src/HOL/Multivariate_Analysis/Integration.thy
 author boehmes Wed Mar 24 14:03:52 2010 +0100 (2010-03-24) changeset 35945 fcd02244e63d parent 35752 c8a8df426666 child 36081 70deefb6c093 permissions -rw-r--r--
inhibit invokation of external SMT solver
```     1
```
```     2 header {* Kurzweil-Henstock gauge integration in many dimensions. *}
```
```     3 (*  Author:                     John Harrison
```
```     4     Translation from HOL light: Robert Himmelmann, TU Muenchen *)
```
```     5
```
```     6 theory Integration
```
```     7   imports Derivative SMT
```
```     8 begin
```
```     9
```
```    10 declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
```
```    11 declare [[smt_record=false]]
```
```    12 declare [[z3_proofs=true]]
```
```    13
```
```    14 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
```
```    15 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
```
```    16 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
```
```    17 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
```
```    18
```
```    19 declare smult_conv_scaleR[simp]
```
```    20
```
```    21 subsection {* Some useful lemmas about intervals. *}
```
```    22
```
```    23 lemma empty_as_interval: "{} = {1..0::real^'n}"
```
```    24   apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
```
```    25   using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
```
```    26
```
```    27 lemma interior_subset_union_intervals:
```
```    28   assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
```
```    29   shows "interior i \<subseteq> interior s" proof-
```
```    30   have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
```
```    31     unfolding assms(1,2) interior_closed_interval by auto
```
```    32   moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
```
```    33     using assms(4) unfolding assms(1,2) by auto
```
```    34   ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
```
```    35     unfolding assms(1,2) interior_closed_interval by auto qed
```
```    36
```
```    37 lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
```
```    38   assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
```
```    39   shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
```
```    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)
```
```    41     unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
```
```    42   have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
```
```    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
```
```    44   thus ?case proof(induct rule:finite_induct)
```
```    45     case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
```
```    46     case (insert i f) guess x using insert(5) .. note x = this
```
```    47     then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
```
```    48     guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
```
```    49     show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
```
```    50       then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
```
```    51       hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
```
```    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
```
```    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
```
```    54     case True show ?thesis proof(cases "x\<in>{a<..<b}")
```
```    55       case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
```
```    56       thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
```
```    57 	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
```
```    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)
```
```    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
```
```    60     hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
```
```    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)
```
```    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
```
```    63 	hence "\<bar>(?z - y) \$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
```
```    64 	hence "y\$k < a\$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
```
```    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
```
```    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
```
```    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)"
```
```    68 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
```
```    69 	  unfolding norm_scaleR norm_basis by auto
```
```    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)
```
```    71 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
```
```    72       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
```
```    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)
```
```    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
```
```    75 	hence "\<bar>(?z - y) \$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
```
```    76 	hence "y\$k > b\$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
```
```    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
```
```    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
```
```    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)"
```
```    80 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
```
```    81 	  unfolding norm_scaleR norm_basis by auto
```
```    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)
```
```    83 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
```
```    84       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
```
```    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
```
```    86     thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
```
```    87   guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
```
```    88   hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
```
```    89   thus False using `t\<in>f` assms(4) by auto qed
```
```    90 subsection {* Bounds on intervals where they exist. *}
```
```    91
```
```    92 definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x\$i = a})"
```
```    93
```
```    94 definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x\$i = a})"
```
```    95
```
```    96 lemma interval_upperbound[simp]: assumes "\<forall>i. a\$i \<le> b\$i" shows "interval_upperbound {a..b} = b"
```
```    97   using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
```
```    98   apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
```
```    99   apply(rule,rule) apply(rule_tac x="b\$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
```
```   100   unfolding mem_interval using assms by auto
```
```   101
```
```   102 lemma interval_lowerbound[simp]: assumes "\<forall>i. a\$i \<le> b\$i" shows "interval_lowerbound {a..b} = a"
```
```   103   using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
```
```   104   apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
```
```   105   apply(rule,rule) apply(rule_tac x="a\$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
```
```   106   unfolding mem_interval using assms by auto
```
```   107
```
```   108 lemmas interval_bounds = interval_upperbound interval_lowerbound
```
```   109
```
```   110 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
```
```   111   using assms unfolding interval_ne_empty by auto
```
```   112
```
```   113 lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
```
```   114   apply(rule interval_upperbound) by auto
```
```   115
```
```   116 lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
```
```   117   apply(rule interval_lowerbound) by auto
```
```   118
```
```   119 lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
```
```   120
```
```   121 subsection {* Content (length, area, volume...) of an interval. *}
```
```   122
```
```   123 definition "content (s::(real^'n) set) =
```
```   124        (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)\$i - (interval_lowerbound s)\$i))"
```
```   125
```
```   126 lemma interval_not_empty:"\<forall>i. a\$i \<le> b\$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
```
```   127   unfolding interval_eq_empty unfolding not_ex not_less by assumption
```
```   128
```
```   129 lemma content_closed_interval: assumes "\<forall>i. a\$i \<le> b\$i"
```
```   130   shows "content {a..b} = (\<Prod>i\<in>UNIV. b\$i - a\$i)"
```
```   131   using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
```
```   132
```
```   133 lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b\$i - a\$i)"
```
```   134   apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
```
```   135
```
```   136 lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
```
```   137   using content_closed_interval[of a b] by auto
```
```   138
```
```   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
```
```   140
```
```   141 lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
```
```   142   have *:"\<forall>i. 0\$i \<le> (1::real^'n::finite)\$i" by auto
```
```   143   have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
```
```   144   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
```
```   145
```
```   146 lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
```
```   147   case False hence *:"\<forall>i. a \$ i \<le> b \$ i" unfolding interval_ne_empty by assumption
```
```   148   have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} \$ i - interval_lowerbound {a..b} \$ i) \<ge> 0"
```
```   149     apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
```
```   150   thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
```
```   151
```
```   152 lemma content_pos_lt: assumes "\<forall>i. a\$i < b\$i" shows "0 < content {a..b}"
```
```   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
```
```   154   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
```
```   155     using assms apply(erule_tac x=x in allE) by auto qed
```
```   156
```
```   157 lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
```
```   158   apply(rule content_pos_lt) by auto
```
```   159
```
```   160 lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b\$i \<le> a\$i)" proof(cases "{a..b} = {}")
```
```   161   case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
```
```   162     apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
```
```   163   guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
```
```   164   show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
```
```   165     apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
```
```   166     apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
```
```   167
```
```   168 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
```
```   169
```
```   170 lemma content_closed_interval_cases:
```
```   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)
```
```   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
```
```   173
```
```   174 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
```
```   175   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
```
```   176
```
```   177 lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
```
```   178   unfolding content_eq_0 by auto
```
```   179
```
```   180 lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a\$i < b\$i)"
```
```   181   apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
```
```   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
```
```   183
```
```   184 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
```
```   185
```
```   186 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
```
```   187   case True thus ?thesis using content_pos_le[of c d] by auto next
```
```   188   case False hence ab_ne:"\<forall>i. a \$ i \<le> b \$ i" unfolding interval_ne_empty by auto
```
```   189   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
```
```   190   have "{c..d} \<noteq> {}" using assms False by auto
```
```   191   hence cd_ne:"\<forall>i. c \$ i \<le> d \$ i" using assms unfolding interval_ne_empty by auto
```
```   192   show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
```
```   193     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
```
```   194     show "0 \<le> b \$ i - a \$ i" using ab_ne[THEN spec[where x=i]] by auto
```
```   195     show "b \$ i - a \$ i \<le> d \$ i - c \$ i"
```
```   196       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
```
```   197       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
```
```   198
```
```   199 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
```
```   200   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
```
```   201
```
```   202 subsection {* The notion of a gauge --- simply an open set containing the point. *}
```
```   203
```
```   204 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
```
```   205
```
```   206 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
```
```   207   using assms unfolding gauge_def by auto
```
```   208
```
```   209 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
```
```   210
```
```   211 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
```
```   212   unfolding gauge_def by auto
```
```   213
```
```   214 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
```
```   215
```
```   216 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
```
```   217
```
```   218 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
```
```   219   unfolding gauge_def by auto
```
```   220
```
```   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-
```
```   222   have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
```
```   223   unfolding gauge_def unfolding *
```
```   224   using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
```
```   225
```
```   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)
```
```   227
```
```   228 subsection {* Divisions. *}
```
```   229
```
```   230 definition division_of (infixl "division'_of" 40) where
```
```   231   "s division_of i \<equiv>
```
```   232         finite s \<and>
```
```   233         (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   234         (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
```
```   235         (\<Union>s = i)"
```
```   236
```
```   237 lemma division_ofD[dest]: assumes  "s division_of i"
```
```   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})"
```
```   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
```
```   240
```
```   241 lemma division_ofI:
```
```   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})"
```
```   243   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
```
```   244   shows "s division_of i" using assms unfolding division_of_def by auto
```
```   245
```
```   246 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
```
```   247   unfolding division_of_def by auto
```
```   248
```
```   249 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
```
```   250   unfolding division_of_def by auto
```
```   251
```
```   252 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
```
```   253
```
```   254 lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
```
```   255   assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s"
```
```   256     ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
```
```   257   ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
```
```   258   assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
```
```   259   { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
```
```   260   moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
```
```   261
```
```   262 lemma elementary_empty: obtains p where "p division_of {}"
```
```   263   unfolding division_of_trivial by auto
```
```   264
```
```   265 lemma elementary_interval: obtains p where  "p division_of {a..b}"
```
```   266   by(metis division_of_trivial division_of_self)
```
```   267
```
```   268 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
```
```   269   unfolding division_of_def by auto
```
```   270
```
```   271 lemma forall_in_division:
```
```   272  "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
```
```   273   unfolding division_of_def by fastsimp
```
```   274
```
```   275 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
```
```   276   apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
```
```   277   show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
```
```   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
```
```   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 }
```
```   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
```
```   281   show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
```
```   282
```
```   283 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
```
```   284
```
```   285 lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
```
```   286   unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
```
```   287   apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
```
```   288
```
```   289 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
```
```   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-
```
```   291 let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
```
```   292 show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
```
```   293   moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
```
```   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
```
```   295     using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
```
```   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
```
```   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
```
```   298   guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
```
```   299   guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
```
```   300   show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
```
```   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
```
```   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
```
```   303   assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
```
```   304   have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
```
```   305       interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
```
```   306       interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
```
```   307       \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
```
```   308   show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
```
```   309     using division_ofD(5)[OF assms(1) k1(2) k2(2)]
```
```   310     using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
```
```   311
```
```   312 lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
```
```   313   shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
```
```   314   case True show ?thesis unfolding True and division_of_trivial by auto next
```
```   315   have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
```
```   316   case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
```
```   317
```
```   318 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
```
```   319   shows "\<exists>p. p division_of (s \<inter> t)"
```
```   320   by(rule,rule division_inter[OF assms])
```
```   321
```
```   322 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
```
```   323   shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
```
```   324 case (insert x f) show ?case proof(cases "f={}")
```
```   325   case True thus ?thesis unfolding True using insert by auto next
```
```   326   case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
```
```   327   moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
```
```   328   show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
```
```   329
```
```   330 lemma division_disjoint_union:
```
```   331   assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
```
```   332   shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI)
```
```   333   note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
```
```   334   show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
```
```   335   show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
```
```   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 = {}"
```
```   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)]]
```
```   338       using assms(3) by blast } moreover
```
```   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)]]
```
```   340       using assms(3) by blast} ultimately
```
```   341   show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
```
```   342   fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
```
```   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
```
```   344
```
```   345 lemma partial_division_extend_1:
```
```   346   assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
```
```   347   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
```
```   348 proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
```
```   349   guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
```
```   350   def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
```
```   351   have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
```
```   352   hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
```
```   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
```
```   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
```
```   355   have "{c..d} \<noteq> {}" using assms by auto
```
```   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)}"
```
```   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)}"
```
```   358   let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
```
```   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)
```
```   360   show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
```
```   361   proof- have "\<And>i. \<pi>' i < Suc n"
```
```   362     proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
```
```   363       hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
```
```   364     qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c \$ i else a \$ i)"
```
```   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)"
```
```   366       unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
```
```   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
```
```   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}"
```
```   369       unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
```
```   370     proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
```
```   371       then guess i unfolding mem_interval not_all .. note i=this
```
```   372       show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
```
```   373         apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
```
```   374     qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
```
```   375     proof- fix x assume x:"x\<in>{a..b}"
```
```   376       { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
```
```   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)}"
```
```   378       assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
```
```   379       hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
```
```   380       hence M:"finite ?M" "?M \<noteq> {}" by auto
```
```   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]]
```
```   382         Min_gr_iff[OF M,unfolded l_def[symmetric]]
```
```   383       have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
```
```   384         apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
```
```   385       proof- assume as:"x \$ \<pi> l < c \$ \<pi> l"
```
```   386         show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
```
```   387         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
```
```   388           thus ?case using as x[unfolded mem_interval,rule_format,of i]
```
```   389             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
```
```   390         qed
```
```   391       next assume as:"x \$ \<pi> l > d \$ \<pi> l"
```
```   392         show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
```
```   393         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
```
```   394           thus ?case using as x[unfolded mem_interval,rule_format,of i]
```
```   395             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
```
```   396         qed qed
```
```   397       thus "x \<in> \<Union>?p" using l(2) by blast
```
```   398     qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
```
```   399
```
```   400     show "finite ?p" by auto
```
```   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
```
```   402     show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
```
```   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
```
```   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)
```
```   405     qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
```
```   406     proof- case goal1 thus ?case using abcd[of x] by auto
```
```   407     next   case goal2 thus ?case using abcd[of x] by auto
```
```   408     qed thus "k \<noteq> {}" using k by auto
```
```   409     show "\<exists>a b. k = {a..b}" using k by auto
```
```   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
```
```   411     { fix k k' l l'
```
```   412       assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
```
```   413       assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
```
```   414       assume "l \<le> l'" fix x
```
```   415       have "x \<notin> interior k \<inter> interior k'"
```
```   416       proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
```
```   417         case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
```
```   418         hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
```
```   419         have ln:"l < n + 1"
```
```   420         proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
```
```   421           hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
```
```   422           hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
```
```   423           thus False using `k\<noteq>k'` k' by auto
```
```   424         qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
```
```   425         have "x \$ \<pi> l < c \$ \<pi> l \<or> d \$ \<pi> l < x \$ \<pi> l" using l(1) apply-
```
```   426         proof(erule disjE)
```
```   427           assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   428           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
```
```   429         next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   430           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
```
```   431         qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
```
```   432           by(auto elim!:allE[where x="\<pi> l"])
```
```   433       next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
```
```   434         hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
```
```   435         note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
```
```   436         assume x:"x \<in> interior k \<inter> interior k'"
```
```   437         show False using l(1) l'(1) apply-
```
```   438         proof(erule_tac[!] disjE)+
```
```   439           assume as:"k = ?p1 l" "k' = ?p1 l'"
```
```   440           note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
```
```   441           have "l \<noteq> l'" using k'(2)[unfolded as] by auto
```
```   442           thus False using * by(smt Cart_lambda_beta \<pi>l)
```
```   443         next assume as:"k = ?p2 l" "k' = ?p2 l'"
```
```   444           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   445           have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
```
```   446           thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
```
```   447             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
```
```   448         next assume as:"k = ?p1 l" "k' = ?p2 l'"
```
```   449           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   450           show False using *[of "\<pi> l"] *[of "\<pi> l'"]
```
```   451             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
```
```   452         next assume as:"k = ?p2 l" "k' = ?p1 l'"
```
```   453           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   454           show False using *[of "\<pi> l"] *[of "\<pi> l'"]
```
```   455             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
```
```   456         qed qed }
```
```   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'"
```
```   458       apply - apply(cases "l' \<le> l") using k'(2) by auto
```
```   459     thus "interior k \<inter> interior k' = {}" by auto
```
```   460 qed qed
```
```   461
```
```   462 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
```
```   463   obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
```
```   464   case True guess q apply(rule elementary_interval[of a b]) .
```
```   465   thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
```
```   466   case False note p = division_ofD[OF assms(1)]
```
```   467   have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
```
```   468     guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
```
```   469     have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
```
```   470     guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
```
```   471   guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
```
```   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-
```
```   473     fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
```
```   474       using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
```
```   475   hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
```
```   476     apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
```
```   477   then guess d .. note d = this
```
```   478   show ?thesis apply(rule that[of "d \<union> p"]) proof-
```
```   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
```
```   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"
```
```   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
```
```   482     show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
```
```   483       apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
```
```   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
```
```   485       show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
```
```   486 	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
```
```   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
```
```   488 	show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
```
```   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}))"
```
```   490 	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
```
```   491
```
```   492 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
```
```   493   unfolding division_of_def by(metis bounded_Union bounded_interval)
```
```   494
```
```   495 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
```
```   496   by(meson elementary_bounded bounded_subset_closed_interval)
```
```   497
```
```   498 lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
```
```   499   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
```
```   500   case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
```
```   501   case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
```
```   502   have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
```
```   503   case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
```
```   504     using false True assms using interior_subset by auto next
```
```   505   case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
```
```   506   have *:"{u..v} \<subseteq> {c..d}" using uv by auto
```
```   507   guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
```
```   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
```
```   509   show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
```
```   510     apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
```
```   511     unfolding interior_inter[THEN sym] proof-
```
```   512     have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
```
```   513     have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
```
```   514       apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
```
```   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
```
```   516     finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
```
```   517
```
```   518 lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
```
```   519   "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```   520   shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
```
```   521   apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
```
```   522   using division_ofD[OF assms(2)] by auto
```
```   523
```
```   524 lemma elementary_union_interval: assumes "p division_of \<Union>p"
```
```   525   obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
```
```   526   note assm=division_ofD[OF assms]
```
```   527   have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
```
```   528   have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
```
```   529 { presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
```
```   530     "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
```
```   531   thus thesis by auto
```
```   532 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
```
```   533   thus thesis apply(rule_tac that[of p]) unfolding as by auto
```
```   534 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
```
```   535 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
```
```   536   show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
```
```   537     unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
```
```   538     using assm(2-4) as apply- by(fastsimp dest: assm(5))+
```
```   539 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
```
```   540   have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
```
```   541     from assm(4)[OF this] guess c .. then guess d ..
