.482 + 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
1.483 + 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"
1.484 + 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
1.485 + show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
1.486 + apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
1.487 + 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
1.488 + show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
1.489 + defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
1.490 + show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
1.491 + show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
1.492 + 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}))"
1.493 + apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
1.494 +
1.495 +lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
1.496 + unfolding division_of_def by(metis bounded_Union bounded_interval)
1.497 +
1.498 +lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
1.499 + by(meson elementary_bounded bounded_subset_closed_interval)
1.500 +
1.501 +lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
1.502 + obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
1.503 + case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
1.504 + case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
1.505 + have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
1.506 + case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
1.507 + using false True assms using interior_subset by auto next
1.508 + case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
1.509 + have *:"{u..v} \<subseteq> {c..d}" using uv by auto
1.510 + guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
1.511 + 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
1.512 + show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
1.513 + apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
1.514 + unfolding interior_inter[THEN sym] proof-
1.515 + have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
1.516 + have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
1.517 + apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
1.518 + also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
1.519 + finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
1.520 +
1.521 +lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
1.522 + "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
1.523 + shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
1.524 + apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
1.525 + using division_ofD[OF assms(2)] by auto
1.526 +
1.527 +lemma elementary_union_interval: assumes "p division_of \<Union>p"
1.528 + obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
1.529 + note assm=division_ofD[OF assms]
1.530 + have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
1.531 + have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
1.532 +{ presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
1.533 + "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
1.534 + thus thesis by auto
1.535 +next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
1.536 + thus thesis apply(rule_tac that[of p]) unfolding as by auto
1.537 +next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
1.538 +next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
1.539 + show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
1.540 + unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
1.541 + using assm(2-4) as apply- by(fastsimp dest: assm(5))+
1.542 +next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
1.543 + have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
1.544 + from assm(4)[OF this] guess c .. then guess d ..
1.545 + thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
1.546 + qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
1.547 + let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
1.548 + show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
1.549 + have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
1.550 + show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
1.551 + show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
1.552 + using q(6) by auto
1.553 + fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
1.554 + show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
1.555 + fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
1.556 + obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto
1.557 + obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
1.558 + show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
1.559 + case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
1.560 + next case False
1.561 + { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
1.562 + "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
1.563 + thus ?thesis by auto }
1.564 + { assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
1.565 + { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto }
1.566 + assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
1.567 + guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
1.568 + have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto
1.569 + hence "interior k \<subseteq> interior x" apply-
1.570 + apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
1.571 + guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
1.572 + have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
1.573 + hence "interior k' \<subseteq> interior x'" apply-
1.574 + apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
1.575 + ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
1.576 + qed qed } qed
1.577 +
1.578 +lemma elementary_unions_intervals:
1.579 + assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
1.580 + obtains p where "p division_of (\<Union>f)" proof-
1.581 + have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
1.582 + show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
1.583 + fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
1.584 + from this(3) guess p .. note p=this
1.585 + from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
1.586 + have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
1.587 + show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
1.588 + unfolding Union_insert ab * by auto
1.589 + qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
1.590 +
1.591 +lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
1.592 + obtains p where "p division_of (s \<union> t)"
1.593 +proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
1.594 + hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
1.595 + show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
1.596 + unfolding * prefer 3 apply(rule_tac p=p in that)
1.597 + using assms[unfolded division_of_def] by auto qed
1.598 +
1.599 +lemma partial_division_extend: fixes t::"(real^'n) set"
1.600 + assumes "p division_of s" "q division_of t" "s \<subseteq> t"
1.601 + obtains r where "p \<subseteq> r" "r division_of t" proof-
1.602 + note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
1.603 + obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
1.604 + guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
1.605 + apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)]
1.606 + guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
1.607 + then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
1.608 + apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
1.609 + { fix x assume x:"x\<in>t" "x\<notin>s"
1.610 + hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
1.611 + then guess r unfolding Union_iff .. note r=this moreover
1.612 + have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
1.613 + thus False using x by auto qed
1.614 + ultimately have "x\<in>\<Union>(r1 - p)" by auto }
1.615 + hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
1.616 + show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
1.617 + unfolding divp(6) apply(rule assms r2)+
1.618 + proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
1.619 + proof(rule inter_interior_unions_intervals)
1.620 + show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
1.621 + have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
1.622 + show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
1.623 + fix m x assume as:"m\<in>r1-p"
1.624 + have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
1.625 + show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
1.626 + show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
1.627 + qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
1.628 + qed qed
1.629 + thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
1.630 + qed auto qed
1.631 +
1.632 +subsection {* Tagged (partial) divisions. *}
1.633 +
1.634 +definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
1.635 + "(s tagged_partial_division_of i) \<equiv>
1.636 + finite s \<and>
1.637 + (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
1.638 + (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
1.639 + \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
1.640 +
1.641 +lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
1.642 + 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"
1.643 + "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
1.644 + "\<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) = {}"
1.645 + using assms unfolding tagged_partial_division_of_def apply- by blast+
1.646 +
1.647 +definition tagged_division_of (infixr "tagged'_division'_of" 40) where
1.648 + "(s tagged_division_of i) \<equiv>
1.649 + (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
1.650 +
1.651 +lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
1.652 + unfolding tagged_division_of_def tagged_partial_division_of_def by auto
1.653 +
1.654 +lemma tagged_division_of:
1.655 + "(s tagged_division_of i) \<longleftrightarrow>
1.656 + finite s \<and>
1.657 + (\<forall>x k. (x,k) \<in> s
1.658 + \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
1.659 + (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
1.660 + \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
1.661 + (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
1.662 + unfolding tagged_division_of_def tagged_partial_division_of_def by auto
1.663 +
1.664 +lemma tagged_division_ofI: assumes
1.665 + "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}"
1.666 + "\<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) = {})"
1.667 + "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
1.668 + shows "s tagged_division_of i"
1.669 + unfolding tagged_division_of apply(rule) defer apply rule
1.670 + apply(rule allI impI conjI assms)+ apply assumption
1.671 + apply(rule, rule assms, assumption) apply(rule assms, assumption)
1.672 + using assms(1,5-) apply- by blast+
1.673 +
1.674 +lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
1.675 + 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}"
1.676 + "\<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) = {})"
1.677 + "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
1.678 +
1.679 +lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i"
1.680 +proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
1.681 + show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
1.682 + fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
1.683 + thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
1.684 + fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
1.685 + thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
1.686 +qed
1.687 +
1.688 +lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
1.689 + shows "(snd ` s) division_of \<Union>(snd ` s)"
1.690 +proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
1.691 + show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
1.692 + fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
1.693 + thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
1.694 + fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
1.695 + thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
1.696 +qed
1.697 +
1.698 +lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
1.699 + shows "t tagged_partial_division_of i"
1.700 + using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
1.701 +
1.702 +lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
1.703 + assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
1.704 + shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
1.705 +proof- note assm=tagged_division_ofD[OF assms(1)]
1.706 + have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
1.707 + show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
1.708 + show "finite p" using assm by auto
1.709 + fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
1.710 + obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
1.711 + have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
1.712 + hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
1.713 + hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
1.714 + 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
1.715 + thus "d (snd x) = 0" unfolding ab by auto qed qed
1.716 +
1.717 +lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
1.718 +
1.719 +lemma tagged_division_of_empty: "{} tagged_division_of {}"
1.720 + unfolding tagged_division_of by auto
1.721 +
1.722 +lemma tagged_partial_division_of_trivial[simp]:
1.723 + "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
1.724 + unfolding tagged_partial_division_of_def by auto
1.725 +
1.726 +lemma tagged_division_of_trivial[simp]:
1.727 + "p tagged_division_of {} \<longleftrightarrow> p = {}"
1.728 + unfolding tagged_division_of by auto
1.729 +
1.730 +lemma tagged_division_of_self:
1.731 + "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
1.732 + apply(rule tagged_division_ofI) by auto
1.733 +
1.734 +lemma tagged_division_union:
1.735 + assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
1.736 + shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
1.737 +proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
1.738 + show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
1.739 + show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
1.740 + 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
1.741 + show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
1.742 + fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
1.743 + have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
1.744 + show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
1.745 + apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
1.746 + using p1(3) p2(3) using xk xk' by auto qed
1.747 +
1.748 +lemma tagged_division_unions:
1.749 + assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
1.750 + "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
1.751 + shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
1.752 +proof(rule tagged_division_ofI)
1.753 + note assm = tagged_division_ofD[OF assms(2)[rule_format]]
1.754 + show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
1.755 + 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
1.756 + also have "\<dots> = \<Union>iset" using assm(6) by auto
1.757 + finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" .
