src/HOL/Multivariate_Analysis/Integration.thy
 author huffman Wed Aug 10 16:35:50 2011 -0700 (2011-08-10) changeset 44140 2c10c35dd4be parent 44125 230a8665c919 child 44167 e81d676d598e permissions -rw-r--r--
remove several redundant and unused theorems about derivatives
```     1 header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
```
```     2 (*  Author:                     John Harrison
```
```     3     Translation from HOL light: Robert Himmelmann, TU Muenchen *)
```
```     4
```
```     5 theory Integration
```
```     6 imports
```
```     7   Derivative
```
```     8   "~~/src/HOL/Library/Indicator_Function"
```
```     9 begin
```
```    10
```
```    11 declare [[smt_certificates="Integration.certs"]]
```
```    12 declare [[smt_fixed=true]]
```
```    13 declare [[smt_oracle=false]]
```
```    14
```
```    15 (*declare not_less[simp] not_le[simp]*)
```
```    16
```
```    17 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
```
```    18   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
```
```    19   scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one
```
```    20
```
```    21 lemma real_arch_invD:
```
```    22   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```    23   by(subst(asm) real_arch_inv)
```
```    24 subsection {* Sundries *}
```
```    25
```
```    26 (*declare basis_component[simp]*)
```
```    27
```
```    28 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
```
```    29 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
```
```    30 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
```
```    31 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
```
```    32
```
```    33 declare norm_triangle_ineq4[intro]
```
```    34
```
```    35 lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast
```
```    36
```
```    37 lemma linear_simps:  assumes "bounded_linear f"
```
```    38   shows "f (a + b) = f a + f b" "f (a - b) = f a - f b" "f 0 = 0" "f (- a) = - f a" "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
```
```    39   apply(rule_tac[!] additive.add additive.minus additive.diff additive.zero bounded_linear.scaleR)
```
```    40   using assms unfolding bounded_linear_def additive_def by auto
```
```    41
```
```    42 lemma bounded_linearI:assumes "\<And>x y. f (x + y) = f x + f y"
```
```    43   "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K"
```
```    44   shows "bounded_linear f"
```
```    45   unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto
```
```    46
```
```    47 lemma real_le_inf_subset:
```
```    48   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s" shows "Inf s <= Inf (t::real set)"
```
```    49   apply(rule isGlb_le_isLb) apply(rule Inf[OF assms(1)])
```
```    50   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
```
```    51   unfolding isLb_def setge_def by auto
```
```    52
```
```    53 lemma real_ge_sup_subset:
```
```    54   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b" shows "Sup s >= Sup (t::real set)"
```
```    55   apply(rule isLub_le_isUb) apply(rule Sup[OF assms(1)])
```
```    56   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
```
```    57   unfolding isUb_def setle_def by auto
```
```    58
```
```    59 lemma bounded_linear_component[intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \$\$ k)"
```
```    60   apply(rule bounded_linearI[where K=1])
```
```    61   using component_le_norm[of _ k] unfolding real_norm_def by auto
```
```    62
```
```    63 lemma transitive_stepwise_lt_eq:
```
```    64   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
```
```    65   shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r")
```
```    66 proof(safe) assume ?r fix n m::nat assume "m < n" thus "R m n" apply-
```
```    67   proof(induct n arbitrary: m) case (Suc n) show ?case
```
```    68     proof(cases "m < n") case True
```
```    69       show ?thesis apply(rule assms[OF Suc(1)[OF True]]) using `?r` by auto
```
```    70     next case False hence "m = n" using Suc(2) by auto
```
```    71       thus ?thesis using `?r` by auto
```
```    72     qed qed auto qed auto
```
```    73
```
```    74 lemma transitive_stepwise_gt:
```
```    75   assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
```
```    76   shows "\<forall>n>m. R m n"
```
```    77 proof- have "\<forall>m. \<forall>n>m. R m n" apply(subst transitive_stepwise_lt_eq)
```
```    78     apply(rule assms) apply(assumption,assumption) using assms(2) by auto
```
```    79   thus ?thesis by auto qed
```
```    80
```
```    81 lemma transitive_stepwise_le_eq:
```
```    82   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
```
```    83   shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r")
```
```    84 proof safe assume ?r fix m n::nat assume "m\<le>n" thus "R m n" apply-
```
```    85   proof(induct n arbitrary: m) case (Suc n) show ?case
```
```    86     proof(cases "m \<le> n") case True show ?thesis apply(rule assms(2))
```
```    87         apply(rule Suc(1)[OF True]) using `?r` by auto
```
```    88     next case False hence "m = Suc n" using Suc(2) by auto
```
```    89       thus ?thesis using assms(1) by auto
```
```    90     qed qed(insert assms(1), auto) qed auto
```
```    91
```
```    92 lemma transitive_stepwise_le:
```
```    93   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
```
```    94   shows "\<forall>n\<ge>m. R m n"
```
```    95 proof- have "\<forall>m. \<forall>n\<ge>m. R m n" apply(subst transitive_stepwise_le_eq)
```
```    96     apply(rule assms) apply(rule assms,assumption,assumption) using assms(3) by auto
```
```    97   thus ?thesis by auto qed
```
```    98
```
```    99 subsection {* Some useful lemmas about intervals. *}
```
```   100
```
```   101 abbreviation One  where "One \<equiv> ((\<chi>\<chi> i. 1)::_::ordered_euclidean_space)"
```
```   102
```
```   103 lemma empty_as_interval: "{} = {One..0}"
```
```   104   apply(rule set_eqI,rule) defer unfolding mem_interval
```
```   105   using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
```
```   106
```
```   107 lemma interior_subset_union_intervals:
```
```   108   assumes "i = {a..b::'a::ordered_euclidean_space}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
```
```   109   shows "interior i \<subseteq> interior s" proof-
```
```   110   have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
```
```   111     unfolding assms(1,2) interior_closed_interval by auto
```
```   112   moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
```
```   113     using assms(4) unfolding assms(1,2) by auto
```
```   114   ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
```
```   115     unfolding assms(1,2) interior_closed_interval by auto qed
```
```   116
```
```   117 lemma inter_interior_unions_intervals: fixes f::"('a::ordered_euclidean_space) set set"
```
```   118   assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
```
```   119   shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
```
```   120   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)
```
```   121     unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
```
```   122   have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
```
```   123   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
```
```   124   thus ?case proof(induct rule:finite_induct)
```
```   125     case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
```
```   126     case (insert i f) guess x using insert(5) .. note x = this
```
```   127     then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
```
```   128     guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
```
```   129     show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
```