```
```   542     thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
```
```   543   qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
```
```   544   let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
```
```   545   show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
```
```   546     have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
```
```   547     show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
```
```   548     show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
```
```   549       using q(6) by auto
```
```   550     fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
```
```   551     show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
```
```   552     fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
```
```   553     obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
```
```   554     obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
```
```   555     show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
```
```   556       case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
```
```   557     next case False
```
```   558       { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
```
```   559         "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
```
```   560         thus ?thesis by auto }
```
```   561       { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
```
```   562       { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
```
```   563       assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
```
```   564       guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
```
```   565       have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
```
```   566       hence "interior k \<subseteq> interior x" apply-
```
```   567         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
```
```   568       guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
```
```   569       have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
```
```   570       hence "interior k' \<subseteq> interior x'" apply-
```
```   571         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
```
```   572       ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
```
```   573     qed qed } qed
```
```   574
```
```   575 lemma elementary_unions_intervals:
```
```   576   assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
```
```   577   obtains p where "p division_of (\<Union>f)" proof-
```
```   578   have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
```
```   579     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
```
```   580     fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
```
```   581     from this(3) guess p .. note p=this
```
```   582     from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
```
```   583     have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
```
```   584     show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
```
```   585       unfolding Union_insert ab * by auto
```
```   586   qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
```
```   587
```
```   588 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
```
```   589   obtains p where "p division_of (s \<union> t)"
```
```   590 proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
```
```   591   hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
```
```   592   show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
```
```   593     unfolding * prefer 3 apply(rule_tac p=p in that)
```
```   594     using assms[unfolded division_of_def] by auto qed
```
```   595
```
```   596 lemma partial_division_extend: fixes t::"(real^'n) set"
```
```   597   assumes "p division_of s" "q division_of t" "s \<subseteq> t"
```
```   598   obtains r where "p \<subseteq> r" "r division_of t" proof-
```
```   599   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
```
```   600   obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
```
```   601   guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
```
```   602     apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
```
```   603   guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
```
```   604   then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
```
```   605     apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
```
```   606   { fix x assume x:"x\<in>t" "x\<notin>s"
```
```   607     hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
```
```   608     then guess r unfolding Union_iff .. note r=this moreover
```
```   609     have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
```
```   610       thus False using x by auto qed
```
```   611     ultimately have "x\<in>\<Union>(r1 - p)" by auto }
```
```   612   hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
```
```   613   show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
```
```   614     unfolding divp(6) apply(rule assms r2)+
```
```   615   proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
```
```   616     proof(rule inter_interior_unions_intervals)
```
```   617       show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
```
```   618       have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
```
```   619       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
```
```   620         fix m x assume as:"m\<in>r1-p"
```
```   621         have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
```
```   622           show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
```
```   623           show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
```
```   624         qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
```
```   625       qed qed
```
```   626     thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
```
```   627   qed auto qed
```
```   628
```
```   629 subsection {* Tagged (partial) divisions. *}
```
```   630
```
```   631 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
```
```   632   "(s tagged_partial_division_of i) \<equiv>
```
```   633         finite s \<and>
```
```   634         (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   635         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
```
```   636                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
```
```   637
```
```   638 lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
```
```   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"
```
```   640   "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
```
```   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) = {}"
```
```   642   using assms unfolding tagged_partial_division_of_def  apply- by blast+
```
```   643
```
```   644 definition tagged_division_of (infixr "tagged'_division'_of" 40) where
```
```   645   "(s tagged_division_of i) \<equiv>
```
```   646         (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   647
```
```   648 lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
```
```   649   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```   650
```
```   651 lemma tagged_division_of:
```
```   652  "(s tagged_division_of i) \<longleftrightarrow>
```
```   653         finite s \<and>
```
```   654         (\<forall>x k. (x,k) \<in> s
```
```   655                \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   656         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
```
```   657                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
```
```   658         (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   659   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```   660
```
```   661 lemma tagged_division_ofI: assumes
```
```   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}"
```
```   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) = {})"
```
```   664   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   665   shows "s tagged_division_of i"
```
```   666   unfolding tagged_division_of apply(rule) defer apply rule
```
```   667   apply(rule allI impI conjI assms)+ apply assumption
```
```   668   apply(rule, rule assms, assumption) apply(rule assms, assumption)
```
```   669   using assms(1,5-) apply- by blast+
```
```   670
```
```   671 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
```
```   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}"
```
```   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) = {})"
```
```   674   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
```
```   675
```
```   676 lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
```
```   677 proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
```
```   678   show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
```
```   679   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
```
```   680   thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
```
```   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
```
```   682   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
```
```   683 qed
```
```   684
```
```   685 lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
```
```   686   shows "(snd ` s) division_of \<Union>(snd ` s)"
```
```   687 proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
```
```   688   show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
```
```   689   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
```
```   690   thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
```
```   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
```
```   692   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
```
```   693 qed
```
```   694
```
```   695 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
```
```   696   shows "t tagged_partial_division_of i"
```
```   697   using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
```
```   698
```
```   699 lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
```
```   700   assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
```
```   701   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
```
```   702 proof- note assm=tagged_division_ofD[OF assms(1)]
```
```   703   have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
```
```   704   show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
```
```   705     show "finite p" using assm by auto
```
```   706     fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
```
```   707     obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
```
```   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
```
```   709     hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
```
```   710     hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
```
```   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
```
```   712     thus "d (snd x) = 0" unfolding ab by auto qed qed
```
```   713
```
```   714 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
```
```   715
```
```   716 lemma tagged_division_of_empty: "{} tagged_division_of {}"
```
```   717   unfolding tagged_division_of by auto
```
```   718
```
```   719 lemma tagged_partial_division_of_trivial[simp]:
```
```   720  "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
```
```   721   unfolding tagged_partial_division_of_def by auto
```
```   722
```
```   723 lemma tagged_division_of_trivial[simp]:
```
```   724  "p tagged_division_of {} \<longleftrightarrow> p = {}"
```
```   725   unfolding tagged_division_of by auto
```
```   726
```
```   727 lemma tagged_division_of_self:
```
```   728  "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
```
```   729   apply(rule tagged_division_ofI) by auto
```
```   730
```
```   731 lemma tagged_division_union:
```
```   732   assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
```
```   733   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
```
```   734 proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
```
```   735   show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
```
```   736   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
```
```   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
```
```   738   show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
```
```   739   fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
```
```   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
```
```   741   show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
```
```   742     apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
```
```   743     using p1(3) p2(3) using xk xk' by auto qed
```
```   744
```
```   745 lemma tagged_division_unions:
```
```   746   assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
```
```   747   "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
```
```   748   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
```
```   749 proof(rule tagged_division_ofI)
```
```   750   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
```
```   751   show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
```
```   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
```
```   753   also have "\<dots> = \<Union>iset" using assm(6) by auto
```
```   754   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" .
```
```   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
```
```   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
```
```   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
```
```   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)
```
```   759     using assms(3)[rule_format] subset_interior by blast
```
```   760   show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
```
```   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
```
```   762 qed
```
```   763
```
```   764 lemma tagged_partial_division_of_union_self:
```
```   765   assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
```
```   766   apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
```
```   767
```
```   768 lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
```
```   769   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```   770   apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
```
```   771
```
```   772 subsection {* Fine-ness of a partition w.r.t. a gauge. *}
```
```   773
```
```   774 definition fine (infixr "fine" 46) where
```
```   775   "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
```
```   776
```
```   777 lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```   778   shows "d fine s" using assms unfolding fine_def by auto
```
```   779
```
```   780 lemma fineD[dest]: assumes "d fine s"
```
```   781   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
```
```   782
```
```   783 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
```
```   784   unfolding fine_def by auto
```
```   785
```
```   786 lemma fine_inters:
```
```   787  "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
```
```   788   unfolding fine_def by blast
```
```   789
```
```   790 lemma fine_union:
```
```   791   "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
```
```   792   unfolding fine_def by blast
```
```   793
```
```   794 lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
```
```   795   unfolding fine_def by auto
```
```   796
```
```   797 lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
```
```   798   unfolding fine_def by blast
```
```   799
```
```   800 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
```
```   801
```
```   802 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
```
```   803   "(f has_integral_compact_interval y) i \<equiv>
```
```   804         (\<forall>e>0. \<exists>d. gauge d \<and>
```
```   805           (\<forall>p. p tagged_division_of i \<and> d fine p
```
```   806                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
```
```   807
```
```   808 definition has_integral (infixr "has'_integral" 46) where
```
```   809 "((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
```
```   810         if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
```
```   811         else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```   812               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
```
```   813                                        norm(z - y) < e))"
```
```   814
```
```   815 lemma has_integral:
```
```   816  "(f has_integral y) ({a..b}) \<longleftrightarrow>
```
```   817         (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
```
```   818                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```   819   unfolding has_integral_def has_integral_compact_interval_def by auto
```
```   820
```
```   821 lemma has_integralD[dest]: assumes
```
```   822  "(f has_integral y) ({a..b})" "e>0"
```
```   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>
```
```   829       (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
```
```   830        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```   831                                \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
```
```   832                                         has_integral z) ({a..b}) \<and>
```
```   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"
```
```   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)"
```
```   839   using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
```
```   840
```
```   841 definition integrable_on (infixr "integrable'_on" 46) where
```
```   842   "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
```
```   843
```
```   844 definition "integral i f \<equiv> SOME y. (f has_integral y) i"
```
```   845
```
```   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)
```
```   849
```
```   850 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
```
```   851   unfolding integrable_on_def by auto
```
```   852
```
```   853 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
```
```   854   by auto
```
```   855
```
```   856 lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
```
```   857   shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
```
```   858 proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
```
```   859     unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
```
```   860   show ?thesis using assms unfolding has_integral apply safe
```
```   861     apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
```
```   862     apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
```
```   863
```
```   864 lemma setsum_content_null:
```
```   865   assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
```
```   866   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
```
```   867 proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
```
```   868   obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
```
```   869   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
```
```   870   from this(2) guess c .. then guess d .. note c_d=this
```
```   871   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
```
```   872   also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
```
```   873     unfolding assms(1) c_d by auto
```
```   874   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
```
```   875 qed
```
```   876
```
```   877 subsection {* Some basic combining lemmas. *}
```
```   878
```
```   879 lemma tagged_division_unions_exists:
```
```   880   assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
```
```   881   "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
```
```   882    obtains p where "p tagged_division_of i" "d fine p"
```
```   883 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
```
```   884   show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
```
```   885     apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
```
```   886     apply(rule fine_unions) using pfn by auto
```
```   887 qed
```
```   888
```
```   889 subsection {* The set we're concerned with must be closed. *}
```
```   890
```
```   891 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
```
```   892   unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
```
```   893
```
```   894 subsection {* General bisection principle for intervals; might be useful elsewhere. *}
```
```   895
```
```   896 lemma interval_bisection_step:
```
```   897   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})"
```
```   898   obtains c d where "~(P{c..d})"
```
```   899   "\<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"
```
```   900 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
```
```   901   note ab=this[unfolded interval_eq_empty not_ex not_less]
```
```   902   { fix f have "finite f \<Longrightarrow>
```
```   903         (\<forall>s\<in>f. P s) \<Longrightarrow>
```
```   904         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
```
```   905         (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
```
```   906     proof(induct f rule:finite_induct)
```
```   907       case empty show ?case using assms(1) by auto
```
```   908     next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
```
```   909         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
```
```   910         using insert by auto
```
```   911     qed } note * = this
```
```   912   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)}"
```
```   913   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"
```
```   914   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
```
```   915     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
```
```   916   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
```
```   917   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
```
```   918     let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a\$i else (a\$i + b\$i) / 2) ..
```
```   919       (\<chi> i. if i \<in> s then (a\$i + b\$i) / 2 else b\$i)}) ` {s. s \<subseteq> UNIV}"
```
```   920     have "?A \<subseteq> ?B" proof case goal1
```
```   921       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
```
```   922       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
```
```   923       show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c\$i = a\$i}" in bexI)
```
```   924         unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
```
```   925       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)"
```
```   926           "d \$ i = (if i \<in> {i. c \$ i = a \$ i} then (a \$ i + b \$ i) / 2 else b \$ i)"
```
```   927           using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
```
```   928       qed auto qed
```
```   929     thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
```
```   930     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
```
```   931     note c_d=this[rule_format]
```
```   932     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case
```
```   933         using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
```
```   934     show "\<exists>a b. s = {a..b}" unfolding c_d by auto
```
```   935     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
```
```   936     note e_f=this[rule_format]
```
```   937     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
```
```   938     then obtain i where "c\$i \<noteq> e\$i \<or> d\$i \<noteq> f\$i" unfolding de_Morgan_conj Cart_eq by auto
```
```   939     hence i:"c\$i \<noteq> e\$i" "d\$i \<noteq> f\$i" apply- apply(erule_tac[!] disjE)
```
```   940     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
```
```   941     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
```
```   942     qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
```
```   943     show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
```
```   944       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
```
```   945       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
```
```   946       show False using c_d(2)[of i] apply- apply(erule_tac disjE)
```
```   947       proof(erule_tac[!] conjE) assume as:"c \$ i = a \$ i" "d \$ i = (a \$ i + b \$ i) / 2"
```
```   948         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
```
```   949       next assume as:"c \$ i = (a \$ i + b \$ i) / 2" "d \$ i = b \$ i"
```
```   950         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
```
```   951       qed qed qed
```
```   952   also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
```
```   953     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
```
```   954     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
```
```   955     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
```
```   956     show "x\<in>{a..b}" unfolding mem_interval proof
```
```   957       fix i show "a \$ i \<le> x \$ i \<and> x \$ i \<le> b \$ i"
```
```   958         using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
```
```   959   next fix x assume x:"x\<in>{a..b}"
```
```   960     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"
```
```   961       (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
```
```   962       have "?P i (a\$i) ((a \$ i + b \$ i) / 2) \<or> ?P i ((a \$ i + b \$ i) / 2) (b\$i)"
```
```   963         using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
```
```   964     qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
```
```   965       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
```
```   966   qed finally show False using assms by auto qed
```
```   967
```
```   968 lemma interval_bisection:
```
```   969   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}"
```
```   970   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})"
```
```   971 proof-
```
```   972   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>
```
```   973                            2 * (snd y\$i - fst y\$i) \<le> snd x\$i - fst x\$i))" proof case goal1 thus ?case proof-
```
```   974       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
```
```   975       thus ?thesis apply(cases "P {fst x..snd x}") by auto
```
```   976     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
```
```   977       thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
```
```   978     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
```
```   979   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
```
```   980   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
```
```   981     (\<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>
```
```   982     2 * (B(Suc n)\$i - A(Suc n)\$i) \<le> B(n)\$i - A(n)\$i)" (is "\<And>n. ?P n")
```
```   983   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
```
```   984     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
```
```   985     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
```
```   986     next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
```
```   987     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
```
```   988
```
```   989   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"
```
```   990   proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b\$i - a\$i) UNIV) / e"] .. note n=this
```
```   991     show ?case apply(rule_tac x=n in exI) proof(rule,rule)
```
```   992       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
```
```   993       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
```
```   994       also have "\<dots> \<le> setsum (\<lambda>i. B n\$i - A n\$i) UNIV"
```
```   995       proof(rule setsum_mono) fix i show "\<bar>(x - y) \$ i\<bar> \<le> B n \$ i - A n \$ i"
```
```   996           using xy[unfolded mem_interval,THEN spec[where x=i]]
```
```   997           unfolding vector_minus_component by auto qed
```
```   998       also have "\<dots> \<le> setsum (\<lambda>i. b\$i - a\$i) UNIV / 2^n" unfolding setsum_divide_distrib
```
```   999       proof(rule setsum_mono) case goal1 thus ?case
```
```  1000         proof(induct n) case 0 thus ?case unfolding AB by auto
```
```  1001         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
```
```  1002           also have "\<dots> \<le> (b \$ i - a \$ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
```
```  1003         qed qed
```
```  1004       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
```
```  1005     qed qed
```
```  1006   { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
```
```  1007     have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
```
```  1008     proof(induct d) case 0 thus ?case by auto
```
```  1009     next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
```
```  1010         apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
```
```  1011       proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
```
```  1012       qed qed } note ABsubset = this
```
```  1013   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
```
```  1014   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
```
```  1015   then guess x0 .. note x0=this[rule_format]
```
```  1016   show thesis proof(rule that[rule_format,of x0])
```
```  1017     show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
```
```  1018     fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
```
```  1019     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}"
```
```  1020       apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
```
```  1021     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
```
```  1022       show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
```
```  1023       show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
```
```  1024     qed qed qed
```
```  1025
```
```  1026 subsection {* Cousin's lemma. *}
```
```  1027
```
```  1028 lemma fine_division_exists: assumes "gauge g"
```
```  1029   obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
```
```  1030 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
```
```  1031   then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
```
```  1032 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
```
```  1033   guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
```
```  1034     apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
```
```  1035   proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
```
```  1036     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 = {}"
```
```  1037     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
```
```  1038       apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
```
```  1039   qed note x=this
```
```  1040   obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
```
```  1041   from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
```
```  1042   have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
```
```  1043   thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
```
```  1044
```
```  1045 subsection {* Basic theorems about integrals. *}
```
```  1046
```
```  1047 lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1048   assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
```
```  1049 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
```
```  1050   have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.
```
```  1051     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
```
```  1052   proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
```
```  1053     guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
```
```  1054     guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
```
```  1055     guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
```
```  1056     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
```
```  1057       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)
```
```  1058     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  1059       apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
```
```  1060     finally show False by auto
```
```  1061   qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
```
```  1062     thus False apply-apply(cases "\<exists>a b. i = {a..b}")
```
```  1063       using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
```
```  1064   assume as:"\<not> (\<exists>a b. i = {a..b})"
```
```  1065   guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
```
```  1066   guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
```
```  1067   have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
```
```  1068     using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
```
```  1069   note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  1070   guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
```
```  1071   guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
```
```  1072   have "z = w" using lem[OF w(1) z(1)] by auto
```
```  1073   hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
```
```  1074     using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
```
```  1075   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
```
```  1076   finally show False by auto qed
```
```  1077
```
```  1078 lemma integral_unique[intro]:
```
```  1079   "(f has_integral y) k \<Longrightarrow> integral k f = y"
```
```  1080   unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
```
```  1081
```
```  1082 lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1083   assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
```
```  1084 proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
```
```  1085     (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
```
```  1086   proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
```
```  1087     assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
```
```  1088     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)"
```
```  1089       apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
```
```  1090     proof(rule,rule,erule conjE) case goal1
```
```  1091       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
```
```  1092         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
```
```  1093         thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
```
```  1094       qed thus ?case using as by auto
```
```  1095     qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1096     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
```
```  1097       using assms by(auto simp add:has_integral intro:lem) }
```
```  1098   have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
```
```  1099   assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
```
```  1100   apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
```
```  1101   proof- fix e::real and a b assume "e>0"
```
```  1102     thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
```
```  1103       apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
```
```  1104   qed auto qed
```
```  1105
```
```  1106 lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
```
```  1107   apply(rule has_integral_is_0) by auto
```
```  1108
```
```  1109 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
```
```  1110   using has_integral_unique[OF has_integral_0] by auto
```
```  1111
```
```  1112 lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1113   assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
```
```  1114 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
```
```  1115   have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
```
```  1116     (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
```
```  1117   proof(subst has_integral,rule,rule) case goal1
```
```  1118     from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
```
```  1119     have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
```
```  1120     guess g using has_integralD[OF goal1(1) *] . note g=this
```
```  1121     show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
```
```  1122     proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
```
```  1123       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
```
```  1124       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"
```
```  1125         unfolding o_def unfolding scaleR[THEN sym] * by simp
```
```  1126       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
```
```  1127       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)" .