1.758 + 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
1.759 + 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
1.760 + 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
1.761 + 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)
1.762 + using assms(3)[rule_format] subset_interior by blast
1.763 + show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
1.764 + using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
1.765 +qed
1.766 +
1.767 +lemma tagged_partial_division_of_union_self:
1.768 + assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
1.769 + apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
1.770 +
1.771 +lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
1.772 + shows "p tagged_division_of (\<Union>(snd ` p))"
1.773 + apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
1.774 +
1.775 +subsection {* Fine-ness of a partition w.r.t. a gauge. *}
1.776 +
1.777 +definition fine (infixr "fine" 46) where
1.778 + "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
1.779 +
1.780 +lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
1.781 + shows "d fine s" using assms unfolding fine_def by auto
1.782 +
1.783 +lemma fineD[dest]: assumes "d fine s"
1.784 + shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
1.785 +
1.786 +lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
1.787 + unfolding fine_def by auto
1.788 +
1.789 +lemma fine_inters:
1.790 + "(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
1.791 + unfolding fine_def by blast
1.792 +
1.793 +lemma fine_union:
1.794 + "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
1.795 + unfolding fine_def by blast
1.796 +
1.797 +lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
1.798 + unfolding fine_def by auto
1.799 +
1.800 +lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
1.801 + unfolding fine_def by blast
1.802 +
1.803 +subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
1.804 +
1.805 +definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
1.806 + "(f has_integral_compact_interval y) i \<equiv>
1.807 + (\<forall>e>0. \<exists>d. gauge d \<and>
1.808 + (\<forall>p. p tagged_division_of i \<and> d fine p
1.809 + \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
1.810 +
1.811 +definition has_integral (infixr "has'_integral" 46) where
1.812 +"((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
1.813 + if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
1.814 + else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
1.815 + \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
1.816 + norm(z - y) < e))"
1.817 +
1.818 +lemma has_integral:
1.819 + "(f has_integral y) ({a..b}) \<longleftrightarrow>
1.820 + (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
1.821 + \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
1.822 + unfolding has_integral_def has_integral_compact_interval_def by auto
1.823 +
1.824 +lemma has_integralD[dest]: assumes
1.825 + "(f has_integral y) ({a..b})" "e>0"
1.826 + obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
1.827 + \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
1.828 + using assms unfolding has_integral by auto
1.829 +
1.830 +lemma has_integral_alt:
1.831 + "(f has_integral y) i \<longleftrightarrow>
1.832 + (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
1.833 + else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
1.834 + \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
1.835 + has_integral z) ({a..b}) \<and>
1.836 + norm(z - y) < e)))"
1.837 + unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
1.838 +
1.839 +lemma has_integral_altD:
1.840 + assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
1.841 + 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)"
1.842 + using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
1.843 +
1.844 +definition integrable_on (infixr "integrable'_on" 46) where
1.845 + "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
1.846 +
1.847 +definition "integral i f \<equiv> SOME y. (f has_integral y) i"
1.848 +
1.849 +lemma integrable_integral[dest]:
1.850 + "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
1.851 + unfolding integrable_on_def integral_def by(rule someI_ex)
1.852 +
1.853 +lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
1.854 + unfolding integrable_on_def by auto
1.855 +
1.856 +lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
1.857 + by auto
1.858 +
1.859 +lemma setsum_content_null:
1.860 + assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
1.861 + shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
1.862 +proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
1.863 + obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
1.864 + note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
1.865 + from this(2) guess c .. then guess d .. note c_d=this
1.866 + have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
1.867 + also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
1.868 + unfolding assms(1) c_d by auto
1.869 + finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
1.870 +qed
1.871 +
1.872 +subsection {* Some basic combining lemmas. *}
1.873 +
1.874 +lemma tagged_division_unions_exists:
1.875 + assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
1.876 + "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
1.877 + obtains p where "p tagged_division_of i" "d fine p"
1.878 +proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
1.879 + show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
1.880 + apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
1.881 + apply(rule fine_unions) using pfn by auto
1.882 +qed
1.883 +
1.884 +subsection {* The set we're concerned with must be closed. *}
1.885 +
1.886 +lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
1.887 + unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
1.888 +
1.889 +subsection {* General bisection principle for intervals; might be useful elsewhere. *}
1.890 +
1.891 +lemma interval_bisection_step:
1.892 + 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})"
1.893 + obtains c d where "~(P{c..d})"
1.894 + "\<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"
1.895 +proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
1.896 + note ab=this[unfolded interval_eq_empty not_ex not_less]
1.897 + { fix f have "finite f \<Longrightarrow>
1.898 + (\<forall>s\<in>f. P s) \<Longrightarrow>
1.899 + (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
1.900 + (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
1.901 + proof(induct f rule:finite_induct)
1.902 + case empty show ?case using assms(1) by auto
1.903 + next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
1.904 + apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
1.905 + using insert by auto
1.906 + qed } note * = this
1.907 + 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)}"
1.908 + 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"
1.909 + { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
1.910 + thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
1.911 + assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
1.912 + have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
1.913 + let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
1.914 + (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
1.915 + have "?A \<subseteq> ?B" proof case goal1
1.916 + then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
1.917 + have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
1.918 + show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
1.919 + unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
1.920 + 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)"
1.921 + "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
1.922 + using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
1.923 + qed auto qed
1.924 + thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
1.925 + fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
1.926 + note c_d=this[rule_format]
1.927 + show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case
1.928 + using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
1.929 + show "\<exists>a b. s = {a..b}" unfolding c_d by auto
1.930 + fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
1.931 + note e_f=this[rule_format]
1.932 + assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
1.933 + then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
1.934 + hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
1.935 + 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
1.936 + 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
1.937 + qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
1.938 + show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
1.939 + fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
1.940 + 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
1.941 + show False using c_d(2)[of i] apply- apply(erule_tac disjE)
1.942 + proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
1.943 + show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
1.944 + next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
1.945 + show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
1.946 + qed qed qed
1.947 + also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
1.948 + fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
1.949 + from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
1.950 + note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
1.951 + show "x\<in>{a..b}" unfolding mem_interval proof
1.952 + fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
1.953 + using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
1.954 + next fix x assume x:"x\<in>{a..b}"
1.955 + 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"
1.956 + (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
1.957 + have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
1.958 + using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
1.959 + qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
1.960 + apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
1.961 + qed finally show False using assms by auto qed
1.962 +
1.963 +lemma interval_bisection:
1.964 + assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
1.965 + obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
1.966 +proof-
1.967 + have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
1.968 + 2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
1.969 + presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
1.970 + thus ?thesis apply(cases "P {fst x..snd x}") by auto
1.971 + next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
1.972 + thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
1.973 + qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
1.974 + def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
1.975 + have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
1.976 + (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and>
1.977 + 2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
1.978 + proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
1.979 + case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
1.980 + proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
1.981 + next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
1.982 + qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
1.983 +
1.984 + have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
1.985 + proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
1.986 + show ?case apply(rule_tac x=n in exI) proof(rule,rule)
1.987 + fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
1.988 + have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
1.989 + also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
1.990 + proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
1.991 + using xy[unfolded mem_interval,THEN spec[where x=i]]
1.992 + unfolding vector_minus_component by auto qed
1.993 + also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
1.994 + proof(rule setsum_mono) case goal1 thus ?case
1.995 + proof(induct n) case 0 thus ?case unfolding AB by auto
1.996 + 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
1.997 + also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
1.998 + qed qed
1.999 + also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
1.1000 + qed qed
1.1001 + { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
1.1002 + have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
1.1003 + proof(induct d) case 0 thus ?case by auto
1.1004 + next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
1.1005 + apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
1.1006 + proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
1.1007 + qed qed } note ABsubset = this
1.1008 + have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
1.1009 + 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
1.1010 + then guess x0 .. note x0=this[rule_format]
1.1011 + show thesis proof(rule that[rule_format,of x0])
1.1012 + show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
1.1013 + fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
1.1014 + 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}"
1.1015 + apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
1.1016 + 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
1.1017 + show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
1.1018 + show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
1.1019 + qed qed qed
1.1020 +
1.1021 +subsection {* Cousin's lemma. *}
1.1022 +
1.1023 +lemma fine_division_exists: assumes "gauge g"
1.1024 + obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
1.1025 +proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
1.1026 + then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
1.1027 +next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
1.1028 + guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
1.1029 + apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
1.1030 + proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
1.1031 + 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 = {}"
1.1032 + 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
1.1033 + apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
1.1034 + qed note x=this
1.1035 + obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
1.1036 + from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
1.1037 + have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
1.1038 + thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
1.1039 +
1.1040 +subsection {* Basic theorems about integrals. *}
1.1041 +
1.1042 +lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.1043 + assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
1.1044 +proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
1.1045 + have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> a b k1 k2.
1.1046 + (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
1.1047 + proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
1.1048 + guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
1.1049 + guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
1.1050 + guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
1.1051 + let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
1.1052 + using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
1.1053 + also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
1.1054 + apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
1.1055 + finally show False by auto
1.1056 + qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
1.1057 + thus False apply-apply(cases "\<exists>a b. i = {a..b}")
1.1058 + using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
1.1059 + assume as:"\<not> (\<exists>a b. i = {a..b})"
1.1060 + guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
1.1061 + guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
1.1062 + have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
1.1063 + using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
1.1064 + note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
1.1065 + guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
1.1066 + guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
1.1067 + have "z = w" using lem[OF w(1) z(1)] by auto
1.1068 + hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
1.1069 + using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
1.1070 + also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
1.1071 + finally show False by auto qed
1.1072 +
1.1073 +lemma integral_unique[intro]:
1.1074 + "(f has_integral y) k \<Longrightarrow> integral k f = y"
1.1075 + unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
1.1076 +
1.1077 +lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.1078 + assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
1.1079 +proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
1.1080 + (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
1.1081 + proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
1.1082 + assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
1.1083 + 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)"
1.1084 + apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
1.1085 + proof(rule,rule,erule conjE) case goal1
1.1086 + have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
1.1087 + 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
1.1088 + thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
1.1089 + qed thus ?case using as by auto
1.1090 + qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
1.1091 + thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
1.1092 + using assms by(auto simp add:has_integral intro:lem) }
1.1093 + have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
1.1094 + assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
1.1095 + apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
1.1096 + proof- fix e::real and a b assume "e>0"
1.1097 + thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
1.1098 + apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
1.1099 + qed auto qed
1.1100 +
1.1101 +lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
1.1102 + apply(rule has_integral_is_0) by auto
1.1103 +
1.1104 +lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
1.1105 + using has_integral_unique[OF has_integral_0] by auto
1.1106 +
1.1107 +lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.1108 + assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
1.1109 +proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
1.1110 + have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
1.1111 + (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
1.1112 + proof(subst has_integral,rule,rule) case goal1
1.1113 + from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
1.1114 + have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
1.1115 + guess g using has_integralD[OF goal1(1) *] . note g=this
1.1116 + show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
1.1117 + proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
1.1118 + 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
1.1119 + 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"
1.1120 + unfolding o_def unfolding scaleR[THEN sym] * by simp
1.1121 + 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
1.1122 + 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)" .
1.1123 + show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
1.1124 + apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
1.1125 + qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
1.1126 + thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
1.1127 + assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
1.1128 + proof(rule,rule) fix e::real assume e:"0<e"
1.1129 + have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
1.1130 + guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
1.1131 + 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)"
1.1132 + apply(rule_tac x=M in exI) apply(rule,rule M(1))
1.1133 + proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
1.1134 + 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)"
1.1135 + unfolding o_def apply(rule ext) using zero by auto
1.1136 + show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
1.1137 + apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
1.1138 + qed qed qed
1.1139 +
1.1140 +lemma has_integral_cmul:
1.1141 + shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
1.1142 + unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
1.1143 + by(rule scaleR.bounded_linear_right)
1.1144 +
1.1145 +lemma has_integral_neg:
1.1146 + shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
1.1147 + apply(drule_tac c="-1" in has_integral_cmul) by auto
1.1148 +
1.1149 +lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.1150 + assumes "(f has_integral k) s" "(g has_integral l) s"
1.1151 + shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
1.1152 +proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
1.1153 + (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
1.1154 + ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
1.1155 + show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
1.1156 + guess d1 using has_integralD[OF goal1(1) *] . note d1=this
1.1157 + guess d2 using has_integralD[OF goal1(2) *] . note d2=this
1.1158 + 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)"
1.1159 + apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
1.1160 + proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
1.1161 + 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)"
1.1162 + 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]
1.1163 + by(rule setsum_cong2,auto)
1.1164 + 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))"
1.1165 + unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
1.1166 + from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
1.1167 + have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
1.1168 + apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
1.1169 + finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
1.1170 + qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
1.1171 + thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
1.1172 + assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
1.1173 + proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
1.1174 + from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
1.1175 + from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
1.1176 + show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
1.1177 + proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
1.1178 + hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
1.1179 + guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
1.1180 + guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
1.1181 + 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
1.1182 + 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"
1.1183 + apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
1.1184 + using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
1.1185 + qed qed qed
1.1186 +
1.1187 +lemma has_integral_sub:
1.1188 + shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
1.1189 + using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
1.1190 +
1.1191 +lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
1.1192 + by(rule integral_unique has_integral_0)+
1.1193 +
1.1194 +lemma integral_add:
1.1195 + shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
1.1196 + integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
1.1197 + apply(rule integral_unique) apply(drule integrable_integral)+
1.1198 + apply(rule has_integral_add) by assumption+
1.1199 +
1.1200 +lemma integral_cmul:
1.1201 + shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
1.1202 + apply(rule integral_unique) apply(drule integrable_integral)+
1.1203 + apply(rule has_integral_cmul) by assumption+
1.1204 +
1.1205 +lemma integral_neg:
1.1206 + shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
1.1207 + apply(rule integral_unique) apply(drule integrable_integral)+
1.1208 + apply(rule has_integral_neg) by assumption+
1.1209 +
1.1210 +lemma integral_sub:
1.1211 + 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"
1.1212 + apply(rule integral_unique) apply(drule integrable_integral)+
1.1213 + apply(rule has_integral_sub) by assumption+
1.1214 +
1.1215 +lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
1.1216 + unfolding integrable_on_def using has_integral_0 by auto
1.1217 +
1.1218 +lemma integrable_add:
1.1219 + shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
1.1220 + unfolding integrable_on_def by(auto intro: has_integral_add)
1.1221 +
1.1222 +lemma integrable_cmul:
1.1223 + shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
1.1224 + unfolding integrable_on_def by(auto intro: has_integral_cmul)
1.1225 +
1.1226 +lemma integrable_neg:
1.1227 + shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
1.1228 + unfolding integrable_on_def by(auto intro: has_integral_neg)
1.1229 +
1.1230 +lemma integrable_sub:
1.1231 + shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
1.1232 + unfolding integrable_on_def by(auto intro: has_integral_sub)
1.1233 +
1.1234 +lemma integrable_linear:
1.1235 + shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
1.1236 + unfolding integrable_on_def by(auto intro: has_integral_linear)
1.1237 +
1.1238 +lemma integral_linear:
1.1239 + shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
1.1240 + apply(rule has_integral_unique) defer unfolding has_integral_integral
1.1241 + apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
1.1242 + apply(rule integrable_linear) by assumption+
1.1243 +
1.1244 +lemma has_integral_setsum:
1.1245 + assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
1.1246 + shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
1.1247 +proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
1.1248 + case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
1.1249 + apply(rule has_integral_add) using insert assms by auto
1.1250 +qed auto
1.1251 +
1.1252 +lemma integral_setsum:
1.1253 + shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
1.1254 + integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
1.1255 + apply(rule integral_unique) apply(rule has_integral_setsum)
1.1256 + using integrable_integral by auto
1.1257 +
1.1258 +lemma integrable_setsum:
1.1259 + 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"
1.1260 + unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
1.1261 +
1.1262 +lemma has_integral_eq:
1.1263 + assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
1.1264 + using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
1.1265 + using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
1.1266 +
1.1267 +lemma integrable_eq:
1.1268 + shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
1.1269 + unfolding integrable_on_def using has_integral_eq[of s f g] by auto
1.1270 +
1.1271 +lemma has_integral_eq_eq:
1.1272 + shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
1.1273 + using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto
1.1274 +
1.1275 +lemma has_integral_null[dest]:
1.1276 + assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})"
1.1277 + unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
1.1278 +proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
1.1279 + fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
1.1280 + have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
1.1281 + using setsum_content_null[OF assms(1) p, of f] .