```   130       then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
```
```   131       hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" unfolding ab ball_min_Int by auto
```
```   132       hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 unfolding  ball_min_Int by auto
```
```   133       hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
```
```   134       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
```
```   135     case True show ?thesis proof(cases "x\<in>{a<..<b}")
```
```   136       case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
```
```   137       thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
```
```   138         unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
```
```   139     case False then obtain k where "x\$\$k \<le> a\$\$k \<or> x\$\$k \<ge> b\$\$k" and k:"k<DIM('a)" unfolding mem_interval by(auto simp add:not_less)
```
```   140     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
```
```   141     hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
```
```   142       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)
```
```   143         fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
```
```   144         hence "\<bar>(?z - y) \$\$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
```
```   145         hence "y\$\$k < a\$\$k" using e[THEN conjunct1] k by(auto simp add:field_simps basis_component as)
```
```   146         hence "y \<notin> i" unfolding ab mem_interval not_all apply(rule_tac x=k in exI) using k by auto thus False using yi by auto qed
```
```   147       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
```
```   148         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)"
```
```   149            apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
```
```   150           unfolding norm_scaleR norm_basis by auto
```
```   151         also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
```
```   152         finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed
```
```   153       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
```
```   154     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)
```
```   155         fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
```
```   156         hence "\<bar>(?z - y) \$\$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
```
```   157         hence "y\$\$k > b\$\$k" using e[THEN conjunct1] k by(auto simp add:field_simps as)
```
```   158         hence "y \<notin> i" unfolding ab mem_interval not_all using k by(rule_tac x=k in exI,auto) thus False using yi by auto qed
```
```   159       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
```
```   160         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)"
```
```   161            apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
```
```   162           unfolding norm_scaleR by auto
```
```   163         also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
```
```   164         finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed
```
```   165       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed
```
```   166     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
```
```   167     thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
```
```   168   guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
```
```   169   hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
```
```   170   thus False using `t\<in>f` assms(4) by auto qed
```
```   171
```
```   172 subsection {* Bounds on intervals where they exist. *}
```
```   173
```
```   174 definition "interval_upperbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Sup {a. \<exists>x\<in>s. x\$\$i = a})::'a)"
```
```   175
```
```   176 definition "interval_lowerbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Inf {a. \<exists>x\<in>s. x\$\$i = a})::'a)"
```
```   177
```
```   178 lemma interval_upperbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a\$\$i \<le> (b::'a)\$\$i" shows "interval_upperbound {a..b} = b"
```
```   179   using assms unfolding interval_upperbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
```
```   180   unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
```
```   181   apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
```
```   182   apply(rule,rule) apply(rule_tac x="b\$\$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
```
```   183   unfolding mem_interval using assms by auto
```
```   184
```
```   185 lemma interval_lowerbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a\$\$i \<le> (b::'a)\$\$i" shows "interval_lowerbound {a..b} = a"
```
```   186   using assms unfolding interval_lowerbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
```
```   187   unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
```
```   188   apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
```
```   189   apply(rule,rule) apply(rule_tac x="a\$\$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
```
```   190   unfolding mem_interval using assms by auto
```
```   191
```
```   192 lemmas interval_bounds = interval_upperbound interval_lowerbound
```
```   193
```
```   194 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
```
```   195   using assms unfolding interval_ne_empty by auto
```
```   196
```
```   197 subsection {* Content (length, area, volume...) of an interval. *}
```
```   198
```
```   199 definition "content (s::('a::ordered_euclidean_space) set) =
```
```   200        (if s = {} then 0 else (\<Prod>i<DIM('a). (interval_upperbound s)\$\$i - (interval_lowerbound s)\$\$i))"
```
```   201
```
```   202 lemma interval_not_empty:"\<forall>i<DIM('a). a\$\$i \<le> b\$\$i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
```
```   203   unfolding interval_eq_empty unfolding not_ex not_less by auto
```
```   204
```
```   205 lemma content_closed_interval: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a\$\$i \<le> b\$\$i"
```
```   206   shows "content {a..b} = (\<Prod>i<DIM('a). b\$\$i - a\$\$i)"
```
```   207   using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
```
```   208
```
```   209 lemma content_closed_interval': fixes a::"'a::ordered_euclidean_space" assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i<DIM('a). b\$\$i - a\$\$i)"
```
```   210   apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
```
```   211
```
```   212 lemma content_real:assumes "a\<le>b" shows "content {a..b} = b-a"
```
```   213 proof- have *:"{..<Suc 0} = {0}" by auto
```
```   214   show ?thesis unfolding content_def using assms by(auto simp: *)
```
```   215 qed
```
```   216
```
```   217 lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1" proof-
```
```   218   have *:"\<forall>i<DIM('a). (0::'a)\$\$i \<le> (One::'a)\$\$i" by auto
```
```   219   have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
```
```   220   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
```
```   221
```
```   222 lemma content_pos_le[intro]: fixes a::"'a::ordered_euclidean_space" shows "0 \<le> content {a..b}" proof(cases "{a..b}={}")
```
```   223   case False hence *:"\<forall>i<DIM('a). a \$\$ i \<le> b \$\$ i" unfolding interval_ne_empty by assumption
```
```   224   have "(\<Prod>i<DIM('a). interval_upperbound {a..b} \$\$ i - interval_lowerbound {a..b} \$\$ i) \<ge> 0"
```
```   225     apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
```
```   226   thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
```
```   227
```
```   228 lemma content_pos_lt: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a\$\$i < b\$\$i" shows "0 < content {a..b}"
```
```   229 proof- have help_lemma1: "\<forall>i<DIM('a). a\$\$i < b\$\$i \<Longrightarrow> \<forall>i<DIM('a). a\$\$i \<le> ((b\$\$i)::real)" apply(rule,erule_tac x=i in allE) by auto