```
```  1128       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
```
```  1129         apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
```
```  1130     qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1131     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
```
```  1132   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
```
```  1133   proof(rule,rule) fix e::real  assume e:"0<e"
```
```  1134     have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
```
```  1135     guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
```
```  1136     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)"
```
```  1137       apply(rule_tac x=M in exI) apply(rule,rule M(1))
```
```  1138     proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
```
```  1139       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)"
```
```  1140         unfolding o_def apply(rule ext) using zero by auto
```
```  1141       show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
```
```  1142         apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
```
```  1143     qed qed qed
```
```  1144
```
```  1145 lemma has_integral_cmul:
```
```  1146   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
```
```  1147   unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
```
```  1148   by(rule scaleR.bounded_linear_right)
```
```  1149
```
```  1150 lemma has_integral_neg:
```
```  1151   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
```
```  1152   apply(drule_tac c="-1" in has_integral_cmul) by auto
```
```  1153
```
```  1154 lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1155   assumes "(f has_integral k) s" "(g has_integral l) s"
```
```  1156   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
```
```  1157 proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
```
```  1158     (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
```
```  1159      ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
```
```  1160     show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
```
```  1161       guess d1 using has_integralD[OF goal1(1) *] . note d1=this
```
```  1162       guess d2 using has_integralD[OF goal1(2) *] . note d2=this
```
```  1163       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)"
```
```  1164         apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
```
```  1165       proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  1166         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)"
```
```  1167           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]
```
```  1168           by(rule setsum_cong2,auto)
```
```  1169         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))"
```
```  1170           unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"
```
```  1171         from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
```
```  1172         have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
```
```  1173           apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
```
```  1174         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
```
```  1175       qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1176     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
```
```  1177   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
```
```  1178   proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
```
```  1179     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
```
```  1180     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
```
```  1181     show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
```
```  1182     proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
```
```  1183       hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
```
```  1184       guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
```
```  1185       guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
```
```  1186       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
```
```  1187       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"
```
```  1188         apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
```
```  1189         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
```
```  1190     qed qed qed
```
```  1191
```
```  1192 lemma has_integral_sub:
```
```  1193   shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
```
```  1194   using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto
```
```  1195
```
```  1196 lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
```
```  1197   by(rule integral_unique has_integral_0)+
```
```  1198
```
```  1199 lemma integral_add:
```
```  1200   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  1201    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
```
```  1202   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1203   apply(rule has_integral_add) by assumption+
```
```  1204
```
```  1205 lemma integral_cmul:
```
```  1206   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
```
```  1207   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1208   apply(rule has_integral_cmul) by assumption+
```
```  1209
```
```  1210 lemma integral_neg:
```
```  1211   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
```
```  1212   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1213   apply(rule has_integral_neg) by assumption+
```
```  1214
```
```  1215 lemma integral_sub:
```
```  1216   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"
```
```  1217   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1218   apply(rule has_integral_sub) by assumption+
```
```  1219
```
```  1220 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
```
```  1221   unfolding integrable_on_def using has_integral_0 by auto
```
```  1222
```
```  1223 lemma integrable_add:
```
```  1224   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
```
```  1225   unfolding integrable_on_def by(auto intro: has_integral_add)
```
```  1226
```
```  1227 lemma integrable_cmul:
```
```  1228   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
```
```  1229   unfolding integrable_on_def by(auto intro: has_integral_cmul)
```
```  1230
```
```  1231 lemma integrable_neg:
```
```  1232   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
```
```  1233   unfolding integrable_on_def by(auto intro: has_integral_neg)
```
```  1234
```
```  1235 lemma integrable_sub:
```
```  1236   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
```
```  1237   unfolding integrable_on_def by(auto intro: has_integral_sub)
```
```  1238
```
```  1239 lemma integrable_linear:
```
```  1240   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
```
```  1241   unfolding integrable_on_def by(auto intro: has_integral_linear)
```
```  1242
```
```  1243 lemma integral_linear:
```
```  1244   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
```
```  1245   apply(rule has_integral_unique) defer unfolding has_integral_integral
```
```  1246   apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
```
```  1247   apply(rule integrable_linear) by assumption+
```
```  1248
```
```  1249 lemma has_integral_setsum:
```
```  1250   assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
```
```  1251   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
```
```  1252 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
```
```  1253   case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
```
```  1254     apply(rule has_integral_add) using insert assms by auto
```
```  1255 qed auto
```
```  1256
```
```  1257 lemma integral_setsum:
```
```  1258   shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
```
```  1259   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
```
```  1260   apply(rule integral_unique) apply(rule has_integral_setsum)
```
```  1261   using integrable_integral by auto
```
```  1262
```
```  1263 lemma integrable_setsum:
```
```  1264   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"
```
```  1265   unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
```
```  1266
```
```  1267 lemma has_integral_eq:
```
```  1268   assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
```
```  1269   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
```
```  1270   using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
```
```  1271
```
```  1272 lemma integrable_eq:
```
```  1273   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
```
```  1274   unfolding integrable_on_def using has_integral_eq[of s f g] by auto
```
```  1275
```
```  1276 lemma has_integral_eq_eq:
```
```  1277   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
```
```  1278   using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto
```
```  1279
```
```  1280 lemma has_integral_null[dest]:
```
```  1281   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
```
```  1282   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
```
```  1283 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
```
```  1284   fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
```
```  1285   have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
```
```  1286     using setsum_content_null[OF assms(1) p, of f] .
```
```  1287   thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
```
```  1288
```
```  1289 lemma has_integral_null_eq[simp]:
```
```  1290   shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
```
```  1291   apply rule apply(rule has_integral_unique,assumption)
```
```  1292   apply(drule has_integral_null,assumption)
```
```  1293   apply(drule has_integral_null) by auto
```
```  1294
```
```  1295 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
```
```  1296   by(rule integral_unique,drule has_integral_null)
```
```  1297
```
```  1298 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
```
```  1299   unfolding integrable_on_def apply(drule has_integral_null) by auto
```
```  1300
```
```  1301 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
```
```  1302   unfolding empty_as_interval apply(rule has_integral_null)
```
```  1303   using content_empty unfolding empty_as_interval .
```
```  1304
```
```  1305 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
```
```  1306   apply(rule,rule has_integral_unique,assumption) by auto
```
```  1307
```
```  1308 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
```
```  1309
```
```  1310 lemma integral_empty[simp]: shows "integral {} f = 0"
```
```  1311   apply(rule integral_unique) using has_integral_empty .
```
```  1312
```
```  1313 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a}"
```
```  1314 proof- have *:"{a} = {a..a}" apply(rule set_ext) unfolding mem_interval singleton_iff Cart_eq
```
```  1315     apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps)
```
```  1316   show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
```
```  1317     apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
```
```  1318     unfolding interior_closed_interval using interval_sing by auto qed
```
```  1319
```
```  1320 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
```
```  1321
```
```  1322 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
```
```  1323
```
```  1324 subsection {* Cauchy-type criterion for integrability. *}
```
```  1325
```
```  1326 lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}"
```
```  1327   shows "f integrable_on {a..b} \<longleftrightarrow>
```
```  1328   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
```
```  1329                             p2 tagged_division_of {a..b} \<and> d fine p2
```
```  1330                             \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
```
```  1331                                      setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
```
```  1332 proof assume ?l
```
```  1333   then guess y unfolding integrable_on_def has_integral .. note y=this
```
```  1334   show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
```
```  1335     then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
```
```  1336     show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
```
```  1337     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
```
```  1338       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"
```
```  1339         apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
```
```  1340         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
```
```  1341     qed qed
```
```  1342 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
```
```  1343   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  1344   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
```
```  1345   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-
```
```  1346   proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
```
```  1347   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
```
```  1348   have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
```
```  1349   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
```
```  1350   proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
```
```  1351     show ?case apply(rule_tac x=N in exI)
```
```  1352     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
```
```  1353       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"
```
```  1354         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
```
```  1355         using dp p(1) using mn by auto
```
```  1356     qed qed
```
```  1357   then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
```
```  1358   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
```
```  1359   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
```
```  1360     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
```
```  1361     guess N2 using y[OF *] .. note N2=this
```
```  1362     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)"
```
```  1363       apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
```
```  1364     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
```
```  1365       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
```
```  1366       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
```
```  1367       show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
```
```  1368         apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
```
```  1369         using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
```
```  1370         using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
```
```  1371
```
```  1372 subsection {* Additivity of integral on abutting intervals. *}
```
```  1373
```
```  1374 lemma interval_split:
```
```  1375   "{a..b::real^'n} \<inter> {x. x\$k \<le> c} = {a .. (\<chi> i. if i = k then min (b\$k) c else b\$i)}"
```
```  1376   "{a..b} \<inter> {x. x\$k \<ge> c} = {(\<chi> i. if i = k then max (a\$k) c else a\$i) .. b}"
```
```  1377   apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
```
```  1378   unfolding Cart_lambda_beta by auto
```
```  1379
```
```  1380 lemma content_split:
```
```  1381   "content {a..b::real^'n} = content({a..b} \<inter> {x. x\$k \<le> c}) + content({a..b} \<inter> {x. x\$k >= c})"
```
```  1382 proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
```
```  1383   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
```
```  1384   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
```
```  1385   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))"
```
```  1386     "(\<Prod>i\<in>UNIV. b\$i - a\$i) = (\<Prod>i\<in>UNIV-{k}. b\$i - a\$i) * (b\$k - a\$k)"
```
```  1387     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
```
```  1388   assume as:"a\<le>b" moreover have "\<And>x. min (b \$ k) c = max (a \$ k) c
```
```  1389     \<Longrightarrow> x* (b\$k - a\$k) = x*(max (a \$ k) c - a \$ k) + x*(b \$ k - max (a \$ k) c)"
```
```  1390     by  (auto simp add:field_simps)
```
```  1391   moreover have "\<not> a \$ k \<le> c \<Longrightarrow> \<not> c \<le> b \$ k \<Longrightarrow> False"
```
```  1392     unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
```
```  1393   ultimately show ?thesis
```
```  1394     unfolding simps unfolding *(1)[of "\<lambda>i x. b\$i - x"] *(1)[of "\<lambda>i x. x - a\$i"] *(2) by(auto)
```
```  1395 qed
```
```  1396
```
```  1397 lemma division_split_left_inj:
```
```  1398   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
```
```  1399   "k1 \<inter> {x::real^'n. x\$k \<le> c} = k2 \<inter> {x. x\$k \<le> c}"
```
```  1400   shows "content(k1 \<inter> {x. x\$k \<le> c}) = 0"
```
```  1401 proof- note d=division_ofD[OF assms(1)]
```
```  1402   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}) = {})"
```
```  1403     unfolding interval_split content_eq_0_interior by auto
```
```  1404   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
```
```  1405   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
```
```  1406   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
```
```  1407   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
```
```  1408     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
```
```  1409
```
```  1410 lemma division_split_right_inj:
```
```  1411   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
```
```  1412   "k1 \<inter> {x::real^'n. x\$k \<ge> c} = k2 \<inter> {x. x\$k \<ge> c}"
```
```  1413   shows "content(k1 \<inter> {x. x\$k \<ge> c}) = 0"
```
```  1414 proof- note d=division_ofD[OF assms(1)]
```
```  1415   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}) = {})"
```
```  1416     unfolding interval_split content_eq_0_interior by auto
```
```  1417   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
```
```  1418   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
```
```  1419   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
```
```  1420   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
```
```  1421     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
```
```  1422
```
```  1423 lemma tagged_division_split_left_inj:
```
```  1424   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}"
```
```  1425   shows "content(k1 \<inter> {x. x\$k \<le> c}) = 0"
```
```  1426 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
```
```  1427   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
```
```  1428     apply(rule_tac[1-2] *) using assms(2-) by auto qed
```
```  1429
```
```  1430 lemma tagged_division_split_right_inj:
```
```  1431   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}"
```
```  1432   shows "content(k1 \<inter> {x. x\$k \<ge> c}) = 0"
```
```  1433 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
```
```  1434   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
```
```  1435     apply(rule_tac[1-2] *) using assms(2-) by auto qed
```
```  1436
```
```  1437 lemma division_split:
```
```  1438   assumes "p division_of {a..b::real^'n}"
```
```  1439   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
```
```  1440         "{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")
```
```  1441 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
```
```  1442   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
```
```  1443   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
```
```  1444     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
```
```  1445     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
```
```  1446       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
```
```  1447     fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
```
```  1448     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
```
```  1449   { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
```
```  1450     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
```
```  1451     show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
```
```  1452       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
```
```  1453     fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
```
```  1454     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
```
```  1455 qed
```
```  1456
```
```  1457 lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1458   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})"
```
```  1459   shows "(f has_integral (i + j)) ({a..b})"
```
```  1460 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
```
```  1461   guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
```
```  1462   guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
```
```  1463   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"
```
```  1464   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
```
```  1465   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
```
```  1466     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
```
```  1467     have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\$k \<le> c} = {}) \<Longrightarrow> x\$k \<le> c"
```
```  1468          "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\$k \<ge> c} = {}) \<Longrightarrow> x\$k \<ge> c"
```
```  1469     proof- fix x kk assume as:"(x,kk)\<in>p"
```
```  1470       show "~(kk \<inter> {x. x\$k \<le> c} = {}) \<Longrightarrow> x\$k \<le> c"
```
```  1471       proof(rule ccontr) case goal1
```
```  1472         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \$ k - c\<bar>"
```
```  1473           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  1474         hence "\<exists>y. y \<in> ball x \<bar>x \$ k - c\<bar> \<inter> {x. x \$ k \<le> c}" using goal1(1) by blast
```
```  1475         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)
```
```  1476           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
```
```  1477         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
```
```  1478       qed
```
```  1479       show "~(kk \<inter> {x. x\$k \<ge> c} = {}) \<Longrightarrow> x\$k \<ge> c"
```
```  1480       proof(rule ccontr) case goal1
```
```  1481         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \$ k - c\<bar>"
```
```  1482           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  1483         hence "\<exists>y. y \<in> ball x \<bar>x \$ k - c\<bar> \<inter> {x. x \$ k \<ge> c}" using goal1(1) by blast
```
```  1484         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)
```
```  1485           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
```
```  1486         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
```
```  1487       qed
```
```  1488     qed
```
```  1489
```
```  1490     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
```
```  1491     have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
```
```  1492     proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
```
```  1493     have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
```
```  1494       setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
```
```  1495                = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
```
```  1496       apply(rule setsum_mono_zero_left) prefer 3
```
```  1497     proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
```
```  1498       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  1499       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
```
```  1500       have "content (g k) = 0" using xk using content_empty by auto
```
```  1501       thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
```
```  1502     qed auto
```
```  1503     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
```
```  1504
```
```  1505     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> {}}"
```
```  1506     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
```
```  1507       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
```
```  1508     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
```
```  1509       fix x l assume xl:"(x,l)\<in>?M1"
```
```  1510       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
```
```  1511       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
```
```  1512       thus "l \<subseteq> d1 x" unfolding xl' by auto
```
```  1513       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)
```
```  1514         using lem0(1)[OF xl'(3-4)] by auto
```
```  1515       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])
```
```  1516       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
```
```  1517       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
```
```  1518       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
```
```  1519       proof(cases "l' = r' \<longrightarrow> x' = y'")
```
```  1520         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1521       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
```
```  1522         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1523       qed qed moreover
```
```  1524
```
```  1525     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> {}}"
```
```  1526     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
```
```  1527       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
```
```  1528     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
```
```  1529       fix x l assume xl:"(x,l)\<in>?M2"
```
```  1530       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
```
```  1531       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
```
```  1532       thus "l \<subseteq> d2 x" unfolding xl' by auto
```
```  1533       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)
```
```  1534         using lem0(2)[OF xl'(3-4)] by auto
```
```  1535       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])
```
```  1536       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
```
```  1537       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
```
```  1538       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
```
```  1539       proof(cases "l' = r' \<longrightarrow> x' = y'")
```
```  1540         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1541       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
```
```  1542         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1543       qed qed ultimately
```
```  1544
```
```  1545     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"
```
```  1546       apply- apply(rule norm_triangle_lt) by auto
```
```  1547     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
```
```  1548       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)
```
```  1549        = (\<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
```
```  1550       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)"
```
```  1551         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
```
```  1552         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
```
```  1553       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
```
```  1554       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
```
```  1555       qed also note setsum_addf[THEN sym]
```
```  1556       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
```
```  1557         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
```
```  1558       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  1559         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"
```
```  1560           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
```
```  1561       qed note setsum_cong2[OF this]
```
```  1562       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 +
```
```  1563         ((\<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) =
```
```  1564         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
```
```  1565     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
```
```  1566
```
```  1567 subsection {* A sort of converse, integrability on subintervals. *}
```
```  1568
```
```  1569 lemma tagged_division_union_interval:
```
```  1570   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})"
```
```  1571   shows "(p1 \<union> p2) tagged_division_of ({a..b})"
```
```  1572 proof- have *:"{a..b} = ({a..b} \<inter> {x. x\$k \<le> c}) \<union> ({a..b} \<inter> {x. x\$k \<ge> c})" by auto
```
```  1573   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
```
```  1574     unfolding interval_split interior_closed_interval
```
```  1575     by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
```
```  1576
```
```  1577 lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
```
```  1578   assumes "(f has_integral i) ({a..b})" "e>0"
```
```  1579   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>
```