1.1282 + thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
1.1283 +
1.1284 +lemma has_integral_null_eq[simp]:
1.1285 + shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
1.1286 + apply rule apply(rule has_integral_unique,assumption)
1.1287 + apply(drule has_integral_null,assumption)
1.1288 + apply(drule has_integral_null) by auto
1.1289 +
1.1290 +lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
1.1291 + by(rule integral_unique,drule has_integral_null)
1.1292 +
1.1293 +lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
1.1294 + unfolding integrable_on_def apply(drule has_integral_null) by auto
1.1295 +
1.1296 +lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
1.1297 + unfolding empty_as_interval apply(rule has_integral_null)
1.1298 + using content_empty unfolding empty_as_interval .
1.1299 +
1.1300 +lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
1.1301 + apply(rule,rule has_integral_unique,assumption) by auto
1.1302 +
1.1303 +lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
1.1304 +
1.1305 +lemma integral_empty[simp]: shows "integral {} f = 0"
1.1306 + apply(rule integral_unique) using has_integral_empty .
1.1307 +
1.1308 +lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"
1.1309 + apply(rule has_integral_null) unfolding content_eq_0_interior
1.1310 + unfolding interior_closed_interval using interval_sing by auto
1.1311 +
1.1312 +lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
1.1313 +
1.1314 +lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
1.1315 +
1.1316 +subsection {* Cauchy-type criterion for integrability. *}
1.1317 +
1.1318 +lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}"
1.1319 + shows "f integrable_on {a..b} \<longleftrightarrow>
1.1320 + (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
1.1321 + p2 tagged_division_of {a..b} \<and> d fine p2
1.1322 + \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
1.1323 + setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
1.1324 +proof assume ?l
1.1325 + then guess y unfolding integrable_on_def has_integral .. note y=this
1.1326 + show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
1.1327 + then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
1.1328 + show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
1.1329 + proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
1.1330 + 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"
1.1331 + apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
1.1332 + using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
1.1333 + qed qed
1.1334 +next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
1.1335 + from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
1.1336 + have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
1.1337 + 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-
1.1338 + proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
1.1339 + from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
1.1340 + have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
1.1341 + have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
1.1342 + proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
1.1343 + show ?case apply(rule_tac x=N in exI)
1.1344 + 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
1.1345 + 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"
1.1346 + apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
1.1347 + using dp p(1) using mn by auto
1.1348 + qed qed
1.1349 + then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
1.1350 + show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
1.1351 + proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
1.1352 + then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
1.1353 + guess N2 using y[OF *] .. note N2=this
1.1354 + 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)"
1.1355 + apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
1.1356 + proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
1.1357 + fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
1.1358 + have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
1.1359 + show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
1.1360 + apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
1.1361 + using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
1.1362 + using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
1.1363 +
1.1364 +subsection {* Additivity of integral on abutting intervals. *}
1.1365 +
1.1366 +lemma interval_split:
1.1367 + "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
1.1368 + "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
1.1369 + apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
1.1370 + unfolding Cart_lambda_beta by auto
1.1371 +
1.1372 +lemma content_split:
1.1373 + "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
1.1374 +proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
1.1375 + { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
1.1376 + have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
1.1377 + 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))"
1.1378 + "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)"
1.1379 + apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
1.1380 + assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
1.1381 + \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
1.1382 + by (auto simp add:field_simps)
1.1383 + moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
1.1384 + unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
1.1385 + ultimately show ?thesis
1.1386 + unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
1.1387 +qed
1.1388 +
1.1389 +lemma division_split_left_inj:
1.1390 + assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
1.1391 + "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
1.1392 + shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
1.1393 +proof- note d=division_ofD[OF assms(1)]
1.1394 + have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
1.1395 + unfolding interval_split content_eq_0_interior by auto
1.1396 + guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
1.1397 + guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
1.1398 + have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
1.1399 + show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
1.1400 + defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
1.1401 +
1.1402 +lemma division_split_right_inj:
1.1403 + assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2"
1.1404 + "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
1.1405 + shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
1.1406 +proof- note d=division_ofD[OF assms(1)]
1.1407 + 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}) = {})"
1.1408 + unfolding interval_split content_eq_0_interior by auto
1.1409 + guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
1.1410 + guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
1.1411 + have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
1.1412 + show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
1.1413 + defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
1.1414 +
1.1415 +lemma tagged_division_split_left_inj:
1.1416 + 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}"
1.1417 + shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
1.1418 +proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
1.1419 + show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
1.1420 + apply(rule_tac[1-2] *) using assms(2-) by auto qed
1.1421 +
1.1422 +lemma tagged_division_split_right_inj:
1.1423 + assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
1.1424 + shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
1.1425 +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
1.1426 + show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
1.1427 + apply(rule_tac[1-2] *) using assms(2-) by auto qed
1.1428 +
1.1429 +lemma division_split:
1.1430 + assumes "p division_of {a..b::real^'n}"
1.1431 + 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$k \<le> c})" (is "?p1 division_of ?I1") and
1.1432 + "{l \<inter> {x. x$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")
1.1433 +proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
1.1434 + show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
1.1435 + { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
1.1436 + guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
1.1437 + show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
1.1438 + using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
1.1439 + fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
1.1440 + assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
1.1441 + { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
1.1442 + guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
1.1443 + show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
1.1444 + using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
1.1445 + fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
1.1446 + assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
1.1447 +qed
1.1448 +
1.1449 +lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.1450 + 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})"
1.1451 + shows "(f has_integral (i + j)) ({a..b})"
1.1452 +proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
1.1453 + guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
1.1454 + guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
1.1455 + 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"
1.1456 + show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
1.1457 + proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
1.1458 + fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
1.1459 + have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
1.1460 + "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
1.1461 + proof- fix x kk assume as:"(x,kk)\<in>p"
1.1462 + show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
1.1463 + proof(rule ccontr) case goal1
1.1464 + from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
1.1465 + using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
1.1466 + hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast
1.1467 + then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
1.1468 + using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
1.1469 + thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
1.1470 + qed
1.1471 + show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
1.1472 + proof(rule ccontr) case goal1
1.1473 + from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
1.1474 + using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
1.1475 + hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast
1.1476 + then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)
1.1477 + using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
1.1478 + thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
1.1479 + qed
1.1480 + qed
1.1481 +
1.1482 + have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
1.1483 + have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
1.1484 + proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
1.1485 + have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
1.1486 + setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
1.1487 + = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
1.1488 + apply(rule setsum_mono_zero_left) prefer 3
1.1489 + proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
1.1490 + assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
1.1491 + then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
1.1492 + have "content (g k) = 0" using xk using content_empty by auto
1.1493 + thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
1.1494 + qed auto
1.1495 + have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
1.1496 +
1.1497 + let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
1.1498 + have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
1.1499 + apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
1.1500 + proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
1.1501 + fix x l assume xl:"(x,l)\<in>?M1"
1.1502 + then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
1.1503 + have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
1.1504 + thus "l \<subseteq> d1 x" unfolding xl' by auto
1.1505 + 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)
1.1506 + using lem0(1)[OF xl'(3-4)] by auto
1.1507 + 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])
1.1508 + fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
1.1509 + then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
1.1510 + assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
1.1511 + proof(cases "l' = r' \<longrightarrow> x' = y'")
1.1512 + case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
1.1513 + next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
1.1514 + thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
1.1515 + qed qed moreover
1.1516 +
1.1517 + 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> {}}"
1.1518 + have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
1.1519 + apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
1.1520 + 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
1.1521 + fix x l assume xl:"(x,l)\<in>?M2"
1.1522 + then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this
1.1523 + have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
1.1524 + thus "l \<subseteq> d2 x" unfolding xl' by auto
1.1525 + 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)
1.1526 + using lem0(2)[OF xl'(3-4)] by auto
1.1527 + 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])
1.1528 + fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
1.1529 + then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this
1.1530 + assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
1.1531 + proof(cases "l' = r' \<longrightarrow> x' = y'")
1.1532 + case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
1.1533 + next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
1.1534 + thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
1.1535 + qed qed ultimately
1.1536 +
1.1537 + 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"
1.1538 + apply- apply(rule norm_triangle_lt) by auto
1.1539 + also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
1.1540 + 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)
1.1541 + = (\<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
1.1542 + 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)"
1.1543 + unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
1.1544 + defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
1.1545 + proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
1.1546 + next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
1.1547 + qed also note setsum_addf[THEN sym]
1.1548 + 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
1.1549 + = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
1.1550 + proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
1.1551 + 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"
1.1552 + unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
1.1553 + qed note setsum_cong2[OF this]
1.1554 + 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 +
1.1555 + ((\<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) =
1.1556 + (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
1.1557 + finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
1.1558 +
1.1559 +subsection {* A sort of converse, integrability on subintervals. *}
1.1560 +
1.1561 +lemma tagged_division_union_interval:
1.1562 + 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})"
1.1563 + shows "(p1 \<union> p2) tagged_division_of ({a..b})"
1.1564 +proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
1.1565 + show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
1.1566 + unfolding interval_split interior_closed_interval
1.1567 + by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
1.1568 +
1.1569 +lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
1.1570 + assumes "(f has_integral i) ({a..b})" "e>0"
1.1571 + obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
1.1572 + p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
1.1573 + \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
1.1574 + setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
1.1575 +proof- guess d using has_integralD[OF assms] . note d=this
1.1576 + show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
1.1577 + 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
1.1578 + 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
1.1579 + note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
1.1580 + 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)"
1.1581 + apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
1.1582 + proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
1.1583 + have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
1.1584 + have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
1.1585 + moreover have "interior {x. x $ k = c} = {}"
1.1586 + proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
1.1587 + then guess e unfolding mem_interior .. note e=this
1.1588 + have x:"x$k = c" using x interior_subset by fastsimp
1.1589 + 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
1.1590 + have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm
1.1591 + apply(rule le_less_trans[OF norm_le_l1]) unfolding *
1.1592 + unfolding setsum_delta[OF finite_UNIV] using e by auto
1.1593 + hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
1.1594 + thus False unfolding mem_Collect_eq using e x by auto
1.1595 + qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
1.1596 + thus "content b *\<^sub>R f a = 0" by auto
1.1597 + qed auto
1.1598 + also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
1.1599 + finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
1.1600 +
1.1601 +lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
1.1602 + shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2)
1.1603 +proof- guess y using assms unfolding integrable_on_def .. note y=this
1.1604 + def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
1.1605 + and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
1.1606 + show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
1.1607 + proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
1.1608 + from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
1.1609 + 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>
1.1610 + norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
1.1611 + show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
1.1612 + 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"
1.1613 + 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"
1.1614 + proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
1.1615 + show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
1.1616 + using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
1.1617 + using p using assms by(auto simp add:group_simps)
1.1618 + qed qed
1.1619 + show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
1.1620 + 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"
1.1621 + 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"
1.1622 + proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
1.1623 + show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
1.1624 + using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
1.1625 + using p using assms by(auto simp add:group_simps) qed qed qed qed
1.1626 +
1.1627 +subsection {* Generalized notion of additivity. *}
1.1628 +
1.1629 +definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
1.1630 +
1.1631 +definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
1.1632 + "operative opp f \<equiv>
1.1633 + (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
1.1634 + (\<forall>a b c k. f({a..b}) =
1.1635 + opp (f({a..b} \<inter> {x. x$k \<le> c}))
1.1636 + (f({a..b} \<inter> {x. x$k \<ge> c})))"
1.1637 +
1.1638 +lemma operativeD[dest]: assumes "operative opp f"
1.1639 + shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
1.1640 + "\<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}))"
1.1641 + using assms unfolding operative_def by auto
1.1642 +