```
```   230   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
```
```   231     using assms apply(erule_tac x=x in allE) by auto qed
```
```   232
```
```   233 lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i<DIM('a). b\$\$i \<le> a\$\$i)" proof(cases "{a..b} = {}")
```
```   234   case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
```
```   235     apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
```
```   236   case False note this[unfolded interval_eq_empty not_ex not_less]
```
```   237   hence as:"\<forall>i<DIM('a). b \$\$ i \<ge> a \$\$ i" by fastsimp
```
```   238   show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_lessThan]
```
```   239     apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
```
```   240     apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
```
```   241
```
```   242 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
```
```   243
```
```   244 lemma content_closed_interval_cases:
```
```   245   "content {a..b::'a::ordered_euclidean_space} = (if \<forall>i<DIM('a). a\$\$i \<le> b\$\$i then setprod (\<lambda>i. b\$\$i - a\$\$i) {..<DIM('a)} else 0)" apply(rule cond_cases)
```
```   246   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
```
```   247
```
```   248 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
```
```   249   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
```
```   250
```
```   251 (*lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
```
```   252   unfolding content_eq_0 by auto*)
```
```   253
```
```   254 lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i<DIM('a). a\$\$i < b\$\$i)"
```
```   255   apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
```
```   256   hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i<DIM('a). a\$\$i < b\$\$i" unfolding content_eq_0 not_ex not_le by fastsimp qed
```
```   257
```
```   258 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
```
```   259
```
```   260 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}" proof(cases "{a..b}={}")
```
```   261   case True thus ?thesis using content_pos_le[of c d] by auto next
```
```   262   case False hence ab_ne:"\<forall>i<DIM('a). a \$\$ i \<le> b \$\$ i" unfolding interval_ne_empty by auto
```
```   263   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
```
```   264   have "{c..d} \<noteq> {}" using assms False by auto
```
```   265   hence cd_ne:"\<forall>i<DIM('a). c \$\$ i \<le> d \$\$ i" using assms unfolding interval_ne_empty by auto
```
```   266   show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
```
```   267     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof
```
```   268     fix i assume i:"i\<in>{..<DIM('a)}"
```
```   269     show "0 \<le> b \$\$ i - a \$\$ i" using ab_ne[THEN spec[where x=i]] i by auto
```
```   270     show "b \$\$ i - a \$\$ i \<le> d \$\$ i - c \$\$ i"
```
```   271       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
```
```   272       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] using i by auto qed qed
```
```   273
```
```   274 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
```
```   275   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastsimp
```
```   276
```
```   277 subsection {* The notion of a gauge --- simply an open set containing the point. *}
```
```   278
```
```   279 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
```
```   280
```
```   281 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
```
```   282   using assms unfolding gauge_def by auto
```
```   283
```
```   284 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
```
```   285
```
```   286 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
```
```   287   unfolding gauge_def by auto
```
```   288
```
```   289 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto
```
```   290
```
```   291 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
```
```   292
```
```   293 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
```
```   294   unfolding gauge_def by auto
```
```   295
```
```   296 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-
```
```   297   have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
```
```   298   unfolding gauge_def unfolding *
```
```   299   using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
```
```   300
```
```   301 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)
```
```   302
```
```   303 subsection {* Divisions. *}
```
```   304
```
```   305 definition division_of (infixl "division'_of" 40) where
```
```   306   "s division_of i \<equiv>
```
```   307         finite s \<and>
```
```   308         (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   309         (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
```
```   310         (\<Union>s = i)"
```
```   311
```
```   312 lemma division_ofD[dest]: assumes  "s division_of i"
```
```   313   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})"
```
```   314   "\<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
```
```   315
```
```   316 lemma division_ofI:
```
```   317   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})"
```
```   318   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
```
```   319   shows "s division_of i" using assms unfolding division_of_def by auto
```
```   320
```
```   321 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
```
```   322   unfolding division_of_def by auto
```
```   323
```
```   324 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
```
```   325   unfolding division_of_def by auto
```
```   326
```
```   327 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto
```
```   328
```
```   329 lemma division_of_sing[simp]: "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
```
```   330   assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s"
```
```   331     ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing by auto }
```
```   332   ultimately show ?l unfolding division_of_def interval_sing by auto next
```
```   333   assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing]]
```
```   334   { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
```
```   335   moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing by auto qed
```
```   336
```
```   337 lemma elementary_empty: obtains p where "p division_of {}"
```
```   338   unfolding division_of_trivial by auto
```
```   339
```
```   340 lemma elementary_interval: obtains p where  "p division_of {a..b}"
```
```   341   by(metis division_of_trivial division_of_self)
```
```   342
```
```   343 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
```
```   344   unfolding division_of_def by auto
```
```   345
```
```   346 lemma forall_in_division:
```
```   347  "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
```
```   348   unfolding division_of_def by fastsimp
```
```   349
```
```   350 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
```
```   351   apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
```
```   352   show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
```
```   353   { 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
```
```   354   show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
```
```   355   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
```
```   356   show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
```
```   357
```
```   358 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
```
```   359
```
```   360 lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
```
```   361   unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
```
```   362   apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
```
```   363
```
