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1 |
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2 header {* Kurzweil-Henstock gauge integration in many dimensions. *} |
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3 (* Author: John Harrison |
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4 Translation from HOL light: Robert Himmelmann, TU Muenchen *) |
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5 |
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6 theory Integration_Aleph |
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7 imports Derivative SMT |
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8 begin |
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9 |
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10 declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]] |
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11 declare [[smt_record=true]] |
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12 declare [[z3_proofs=true]] |
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13 |
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14 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto |
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15 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto |
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16 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto |
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17 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto |
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18 |
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19 declare smult_conv_scaleR[simp] |
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20 |
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21 subsection {* Some useful lemmas about intervals. *} |
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22 |
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23 lemma empty_as_interval: "{} = {1..0::real^'n}" |
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24 apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval |
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25 using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto |
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26 |
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27 lemma interior_subset_union_intervals: |
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28 assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}" |
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29 shows "interior i \<subseteq> interior s" proof- |
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30 have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5) |
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31 unfolding assms(1,2) interior_closed_interval by auto |
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32 moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed) |
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33 using assms(4) unfolding assms(1,2) by auto |
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34 ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior) |
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35 unfolding assms(1,2) interior_closed_interval by auto qed |
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36 |
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37 lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set" |
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38 assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}" |
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39 shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1 |
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40 have lem1:"\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U" apply rule defer apply(rule_tac Int_greatest) |
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41 unfolding open_subset_interior[OF open_ball] using interior_subset by auto |
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42 have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto |
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43 have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow> (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)" proof- case goal1 |
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44 thus ?case proof(induct rule:finite_induct) |
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45 case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next |
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46 case (insert i f) guess x using insert(5) .. note x = this |
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47 then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this |
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48 guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this |
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49 show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto |
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50 then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] .. |
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51 hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto |
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52 hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 by auto hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto |
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53 hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t" apply-apply(rule insert(3)) using insert(4) by auto thus ?thesis by auto next |
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54 case True show ?thesis proof(cases "x\<in>{a<..<b}") |
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55 case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] .. |
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56 thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI) |
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57 unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next |
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58 case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) |
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59 hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto |
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60 hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE) |
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61 let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE) |
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62 fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto |
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63 hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto |
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64 hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps) |
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65 hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed |
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66 moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof |
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67 fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)" |
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68 apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"]) |
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69 unfolding norm_scaleR norm_basis by auto |
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70 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) |
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71 finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed |
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72 ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto |
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73 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) |
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74 fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto |
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75 hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto |
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76 hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps) |
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77 hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed |
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78 moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof |
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79 fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)" |
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80 apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"]) |
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81 unfolding norm_scaleR norm_basis by auto |
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82 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) |
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83 finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed |
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84 ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed |
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85 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 |
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86 thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this |
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87 guess t using *[OF assms(1,3) goal1] .. from this(2) guess x .. then guess e .. |
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88 hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto |
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89 thus False using `t\<in>f` assms(4) by auto qed |
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90 subsection {* Bounds on intervals where they exist. *} |
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91 |
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92 definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})" |
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93 |
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94 definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})" |
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95 |
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96 lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b" |
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97 using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE) |
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98 apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer |
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99 apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI) |
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100 unfolding mem_interval using assms by auto |
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101 |
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102 lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a" |
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103 using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE) |
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104 apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer |
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105 apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI) |
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106 unfolding mem_interval using assms by auto |
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107 |
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108 lemmas interval_bounds = interval_upperbound interval_lowerbound |
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109 |
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110 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a" |
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111 using assms unfolding interval_ne_empty by auto |
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112 |
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113 lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)" |
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114 apply(rule interval_upperbound) by auto |
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115 |
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116 lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)" |
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117 apply(rule interval_lowerbound) by auto |
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118 |
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119 lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1 |
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120 |
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121 subsection {* Content (length, area, volume...) of an interval. *} |
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122 |
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123 definition "content (s::(real^'n) set) = |
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124 (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))" |
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125 |
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126 lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}" |
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127 unfolding interval_eq_empty unfolding not_ex not_less by assumption |
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128 |
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129 lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i" |
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130 shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)" |
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131 using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto |
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132 |
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133 lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)" |
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134 apply(rule content_closed_interval) using assms unfolding interval_ne_empty . |
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135 |
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136 lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a" |
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137 using content_closed_interval[of a b] by auto |
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138 |
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139 lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto |
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140 |
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141 lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof- |
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142 have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto |
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143 have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto |
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144 thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed |
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145 |
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146 lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}") |
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147 case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption |
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148 have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0" |
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149 apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto |
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150 thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto) |
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151 |
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152 lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}" |
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153 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 |
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154 show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos) |
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155 using assms apply(erule_tac x=x in allE) by auto qed |
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156 |
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157 lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})" |
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158 apply(rule content_pos_lt) by auto |
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159 |
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160 lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}") |
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161 case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply- |
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162 apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next |
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163 guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less] |
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164 show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV] |
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165 apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer |
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166 apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed |
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167 |
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168 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto |
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169 |
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170 lemma content_closed_interval_cases: |
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171 "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) |
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172 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 |
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173 |
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174 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}" |
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175 unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto |
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176 |
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177 lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a" |
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178 unfolding content_eq_0 by auto |
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179 |
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180 lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)" |
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181 apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}" |
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182 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 |
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183 |
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184 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto |
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185 |
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186 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}") |
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187 case True thus ?thesis using content_pos_le[of c d] by auto next |
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188 case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto |
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189 hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto |
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190 have "{c..d} \<noteq> {}" using assms False by auto |
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191 hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto |
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192 show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne] |
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193 unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n |
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194 show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto |
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195 show "b $ i - a $ i \<le> d $ i - c $ i" |
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196 using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i] |
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197 using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed |
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198 |
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199 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0" |
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200 unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto |
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201 |
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202 subsection {* The notion of a gauge --- simply an open set containing the point. *} |
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203 |
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204 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))" |
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205 |
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206 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g" |
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207 using assms unfolding gauge_def by auto |
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208 |
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209 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto |
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210 |
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211 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))" |
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212 unfolding gauge_def by auto |
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213 |
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214 lemma gauge_ball[intro?]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto |
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215 |
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216 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto |
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217 |
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218 lemma gauge_inter: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))" |
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219 unfolding gauge_def by auto |
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220 |
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221 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- |
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222 have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis |
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223 unfolding gauge_def unfolding * |
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224 using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed |
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225 |
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226 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) |
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227 |
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228 subsection {* Divisions. *} |
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229 |
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230 definition division_of (infixl "division'_of" 40) where |
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231 "s division_of i \<equiv> |
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232 finite s \<and> |
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233 (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and> |
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234 (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and> |
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235 (\<Union>s = i)" |
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236 |
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237 lemma division_ofD[dest]: assumes "s division_of i" |
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238 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})" |
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239 "\<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 |
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240 |
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241 lemma division_ofI: |
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242 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})" |
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243 "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" |
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244 shows "s division_of i" using assms unfolding division_of_def by auto |
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245 |
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246 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s" |
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247 unfolding division_of_def by auto |
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248 |
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249 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}" |
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250 unfolding division_of_def by auto |
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251 |
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252 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto |
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253 |
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254 lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof |
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255 assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" |
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256 ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto } |
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257 ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next |
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258 assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]] |
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259 { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto } |
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260 moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed |
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261 |
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262 lemma elementary_empty: obtains p where "p division_of {}" |
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263 unfolding division_of_trivial by auto |
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264 |
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265 lemma elementary_interval: obtains p where "p division_of {a..b}" |
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266 by(metis division_of_trivial division_of_self) |
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267 |
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268 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k" |
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269 unfolding division_of_def by auto |
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270 |
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271 lemma forall_in_division: |
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272 "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))" |
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273 unfolding division_of_def by fastsimp |
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274 |
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275 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)" |
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276 apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)] |
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277 show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto |
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278 { 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 |
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279 show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto } |
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280 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 |
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281 show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto |
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282 |
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283 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto |
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284 |
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285 lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0" |
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286 unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)]) |
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287 apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed |
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288 |
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289 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)" |
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290 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- |
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291 let ?A = "{s. s \<in> (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto |
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292 show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto |
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293 moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto |
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294 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 |
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295 using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto |
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296 { 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 |
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297 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 |
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298 guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this |
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299 guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this |
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300 show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2 |
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301 assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto |
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302 assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto |
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303 assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto |
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304 have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow> |
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305 interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow> |
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306 interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2) |
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307 \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto |
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308 show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior) |
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309 using division_ofD(5)[OF assms(1) k1(2) k2(2)] |
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310 using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed |
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311 |
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312 lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i" |
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313 shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}") |
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314 case True show ?thesis unfolding True and division_of_trivial by auto next |
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315 have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto |
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316 case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed |
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317 |
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318 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)" |
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319 shows "\<exists>p. p division_of (s \<inter> t)" |
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320 by(rule,rule division_inter[OF assms]) |
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321 |
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322 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)" |
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323 shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct) |
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324 case (insert x f) show ?case proof(cases "f={}") |
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325 case True thus ?thesis unfolding True using insert by auto next |
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326 case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] .. |
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327 moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately |
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328 show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto |
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329 |
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330 lemma division_disjoint_union: |
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331 assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}" |
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332 shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) |
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333 note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)] |
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334 show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto |
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335 show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto |
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336 { 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 = {}" |
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337 { 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)]] |
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338 using assms(3) by blast } moreover |
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339 { 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)]] |
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340 using assms(3) by blast} ultimately |
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341 show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto } |
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342 fix k assume k:"k \<in> p1 \<union> p2" show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto |
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343 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 |
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344 |
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345 lemma partial_division_extend_1: |
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346 assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}" |
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347 obtains p where "p division_of {a..b}" "{c..d} \<in> p" |
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348 proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto |
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349 guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this |
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350 def \<pi>' \<equiv> "inv_into {1..n} \<pi>" |
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351 have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto |
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352 hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto |
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353 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 |
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354 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 |
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355 have "{c..d} \<noteq> {}" using assms by auto |
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356 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$i)}" |
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357 let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$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)}" |
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358 let ?p = "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}" |
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359 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) |
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360 show ?thesis apply(rule that[of ?p]) apply(rule division_ofI) |
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361 proof- have "\<And>i. \<pi>' i < Suc n" |
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362 proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i" |
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363 hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto |
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364 qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)" |
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365 "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)" |
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366 unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto |
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367 thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto |
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368 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}" |
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369 unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr) |
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370 proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}" |
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371 then guess i unfolding mem_interval not_all .. note i=this |
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372 show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE) |
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373 apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto |
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374 qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p" |
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375 proof- fix x assume x:"x\<in>{a..b}" |
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376 { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast } |
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377 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)}" |
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378 assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all .. |
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379 hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI) |
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380 hence M:"finite ?M" "?M \<noteq> {}" by auto |
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381 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]] |
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382 Min_gr_iff[OF M,unfolded l_def[symmetric]] |
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383 have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le |
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384 apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2) |
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385 proof- assume as:"x $ \<pi> l < c $ \<pi> l" |
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386 show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta |
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387 proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto |
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388 thus ?case using as x[unfolded mem_interval,rule_format,of i] |
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389 apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"]) |
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390 qed |
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391 next assume as:"x $ \<pi> l > d $ \<pi> l" |
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392 show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta |
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393 proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto |
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394 thus ?case using as x[unfolded mem_interval,rule_format,of i] |
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395 apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"]) |
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396 qed qed |
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397 thus "x \<in> \<Union>?p" using l(2) by blast |
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398 qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast) |
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399 |
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400 show "finite ?p" by auto |
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401 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 |
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402 show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) |
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403 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 |
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404 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) |
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405 qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI) |
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406 proof- case goal1 thus ?case using abcd[of x] by auto |
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407 next case goal2 thus ?case using abcd[of x] by auto |
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408 qed thus "k \<noteq> {}" using k by auto |
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409 show "\<exists>a b. k = {a..b}" using k by auto |
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410 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 |
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411 { fix k k' l l' |
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412 assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" |
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413 assume k':"k' \<in> ?p" "k \<noteq> k'" and l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" |
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414 assume "l \<le> l'" fix x |
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415 have "x \<notin> interior k \<inter> interior k'" |
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416 proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'" |
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417 case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le) |
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418 hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse) |
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419 have ln:"l < n + 1" |
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420 proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto |
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421 hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le) |
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422 hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse) |
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423 thus False using `k\<noteq>k'` k' by auto |
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424 qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto |
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425 have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply- |
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426 proof(erule disjE) |
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427 assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
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428 show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto |
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429 next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
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430 show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto |
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431 qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval |
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432 by(auto elim!:allE[where x="\<pi> l"]) |
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433 next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto |
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434 hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto |
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435 note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)] |
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436 assume x:"x \<in> interior k \<inter> interior k'" |
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437 show False using l(1) l'(1) apply- |
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438 proof(erule_tac[!] disjE)+ |
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439 assume as:"k = ?p1 l" "k' = ?p1 l'" |
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440 note * = x[unfolded as Int_iff interior_closed_interval mem_interval] |
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441 have "l \<noteq> l'" using k'(2)[unfolded as] by auto |
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442 thus False using * by(smt Cart_lambda_beta \<pi>l) |
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443 next assume as:"k = ?p2 l" "k' = ?p2 l'" |
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444 note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
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445 have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto |
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446 thus False using *[of "\<pi> l"] *[of "\<pi> l'"] |
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447 unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto |
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448 next assume as:"k = ?p1 l" "k' = ?p2 l'" |
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449 note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
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450 show False using *[of "\<pi> l"] *[of "\<pi> l'"] |
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451 unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt |
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452 next assume as:"k = ?p2 l" "k' = ?p1 l'" |
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453 note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format] |
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454 show False using *[of "\<pi> l"] *[of "\<pi> l'"] |
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455 unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt |
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456 qed qed } |
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457 from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'" |
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458 apply - apply(cases "l' \<le> l") using k'(2) by auto |
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459 thus "interior k \<inter> interior k' = {}" by auto |
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460 qed qed |
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461 |
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462 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}" |
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463 obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}") |
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464 case True guess q apply(rule elementary_interval[of a b]) . |
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465 thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next |
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466 case False note p = division_ofD[OF assms(1)] |
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467 have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1 |
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468 guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this |
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469 have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto |
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470 guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed |
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471 guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]] |
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472 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- |
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473 fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI) |
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474 using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed |
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475 hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters) |
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476 apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto |
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477 then guess d .. note d = this |
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478 show ?thesis apply(rule that[of "d \<union> p"]) proof- |
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479 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 |
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480 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" |
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481 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 |
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482 show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)]) |
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483 apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule) |
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484 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 |
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485 show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"]) |
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486 defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]] |
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487 show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto |
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488 show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto |
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489 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}))" |
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490 apply(rule subset_interior *)+ using k by auto qed qed qed auto qed |
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491 |
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492 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)" |
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493 unfolding division_of_def by(metis bounded_Union bounded_interval) |
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494 |
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495 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}" |
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496 by(meson elementary_bounded bounded_subset_closed_interval) |
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497 |
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498 lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}" |
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499 obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}") |
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500 case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next |
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501 case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}") |
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502 have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto |
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503 case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union) |
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504 using false True assms using interior_subset by auto next |
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505 case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto |
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506 have *:"{u..v} \<subseteq> {c..d}" using uv by auto |
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507 guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)] |
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508 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 |
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509 show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union) |
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510 apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer |
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511 unfolding interior_inter[THEN sym] proof- |
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512 have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto |
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513 have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" |
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514 apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto |
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515 also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto |
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516 finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed |
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517 |
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518 lemma division_of_unions: assumes "finite f" "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)" |
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519 "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}" |
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520 shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+ |
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521 apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)]) |
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522 using division_ofD[OF assms(2)] by auto |
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523 |
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524 lemma elementary_union_interval: assumes "p division_of \<Union>p" |
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525 obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof- |
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526 note assm=division_ofD[OF assms] |
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527 have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto |
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528 have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto |
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529 { presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis" |
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530 "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis" |
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531 thus thesis by auto |
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532 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) . |
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533 thus thesis apply(rule_tac that[of p]) unfolding as by auto |
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534 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto |
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535 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}" |
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536 show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI) |
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537 unfolding finite_insert apply(rule assm(1)) unfolding Union_insert |
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538 using assm(2-4) as apply- by(fastsimp dest: assm(5))+ |
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539 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}" |
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540 have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1 |
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541 from assm(4)[OF this] guess c .. then guess d .. |
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542 thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto |
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543 qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]] |
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544 let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}" |
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545 show thesis apply(rule that[of "?D"]) proof(rule division_ofI) |
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546 have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto |
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547 show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto |
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548 show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym] |
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549 using q(6) by auto |
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550 fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto |
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551 show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto |
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552 fix k' assume k':"k'\<in>?D" "k\<noteq>k'" |
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553 obtain x where x: "k \<in>insert {a..b} (q x)" "x\<in>p" using k by auto |
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554 obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto |
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555 show "interior k \<inter> interior k' = {}" proof(cases "x=x'") |
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556 case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto |
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557 next case False |
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558 { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" |
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559 "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis" |
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560 thus ?thesis by auto } |
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561 { assume as':"k = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto } |
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562 { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x k'(2) unfolding as' by auto } |
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563 assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}" |
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564 guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this |
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565 have "interior k \<inter> interior {a..b} = {}" apply(rule q(5)) using x k'(2) using as' by auto |
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566 hence "interior k \<subseteq> interior x" apply- |
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567 apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover |
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568 guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this |
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569 have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto |
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570 hence "interior k' \<subseteq> interior x'" apply- |
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571 apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto |
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572 ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto |
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573 qed qed } qed |
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574 |
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575 lemma elementary_unions_intervals: |
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576 assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}" |
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577 obtains p where "p division_of (\<Union>f)" proof- |
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578 have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) |
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579 show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto |
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580 fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f" |
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581 from this(3) guess p .. note p=this |
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582 from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this |