1.1643 +lemma operative_trivial:
1.1644 + "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
1.1645 + unfolding operative_def by auto
1.1646 +
1.1647 +lemma property_empty_interval:
1.1648 + "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
1.1649 + using content_empty unfolding empty_as_interval by auto
1.1650 +
1.1651 +lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
1.1652 + unfolding operative_def apply(rule property_empty_interval) by auto
1.1653 +
1.1654 +subsection {* Using additivity of lifted function to encode definedness. *}
1.1655 +
1.1656 +lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
1.1657 + by (metis map_of.simps option.nchotomy)
1.1658 +
1.1659 +lemma exists_option:
1.1660 + "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
1.1661 + by (metis map_of.simps option.nchotomy)
1.1662 +
1.1663 +fun lifted where
1.1664 + "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
1.1665 + "lifted opp None _ = (None::'b option)" |
1.1666 + "lifted opp _ None = None"
1.1667 +
1.1668 +lemma lifted_simp_1[simp]: "lifted opp v None = None"
1.1669 + apply(induct v) by auto
1.1670 +
1.1671 +definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and>
1.1672 + (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
1.1673 + (\<forall>x. opp (neutral opp) x = x)"
1.1674 +
1.1675 +lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
1.1676 + "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
1.1677 + "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
1.1678 + unfolding monoidal_def using assms by fastsimp
1.1679 +
1.1680 +lemma monoidal_ac: assumes "monoidal opp"
1.1681 + shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
1.1682 + "opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)"
1.1683 + using assms unfolding monoidal_def apply- by metis+
1.1684 +
1.1685 +lemma monoidal_simps[simp]: assumes "monoidal opp"
1.1686 + shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
1.1687 + using monoidal_ac[OF assms] by auto
1.1688 +
1.1689 +lemma neutral_lifted[cong]: assumes "monoidal opp"
1.1690 + shows "neutral (lifted opp) = Some(neutral opp)"
1.1691 + apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
1.1692 +proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
1.1693 + thus "x = Some (neutral opp)" apply(induct x) defer
1.1694 + apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
1.1695 + apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
1.1696 +qed(auto simp add:monoidal_ac[OF assms])
1.1697 +
1.1698 +lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
1.1699 + unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
1.1700 +
1.1701 +definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
1.1702 +definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
1.1703 +definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
1.1704 +
1.1705 +lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
1.1706 +lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
1.1707 +
1.1708 +lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
1.1709 + unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
1.1710 +
1.1711 +lemma support_clauses:
1.1712 + "\<And>f g s. support opp f {} = {}"
1.1713 + "\<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))"
1.1714 + "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
1.1715 + "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
1.1716 + "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
1.1717 + "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
1.1718 + "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
1.1719 +unfolding support_def by auto
1.1720 +
1.1721 +lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
1.1722 + unfolding support_def by auto
1.1723 +
1.1724 +lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
1.1725 + unfolding iterate_def fold'_def by auto
1.1726 +
1.1727 +lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
1.1728 + shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
1.1729 +proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
1.1730 + show ?thesis unfolding iterate_def if_P[OF True] * by auto
1.1731 +next case False note x=this
1.1732 + note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
1.1733 + show ?thesis proof(cases "f x = neutral opp")
1.1734 + case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
1.1735 + unfolding True monoidal_simps[OF assms(1)] by auto
1.1736 + next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False]
1.1737 + apply(subst fun_left_comm.fold_insert[OF * finite_support])
1.1738 + using `finite s` unfolding support_def using False x by auto qed qed
1.1739 +
1.1740 +lemma iterate_some:
1.1741 + assumes "monoidal opp" "finite s"
1.1742 + shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
1.1743 +proof(induct s) case empty thus ?case using assms by auto
1.1744 +next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
1.1745 + defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
1.1746 +
1.1747 +subsection {* Two key instances of additivity. *}
1.1748 +
1.1749 +lemma neutral_add[simp]:
1.1750 + "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
1.1751 + apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
1.1752 +
1.1753 +lemma operative_content[intro]: "operative (op +) content"
1.1754 + unfolding operative_def content_split[THEN sym] neutral_add by auto
1.1755 +
1.1756 +lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
1.1757 + unfolding neutral_def apply(rule some_equality) defer
1.1758 + apply(erule_tac x=0 in allE) by auto
1.1759 +
1.1760 +lemma monoidal_monoid[intro]:
1.1761 + shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
1.1762 + unfolding monoidal_def neutral_monoid by(auto simp add: group_simps)
1.1763 +
1.1764 +lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.1765 + shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
1.1766 + unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
1.1767 + apply(rule,rule,rule,rule) defer apply(rule allI)+
1.1768 +proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
1.1769 + 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)
1.1770 + (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)"
1.1771 + proof(cases "f integrable_on {a..b}")
1.1772 + case True show ?thesis unfolding if_P[OF True]
1.1773 + unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
1.1774 + unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split)
1.1775 + apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
1.1776 + 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}))"
1.1777 + proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
1.1778 + 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)
1.1779 + apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
1.1780 + thus False using False by auto
1.1781 + qed thus ?thesis using False by auto
1.1782 + qed next
1.1783 + fix a b assume as:"content {a..b::real^'n} = 0"
1.1784 + thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
1.1785 + unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
1.1786 +
1.1787 +subsection {* Points of division of a partition. *}
1.1788 +
1.1789 +definition "division_points (k::(real^'n) set) d =
1.1790 + {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
1.1791 + (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
1.1792 +
1.1793 +lemma division_points_finite: assumes "d division_of i"
1.1794 + shows "finite (division_points i d)"
1.1795 +proof- note assm = division_ofD[OF assms]
1.1796 + let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
1.1797 + (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
1.1798 + have *:"division_points i d = \<Union>(?M ` UNIV)"
1.1799 + unfolding division_points_def by auto
1.1800 + show ?thesis unfolding * using assm by auto qed
1.1801 +
1.1802 +lemma division_points_subset:
1.1803 + assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
1.1804 + 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$k \<le> c} = {})}
1.1805 + \<subseteq> division_points ({a..b}) d" (is ?t1) and
1.1806 + "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
1.1807 + \<subseteq> division_points ({a..b}) d" (is ?t2)
1.1808 +proof- note assm = division_ofD[OF assms(1)]
1.1809 + 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"
1.1810 + "\<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"
1.1811 + using assms using less_imp_le by auto
1.1812 + show ?t1 unfolding division_points_def interval_split[of a b]
1.1813 + unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
1.1814 + unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
1.1815 + proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
1.1816 + "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> {}"
1.1817 + from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
1.1818 + 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 .
1.1819 + have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
1.1820 + 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)"
1.1821 + using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
1.1822 + apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
1.1823 + apply(case_tac[!] "fst x = k") using assms by auto
1.1824 + qed
1.1825 + show ?t2 unfolding division_points_def interval_split[of a b]
1.1826 + unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
1.1827 + unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
1.1828 + 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"
1.1829 + "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
1.1830 + from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
1.1831 + 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 .
1.1832 + have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
1.1833 + 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)"
1.1834 + using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
1.1835 + apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
1.1836 + apply(case_tac[!] "fst x = k") using assms by auto qed qed
1.1837 +
1.1838 +lemma division_points_psubset:
1.1839 + assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k"
1.1840 + "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
1.1841 + 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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
1.1842 + "division_points ({a..b} \<inter> {x. x$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")
1.1843 +proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
1.1844 + guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
1.1845 + 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
1.1846 + unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
1.1847 + have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
1.1848 + "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
1.1849 + unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
1.1850 + unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
1.1851 + have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
1.1852 + apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
1.1853 + apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
1.1854 + unfolding division_points_def unfolding interval_bounds[OF ab]
1.1855 + apply (auto simp add:interval_bounds) unfolding * by auto
1.1856 + thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
1.1857 +
1.1858 + have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
1.1859 + "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
1.1860 + unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
1.1861 + unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
1.1862 + have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
1.1863 + apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
1.1864 + apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
1.1865 + unfolding division_points_def unfolding interval_bounds[OF ab]
1.1866 + apply (auto simp add:interval_bounds) unfolding * by auto
1.1867 + thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
1.1868 +
1.1869 +subsection {* Preservation by divisions and tagged divisions. *}
1.1870 +
1.1871 +lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
1.1872 + unfolding support_def by auto
1.1873 +
1.1874 +lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
1.1875 + unfolding iterate_def support_support by auto
1.1876 +
1.1877 +lemma iterate_expand_cases:
1.1878 + "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
1.1879 + apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
1.1880 +
1.1881 +lemma iterate_image: assumes "monoidal opp" "inj_on f s"
1.1882 + shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
1.1883 +proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
1.1884 + iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
1.1885 + proof- case goal1 show ?case using goal1
1.1886 + proof(induct s) case empty thus ?case using assms(1) by auto
1.1887 + next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
1.1888 + unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
1.1889 + unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
1.1890 + apply(rule finite_imageI insert)+ apply(subst if_not_P)
1.1891 + unfolding image_iff o_def using insert(2,4) by auto
1.1892 + qed qed
1.1893 + show ?thesis
1.1894 + apply(cases "finite (support opp g (f ` s))")
1.1895 + apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
1.1896 + unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
1.1897 + apply(rule subset_inj_on[OF assms(2) support_subset])+
1.1898 + apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
1.1899 + apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
1.1900 +
1.1901 +
1.1902 +(* This lemma about iterations comes up in a few places. *)
1.1903 +lemma iterate_nonzero_image_lemma:
1.1904 + assumes "monoidal opp" "finite s" "g(a) = neutral opp"
1.1905 + "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
1.1906 + shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
1.1907 +proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
1.1908 + have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
1.1909 + unfolding support_def using assms(3) by auto
1.1910 + show ?thesis unfolding *
1.1911 + apply(subst iterate_support[THEN sym]) unfolding support_clauses
1.1912 + apply(subst iterate_image[OF assms(1)]) defer
1.1913 + apply(subst(2) iterate_support[THEN sym]) apply(subst **)
1.1914 + unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
1.1915 +
1.1916 +lemma iterate_eq_neutral:
1.1917 + assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)"
1.1918 + shows "(iterate opp s f = neutral opp)"
1.1919 +proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
1.1920 + show ?thesis apply(subst iterate_support[THEN sym])
1.1921 + unfolding * using assms(1) by auto qed
1.1922 +
1.1923 +lemma iterate_op: assumes "monoidal opp" "finite s"
1.1924 + shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
1.1925 +proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
1.1926 +next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
1.1927 + unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
1.1928 +
1.1929 +lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
1.1930 + shows "iterate opp s f = iterate opp s g"
1.1931 +proof- have *:"support opp g s = support opp f s"
1.1932 + unfolding support_def using assms(2) by auto
1.1933 + show ?thesis
1.1934 + proof(cases "finite (support opp f s)")
1.1935 + case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
1.1936 + unfolding * by auto
1.1937 + next def su \<equiv> "support opp f s"
1.1938 + case True note support_subset[of opp f s]
1.1939 + thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
1.1940 + unfolding su_def[symmetric]
1.1941 + proof(induct su) case empty show ?case by auto
1.1942 + next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
1.1943 + unfolding if_not_P[OF insert(2)] apply(subst insert(3))
1.1944 + defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
1.1945 +
1.1946 +lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
1.1947 +
1.1948 +lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
1.1949 + assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
1.1950 + shows "iterate opp d f = f {a..b}"
1.1951 +proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
1.1952 + proof(induct C arbitrary:a b d rule:full_nat_induct)
1.1953 + case goal1
1.1954 + { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
1.1955 + thus ?case apply-apply(cases) defer apply assumption
1.1956 + proof- assume as:"content {a..b} = 0"
1.1957 + show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
1.1958 + proof fix x assume x:"x\<in>d"
1.1959 + then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
1.1960 + thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
1.1961 + using operativeD(1)[OF assms(2)] x by auto
1.1962 + qed qed }
1.1963 + assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
1.1964 + hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case
1.1965 + proof(cases "division_points {a..b} d = {}")
1.1966 + case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
1.1967 + (\<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)"
1.1968 + unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
1.1969 + apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
1.1970 + proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
1.1971 + hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
1.1972 + 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
1.1973 + have "(j, u$j) \<notin> division_points {a..b} d"
1.1974 + "(j, v$j) \<notin> division_points {a..b} d" using True by auto
1.1975 + note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
1.1976 + note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
1.1977 + moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as]
1.1978 + unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
1.1979 + unfolding interval_ne_empty mem_interval by auto
1.1980 + 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"
1.1981 + unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
1.1982 + qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
1.1983 + note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
1.1984 + then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
1.1985 + have "{a..b} \<in> d"
1.1986 + proof- { presume "i = {a..b}" thus ?thesis using i by auto }
1.1987 + { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
1.1988 + show "u = a" "v = b" unfolding Cart_eq
1.1989 + proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
1.1990 + thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
1.1991 + qed qed
1.1992 + hence *:"d = insert {a..b} (d - {{a..b}})" by auto
1.1993 + have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
1.1994 + proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
1.1995 + then guess u v apply-by(erule exE conjE)+ note uv=this
1.1996 + have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
1.1997 + then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
1.1998 + hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
1.1999 + hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
1.2000 + thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
1.2001 + qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
1.2002 + apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
1.2003 + next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
1.2004 + then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
1.2005 + by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
1.2006 + from this(3) guess j .. note j=this
1.2007 + def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
1.2008 + def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
1.2009 + 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)"
1.2010 + note division_points_psubset[OF goal1(4) ab kc(1-2) j]
1.2011 + note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
1.2012 + 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})"
1.2013 + apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
1.2014 + using division_split[OF goal1(4), where k=k and c=c]
1.2015 + unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
1.2016 + using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
1.2017 + have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
1.2018 + unfolding * apply(rule operativeD(2)) using goal1(3) .