```  1580                                 p2 tagged_division_of ({a..b} \<inter> {x. x\$k \<ge> c}) \<and> d fine p2
```
```  1581                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
```
```  1582                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
```
```  1583 proof- guess d using has_integralD[OF assms] . note d=this
```
```  1584   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
```
```  1585   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
```
```  1586                    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
```
```  1587     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
```
```  1588     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)"
```
```  1589       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
```
```  1590     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
```
```  1591       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
```
```  1592       have "b \<subseteq> {x. x\$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
```
```  1593       moreover have "interior {x. x \$ k = c} = {}"
```
```  1594       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x\$k = c}" by auto
```
```  1595         then guess e unfolding mem_interior .. note e=this
```
```  1596         have x:"x\$k = c" using x interior_subset by fastsimp
```
```  1597         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
```
```  1598         have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm
```
```  1599           apply(rule le_less_trans[OF norm_le_l1]) unfolding *
```
```  1600           unfolding setsum_delta[OF finite_UNIV] using e by auto
```
```  1601         hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x\$k = c}" using e by auto
```
```  1602         thus False unfolding mem_Collect_eq using e x by auto
```
```  1603       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
```
```  1604       thus "content b *\<^sub>R f a = 0" by auto
```
```  1605     qed auto
```
```  1606     also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
```
```  1607     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
```
```  1608
```
```  1609 lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
```
```  1610   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)
```
```  1611 proof- guess y using assms unfolding integrable_on_def .. note y=this
```
```  1612   def b' \<equiv> "(\<chi> i. if i = k then min (b\$k) c else b\$i)::real^'n"
```
```  1613   and a' \<equiv> "(\<chi> i. if i = k then max (a\$k) c else a\$i)::real^'n"
```
```  1614   show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
```
```  1615   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
```
```  1616     from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
```
```  1617     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>
```
```  1618                               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     show "?P {x. x \$ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
```
```  1620     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"
```
```  1621       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
```
```  1622       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
```
```  1623         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
```
```  1624           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
```
```  1625           using p using assms by(auto simp add:group_simps)
```
```  1626       qed qed
```
```  1627     show "?P {x. x \$ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
```
```  1628     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"
```
```  1629       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"
```
```  1630       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
```
```  1631         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
```
```  1632           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
```
```  1633           using p using assms by(auto simp add:group_simps) qed qed qed qed
```
```  1634
```
```  1635 subsection {* Generalized notion of additivity. *}
```
```  1636
```
```  1637 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
```
```  1638
```
```  1639 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```  1640   "operative opp f \<equiv>
```
```  1641     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
```
```  1642     (\<forall>a b c k. f({a..b}) =
```
```  1643                    opp (f({a..b} \<inter> {x. x\$k \<le> c}))
```
```  1644                        (f({a..b} \<inter> {x. x\$k \<ge> c})))"
```
```  1645
```
```  1646 lemma operativeD[dest]: assumes "operative opp f"
```
```  1647   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
```
```  1648   "\<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}))"
```
```  1649   using assms unfolding operative_def by auto
```
```  1650
```
```  1651 lemma operative_trivial:
```
```  1652  "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
```
```  1653   unfolding operative_def by auto
```
```  1654
```
```  1655 lemma property_empty_interval:
```
```  1656  "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
```
```  1657   using content_empty unfolding empty_as_interval by auto
```
```  1658
```
```  1659 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
```
```  1660   unfolding operative_def apply(rule property_empty_interval) by auto
```
```  1661
```
```  1662 subsection {* Using additivity of lifted function to encode definedness. *}
```
```  1663
```
```  1664 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
```
```  1665   by (metis map_of.simps option.nchotomy)
```
```  1666
```
```  1667 lemma exists_option:
```
```  1668  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
```
```  1669   by (metis map_of.simps option.nchotomy)
```
```  1670
```
```  1671 fun lifted where
```
```  1672   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
```
```  1673   "lifted opp None _ = (None::'b option)" |
```
```  1674   "lifted opp _ None = None"
```
```  1675
```
```  1676 lemma lifted_simp_1[simp]: "lifted opp v None = None"
```
```  1677   apply(induct v) by auto
```
```  1678
```
```  1679 definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
```
```  1680                    (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
```
```  1681                    (\<forall>x. opp (neutral opp) x = x)"
```
```  1682
```
```  1683 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
```
```  1684   "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
```
```  1685   "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
```
```  1686   unfolding monoidal_def using assms by fastsimp
```
```  1687
```
```  1688 lemma monoidal_ac: assumes "monoidal opp"
```
```  1689   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
```
```  1690   "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
```
```  1691   using assms unfolding monoidal_def apply- by metis+
```
```  1692
```
```  1693 lemma monoidal_simps[simp]: assumes "monoidal opp"
```
```  1694   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
```
```  1695   using monoidal_ac[OF assms] by auto
```
```  1696
```
```  1697 lemma neutral_lifted[cong]: assumes "monoidal opp"
```
```  1698   shows "neutral (lifted opp) = Some(neutral opp)"
```
```  1699   apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
```
```  1700 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
```
```  1701   thus "x = Some (neutral opp)" apply(induct x) defer
```
```  1702     apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
```
```  1703     apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
```
```  1704 qed(auto simp add:monoidal_ac[OF assms])
```
```  1705
```
```  1706 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
```
```  1707   unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
```
```  1708
```
```  1709 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
```
```  1710 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
```
```  1711 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
```
```  1712
```
```  1713 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
```
```  1714 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
```
```  1715
```
```  1716 lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
```
```  1717   unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
```
```  1718
```
```  1719 lemma support_clauses:
```
```  1720   "\<And>f g s. support opp f {} = {}"
```
```  1721   "\<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))"
```
```  1722   "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
```
```  1723   "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
```
```  1724   "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
```
```  1725   "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
```
```  1726   "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
```
```  1727 unfolding support_def by auto
```
```  1728
```
```  1729 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
```
```  1730   unfolding support_def by auto
```
```  1731
```
```  1732 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
```
```  1733   unfolding iterate_def fold'_def by auto
```
```  1734
```
```  1735 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
```
```  1736   shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
```
```  1737 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
```
```  1738   show ?thesis unfolding iterate_def if_P[OF True] * by auto
```
```  1739 next case False note x=this
```
```  1740   note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
```
```  1741   show ?thesis proof(cases "f x = neutral opp")
```
```  1742     case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
```
```  1743       unfolding True monoidal_simps[OF assms(1)] by auto
```
```  1744   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
```
```  1745       apply(subst fun_left_comm.fold_insert[OF * finite_support])
```
```  1746       using `finite s` unfolding support_def using False x by auto qed qed
```
```  1747
```
```  1748 lemma iterate_some:
```
```  1749   assumes "monoidal opp"  "finite s"
```
```  1750   shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
```
```  1751 proof(induct s) case empty thus ?case using assms by auto
```
```  1752 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
```
```  1753     defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
```
```  1754
```
```  1755 subsection {* Two key instances of additivity. *}
```
```  1756
```
```  1757 lemma neutral_add[simp]:
```
```  1758   "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
```
```  1759   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
```
```  1760
```
```  1761 lemma operative_content[intro]: "operative (op +) content"
```
```  1762   unfolding operative_def content_split[THEN sym] neutral_add by auto
```
```  1763
```
```  1764 lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
```
```  1765   unfolding neutral_def apply(rule some_equality) defer
```
```  1766   apply(erule_tac x=0 in allE) by auto
```
```  1767
```
```  1768 lemma monoidal_monoid[intro]:
```
```  1769   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  1770   unfolding monoidal_def neutral_monoid by(auto simp add: group_simps)
```
```  1771
```
```  1772 lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  1773   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
```
```  1774   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
```
```  1775   apply(rule,rule,rule,rule) defer apply(rule allI)+
```
```  1776 proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
```
```  1777               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)
```
```  1778                (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)"
```
```  1779   proof(cases "f integrable_on {a..b}")
```
```  1780     case True show ?thesis unfolding if_P[OF True]
```
```  1781       unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
```
```  1782       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split)
```
```  1783       apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
```
```  1784   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}))"
```
```  1785     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
```
```  1786         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)
```
```  1787         apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
```
```  1788       thus False using False by auto
```
```  1789     qed thus ?thesis using False by auto
```
```  1790   qed next
```
```  1791   fix a b assume as:"content {a..b::real^'n} = 0"
```
```  1792   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
```
```  1793     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
```
```  1794
```
```  1795 subsection {* Points of division of a partition. *}
```
```  1796
```
```  1797 definition "division_points (k::(real^'n) set) d =
```
```  1798     {(j,x). (interval_lowerbound k)\$j < x \<and> x < (interval_upperbound k)\$j \<and>
```
```  1799            (\<exists>i\<in>d. (interval_lowerbound i)\$j = x \<or> (interval_upperbound i)\$j = x)}"
```
```  1800
```
```  1801 lemma division_points_finite: assumes "d division_of i"
```
```  1802   shows "finite (division_points i d)"
```
```  1803 proof- note assm = division_ofD[OF assms]
```
```  1804   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\$j < x \<and> x < (interval_upperbound i)\$j \<and>
```
```  1805            (\<exists>i\<in>d. (interval_lowerbound i)\$j = x \<or> (interval_upperbound i)\$j = x)}"
```
```  1806   have *:"division_points i d = \<Union>(?M ` UNIV)"
```
```  1807     unfolding division_points_def by auto
```
```  1808   show ?thesis unfolding * using assm by auto qed
```
```  1809
```
```  1810 lemma division_points_subset:
```
```  1811   assumes "d division_of {a..b}" "\<forall>i. a\$i < b\$i"  "a\$k < c" "c < b\$k"
```
```  1812   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} = {})}
```
```  1813                   \<subseteq> division_points ({a..b}) d" (is ?t1) and
```
```  1814         "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} = {})}
```
```  1815                   \<subseteq> division_points ({a..b}) d" (is ?t2)
```
```  1816 proof- note assm = division_ofD[OF assms(1)]
```
```  1817   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"
```
```  1818     "\<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"
```
```  1819     using assms using less_imp_le by auto
```
```  1820   show ?t1 unfolding division_points_def interval_split[of a b]
```
```  1821     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
```
```  1822     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
```
```  1823   proof- fix i l x assume as:"a \$ fst x < snd x" "snd x < (if fst x = k then c else b \$ fst x)"
```
```  1824       "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> {}"
```
```  1825     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1826     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 .
```
```  1827     have **:"\<forall>i. u\$i \<le> v\$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
```
```  1828     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)"
```
```  1829       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
```
```  1830       apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
```
```  1831       apply(case_tac[!] "fst x = k") using assms by auto
```
```  1832   qed
```
```  1833   show ?t2 unfolding division_points_def interval_split[of a b]
```
```  1834     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
```
```  1835     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
```
```  1836   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"
```
```  1837       "i = l \<inter> {x. c \<le> x \$ k}" "l \<in> d" "l \<inter> {x. c \<le> x \$ k} \<noteq> {}"
```
```  1838     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1839     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 .
```
```  1840     have **:"\<forall>i. u\$i \<le> v\$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
```
```  1841     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)"
```
```  1842       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
```
```  1843       apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
```
```  1844       apply(case_tac[!] "fst x = k") using assms by auto qed qed
```
```  1845
```
```  1846 lemma division_points_psubset:
```
```  1847   assumes "d division_of {a..b}"  "\<forall>i. a\$i < b\$i"  "a\$k < c" "c < b\$k"
```
```  1848   "l \<in> d" "interval_lowerbound l\$k = c \<or> interval_upperbound l\$k = c"
```
```  1849   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")
```
```  1850         "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")
```
```  1851 proof- have ab:"\<forall>i. a\$i \<le> b\$i" using assms(2) by(auto intro!:less_imp_le)
```
```  1852   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
```
```  1853   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
```
```  1854     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
```
```  1855   have *:"interval_upperbound ({a..b} \<inter> {x. x \$ k \<le> interval_upperbound l \$ k}) \$ k = interval_upperbound l \$ k"
```
```  1856          "interval_upperbound ({a..b} \<inter> {x. x \$ k \<le> interval_lowerbound l \$ k}) \$ k = interval_lowerbound 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 - ?D1" 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 "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
```
```  1865
```
```  1866   have *:"interval_lowerbound ({a..b} \<inter> {x. x \$ k \<ge> interval_lowerbound l \$ k}) \$ k = interval_lowerbound l \$ k"
```
```  1867          "interval_lowerbound ({a..b} \<inter> {x. x \$ k \<ge> interval_upperbound l \$ k}) \$ k = interval_upperbound l \$ k"
```
```  1868     unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
```
```  1869     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
```
```  1870   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
```
```  1871     apply(rule_tac x="(k,(interval_lowerbound l)\$k)" in exI) defer
```
```  1872     apply(rule_tac x="(k,(interval_upperbound l)\$k)" in exI)
```
```  1873     unfolding division_points_def unfolding interval_bounds[OF ab]
```
```  1874     apply (auto simp add:interval_bounds) unfolding * by auto
```
```  1875   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
```
```  1876
```
```  1877 subsection {* Preservation by divisions and tagged divisions. *}
```
```  1878
```
```  1879 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
```
```  1880   unfolding support_def by auto
```
```  1881
```
```  1882 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
```
```  1883   unfolding iterate_def support_support by auto
```
```  1884
```
```  1885 lemma iterate_expand_cases:
```
```  1886   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
```
```  1887   apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
```
```  1888
```
```  1889 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
```
```  1890   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
```
```  1891 proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
```
```  1892      iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
```
```  1893   proof- case goal1 show ?case using goal1
```
```  1894     proof(induct s) case empty thus ?case using assms(1) by auto
```
```  1895     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
```
```  1896         unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
```
```  1897         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
```
```  1898         apply(rule finite_imageI insert)+ apply(subst if_not_P)
```
```  1899         unfolding image_iff o_def using insert(2,4) by auto
```
```  1900     qed qed
```
```  1901   show ?thesis
```
```  1902     apply(cases "finite (support opp g (f ` s))")
```
```  1903     apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
```
```  1904     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
```
```  1905     apply(rule subset_inj_on[OF assms(2) support_subset])+
```
```  1906     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
```
```  1907     apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
```
```  1908
```
```  1909
```
```  1910 (* This lemma about iterations comes up in a few places.                     *)
```
```  1911 lemma iterate_nonzero_image_lemma:
```
```  1912   assumes "monoidal opp" "finite s" "g(a) = neutral opp"
```
```  1913   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
```
```  1914   shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
```
```  1915 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
```
```  1916   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
```
```  1917     unfolding support_def using assms(3) by auto
```
```  1918   show ?thesis unfolding *
```
```  1919     apply(subst iterate_support[THEN sym]) unfolding support_clauses
```
```  1920     apply(subst iterate_image[OF assms(1)]) defer
```
```  1921     apply(subst(2) iterate_support[THEN sym]) apply(subst **)
```
```  1922     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
```
```  1923
```
```  1924 lemma iterate_eq_neutral:
```
```  1925   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
```
```  1926   shows "(iterate opp s f = neutral opp)"
```
```  1927 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
```
```  1928   show ?thesis apply(subst iterate_support[THEN sym])
```
```  1929     unfolding * using assms(1) by auto qed
```
```  1930
```
```  1931 lemma iterate_op: assumes "monoidal opp" "finite s"
```
```  1932   shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
```
```  1933 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
```
```  1934 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
```
```  1935     unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
```
```  1936
```
```  1937 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  1938   shows "iterate opp s f = iterate opp s g"
```
```  1939 proof- have *:"support opp g s = support opp f s"
```
```  1940     unfolding support_def using assms(2) by auto
```
```  1941   show ?thesis
```
```  1942   proof(cases "finite (support opp f s)")
```
```  1943     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
```
```  1944       unfolding * by auto
```
```  1945   next def su \<equiv> "support opp f s"
```
```  1946     case True note support_subset[of opp f s]
```
```  1947     thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
```
```  1948       unfolding su_def[symmetric]
```
```  1949     proof(induct su) case empty show ?case by auto
```
```  1950     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
```
```  1951         unfolding if_not_P[OF insert(2)] apply(subst insert(3))
```
```  1952         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
```
```  1953
```
```  1954 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
```
```  1955
```
```  1956 lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
```
```  1957   assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
```
```  1958   shows "iterate opp d f = f {a..b}"
```
```  1959 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
```
```  1960   proof(induct C arbitrary:a b d rule:full_nat_induct)
```
```  1961     case goal1
```
```  1962     { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
```
```  1963       thus ?case apply-apply(cases) defer apply assumption
```
```  1964       proof- assume as:"content {a..b} = 0"
```
```  1965         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
```
```  1966         proof fix x assume x:"x\<in>d"
```
```  1967           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
```
```  1968           thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
```
```  1969             using operativeD(1)[OF assms(2)] x by auto
```
```  1970         qed qed }
```
```  1971     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
```
```  1972     hence ab':"\<forall>i. a\$i \<le> b\$i" by (auto intro!: less_imp_le) show ?case
```
```  1973     proof(cases "division_points {a..b} d = {}")
```
```  1974       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
```
```  1975         (\<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)"
```
```  1976         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
```
```  1977         apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
```
```  1978       proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
```
```  1979         hence uv:"\<forall>i. u\$i \<le> v\$i" "u\$j \<le> v\$j" unfolding interval_ne_empty by auto
```
```  1980         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
```
```  1981         have "(j, u\$j) \<notin> division_points {a..b} d"
```
```  1982           "(j, v\$j) \<notin> division_points {a..b} d" using True by auto
```
```  1983         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
```
```  1984         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
```
```  1985         moreover have "a\$j \<le> u\$j" "v\$j \<le> b\$j" using division_ofD(2,2,3)[OF goal1(4) as]
```
```  1986           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
```
```  1987           unfolding interval_ne_empty mem_interval by auto
```
```  1988         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"
```
```  1989           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
```
```  1990       qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
```
```  1991       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
```
```  1992       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
```
```  1993       have "{a..b} \<in> d"
```
```  1994       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
```
```  1995         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
```
```  1996         show "u = a" "v = b" unfolding Cart_eq
```
```  1997         proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
```
```  1998           thus "u \$ j = a \$ j" "v \$ j = b \$ j" using uv(2)[rule_format,of j] by auto
```
```  1999         qed qed
```
```  2000       hence *:"d = insert {a..b} (d - {{a..b}})" by auto
```
```  2001       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
```
```  2002       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
```
```  2003         then guess u v apply-by(erule exE conjE)+ note uv=this
```
```  2004         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
```
```  2005         then obtain j where "u\$j \<noteq> a\$j \<or> v\$j \<noteq> b\$j" unfolding Cart_eq by auto
```
```  2006         hence "u\$j = v\$j" using uv(2)[rule_format,of j] by auto
```
```  2007         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
```
```  2008         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
```
```  2009       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
```
```  2010         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
```
```  2011     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
```
```  2012       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
```
```  2013         by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
```
```  2014       from this(3) guess j .. note j=this
```
```  2015       def d1 \<equiv> "{l \<inter> {x. x\$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\$k \<le> c} \<noteq> {}}"
```
```  2016       def d2 \<equiv> "{l \<inter> {x. x\$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\$k \<ge> c} \<noteq> {}}"
```
```  2017       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)"
```
```  2018       note division_points_psubset[OF goal1(4) ab kc(1-2) j]
```
```  2019       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
```
```  2020       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})"
```
```  2021         apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
```
```  2022         using division_split[OF goal1(4), where k=k and c=c]
```
```  2023         unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
```
```  2024         using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
```
```  2025       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
```
```  2026         unfolding * apply(rule operativeD(2)) using goal1(3) .