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583 have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto |
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584 show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b] |
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585 unfolding Union_insert ab * by auto |
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586 qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed |
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587 |
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588 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)" |
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589 obtains p where "p division_of (s \<union> t)" |
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590 proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto |
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591 hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto |
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592 show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"]) |
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593 unfolding * prefer 3 apply(rule_tac p=p in that) |
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594 using assms[unfolded division_of_def] by auto qed |
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595 |
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596 lemma partial_division_extend: fixes t::"(real^'n) set" |
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597 assumes "p division_of s" "q division_of t" "s \<subseteq> t" |
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598 obtains r where "p \<subseteq> r" "r division_of t" proof- |
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599 note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)] |
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600 obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto |
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601 guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]]) |
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602 apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+ note r1 = this division_ofD[OF this(2)] |
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603 guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto |
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604 then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" |
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605 apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto |
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606 { fix x assume x:"x\<in>t" "x\<notin>s" |
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607 hence "x\<in>\<Union>r1" unfolding r1 using ab by auto |
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608 then guess r unfolding Union_iff .. note r=this moreover |
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609 have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto |
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610 thus False using x by auto qed |
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611 ultimately have "x\<in>\<Union>(r1 - p)" by auto } |
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612 hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto |
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613 show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union) |
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614 unfolding divp(6) apply(rule assms r2)+ |
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615 proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}" |
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616 proof(rule inter_interior_unions_intervals) |
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617 show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto |
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618 have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto |
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619 show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule) |
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620 fix m x assume as:"m\<in>r1-p" |
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621 have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals) |
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622 show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto |
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623 show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto |
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624 qed thus "interior s \<inter> interior m = {}" unfolding divp by auto |
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625 qed qed |
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626 thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto |
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627 qed auto qed |
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628 |
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629 subsection {* Tagged (partial) divisions. *} |
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630 |
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631 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where |
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632 "(s tagged_partial_division_of i) \<equiv> |
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633 finite s \<and> |
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634 (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and> |
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635 (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2)) |
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636 \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))" |
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637 |
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638 lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i" |
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639 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" |
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640 "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}" |
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641 "\<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) = {}" |
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642 using assms unfolding tagged_partial_division_of_def apply- by blast+ |
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643 |
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644 definition tagged_division_of (infixr "tagged'_division'_of" 40) where |
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645 "(s tagged_division_of i) \<equiv> |
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646 (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
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647 |
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648 lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s" |
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649 unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
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650 |
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651 lemma tagged_division_of: |
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652 "(s tagged_division_of i) \<longleftrightarrow> |
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653 finite s \<and> |
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654 (\<forall>x k. (x,k) \<in> s |
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655 \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and> |
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656 (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2)) |
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657 \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and> |
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658 (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
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659 unfolding tagged_division_of_def tagged_partial_division_of_def by auto |
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660 |
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661 lemma tagged_division_ofI: assumes |
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662 "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}" |
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663 "\<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) = {})" |
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664 "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" |
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665 shows "s tagged_division_of i" |
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666 unfolding tagged_division_of apply(rule) defer apply rule |
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667 apply(rule allI impI conjI assms)+ apply assumption |
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668 apply(rule, rule assms, assumption) apply(rule assms, assumption) |
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669 using assms(1,5-) apply- by blast+ |
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670 |
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671 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i" |
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672 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}" |
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673 "\<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) = {})" |
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674 "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+ |
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675 |
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676 lemma division_of_tagged_division: assumes"s tagged_division_of i" shows "(snd ` s) division_of i" |
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677 proof(rule division_ofI) note assm=tagged_division_ofD[OF assms] |
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678 show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto |
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679 fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto |
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680 thus "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+ |
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681 fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto |
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682 thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto |
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683 qed |
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684 |
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685 lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i" |
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686 shows "(snd ` s) division_of \<Union>(snd ` s)" |
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687 proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms] |
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688 show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto |
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689 fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto |
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690 thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto |
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691 fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto |
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692 thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto |
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693 qed |
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694 |
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695 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s" |
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696 shows "t tagged_partial_division_of i" |
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697 using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast |
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698 |
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699 lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector" |
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700 assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0" |
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701 shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)" |
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702 proof- note assm=tagged_division_ofD[OF assms(1)] |
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703 have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto |
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704 show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero) |
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705 show "finite p" using assm by auto |
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706 fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" |
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707 obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto |
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708 have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto |
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709 hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto |
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710 hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto |
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711 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 |
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712 thus "d (snd x) = 0" unfolding ab by auto qed qed |
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713 |
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714 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto |
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715 |
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716 lemma tagged_division_of_empty: "{} tagged_division_of {}" |
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717 unfolding tagged_division_of by auto |
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718 |
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719 lemma tagged_partial_division_of_trivial[simp]: |
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720 "p tagged_partial_division_of {} \<longleftrightarrow> p = {}" |
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721 unfolding tagged_partial_division_of_def by auto |
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722 |
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723 lemma tagged_division_of_trivial[simp]: |
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724 "p tagged_division_of {} \<longleftrightarrow> p = {}" |
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725 unfolding tagged_division_of by auto |
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726 |
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727 lemma tagged_division_of_self: |
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728 "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}" |
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729 apply(rule tagged_division_ofI) by auto |
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730 |
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731 lemma tagged_division_union: |
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732 assumes "p1 tagged_division_of s1" "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}" |
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733 shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)" |
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734 proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)] |
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735 show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto |
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736 show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast |
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737 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 |
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738 show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast |
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739 fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')" |
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740 have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast |
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741 show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1") |
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742 apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5)) |
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743 using p1(3) p2(3) using xk xk' by auto qed |
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744 |
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745 lemma tagged_division_unions: |
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746 assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)" |
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747 "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" |
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748 shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)" |
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749 proof(rule tagged_division_ofI) |
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750 note assm = tagged_division_ofD[OF assms(2)[rule_format]] |
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751 show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto |
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752 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 |
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753 also have "\<dots> = \<Union>iset" using assm(6) by auto |
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754 finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . |
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755 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 |
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756 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 |
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757 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 |
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758 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) |
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759 using assms(3)[rule_format] subset_interior by blast |
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760 show "interior k \<inter> interior k' = {}" apply(cases "i=i'") |
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761 using assm(5)[OF i _ xk'(2)] i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto |
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762 qed |
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763 |
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764 lemma tagged_partial_division_of_union_self: |
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765 assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))" |
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766 apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto |
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767 |
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768 lemma tagged_division_of_union_self: assumes "p tagged_division_of s" |
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769 shows "p tagged_division_of (\<Union>(snd ` p))" |
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770 apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto |
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771 |
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772 subsection {* Fine-ness of a partition w.r.t. a gauge. *} |
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773 |
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774 definition fine (infixr "fine" 46) where |
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775 "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))" |
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776 |
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777 lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" |
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778 shows "d fine s" using assms unfolding fine_def by auto |
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779 |
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780 lemma fineD[dest]: assumes "d fine s" |
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781 shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto |
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782 |
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783 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p" |
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784 unfolding fine_def by auto |
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785 |
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786 lemma fine_inters: |
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787 "(\<lambda>x. \<Inter> {f d x | d. d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)" |
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788 unfolding fine_def by blast |
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789 |
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790 lemma fine_union: |
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791 "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)" |
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792 unfolding fine_def by blast |
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793 |
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794 lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)" |
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795 unfolding fine_def by auto |
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796 |
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797 lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p" |
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798 unfolding fine_def by blast |
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799 |
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800 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *} |
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801 |
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802 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where |
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803 "(f has_integral_compact_interval y) i \<equiv> |
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804 (\<forall>e>0. \<exists>d. gauge d \<and> |
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805 (\<forall>p. p tagged_division_of i \<and> d fine p |
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806 \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))" |
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807 |
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808 definition has_integral (infixr "has'_integral" 46) where |
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809 "((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv> |
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810 if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i |
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811 else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} |
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812 \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and> |
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813 norm(z - y) < e))" |
|
814 |
|
815 lemma has_integral: |
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816 "(f has_integral y) ({a..b}) \<longleftrightarrow> |
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817 (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p |
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818 \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))" |
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819 unfolding has_integral_def has_integral_compact_interval_def by auto |
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820 |
|
821 lemma has_integralD[dest]: assumes |
|
822 "(f has_integral y) ({a..b})" "e>0" |
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823 obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p |
|
824 \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e" |
|
825 using assms unfolding has_integral by auto |
|
826 |
|
827 lemma has_integral_alt: |
|
828 "(f has_integral y) i \<longleftrightarrow> |
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829 (if (\<exists>a b. i = {a..b}) then (f has_integral y) i |
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830 else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} |
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831 \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) |
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832 has_integral z) ({a..b}) \<and> |
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833 norm(z - y) < e)))" |
|
834 unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto |
|
835 |
|
836 lemma has_integral_altD: |
|
837 assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0" |
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838 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)" |
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839 using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto |
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840 |
|
841 definition integrable_on (infixr "integrable'_on" 46) where |
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842 "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i" |
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843 |
|
844 definition "integral i f \<equiv> SOME y. (f has_integral y) i" |
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845 |
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846 lemma integrable_integral[dest]: |
|
847 "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i" |
|
848 unfolding integrable_on_def integral_def by(rule someI_ex) |
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849 |
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850 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s" |
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851 unfolding integrable_on_def by auto |
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852 |
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853 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s" |
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854 by auto |
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855 |
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856 lemma setsum_content_null: |
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857 assumes "content({a..b}) = 0" "p tagged_division_of {a..b}" |
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858 shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)" |
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859 proof(rule setsum_0',rule) fix y assume y:"y\<in>p" |
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860 obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast |
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861 note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]] |
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862 from this(2) guess c .. then guess d .. note c_d=this |
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863 have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto |
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864 also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d] |
|
865 unfolding assms(1) c_d by auto |
|
866 finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" . |
|
867 qed |
|
868 |
|
869 subsection {* Some basic combining lemmas. *} |
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870 |
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871 lemma tagged_division_unions_exists: |
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872 assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p" |
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873 "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)" |
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874 obtains p where "p tagged_division_of i" "d fine p" |
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875 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]] |
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876 show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym] |
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877 apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer |
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878 apply(rule fine_unions) using pfn by auto |
|
879 qed |
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880 |
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881 subsection {* The set we're concerned with must be closed. *} |
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882 |
|
883 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)" |
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884 unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval) |
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885 |
|
886 subsection {* General bisection principle for intervals; might be useful elsewhere. *} |
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887 |
|
888 lemma interval_bisection_step: |
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889 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})" |
|
890 obtains c d where "~(P{c..d})" |
|
891 "\<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" |
|
892 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto |
|
893 note ab=this[unfolded interval_eq_empty not_ex not_less] |
|
894 { fix f have "finite f \<Longrightarrow> |
|
895 (\<forall>s\<in>f. P s) \<Longrightarrow> |
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896 (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow> |
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897 (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)" |
|
898 proof(induct f rule:finite_induct) |
|
899 case empty show ?case using assms(1) by auto |
|
900 next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format]) |
|
901 apply rule defer apply rule defer apply(rule inter_interior_unions_intervals) |
|
902 using insert by auto |
|
903 qed } note * = this |
|
904 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)}" |
|
905 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" |
|