1.2019 + also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
1.2020 + unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
1.2021 + unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
1.2022 + unfolding empty_as_interval[THEN sym] apply(rule content_empty)
1.2023 + 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"
1.2024 + from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
1.2025 + show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
1.2026 + apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
1.2027 + apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
1.2028 + qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
1.2029 + unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
1.2030 + unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
1.2031 + unfolding empty_as_interval[THEN sym] apply(rule content_empty)
1.2032 + 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"
1.2033 + from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
1.2034 + show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
1.2035 + apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
1.2036 + apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
1.2037 + 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}))"
1.2038 + unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
1.2039 + 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})))
1.2040 + = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
1.2041 + apply(rule iterate_op[THEN sym]) using goal1 by auto
1.2042 + finally show ?thesis by auto
1.2043 + qed qed qed
1.2044 +
1.2045 +lemma iterate_image_nonzero: assumes "monoidal opp"
1.2046 + "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
1.2047 + shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
1.2048 +proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
1.2049 + case goal1 show ?case using assms(1) by auto
1.2050 +next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
1.2051 + show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
1.2052 + apply(rule finite_imageI goal2)+
1.2053 + apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
1.2054 + apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
1.2055 + apply(subst iterate_insert[OF assms(1) goal2(1)])
1.2056 + unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
1.2057 + apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
1.2058 + using goal2 unfolding o_def by auto qed
1.2059 +
1.2060 +lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
1.2061 + shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
1.2062 +proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
1.2063 + have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
1.2064 + apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
1.2065 + unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
1.2066 + proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
1.2067 + guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
1.2068 + show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
1.2069 + unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
1.2070 + unfolding as(4)[THEN sym] uv by auto
1.2071 + qed also have "\<dots> = f {a..b}"
1.2072 + using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
1.2073 + finally show ?thesis . qed
1.2074 +
1.2075 +subsection {* Additivity of content. *}
1.2076 +
1.2077 +lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
1.2078 +proof- have *:"setsum f s = setsum f (support op + f s)"
1.2079 + apply(rule setsum_mono_zero_right)
1.2080 + unfolding support_def neutral_monoid using assms by auto
1.2081 + thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
1.2082 + unfolding neutral_monoid . qed
1.2083 +
1.2084 +lemma additive_content_division: assumes "d division_of {a..b}"
1.2085 + shows "setsum content d = content({a..b})"
1.2086 + unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
1.2087 + apply(subst setsum_iterate) using assms by auto
1.2088 +
1.2089 +lemma additive_content_tagged_division:
1.2090 + assumes "d tagged_division_of {a..b}"
1.2091 + shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
1.2092 + unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
1.2093 + apply(subst setsum_iterate) using assms by auto
1.2094 +
1.2095 +subsection {* Finally, the integral of a constant\<forall> *}
1.2096 +
1.2097 +lemma has_integral_const[intro]:
1.2098 + "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
1.2099 + unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
1.2100 + apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
1.2101 + unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
1.2102 + defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
1.2103 +
1.2104 +subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
1.2105 +
1.2106 +lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
1.2107 + shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
1.2108 + apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
1.2109 + apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
1.2110 + apply(subst real_mult_commute) apply(rule mult_left_mono)
1.2111 + apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
1.2112 + apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
1.2113 +proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
1.2114 + fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
1.2115 + thus "0 \<le> content x" using content_pos_le by auto
1.2116 +qed(insert assms,auto)
1.2117 +
1.2118 +lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
1.2119 + shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
1.2120 +proof(cases "{a..b} = {}") case True
1.2121 + show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
1.2122 +next case False show ?thesis
1.2123 + apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
1.2124 + apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
1.2125 + unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
1.2126 + apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
1.2127 + apply(subst o_def, rule abs_of_nonneg)
1.2128 + proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
1.2129 + unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
1.2130 + guess w using nonempty_witness[OF False] .
1.2131 + thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
1.2132 + fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
1.2133 + from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
1.2134 + show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
1.2135 + 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
1.2136 + qed(insert assms,auto) qed
1.2137 +
1.2138 +lemma rsum_diff_bound:
1.2139 + assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
1.2140 + 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})"
1.2141 + apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
1.2142 + unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
1.2143 +
1.2144 +lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.2145 + assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
1.2146 + shows "norm i \<le> B * content {a..b}"
1.2147 +proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
1.2148 + thus ?thesis proof(cases ?P) case False
1.2149 + hence *:"content {a..b} = 0" using content_lt_nz by auto
1.2150 + hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
1.2151 + show ?thesis unfolding * ** using assms(1) by auto
1.2152 + qed auto } assume ab:?P
1.2153 + { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
1.2154 + assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
1.2155 + from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
1.2156 + from fine_division_exists[OF this(1), of a b] guess p . note p=this
1.2157 + have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
1.2158 + proof- case goal1 thus ?case unfolding not_less
1.2159 + using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
1.2160 + qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
1.2161 +
1.2162 +subsection {* Similar theorems about relationship among components. *}
1.2163 +
1.2164 +lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
1.2165 + assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
1.2166 + 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"
1.2167 + unfolding setsum_component apply(rule setsum_mono)
1.2168 + apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
1.2169 +proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
1.2170 + from this(3) guess u v apply-by(erule exE)+ note b=this
1.2171 + show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
1.2172 + unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
1.2173 + defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
1.2174 +
1.2175 +lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
1.2176 + assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
1.2177 + shows "i$k \<le> j$k"
1.2178 +proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow>
1.2179 + (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"
1.2180 + proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
1.2181 + guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
1.2182 + guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
1.2183 + guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
1.2184 + note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k]
1.2185 + note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
1.2186 + thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
1.2187 + qed let ?P = "\<exists>a b. s = {a..b}"
1.2188 + { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
1.2189 + case True then guess a b apply-by(erule exE)+ note s=this
1.2190 + show ?thesis apply(rule lem) using assms[unfolded s] by auto
1.2191 + qed auto } assume as:"\<not> ?P"
1.2192 + { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
1.2193 + assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
1.2194 + note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
1.2195 + have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
1.2196 + from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
1.2197 + note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
1.2198 + guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
1.2199 + guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
1.2200 + 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*)
1.2201 + note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
1.2202 + have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
1.2203 + show False unfolding Cart_nth.diff by(rule *) qed
1.2204 +
1.2205 +lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
1.2206 + assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
1.2207 + shows "(integral s f)$k \<le> (integral s g)$k"
1.2208 + apply(rule has_integral_component_le) using integrable_integral assms by auto
1.2209 +
1.2210 +lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
1.2211 + assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
1.2212 + shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
1.2213 + using assms(3) unfolding vector_le_def by auto
1.2214 +
1.2215 +lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
1.2216 + assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
1.2217 + shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
1.2218 + apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
1.2219 +
1.2220 +lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
1.2221 + assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
1.2222 + using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
1.2223 +
1.2224 +lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
1.2225 + assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
1.2226 + apply(rule has_integral_component_pos) using assms by auto
1.2227 +
1.2228 +lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
1.2229 + assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
1.2230 + using has_integral_component_pos[OF assms(1), of 1]
1.2231 + using assms(2) unfolding vector_le_def by auto
1.2232 +
1.2233 +lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
1.2234 + assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
1.2235 + apply(rule has_integral_dest_vec1_pos) using assms by auto
1.2236 +
1.2237 +lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
1.2238 + assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
1.2239 + using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
1.2240 +
1.2241 +lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
1.2242 + assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
1.2243 + using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
1.2244 +
1.2245 +lemma has_integral_component_lbound:
1.2246 + 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"
1.2247 + using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
1.2248 + unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
1.2249 +
1.2250 +lemma has_integral_component_ubound:
1.2251 + assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
1.2252 + shows "i$k \<le> B * content({a..b})"
1.2253 + using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
1.2254 + unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
1.2255 +
1.2256 +lemma integral_component_lbound:
1.2257 + assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
1.2258 + shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
1.2259 + apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
1.2260 +
1.2261 +lemma integral_component_ubound:
1.2262 + assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
1.2263 + shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
1.2264 + apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
1.2265 +
1.2266 +subsection {* Uniform limit of integrable functions is integrable. *}
1.2267 +
1.2268 +lemma real_arch_invD:
1.2269 + "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
1.2270 + by(subst(asm) real_arch_inv)
1.2271 +
1.2272 +lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2273 + assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
1.2274 + shows "f integrable_on {a..b}"
1.2275 +proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
1.2276 + show ?thesis apply cases apply(rule *,assumption)
1.2277 + unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
1.2278 + assume as:"content {a..b} > 0"
1.2279 + have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
1.2280 + from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
1.2281 + from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
1.2282 +
1.2283 + have "Cauchy i" unfolding Cauchy_def
1.2284 + proof(rule,rule) fix e::real assume "e>0"
1.2285 + hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
1.2286 + then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
1.2287 + 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)
1.2288 + proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
1.2289 + from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
1.2290 + from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
1.2291 + from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
1.2292 + 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"
1.2293 + proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
1.2294 + using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
1.2295 + using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
1.2296 + also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
1.2297 + finally show ?case .
1.2298 + qed
1.2299 + show ?case unfolding vector_dist_norm apply(rule lem2) defer
1.2300 + apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
1.2301 + using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
1.2302 + apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
1.2303 + proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
1.2304 + using M as by(auto simp add:field_simps)
1.2305 + fix x assume x:"x \<in> {a..b}"
1.2306 + have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
1.2307 + using g(1)[OF x, of n] g(1)[OF x, of m] by auto
1.2308 + also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
1.2309 + apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
1.2310 + also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
1.2311 + finally show "norm (g n x - g m x) \<le> 2 / real M"
1.2312 + using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
1.2313 + by(auto simp add:group_simps simp add:norm_minus_commute)
1.2314 + qed qed qed
1.2315 + from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
1.2316 +
1.2317 + show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
1.2318 + proof(rule,rule)
1.2319 + case goal1 hence *:"e/3 > 0" by auto
1.2320 + from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
1.2321 + from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
1.2322 + from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
1.2323 + from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
1.2324 + 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"
1.2325 + proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
1.2326 + using norm_triangle_ineq[of "sf - sg" "sg - s"]
1.2327 + using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:group_simps)
1.2328 + also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
1.2329 + finally show ?case .