```
```  2027       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\$k \<le> c}))"
```
```  2028         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
```
```  2029         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
```
```  2030         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
```
```  2031       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"
```
```  2032         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
```
```  2033         show "f (l \<inter> {x. x \$ k \<le> c}) = neutral opp" unfolding l interval_split
```
```  2034           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
```
```  2035           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
```
```  2036       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\$k \<ge> c}))"
```
```  2037         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
```
```  2038         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
```
```  2039         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
```
```  2040       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"
```
```  2041         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
```
```  2042         show "f (l \<inter> {x. x \$ k \<ge> c}) = neutral opp" unfolding l interval_split
```
```  2043           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
```
```  2044           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
```
```  2045       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}))"
```
```  2046         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
```
```  2047       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})))
```
```  2048         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
```
```  2049         apply(rule iterate_op[THEN sym]) using goal1 by auto
```
```  2050       finally show ?thesis by auto
```
```  2051     qed qed qed
```
```  2052
```
```  2053 lemma iterate_image_nonzero: assumes "monoidal opp"
```
```  2054   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
```
```  2055   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
```
```  2056 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
```
```  2057   case goal1 show ?case using assms(1) by auto
```
```  2058 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
```
```  2059   show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
```
```  2060     apply(rule finite_imageI goal2)+
```
```  2061     apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
```
```  2062     apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
```
```  2063     apply(subst iterate_insert[OF assms(1) goal2(1)])
```
```  2064     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
```
```  2065     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
```
```  2066     using goal2 unfolding o_def by auto qed
```
```  2067
```
```  2068 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
```
```  2069   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
```
```  2070 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
```
```  2071   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
```
```  2072     apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
```
```  2073     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
```
```  2074   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
```
```  2075     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
```
```  2076     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
```
```  2077       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
```
```  2078       unfolding as(4)[THEN sym] uv by auto
```
```  2079   qed also have "\<dots> = f {a..b}"
```
```  2080     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
```
```  2081   finally show ?thesis . qed
```
```  2082
```
```  2083 subsection {* Additivity of content. *}
```
```  2084
```
```  2085 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
```
```  2086 proof- have *:"setsum f s = setsum f (support op + f s)"
```
```  2087     apply(rule setsum_mono_zero_right)
```
```  2088     unfolding support_def neutral_monoid using assms by auto
```
```  2089   thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
```
```  2090     unfolding neutral_monoid . qed
```
```  2091
```
```  2092 lemma additive_content_division: assumes "d division_of {a..b}"
```
```  2093   shows "setsum content d = content({a..b})"
```
```  2094   unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
```
```  2095   apply(subst setsum_iterate) using assms by auto
```
```  2096
```
```  2097 lemma additive_content_tagged_division:
```
```  2098   assumes "d tagged_division_of {a..b}"
```
```  2099   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
```
```  2100   unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
```
```  2101   apply(subst setsum_iterate) using assms by auto
```
```  2102
```
```  2103 subsection {* Finally, the integral of a constant\<forall> *}
```
```  2104
```
```  2105 lemma has_integral_const[intro]:
```
```  2106   "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
```
```  2107   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
```
```  2108   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
```
```  2109   unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
```
```  2110   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
```
```  2111
```
```  2112 subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
```
```  2113
```
```  2114 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
```
```  2115   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
```
```  2116   apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
```
```  2117   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
```
```  2118   apply(subst real_mult_commute) apply(rule mult_left_mono)
```
```  2119   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
```
```  2120   apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
```
```  2121 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
```
```  2122   fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
```
```  2123   thus "0 \<le> content x" using content_pos_le by auto
```
```  2124 qed(insert assms,auto)
```
```  2125
```
```  2126 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
```
```  2127   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
```
```  2128 proof(cases "{a..b} = {}") case True
```
```  2129   show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
```
```  2130 next case False show ?thesis
```
```  2131     apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
```
```  2132     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
```
```  2133     unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
```
```  2134     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
```
```  2135     apply(subst o_def, rule abs_of_nonneg)
```
```  2136   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
```
```  2137       unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
```
```  2138     guess w using nonempty_witness[OF False] .
```
```  2139     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
```
```  2140     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
```
```  2141     from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
```
```  2142     show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
```
```  2143     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
```
```  2144   qed(insert assms,auto) qed
```
```  2145
```
```  2146 lemma rsum_diff_bound:
```
```  2147   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
```
```  2148   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})"
```
```  2149   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
```
```  2150   unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
```
```  2151
```
```  2152 lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  2153   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
```
```  2154   shows "norm i \<le> B * content {a..b}"
```
```  2155 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
```
```  2156     thus ?thesis proof(cases ?P) case False
```
```  2157       hence *:"content {a..b} = 0" using content_lt_nz by auto
```
```  2158       hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
```
```  2159       show ?thesis unfolding * ** using assms(1) by auto
```
```  2160     qed auto } assume ab:?P
```
```  2161   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
```
```  2162   assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
```
```  2163   from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
```
```  2164   from fine_division_exists[OF this(1), of a b] guess p . note p=this
```
```  2165   have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
```
```  2166   proof- case goal1 thus ?case unfolding not_less
```
```  2167     using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
```
```  2168   qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
```
```  2169
```
```  2170 subsection {* Similar theorems about relationship among components. *}
```
```  2171
```
```  2172 lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2173   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)\$i \<le> (g x)\$i"
```
```  2174   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"
```
```  2175   unfolding setsum_component apply(rule setsum_mono)
```
```  2176   apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
```
```  2177 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
```
```  2178   from this(3) guess u v apply-by(erule exE)+ note b=this
```
```  2179   show "(content b *\<^sub>R f a) \$ i \<le> (content b *\<^sub>R g a) \$ i" unfolding b
```
```  2180     unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
```
```  2181     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
```
```  2182
```
```  2183 lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2184   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)\$k \<le> (g x)\$k"
```
```  2185   shows "i\$k \<le> j\$k"
```
```  2186 proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow>
```
```  2187     (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"
```
```  2188   proof(rule ccontr) case goal1 hence *:"0 < (i\$k - j\$k) / 3" by auto
```
```  2189     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
```
```  2190     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
```
```  2191     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
```
```  2192     note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]
```
```  2193     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
```
```  2194     thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
```
```  2195   qed let ?P = "\<exists>a b. s = {a..b}"
```
```  2196   { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
```
```  2197       case True then guess a b apply-by(erule exE)+ note s=this
```
```  2198       show ?thesis apply(rule lem) using assms[unfolded s] by auto
```
```  2199     qed auto } assume as:"\<not> ?P"
```
```  2200   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
```
```  2201   assume "\<not> i\$k \<le> j\$k" hence ij:"(i\$k - j\$k) / 3 > 0" by auto
```
```  2202   note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
```
```  2203   have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
```
```  2204   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
```
```  2205   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  2206   guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
```
```  2207   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
```
```  2208   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*)
```
```  2209   note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
```
```  2210   have "w1\$k \<le> w2\$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
```
```  2211   show False unfolding Cart_nth.diff by(rule *) qed
```
```  2212
```
```  2213 lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2214   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)\$k \<le> (g x)\$k"
```
```  2215   shows "(integral s f)\$k \<le> (integral s g)\$k"
```
```  2216   apply(rule has_integral_component_le) using integrable_integral assms by auto
```
```  2217
```
```  2218 lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2219   assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
```
```  2220   shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
```
```  2221   using assms(3) unfolding vector_le_def by auto
```
```  2222
```
```  2223 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2224   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
```
```  2225   shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
```
```  2226   apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
```
```  2227
```
```  2228 lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2229   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\$k" shows "0 \<le> i\$k"
```
```  2230   using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
```
```  2231
```
```  2232 lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2233   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)\$k" shows "0 \<le> (integral s f)\$k"
```
```  2234   apply(rule has_integral_component_pos) using assms by auto
```
```  2235
```
```  2236 lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2237   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
```
```  2238   using has_integral_component_pos[OF assms(1), of 1]
```
```  2239   using assms(2) unfolding vector_le_def by auto
```
```  2240
```
```  2241 lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2242   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
```
```  2243   apply(rule has_integral_dest_vec1_pos) using assms by auto
```
```  2244
```
```  2245 lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  2246   assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)\$k \<le> 0" shows "i\$k \<le> 0"
```
```  2247   using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
```
```  2248
```
```  2249 lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2250   assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
```
```  2251   using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
```
```  2252
```
```  2253 lemma has_integral_component_lbound:
```
```  2254   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"
```
```  2255   using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
```
```  2256   unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
```
```  2257
```
```  2258 lemma has_integral_component_ubound:
```
```  2259   assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x\$k \<le> B"
```
```  2260   shows "i\$k \<le> B * content({a..b})"
```
```  2261   using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
```
```  2262   unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
```
```  2263
```
```  2264 lemma integral_component_lbound:
```
```  2265   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)\$k"
```
```  2266   shows "B * content({a..b}) \<le> (integral({a..b}) f)\$k"
```
```  2267   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
```
```  2268
```
```  2269 lemma integral_component_ubound:
```
```  2270   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\$k \<le> B"
```
```  2271   shows "(integral({a..b}) f)\$k \<le> B * content({a..b})"
```
```  2272   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
```
```  2273
```
```  2274 subsection {* Uniform limit of integrable functions is integrable. *}
```
```  2275
```
```  2276 lemma real_arch_invD:
```
```  2277   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```  2278   by(subst(asm) real_arch_inv)
```
```  2279
```
```  2280 lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2281   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
```
```  2282   shows "f integrable_on {a..b}"
```
```  2283 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
```
```  2284     show ?thesis apply cases apply(rule *,assumption)
```
```  2285       unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
```
```  2286   assume as:"content {a..b} > 0"
```
```  2287   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
```
```  2288   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
```
```  2289   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
```
```  2290
```
```  2291   have "Cauchy i" unfolding Cauchy_def
```
```  2292   proof(rule,rule) fix e::real assume "e>0"
```
```  2293     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
```
```  2294     then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
```
```  2295     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)
```
```  2296     proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
```
```  2297       from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
```
```  2298       from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
```
```  2299       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
```
```  2300       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"
```
```  2301       proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
```
```  2302           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
```
```  2303           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)
```
```  2304         also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
```
```  2305         finally show ?case .
```
```  2306       qed
```
```  2307       show ?case unfolding vector_dist_norm apply(rule lem2) defer
```
```  2308         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
```
```  2309         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
```
```  2310         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
```
```  2311       proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
```
```  2312           using M as by(auto simp add:field_simps)
```
```  2313         fix x assume x:"x \<in> {a..b}"
```
```  2314         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
```
```  2315             using g(1)[OF x, of n] g(1)[OF x, of m] by auto
```
```  2316         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
```
```  2317           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
```
```  2318         also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
```
```  2319         finally show "norm (g n x - g m x) \<le> 2 / real M"
```
```  2320           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
```
```  2321           by(auto simp add:group_simps simp add:norm_minus_commute)
```
```  2322       qed qed qed
```
```  2323   from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
```
```  2324
```
```  2325   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
```
```  2326   proof(rule,rule)
```
```  2327     case goal1 hence *:"e/3 > 0" by auto
```
```  2328     from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
```
```  2329     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
```
```  2330     from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
```
```  2331     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
```
```  2332     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"
```
```  2333     proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
```
```  2334         using norm_triangle_ineq[of "sf - sg" "sg - s"]
```
```  2335         using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:group_simps)
```
```  2336       also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)
```
```  2337       finally show ?case .
```
```  2338     qed
```
```  2339     show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
```
```  2340     proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
```
```  2341       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
```
```  2342         apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
```
```  2343       proof- have "content {a..b} < e / 3 * (real N2)"
```
```  2344           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
```
```  2345         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
```
```  2346           apply-apply(rule less_le_trans,assumption) using `e>0` by auto
```
```  2347         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
```
```  2348           unfolding inverse_eq_divide by(auto simp add:field_simps)
```
```  2349         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
```
```  2350       qed qed qed qed
```
```  2351
```
```  2352 subsection {* Negligible sets. *}
```
```  2353
```
```  2354 definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
```
```  2355
```
```  2356 lemma dest_vec1_indicator:
```
```  2357  "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
```
```  2358
```
```  2359 lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
```
```  2360
```
```  2361 lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
```
```  2362
```
```  2363 lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
```
```  2364   unfolding indicator_def by auto
```
```  2365
```
```  2366 definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
```
```  2367
```
```  2368 lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
```
```  2369   unfolding indicator_def by auto
```
```  2370
```
```  2371 subsection {* Negligibility of hyperplane. *}
```
```  2372
```
```  2373 lemma vsum_nonzero_image_lemma:
```
```  2374   assumes "finite s" "g(a) = 0"
```
```  2375   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
```
```  2376   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
```
```  2377   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
```
```  2378   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
```
```  2379   unfolding assms using neutral_add unfolding neutral_add using assms by auto
```
```  2380
```
```  2381 lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x\$k - c) \<le> (e::real)} =
```
```  2382   {(\<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)}"
```
```  2383 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
```
```  2384   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
```
```  2385   show ?thesis unfolding * ** interval_split by(rule refl) qed
```
```  2386
```
```  2387 lemma division_doublesplit: assumes "p division_of {a..b::real^'n}"
```
```  2388   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})"
```
```  2389 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
```
```  2390   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
```
```  2391   note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
```
```  2392   note division_split(2)[OF this, where c="c-e" and k=k]
```
```  2393   thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
```
```  2394     apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
```
```  2395     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x \$ k}" in exI) apply rule defer apply rule
```
```  2396     apply(rule_tac x=l in exI) by blast+ qed
```
```  2397
```
```  2398 lemma content_doublesplit: assumes "0 < e"
```
```  2399   obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x\$k - c) \<le> d}) < e"
```
```  2400 proof(cases "content {a..b} = 0")
```
```  2401   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
```
```  2402     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
```
```  2403     unfolding interval_doublesplit[THEN sym] using assms by auto
```
```  2404 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b\$i - a\$i) (UNIV - {k})"
```
```  2405   note False[unfolded content_eq_0 not_ex not_le, rule_format]
```
```  2406   hence prod0:"0 < setprod (\<lambda>i. b\$i - a\$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
```
```  2407   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
```
```  2408   proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
```
```  2409     have **:"{a..b} \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
```
```  2410       (\<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)
```
```  2411       = (\<Prod>i\<in>UNIV - {k}. b\$i - a\$i)" apply(rule setprod_cong,rule refl)
```
```  2412       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
```
```  2413     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)
```
```  2414       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
```
```  2415       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
```
```  2416     proof- have "(min (b \$ k) (c + d) - max (a \$ k) (c - d)) \<le> 2 * d" by auto
```
```  2417       also have "... < e / (\<Prod>i\<in>UNIV - {k}. b \$ i - a \$ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
```
```  2418       finally show "(min (b \$ k) (c + d) - max (a \$ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b \$ i - a \$ i) < e"
```
```  2419         unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
```
```  2420
```
```  2421 lemma negligible_standard_hyperplane[intro]: "negligible {x. x\$k = (c::real)}"
```
```  2422   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
```
```  2423 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}"
```
```  2424   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
```
```  2425   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
```
```  2426     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)"
```
```  2427       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
```
```  2428       apply(cases,rule disjI1,assumption,rule disjI2)
```
```  2429     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)
```
```  2430       show "content l = content (l \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
```
```  2431         apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
```
```  2432       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
```
```  2433         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]
```
```  2434         thus "\<bar>y \$ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
```
```  2435       qed auto qed
```
```  2436     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
```
```  2437     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
```
```  2438       apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
```
```  2439       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
```
```  2440       prefer 2 apply(subst(asm) eq_commute) apply assumption
```
```  2441       apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
```
```  2442     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}))"
```
```  2443         apply(rule setsum_mono) unfolding split_paired_all split_conv
```
```  2444         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
```
```  2445       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
```
```  2446       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d}) \<le> content {u..v}"
```
```  2447           unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
```
```  2448         thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
```
```  2449       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"
```
```  2450           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
```
```  2451         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)"
```
```  2452           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
```
```  2453           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
```
```  2454         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
```
```  2455         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
```
```  2456         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)]
```
```  2457         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"
```
```  2458           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
```
```  2459           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
```
```  2460         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
```
```  2461           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}"
```
```  2462           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
```
```  2463           note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
```
```  2464           hence "interior ({m..n} \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
```
```  2465           thus "content ({m..n} \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
```
```  2466         qed qed
```
```  2467       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \$ k - c\<bar> \<le> d}) * ?i x) < e" .