906 { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False" |
|
907 thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto } |
|
908 assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}" |
|
909 have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) |
|
910 let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) .. |
|
911 (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}" |
|
912 have "?A \<subseteq> ?B" proof case goal1 |
|
913 then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format] |
|
914 have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto |
|
915 show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI) |
|
916 unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta |
|
917 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)" |
|
918 "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)" |
|
919 using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps) |
|
920 qed auto qed |
|
921 thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto |
|
922 fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) |
|
923 note c_d=this[rule_format] |
|
924 show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case |
|
925 using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed |
|
926 show "\<exists>a b. s = {a..b}" unfolding c_d by auto |
|
927 fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+) |
|
928 note e_f=this[rule_format] |
|
929 assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto |
|
930 then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto |
|
931 hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE) |
|
932 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 |
|
933 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 |
|
934 qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto |
|
935 show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *) |
|
936 fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}" |
|
937 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 |
|
938 show False using c_d(2)[of i] apply- apply(erule_tac disjE) |
|
939 proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2" |
|
940 show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps) |
|
941 next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i" |
|
942 show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps) |
|
943 qed qed qed |
|
944 also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule) |
|
945 fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff .. |
|
946 from this(1) guess c unfolding mem_Collect_eq .. then guess d .. |
|
947 note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]] |
|
948 show "x\<in>{a..b}" unfolding mem_interval proof |
|
949 fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" |
|
950 using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed |
|
951 next fix x assume x:"x\<in>{a..b}" |
|
952 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" |
|
953 (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i |
|
954 have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)" |
|
955 using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast |
|
956 qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq |
|
957 apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto |
|
958 qed finally show False using assms by auto qed |
|
959 |
|
960 lemma interval_bisection: |
|
961 assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}" |
|
962 obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})" |
|
963 proof- |
|
964 have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and> |
|
965 2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof- |
|
966 presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis" |
|
967 thus ?thesis apply(cases "P {fst x..snd x}") by auto |
|
968 next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . |
|
969 thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto |
|
970 qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this |
|
971 def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def |
|
972 have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and> |
|
973 (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> |
|
974 2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n") |
|
975 proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto |
|
976 case goal3 note S = ab_def funpow.simps o_def id_apply show ?case |
|
977 proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto |
|
978 next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto |
|
979 qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format] |
|
980 |
|
981 have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e" |
|
982 proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this |
|
983 show ?case apply(rule_tac x=n in exI) proof(rule,rule) |
|
984 fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}" |
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985 have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1) |
|
986 also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV" |
|
987 proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i" |
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988 using xy[unfolded mem_interval,THEN spec[where x=i]] |
|
989 unfolding vector_minus_component by auto qed |
|
990 also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib |
|
991 proof(rule setsum_mono) case goal1 thus ?case |
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992 proof(induct n) case 0 thus ?case unfolding AB by auto |
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993 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 |
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994 also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case . |
|
995 qed qed |
|
996 also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" . |
|
997 qed qed |
|
998 { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this |
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999 have "{A n..B n} \<subseteq> {A m..B m}" unfolding d |
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1000 proof(induct d) case 0 thus ?case by auto |
|
1001 next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc]) |
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1002 apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE) |
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1003 proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps) |
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1004 qed qed } note ABsubset = this |
|
1005 have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv]) |
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1006 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 |
|
1007 then guess x0 .. note x0=this[rule_format] |
|
1008 show thesis proof(rule that[rule_format,of x0]) |
|
1009 show "x0\<in>{a..b}" using x0[of 0] unfolding AB . |
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1010 fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this |
|
1011 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}" |
|
1012 apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer |
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1013 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 |
|
1014 show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto |
|
1015 show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto |
|
1016 qed qed qed |
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1017 |
|
1018 subsection {* Cousin's lemma. *} |
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1019 |
|
1020 lemma fine_division_exists: assumes "gauge g" |
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1021 obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p" |
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1022 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False" |
|
1023 then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto |
|
1024 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)" |
|
1025 guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as]) |
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1026 apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+ |
|
1027 proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto |
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1028 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 = {}" |
|
1029 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 |
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1030 apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto |
|
1031 qed note x=this |
|
1032 obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto |
|
1033 from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this |
|
1034 have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto |
|
1035 thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed |
|
1036 |
|
1037 subsection {* Basic theorems about integrals. *} |
|
1038 |
|
1039 lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1040 assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2" |
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1041 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto |
|
1042 have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> a b k1 k2. |
|
1043 (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False" |
|
1044 proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto |
|
1045 guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this |
|
1046 guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this |
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1047 guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this |
|
1048 let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)" |
|
1049 using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute) |
|
1050 also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" |
|
1051 apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto |
|
1052 finally show False by auto |
|
1053 qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False" |
|
1054 thus False apply-apply(cases "\<exists>a b. i = {a..b}") |
|
1055 using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) } |
|
1056 assume as:"\<not> (\<exists>a b. i = {a..b})" |
|
1057 guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format] |
|
1058 guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format] |
|
1059 have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval) |
|
1060 using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+ |
|
1061 note ab=conjunctD2[OF this[unfolded Un_subset_iff]] |
|
1062 guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this] |
|
1063 guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this] |
|
1064 have "z = w" using lem[OF w(1) z(1)] by auto |
|
1065 hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)" |
|
1066 using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) |
|
1067 also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2)) |
|
1068 finally show False by auto qed |
|
1069 |
|
1070 lemma integral_unique[intro]: |
|
1071 "(f has_integral y) k \<Longrightarrow> integral k f = y" |
|
1072 unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) |
|
1073 |
|
1074 lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1075 assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s" |
|
1076 proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a. |
|
1077 (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral |
|
1078 proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a" |
|
1079 assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)" |
|
1080 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)" |
|
1081 apply(rule_tac x="\<lambda>x. ball x 1" in exI) apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball) |
|
1082 proof(rule,rule,erule conjE) case goal1 |
|
1083 have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule) |
|
1084 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 |
|
1085 thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto |
|
1086 qed thus ?case using as by auto |
|
1087 qed auto qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1088 thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") |
|
1089 using assms by(auto simp add:has_integral intro:lem) } |
|
1090 have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto |
|
1091 assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P * |
|
1092 apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule) |
|
1093 proof- fix e::real and a b assume "e>0" |
|
1094 thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e" |
|
1095 apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto |
|
1096 qed auto qed |
|
1097 |
|
1098 lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s" |
|
1099 apply(rule has_integral_is_0) by auto |
|
1100 |
|
1101 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0" |
|
1102 using has_integral_unique[OF has_integral_0] by auto |
|
1103 |
|
1104 lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1105 assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s" |
|
1106 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format] |
|
1107 have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b. |
|
1108 (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})" |
|
1109 proof(subst has_integral,rule,rule) case goal1 |
|
1110 from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format] |
|
1111 have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto |
|
1112 guess g using has_integralD[OF goal1(1) *] . note g=this |
|
1113 show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1)) |
|
1114 proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" |
|
1115 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 |
|
1116 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" |
|
1117 unfolding o_def unfolding scaleR[THEN sym] * by simp |
|
1118 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 |
|
1119 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)" . |
|
1120 show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym] |
|
1121 apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps) |
|
1122 qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1123 thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) } |
|
1124 assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P |
|
1125 proof(rule,rule) fix e::real assume e:"0<e" |
|
1126 have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1)) |
|
1127 guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this |
|
1128 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)" |
|
1129 apply(rule_tac x=M in exI) apply(rule,rule M(1)) |
|
1130 proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this] |
|
1131 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)" |
|
1132 unfolding o_def apply(rule ext) using zero by auto |
|
1133 show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym] |
|
1134 apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps) |
|
1135 qed qed qed |
|
1136 |
|
1137 lemma has_integral_cmul: |
|
1138 shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s" |
|
1139 unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption) |
|
1140 by(rule scaleR.bounded_linear_right) |
|
1141 |
|
1142 lemma has_integral_neg: |
|
1143 shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s" |
|
1144 apply(drule_tac c="-1" in has_integral_cmul) by auto |
|
1145 |
|
1146 lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1147 assumes "(f has_integral k) s" "(g has_integral l) s" |
|
1148 shows "((\<lambda>x. f x + g x) has_integral (k + l)) s" |
|
1149 proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l. |
|
1150 (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow> |
|
1151 ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1 |
|
1152 show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto |
|
1153 guess d1 using has_integralD[OF goal1(1) *] . note d1=this |
|
1154 guess d2 using has_integralD[OF goal1(2) *] . note d2=this |
|
1155 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)" |
|
1156 apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)]) |
|
1157 proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p" |
|
1158 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)" |
|
1159 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] |
|
1160 by(rule setsum_cong2,auto) |
|
1161 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))" |
|
1162 unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>" |
|
1163 from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto |
|
1164 have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq]) |
|
1165 apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto |
|
1166 finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto |
|
1167 qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis" |
|
1168 thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) } |
|
1169 assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P |
|
1170 proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto |
|
1171 from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format] |
|
1172 from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format] |
|
1173 show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1) |
|
1174 proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}" |
|
1175 hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto |
|
1176 guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this] |
|
1177 guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this] |
|
1178 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 |
|
1179 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" |
|
1180 apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]]) |
|
1181 using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps) |
|
1182 qed qed qed |
|
1183 |
|
1184 lemma has_integral_sub: |
|
1185 shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s" |
|
1186 using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto |
|
1187 |
|
1188 lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0" |
|
1189 by(rule integral_unique has_integral_0)+ |
|
1190 |
|
1191 lemma integral_add: |
|
1192 shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> |
|
1193 integral s (\<lambda>x. f x + g x) = integral s f + integral s g" |
|
1194 apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1195 apply(rule has_integral_add) by assumption+ |
|
1196 |
|
1197 lemma integral_cmul: |
|
1198 shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f" |
|
1199 apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1200 apply(rule has_integral_cmul) by assumption+ |
|
1201 |
|
1202 lemma integral_neg: |
|
1203 shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f" |
|
1204 apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1205 apply(rule has_integral_neg) by assumption+ |
|
1206 |
|
1207 lemma integral_sub: |
|
1208 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" |
|
1209 apply(rule integral_unique) apply(drule integrable_integral)+ |
|
1210 apply(rule has_integral_sub) by assumption+ |
|
1211 |
|
1212 lemma integrable_0: "(\<lambda>x. 0) integrable_on s" |
|
1213 unfolding integrable_on_def using has_integral_0 by auto |
|
1214 |
|
1215 lemma integrable_add: |
|
1216 shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s" |
|
1217 unfolding integrable_on_def by(auto intro: has_integral_add) |
|
1218 |
|
1219 lemma integrable_cmul: |
|
1220 shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s" |
|
1221 unfolding integrable_on_def by(auto intro: has_integral_cmul) |
|
1222 |
|
1223 lemma integrable_neg: |
|
1224 shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s" |
|
1225 unfolding integrable_on_def by(auto intro: has_integral_neg) |
|
1226 |
|
1227 lemma integrable_sub: |
|
1228 shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s" |
|
1229 unfolding integrable_on_def by(auto intro: has_integral_sub) |
|
1230 |
|
1231 lemma integrable_linear: |
|
1232 shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s" |
|
1233 unfolding integrable_on_def by(auto intro: has_integral_linear) |
|
1234 |
|
1235 lemma integral_linear: |
|
1236 shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)" |
|
1237 apply(rule has_integral_unique) defer unfolding has_integral_integral |
|
1238 apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym] |
|
1239 apply(rule integrable_linear) by assumption+ |
|
1240 |
|
1241 lemma has_integral_setsum: |
|
1242 assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s" |
|
1243 shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s" |
|
1244 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct) |
|
1245 case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)] |
|
1246 apply(rule has_integral_add) using insert assms by auto |
|
1247 qed auto |
|
1248 |
|
1249 lemma integral_setsum: |
|
1250 shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow> |
|
1251 integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t" |
|
1252 apply(rule integral_unique) apply(rule has_integral_setsum) |
|
1253 using integrable_integral by auto |
|
1254 |
|
1255 lemma integrable_setsum: |
|
1256 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" |
|
1257 unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto |
|
1258 |
|
1259 lemma has_integral_eq: |
|
1260 assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s" |
|
1261 using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0] |
|
1262 using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto |
|
1263 |
|
1264 lemma integrable_eq: |
|
1265 shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s" |
|
1266 unfolding integrable_on_def using has_integral_eq[of s f g] by auto |
|
1267 |
|
1268 lemma has_integral_eq_eq: |
|
1269 shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)" |
|
1270 using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto |
|
1271 |
|
1272 lemma has_integral_null[dest]: |
|
1273 assumes "content({a..b}) = 0" shows "(f has_integral 0) ({a..b})" |
|
1274 unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer |
|
1275 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto |
|
1276 fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*) |
|
1277 have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right |
|
1278 using setsum_content_null[OF assms(1) p, of f] . |
|
1279 thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed |
|
1280 |
|
1281 lemma has_integral_null_eq[simp]: |
|
1282 shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)" |
|
1283 apply rule apply(rule has_integral_unique,assumption) |
|
1284 apply(drule has_integral_null,assumption) |
|
1285 apply(drule has_integral_null) by auto |
|
1286 |
|
1287 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0" |
|
1288 by(rule integral_unique,drule has_integral_null) |
|
1289 |
|
1290 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}" |
|
1291 unfolding integrable_on_def apply(drule has_integral_null) by auto |
|
1292 |
|
1293 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}" |
|
1294 unfolding empty_as_interval apply(rule has_integral_null) |
|
1295 using content_empty unfolding empty_as_interval . |
|
1296 |
|
1297 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0" |
|
1298 apply(rule,rule has_integral_unique,assumption) by auto |
|
1299 |
|
1300 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto |
|
1301 |
|
1302 lemma integral_empty[simp]: shows "integral {} f = 0" |
|
1303 apply(rule integral_unique) using has_integral_empty . |
|
1304 |
|
1305 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" |
|
1306 apply(rule has_integral_null) unfolding content_eq_0_interior |
|
1307 unfolding interior_closed_interval using interval_sing by auto |
|
1308 |
|
1309 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto |
|
1310 |
|
1311 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto |
|
1312 |
|
1313 subsection {* Cauchy-type criterion for integrability. *} |
|
1314 |
|
1315 lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" |
|
1316 shows "f integrable_on {a..b} \<longleftrightarrow> |
|
1317 (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and> |
|
1318 p2 tagged_division_of {a..b} \<and> d fine p2 |
|
1319 \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 - |
|
1320 setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)") |
|
1321 proof assume ?l |
|
1322 then guess y unfolding integrable_on_def has_integral .. note y=this |
|
1323 show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto |
|
1324 then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format] |
|
1325 show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+) |
|
1326 proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2" |
|
1327 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" |
|
1328 apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm]) |
|
1329 using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] . |
|
1330 qed qed |
|
1331 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto |
|
1332 from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format] |
|
1333 have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto |
|
1334 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- |
|
1335 proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed |
|
1336 from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]] |
|
1337 have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto |
|
1338 have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))" |
|
1339 proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this |
|
1340 show ?case apply(rule_tac x=N in exI) |
|
1341 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 |
|
1342 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" |
|
1343 apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2)) |
|
1344 using dp p(1) using mn by auto |
|
1345 qed qed |
|
1346 then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format] |
|
1347 show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI) |
|
1348 proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto |
|
1349 then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto |
|
1350 guess N2 using y[OF *] .. note N2=this |
|
1351 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)" |
|
1352 apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer |
|
1353 proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto |
|
1354 fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q" |
|
1355 have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto |
|
1356 show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r) |
|
1357 apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer |
|
1358 using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"] |
|
1359 using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed |
|
1360 |
|
1361 subsection {* Additivity of integral on abutting intervals. *} |
|
1362 |
|
1363 lemma interval_split: |
|
1364 "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}" |
|
1365 "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}" |
|
1366 apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq |
|
1367 unfolding Cart_lambda_beta by auto |
|
1368 |
|
1369 lemma content_split: |
|
1370 "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})" |
|
1371 proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def |
|
1372 { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto } |
|
1373 have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto |
|
1374 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))" |
|
1375 "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" |
|
1376 apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto |
|
1377 assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c |
|
1378 \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)" |
|
1379 by (auto simp add:field_simps) |
|
1380 moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False" |
|
1381 unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto |
|
1382 ultimately show ?thesis |
|
1383 unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto) |
|
1384 qed |
|
1385 |
|
1386 lemma division_split_left_inj: |
|
1387 assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2" |
|
1388 "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}" |
|
1389 shows "content(k1 \<inter> {x. x$k \<le> c}) = 0" |
|
1390 proof- note d=division_ofD[OF assms(1)] |
|
1391 have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})" |
|
1392 unfolding interval_split content_eq_0_interior by auto |
|
1393 guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this |
|
1394 guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this |
|
1395 have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto |
|
1396 show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]]) |
|
1397 defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed |
|
1398 |
|
1399 lemma division_split_right_inj: |
|
1400 assumes "d division_of i" "k1 \<in> d" "k2 \<in> d" "k1 \<noteq> k2" |
|
1401 "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" |
|
1402 shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0" |
|
1403 proof- note d=division_ofD[OF assms(1)] |
|
1404 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}) = {})" |
|
1405 unfolding interval_split content_eq_0_interior by auto |
|
1406 guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this |
|
1407 guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this |
|
1408 have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto |
|
1409 show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]]) |
|
1410 defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed |
|
1411 |
|
1412 lemma tagged_division_split_left_inj: |
|
1413 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}" |
|
1414 shows "content(k1 \<inter> {x. x$k \<le> c}) = 0" |
|
1415 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto |
|
1416 show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]]) |
|
1417 apply(rule_tac[1-2] *) using assms(2-) by auto qed |
|
1418 |
|
1419 lemma tagged_division_split_right_inj: |
|
1420 assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" |
|
1421 shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0" |
|
1422 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 |
|
1423 show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]]) |
|
1424 apply(rule_tac[1-2] *) using assms(2-) by auto qed |
|
1425 |
|
1426 lemma division_split: |
|
1427 assumes "p division_of {a..b::real^'n}" |
|
1428 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$k \<le> c})" (is "?p1 division_of ?I1") and |
|
1429 "{l \<inter> {x. x$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") |
|
1430 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms] |
|
1431 show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto |
|
1432 { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this |
|
1433 guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this |
|
1434 show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l |
|
1435 using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto |
|
1436 fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this |
|
1437 assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto } |
|
1438 { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this |
|
1439 guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this |
|
1440 show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l |
|
1441 using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto |
|
1442 fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this |
|
1443 assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto } |
|
1444 qed |
|
1445 |
|
1446 lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
1447 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})" |
|
1448 shows "(f has_integral (i + j)) ({a..b})" |
|
1449 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto |
|
1450 guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]] |
|
1451 guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]] |
|
1452 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" |
|
1453 show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+) |
|
1454 proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto |
|
1455 fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)] |
|
1456 have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c" |
|
1457 "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c" |
|
1458 proof- fix x kk assume as:"(x,kk)\<in>p" |
|
1459 show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c" |
|
1460 proof(rule ccontr) case goal1 |
|
1461 from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>" |
|
1462 using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto |
|
1463 hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast |
|
1464 then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans) |
|
1465 using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm) |
|
1466 thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps) |
|
1467 qed |
|
1468 show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c" |
|
1469 proof(rule ccontr) case goal1 |
|
1470 from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>" |
|
1471 using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto |
|
1472 hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast |
|
1473 then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans) |
|
1474 using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm) |
|
1475 thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps) |
|
1476 qed |
|
1477 qed |
|
1478 |
|
1479 have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto |
|
1480 have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}" |
|
1481 proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed |
|
1482 have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow> |
|
1483 setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})} |
|
1484 = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)" |
|
1485 apply(rule setsum_mono_zero_left) prefer 3 |
|
1486 proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)" |
|
1487 assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" |
|
1488 then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto |
|
1489 have "content (g k) = 0" using xk using content_empty by auto |
|
1490 thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto |
|
1491 qed auto |
|
1492 have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto |
|
1493 |
|
1494 let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}" |
|
1495 have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI) |
|
1496 apply(rule lem2 p(3))+ prefer 6 apply(rule fineI) |
|
1497 proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto |
|
1498 fix x l assume xl:"(x,l)\<in>?M1" |
|
1499 then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this |
|
1500 have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto |
|
1501 thus "l \<subseteq> d1 x" unfolding xl' by auto |
|
1502 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) |
|
1503 using lem0(1)[OF xl'(3-4)] by auto |
|
1504 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]) |
|
1505 fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1" |
|
1506 then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this |
|
1507 assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}" |
|
1508 proof(cases "l' = r' \<longrightarrow> x' = y'") |
|
1509 case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1510 next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto |
|
1511 thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1512 qed qed moreover |
|
1513 |
|
1514 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> {}}" |
|
1515 have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI) |
|
1516 apply(rule lem2 p(3))+ prefer 6 apply(rule fineI) |
|
1517 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 |
|
1518 fix x l assume xl:"(x,l)\<in>?M2" |
|
1519 then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note xl'=this |
|
1520 have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto |
|
1521 thus "l \<subseteq> d2 x" unfolding xl' by auto |
|
1522 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) |
|
1523 using lem0(2)[OF xl'(3-4)] by auto |
|
1524 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]) |
|
1525 fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2" |
|
1526 then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) . note yr'=this |
|
1527 assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}" |
|
1528 proof(cases "l' = r' \<longrightarrow> x' = y'") |
|
1529 case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1530 next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto |
|
1531 thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto |
|
1532 qed qed ultimately |
|
1533 |
|
1534 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" |
|
1535 apply- apply(rule norm_triangle_lt) by auto |
|
1536 also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto |
|
1537 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) |
|
1538 = (\<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 |
|
1539 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)" |
|
1540 unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)]) |
|
1541 defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *) |
|
1542 proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto |
|
1543 next case goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto |
|
1544 qed also note setsum_addf[THEN sym] |
|
1545 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 |
|
1546 = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv |
|
1547 proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this |
|
1548 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" |
|
1549 unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto |
|
1550 qed note setsum_cong2[OF this] |
|
1551 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 + |
|
1552 ((\<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) = |
|
1553 (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto } |
|
1554 finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed |
|
1555 |
|
1556 subsection {* A sort of converse, integrability on subintervals. *} |
|
1557 |
|
1558 lemma tagged_division_union_interval: |
|
1559 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})" |
|
1560 shows "(p1 \<union> p2) tagged_division_of ({a..b})" |
|
1561 proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto |
|
1562 show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms]) |
|
1563 unfolding interval_split interior_closed_interval |
|
1564 by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed |
|
1565 |
|
1566 lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector" |
|
1567 assumes "(f has_integral i) ({a..b})" "e>0" |
|
1568 obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and> |
|
1569 p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2 |
|
1570 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + |
|
1571 setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)" |
|
1572 proof- guess d using has_integralD[OF assms] . note d=this |
|
1573 show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+) |
|
1574 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 |
|
1575 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 |
|
1576 note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this |
|
1577 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)" |
|
1578 apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv |
|
1579 proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2" |
|
1580 have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this |
|
1581 have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp |
|
1582 moreover have "interior {x. x $ k = c} = {}" |
|
1583 proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto |
|
1584 then guess e unfolding mem_interior .. note e=this |
|
1585 have x:"x$k = c" using x interior_subset by fastsimp |
|
1586 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 |
|
1587 have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm |
|
1588 apply(rule le_less_trans[OF norm_le_l1]) unfolding * |
|
1589 unfolding setsum_delta[OF finite_UNIV] using e by auto |
|
1590 hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto |
|
1591 thus False unfolding mem_Collect_eq using e x by auto |
|
1592 qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto |
|
1593 thus "content b *\<^sub>R f a = 0" by auto |
|
1594 qed auto |
|
1595 also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+ |
|
1596 finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed |
|
1597 |
|
1598 lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}" |
|
1599 shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) |
|
1600 proof- guess y using assms unfolding integrable_on_def .. note y=this |
|
1601 def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n" |
|
1602 and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n" |
|
1603 show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym] |
|
1604 proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto |
|
1605 from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format] |
|
1606 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> |
|
1607 norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)" |
|
1608 show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule) |
|
1609 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" |
|
1610 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" |
|
1611 proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this |
|
1612 show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]] |
|
1613 using as unfolding interval_split b'_def[symmetric] a'_def[symmetric] |
|
1614 using p using assms by(auto simp add:group_simps) |
|
1615 qed qed |
|
1616 show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule) |
|
1617 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" |
|
1618 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" |
|
1619 proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this |
|
1620 show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]] |
|
1621 using as unfolding interval_split b'_def[symmetric] a'_def[symmetric] |
|
1622 using p using assms by(auto simp add:group_simps) qed qed qed qed |
|
1623 |
|
1624 subsection {* Generalized notion of additivity. *} |
|
1625 |
|
1626 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)" |
|
1627 |
|
1628 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where |
|
1629 "operative opp f \<equiv> |
|
1630 (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and> |
|
1631 (\<forall>a b c k. f({a..b}) = |
|
1632 opp (f({a..b} \<inter> {x. x$k \<le> c})) |
|
1633 (f({a..b} \<inter> {x. x$k \<ge> c})))" |
|
1634 |
|
1635 lemma operativeD[dest]: assumes "operative opp f" |
|
1636 shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)" |
|
1637 "\<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}))" |
|
1638 using assms unfolding operative_def by auto |
|
1639 |
|
1640 lemma operative_trivial: |
|
1641 "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp" |
|
1642 unfolding operative_def by auto |
|
1643 |
|
1644 lemma property_empty_interval: |
|
1645 "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" |
|
1646 using content_empty unfolding empty_as_interval by auto |
|
1647 |
|
1648 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp" |
|
1649 unfolding operative_def apply(rule property_empty_interval) by auto |
|
1650 |
|
1651 subsection {* Using additivity of lifted function to encode definedness. *} |
|
1652 |
|
1653 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))" |
|
1654 by (metis map_of.simps option.nchotomy) |
|
1655 |
|
1656 lemma exists_option: |
|
1657 "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" |
|
1658 by (metis map_of.simps option.nchotomy) |
|
1659 |
|
1660 fun lifted where |
|
1661 "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" | |
|
1662 "lifted opp None _ = (None::'b option)" | |
|
1663 "lifted opp _ None = None" |
|
1664 |
|
1665 lemma lifted_simp_1[simp]: "lifted opp v None = None" |
|
1666 apply(induct v) by auto |
|
1667 |
|
1668 definition "monoidal opp \<equiv> (\<forall>x y. opp x y = opp y x) \<and> |
|
1669 (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and> |
|
1670 (\<forall>x. opp (neutral opp) x = x)" |
|
1671 |
|
1672 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x" |
|
1673 "\<And>x y z. opp x (opp y z) = opp (opp x y) z" |
|
1674 "\<And>x. opp (neutral opp) x = x" shows "monoidal opp" |
|
1675 unfolding monoidal_def using assms by fastsimp |
|
1676 |
|
1677 lemma monoidal_ac: assumes "monoidal opp" |
|
1678 shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a" |
|
1679 "opp (opp a b) c = opp a (opp b c)" "opp a (opp b c) = opp b (opp a c)" |
|
1680 using assms unfolding monoidal_def apply- by metis+ |
|
1681 |
|
1682 lemma monoidal_simps[simp]: assumes "monoidal opp" |
|
1683 shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" |
|
1684 using monoidal_ac[OF assms] by auto |
|
1685 |
|
1686 lemma neutral_lifted[cong]: assumes "monoidal opp" |
|
1687 shows "neutral (lifted opp) = Some(neutral opp)" |
|
1688 apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3 |
|
1689 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y" |
|
1690 thus "x = Some (neutral opp)" apply(induct x) defer |
|
1691 apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality) |
|
1692 apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto |
|
1693 qed(auto simp add:monoidal_ac[OF assms]) |
|
1694 |
|
1695 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)" |
|
1696 unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto |
|
1697 |
|
1698 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}" |
|
1699 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)" |
|
1700 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)" |
|
1701 |
|
1702 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto |
|
1703 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto |
|
1704 |
|
1705 lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp" |
|
1706 unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto |
|
1707 |
|
1708 lemma support_clauses: |
|
1709 "\<And>f g s. support opp f {} = {}" |
|
1710 "\<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))" |
|
1711 "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}" |
|
1712 "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)" |
|
1713 "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)" |
|
1714 "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)" |
|
1715 "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)" |
|
1716 unfolding support_def by auto |
|
1717 |
|
1718 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)" |
|
1719 unfolding support_def by auto |
|
1720 |
|
1721 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp" |
|
1722 unfolding iterate_def fold'_def by auto |
|
1723 |
|
1724 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s" |
|
1725 shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" |
|
1726 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto |
|
1727 show ?thesis unfolding iterate_def if_P[OF True] * by auto |
|
1728 next case False note x=this |
|
1729 note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]] |
|
1730 show ?thesis proof(cases "f x = neutral opp") |
|
1731 case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True] |
|
1732 unfolding True monoidal_simps[OF assms(1)] by auto |
|
1733 next case False show ?thesis unfolding iterate_def fold'_def if_not_P[OF x] support_clauses if_not_P[OF False] |
|
1734 apply(subst fun_left_comm.fold_insert[OF * finite_support]) |
|
1735 using `finite s` unfolding support_def using False x by auto qed qed |
|
1736 |
|
1737 lemma iterate_some: |
|
1738 assumes "monoidal opp" "finite s" |
|
1739 shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2) |
|
1740 proof(induct s) case empty thus ?case using assms by auto |
|
1741 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P) |
|
1742 defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed |
|
1743 |
|
1744 subsection {* Two key instances of additivity. *} |
|
1745 |
|
1746 lemma neutral_add[simp]: |
|
1747 "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def |
|
1748 apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto |
|
1749 |
|
1750 lemma operative_content[intro]: "operative (op +) content" |
|
1751 unfolding operative_def content_split[THEN sym] neutral_add by auto |
|
1752 |
|
1753 lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0" |
|
1754 unfolding neutral_def apply(rule some_equality) defer |
|
1755 apply(erule_tac x=0 in allE) by auto |
|
1756 |
|
1757 lemma monoidal_monoid[intro]: |
|
1758 shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)" |
|
1759 unfolding monoidal_def neutral_monoid by(auto simp add: group_simps) |
|
1760 |
|
1761 lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
1762 shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)" |
|
1763 unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add |
|
1764 apply(rule,rule,rule,rule) defer apply(rule allI)+ |
|
1765 proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = |
|
1766 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) |
|
1767 (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)" |
|
1768 proof(cases "f integrable_on {a..b}") |
|
1769 case True show ?thesis unfolding if_P[OF True] |
|
1770 unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]] |
|
1771 unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) |
|
1772 apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+ |
|
1773 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}))" |
|
1774 proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def |
|
1775 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) |
|
1776 apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto |
|
1777 thus False using False by auto |
|
1778 qed thus ?thesis using False by auto |
|
1779 qed next |
|
1780 fix a b assume as:"content {a..b::real^'n} = 0" |
|
1781 thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0" |
|
1782 unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed |
|
1783 |
|
1784 subsection {* Points of division of a partition. *} |
|
1785 |
|
1786 definition "division_points (k::(real^'n) set) d = |
|
1787 {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and> |
|
1788 (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}" |
|
1789 |
|
1790 lemma division_points_finite: assumes "d division_of i" |
|
1791 shows "finite (division_points i d)" |
|
1792 proof- note assm = division_ofD[OF assms] |
|
1793 let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and> |
|
1794 (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}" |
|
1795 have *:"division_points i d = \<Union>(?M ` UNIV)" |
|
1796 unfolding division_points_def by auto |
|
1797 show ?thesis unfolding * using assm by auto qed |
|
1798 |
|
1799 lemma division_points_subset: |
|
1800 assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k" |
|
1801 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$k \<le> c} = {})} |
|
1802 \<subseteq> division_points ({a..b}) d" (is ?t1) and |
|
1803 "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})} |
|
1804 \<subseteq> division_points ({a..b}) d" (is ?t2) |
|
1805 proof- note assm = division_ofD[OF assms(1)] |
|
1806 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" |
|
1807 "\<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" |
|
1808 using assms using less_imp_le by auto |
|
1809 show ?t1 unfolding division_points_def interval_split[of a b] |
|
1810 unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding * |
|
1811 unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+ |
|
1812 proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)" |
|
1813 "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> {}" |
|
1814 from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this |
|
1815 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 . |
|
1816 have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto |
|
1817 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)" |
|
1818 using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply- |
|
1819 apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **] |
|
1820 apply(case_tac[!] "fst x = k") using assms by auto |
|
1821 qed |
|
1822 show ?t2 unfolding division_points_def interval_split[of a b] |
|
1823 unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding * |
|
1824 unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+ |
|
1825 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" |
|
1826 "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}" |
|
1827 from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this |
|
1828 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 . |
|
1829 have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto |
|
1830 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)" |
|
1831 using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply- |
|
1832 apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **] |
|
1833 apply(case_tac[!] "fst x = k") using assms by auto qed qed |
|
1834 |
|
1835 lemma division_points_psubset: |
|
1836 assumes "d division_of {a..b}" "\<forall>i. a$i < b$i" "a$k < c" "c < b$k" |
|
1837 "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c" |
|
1838 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$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") |
|
1839 "division_points ({a..b} \<inter> {x. x$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") |
|
1840 proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le) |
|
1841 guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this |
|
1842 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 |
|
1843 unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto |
|
1844 have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k" |
|
1845 "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k" |
|
1846 unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds) |
|
1847 unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto |
|
1848 have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE) |
|
1849 apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer |
|
1850 apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI) |
|
1851 unfolding division_points_def unfolding interval_bounds[OF ab] |
|
1852 apply (auto simp add:interval_bounds) unfolding * by auto |
|
1853 thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto |
|
1854 |
|
1855 have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k" |
|
1856 "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k" |
|
1857 unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds) |
|
1858 unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto |
|
1859 have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE) |
|
1860 apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer |
|
1861 apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI) |
|
1862 unfolding division_points_def unfolding interval_bounds[OF ab] |
|
1863 apply (auto simp add:interval_bounds) unfolding * by auto |
|
1864 thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed |
|
1865 |
|
1866 subsection {* Preservation by divisions and tagged divisions. *} |
|
1867 |
|
1868 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s" |
|
1869 unfolding support_def by auto |
|
1870 |
|
1871 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f" |
|
1872 unfolding iterate_def support_support by auto |
|
1873 |
|
1874 lemma iterate_expand_cases: |
|
1875 "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)" |
|
1876 apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto |
|
1877 |
|
1878 lemma iterate_image: assumes "monoidal opp" "inj_on f s" |
|
1879 shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" |
|
1880 proof- have *:"\<And>s. finite s \<Longrightarrow> \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow> |
|
1881 iterate opp (f ` s) g = iterate opp s (g \<circ> f)" |
|
1882 proof- case goal1 show ?case using goal1 |
|
1883 proof(induct s) case empty thus ?case using assms(1) by auto |
|
1884 next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] |
|
1885 unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym]) |
|
1886 unfolding image_insert defer apply(subst iterate_insert[OF assms(1)]) |
|
1887 apply(rule finite_imageI insert)+ apply(subst if_not_P) |
|
1888 unfolding image_iff o_def using insert(2,4) by auto |
|
1889 qed qed |
|
1890 show ?thesis |
|
1891 apply(cases "finite (support opp g (f ` s))") |
|
1892 apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym]) |
|
1893 unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric] |
|
1894 apply(rule subset_inj_on[OF assms(2) support_subset])+ |
|
1895 apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False) |
|
1896 apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed |
|
1897 |
|
1898 |
|
1899 (* This lemma about iterations comes up in a few places. *) |
|
1900 lemma iterate_nonzero_image_lemma: |
|
1901 assumes "monoidal opp" "finite s" "g(a) = neutral opp" |
|
1902 "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp" |
|
1903 shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)" |
|
1904 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto |
|
1905 have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s" |
|
1906 unfolding support_def using assms(3) by auto |
|
1907 show ?thesis unfolding * |
|
1908 apply(subst iterate_support[THEN sym]) unfolding support_clauses |
|
1909 apply(subst iterate_image[OF assms(1)]) defer |
|
1910 apply(subst(2) iterate_support[THEN sym]) apply(subst **) |
|
1911 unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed |
|
1912 |
|
1913 lemma iterate_eq_neutral: |
|
1914 assumes "monoidal opp" "\<forall>x \<in> s. (f(x) = neutral opp)" |
|
1915 shows "(iterate opp s f = neutral opp)" |
|
1916 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto |
|
1917 show ?thesis apply(subst iterate_support[THEN sym]) |
|
1918 unfolding * using assms(1) by auto qed |
|
1919 |
|
1920 lemma iterate_op: assumes "monoidal opp" "finite s" |
|
1921 shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2) |
|
1922 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto |
|
1923 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3) |
|
1924 unfolding monoidal_ac[OF assms(1)] by(rule refl) qed |
|
1925 |
|
1926 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x" |
|
1927 shows "iterate opp s f = iterate opp s g" |
|
1928 proof- have *:"support opp g s = support opp f s" |
|
1929 unfolding support_def using assms(2) by auto |
|
1930 show ?thesis |
|
1931 proof(cases "finite (support opp f s)") |
|
1932 case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases) |
|
1933 unfolding * by auto |
|
1934 next def su \<equiv> "support opp f s" |
|
1935 case True note support_subset[of opp f s] |
|
1936 thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True |
|
1937 unfolding su_def[symmetric] |
|
1938 proof(induct su) case empty show ?case by auto |
|
1939 next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] |
|
1940 unfolding if_not_P[OF insert(2)] apply(subst insert(3)) |
|
1941 defer apply(subst assms(2)[of x]) using insert by auto qed qed qed |
|
1942 |
|
1943 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto |
|
1944 |
|
1945 lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a" |
|
1946 assumes "monoidal opp" "operative opp f" "d division_of {a..b}" |
|
1947 shows "iterate opp d f = f {a..b}" |
|
1948 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms |
|
1949 proof(induct C arbitrary:a b d rule:full_nat_induct) |
|
1950 case goal1 |
|
1951 { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case" |
|
1952 thus ?case apply-apply(cases) defer apply assumption |
|
1953 proof- assume as:"content {a..b} = 0" |
|
1954 show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)]) |
|
1955 proof fix x assume x:"x\<in>d" |
|
1956 then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+ |
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1957 thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] |
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1958 using operativeD(1)[OF assms(2)] x by auto |
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1959 qed qed } |
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1960 assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq] |
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1961 hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case |
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1962 proof(cases "division_points {a..b} d = {}") |
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1963 case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and> |
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1964 (\<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)" |
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1965 unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule) |
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1966 apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule) |
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1967 proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this] |
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1968 hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto |
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1969 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 |
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1970 have "(j, u$j) \<notin> division_points {a..b} d" |
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1971 "(j, v$j) \<notin> division_points {a..b} d" using True by auto |
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1972 note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps] |