```   364 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::('a::ordered_euclidean_space) set)"
```
```   365   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-
```
```   366 let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
```
```   367 show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
```
```   368   moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
```
```   369   have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto show "\<Union>?A = s1 \<inter> s2" apply(rule set_eqI) unfolding * and Union_image_eq UN_iff
```
```   370     using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
```
```   371   { 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
```
```   372   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
```
```   373   guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
```
```   374   guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
```
```   375   show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
```
```   376   assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
```
```   377   assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
```
```   378   assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
```
```   379   have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
```
```   380       interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
```
```   381       interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
```
```   382       \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
```
```   383   show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
```
```   384     using division_ofD(5)[OF assms(1) k1(2) k2(2)]
```
```   385     using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
```
```   386
```
```   387 lemma division_inter_1: assumes "d division_of i" "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
```
```   388   shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
```
```   389   case True show ?thesis unfolding True and division_of_trivial by auto next
```
```   390   have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto
```
```   391   case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
```
```   392
```
```   393 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::('a::ordered_euclidean_space) set)"
```
```   394   shows "\<exists>p. p division_of (s \<inter> t)"
```
```   395   by(rule,rule division_inter[OF assms])
```
```   396
```
```   397 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
```
```   398   shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
```
```   399 case (insert x f) show ?case proof(cases "f={}")
```
```   400   case True thus ?thesis unfolding True using insert by auto next
```
```   401   case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
```
```   402   moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
```
```   403   show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
```
```   404
```
```   405 lemma division_disjoint_union:
```
```   406   assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
```
```   407   shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI)
```
```   408   note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
```
```   409   show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
```
```   410   show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
```
```   411   { 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 = {}"
```
```   412   { 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)]]
```
```   413       using assms(3) by blast } moreover
```
```   414   { 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)]]
```
```   415       using assms(3) by blast} ultimately
```
```   416   show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
```
```   417   fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
```
```   418   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
```
```   419
```
```   420 (* move *)
```
```   421 lemma Eucl_nth_inverse[simp]: fixes x::"'a::euclidean_space" shows "(\<chi>\<chi> i. x \$\$ i) = x"
```
```   422   apply(subst euclidean_eq) by auto
```
```   423
```
```   424 lemma partial_division_extend_1:
```
```   425   assumes "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}" "{c..d} \<noteq> {}"
```
```   426   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
```
```   427 proof- def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def using DIM_positive[where 'a='a] by auto
```
```   428   guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_lessThan[of "DIM('a)"]] .. note \<pi>=this
```
```   429   def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
```
```   430   have \<pi>':"bij_betw \<pi>' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
```
```   431   hence \<pi>'i:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto
```
```   432   have \<pi>i:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi> i <DIM('a)" using \<pi> unfolding bij_betw_def n_def by auto
```
```   433   have \<pi>\<pi>'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi> (\<pi>' i) = i" unfolding \<pi>'_def
```
```   434     apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
```
```   435   have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq)
```
```   436     using \<pi> unfolding n_def bij_betw_def by auto
```
```   437   have "{c..d} \<noteq> {}" using assms by auto
```
```   438   let ?p1 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c\$\$i else a\$\$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d\$\$i else if \<pi>' i = l then c\$\$\<pi> l else b\$\$i)}"
```
```   439   let ?p2 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c\$\$i else if \<pi>' i = l then d\$\$\<pi> l else a\$\$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d\$\$i else b\$\$i)}"
```
```   440   let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
```
```   441   have abcd:"\<And>i. i<DIM('a) \<Longrightarrow> a \$\$ i \<le> c \$\$ i \<and> c\$\$i \<le> d\$\$i \<and> d \$\$ i \<le> b \$\$ i" using assms
```
```   442     unfolding subset_interval interval_eq_empty by auto
```
```   443   show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
```
```   444   proof- have "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < Suc n"
```
```   445     proof(rule ccontr,unfold not_less) fix i assume i:"i<DIM('a)" and "Suc n \<le> \<pi>' i"
```
```   446       hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' i unfolding bij_betw_def by auto
```
```   447     qed hence "c = (\<chi>\<chi> i. if \<pi>' i < Suc n then c \$\$ i else a \$\$ i)"
```
```   448         "d = (\<chi>\<chi> i. if \<pi>' i < Suc n then d \$\$ i else if \<pi>' i = n + 1 then c \$\$ \<pi> (n + 1) else b \$\$ i)"
```
```   449       unfolding euclidean_eq[where 'a='a] using \<pi>' unfolding bij_betw_def by auto
```
```   450     thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
```
```   451     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}"
```
```   452       unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
```
```   453     proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
```
```   454       then guess i unfolding mem_interval not_all not_imp .. note i=conjunctD2[OF this]
```
```   455       show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
```
```   456         apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto
```
```   457     qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
```
```   458     proof- fix x assume x:"x\<in>{a..b}"
```
```   459       { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
```
```   460       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)}"
```
```   461       assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all not_imp ..