```
```  2468     qed qed qed
```
```  2469
```
```  2470 subsection {* A technical lemma about "refinement" of division. *}
```
```  2471
```
```  2472 lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
```
```  2473   assumes "p tagged_division_of {a..b}" "gauge d"
```
```  2474   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"
```
```  2475 proof-
```
```  2476   let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
```
```  2477     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
```
```  2478                    (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
```
```  2479   { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
```
```  2480     presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
```
```  2481     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
```
```  2482   } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
```
```  2483   show "?P p" apply(rule,rule) using as proof(induct p)
```
```  2484     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
```
```  2485   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
```
```  2486     note tagged_partial_division_subset[OF insert(4) subset_insertI]
```
```  2487     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
```
```  2488     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
```
```  2489     note p = tagged_partial_division_ofD[OF insert(4)]
```
```  2490     from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
```
```  2491
```
```  2492     have "finite {k. \<exists>x. (x, k) \<in> p}"
```
```  2493       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
```
```  2494       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
```
```  2495     hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
```
```  2496       apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
```
```  2497       unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
```
```  2498       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
```
```  2499       using insert(2) unfolding uv xk by auto
```
```  2500
```
```  2501     show ?case proof(cases "{u..v} \<subseteq> d x")
```
```  2502       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
```
```  2503         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
```
```  2504         apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int)
```
```  2505         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
```
```  2506         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
```
```  2507         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
```
```  2508     next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
```
```  2509       show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
```
```  2510         apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
```
```  2511         unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
```
```  2512         apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
```
```  2513         apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
```
```  2514     qed qed qed
```
```  2515
```
```  2516 subsection {* Hence the main theorem about negligible sets. *}
```
```  2517
```
```  2518 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
```
```  2519   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
```
```  2520 proof(induct) case (insert x s)
```
```  2521   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
```
```  2522   show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
```
```  2523
```
```  2524 lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
```
```  2525   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
```
```  2526 proof(induct) case (insert a s)
```
```  2527   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
```
```  2528   show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
```
```  2529     prefer 4 apply(subst insert(3)) unfolding add_right_cancel
```
```  2530   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
```
```  2531     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
```
```  2532   qed(insert insert, auto) qed auto
```
```  2533
```
```  2534 lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  2535   assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
```
```  2536   shows "(f has_integral 0) t"
```
```  2537 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})"
```
```  2538   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
```
```  2539   show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
```
```  2540     apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
```
```  2541   proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
```
```  2542     show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
```
```  2543   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)"
```
```  2544       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
```
```  2545       apply(rule,rule P) using assms(2) by auto
```
```  2546   qed
```
```  2547 next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  2548   show "(f has_integral 0) {a..b}" unfolding has_integral
```
```  2549   proof(safe) case goal1
```
```  2550     hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
```
```  2551       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
```
```  2552     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
```
```  2553     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
```
```  2554     show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
```
```  2555     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
```
```  2556       fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
```
```  2557       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  2558       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
```
```  2559       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
```
```  2560       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
```
```  2561       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)"
```
```  2562         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
```
```  2563       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
```
```  2564       have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe)
```
```  2565         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
```
```  2566       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"
```
```  2567       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
```
```  2568           apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
```
```  2569       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
```
```  2570                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
```
```  2571         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
```
```  2572         apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
```
```  2573       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
```
```  2574         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
```
```  2575           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
```
```  2576           using tagged_division_ofD(4)[OF q(1) as''] by auto
```
```  2577       next fix i::nat show "finite (q i)" using q by auto
```
```  2578       next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
```
```  2579         have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
```
```  2580         have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
```
```  2581         hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
```
```  2582         moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
```
```  2583         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
```
```  2584         moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
```
```  2585         proof(cases "x\<in>s") case False thus ?thesis using assm by auto
```
```  2586         next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
```
```  2587           moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
```
```  2588           ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
```
```  2589         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)"
```
```  2590           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
```
```  2591       qed(insert as, auto)
```
```  2592       also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
```
```  2593       proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
```
```  2594           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
```
```  2595       qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
```
```  2596         apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
```
```  2597         apply(subst sumr_geometric) using goal1 by auto
```
```  2598       finally show "?goal" by auto qed qed qed
```
```  2599
```
```  2600 lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  2601   assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
```
```  2602   shows "(g has_integral y) t"
```
```  2603 proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
```
```  2604     assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
```
```  2605     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
```
```  2606       apply(rule has_integral_negligible[OF assms(1)]) using as by auto
```
```  2607     hence "(g has_integral y) {a..b}" by auto } note * = this
```
```  2608   show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
```
```  2609     apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
```
```  2610     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)
```
```  2611     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
```
```  2612
```
```  2613 lemma has_integral_spike_eq:
```
```  2614   assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
```
```  2615   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2616   apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
```
```  2617
```
```  2618 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
```
```  2619   shows "g integrable_on  t"
```
```  2620   using assms unfolding integrable_on_def apply-apply(erule exE)
```
```  2621   apply(rule,rule has_integral_spike) by fastsimp+
```
```  2622
```
```  2623 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
```
```  2624   shows "integral t f = integral t g"
```
```  2625   unfolding integral_def using has_integral_spike_eq[OF assms] by auto
```
```  2626
```
```  2627 subsection {* Some other trivialities about negligible sets. *}
```
```  2628
```
```  2629 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
```
```  2630 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
```
```  2631     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
```
```  2632     using assms(2) unfolding indicator_def by auto qed
```
```  2633
```
```  2634 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
```
```  2635
```
```  2636 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
```
```  2637
```
```  2638 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
```
```  2639 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
```
```  2640   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
```
```  2641     defer apply assumption unfolding indicator_def by auto qed
```
```  2642
```
```  2643 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
```
```  2644   using negligible_union by auto
```
```  2645
```
```  2646 lemma negligible_sing[intro]: "negligible {a::real^'n}"
```
```  2647 proof- guess x using UNIV_witness[where 'a='n] ..
```
```  2648   show ?thesis using negligible_standard_hyperplane[of x "a\$x"] by auto qed
```
```  2649
```
```  2650 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
```
```  2651   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
```
```  2652
```
```  2653 lemma negligible_empty[intro]: "negligible {}" by auto
```
```  2654
```
```  2655 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
```
```  2656   using assms apply(induct s) by auto
```
```  2657
```
```  2658 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
```
```  2659   using assms by(induct,auto)
```
```  2660
```
```  2661 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
```
```  2662   apply safe defer apply(subst negligible_def)
```
```  2663 proof- fix t::"(real^'n) set" assume as:"negligible s"
```
```  2664   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)
```
```  2665   show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
```
```  2666     apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
```
```  2667     apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
```
```  2668     using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
```
```  2669
```
```  2670 subsection {* Finite case of the spike theorem is quite commonly needed. *}
```
```  2671
```
```  2672 lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
```
```  2673   "(f has_integral y) t" shows "(g has_integral y) t"
```
```  2674   apply(rule has_integral_spike) using assms by auto
```
```  2675
```
```  2676 lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
```
```  2677   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2678   apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
```
```  2679
```
```  2680 lemma integrable_spike_finite:
```
```  2681   assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
```
```  2682   using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
```
```  2683   apply(rule has_integral_spike_finite) by auto
```
```  2684
```
```  2685 subsection {* In particular, the boundary of an interval is negligible. *}
```
```  2686
```
```  2687 lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
```
```  2688 proof- let ?A = "\<Union>((\<lambda>k. {x. x\$k = a\$k} \<union> {x. x\$k = b\$k}) ` UNIV)"
```
```  2689   have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
```
```  2690     apply(erule conjE exE)+ apply(rule_tac X="{x. x \$ xa = a \$ xa} \<union> {x. x \$ xa = b \$ xa}" in UnionI)
```
```  2691     apply(erule_tac[!] x=xa in allE) by auto
```
```  2692   thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
```
```  2693
```
```  2694 lemma has_integral_spike_interior:
```
```  2695   assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
```
```  2696   apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
```
```  2697
```
```  2698 lemma has_integral_spike_interior_eq:
```
```  2699   assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
```
```  2700   apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
```
```  2701
```
```  2702 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}"
```
```  2703   using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
```
```  2704
```
```  2705 subsection {* Integrability of continuous functions. *}
```
```  2706
```
```  2707 lemma neutral_and[simp]: "neutral op \<and> = True"
```
```  2708   unfolding neutral_def apply(rule some_equality) by auto
```
```  2709
```
```  2710 lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
```
```  2711
```
```  2712 lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
```
```  2713 apply induct unfolding iterate_insert[OF monoidal_and] by auto
```
```  2714
```
```  2715 lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
```
```  2716   shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
```
```  2717   using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
```
```  2718
```
```  2719 lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2720   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
```
```  2721 proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
```
```  2722     thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
```
```  2723       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
```
```  2724   { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
```
```  2725     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}"
```
```  2726       "\<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}"
```
```  2727       apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
```
```  2728   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}"
```
```  2729                           "\<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}"
```
```  2730   let ?g = "\<lambda>x. if x\$k = c then f x else if x\$k \<le> c then g1 x else g2 x"
```
```  2731   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)
```
```  2732   proof safe case goal1 thus ?case apply- apply(cases "x\$k=c", case_tac "x\$k < c") using as assms by auto
```
```  2733   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}"
```
```  2734     then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
```
```  2735     show ?case unfolding integrable_on_def by auto
```
```  2736   next show "?g integrable_on {a..b} \<inter> {x. x \$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \$ k \<ge> c}"
```
```  2737       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
```
```  2738
```
```  2739 lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2740   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"
```
```  2741   obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
```
```  2742 proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
```
```  2743   note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
```
```  2744   guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
```
```  2745
```
```  2746 lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2747   assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
```
```  2748 proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
```
```  2749   from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
```
```  2750   note d=conjunctD2[OF this,rule_format]
```
```  2751   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
```
```  2752   note p' = tagged_division_ofD[OF p(1)]
```
```  2753   have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  2754   proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
```
```  2755     from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
```
```  2756     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)
```
```  2757     proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
```
```  2758       fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
```
```  2759       note d(2)[OF _ _ this[unfolded mem_ball]]
```
```  2760       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
```
```  2761   from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
```
```  2762   thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
```
```  2763
```
```  2764 subsection {* Specialization of additivity to one dimension. *}
```
```  2765
```
```  2766 lemma operative_1_lt: assumes "monoidal opp"
```
```  2767   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
```
```  2768                 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
```
```  2769   unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
```
```  2770 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"
```
```  2771     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
```
```  2772     thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c\$1"] by auto
```
```  2773 next fix a b::"real^1" and c::real
```
```  2774   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}"
```
```  2775   show "f {a..b} = opp (f ({a..b} \<inter> {x. x \$ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x \$ 1}))"
```
```  2776   proof(cases "c \<in> {a\$1 .. b\$1}")
```
```  2777     case False hence "c<a\$1 \<or> c>b\$1" by auto
```
```  2778     thus ?thesis apply-apply(erule disjE)
```
```  2779     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
```
```  2780       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
```
```  2781     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
```
```  2782       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
```
```  2783     qed
```
```  2784   next case True hence *:"min (b \$ 1) c = c" "max (a \$ 1) c = c" by auto
```
```  2785     show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
```
```  2786     proof(cases "c = a\$1 \<or> c = b\$1")
```
```  2787       case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
```
```  2788         apply-apply(subst as(2)[rule_format]) using True by auto
```
```  2789     next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
```
```  2790       proof(erule disjE) assume "c=a\$1" hence *:"a = vec1 c" unfolding Cart_eq by auto
```
```  2791         hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2792         thus ?thesis using assms unfolding * by auto
```
```  2793       next assume "c=b\$1" hence *:"b = vec1 c" unfolding Cart_eq by auto
```
```  2794         hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2795         thus ?thesis using assms unfolding * by auto qed qed qed qed
```
```  2796
```
```  2797 lemma operative_1_le: assumes "monoidal opp"
```
```  2798   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
```
```  2799                 (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
```
```  2800 unfolding operative_1_lt[OF assms]
```
```  2801 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"
```
```  2802   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
```
```  2803 next fix a b c ::"real^1"
```
```  2804   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"
```
```  2805   note as = this[rule_format]
```
```  2806   show "opp (f {a..c}) (f {c..b}) = f {a..b}"
```
```  2807   proof(cases "c = a \<or> c = b")
```
```  2808     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)
```
```  2809     next case True thus ?thesis apply-
```
```  2810       proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2811         thus ?thesis using assms unfolding * by auto
```
```  2812       next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2813         thus ?thesis using assms unfolding * by auto qed qed qed
```
```  2814
```
```  2815 subsection {* Special case of additivity we need for the FCT. *}
```
```  2816
```
```  2817 lemma interval_bound_sing[simp]: "interval_upperbound {a} = a"  "interval_lowerbound {a} = a"
```
```  2818   unfolding interval_upperbound_def interval_lowerbound_def unfolding Cart_eq by auto
```
```  2819
```
```  2820 lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
```
```  2821   assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
```
```  2822   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
```
```  2823 proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
```
```  2824   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
```
```  2825     by(auto simp add:not_less interval_bound_1 vector_less_def)
```
```  2826   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
```
```  2827   note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
```
```  2828   show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
```
```  2829     apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
```
```  2830
```
```  2831 subsection {* A useful lemma allowing us to factor out the content size. *}
```
```  2832
```
```  2833 lemma has_integral_factor_content:
```
```  2834   "(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
```
```  2835     \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
```
```  2836 proof(cases "content {a..b} = 0")
```
```  2837   case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
```
```  2838     apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
```
```  2839     apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
```
```  2840     apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
```
```  2841 next case False note F = this[unfolded content_lt_nz[THEN sym]]
```
```  2842   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)"
```
```  2843   show ?thesis apply(subst has_integral)
```
```  2844   proof safe fix e::real assume e:"e>0"
```
```  2845     { 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)
```
```  2846         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
```
```  2847         using F e by(auto simp add:field_simps intro:mult_pos_pos) }
```
```  2848     {  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)
```
```  2849         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
```
```  2850         using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
```
```  2851
```
```  2852 subsection {* Fundamental theorem of calculus. *}
```
```  2853
```
```  2854 lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  2855   assumes "a \<le> b"  "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
```
```  2856   shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
```
```  2857 unfolding has_integral_factor_content
```
```  2858 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
```
```  2859   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
```
```  2860   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
```
```  2861   note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
```
```  2862   guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  2863   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
```
```  2864                  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})"
```
```  2865     apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
```
```  2866     apply(rule gauge_ball_dependent,rule,rule d(1))
```
```  2867   proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
```
```  2868     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}"
```
```  2869       unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
```
```  2870       unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
```
```  2871       apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
```
```  2872     proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
```
```  2873       note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
```
```  2874       have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
```
```  2875       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] .
```
```  2876       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)"
```
```  2877         apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
```
```  2878         unfolding scaleR.diff_left by(auto simp add:group_simps)
```
```  2879       also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
```
```  2880         apply(rule add_mono) apply(rule d(2)[of "x\$1" "u\$1",unfolded o_def vec1_dest_vec1]) prefer 4
```
```  2881         apply(rule d(2)[of "x\$1" "v\$1",unfolded o_def vec1_dest_vec1])
```
```  2882         using ball[rule_format,of u] ball[rule_format,of v]
```
```  2883         using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
```
```  2884       also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
```
```  2885         unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
```
```  2886       finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
```
```  2887         e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
```
```  2888     qed(insert as, auto) qed qed
```
```  2889
```
```  2890 subsection {* Attempt a systematic general set of "offset" results for components. *}
```
```  2891
```
```  2892 lemma gauge_modify:
```
```  2893   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
```
```  2894   shows "gauge (\<lambda>x y. d (f x) (f y))"
```
```  2895   using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
```
```  2896   apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
```
```  2897
```
```  2898 subsection {* Only need trivial subintervals if the interval itself is trivial. *}
```
```  2899
```
```  2900 lemma division_of_nontrivial: fixes s::"(real^'n) set set"
```
```  2901   assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
```
```  2902   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
```
```  2903 proof(induct "card s" arbitrary:s rule:nat_less_induct)
```
```  2904   fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
```
```  2905     "\<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}"
```
```  2906   note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
```
```  2907   { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
```
```  2908     show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
```
```  2909   assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
```
```  2910   then obtain k where k:"k\<in>s" "content k = 0" by auto
```
```  2911   from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
```
```  2912   from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
```
```  2913   hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
```
```  2914   have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
```
```  2915     apply safe apply(rule closed_interval) using assm(1) by auto
```
```  2916   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
```
```  2917   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
```
```  2918     from k(2)[unfolded k content_eq_0] guess i ..