1.2330 + qed
1.2331 + show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
1.2332 + proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
1.2333 + show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
1.2334 + apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
1.2335 + proof- have "content {a..b} < e / 3 * (real N2)"
1.2336 + using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
1.2337 + hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
1.2338 + apply-apply(rule less_le_trans,assumption) using `e>0` by auto
1.2339 + thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
1.2340 + unfolding inverse_eq_divide by(auto simp add:field_simps)
1.2341 + show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
1.2342 + qed qed qed qed
1.2343 +
1.2344 +subsection {* Negligible sets. *}
1.2345 +
1.2346 +definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
1.2347 +
1.2348 +lemma dest_vec1_indicator:
1.2349 + "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
1.2350 +
1.2351 +lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
1.2352 +
1.2353 +lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
1.2354 +
1.2355 +lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
1.2356 + unfolding indicator_def by auto
1.2357 +
1.2358 +definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
1.2359 +
1.2360 +lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
1.2361 + unfolding indicator_def by auto
1.2362 +
1.2363 +subsection {* Negligibility of hyperplane. *}
1.2364 +
1.2365 +lemma vsum_nonzero_image_lemma:
1.2366 + assumes "finite s" "g(a) = 0"
1.2367 + "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
1.2368 + shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
1.2369 + unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
1.2370 + apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
1.2371 + unfolding assms using neutral_add unfolding neutral_add using assms by auto
1.2372 +
1.2373 +lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
1.2374 + {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
1.2375 +proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
1.2376 + have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
1.2377 + show ?thesis unfolding * ** interval_split by(rule refl) qed
1.2378 +
1.2379 +lemma division_doublesplit: assumes "p division_of {a..b::real^'n}"
1.2380 + shows "{l \<inter> {x. abs(x$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})"
1.2381 +proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
1.2382 + have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
1.2383 + note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
1.2384 + note division_split(2)[OF this, where c="c-e" and k=k]
1.2385 + thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
1.2386 + apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
1.2387 + apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
1.2388 + apply(rule_tac x=l in exI) by blast+ qed
1.2389 +
1.2390 +lemma content_doublesplit: assumes "0 < e"
1.2391 + obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
1.2392 +proof(cases "content {a..b} = 0")
1.2393 + case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
1.2394 + apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
1.2395 + unfolding interval_doublesplit[THEN sym] using assms by auto
1.2396 +next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
1.2397 + note False[unfolded content_eq_0 not_ex not_le, rule_format]
1.2398 + hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
1.2399 + hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
1.2400 + proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
1.2401 + have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
1.2402 + (\<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)
1.2403 + = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
1.2404 + unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
1.2405 + 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)
1.2406 + unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
1.2407 + unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
1.2408 + proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
1.2409 + also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
1.2410 + finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
1.2411 + unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
1.2412 +
1.2413 +lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}"
1.2414 + unfolding negligible_def has_integral apply(rule,rule,rule,rule)
1.2415 +proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
1.2416 + show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
1.2417 + proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
1.2418 + have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
1.2419 + apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
1.2420 + apply(cases,rule disjI1,assumption,rule disjI2)
1.2421 + proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
1.2422 + show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
1.2423 + apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
1.2424 + proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
1.2425 + note this[unfolded subset_eq mem_ball vector_dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
1.2426 + thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
1.2427 + qed auto qed
1.2428 + note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
1.2429 + 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
1.2430 + apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
1.2431 + apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
1.2432 + prefer 2 apply(subst(asm) eq_commute) apply assumption
1.2433 + apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
1.2434 + 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}))"
1.2435 + apply(rule setsum_mono) unfolding split_paired_all split_conv
1.2436 + apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
1.2437 + also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
1.2438 + proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
1.2439 + unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
1.2440 + thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
1.2441 + 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"
1.2442 + apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
1.2443 + 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)"
1.2444 + guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
1.2445 + show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
1.2446 + qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
1.2447 + note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
1.2448 + note this[unfolded real_scaleR_def real_norm_def class_semiring.semiring_rules, of k c d] note le_less_trans[OF this d(2)]
1.2449 + 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"
1.2450 + apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
1.2451 + apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
1.2452 + proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
1.2453 + 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}"
1.2454 + 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
1.2455 + note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
1.2456 + hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
1.2457 + thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
1.2458 + qed qed
1.2459 + finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
1.2460 + qed qed qed
1.2461 +
1.2462 +subsection {* A technical lemma about "refinement" of division. *}
1.2463 +
1.2464 +lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
1.2465 + assumes "p tagged_division_of {a..b}" "gauge d"
1.2466 + 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"
1.2467 +proof-
1.2468 + let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
1.2469 + (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
1.2470 + (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
1.2471 + { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
1.2472 + presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
1.2473 + thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
1.2474 + } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
1.2475 + show "?P p" apply(rule,rule) using as proof(induct p)
1.2476 + case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
1.2477 + next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
1.2478 + note tagged_partial_division_subset[OF insert(4) subset_insertI]
1.2479 + from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
1.2480 + 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
1.2481 + note p = tagged_partial_division_ofD[OF insert(4)]
1.2482 + from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
1.2483 +
1.2484 + have "finite {k. \<exists>x. (x, k) \<in> p}"
1.2485 + apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
1.2486 + apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
1.2487 + hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
1.2488 + apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
1.2489 + unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
1.2490 + apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
1.2491 + using insert(2) unfolding uv xk by auto
1.2492 +
1.2493 + show ?case proof(cases "{u..v} \<subseteq> d x")
1.2494 + case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
1.2495 + unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
1.2496 + apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int)
1.2497 + apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
1.2498 + unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
1.2499 + apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
1.2500 + next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
1.2501 + show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
1.2502 + apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
1.2503 + unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
1.2504 + apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
1.2505 + apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
1.2506 + qed qed qed
1.2507 +
1.2508 +subsection {* Hence the main theorem about negligible sets. *}
1.2509 +
1.2510 +lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
1.2511 + shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
1.2512 +proof(induct) case (insert x s)
1.2513 + 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
1.2514 + show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
1.2515 +
1.2516 +lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
1.2517 + 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
1.2518 +proof(induct) case (insert a s)
1.2519 + 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
1.2520 + show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
1.2521 + prefer 4 apply(subst insert(3)) unfolding add_right_cancel
1.2522 + 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
1.2523 + show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
1.2524 + qed(insert insert, auto) qed auto
1.2525 +
1.2526 +lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.2527 + assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
1.2528 + shows "(f has_integral 0) t"
1.2529 +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})"
1.2530 + let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
1.2531 + show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
1.2532 + apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
1.2533 + proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
1.2534 + show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
1.2535 + 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)"
1.2536 + apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
1.2537 + apply(rule,rule P) using assms(2) by auto
1.2538 + qed
1.2539 +next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
1.2540 + show "(f has_integral 0) {a..b}" unfolding has_integral
1.2541 + proof(safe) case goal1
1.2542 + hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
1.2543 + apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
1.2544 + note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
1.2545 + from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
1.2546 + show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
1.2547 + proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
1.2548 + fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
1.2549 + let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
1.2550 + { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto }
1.2551 + assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
1.2552 + 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
1.2553 + 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)"
1.2554 + apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
1.2555 + from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
1.2556 + have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe)
1.2557 + unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
1.2558 + 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"
1.2559 + proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
1.2560 + apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
1.2561 + have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
1.2562 + norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
1.2563 + unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
1.2564 + apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
1.2565 + 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
1.2566 + fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
1.2567 + unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
1.2568 + using tagged_division_ofD(4)[OF q(1) as''] by auto
1.2569 + next fix i::nat show "finite (q i)" using q by auto
1.2570 + next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
1.2571 + have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
1.2572 + have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
1.2573 + hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
1.2574 + moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
1.2575 + note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
1.2576 + moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
1.2577 + proof(cases "x\<in>s") case False thus ?thesis using assm by auto
1.2578 + next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
1.2579 + moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
1.2580 + ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
1.2581 + 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)"
1.2582 + 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
1.2583 + qed(insert as, auto)
1.2584 + also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
1.2585 + proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
1.2586 + using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
1.2587 + qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
1.2588 + apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
1.2589 + apply(subst sumr_geometric) using goal1 by auto
1.2590 + finally show "?goal" by auto qed qed qed
1.2591 +
1.2592 +lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
1.2593 + assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
1.2594 + shows "(g has_integral y) t"
1.2595 +proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
1.2596 + assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
1.2597 + have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
1.2598 + apply(rule has_integral_negligible[OF assms(1)]) using as by auto
1.2599 + hence "(g has_integral y) {a..b}" by auto } note * = this
1.2600 + show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
1.2601 + apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
1.2602 + 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)
1.2603 + 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
1.2604 +
1.2605 +lemma has_integral_spike_eq:
1.2606 + assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
1.2607 + shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
1.2608 + apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
1.2609 +
1.2610 +lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
1.2611 + shows "g integrable_on t"
1.2612 + using assms unfolding integrable_on_def apply-apply(erule exE)
1.2613 + apply(rule,rule has_integral_spike) by fastsimp+
1.2614 +
1.2615 +lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
1.2616 + shows "integral t f = integral t g"
1.2617 + unfolding integral_def using has_integral_spike_eq[OF assms] by auto
1.2618 +
1.2619 +subsection {* Some other trivialities about negligible sets. *}
1.2620 +
1.2621 +lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
1.2622 +proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
1.2623 + apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
1.2624 + using assms(2) unfolding indicator_def by auto qed
1.2625 +
1.2626 +lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
1.2627 +
1.2628 +lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
1.2629 +
1.2630 +lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
1.2631 +proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
1.2632 + thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
1.2633 + defer apply assumption unfolding indicator_def by auto qed
1.2634 +
1.2635 +lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
1.2636 + using negligible_union by auto
1.2637 +
1.2638 +lemma negligible_sing[intro]: "negligible {a::real^'n}"
1.2639 +proof- guess x using UNIV_witness[where 'a='n] ..