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1973 note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]] |
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1974 moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] |
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1975 unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) |
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1976 unfolding interval_ne_empty mem_interval by auto |
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1977 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" |
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1978 unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto |
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1979 qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le) |
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1980 note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff] |
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1981 then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this |
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1982 have "{a..b} \<in> d" |
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1983 proof- { presume "i = {a..b}" thus ?thesis using i by auto } |
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1984 { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto } |
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1985 show "u = a" "v = b" unfolding Cart_eq |
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1986 proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j] |
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1987 thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto |
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1988 qed qed |
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1989 hence *:"d = insert {a..b} (d - {{a..b}})" by auto |
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1990 have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)]) |
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1991 proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this] |
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1992 then guess u v apply-by(erule exE conjE)+ note uv=this |
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1993 have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto |
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1994 then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto |
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1995 hence "u$j = v$j" using uv(2)[rule_format,of j] by auto |
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1996 hence "content {u..v} = 0" unfolding content_eq_0 apply(rule_tac x=j in exI) by auto |
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1997 thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)]) |
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1998 qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) |
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1999 apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto |
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2000 next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto |
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2001 then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv |
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2002 by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']] |
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2003 from this(3) guess j .. note j=this |
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2004 def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}" |
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2005 def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}" |
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2006 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)" |
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2007 note division_points_psubset[OF goal1(4) ab kc(1-2) j] |
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2008 note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)] |
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2009 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})" |
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2010 apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format]) |
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2011 using division_split[OF goal1(4), where k=k and c=c] |
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2012 unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono |
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2013 using goal1(2-3) using division_points_finite[OF goal1(4)] by auto |
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2014 have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev") |
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2015 unfolding * apply(rule operativeD(2)) using goal1(3) . |
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2016 also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))" |
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2017 unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def]) |
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2018 unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+ |
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2019 unfolding empty_as_interval[THEN sym] apply(rule content_empty) |
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2020 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" |
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2021 from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this |
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2022 show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split |
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2023 apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj) |
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2024 apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+ |
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2025 qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))" |
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2026 unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def]) |
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2027 unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+ |
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2028 unfolding empty_as_interval[THEN sym] apply(rule content_empty) |
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2029 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" |
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2030 from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this |
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2031 show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split |
|
2032 apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj) |
|
2033 apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+ |
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2034 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}))" |
|
2035 unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) . |
|
2036 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}))) |
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2037 = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3 |
|
2038 apply(rule iterate_op[THEN sym]) using goal1 by auto |
|
2039 finally show ?thesis by auto |
|
2040 qed qed qed |
|
2041 |
|
2042 lemma iterate_image_nonzero: assumes "monoidal opp" |
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2043 "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp" |
|
2044 shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms |
|
2045 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl]) |
|
2046 case goal1 show ?case using assms(1) by auto |
|
2047 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto |
|
2048 show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)]) |
|
2049 apply(rule finite_imageI goal2)+ |
|
2050 apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer |
|
2051 apply(subst iterate_insert[OF assms(1) goal2(1)]) defer |
|
2052 apply(subst iterate_insert[OF assms(1) goal2(1)]) |
|
2053 unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE) |
|
2054 apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format]) |
|
2055 using goal2 unfolding o_def by auto qed |
|
2056 |
|
2057 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}" |
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2058 shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}" |
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2059 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)] |
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2060 have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding * |
|
2061 apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ |
|
2062 unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE) |
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2063 proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba" |
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2064 guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this |
|
2065 show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)]) |
|
2066 unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)] |
|
2067 unfolding as(4)[THEN sym] uv by auto |
|
2068 qed also have "\<dots> = f {a..b}" |
|
2069 using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] . |
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2070 finally show ?thesis . qed |
|
2071 |
|
2072 subsection {* Additivity of content. *} |
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2073 |
|
2074 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f" |
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2075 proof- have *:"setsum f s = setsum f (support op + f s)" |
|
2076 apply(rule setsum_mono_zero_right) |
|
2077 unfolding support_def neutral_monoid using assms by auto |
|
2078 thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def |
|
2079 unfolding neutral_monoid . qed |
|
2080 |
|
2081 lemma additive_content_division: assumes "d division_of {a..b}" |
|
2082 shows "setsum content d = content({a..b})" |
|
2083 unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym] |
|
2084 apply(subst setsum_iterate) using assms by auto |
|
2085 |
|
2086 lemma additive_content_tagged_division: |
|
2087 assumes "d tagged_division_of {a..b}" |
|
2088 shows "setsum (\<lambda>(x,l). content l) d = content({a..b})" |
|
2089 unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym] |
|
2090 apply(subst setsum_iterate) using assms by auto |
|
2091 |
|
2092 subsection {* Finally, the integral of a constant\<forall> *} |
|
2093 |
|
2094 lemma has_integral_const[intro]: |
|
2095 "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})" |
|
2096 unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI) |
|
2097 apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE) |
|
2098 unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def]) |
|
2099 defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto |
|
2100 |
|
2101 subsection {* Bounds on the norm of Riemann sums and the integral itself. *} |
|
2102 |
|
2103 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e" |
|
2104 shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r") |
|
2105 apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym] |
|
2106 apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero) |
|
2107 apply(subst real_mult_commute) apply(rule mult_left_mono) |
|
2108 apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2) |
|
2109 apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)] |
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2110 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" . |
|
2111 fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+ |
|
2112 thus "0 \<le> content x" using content_pos_le by auto |
|
2113 qed(insert assms,auto) |
|
2114 |
|
2115 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e" |
|
2116 shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})" |
|
2117 proof(cases "{a..b} = {}") case True |
|
2118 show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto |
|
2119 next case False show ?thesis |
|
2120 apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR |
|
2121 apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer |
|
2122 unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono) |
|
2123 apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2) |
|
2124 apply(subst o_def, rule abs_of_nonneg) |
|
2125 proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl) |
|
2126 unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto |
|
2127 guess w using nonempty_witness[OF False] . |
|
2128 thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto |
|
2129 fix xk assume *:"xk\<in>p" guess x k using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this] |
|
2130 from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this |
|
2131 show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le) |
|
2132 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 |
|
2133 qed(insert assms,auto) qed |
|
2134 |
|
2135 lemma rsum_diff_bound: |
|
2136 assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e" |
|
2137 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})" |
|
2138 apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm]) |
|
2139 unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto |
|
2140 |
|
2141 lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2142 assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B" |
|
2143 shows "norm i \<le> B * content {a..b}" |
|
2144 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis" |
|
2145 thus ?thesis proof(cases ?P) case False |
|
2146 hence *:"content {a..b} = 0" using content_lt_nz by auto |
|
2147 hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto |
|
2148 show ?thesis unfolding * ** using assms(1) by auto |
|
2149 qed auto } assume ab:?P |
|
2150 { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto } |
|
2151 assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto |
|
2152 from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format] |
|
2153 from fine_division_exists[OF this(1), of a b] guess p . note p=this |
|
2154 have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)" |
|
2155 proof- case goal1 thus ?case unfolding not_less |
|
2156 using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto |
|
2157 qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed |
|
2158 |
|
2159 subsection {* Similar theorems about relationship among components. *} |
|
2160 |
|
2161 lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2162 assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i" |
|
2163 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" |
|
2164 unfolding setsum_component apply(rule setsum_mono) |
|
2165 apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv |
|
2166 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab] |
|
2167 from this(3) guess u v apply-by(erule exE)+ note b=this |
|
2168 show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b |
|
2169 unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono) |
|
2170 defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed |
|
2171 |
|
2172 lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2173 assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k" |
|
2174 shows "i$k \<le> j$k" |
|
2175 proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> |
|
2176 (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" |
|
2177 proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto |
|
2178 guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format] |
|
2179 guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format] |
|
2180 guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter . |
|
2181 note p = this(1) conjunctD2[OF this(2)] note le_less_trans[OF component_le_norm, of _ _ k] |
|
2182 note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]] |
|
2183 thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt |
|
2184 qed let ?P = "\<exists>a b. s = {a..b}" |
|
2185 { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P) |
|
2186 case True then guess a b apply-by(erule exE)+ note s=this |
|
2187 show ?thesis apply(rule lem) using assms[unfolded s] by auto |
|
2188 qed auto } assume as:"\<not> ?P" |
|
2189 { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto } |
|
2190 assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto |
|
2191 note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format] |
|
2192 have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+ |
|
2193 from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+ |
|
2194 note ab = conjunctD2[OF this[unfolded Un_subset_iff]] |
|
2195 guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this] |
|
2196 guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this] |
|
2197 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*) |
|
2198 note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover |
|
2199 have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately |
|
2200 show False unfolding Cart_nth.diff by(rule *) qed |
|
2201 |
|
2202 lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2203 assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. (f x)$k \<le> (g x)$k" |
|
2204 shows "(integral s f)$k \<le> (integral s g)$k" |
|
2205 apply(rule has_integral_component_le) using integrable_integral assms by auto |
|
2206 |
|
2207 lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1" |
|
2208 assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x" |
|
2209 shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)]) |
|
2210 using assms(3) unfolding vector_le_def by auto |
|
2211 |
|
2212 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1" |
|
2213 assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x" |
|
2214 shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)" |
|
2215 apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto |
|
2216 |
|
2217 lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m" |
|
2218 assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k" |
|
2219 using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto |
|
2220 |
|
2221 lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m" |
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2222 assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k" |
|
2223 apply(rule has_integral_component_pos) using assms by auto |
|
2224 |
|
2225 lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1" |
|
2226 assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" |
|
2227 using has_integral_component_pos[OF assms(1), of 1] |
|
2228 using assms(2) unfolding vector_le_def by auto |
|
2229 |
|
2230 lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1" |
|
2231 assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" |
|
2232 apply(rule has_integral_dest_vec1_pos) using assms by auto |
|
2233 |
|
2234 lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m" |
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2235 assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0" |
|
2236 using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto |
|
2237 |
|
2238 lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1" |
|
2239 assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0" |
|
2240 using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto |
|
2241 |
|
2242 lemma has_integral_component_lbound: |
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2243 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" |
|
2244 using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2) |
|
2245 unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps) |
|
2246 |
|
2247 lemma has_integral_component_ubound: |
|
2248 assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B" |
|
2249 shows "i$k \<le> B * content({a..b})" |
|
2250 using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"] |
|
2251 unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps) |
|
2252 |
|
2253 lemma integral_component_lbound: |
|
2254 assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k" |
|
2255 shows "B * content({a..b}) \<le> (integral({a..b}) f)$k" |
|
2256 apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto |
|
2257 |
|
2258 lemma integral_component_ubound: |
|
2259 assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B" |
|
2260 shows "(integral({a..b}) f)$k \<le> B * content({a..b})" |
|
2261 apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto |
|
2262 |
|
2263 subsection {* Uniform limit of integrable functions is integrable. *} |
|
2264 |
|
2265 lemma real_arch_invD: |
|
2266 "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)" |
|
2267 by(subst(asm) real_arch_inv) |
|
2268 |
|
2269 lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach" |
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2270 assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}" |
|
2271 shows "f integrable_on {a..b}" |
|
2272 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis" |
|
2273 show ?thesis apply cases apply(rule *,assumption) |
|
2274 unfolding content_lt_nz integrable_on_def using has_integral_null by auto } |
|
2275 assume as:"content {a..b} > 0" |
|
2276 have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto |
|
2277 from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format] |
|
2278 from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format] |
|
2279 |
|
2280 have "Cauchy i" unfolding Cauchy_def |
|
2281 proof(rule,rule) fix e::real assume "e>0" |
|
2282 hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps) |
|
2283 then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this |
|
2284 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) |
|
2285 proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this] |
|
2286 from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format] |
|
2287 from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format] |
|
2288 from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this |
|
2289 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" |
|
2290 proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)" |
|
2291 using norm_triangle_ineq[of "i1 - s1" "s1 - i2"] |
|
2292 using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps) |
|
2293 also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps) |
|
2294 finally show ?case . |
|
2295 qed |
|
2296 show ?case unfolding vector_dist_norm apply(rule lem2) defer |
|
2297 apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]]) |
|
2298 using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans) |
|
2299 apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"]) |
|
2300 proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse |
|
2301 using M as by(auto simp add:field_simps) |
|
2302 fix x assume x:"x \<in> {a..b}" |
|
2303 have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" |
|
2304 using g(1)[OF x, of n] g(1)[OF x, of m] by auto |
|
2305 also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono) |
|
2306 apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto |
|
2307 also have "\<dots> = 2 / real M" unfolding real_divide_def by auto |
|
2308 finally show "norm (g n x - g m x) \<le> 2 / real M" |
|
2309 using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"] |
|
2310 by(auto simp add:group_simps simp add:norm_minus_commute) |
|
2311 qed qed qed |
|
2312 from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this |
|
2313 |
|
2314 show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral |
|
2315 proof(rule,rule) |
|
2316 case goal1 hence *:"e/3 > 0" by auto |
|
2317 from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this |
|
2318 from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps) |
|
2319 from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this |
|
2320 from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format] |
|
2321 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" |
|
2322 proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)" |
|
2323 using norm_triangle_ineq[of "sf - sg" "sg - s"] |
|
2324 using norm_triangle_ineq[of "sg - i" " i - s"] by(auto simp add:group_simps) |
|
2325 also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps) |
|
2326 finally show ?case . |
|
2327 qed |
|
2328 show ?case apply(rule_tac x=g' in exI) apply(rule,rule g') |
|
2329 proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this] |
|
2330 show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *]) |
|
2331 apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption) |
|
2332 proof- have "content {a..b} < e / 3 * (real N2)" |
|
2333 using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps) |
|
2334 hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)" |
|
2335 apply-apply(rule less_le_trans,assumption) using `e>0` by auto |
|
2336 thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3" |
|
2337 unfolding inverse_eq_divide by(auto simp add:field_simps) |
|
2338 show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto) |
|
2339 qed qed qed qed |
|
2340 |
|
2341 subsection {* Negligible sets. *} |
|
2342 |
|
2343 definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))" |
|
2344 |
|
2345 lemma dest_vec1_indicator: |
|
2346 "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto |
|
2347 |
|
2348 lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto |
|
2349 |
|
2350 lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto |
|
2351 |
|
2352 lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1" |
|
2353 unfolding indicator_def by auto |
|
2354 |
|
2355 definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})" |
|
2356 |
|
2357 lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0" |
|
2358 unfolding indicator_def by auto |
|
2359 |
|
2360 subsection {* Negligibility of hyperplane. *} |
|
2361 |
|
2362 lemma vsum_nonzero_image_lemma: |
|
2363 assumes "finite s" "g(a) = 0" |
|
2364 "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0" |
|
2365 shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s" |
|
2366 unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer |
|
2367 apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+ |
|
2368 unfolding assms using neutral_add unfolding neutral_add using assms by auto |
|
2369 |
|
2370 lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} = |
|
2371 {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}" |
|
2372 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto |
|
2373 have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast |
|
2374 show ?thesis unfolding * ** interval_split by(rule refl) qed |
|
2375 |
|
2376 lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" |
|
2377 shows "{l \<inter> {x. abs(x$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})" |
|
2378 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto |
|
2379 have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto |
|
2380 note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split] |
|
2381 note division_split(2)[OF this, where c="c-e" and k=k] |
|
2382 thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit |
|
2383 apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer |
|
2384 apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule |
|
2385 apply(rule_tac x=l in exI) by blast+ qed |
|
2386 |
|
2387 lemma content_doublesplit: assumes "0 < e" |
|
2388 obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e" |
|
2389 proof(cases "content {a..b} = 0") |
|
2390 case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit |
|
2391 apply(rule le_less_trans[OF content_subset]) defer apply(subst True) |
|
2392 unfolding interval_doublesplit[THEN sym] using assms by auto |
|
2393 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" |
|
2394 note False[unfolded content_eq_0 not_ex not_le, rule_format] |
|
2395 hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt |
|
2396 hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis |
|
2397 proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto |
|
2398 have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> |
|
2399 (\<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) |
|
2400 = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl) |
|
2401 unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto |
|
2402 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) |
|
2403 unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding ** |
|
2404 unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl] |
|
2405 proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto |
|
2406 also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps) |
|
2407 finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e" |
|
2408 unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed |
|
2409 |
|
2410 lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" |
|
2411 unfolding negligible_def has_integral apply(rule,rule,rule,rule) |
|
2412 proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}" |
|
2413 show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d) |
|
2414 proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p" |
|
2415 have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)" |
|
2416 apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv |
|
2417 apply(cases,rule disjI1,assumption,rule disjI2) |
|
2418 proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto) |
|
2419 show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content]) |
|
2420 apply(rule set_ext,rule,rule) unfolding mem_Collect_eq |
|
2421 proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv] |
|
2422 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] |
|
2423 thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto |
|
2424 qed auto qed |
|
2425 note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]] |
|
2426 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 |
|
2427 apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv |
|
2428 apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst) |
|
2429 prefer 2 apply(subst(asm) eq_commute) apply assumption |
|
2430 apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le) |
|
2431 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}))" |
|
2432 apply(rule setsum_mono) unfolding split_paired_all split_conv |
|
2433 apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le) |
|
2434 also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]]) |
|
2435 proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}" |
|
2436 unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto |
|
2437 thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt |
|
2438 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" |
|
2439 apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all |
|
2440 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)" |
|
2441 guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this |