```
```  2919     hence i:"c\$i = d\$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
```
```  2920     hence xi:"x\$i = d\$i" using as unfolding k mem_interval by smt
```
```  2921     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)"
```
```  2922     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
```
```  2923     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)
```
```  2924       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]]
```
```  2925       hence xyi:"y\$i \<noteq> x\$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
```
```  2926         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+
```
```  2927       thus "y \<noteq> x" unfolding Cart_eq by auto
```
```  2928       have *:"UNIV = insert i (UNIV - {i})" by auto
```
```  2929       have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
```
```  2930         apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
```
```  2931       proof- show "\<bar>(y - x) \$ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
```
```  2932           apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
```
```  2933         show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) \$ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto
```
```  2934       qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
```
```  2935       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
```
```  2936       moreover have "y \<in> \<Union>s" unfolding s mem_interval
```
```  2937       proof note simps = y_def Cart_lambda_beta if_not_P
```
```  2938         fix j::'n show "a \$ j \<le> y \$ j \<and> y \$ j \<le> b \$ j"
```
```  2939         proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
```
```  2940           thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
```
```  2941         next case True note T = this show ?thesis
```
```  2942           proof(cases "c \$ i \<le> (a \$ i + b \$ i) / 2")
```
```  2943             case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
```
```  2944               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
```
```  2945           next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
```
```  2946               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
```
```  2947           qed qed qed
```
```  2948       ultimately show "y \<in> \<Union>(s - {k})" by auto
```
```  2949     qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
```
```  2950   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
```
```  2951     apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
```
```  2952   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
```
```  2953
```
```  2954 subsection {* Integrabibility on subintervals. *}
```
```  2955
```
```  2956 lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
```
```  2957   "operative op \<and> (\<lambda>i. f integrable_on i)"
```
```  2958   unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
```
```  2959   unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
```
```  2960   unfolding integrable_on_def by(auto intro: has_integral_split)
```
```  2961
```
```  2962 lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2963   assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
```
```  2964   apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
```
```  2965   using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
```
```  2966
```
```  2967 subsection {* Combining adjacent intervals in 1 dimension. *}
```
```  2968
```
```  2969 lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
```
```  2970   "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
```
```  2971   shows "(f has_integral (i + j)) {a..b}"
```
```  2972 proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
```
```  2973   note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
```
```  2974   hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
```
```  2975     apply(subst(asm) if_P) using assms(3-) by auto
```
```  2976   with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
```
```  2977     unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
```
```  2978
```
```  2979 lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  2980   assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
```
```  2981   shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
```
```  2982   apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
```
```  2983   apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
```
```  2984
```
```  2985 lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  2986   assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
```
```  2987   shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
```
```  2988
```
```  2989 subsection {* Reduce integrability to "local" integrability. *}
```
```  2990
```
```  2991 lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  2992   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}"
```
```  2993   shows "f integrable_on {a..b}"
```
```  2994 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})"
```
```  2995     using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
```
```  2996   guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
```
```  2997   note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
```
```  2998   show ?thesis unfolding * apply safe unfolding snd_conv
```
```  2999   proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
```
```  3000     thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
```
```  3001
```
```  3002 subsection {* Second FCT or existence of antiderivative. *}
```
```  3003
```
```  3004 lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
```
```  3005   unfolding integrable_on_def by(rule,rule has_integral_const)
```
```  3006
```
```  3007 lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3008   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
```
```  3009   shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
```
```  3010   unfolding has_vector_derivative_def has_derivative_within_alt
```
```  3011 apply safe apply(rule scaleR.bounded_linear_left)
```
```  3012 proof- fix e::real assume e:"e>0"
```
```  3013   note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
```
```  3014   from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
```
```  3015   let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
```
```  3016   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)"
```
```  3017   proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
```
```  3018       case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" 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 y - ?I a x = ?I x y" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
```
```  3021         using False unfolding not_less using assms(2) goal1 by auto
```
```  3022       have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
```
```  3023       show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
```
```  3024         defer apply(rule has_integral_sub) apply(rule integrable_integral)
```
```  3025         apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
```
```  3026       proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
```
```  3027         have *:"y - x = norm(y - x)" using False by auto
```
```  3028         show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
```
```  3029         show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
```
```  3030           apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
```
```  3031       qed(insert e,auto)
```
```  3032     next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
```
```  3033         apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
```
```  3034       hence *:"?I a x - ?I a y = ?I y x" unfolding group_simps apply(subst eq_commute) apply(rule integral_combine)
```
```  3035         using True using assms(2) goal1 by auto
```
```  3036       have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
```
```  3037       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
```
```  3038       show ?thesis apply(subst ***) unfolding norm_minus_cancel **
```
```  3039         apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
```
```  3040         defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
```
```  3041         apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
```
```  3042         apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
```
```  3043       proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
```
```  3044         have *:"x - y = norm(y - x)" using True by auto
```
```  3045         show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
```
```  3046         show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
```
```  3047           apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
```
```  3048       qed(insert e,auto) qed qed qed
```
```  3049
```
```  3050 lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  3051   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
```
```  3052   shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x\$1) within {a\$1..b\$1})"
```
```  3053   using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x\$1"]
```
```  3054   unfolding o_def vec1_dest_vec1 using assms(2) by auto
```
```  3055
```
```  3056 lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
```
```  3057   obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
```
```  3058   apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
```
```  3059
```
```  3060 subsection {* Combined fundamental theorem of calculus. *}
```
```  3061
```
```  3062 lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
```
```  3063   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}"
```
```  3064 proof- from antiderivative_continuous[OF assms] guess g . note g=this
```
```  3065   show ?thesis apply(rule that[of g])
```
```  3066   proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
```
```  3067       apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
```
```  3068     thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
```
```  3069       unfolding o_def vec1_dest_vec1 by auto qed qed
```
```  3070
```
```  3071 subsection {* General "twiddling" for interval-to-interval function image. *}
```
```  3072
```
```  3073 lemma has_integral_twiddle:
```
```  3074   assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
```
```  3075   "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
```
```  3076   "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
```
```  3077   "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
```
```  3078   "(f has_integral i) {a..b}"
```
```  3079   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
```
```  3080 proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
```
```  3081     show ?thesis apply cases defer apply(rule *,assumption)
```
```  3082     proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
```
```  3083   assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
```
```  3084   have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
```
```  3085     using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
```
```  3086     using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
```
```  3087   show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
```
```  3088   proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
```
```  3089     from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
```
```  3090     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)
```
```  3091     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)"
```
```  3092     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
```
```  3093       fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
```
```  3094       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
```
```  3095       proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
```
```  3096         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
```
```  3097         fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
```
```  3098         show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
```
```  3099         { 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)"]
```
```  3100             using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
```
```  3101         fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
```
```  3102         hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
```
```  3103         have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
```
```  3104         proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
```
```  3105           hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
```
```  3106             unfolding image_Int[OF inj(1)] by auto thus False using as by blast
```
```  3107         qed thus "g x = g x'" by auto
```
```  3108         { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
```
```  3109         { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
```
```  3110       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
```
```  3111         then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
```
```  3112         thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
```
```  3113           apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
```
```  3114           using X(2) assms(3)[rule_format,of x] by auto
```
```  3115       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
```
```  3116        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
```
```  3117         unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
```
```  3118         apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
```
```  3119       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]
```
```  3120         unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
```
```  3121       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
```
```  3122         using assms(1) by(auto simp add:field_simps) qed qed qed
```
```  3123
```
```  3124 subsection {* Special case of a basic affine transformation. *}
```
```  3125
```
```  3126 lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
```
```  3127   unfolding image_affinity_interval by auto
```
```  3128
```
```  3129 lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
```
```  3130    Cart_eq vector_le_def vector_less_def
```
```  3131
```
```  3132 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
```
```  3133   apply(rule setprod_cong) using assms by auto
```
```  3134
```
```  3135 lemma content_image_affinity_interval:
```
```  3136  "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
```
```  3137 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
```
```  3138       unfolding not_not using content_empty by auto }
```
```  3139   assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
```
```  3140     case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
```
```  3141       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
```
```  3142       defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
```
```  3143       apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le
```
```  3144       by(auto simp add:field_simps intro:mult_left_mono)
```
```  3145   next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
```
```  3146       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval')
```
```  3147       defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
```
```  3148       apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le
```
```  3149       by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
```
```  3150
```
```  3151 lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
```
```  3152   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})"
```
```  3153   apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
```
```  3154   defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
```
```  3155   apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
```
```  3156
```
```  3157 lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
```
```  3158   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})"
```
```  3159   using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
```
```  3160
```
```  3161 subsection {* Special case of stretching coordinate axes separately. *}
```
```  3162
```
```  3163 lemma image_stretch_interval:
```
```  3164   "(\<lambda>x. \<chi> k. m k * x\$k) ` {a..b::real^'n} =
```
```  3165   (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")
```
```  3166 proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
```
```  3167 next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
```
```  3168   case False note ab = this[unfolded interval_ne_empty]
```
```  3169   show ?thesis apply-apply(rule set_ext)
```
```  3170   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
```
```  3171     show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
```
```  3172       unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
```
```  3173       unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a \$ i \<le> xa \<and> xa \<le> b \$ i) \<and> x \$ i = m i * xa"]
```
```  3174     proof(rule **,rule) fix i::'n show "(\<exists>xa. (a \$ i \<le> xa \<and> xa \<le> b \$ i) \<and> x \$ i = m i * xa) =
```
```  3175         (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))"
```
```  3176       proof(cases "m i = 0") case True thus ?thesis using ab by auto
```
```  3177       next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
```
```  3178         proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a \$ i) (m i * b \$ i) = m i * a \$ i"
```
```  3179             "max (m i * a \$ i) (m i * b \$ i) = m i * b \$ i" using ab unfolding min_def max_def by auto
```
```  3180           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x\$i" in exI)
```
```  3181             using as by(auto simp add:field_simps)
```
```  3182         next assume as:"0 > m i" hence *:"max (m i * a \$ i) (m i * b \$ i) = m i * a \$ i"
```
```  3183             "min (m i * a \$ i) (m i * b \$ i) = m i * b \$ i" using ab as unfolding min_def max_def
```
```  3184             by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
```
```  3185           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x\$i" in exI)
```
```  3186             using as by(auto simp add:field_simps) qed qed qed qed qed
```
```  3187
```
```  3188 lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x\$k) ` {a..b::real^'n} = {u..v}"
```
```  3189   unfolding image_stretch_interval by auto
```
```  3190
```
```  3191 lemma content_image_stretch_interval:
```
```  3192   "content((\<lambda>x::real^'n. \<chi> k. m k * x\$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
```
```  3193 proof(cases "{a..b} = {}") case True thus ?thesis
```
```  3194     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
```
```  3195 next case False hence "(\<lambda>x. \<chi> k. m k * x \$ k) ` {a..b} \<noteq> {}" by auto
```
```  3196   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
```
```  3197     unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
```
```  3198   proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
```
```  3199     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)"
```
```  3200       apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i]
```
```  3201       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
```
```  3202
```
```  3203 lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
```
```  3204   shows "((\<lambda>x. f(\<chi> k. m k * x\$k)) has_integral
```
```  3205              ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x\$k) ` {a..b})"
```
```  3206   apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
```
```  3207   unfolding image_stretch_interval empty_as_interval Cart_eq using assms
```
```  3208 proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x \$ k)"
```
```  3209    apply(rule,rule linear_continuous_at) unfolding linear_linear
```
```  3210    unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
```
```  3211
```
```  3212 lemma integrable_stretch:
```
```  3213   assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
```
```  3214   shows "(\<lambda>x. f(\<chi> k. m k * x\$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x\$k) ` {a..b})"
```
```  3215   using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
```
```  3216
```
```  3217 subsection {* even more special cases. *}
```
```  3218
```
```  3219 lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
```
```  3220   apply(rule set_ext,rule) defer unfolding image_iff
```
```  3221   apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
```
```  3222
```
```  3223 lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
```
```  3224   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
```
```  3225   using has_integral_affinity[OF assms, of "-1" 0] by auto
```
```  3226
```
```  3227 lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
```
```  3228   apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
```
```  3229
```
```  3230 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
```
```  3231   unfolding integrable_on_def by auto
```
```  3232
```
```  3233 lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
```
```  3234   unfolding integral_def by auto
```
```  3235
```
```  3236 subsection {* Stronger form of FCT; quite a tedious proof. *}
```
```  3237
```
```  3238 (** move this **)
```
```  3239 declare norm_triangle_ineq4[intro]
```
```  3240
```
```  3241 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)
```
```  3242
```
```  3243 lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
```
```  3244   assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
```
```  3245   shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
```
```  3246   using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
```
```  3247   unfolding o_def vec1_dest_vec1 using assms(1) by auto
```
```  3248
```
```  3249 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"
```
```  3250   unfolding split_def by(rule refl)
```
```  3251
```
```  3252 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
```
```  3253   apply(subst(asm)(2) norm_minus_cancel[THEN sym])
```
```  3254   apply(drule norm_triangle_le) by(auto simp add:group_simps)
```
```  3255
```
```  3256 lemma fundamental_theorem_of_calculus_interior:
```
```  3257   assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
```
```  3258   shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
```
```  3259 proof- { presume *:"a < b \<Longrightarrow> ?thesis"
```
```  3260     show ?thesis proof(cases,rule *,assumption)
```
```  3261       assume "\<not> a < b" hence "a = b" using assms(1) by auto
```
```  3262       hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
```
```  3263       show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
```
```  3264     qed } assume ab:"a < b"
```
```  3265   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
```
```  3266                    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})"
```
```  3267   { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
```
```  3268   fix e::real assume e:"e>0"
```
```  3269   note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
```
```  3270   note conjunctD2[OF this] note bounded=this(1) and this(2)
```
```  3271   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)"
```
```  3272     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]
```
```  3273   from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
```
```  3274   have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
```
```  3275   from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
```
```  3276
```
```  3277   have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
```
```  3278     \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
```
```  3279   proof- have "a\<in>{a..b}" using ab by auto
```
```  3280     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3281     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
```
```  3282     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3283     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
```
```  3284     proof(cases "f' a = 0") case True
```
```  3285       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
```
```  3286     next case False thus ?thesis
```
```  3287         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
```
```  3288         using ab e by(auto simp add:field_simps)
```
```  3289     qed then guess l .. note l = conjunctD2[OF this]
```
```  3290     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
```
```  3291     proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
```
```  3292       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
```
```  3293       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)
```
```  3294       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3295       proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
```
```  3296         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
```
```  3297       next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
```
```  3298           apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
```
```  3299       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
```
```  3300     qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
```
```  3301
```
```  3302   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"
```
```  3303   proof- have "b\<in>{a..b}" using ab by auto
```
```  3304     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3305     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
```
```  3306     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3307     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
```
```  3308     proof(cases "f' b = 0") case True
```
```  3309       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
```
```  3310     next case False thus ?thesis
```
```  3311         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
```
```  3312         using ab e by(auto simp add:field_simps)
```
```  3313     qed then guess l .. note l = conjunctD2[OF this]
```
```  3314     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
```
```  3315     proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
```
```  3316       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
```
```  3317       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)
```
```  3318       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3319       proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
```
```  3320         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
```
```  3321       next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
```
```  3322           apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
```
```  3323       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
```
```  3324     qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
```
```  3325
```
```  3326   let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
```
```  3327   show "?P e" apply(rule_tac x="?d" in exI)
```
```  3328   proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
```
```  3329   next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
```
```  3330     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
```
```  3331     note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
```
```  3332     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
```
```  3333     show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
```
```  3334       unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
```
```  3335     proof(rule norm_triangle_le,rule **)
```
```  3336       case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
```
```  3337       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"
```
```  3338           "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
```
```  3339           < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
```
```  3340         from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
```
```  3341         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]
```
```  3342         note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
```
```  3343
```
```  3344         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]
```
```  3345         have "norm ((v\$1 - u\$1) *\<^sub>R f' (x\$1) - (f (v\$1) - f (u\$1))) =
```
```  3346           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)))"
```
```  3347           apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
```
```  3348         also have "... \<le> e / 2 * norm (u\$1 - x\$1) + e / 2 * norm (v\$1 - x\$1)" apply(rule norm_triangle_le_sub)
```
```  3349           apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
```
```  3350           apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
```
```  3351         also have "... \<le> e / 2 * norm (v\$1 - u\$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
```
```  3352         finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
```
```  3353           apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
```
```  3354
```
```  3355     next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
```
```  3356       case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
```
```  3357         defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
```
```  3358         apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
```
```  3359       proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
```
```  3360         from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  3361         with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
```
```  3362         thus "0 \<le> e * ((interval_upperbound k)\$1 - (interval_lowerbound k)\$1)"
```
```  3363           unfolding uv using e by(auto simp add:field_simps)
```
```  3364       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
```
```  3365         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
```
```  3366           (f ((interval_upperbound k)\$1) - f ((interval_lowerbound k)\$1))) \<le> e * (b - a) / 2"
```
```  3367           apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
```
```  3368           apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
```
```  3369         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}"
```
```  3370           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
```
```  3371           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
```
```  3372           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
```
```  3373         next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} =
```
```  3374             {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
```
```  3375           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"
```
```  3376           proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
```
```  3377             thus ?case using `x\<in>s` goal2(2) by auto
```
```  3378           qed auto
```
```  3379           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"])
```
```  3380             apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
```
```  3381           proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
```
```  3382             have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v"
```
```  3383             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
```
```  3384               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
```
```  3385               have u:"u = vec1 a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
```
```  3386                 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
```
```  3387                 have "u > vec1 a" unfolding Cart_simps by auto
```
```  3388                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
```
```  3389               qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
```
```  3390             qed
```
```  3391             have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v"
```
```  3392             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
```
```  3393               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
```
```  3394               have u:"v = vec1 b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
```
```  3395                 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
```
```  3396                 have "v < vec1 b" unfolding Cart_simps by auto
```
```  3397                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
```
```  3398               qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
```
```  3399             qed
```
```  3400
```
```  3401             show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
```
```  3402               unfolding mem_Collect_eq fst_conv snd_conv apply safe
```
```  3403             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"
```
```  3404               guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
```
```  3405               guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v\$1) (v'\$1))"
```
```  3406               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]
```
```  3407               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)
```
```  3408               ultimately have "vec1 ((a + ?v\$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
```
```  3409               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
```
```  3410               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
```
```  3411             qed
```
```  3412             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
```
```  3413               unfolding mem_Collect_eq fst_conv snd_conv apply safe
```
```  3414             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"
```
```  3415               guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
```