```
```   462       hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
```
```   463       hence M:"finite ?M" "?M \<noteq> {}" by auto
```
```   464       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]]
```
```   465         Min_gr_iff[OF M,unfolded l_def[symmetric]]
```
```   466       have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
```
```   467         apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
```
```   468       proof- assume as:"x \$\$ \<pi> l < c \$\$ \<pi> l"
```
```   469         show "x \<in> ?p1 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
```
```   470         proof- case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal1 by auto
```
```   471           thus ?case using as x[unfolded mem_interval,rule_format,of i]
```
```   472             apply auto using l(3)[of "\<pi>' i"] using goal1 by(auto elim!:ballE[where x="\<pi>' i"])
```
```   473         next case goal2 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal2 by auto
```
```   474           thus ?case using as x[unfolded mem_interval,rule_format,of i]
```
```   475             apply auto using l(3)[of "\<pi>' i"] using goal2 by(auto elim!:ballE[where x="\<pi>' i"])
```
```   476         qed
```
```   477       next assume as:"x \$\$ \<pi> l > d \$\$ \<pi> l"
```
```   478         show "x \<in> ?p2 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
```
```   479         proof- fix i assume i:"i<DIM('a)"
```
```   480           have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using i by auto
```
```   481           thus "(if \<pi>' i < l then c \$\$ i else if \<pi>' i = l then d \$\$ \<pi> l else a \$\$ i) \<le> x \$\$ i"
```
```   482             "x \$\$ i \<le> (if \<pi>' i < l then d \$\$ i else b \$\$ i)"
```
```   483             using as x[unfolded mem_interval,rule_format,of i]
```
```   484             apply auto using l(3)[of "\<pi>' i"] i by(auto elim!:ballE[where x="\<pi>' i"])
```
```   485         qed qed
```
```   486       thus "x \<in> \<Union>?p" using l(2) by blast
```
```   487     qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
```
```   488
```
```   489     show "finite ?p" by auto
```
```   490     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
```
```   491     show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule)
```
```   492     proof fix i x assume i:"i<DIM('a)" assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
```
```   493       ultimately show "a\$\$i \<le> x\$\$i" "x\$\$i \<le> b\$\$i" using abcd[of i] using l using i
```
```   494         by(auto elim:disjE elim!:allE[where x=i] simp add:eucl_le[where 'a='a])
```
```   495     qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
```
```   496     proof- case goal1 thus ?case using abcd[of x] by auto
```
```   497     next   case goal2 thus ?case using abcd[of x] by auto
```
```   498     qed thus "k \<noteq> {}" using k by auto
```
```   499     show "\<exists>a b. k = {a..b}" using k by auto
```
```   500     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
```
```   501     { fix k k' l l'
```
```   502       assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}"
```
```   503       assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}"
```
```   504       assume "l \<le> l'" fix x
```
```   505       have "x \<notin> interior k \<inter> interior k'"
```
```   506       proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
```
```   507         case True hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l'" using \<pi>'i using l' by(auto simp add:less_Suc_eq_le)
```
```   508         hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l' then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
```
```   509         hence k':"k' = {c..d}" using l'(1) unfolding * by auto
```
```   510         have ln:"l < n + 1"
```
```   511         proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
```
```   512           hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
```
```   513           hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
```
```   514           hence "k = {c..d}" using l(1) \<pi>'i unfolding * by(auto)
```
```   515           thus False using `k\<noteq>k'` k' by auto
```
```   516         qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
```
```   517         have "x \$\$ \<pi> l < c \$\$ \<pi> l \<or> d \$\$ \<pi> l < x \$\$ \<pi> l" using l(1) apply-
```
```   518         proof(erule disjE)
```
```   519           assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   520           show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] by(auto simp add:** not_less)
```
```   521         next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   522           show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] unfolding ** by auto
```
```   523         qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
```
```   524           by(auto elim!:allE[where x="\<pi> l"])
```
```   525       next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
```
```   526         hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
```
```   527         note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
```
```   528         assume x:"x \<in> interior k \<inter> interior k'"
```
```   529         show False using l(1) l'(1) apply-
```
```   530         proof(erule_tac[!] disjE)+
```
```   531           assume as:"k = ?p1 l" "k' = ?p1 l'"
```
```   532           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   533           have "l \<noteq> l'" using k'(2)[unfolded as] by auto
```
```   534           thus False using *[of "\<pi> l'"] *[of "\<pi> l"] ln using \<pi>i[OF ln(1)] \<pi>i[OF ln(2)] apply(cases "l<l'")
```
```   535             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
```
```   536         next assume as:"k = ?p2 l" "k' = ?p2 l'"
```
```   537           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   538           have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
```
```   539           thus False using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
```
```   540         next assume as:"k = ?p1 l" "k' = ?p2 l'"
```
```   541           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   542           show False using abcd[of "\<pi> l'"] using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln apply(cases "l=l'")
```
```   543             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
```
```   544         next assume as:"k = ?p2 l" "k' = ?p1 l'"
```
```   545           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
```
```   546           show False using *[of "\<pi> l"] *[of "\<pi> l'"] ln `l \<le> l'` apply(cases "l=l'") using abcd[of "\<pi> l'"]
```
```   547             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
```
```   548         qed qed }
```
```   549     from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
```
```   550       apply - apply(cases "l' \<le> l") using k'(2) by auto
```
```   551     thus "interior k \<inter> interior k' = {}" by auto
```
```   552 qed qed
```
```   553
```
```   554 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
```
```   555   obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}" proof(cases "p = {}")
```
```   556   case True guess q apply(rule elementary_interval[of a b]) .