|
2442 show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le) |
|
2443 qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit] |
|
2444 note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]] |
|
2445 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)] |
|
2446 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" |
|
2447 apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym]) |
|
2448 apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p''] |
|
2449 proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v |
|
2450 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}" |
|
2451 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 |
|
2452 note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]] |
|
2453 hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto |
|
2454 thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] . |
|
2455 qed qed |
|
2456 finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" . |
|
2457 qed qed qed |
|
2458 |
|
2459 subsection {* A technical lemma about "refinement" of division. *} |
|
2460 |
|
2461 lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set" |
|
2462 assumes "p tagged_division_of {a..b}" "gauge d" |
|
2463 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" |
|
2464 proof- |
|
2465 let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow> |
|
2466 (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and> |
|
2467 (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))" |
|
2468 { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto |
|
2469 presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this |
|
2470 thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto |
|
2471 } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p" |
|
2472 show "?P p" apply(rule,rule) using as proof(induct p) |
|
2473 case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto |
|
2474 next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this |
|
2475 note tagged_partial_division_subset[OF insert(4) subset_insertI] |
|
2476 from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this] |
|
2477 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 |
|
2478 note p = tagged_partial_division_ofD[OF insert(4)] |
|
2479 from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this |
|
2480 |
|
2481 have "finite {k. \<exists>x. (x, k) \<in> p}" |
|
2482 apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq |
|
2483 apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto |
|
2484 hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}" |
|
2485 apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI) |
|
2486 unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption) |
|
2487 apply(rule p(5)) unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption |
|
2488 using insert(2) unfolding uv xk by auto |
|
2489 |
|
2490 show ?case proof(cases "{u..v} \<subseteq> d x") |
|
2491 case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule |
|
2492 unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self) |
|
2493 apply(rule p[unfolded xk uv] insertI1)+ apply(rule q1,rule int) |
|
2494 apply(rule,rule fine_union,subst fine_def) defer apply(rule q1) |
|
2495 unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule) |
|
2496 apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto |
|
2497 next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this |
|
2498 show ?thesis apply(rule_tac x="q2 \<union> q1" in exI) |
|
2499 apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+ |
|
2500 unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union) |
|
2501 apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE) |
|
2502 apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto |
|
2503 qed qed qed |
|
2504 |
|
2505 subsection {* Hence the main theorem about negligible sets. *} |
|
2506 |
|
2507 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)" |
|
2508 shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms |
|
2509 proof(induct) case (insert x s) |
|
2510 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 |
|
2511 show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto |
|
2512 |
|
2513 lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)" |
|
2514 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 |
|
2515 proof(induct) case (insert a s) |
|
2516 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 |
|
2517 show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)] |
|
2518 prefer 4 apply(subst insert(3)) unfolding add_right_cancel |
|
2519 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 |
|
2520 show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto |
|
2521 qed(insert insert, auto) qed auto |
|
2522 |
|
2523 lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2524 assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0" |
|
2525 shows "(f has_integral 0) t" |
|
2526 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})" |
|
2527 let ?f = "(\<lambda>x. if x \<in> t then f x else 0)" |
|
2528 show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl) |
|
2529 apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P |
|
2530 proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this |
|
2531 show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto |
|
2532 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)" |
|
2533 apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI) |
|
2534 apply(rule,rule P) using assms(2) by auto |
|
2535 qed |
|
2536 next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" |
|
2537 show "(f has_integral 0) {a..b}" unfolding has_integral |
|
2538 proof(safe) case goal1 |
|
2539 hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" |
|
2540 apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps) |
|
2541 note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] |
|
2542 from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]] |
|
2543 show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) |
|
2544 proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto |
|
2545 fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" |
|
2546 let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" |
|
2547 { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto } |
|
2548 assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N .. |
|
2549 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 |
|
2550 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)" |
|
2551 apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto |
|
2552 from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]] |
|
2553 have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) |
|
2554 unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto |
|
2555 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" |
|
2556 proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4 |
|
2557 apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed |
|
2558 have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) * |
|
2559 norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}" |
|
2560 unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right |
|
2561 apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 |
|
2562 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 |
|
2563 fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)" |
|
2564 unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg) |
|
2565 using tagged_division_ofD(4)[OF q(1) as''] by auto |
|
2566 next fix i::nat show "finite (q i)" using q by auto |
|
2567 next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>" |
|
2568 have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto |
|
2569 have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto |
|
2570 hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto |
|
2571 moreover note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv] |
|
2572 note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]] |
|
2573 moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)" |
|
2574 proof(cases "x\<in>s") case False thus ?thesis using assm by auto |
|
2575 next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto |
|
2576 moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto |
|
2577 ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps) |
|
2578 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)" |
|
2579 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 |
|
2580 qed(insert as, auto) |
|
2581 also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) |
|
2582 proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym]) |
|
2583 using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps) |
|
2584 qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym] |
|
2585 apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym] |
|
2586 apply(subst sumr_geometric) using goal1 by auto |
|
2587 finally show "?goal" by auto qed qed qed |
|
2588 |
|
2589 lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" |
|
2590 assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t" |
|
2591 shows "(g has_integral y) t" |
|
2592 proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a |
|
2593 assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}" |
|
2594 have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)]) |
|
2595 apply(rule has_integral_negligible[OF assms(1)]) using as by auto |
|
2596 hence "(g has_integral y) {a..b}" by auto } note * = this |
|
2597 show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe) |
|
2598 apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P |
|
2599 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) |
|
2600 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 |
|
2601 |
|
2602 lemma has_integral_spike_eq: |
|
2603 assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" |
|
2604 shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)" |
|
2605 apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto |
|
2606 |
|
2607 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t" |
|
2608 shows "g integrable_on t" |
|
2609 using assms unfolding integrable_on_def apply-apply(erule exE) |
|
2610 apply(rule,rule has_integral_spike) by fastsimp+ |
|
2611 |
|
2612 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" |
|
2613 shows "integral t f = integral t g" |
|
2614 unfolding integral_def using has_integral_spike_eq[OF assms] by auto |
|
2615 |
|
2616 subsection {* Some other trivialities about negligible sets. *} |
|
2617 |
|
2618 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def |
|
2619 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b] |
|
2620 apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption |
|
2621 using assms(2) unfolding indicator_def by auto qed |
|
2622 |
|
2623 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto |
|
2624 |
|
2625 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto |
|
2626 |
|
2627 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def |
|
2628 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b] |
|
2629 thus ?case apply(subst has_integral_spike_eq[OF assms(2)]) |
|
2630 defer apply assumption unfolding indicator_def by auto qed |
|
2631 |
|
2632 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)" |
|
2633 using negligible_union by auto |
|
2634 |
|
2635 lemma negligible_sing[intro]: "negligible {a::real^'n}" |
|
2636 proof- guess x using UNIV_witness[where 'a='n] .. |
|
2637 show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed |
|
2638 |
|
2639 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s" |
|
2640 apply(subst insert_is_Un) unfolding negligible_union_eq by auto |
|
2641 |
|
2642 lemma negligible_empty[intro]: "negligible {}" by auto |
|
2643 |
|
2644 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s" |
|
2645 using assms apply(induct s) by auto |
|
2646 |
|
2647 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)" |
|
2648 using assms by(induct,auto) |
|
2649 |
|
2650 lemma negligible: "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)" |
|
2651 apply safe defer apply(subst negligible_def) |
|
2652 proof- fix t::"(real^'n) set" assume as:"negligible s" |
|
2653 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) |
|
2654 show "(indicator s has_integral 0) t" apply(subst has_integral_alt) |
|
2655 apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format]) |
|
2656 apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI) |
|
2657 using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto |
|
2658 |
|
2659 subsection {* Finite case of the spike theorem is quite commonly needed. *} |
|
2660 |
|
2661 lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" |
|
2662 "(f has_integral y) t" shows "(g has_integral y) t" |
|
2663 apply(rule has_integral_spike) using assms by auto |
|
2664 |
|
2665 lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" |
|
2666 shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)" |
|
2667 apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto |
|
2668 |
|
2669 lemma integrable_spike_finite: |
|
2670 assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on t" |
|
2671 using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI) |
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2672 apply(rule has_integral_spike_finite) by auto |
|
2673 |
|
2674 subsection {* In particular, the boundary of an interval is negligible. *} |
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2675 |
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2676 lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})" |
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2677 proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)" |
|
2678 have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all |
|
2679 apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI) |
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2680 apply(erule_tac[!] x=xa in allE) by auto |
|
2681 thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed |
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2682 |
|
2683 lemma has_integral_spike_interior: |
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2684 assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})" |
|
2685 apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto |
|
2686 |
|
2687 lemma has_integral_spike_interior_eq: |
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2688 assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))" |
|
2689 apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto |
|
2690 |
|
2691 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}" |
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2692 using assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto |
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2693 |
|
2694 subsection {* Integrability of continuous functions. *} |
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2695 |
|
2696 lemma neutral_and[simp]: "neutral op \<and> = True" |
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2697 unfolding neutral_def apply(rule some_equality) by auto |
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2698 |
|
2699 lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto |
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2700 |
|
2701 lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms |
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2702 apply induct unfolding iterate_insert[OF monoidal_and] by auto |
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2703 |
|
2704 lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}" |
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2705 shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}" |
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2706 using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto |
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2707 |
|
2708 lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach" |
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2709 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 |
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2710 proof safe fix a b::"real^'n" { assume "content {a..b} = 0" |
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2711 thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" |
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2712 apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) } |
|
2713 { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" |
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2714 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}" |
|
2715 "\<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}" |
|
2716 apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto } |
|
2717 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}" |
|
2718 "\<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}" |
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2719 let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x" |
|
2720 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) |
|
2721 proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto |
|
2722 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}" |
|
2723 then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this] |
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2724 show ?case unfolding integrable_on_def by auto |
|
2725 next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}" |
|
2726 apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed |
|
2727 |
|
2728 lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach" |
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2729 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" |
|
2730 obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" |
|
2731 proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)] |
|
2732 note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]] |
|
2733 guess g .. thus thesis apply-apply(rule that[of g]) by auto qed |
|
2734 |
|
2735 lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach" |
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2736 assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}" |
|
2737 proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e" |
|
2738 from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d .. |
|
2739 note d=conjunctD2[OF this,rule_format] |
|
2740 from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this |
|
2741 note p' = tagged_division_ofD[OF p(1)] |
|
2742 have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i" |
|
2743 proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" |
|
2744 from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this |
|
2745 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) |
|
2746 proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const) |
|
2747 fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this] |
|
2748 note d(2)[OF _ _ this[unfolded mem_ball]] |
|
2749 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 |
|
2750 from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g . |
|
2751 thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed |
|
2752 |
|
2753 subsection {* Specialization of additivity to one dimension. *} |
|
2754 |
|
2755 lemma operative_1_lt: assumes "monoidal opp" |
|
2756 shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and> |
|
2757 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))" |
|
2758 unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def |
|
2759 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" |
|
2760 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 |
|
2761 thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto |
|
2762 next fix a b::"real^1" and c::real |
|
2763 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}" |
|
2764 show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))" |
|
2765 proof(cases "c \<in> {a$1 .. b$1}") |
|
2766 case False hence "c<a$1 \<or> c>b$1" by auto |
|
2767 thus ?thesis apply-apply(erule disjE) |
|
2768 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 |
|
2769 show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto |
|
2770 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 |
|
2771 show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto |
|
2772 qed |
|
2773 next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto |
|
2774 show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym] |
|
2775 proof(cases "c = a$1 \<or> c = b$1") |
|
2776 case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" |
|
2777 apply-apply(subst as(2)[rule_format]) using True by auto |
|
2778 next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply- |
|
2779 proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto |
|
2780 hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2781 thus ?thesis using assms unfolding * by auto |
|
2782 next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto |
|
2783 hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2784 thus ?thesis using assms unfolding * by auto qed qed qed qed |
|
2785 |
|
2786 lemma operative_1_le: assumes "monoidal opp" |
|
2787 shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and> |
|
2788 (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))" |
|
2789 unfolding operative_1_lt[OF assms] |
|
2790 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" |
|
2791 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 |
|
2792 next fix a b c ::"real^1" |
|
2793 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" |
|
2794 note as = this[rule_format] |
|
2795 show "opp (f {a..c}) (f {c..b}) = f {a..b}" |
|
2796 proof(cases "c = a \<or> c = b") |
|
2797 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) |
|
2798 next case True thus ?thesis apply- |
|
2799 proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2800 thus ?thesis using assms unfolding * by auto |
|
2801 next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto |
|
2802 thus ?thesis using assms unfolding * by auto qed qed qed |
|
2803 |
|
2804 subsection {* Special case of additivity we need for the FCT. *} |
|
2805 |
|
2806 lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector" |
|
2807 assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}" |
|
2808 shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a" |
|
2809 proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))" |
|
2810 have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1 |
|
2811 by(auto simp add:not_less interval_bound_1 vector_less_def) |
|
2812 have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)] |
|
2813 note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ] |
|
2814 show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer |
|
2815 apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed |
|
2816 |
|
2817 subsection {* A useful lemma allowing us to factor out the content size. *} |
|
2818 |
|
2819 lemma has_integral_factor_content: |
|
2820 "(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 |
|
2821 \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))" |
|
2822 proof(cases "content {a..b} = 0") |
|
2823 case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe |
|
2824 apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer |
|
2825 apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption) |
|
2826 apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto |
|
2827 next case False note F = this[unfolded content_lt_nz[THEN sym]] |
|
2828 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)" |
|
2829 show ?thesis apply(subst has_integral) |
|
2830 proof safe fix e::real assume e:"e>0" |
|
2831 { 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) |
|
2832 apply(erule impE) defer apply(erule exE,rule_tac x=d in exI) |
|
2833 using F e by(auto simp add:field_simps intro:mult_pos_pos) } |
|
2834 { 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) |
|
2835 apply(erule impE) defer apply(erule exE,rule_tac x=d in exI) |
|
2836 using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed |
|
2837 |
|
2838 subsection {* Fundamental theorem of calculus. *} |
|
2839 |
|
2840 lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
2841 assumes "a \<le> b" "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})" |
|
2842 shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})" |
|
2843 unfolding has_integral_factor_content |
|
2844 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 |
|
2845 note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt] |
|
2846 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 |
|
2847 note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]] |
|
2848 guess d .. note d=conjunctD2[OF this[rule_format],rule_format] |
|
2849 show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow> |
|
2850 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})" |
|
2851 apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe) |
|
2852 apply(rule gauge_ball_dependent,rule,rule d(1)) |
|
2853 proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p" |
|
2854 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}" |
|
2855 unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym] |
|
2856 unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym] |
|
2857 apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] |
|
2858 proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p" |
|
2859 note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this |
|
2860 have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto |
|
2861 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] . |
|
2862 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)" |
|
2863 apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm]) |
|
2864 unfolding scaleR.diff_left by(auto simp add:group_simps) |
|
2865 also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)" |
|
2866 apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4 |
|
2867 apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1]) |
|
2868 using ball[rule_format,of u] ball[rule_format,of v] |
|
2869 using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real) |
|
2870 also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" |
|
2871 unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps) |
|
2872 finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le> |
|
2873 e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] . |
|
2874 qed(insert as, auto) qed qed |
|
2875 |
|
2876 subsection {* Attempt a systematic general set of "offset" results for components. *} |
|
2877 |
|
2878 lemma gauge_modify: |
|
2879 assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d" |
|
2880 shows "gauge (\<lambda>x y. d (f x) (f y))" |
|
2881 using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE) |
|
2882 apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto |
|
2883 |
|
2884 subsection {* Only need trivial subintervals if the interval itself is trivial. *} |
|
2885 |
|
2886 lemma division_of_nontrivial: fixes s::"(real^'n) set set" |
|
2887 assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0" |
|
2888 shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply- |
|
2889 proof(induct "card s" arbitrary:s rule:nat_less_induct) |
|
2890 fix s::"(real^'n) set set" assume assm:"s division_of {a..b}" |
|
2891 "\<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}" |
|
2892 note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}" |
|
2893 { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis" |
|
2894 show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto } |
|
2895 assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s" |
|
2896 then obtain k where k:"k\<in>s" "content k = 0" by auto |
|
2897 from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this |
|
2898 from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto |
|
2899 hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto |
|
2900 have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4)) |
|
2901 apply safe apply(rule closed_interval) using assm(1) by auto |
|
2902 have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable |
|
2903 proof safe fix x and e::real assume as:"x\<in>k" "e>0" |
|
2904 from k(2)[unfolded k content_eq_0] guess i .. |
|
2905 hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt |
|
2906 hence xi:"x$i = d$i" using as unfolding k mem_interval by smt |
|
2907 def y \<equiv> "(\<chi> j. if j = i then if c$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$j)" |
|
2908 show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) |
|
2909 proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less) |
|
2910 hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]] |
|
2911 hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl] |
|
2912 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+ |
|
2913 thus "y \<noteq> x" unfolding Cart_eq by auto |
|
2914 have *:"UNIV = insert i (UNIV - {i})" by auto |
|
2915 have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1]) |
|
2916 apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono) |
|
2917 proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl] |
|
2918 apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto |
|
2919 show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto |
|
2920 qed auto thus "dist y x < e" unfolding vector_dist_norm by auto |
|
2921 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 |
|
2922 moreover have "y \<in> \<Union>s" unfolding s mem_interval |
|
2923 proof note simps = y_def Cart_lambda_beta if_not_P |
|
2924 fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" |
|
2925 proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto |
|
2926 thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto |
|
2927 next case True note T = this show ?thesis |
|
2928 proof(cases "c $ i \<le> (a $ i + b $ i) / 2") |
|
2929 case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i |
|
2930 using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) |
|
2931 next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i |
|
2932 using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) |
|
2933 qed qed qed |
|
2934 ultimately show "y \<in> \<Union>(s - {k})" by auto |
|
2935 qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto |
|
2936 hence "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl]) |
|
2937 apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto |
|
2938 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 |
|
2939 |
|
2940 subsection {* Integrabibility on subintervals. *} |
|
2941 |
|
2942 lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows |
|
2943 "operative op \<and> (\<lambda>i. f integrable_on i)" |
|
2944 unfolding operative_def neutral_and apply safe apply(subst integrable_on_def) |
|
2945 unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+ |
|
2946 unfolding integrable_on_def by(auto intro: has_integral_split) |
|
2947 |
|
2948 lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2949 assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" |
|
2950 apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption) |
|
2951 using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto |
|
2952 |
|
2953 subsection {* Combining adjacent intervals in 1 dimension. *} |
|
2954 |
|
2955 lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b" |
|
2956 "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}" |
|
2957 shows "(f has_integral (i + j)) {a..b}" |
|
2958 proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]] |
|
2959 note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]] |
|
2960 hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer |
|
2961 apply(subst(asm) if_P) using assms(3-) by auto |
|
2962 with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P) |
|
2963 unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed |
|
2964 |
|
2965 lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
2966 assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})" |
|
2967 shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f" |
|
2968 apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)]) |
|
2969 apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto |
|
2970 |
|
2971 lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
2972 assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}" |
|
2973 shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine) |
|
2974 |
|
2975 subsection {* Reduce integrability to "local" integrability. *} |
|
2976 |
|
2977 lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach" |
|
2978 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}" |
|
2979 shows "f integrable_on {a..b}" |
|
2980 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})" |
|
2981 using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format] |
|
2982 guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2) |
|
2983 note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f] |
|
2984 show ?thesis unfolding * apply safe unfolding snd_conv |
|
2985 proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this] |
|
2986 thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed |
|
2987 |
|
2988 subsection {* Second FCT or existence of antiderivative. *} |
|
2989 |
|
2990 lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}" |
|
2991 unfolding integrable_on_def by(rule,rule has_integral_const) |
|
2992 |
|
2993 lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach" |
|
2994 assumes "continuous_on {a..b} f" "x \<in> {a..b}" |
|
2995 shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})" |
|
2996 unfolding has_vector_derivative_def has_derivative_within_alt |
|
2997 apply safe apply(rule scaleR.bounded_linear_left) |
|
2998 proof- fix e::real assume e:"e>0" |
|
2999 note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def] |
|
3000 from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format] |
|
3001 let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)" |
|
3002 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)" |
|
3003 proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x") |
|
3004 case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous) |
|
3005 apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto |
|
3006 hence *:"?I a y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine) |
|
3007 using False unfolding not_less using assms(2) goal1 by auto |
|
3008 have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto |
|
3009 show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def |
|
3010 defer apply(rule has_integral_sub) apply(rule integrable_integral) |
|
3011 apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+ |
|
3012 proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto |
|
3013 have *:"y - x = norm(y - x)" using False by auto |
|
3014 show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto |
|
3015 show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le) |
|
3016 apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto |
|
3017 qed(insert e,auto) |
|
3018 next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous) |
|
3019 apply(rule continuous_on_o_dest_vec1 assms)+ unfolding not_less using assms(2) goal1 by auto |
|
3020 hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine) |
|
3021 using True using assms(2) goal1 by auto |
|
3022 have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto |
|
3023 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 |
|
3024 show ?thesis apply(subst ***) unfolding norm_minus_cancel ** |
|
3025 apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def |
|
3026 defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus |
|
3027 apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) |
|
3028 apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+ |
|
3029 proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto |
|
3030 have *:"x - y = norm(y - x)" using True by auto |
|
3031 show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto |
|
3032 show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le) |
|
3033 apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto |
|
3034 qed(insert e,auto) qed qed qed |
|
3035 |
|
3036 lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach" |
|
3037 assumes "continuous_on {a..b} f" "x \<in> {a..b}" |
|
3038 shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})" |
|
3039 using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"] |
|
3040 unfolding o_def vec1_dest_vec1 using assms(2) by auto |
|
3041 |
|
3042 lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f" |
|
3043 obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})" |
|
3044 apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto |
|
3045 |
|
3046 subsection {* Combined fundamental theorem of calculus. *} |
|
3047 |
|
3048 lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f" |
|
3049 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}" |
|
3050 proof- from antiderivative_continuous[OF assms] guess g . note g=this |
|
3051 show ?thesis apply(rule that[of g]) |
|
3052 proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})" |
|
3053 apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto |
|
3054 thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"] |
|
3055 unfolding o_def vec1_dest_vec1 by auto qed qed |
|
3056 |
|
3057 subsection {* General "twiddling" for interval-to-interval function image. *} |
|
3058 |
|
3059 lemma has_integral_twiddle: |
|
3060 assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g" |
|
3061 "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}" |
|
3062 "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}" |
|
3063 "\<forall>u v. content(g ` {u..v}) = r * content {u..v}" |
|
3064 "(f has_integral i) {a..b}" |
|
3065 shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})" |
|
3066 proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis" |
|
3067 show ?thesis apply cases defer apply(rule *,assumption) |
|
3068 proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed } |
|
3069 assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this |
|
3070 have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr) |
|
3071 using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer |
|
3072 using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto |
|
3073 show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz) |
|
3074 proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos) |
|
3075 from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format] |
|
3076 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) |
|
3077 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)" |
|
3078 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 |
|
3079 fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] |
|
3080 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 |
|
3081 proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto |
|
3082 show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto |
|
3083 fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto |
|
3084 show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto |
|
3085 { 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)"] |
|
3086 using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto } |
|
3087 fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')" |
|
3088 hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto |
|
3089 have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk']) |
|
3090 proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z . |
|
3091 hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)] |
|
3092 unfolding image_Int[OF inj(1)] by auto thus False using as by blast |
|
3093 qed thus "g x = g x'" by auto |
|
3094 { fix z assume "z \<in> k" thus "g z \<in> g ` k'" using same by auto } |
|
3095 { fix z assume "z \<in> k'" thus "g z \<in> g ` k" using same by auto } |
|
3096 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 |
|
3097 then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq .. |
|
3098 thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply- |
|
3099 apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI) |
|
3100 using X(2) assms(3)[rule_format,of x] by auto |
|
3101 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 |
|
3102 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 |
|
3103 unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv |
|
3104 apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto |
|
3105 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] |
|
3106 unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" . |
|
3107 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 |
|
3108 using assms(1) by(auto simp add:field_simps) qed qed qed |
|
3109 |
|
3110 subsection {* Special case of a basic affine transformation. *} |
|
3111 |
|
3112 lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}" |
|
3113 unfolding image_affinity_interval by auto |
|
3114 |
|
3115 lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta |
|
3116 Cart_eq vector_le_def vector_less_def |
|
3117 |
|
3118 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A" |
|
3119 apply(rule setprod_cong) using assms by auto |
|
3120 |
|
3121 lemma content_image_affinity_interval: |
|
3122 "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r") |
|
3123 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption) |
|
3124 unfolding not_not using content_empty by auto } |
|
3125 assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0") |
|
3126 case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True] |
|
3127 unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') |
|
3128 defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym] |
|
3129 apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le |
|
3130 by(auto simp add:field_simps intro:mult_left_mono) |
|
3131 next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False] |
|
3132 unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') |
|
3133 defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym] |
|
3134 apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le |
|
3135 by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed |
|
3136 |
|
3137 lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0" |
|
3138 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})" |
|
3139 apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power) |
|
3140 defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps) |
|
3141 apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto |
|
3142 |
|
3143 lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0" |
|
3144 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})" |
|
3145 using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto |
|
3146 |
|
3147 subsection {* Special case of stretching coordinate axes separately. *} |
|
3148 |
|
3149 lemma image_stretch_interval: |
|
3150 "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = |
|
3151 (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") |
|
3152 proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto |
|
3153 next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto |
|
3154 case False note ab = this[unfolded interval_ne_empty] |
|
3155 show ?thesis apply-apply(rule set_ext) |
|
3156 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 |
|
3157 show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] |
|
3158 unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq * |
|
3159 unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"] |
|
3160 proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) = |
|
3161 (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))" |
|
3162 proof(cases "m i = 0") case True thus ?thesis using ab by auto |
|
3163 next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply- |
|
3164 proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i" |
|
3165 "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto |
|
3166 show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI) |
|
3167 using as by(auto simp add:field_simps) |
|
3168 next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i" |
|
3169 "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def |
|
3170 by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym) |
|
3171 show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI) |
|
3172 using as by(auto simp add:field_simps) qed qed qed qed qed |
|
3173 |
|
3174 lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}" |
|
3175 unfolding image_stretch_interval by auto |
|
3176 |
|
3177 lemma content_image_stretch_interval: |
|
3178 "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})" |
|
3179 proof(cases "{a..b} = {}") case True thus ?thesis |
|
3180 unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto |
|
3181 next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto |
|
3182 thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P |
|
3183 unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta |
|
3184 proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto |
|
3185 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)" |
|
3186 apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] |
|
3187 by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed |
|
3188 |
|
3189 lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)" |
|
3190 shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral |
|
3191 ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})" |
|
3192 apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval |
|
3193 unfolding image_stretch_interval empty_as_interval Cart_eq using assms |
|
3194 proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)" |
|
3195 apply(rule,rule linear_continuous_at) unfolding linear_linear |
|
3196 unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto |
|
3197 |
|
3198 lemma integrable_stretch: |
|
3199 assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)" |
|
3200 shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})" |
|
3201 using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto |
|
3202 |
|
3203 subsection {* even more special cases. *} |
|
3204 |
|
3205 lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}" |
|
3206 apply(rule set_ext,rule) defer unfolding image_iff |
|
3207 apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff) |
|
3208 |
|
3209 lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}" |
|
3210 shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}" |
|
3211 using has_integral_affinity[OF assms, of "-1" 0] by auto |
|
3212 |
|
3213 lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})" |
|
3214 apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto |
|
3215 |
|
3216 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}" |
|
3217 unfolding integrable_on_def by auto |
|
3218 |
|
3219 lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f" |
|
3220 unfolding integral_def by auto |
|
3221 |
|
3222 subsection {* Stronger form of FCT; quite a tedious proof. *} |
|
3223 |
|
3224 (** move this **) |
|
3225 declare norm_triangle_ineq4[intro] |
|
3226 |
|
3227 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) |
|
3228 |
|
3229 lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector" |
|
3230 assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}" |
|
3231 shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a" |
|
3232 using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"] |
|
3233 unfolding o_def vec1_dest_vec1 using assms(1) by auto |
|
3234 |
|
3235 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" |
|
3236 unfolding split_def by(rule refl) |
|
3237 |
|
3238 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e" |
|
3239 apply(subst(asm)(2) norm_minus_cancel[THEN sym]) |
|
3240 apply(drule norm_triangle_le) by(auto simp add:group_simps) |
|
3241 |
|
3242 lemma fundamental_theorem_of_calculus_interior: |
|
3243 assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)" |
|
3244 shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" |
|
3245 proof- { presume *:"a < b \<Longrightarrow> ?thesis" |
|
3246 show ?thesis proof(cases,rule *,assumption) |
|
3247 assume "\<not> a < b" hence "a = b" using assms(1) by auto |
|
3248 hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt |
|
3249 show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto |
|
3250 qed } assume ab:"a < b" |
|
3251 let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow> |
|
3252 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})" |
|
3253 { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto } |
|
3254 fix e::real assume e:"e>0" |
|
3255 note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib] |
|
3256 note conjunctD2[OF this] note bounded=this(1) and this(2) |
|
3257 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)" |
|
3258 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] |
|
3259 from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format] |
|
3260 have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto |
|
3261 from this[unfolded bounded_pos] guess B .. note B = this[rule_format] |
|
3262 |
|
3263 have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da |
|
3264 \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)" |
|
3265 proof- have "a\<in>{a..b}" using ab by auto |
|
3266 note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this] |
|
3267 note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps) |
|
3268 from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format] |
|
3269 have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8" |
|
3270 proof(cases "f' a = 0") case True |
|
3271 thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) |
|
3272 next case False thus ?thesis |
|
3273 apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) |
|
3274 using ab e by(auto simp add:field_simps) |
|
3275 qed then guess l .. note l = conjunctD2[OF this] |
|
3276 show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+) |
|
3277 proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" |
|
3278 note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval] |
|
3279 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) |
|
3280 also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" |
|
3281 proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto |
|
3282 thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto |
|
3283 next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer |
|
3284 apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps) |
|
3285 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 |
|
3286 qed qed then guess da .. note da=conjunctD2[OF this,rule_format] |
|
3287 |
|
3288 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" |
|
3289 proof- have "b\<in>{a..b}" using ab by auto |
|
3290 note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this] |
|
3291 note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps) |
|
3292 from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format] |
|
3293 have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8" |
|
3294 proof(cases "f' b = 0") case True |
|
3295 thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) |
|
3296 next case False thus ?thesis |
|
3297 apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI) |
|
3298 using ab e by(auto simp add:field_simps) |
|
3299 qed then guess l .. note l = conjunctD2[OF this] |
|
3300 show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+) |
|
3301 proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" |
|
3302 note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval] |
|
3303 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) |
|
3304 also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" |
|
3305 proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto |
|
3306 thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto |
|
3307 next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute) |
|
3308 apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps) |
|
3309 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 |
|
3310 qed qed then guess db .. note db=conjunctD2[OF this,rule_format] |
|
3311 |
|
3312 let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))" |
|
3313 show "?P e" apply(rule_tac x="?d" in exI) |
|
3314 proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto |
|
3315 next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)] |
|
3316 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 |
|
3317 note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym] |
|
3318 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 |
|
3319 show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus |
|
3320 unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)] |
|
3321 proof(rule norm_triangle_le,rule **) |
|
3322 case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum) |
|
3323 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" |
|
3324 "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2 |
|
3325 < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))" |
|
3326 from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this |
|
3327 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] |
|
3328 note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]] |
|
3329 |
|
3330 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] |
|
3331 have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) = |
|
3332 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)))" |
|
3333 apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto |
|
3334 also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub) |
|
3335 apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq |
|
3336 apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real) |
|
3337 also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps) |
|
3338 finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2" |
|
3339 apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed |
|
3340 |
|
3341 next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto |
|
3342 case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv) |
|
3343 defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] |
|
3344 apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms) |
|
3345 proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this] |
|
3346 from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this |
|
3347 with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto |
|
3348 thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)" |
|
3349 unfolding uv using e by(auto simp add:field_simps) |
|
3350 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 |
|
3351 show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x - |
|
3352 (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" |
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3353 apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"]) |
|
3354 apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def |
|
3355 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}" |
|
3356 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 |
|
3357 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 |
|
3358 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 |
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3359 next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = |
|
3360 {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 |
|
3361 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" |
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3362 proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto |
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3363 thus ?case using `x\<in>s` goal2(2) by auto |
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3364 qed auto |
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3365 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"]) |
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3366 apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **) |
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3367 proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}" |
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3368 have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" |
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3369 proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this |
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3370 have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto |
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3371 have u:"u = vec1 a" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto |
|
3372 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 |
|
3373 have "u > vec1 a" unfolding Cart_simps by auto |
|
3374 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps) |
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3375 qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto |
|
3376 qed |
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3377 have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" |
|
3378 proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this |
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3379 have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto |
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3380 have u:"v = vec1 b" proof(rule ccontr) have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto |
|
3381 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 |
|
3382 have "v < vec1 b" unfolding Cart_simps by auto |
|
3383 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps) |
|
3384 qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto |
|
3385 qed |
|
3386 |
|
3387 show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all) |
|
3388 unfolding mem_Collect_eq fst_conv snd_conv apply safe |
|
3389 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" |
|
3390 guess v using pa[OF k(1)] .. note v = conjunctD2[OF this] |
|
3391 guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))" |
|
3392 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] |
|
3393 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) |
|
3394 ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto |
|
3395 hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto |
|
3396 { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . } |
|
3397 qed |
|
3398 show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all) |
|
3399 unfolding mem_Collect_eq fst_conv snd_conv apply safe |
|
3400 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" |
|
3401 guess v using pb[OF k(1)] .. note v = conjunctD2[OF this] |
|
3402 guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))" |
|
3403 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] |
|
3404 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) |
|
3405 ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto |
|
3406 hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto |
|
3407 { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . } |
|
3408 qed |
|
3409 |
|
3410 let ?a = a and ?b = b (* a is something else while proofing the next theorem. *) |
|
3411 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) |
|
3412 \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1 |
|
3413 proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this] |
|
3414 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 |
|
3415 moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)] |
|
3416 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) |
|
3417 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately |
|
3418 show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply- |
|
3419 apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1]) |
|
3420 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto |
|
3421 qed |
|
3422 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) |
|
3423 \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1 |
|
3424 proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this] |
|
3425 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 |
|
3426 moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)] |
|
3427 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab |
|
3428 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately |
|
3429 show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply- |
|
3430 apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1]) |
|
3431 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto |
|
3432 qed |
|
3433 qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed |
|
3434 |
|
3435 subsection {* Stronger form with finite number of exceptional points. *} |
|
3436 |
|
3437 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach" |
|
3438 assumes"finite s" "a \<le> b" "continuous_on {a..b} f" |
|
3439 "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)" |
|
3440 shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply- |
|
3441 proof(induct "card s" arbitrary:s a b) |
|
3442 case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto |
|
3443 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq |
|
3444 apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format] |
|
3445 show ?case proof(cases "c\<in>{a<..<b}") |
|
3446 case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer |
|
3447 apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto |
|
3448 next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto |
|
3449 case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto |
|
3450 thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+ |
|
3451 apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs |
|
3452 proof- show "continuous_on {a..c} f" "continuous_on {c..b} f" |
|
3453 apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto |
|
3454 let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)" |
|
3455 show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto |
|
3456 qed auto qed qed |
|
3457 |
|
3458 lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach" |
|
3459 assumes "finite s" "a \<le> b" "continuous_on {a..b} f" |
|
3460 "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)" |
|
3461 shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}" |
|
3462 apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f']) |
|
3463 using assms(4) by auto |
|
3464 |
|
3465 end |