```  3416               guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v\$1) (v'\$1))"
```
```  3417               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]
```
```  3418               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)
```
```  3419               ultimately have "vec1 ((b + ?v\$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
```
```  3420               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
```
```  3421               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
```
```  3422             qed
```
```  3423
```
```  3424             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
```
```  3425             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)
```
```  3426               \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
```
```  3427             proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
```
```  3428               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
```
```  3429               moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
```
```  3430                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
```
```  3431                 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
```
```  3432               show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
```
```  3433                 apply(rule da(2)[of "v\$1",unfolded vec1_dest_vec1])
```
```  3434                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
```
```  3435             qed
```
```  3436             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)
```
```  3437               \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
```
```  3438             proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
```
```  3439               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
```
```  3440               moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
```
```  3441                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
```
```  3442                 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
```
```  3443               show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
```
```  3444                 apply(rule db(2)[of "v\$1",unfolded vec1_dest_vec1])
```
```  3445                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
```
```  3446             qed
```
```  3447           qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
```
```  3448
```
```  3449 subsection {* Stronger form with finite number of exceptional points. *}
```
```  3450
```
```  3451 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3452   assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
```
```  3453   "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  3454   shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply-
```
```  3455 proof(induct "card s" arbitrary:s a b)
```
```  3456   case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
```
```  3457 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
```
```  3458     apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
```
```  3459   show ?case proof(cases "c\<in>{a<..<b}")
```
```  3460     case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
```
```  3461       apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
```
```  3462   next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
```
```  3463     case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
```
```  3464     thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
```
```  3465       apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
```
```  3466     proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
```
```  3467         apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
```
```  3468       let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
```
```  3469       show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
```
```  3470     qed auto qed qed
```
```  3471
```
```  3472 lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3473   assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
```
```  3474   "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  3475   shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}"
```
```  3476   apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
```
```  3477   using assms(4) by auto
```
```  3478
```
```  3479 lemma indefinite_integral_continuous_left: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  3480   assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e"
```
```  3481   obtains d where "0 < d" "\<forall>t. c\$1 - d < t\$1 \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e"
```
```  3482 proof- have "\<exists>w>0. \<forall>t. c\$1 - w < t\$1 \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3"
```
```  3483   proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)"
```
```  3484       apply-apply(rule divide_pos_pos) using `e>0` by auto
```
```  3485     thus ?thesis apply-apply(rule,rule,assumption,safe)
```
```  3486     proof- fix t assume as:"t < c" and "c\$1 - e / 3 / norm (f c) < t\$(1::1)"
```
```  3487       hence "c\$1 - t\$1 < e / 3 / norm (f c)" by auto
```
```  3488       hence "norm (c - t) < e / 3 / norm (f c)" using as unfolding norm_vector_1 vector_less_def by auto
```
```  3489       thus "norm (f c) * norm (c - t) < e / 3" using False apply-
```
```  3490         apply(subst real_mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
```
```  3491     qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
```
```  3492   qed then guess w .. note w = conjunctD2[OF this,rule_format]
```
```  3493
```
```  3494   have *:"e / 3 > 0" using assms by auto
```
```  3495   have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
```
```  3496   from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
```
```  3497   note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x"
```
```  3498   have "gauge d" unfolding d_def using w(1) d1 by auto
```
```  3499   note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this]
```
```  3500   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
```
```  3501
```
```  3502   let ?d = "min k (c\$1 - a\$1)/2" show ?thesis apply(rule that[of ?d])
```
```  3503   proof safe show "?d > 0" using k(1) using assms(2) unfolding vector_less_def by auto
```
```  3504     fix t assume as:"c\$1 - ?d < t\$1" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
```
```  3505     { presume *:"t < c \<Longrightarrow> ?thesis"
```
```  3506       show ?thesis apply(cases "t = c") defer apply(rule *)
```
```  3507         unfolding vector_less_def apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
```
```  3508
```
```  3509     have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
```
```  3510     from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
```
```  3511     note d2 = conjunctD2[OF this,rule_format]
```
```  3512     def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x"
```
```  3513     have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto
```
```  3514     from fine_division_exists[OF this, of a t] guess p . note p=this
```
```  3515     note p'=tagged_division_ofD[OF this(1)]
```
```  3516     have pt:"\<forall>(x,k)\<in>p. x\$1 \<le> t\$1" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
```
```  3517     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
```
```  3518     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
```
```  3519
```
```  3520     have *:"{a..c} \<inter> {x. x\$1 \<le> t\$1} = {a..t}" "{a..c} \<inter> {x. x\$1 \<ge> t\$1} = {t..c}"
```
```  3521       using assms(2-3) as by(auto simp add:field_simps)
```
```  3522     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
```
```  3523       apply(rule tagged_division_union_interval[of _ _ _ 1 "t\$1"]) unfolding * apply(rule p)
```
```  3524       apply(rule tagged_division_of_self) unfolding fine_def
```
```  3525     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
```
```  3526         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
```
```  3527     next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real
```
```  3528         using as(1) by(auto simp add:field_simps)
```
```  3529       thus "x \<in> d1 c" using k(2) unfolding d_def by auto
```
```  3530     qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
```
```  3531
```
```  3532     have *:"integral{a..c} f - integral {a..t} f = -(((c\$1 - t\$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
```
```  3533         integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c\$1 - t\$1) *\<^sub>R f c"
```
```  3534       "e = (e/3 + e/3) + e/3" by auto
```
```  3535     have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c\$1 - t\$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  3536     proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
```
```  3537       have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
```
```  3538         have "c \<in> {a..t}" by auto thus False using `t<c` unfolding vector_less_def by auto
```
```  3539       qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
```
```  3540         unfolding split_conv defer apply(subst content_1) using as(2) by auto qed
```
```  3541
```
```  3542     have ***:"c\$1 - w < t\$1 \<and> t < c"
```
```  3543     proof- have "c\$1 - k < t\$1" using `k>0` as(1) by(auto simp add:field_simps)
```
```  3544       moreover have "k \<le> w" apply(rule ccontr) using k(2)
```
```  3545         unfolding subset_eq apply(erule_tac x="c + vec ((k + w)/2)" in ballE)
```
```  3546         unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real)
```
```  3547       ultimately show  ?thesis using `t<c` by(auto simp add:field_simps) qed
```
```  3548
```
```  3549     show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
```
```  3550       unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
```
```  3551       using w(2)[OF ***] unfolding norm_scaleR norm_real by(auto simp add:field_simps) qed qed
```
```  3552
```
```  3553 lemma indefinite_integral_continuous_right: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  3554   assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
```
```  3555   obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t\$1 < c\$1 + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e"
```
```  3556 proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a"
```
```  3557     using assms unfolding Cart_simps by auto
```
```  3558   from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b\$1 - c\$1)"
```
```  3559   show ?thesis apply(rule that[of "?d"])
```
```  3560   proof safe show "0 < ?d" using d(1) assms(3) unfolding Cart_simps by auto
```
```  3561     fix t::"_^1" assume as:"c \<le> t" "t\$1 < c\$1 + ?d"
```
```  3562     have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
```
```  3563       "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding group_simps
```
```  3564       apply(rule_tac[!] integral_combine) using assms as unfolding Cart_simps by auto
```
```  3565     have "(- c)\$1 - d < (- t)\$1 \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
```
```  3566     thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
```
```  3567       unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:group_simps) qed qed
```
```  3568
```
```  3569 declare dest_vec1_eq[simp del] not_less[simp] not_le[simp]
```
```  3570
```
```  3571 lemma indefinite_integral_continuous: fixes f::"real^1 \<Rightarrow> 'a::banach"
```
```  3572   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
```
```  3573 proof(unfold continuous_on_def, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
```
```  3574   let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e"
```
```  3575   { presume *:"a<b \<Longrightarrow> ?thesis"
```
```  3576     show ?thesis apply(cases,rule *,assumption)
```
```  3577     proof- case goal1 hence "{a..b} = {x}" using as(1) unfolding Cart_simps
```
```  3578         by(auto simp only:field_simps not_less Cart_eq forall_1 mem_interval)
```
```  3579       thus ?case using `e>0` by auto
```
```  3580     qed } assume "a<b"
```
```  3581   have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add: Cart_simps)
```
```  3582   thus ?thesis apply-apply(erule disjE)+
```
```  3583   proof- assume "x=a" have "a \<le> a" by auto
```
```  3584     from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
```
```  3585     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
```
```  3586       unfolding `x=a` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
```
```  3587   next   assume "x=b" have "b \<le> b" by auto
```
```  3588     from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
```
```  3589     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
```
```  3590       unfolding `x=b` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
```
```  3591   next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add:Cart_simps)
```
```  3592     from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
```
```  3593     from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
```
```  3594     show ?thesis apply(rule_tac x="min d1 d2" in exI)
```
```  3595     proof safe show "0 < min d1 d2" using d1 d2 by auto
```
```  3596       fix y assume "y\<in>{a..b}" "dist y x < min d1 d2"
```
```  3597       thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute)
```
```  3598         apply(cases "y < x") unfolding vector_dist_norm apply(rule d1(2)[rule_format]) defer
```
```  3599         apply(rule d2(2)[rule_format]) unfolding Cart_simps not_less norm_real by(auto simp add:field_simps)
```
```  3600     qed qed qed
```
```  3601
```
```  3602 subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
```
```  3603
```
```  3604 lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3605   assumes "finite k" "continuous_on {a..b} f" "f a = y"
```
```  3606   "\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
```
```  3607   shows "f x = y"
```
```  3608 proof- have ab:"a\<le>b" using assms by auto
```
```  3609   have *:"(\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 = (\<lambda>x. 0)" unfolding o_def by auto have **:"a \<le> x" using assms by auto
```
```  3610   have "((\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 has_integral f x - f a) {vec1 a..vec1 x}"
```
```  3611     apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) ** ])
```
```  3612     apply(rule continuous_on_subset[OF assms(2)]) defer
```
```  3613     apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
```
```  3614     apply assumption apply(rule open_interval_real) apply(rule has_derivative_within_subset[where s="{a..b}"])
```
```  3615     using assms(4) assms(5) by auto note this[unfolded *]
```
```  3616   note has_integral_unique[OF has_integral_0 this]
```
```  3617   thus ?thesis unfolding assms by auto qed
```
```  3618
```
```  3619 subsection {* Generalize a bit to any convex set. *}
```
```  3620
```
```  3621 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
```
```  3622   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
```
```  3623   scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one
```
```  3624
```
```  3625 lemma has_derivative_zero_unique_strong_convex: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3626   assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
```
```  3627   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
```
```  3628   shows "f x = y"
```
```  3629 proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
```
```  3630       unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
```
```  3631   note conv = assms(1)[unfolded convex_alt,rule_format]
```
```  3632   have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
```
```  3633     apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
```
```  3634     apply safe apply(rule conv) using assms(4,7) by auto
```
```  3635   have *:"\<And>t xa. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x \<Longrightarrow> t = xa"
```
```  3636   proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
```
```  3637       unfolding scaleR_simps by(auto simp add:group_simps)
```
```  3638     thus ?case using `x\<noteq>c` by auto qed
```
```  3639   have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
```
```  3640     apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
```
```  3641     apply safe unfolding image_iff apply rule defer apply assumption
```
```  3642     apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
```
```  3643   have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
```
```  3644     apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
```
```  3645     unfolding o_def using assms(5) defer apply-apply(rule)
```
```  3646   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
```
```  3647     have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
```
```  3648       using `x\<in>s` `c\<in>s` as by(auto simp add:scaleR_simps)
```
```  3649     have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
```
```  3650       apply(rule diff_chain_within) apply(rule has_derivative_add)
```
```  3651       unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
```
```  3652       apply(rule has_derivative_vmul_within,rule has_derivative_id)+
```
```  3653       apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
```
```  3654       apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
```
```  3655     thus "((\<lambda>xa. f ((1 - xa) *\<^sub>R c + xa *\<^sub>R x)) has_derivative (\<lambda>h. 0)) (at t within {0..1})" unfolding o_def .
```
```  3656   qed auto thus ?thesis by auto qed
```
```  3657
```
```  3658 subsection {* Also to any open connected set with finite set of exceptions. Could
```
```  3659  generalize to locally convex set with limpt-free set of exceptions. *}
```
```  3660
```
```  3661 lemma has_derivative_zero_unique_strong_connected: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3662   assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
```
```  3663   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
```
```  3664   shows "f x = y"
```
```  3665 proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
```
```  3666     apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
```
```  3667     apply(rule continuous_closed_in_preimage[OF assms(4) closed_sing])
```
```  3668     apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
```
```  3669   proof safe fix x assume "x\<in>s"
```
```  3670     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
```
```  3671     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
```
```  3672     proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
```
```  3673       show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball])
```
```  3674         apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+
```
```  3675         apply(subst centre_in_ball,rule e,rule) apply safe
```
```  3676         apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format])
```
```  3677         using y e by auto qed qed
```
```  3678   thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed
```
```  3679
```
```  3680 subsection {* Integrating characteristic function of an interval. *}
```
```  3681
```
```  3682 lemma has_integral_restrict_open_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3683   assumes "(f has_integral i) {c..d}" "{c..d} \<subseteq> {a..b}"
```
```  3684   shows "((\<lambda>x. if x \<in> {c<..<d} then f x else 0) has_integral i) {a..b}"
```
```  3685 proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
```
```  3686   { presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
```
```  3687     show ?thesis apply(cases,rule *,assumption)
```
```  3688     proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto
```
```  3689       show ?thesis using assms(1) unfolding * using goal1 by auto
```
```  3690     qed } assume "{c..d}\<noteq>{}"
```
```  3691   from partial_division_extend_1[OF assms(2) this] guess p . note p=this
```
```  3692   note mon = monoidal_lifted[OF monoidal_monoid]
```
```  3693   note operat = operative_division[OF this operative_integral p(1), THEN sym]
```
```  3694   let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
```
```  3695   { presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
```
```  3696       apply- apply(cases,subst(asm) if_P,assumption) by auto
```
```  3697     thus ?thesis using integrable_integral unfolding g_def by auto }
```
```  3698
```
```  3699   note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
```
```  3700   note * = this[unfolded neutral_monoid]
```
```  3701   have iterate:"iterate (lifted op +) (p - {{c..d}})
```
```  3702       (\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0"
```
```  3703   proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this]
```
```  3704     from div(3) guess u v apply-by(erule exE)+ note uv=this
```
```  3705     have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto
```
```  3706     hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"])
```
```  3707       unfolding g_def interior_closed_interval by auto thus ?case by auto
```
```  3708   qed
```
```  3709
```
```  3710   have *:"p = insert {c..d} (p - {{c..d}})" using p by auto
```
```  3711   have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f])
```
```  3712     unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto
```
```  3713   moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)])
```
```  3714     apply(rule has_integral_spike_interior[where f=g]) defer
```
```  3715     apply(rule integrable_integral[OF **]) unfolding g_def by auto
```
```  3716   ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+
```
```  3717     unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
```
```  3718
```
```  3719 lemma has_integral_restrict_closed_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3720   assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}"
```
```  3721   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
```
```  3722 proof- note has_integral_restrict_open_subinterval[OF assms]
```
```  3723   note * = has_integral_spike[OF negligible_frontier_interval _ this]
```
```  3724   show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
```
```  3725
```
```  3726 lemma has_integral_restrict_closed_subintervals_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" assumes "{c..d} \<subseteq> {a..b}"
```
```  3727   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b} \<longleftrightarrow> (f has_integral i) {c..d}" (is "?l = ?r")
```
```  3728 proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
```
```  3729   show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
```
```  3730   proof assumption assume ?l hence "?g integrable_on {c..d}"
```
```  3731       apply-apply(rule integrable_subinterval[OF _ assms]) by auto
```
```  3732     hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto
```
```  3733     hence "i = integral {c..d} f" apply-apply(rule has_integral_unique)
```
```  3734       apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto
```
```  3735     thus ?r using * by auto qed qed auto
```
```  3736
```
```  3737 subsection {* Hence we can apply the limit process uniformly to all integrals. *}
```
```  3738
```
```  3739 lemma has_integral': fixes f::"real^'n \<Rightarrow> 'a::banach" shows
```
```  3740  "(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```  3741   \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then f(x) else 0) has_integral z) {a..b} \<and> norm(z - i) < e))" (is "?l \<longleftrightarrow> (\<forall>e>0. ?r e)")
```
```  3742 proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis"
```
```  3743     show ?thesis apply(cases,rule *,assumption)
```
```  3744       apply(subst has_integral_alt) by auto }
```
```  3745   assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this
```
```  3746   from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B ..
```
```  3747   note B = conjunctD2[OF this,rule_format] show ?thesis apply safe
```
```  3748   proof- fix e assume ?l "e>(0::real)"
```
```  3749     show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI)
```
```  3750     proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::real^'n}"
```
```  3751       thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s
```
```  3752         apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s])
```
```  3753         apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
```
```  3754         by(auto simp add:vector_dist_norm)
```
```  3755     qed(insert B `e>0`, auto)
```
```  3756   next assume as:"\<forall>e>0. ?r e"
```
```  3757     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
```
```  3758     def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n"
```
```  3759     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
```
```  3760     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
```
```  3761         by(auto simp add:field_simps) qed
```
```  3762     have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball vector_dist_norm
```
```  3763     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
```
```  3764     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
```
```  3765       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
```
```  3766     then guess y .. note y=this
```
```  3767
```
```  3768     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
```
```  3769       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
```
```  3770       def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n"
```
```  3771       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
```
```  3772       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
```
```  3773           by(auto simp add:field_simps) qed
```
```  3774       have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball vector_dist_norm
```
```  3775       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
```
```  3776       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
```
```  3777       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
```
```  3778       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
```
```  3779       thus False by auto qed
```
```  3780     thus ?l using y unfolding s by auto qed qed
```
```  3781
```
```  3782 subsection {* Hence a general restriction property. *}
```
```  3783
```
```  3784 lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows
```
```  3785   "((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
```
```  3786 proof- have *:"\<And>x. (if x \<in> t then if x \<in> s then f x else 0 else 0) =  (if x\<in>s then f x else 0)" using assms by auto
```
```  3787   show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed
```
```  3788
```
```  3789 lemma has_integral_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
```
```  3790   "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
```
```  3791
```
```  3792 lemma has_integral_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3793   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
```
```  3794   shows "(f has_integral i) t"
```
```  3795 proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
```
```  3796     apply(rule) using assms(1-2) by auto
```
```  3797   thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
```
```  3798   apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
```
```  3799
```
```  3800 lemma integrable_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3801   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
```
```  3802   shows "f integrable_on t"
```
```  3803   using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
```
```  3804
```
```  3805 lemma integral_restrict_univ[intro]: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3806   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
```
```  3807   apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
```
```  3808
```
```  3809 lemma integrable_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
```
```  3810  "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
```
```  3811   unfolding integrable_on_def by auto
```
```  3812
```
```  3813 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r")
```
```  3814 proof assume ?r show ?l unfolding negligible_def
```
```  3815   proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
```
```  3816       unfolding indicator_def by auto qed qed auto
```
```  3817
```
```  3818 lemma has_integral_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3819   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
```
```  3820   unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by auto
```
```  3821
```
```  3822 lemma has_integral_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3823   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
```
```  3824   shows "(f has_integral y) t"
```
```  3825   using assms has_integral_spike_set_eq by auto
```
```  3826
```
```  3827 lemma integrable_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3828   assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
```
```  3829   shows "f integrable_on t" using assms(2) unfolding integrable_on_def
```
```  3830   unfolding has_integral_spike_set_eq[OF assms(1)] .
```
```  3831
```
```  3832 lemma integrable_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3833   assumes "negligible((s - t) \<union> (t - s))"
```
```  3834   shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)"
```
```  3835   apply(rule,rule_tac[!] integrable_spike_set) using assms by auto
```
```  3836
```
```  3837 (*lemma integral_spike_set:
```
```  3838  "\<forall>f:real^M->real^N g s t.
```
```  3839         negligible(s DIFF t \<union> t DIFF s)
```
```  3840         \<longrightarrow> integral s f = integral t f"
```
```  3841 qed  REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
```
```  3842   AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  3843   ASM_MESON_TAC[]);;
```
```  3844
```
```  3845 lemma has_integral_interior:
```
```  3846  "\<forall>f:real^M->real^N y s.
```
```  3847         negligible(frontier s)
```
```  3848         \<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
```
```  3849 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  3850   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
```
```  3851     NEGLIGIBLE_SUBSET)) THEN
```
```  3852   REWRITE_TAC[frontier] THEN
```
```  3853   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
```
```  3854   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
```
```  3855   SET_TAC[]);;
```
```  3856
```
```  3857 lemma has_integral_closure:
```
```  3858  "\<forall>f:real^M->real^N y s.
```
```  3859         negligible(frontier s)
```
```  3860         \<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
```
```  3861 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  3862   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
```
```  3863     NEGLIGIBLE_SUBSET)) THEN
```
```  3864   REWRITE_TAC[frontier] THEN
```
```  3865   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
```
```  3866   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
```
```  3867   SET_TAC[]);;*)
```
```  3868
```
```  3869 subsection {* More lemmas that are useful later. *}
```
```  3870
```
```  3871 lemma has_integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  3872   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\$k"
```
```  3873   shows "i\$k \<le> j\$k"
```
```  3874 proof- note has_integral_restrict_univ[THEN sym, of f]
```
```  3875   note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
```
```  3876   show ?thesis apply(rule *) using assms(1,4) by auto qed
```
```  3877
```
```  3878 lemma integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
```
```  3879   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)\$k"
```
```  3880   shows "(integral s f)\$k \<le> (integral t f)\$k"
```
```  3881   apply(rule has_integral_subset_component_le) using assms by auto
```
```  3882
```
```  3883 lemma has_integral_alt': fixes f::"real^'n \<Rightarrow> 'a::banach"
```
```  3884   shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
```
```  3885   (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e)" (is "?l = ?r")
```
```  3886 proof assume ?r
```
```  3887   show ?l apply- apply(subst has_integral')
```
```  3888   proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
```
```  3889     show ?case apply(rule,rule,rule B,safe)
```
```  3890       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI)
```
```  3891       apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto
```
```  3892   qed next
```
```  3893   assume ?l note as = this[unfolded has_integral'[of f],rule_format]
```
```  3894   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
```
```  3895   show ?r proof safe fix a b::"real^'n"
```
```  3896     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  3897     let ?a = "(\<chi> i. min (a\$i) (-B))::real^'n" and ?b = "(\<chi> i. max (b\$i) B)::real^'n"
```
```  3898     show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
```
```  3899     proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval vector_dist_norm
```
```  3900       proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
```
```  3901       from B(2)[OF this] guess z .. note conjunct1[OF this]
```
```  3902       thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
```
```  3903       show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
```
```  3904
```
```  3905     fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  3906     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
```
```  3907                     norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
```
```  3908     proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
```
```  3909       from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed
```
```  3910
```
```  3911
```
```  3912
```
```  3913 declare [[smt_certificates=""]]
```
```  3914
```
```  3915 end
```