```
```   557   thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
```
```   558   case False note p = division_ofD[OF assms(1)]
```
```   559   have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
```
```   560     guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
```
```   561     have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
```
```   562     guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
```
```   563   guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
```
```   564   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-
```
```   565     fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
```
```   566       using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
```
```   567   hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
```
```   568     apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
```
```   569   then guess d .. note d = this
```
```   570   show ?thesis apply(rule that[of "d \<union> p"]) proof-
```
```   571     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
```
```   572     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"
```
```   573       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
```
```   574     show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
```
```   575       apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
```
```   576       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
```
```   577       show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
```
```   578         defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
```
```   579         show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
```
```   580         show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
```
```   581         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}))"
```
```   582           apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
```
```   583
```
```   584 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::('a::ordered_euclidean_space) set)"
```
```   585   unfolding division_of_def by(metis bounded_Union bounded_interval)
```
```   586
```
```   587 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
```
```   588   by(meson elementary_bounded bounded_subset_closed_interval)
```
```   589
```
```   590 lemma division_union_intervals_exists: assumes "{a..b::'a::ordered_euclidean_space} \<noteq> {}"
```
```   591   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
```
```   592   case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
```
```   593   case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
```
```   594   have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
```
```   595   case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
```
```   596     using false True assms using interior_subset by auto next
```
```   597   case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
```
```   598   have *:"{u..v} \<subseteq> {c..d}" using uv by auto
```
```   599   guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
```
```   600   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
```
```   601   show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
```
```   602     apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
```
```   603     unfolding interior_inter[THEN sym] proof-
```
```   604     have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
```
```   605     have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
```
```   606       apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
```
```   607     also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
```
```   608     finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
```
```   609
```
```   610 lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
```
```   611   "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```   612   shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
```
```   613   apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
```
```   614   using division_ofD[OF assms(2)] by auto
```
```   615
```
```   616 lemma elementary_union_interval: assumes "p division_of \<Union>p"
```
```   617   obtains q where "q division_of ({a..b::'a::ordered_euclidean_space} \<union> \<Union>p)" proof-
```
```   618   note assm=division_ofD[OF assms]
```
```   619   have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
```
```   620   have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
```
```   621 { presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
```
```   622     "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
```
```   623   thus thesis by auto
```
```   624 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
```
```   625   thus thesis apply(rule_tac that[of p]) unfolding as by auto
```
```   626 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
```
```   627 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
```
```   628   show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
```
```   629     unfolding finite_insert apply(rule assm(1)) unfolding Union_insert
```
```   630     using assm(2-4) as apply- by(fastsimp dest: assm(5))+
```
```   631 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
```
```   632   have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
```
```   633     from assm(4)[OF this] guess c .. then guess d ..
```
```   634     thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
```
```   635   qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
```
```   636   let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
```
```   637   show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
```
```   638     have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
```
```   639     show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
```
```   640     show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
```
```   641       using q(6) by auto
```
```   642     fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
```
```   643     show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
```
```   644     fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
```
```   645     obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
```
```   646     obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
```
```   647     show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
```
```   648       case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
```
```   649     next case False
```
```   650       { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis"
```
```   651         "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
```
```   652         thus ?thesis by auto }
```
```   653       { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
```
```   654       { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
```
```   655       assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
```
```   656       guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
```
```   657       have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
```
```   658       hence "interior k \<subseteq> interior x" apply-
```
```   659         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
```
```   660       guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
```
```   661       have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
```
```   662       hence "interior k' \<subseteq> interior x'" apply-
```
```   663         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
```
```   664       ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
```
```   665     qed qed } qed
```
```   666
```
```   667 lemma elementary_unions_intervals:
```
```   668   assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
```
```   669   obtains p where "p division_of (\<Union>f)" proof-
```
```   670   have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct)
```
```   671     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
```
```   672     fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
```
```   673     from this(3) guess p .. note p=this
```
```   674     from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
```
```   675     have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
```
```   676     show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
```
```   677       unfolding Union_insert ab * by auto
```
```   678   qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
```
```   679
```
```   680 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::('a::ordered_euclidean_space) set)"
```
```   681   obtains p where "p division_of (s \<union> t)"
```
```   682 proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
```
```   683   hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
```
```   684   show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
```
```   685     unfolding * prefer 3 apply(rule_tac p=p in that)
```
```   686     using assms[unfolded division_of_def] by auto qed
```
```   687
```
```   688 lemma partial_division_extend: fixes t::"('a::ordered_euclidean_space) set"
```
```   689   assumes "p division_of s" "q division_of t" "s \<subseteq> t"
```
```   690   obtains r where "p \<subseteq> r" "r division_of t" proof-
```
```   691   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
```
```   692   obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