1.2640 + show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
1.2641 +
1.2642 +lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
1.2643 + apply(subst insert_is_Un) unfolding negligible_union_eq by auto
1.2644 +
1.2645 +lemma negligible_empty[intro]: "negligible {}" by auto
1.2646 +
1.2647 +lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
1.2648 + using assms apply(induct s) by auto
1.2649 +
1.2650 +lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
1.2651 + using assms by(induct,auto)
1.2652 +
1.2653 +lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
1.2654 + apply safe defer apply(subst negligible_def)
1.2655 +proof- fix t::"(real^'n) set" assume as:"negligible s"
1.2656 + 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)
1.2657 + show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
1.2658 + apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
1.2659 + apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
1.2660 + using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
1.2661 +
1.2662 +subsection {* Finite case of the spike theorem is quite commonly needed. *}
1.2663 +
1.2664 +lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
1.2665 + "(f has_integral y) t" shows "(g has_integral y) t"
1.2666 + apply(rule has_integral_spike) using assms by auto
1.2667 +
1.2668 +lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
1.2669 + shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
1.2670 + apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
1.2671 +
1.2672 +lemma integrable_spike_finite:
1.2673 + assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t"
1.2674 + using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
1.2675 + apply(rule has_integral_spike_finite) by auto
1.2676 +
1.2677 +subsection {* In particular, the boundary of an interval is negligible. *}
1.2678 +
1.2679 +lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
1.2680 +proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
1.2681 + have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
1.2682 + apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
1.2683 + apply(erule_tac[!] x=xa in allE) by auto
1.2684 + thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
1.2685 +
1.2686 +lemma has_integral_spike_interior:
1.2687 + assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
1.2688 + apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
1.2689 +
1.2690 +lemma has_integral_spike_interior_eq:
1.2691 + assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
1.2692 + apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
1.2693 +
1.2694 +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}"
1.2695 + using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
1.2696 +
1.2697 +subsection {* Integrability of continuous functions. *}
1.2698 +
1.2699 +lemma neutral_and[simp]: "neutral op \<and> = True"
1.2700 + unfolding neutral_def apply(rule some_equality) by auto
1.2701 +
1.2702 +lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
1.2703 +
1.2704 +lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
1.2705 +apply induct unfolding iterate_insert[OF monoidal_and] by auto
1.2706 +
1.2707 +lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
1.2708 + shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
1.2709 + using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
1.2710 +
1.2711 +lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2712 + 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
1.2713 +proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
1.2714 + thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
1.2715 + apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
1.2716 + { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
1.2717 + 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}"
1.2718 + "\<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}"
1.2719 + apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
1.2720 + 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}"
1.2721 + "\<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}"
1.2722 + let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
1.2723 + 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)
1.2724 + proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
1.2725 + 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}"
1.2726 + then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
1.2727 + show ?case unfolding integrable_on_def by auto
1.2728 + next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
1.2729 + apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
1.2730 +
1.2731 +lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2732 + 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"
1.2733 + obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
1.2734 +proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
1.2735 + note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
1.2736 + guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
1.2737 +
1.2738 +lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2739 + assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
1.2740 +proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
1.2741 + from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
1.2742 + note d=conjunctD2[OF this,rule_format]
1.2743 + from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
1.2744 + note p' = tagged_division_ofD[OF p(1)]
1.2745 + have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
1.2746 + proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
1.2747 + from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
1.2748 + 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)
1.2749 + proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
1.2750 + fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
1.2751 + note d(2)[OF _ _ this[unfolded mem_ball]]
1.2752 + thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding vector_dist_norm l norm_minus_commute by fastsimp qed qed
1.2753 + from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
1.2754 + thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
1.2755 +
1.2756 +subsection {* Specialization of additivity to one dimension. *}
1.2757 +
1.2758 +lemma operative_1_lt: assumes "monoidal opp"
1.2759 + shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
1.2760 + (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
1.2761 + unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
1.2762 +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"
1.2763 + 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
1.2764 + thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
1.2765 +next fix a b::"real^1" and c::real
1.2766 + 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}"
1.2767 + show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
1.2768 + proof(cases "c \<in> {a$1 .. b$1}")
1.2769 + case False hence "c<a$1 \<or> c>b$1" by auto
1.2770 + thus ?thesis apply-apply(erule disjE)
1.2771 + 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
1.2772 + show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
1.2773 + 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
1.2774 + show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
1.2775 + qed
1.2776 + next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
1.2777 + show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
1.2778 + proof(cases "c = a$1 \<or> c = b$1")
1.2779 + case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
1.2780 + apply-apply(subst as(2)[rule_format]) using True by auto
1.2781 + next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
1.2782 + proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto
1.2783 + hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
1.2784 + thus ?thesis using assms unfolding * by auto
1.2785 + next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto
1.2786 + hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
1.2787 + thus ?thesis using assms unfolding * by auto qed qed qed qed
1.2788 +
1.2789 +lemma operative_1_le: assumes "monoidal opp"
1.2790 + shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
1.2791 + (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
1.2792 +unfolding operative_1_lt[OF assms]
1.2793 +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"
1.2794 + 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
1.2795 +next fix a b c ::"real^1"
1.2796 + 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"
1.2797 + note as = this[rule_format]
1.2798 + show "opp (f {a..c}) (f {c..b}) = f {a..b}"
1.2799 + proof(cases "c = a \<or> c = b")
1.2800 + 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)
1.2801 + next case True thus ?thesis apply-
1.2802 + proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
1.2803 + thus ?thesis using assms unfolding * by auto
1.2804 + next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
1.2805 + thus ?thesis using assms unfolding * by auto qed qed qed
1.2806 +
1.2807 +subsection {* Special case of additivity we need for the FCT. *}
1.2808 +
1.2809 +lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
1.2810 + assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
1.2811 + shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
1.2812 +proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
1.2813 + have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
1.2814 + by(auto simp add:not_less interval_bound_1 vector_less_def)
1.2815 + have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
1.2816 + note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
1.2817 + show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
1.2818 + apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
1.2819 +
1.2820 +subsection {* A useful lemma allowing us to factor out the content size. *}
1.2821 +
1.2822 +lemma has_integral_factor_content:
1.2823 + "(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
1.2824 + \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
1.2825 +proof(cases "content {a..b} = 0")
1.2826 + case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
1.2827 + apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
1.2828 + apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
1.2829 + apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
1.2830 +next case False note F = this[unfolded content_lt_nz[THEN sym]]
1.2831 + 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)"
1.2832 + show ?thesis apply(subst has_integral)
1.2833 + proof safe fix e::real assume e:"e>0"
1.2834 + { 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)
1.2835 + apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
1.2836 + using F e by(auto simp add:field_simps intro:mult_pos_pos) }
1.2837 + { 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)
1.2838 + apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
1.2839 + using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
1.2840 +
1.2841 +subsection {* Fundamental theorem of calculus. *}
1.2842 +
1.2843 +lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
1.2844 + assumes "a \<le> b" "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
1.2845 + shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
1.2846 +unfolding has_integral_factor_content
1.2847 +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
1.2848 + note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
1.2849 + 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
1.2850 + note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
1.2851 + guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
1.2852 + show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
1.2853 + 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})"
1.2854 + apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
1.2855 + apply(rule gauge_ball_dependent,rule,rule d(1))
1.2856 + proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
1.2857 + 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}"
1.2858 + unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
1.2859 + unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
1.2860 + apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
1.2861 + proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
1.2862 + note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
1.2863 + have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
1.2864 + 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] .
1.2865 + 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)"
1.2866 + apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
1.2867 + unfolding scaleR.diff_left by(auto simp add:group_simps)
1.2868 + also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
1.2869 + apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
1.2870 + apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
1.2871 + using ball[rule_format,of u] ball[rule_format,of v]
1.2872 + using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
1.2873 + also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
1.2874 + unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
1.2875 + finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
1.2876 + e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
1.2877 + qed(insert as, auto) qed qed
1.2878 +
1.2879 +subsection {* Attempt a systematic general set of "offset" results for components. *}
1.2880 +
1.2881 +lemma gauge_modify:
1.2882 + assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
1.2883 + shows "gauge (\<lambda>x y. d (f x) (f y))"
1.2884 + using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
1.2885 + apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
1.2886 +
1.2887 +subsection {* Only need trivial subintervals if the interval itself is trivial. *}
1.2888 +
1.2889 +lemma division_of_nontrivial: fixes s::"(real^'n) set set"
1.2890 + assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
1.2891 + shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
1.2892 +proof(induct "card s" arbitrary:s rule:nat_less_induct)
1.2893 + fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
1.2894 + "\<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}"
1.2895 + note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
1.2896 + { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
1.2897 + show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
1.2898 + assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
1.2899 + then obtain k where k:"k\<in>s" "content k = 0" by auto
1.2900 + from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
1.2901 + from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
1.2902 + hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
1.2903 + have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
1.2904 + apply safe apply(rule closed_interval) using assm(1) by auto
1.2905 + have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
1.2906 + proof safe fix x and e::real assume as:"x\<in>k" "e>0"
1.2907 + from k(2)[unfolded k content_eq_0] guess i ..
1.2908 + hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
1.2909 + hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
1.2910 + def y \<equiv> "(\<chi> j. if j = i then if c$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$j)"
1.2911 + show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
1.2912 + proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
1.2913 + hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
1.2914 + hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
1.2915 + 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+
1.2916 + thus "y \<noteq> x" unfolding Cart_eq by auto
1.2917 + have *:"UNIV = insert i (UNIV - {i})" by auto
1.2918 + have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
1.2919 + apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
1.2920 + proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
1.2921 + apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
1.2922 + show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto
1.2923 + qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
1.2924 + 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
1.2925 + moreover have "y \<in> \<Union>s" unfolding s mem_interval
1.2926 + proof note simps = y_def Cart_lambda_beta if_not_P
1.2927 + fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j"
1.2928 + proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
1.2929 + thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
1.2930 + next case True note T = this show ?thesis
1.2931 + proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
1.2932 + case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
1.2933 + using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
1.2934 + next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
1.2935 + using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
1.2936 + qed qed qed
1.2937 + ultimately show "y \<in> \<Union>(s - {k})" by auto
1.2938 + qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
1.2939 + hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
1.2940 + apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
1.2941 + 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
1.2942 +
1.2943 +subsection {* Integrabibility on subintervals. *}
1.2944 +
1.2945 +lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
1.2946 + "operative op \<and> (\<lambda>i. f integrable_on i)"
1.2947 + unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
1.2948 + unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
1.2949 + unfolding integrable_on_def by(auto intro: has_integral_split)
1.2950 +
1.2951 +lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2952 + assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
1.2953 + apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
1.2954 + using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
1.2955 +
1.2956 +subsection {* Combining adjacent intervals in 1 dimension. *}
1.2957 +
1.2958 +lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
1.2959 + "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
1.2960 + shows "(f has_integral (i + j)) {a..b}"
1.2961 +proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
1.2962 + note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
1.2963 + hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
1.2964 + apply(subst(asm) if_P) using assms(3-) by auto
1.2965 + with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
1.2966 + unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
1.2967 +
1.2968 +lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
1.2969 + assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
1.2970 + shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
1.2971 + apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
1.2972 + apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
1.2973 +
1.2974 +lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
1.2975 + assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
1.2976 + shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
1.2977 +
1.2978 +subsection {* Reduce integrability to "local" integrability. *}
1.2979 +
1.2980 +lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
1.2981 + 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}"
1.2982 + shows "f integrable_on {a..b}"
1.2983 +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})"
1.2984 + using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
1.2985 + guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
1.2986 + note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
1.2987 + show ?thesis unfolding * apply safe unfolding snd_conv
1.2988 + proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
1.2989 + thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
1.2990 +
1.2991 +subsection {* Second FCT or existence of antiderivative. *}
1.2992 +
1.2993 +lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
1.2994 + unfolding integrable_on_def by(rule,rule has_integral_const)
1.2995 +
1.2996 +lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
1.2997 + assumes "continuous_on {a..b} f" "x \<in> {a..b}"
1.2998 + shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
1.2999 + unfolding has_vector_derivative_def has_derivative_within_alt
1.3000 +apply safe apply(rule scaleR.bounded_linear_left)
1.3001 +proof- fix e::real assume e:"e>0"
1.3002 + note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
1.3003 + from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
1.3004 + let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
1.3005 + 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)"
1.3006 + proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
1.3007 + case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
1.3008 + apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
1.3009 + hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
1.3010 + using False unfolding not_less using assms(2) goal1 by auto
1.3011 + have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
1.3012 + show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
1.3013 + defer apply(rule has_integral_sub) apply(rule integrable_integral)
1.3014 + apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
1.3015 + proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
1.3016 + have *:"y - x = norm(y - x)" using False by auto
1.3017 + show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
1.3018 + show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
1.3019 + apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
1.3020 + qed(insert e,auto)
1.3021 + next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
1.3022 + apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto
1.3023 + hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
1.3024 + using True using assms(2) goal1 by auto
1.3025 + have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
1.3026 + 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
1.3027 + show ?thesis apply(subst ***) unfolding norm_minus_cancel **
1.3028 + apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
1.3029 + defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
1.3030 + apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
1.3031 + apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
1.3032 + proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
1.3033 + have *:"x - y = norm(y - x)" using True by auto
1.3034 + show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
1.3035 + show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
1.3036 + apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
1.3037 + qed(insert e,auto) qed qed qed
1.3038 +
1.3039 +lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
1.3040 + assumes "continuous_on {a..b} f" "x \<in> {a..b}"
1.3041 + shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
1.3042 + using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
1.3043 + unfolding o_def vec1_dest_vec1 using assms(2) by auto
1.3044 +
1.3045 +lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
1.3046 + obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
1.3047 + apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
1.3048 +
1.3049 +subsection {* Combined fundamental theorem of calculus. *}
1.3050 +
1.3051 +lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
1.3052 + 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}"
1.3053 +proof- from antiderivative_continuous[OF assms] guess g . note g=this
1.3054 + show ?thesis apply(rule that[of g])
1.3055 + proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
1.3056 + apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