```
```   693   guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
```
```   694     apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
```
```   695   guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto
```
```   696   then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
```
```   697     apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
```
```   698   { fix x assume x:"x\<in>t" "x\<notin>s"
```
```   699     hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
```
```   700     then guess r unfolding Union_iff .. note r=this moreover
```
```   701     have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
```
```   702       thus False using x by auto qed
```
```   703     ultimately have "x\<in>\<Union>(r1 - p)" by auto }
```
```   704   hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
```
```   705   show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
```
```   706     unfolding divp(6) apply(rule assms r2)+
```
```   707   proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
```
```   708     proof(rule inter_interior_unions_intervals)
```
```   709       show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
```
```   710       have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
```
```   711       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
```
```   712         fix m x assume as:"m\<in>r1-p"
```
```   713         have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
```
```   714           show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
```
```   715           show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
```
```   716         qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
```
```   717       qed qed
```
```   718     thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
```
```   719   qed auto qed
```
```   720
```
```   721 subsection {* Tagged (partial) divisions. *}
```
```   722
```
```   723 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
```
```   724   "(s tagged_partial_division_of i) \<equiv>
```
```   725         finite s \<and>
```
```   726         (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   727         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
```
```   728                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
```
```   729
```
```   730 lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
```
```   731   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"
```
```   732   "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
```
```   733   "\<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) = {}"
```
```   734   using assms unfolding tagged_partial_division_of_def  apply- by blast+
```
```   735
```
```   736 definition tagged_division_of (infixr "tagged'_division'_of" 40) where
```
```   737   "(s tagged_division_of i) \<equiv>
```
```   738         (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   739
```
```   740 lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
```
```   741   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```   742
```
```   743 lemma tagged_division_of:
```
```   744  "(s tagged_division_of i) \<longleftrightarrow>
```
```   745         finite s \<and>
```
```   746         (\<forall>x k. (x,k) \<in> s
```
```   747                \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
```
```   748         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
```
```   749                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
```
```   750         (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   751   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```   752
```
```   753 lemma tagged_division_ofI: assumes
```
```   754   "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}"
```
```   755   "\<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) = {})"
```
```   756   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```   757   shows "s tagged_division_of i"
```
```   758   unfolding tagged_division_of apply(rule) defer apply rule
```
```   759   apply(rule allI impI conjI assms)+ apply assumption
```
```   760   apply(rule, rule assms, assumption) apply(rule assms, assumption)
```
```   761   using assms(1,5-) apply- by blast+
```
```   762
```
```   763 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
```
```   764   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}"
```
```   765   "\<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) = {})"
```
```   766   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
```
```   767
```
```   768 lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
```
```   769 proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
```
```   770   show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
```
```   771   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
```
```   772   thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
```
```   773   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
```
```   774   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
```
```   775 qed
```
```   776
```
```   777 lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
```
```   778   shows "(snd ` s) division_of \<Union>(snd ` s)"
```
```   779 proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
```
```   780   show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
```
```   781   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
```
```   782   thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
```
```   783   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
```
```   784   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
```
```   785 qed
```
```   786
```
```   787 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
```
```   788   shows "t tagged_partial_division_of i"
```
```   789   using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
```
```   790
```
```   791 lemma setsum_over_tagged_division_lemma: fixes d::"('m::ordered_euclidean_space) set \<Rightarrow> 'a::real_normed_vector"
```
```   792   assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
```
```   793   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
```
```   794 proof- note assm=tagged_division_ofD[OF assms(1)]
```
```   795   have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
```
```   796   show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
```
```   797     show "finite p" using assm by auto
```
```   798     fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
```
```   799     obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
```
```   800     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
```
```   801     hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto
```
```   802     hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
```
```   803     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
```
```   804     thus "d (snd x) = 0" unfolding ab by auto qed qed
```
```   805
```
```   806 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
```
```   807
```
```   808 lemma tagged_division_of_empty: "{} tagged_division_of {}"
```
```   809   unfolding tagged_division_of by auto
```
```   810
```
```   811 lemma tagged_partial_division_of_trivial[simp]:
```
```   812  "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
```
```   813   unfolding tagged_partial_division_of_def by auto
```
```   814
```
```   815 lemma tagged_division_of_trivial[simp]:
```
```   816  "p tagged_division_of {} \<longleftrightarrow> p = {}"
```
```   817   unfolding tagged_division_of by auto
```
```   818
```
```   819 lemma tagged_division_of_self:
```
```   820  "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
```
```   821   apply(rule tagged_division_ofI) by auto
```
```   822
```
```   823 lemma tagged_division_union:
```
```   824   assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
```
```   825   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
```
```   826 proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
```
```   827   show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
```
```   828   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
```
```   829   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
```
```   830   show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
```
```   831   fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
```
```   832   have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
```
```   833   show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
```
```   834     apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
```
```   835     using p1(3) p2(3) using xk xk' by auto qed
```
```   836
```
```   837 lemma tagged_division_unions:
```
```   838   assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
```
```   839   "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
```
```   840   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
```
```   841 proof(rule tagged_division_ofI)
```
```   842   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
```
```   843   show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
```
```   844   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
```
```   845   also have "\<dots> = \<Union>iset" using assm(6) by auto
```
```   846   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" .