1.3057 + thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
1.3058 + unfolding o_def vec1_dest_vec1 by auto qed qed
1.3059 +
1.3060 +subsection {* General "twiddling" for interval-to-interval function image. *}
1.3061 +
1.3062 +lemma has_integral_twiddle:
1.3063 + assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
1.3064 + "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
1.3065 + "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
1.3066 + "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
1.3067 + "(f has_integral i) {a..b}"
1.3068 + shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
1.3069 +proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
1.3070 + show ?thesis apply cases defer apply(rule *,assumption)
1.3071 + proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
1.3072 + assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
1.3073 + have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
1.3074 + using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
1.3075 + using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
1.3076 + show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
1.3077 + proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
1.3078 + from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
1.3079 + 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)
1.3080 + 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)"
1.3081 + 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
1.3082 + fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
1.3083 + 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
1.3084 + proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
1.3085 + show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
1.3086 + fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
1.3087 + show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
1.3088 + { 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)"]
1.3089 + using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
1.3090 + fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
1.3091 + hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
1.3092 + have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
1.3093 + proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
1.3094 + hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
1.3095 + unfolding image_Int[OF inj(1)] by auto thus False using as by blast
1.3096 + qed thus "g x = g x'" by auto
1.3097 + { fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto }
1.3098 + { fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto }
1.3099 + 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
1.3100 + then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
1.3101 + thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
1.3102 + apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
1.3103 + using X(2) assms(3)[rule_format,of x] by auto
1.3104 + 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
1.3105 + 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 group_simps add_left_cancel
1.3106 + unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
1.3107 + apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
1.3108 + 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]
1.3109 + unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
1.3110 + 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
1.3111 + using assms(1) by(auto simp add:field_simps) qed qed qed
1.3112 +
1.3113 +subsection {* Special case of a basic affine transformation. *}
1.3114 +
1.3115 +lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
1.3116 + unfolding image_affinity_interval by auto
1.3117 +
1.3118 +lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
1.3119 + Cart_eq vector_le_def vector_less_def
1.3120 +
1.3121 +lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
1.3122 + apply(rule setprod_cong) using assms by auto
1.3123 +
1.3124 +lemma content_image_affinity_interval:
1.3125 + "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
1.3126 +proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
1.3127 + unfolding not_not using content_empty by auto }
1.3128 + assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
1.3129 + case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
1.3130 + unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
1.3131 + defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
1.3132 + apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le
1.3133 + by(auto simp add:field_simps intro:mult_left_mono)
1.3134 + next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
1.3135 + unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
1.3136 + defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
1.3137 + apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le
1.3138 + by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
1.3139 +
1.3140 +lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
1.3141 + 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})"
1.3142 + apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
1.3143 + defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
1.3144 + apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
1.3145 +
1.3146 +lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
1.3147 + 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})"
1.3148 + using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
1.3149 +
1.3150 +subsection {* Special case of stretching coordinate axes separately. *}
1.3151 +
1.3152 +lemma image_stretch_interval:
1.3153 + "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
1.3154 + (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")
1.3155 +proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
1.3156 +next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
1.3157 + case False note ab = this[unfolded interval_ne_empty]
1.3158 + show ?thesis apply-apply(rule set_ext)
1.3159 + 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
1.3160 + show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
1.3161 + unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
1.3162 + unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
1.3163 + proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
1.3164 + (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))"
1.3165 + proof(cases "m i = 0") case True thus ?thesis using ab by auto
1.3166 + next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
1.3167 + proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
1.3168 + "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
1.3169 + show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
1.3170 + using as by(auto simp add:field_simps)
1.3171 + next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
1.3172 + "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def
1.3173 + by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
1.3174 + show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
1.3175 + using as by(auto simp add:field_simps) qed qed qed qed qed
1.3176 +
1.3177 +lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
1.3178 + unfolding image_stretch_interval by auto
1.3179 +
1.3180 +lemma content_image_stretch_interval:
1.3181 + "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
1.3182 +proof(cases "{a..b} = {}") case True thus ?thesis
1.3183 + unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
1.3184 +next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
1.3185 + thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
1.3186 + unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
1.3187 + proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
1.3188 + 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)"
1.3189 + apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i]
1.3190 + by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
1.3191 +
1.3192 +lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
1.3193 + shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
1.3194 + ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
1.3195 + apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
1.3196 + unfolding image_stretch_interval empty_as_interval Cart_eq using assms
1.3197 +proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
1.3198 + apply(rule,rule linear_continuous_at) unfolding linear_linear
1.3199 + unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
1.3200 +
1.3201 +lemma integrable_stretch:
1.3202 + assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
1.3203 + shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
1.3204 + using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
1.3205 +
1.3206 +subsection {* even more special cases. *}
1.3207 +
1.3208 +lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
1.3209 + apply(rule set_ext,rule) defer unfolding image_iff
1.3210 + apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
1.3211 +
1.3212 +lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
1.3213 + shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
1.3214 + using has_integral_affinity[OF assms, of "-1" 0] by auto
1.3215 +
1.3216 +lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
1.3217 + apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
1.3218 +
1.3219 +lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
1.3220 + unfolding integrable_on_def by auto
1.3221 +
1.3222 +lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
1.3223 + unfolding integral_def by auto
1.3224 +
1.3225 +subsection {* Stronger form of FCT; quite a tedious proof. *}
1.3226 +
1.3227 +(** move this **)
1.3228 +declare norm_triangle_ineq4[intro]
1.3229 +
1.3230 +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)
1.3231 +
1.3232 +lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
1.3233 + assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
1.3234 + shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
1.3235 + using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
1.3236 + unfolding o_def vec1_dest_vec1 using assms(1) by auto
1.3237 +
1.3238 +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"
1.3239 + unfolding split_def by(rule refl)
1.3240 +
1.3241 +lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
1.3242 + apply(subst(asm)(2) norm_minus_cancel[THEN sym])
1.3243 + apply(drule norm_triangle_le) by(auto simp add:group_simps)
1.3244 +
1.3245 +lemma fundamental_theorem_of_calculus_interior:
1.3246 + assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
1.3247 + shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
1.3248 +proof- { presume *:"a < b \<Longrightarrow> ?thesis"
1.3249 + show ?thesis proof(cases,rule *,assumption)
1.3250 + assume "\<not> a < b" hence "a = b" using assms(1) by auto
1.3251 + hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
1.3252 + show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
1.3253 + qed } assume ab:"a < b"
1.3254 + let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
1.3255 + 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})"
1.3256 + { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
1.3257 + fix e::real assume e:"e>0"
1.3258 + note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
1.3259 + note conjunctD2[OF this] note bounded=this(1) and this(2)
1.3260 + 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)"
1.3261 + 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]
1.3262 + from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
1.3263 + have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
1.3264 + from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
1.3265 +
1.3266 + have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
1.3267 + \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
1.3268 + proof- have "a\<in>{a..b}" using ab by auto
1.3269 + note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
1.3270 + note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
1.3271 + from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
1.3272 + have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
1.3273 + proof(cases "f' a = 0") case True
1.3274 + thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
1.3275 + next case False thus ?thesis
1.3276 + apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
1.3277 + using ab e by(auto simp add:field_simps)
1.3278 + qed then guess l .. note l = conjunctD2[OF this]
1.3279 + show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
1.3280 + proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
1.3281 + note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
1.3282 + 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)
1.3283 + also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
1.3284 + proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
1.3285 + thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
1.3286 + next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
1.3287 + apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
1.3288 + 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
1.3289 + qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
1.3290 +
1.3291 + 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"
1.3292 + proof- have "b\<in>{a..b}" using ab by auto
1.3293 + note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
1.3294 + note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
1.3295 + from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
1.3296 + have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
1.3297 + proof(cases "f' b = 0") case True
1.3298 + thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
1.3299 + next case False thus ?thesis
1.3300 + apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
1.3301 + using ab e by(auto simp add:field_simps)
1.3302 + qed then guess l .. note l = conjunctD2[OF this]
1.3303 + show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
1.3304 + proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
1.3305 + note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
1.3306 + 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)
1.3307 + also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
1.3308 + proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
1.3309 + thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
1.3310 + next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
1.3311 + apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
1.3312 + 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
1.3313 + qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
1.3314 +
1.3315 + let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
1.3316 + show "?P e" apply(rule_tac x="?d" in exI)
1.3317 + proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
1.3318 + next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
1.3319 + 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
1.3320 + note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
1.3321 + 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
1.3322 + show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
1.3323 + unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
1.3324 + proof(rule norm_triangle_le,rule **)
1.3325 + case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
1.3326 + 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"
1.3327 + "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
1.3328 + < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
1.3329 + from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
1.3330 + 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]
1.3331 + note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
1.3332 +
1.3333 + 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_simps) note * = d(2)[OF this]
1.3334 + have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
1.3335 + 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)))"
1.3336 + apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
1.3337 + also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
1.3338 + apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
1.3339 + apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
1.3340 + also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
1.3341 + finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
1.3342 + apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
1.3343 +
1.3344 + next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
1.3345 + case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
1.3346 + defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
1.3347 + apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
1.3348 + proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
1.3349 + from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
1.3350 + with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
1.3351 + thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
1.3352 + unfolding uv using e by(auto simp add:field_simps)
1.3353 + 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
1.3354 + show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
1.3355 + (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2"
1.3356 + apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
1.3357 + apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
1.3358 + 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}"
1.3359 + 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
1.3360 + 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
1.3361 + 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
1.3362 + next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} =
1.3363 + {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
1.3364 + 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"
1.3365 + proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
1.3366 + thus ?case using `x\<in>s` goal2(2) by auto
1.3367 + qed auto
1.3368 + 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"])
1.3369 + apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
1.3370 + proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
1.3371 + have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v"
1.3372 + proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
1.3373 + have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
1.3374 + have u:"u = vec1 a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
1.3375 + 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
1.3376 + have "u > vec1 a" unfolding Cart_simps by auto
1.3377 + thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
1.3378 + qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
1.3379 + qed
1.3380 + have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v"
1.3381 + proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
1.3382 + have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
1.3383 + have u:"v = vec1 b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
1.3384 + 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
1.3385 + have "v < vec1 b" unfolding Cart_simps by auto
1.3386 + thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
1.3387 + qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
1.3388 + qed
1.3389 +
1.3390 + show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
1.3391 + unfolding mem_Collect_eq fst_conv snd_conv apply safe
1.3392 + 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"
1.3393 + guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
1.3394 + guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
1.3395 + 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]
1.3396 + 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)
1.3397 + ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
1.3398 + hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
1.3399 + { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
1.3400 + qed
1.3401 + show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
1.3402 + unfolding mem_Collect_eq fst_conv snd_conv apply safe
1.3403 + 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"
1.3404 + guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
1.3405 + guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
1.3406 + 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]
1.3407 + 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)
1.3408 + ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
1.3409 + hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
1.3410 + { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
1.3411 + qed
1.3412 +
1.3413 + let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
1.3414 + 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)
1.3415 + \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
1.3416 + proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
1.3417 + 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
1.3418 + moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
1.3419 + apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
1.3420 + by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
1.3421 + show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
1.3422 + apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1])
1.3423 + using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
1.3424 + qed
1.3425 + 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)
1.3426 + \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
1.3427 + proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
1.3428 + 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
1.3429 + moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
1.3430 + apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
1.3431 + by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
1.3432 + show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
1.3433 + apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1])
1.3434 + using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
1.3435 + qed
1.3436 + qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
1.3437 +
1.3438 +subsection {* Stronger form with finite number of exceptional points. *}
1.3439 +
1.3440 +lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
1.3441 + assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
1.3442 + "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
1.3443 + shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply-
1.3444 +proof(induct "card s" arbitrary:s a b)
1.3445 + case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
1.3446 +next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
1.3447 + apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
1.3448 + show ?case proof(cases "c\<in>{a<..<b}")
1.3449 + case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
1.3450 + apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
1.3451 + next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
1.3452 + case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
1.3453 + thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
1.3454 + apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
1.3455 + proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
1.3456 + apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
1.3457 + let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
1.3458 + show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
1.3459 + qed auto qed qed
1.3460 +
1.3461 +lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
1.3462 + assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
1.3463 + "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
1.3464 + shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}"
1.3465 + apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
1.3466 + using assms(4) by auto
1.3467 +
1.3468 +end
1.3469 \ No newline at end of file