```
```   847   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
```
```   848   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
```
```   849   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
```
```   850   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)
```
```   851     using assms(3)[rule_format] subset_interior by blast
```
```   852   show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
```
```   853     using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
```
```   854 qed
```
```   855
```
```   856 lemma tagged_partial_division_of_union_self:
```
```   857   assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
```
```   858   apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
```
```   859
```
```   860 lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
```
```   861   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```   862   apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
```
```   863
```
```   864 subsection {* Fine-ness of a partition w.r.t. a gauge. *}
```
```   865
```
```   866 definition fine (infixr "fine" 46) where
```
```   867   "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
```
```   868
```
```   869 lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```   870   shows "d fine s" using assms unfolding fine_def by auto
```
```   871
```
```   872 lemma fineD[dest]: assumes "d fine s"
```
```   873   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
```
```   874
```
```   875 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
```
```   876   unfolding fine_def by auto
```
```   877
```
```   878 lemma fine_inters:
```
```   879  "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
```
```   880   unfolding fine_def by blast
```
```   881
```
```   882 lemma fine_union:
```
```   883   "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
```
```   884   unfolding fine_def by blast
```
```   885
```
```   886 lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
```
```   887   unfolding fine_def by auto
```
```   888
```
```   889 lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
```
```   890   unfolding fine_def by blast
```
```   891
```
```   892 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
```
```   893
```
```   894 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
```
```   895   "(f has_integral_compact_interval y) i \<equiv>
```
```   896         (\<forall>e>0. \<exists>d. gauge d \<and>
```
```   897           (\<forall>p. p tagged_division_of i \<and> d fine p
```
```   898                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
```
```   899
```
```   900 definition has_integral (infixr "has'_integral" 46) where
```
```   901 "((f::('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
```
```   902         if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
```
```   903         else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```   904               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
```
```   905                                        norm(z - y) < e))"
```
```   906
```
```   907 lemma has_integral:
```
```   908  "(f has_integral y) ({a..b}) \<longleftrightarrow>
```
```   909         (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
```
```   910                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```   911   unfolding has_integral_def has_integral_compact_interval_def by auto
```
```   912
```
```   913 lemma has_integralD[dest]: assumes
```
```   914  "(f has_integral y) ({a..b})" "e>0"
```
```   915   obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
```
```   916                         \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
```
```   917   using assms unfolding has_integral by auto
```
```   918
```
```   919 lemma has_integral_alt:
```
```   920  "(f has_integral y) i \<longleftrightarrow>
```
```   921       (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
```
```   922        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```   923                                \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
```
```   924                                         has_integral z) ({a..b}) \<and>
```
```   925                                        norm(z - y) < e)))"
```
```   926   unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
```
```   927
```
```   928 lemma has_integral_altD:
```
```   929   assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
```
```   930   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)"
```
```   931   using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
```
```   932
```
```   933 definition integrable_on (infixr "integrable'_on" 46) where
```
```   934   "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
```
```   935
```
```   936 definition "integral i f \<equiv> SOME y. (f has_integral y) i"
```
```   937
```
```   938 lemma integrable_integral[dest]:
```
```   939  "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
```
```   940   unfolding integrable_on_def integral_def by(rule someI_ex)
```
```   941
```
```   942 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
```
```   943   unfolding integrable_on_def by auto
```
```   944
```
```   945 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
```
```   946   by auto
```
```   947
```
```   948 lemma setsum_content_null:
```
```   949   assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
```
```   950   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
```
```   951 proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
```
```   952   obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
```
```   953   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
```
```   954   from this(2) guess c .. then guess d .. note c_d=this
```
```   955   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
```
```   956   also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
```
```   957     unfolding assms(1) c_d by auto
```
```   958   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
```
```   959 qed
```
```   960
```
```   961 subsection {* Some basic combining lemmas. *}
```
```   962
```
```   963 lemma tagged_division_unions_exists:
```
```   964   assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
```
```   965   "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
```
```   966    obtains p where "p tagged_division_of i" "d fine p"
```
```   967 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
```
```   968   show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
```
```   969     apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer
```
```   970     apply(rule fine_unions) using pfn by auto
```
```   971 qed
```
```   972
```
```   973 subsection {* The set we're concerned with must be closed. *}
```
```   974
```
```   975 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::('n::ordered_euclidean_space) set)"
```
```   976   unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
```
```   977
```
```   978 subsection {* General bisection principle for intervals; might be useful elsewhere. *}
```
```   979
```
```   980 lemma interval_bisection_step:  fixes type::"'a::ordered_euclidean_space"
```
```   981   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::'a})"
```
```   982   obtains c d where "~(P{c..d})"
```
```   983   "\<forall>i<DIM('a). a\$\$i \<le> c\$\$i \<and> c\$\$i \<le> d\$\$i \<and> d\$\$i \<le> b\$\$i \<and> 2 * (d\$\$i - c\$\$i) \<le> b\$\$i - a\$\$i"
```
```   984 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
```
```   985   note ab=this[unfolded interval_eq_empty not_ex not_less]
```
```   986   { fix f have "finite f \<Longrightarrow>
```
```   987         (\<forall>s\<in>f. P s) \<Longrightarrow>
```
```   988         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
```
```   989         (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
```
```   990     proof(induct f rule:finite_induct)
```
```   991       case empty show ?case using assms(1) by auto
```
```   992     next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
```
```   993         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
```
```   994         using insert by auto
```
```   995     qed } note * = this
```
```   996   let ?A = "{{c..d} | c d::'a. \<forall>i<DIM('a). (c\$\$i = a\$\$i) \<and> (d\$\$i = (a\$\$i + b\$\$i) / 2) \<or> (c\$\$i = (a\$\$i + b\$\$i) / 2) \<and> (d\$\$i = b\$\$i)}"
```
```   997   let ?PP = "\<lambda>c d. \<forall>i<DIM('a). a\$\$i \<le> c\$\$i \<and> c\$\$i \<le> d\$\$i \<and> d\$\$i \<le> b\$\$i \<and> 2 * (d\$\$i - c\$\$i) \<le> b\$\$i - a\$\$i"
```
```   998   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
```
```   999     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
```
```  1000   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
```
```  1001   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI)
```
```  1002     let ?B = "(\<lambda>s.{(\<chi>\<chi> i. if i \<in> s then a\$\$i else (a\$\$i + b\$\$i) / 2)::'a ..
```
```  1003       (\<chi>\<chi> i. if i \<in> s then (a\$\$i + b\$\$i) / 2 else b\$\$i)}) ` {s. s \<subseteq> {..<DIM('a)}}"
```
```  1004     have "?A \<subseteq> ?B" proof case goal1
```
```  1005       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
```
```  1006       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
```
```  1007       show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. i<DIM('a) \<and> c\$\$i = a\$\$i}" in bexI)
```
```  1008         unfolding c_d apply(rule * ) unfolding euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta' mem_Collect_eq
```
```  1009       proof- fix i assume "i<DIM('a)" thus " c \$\$ i = (if i < DIM('a) \<and> c \$\$ i = a \$\$ i then a \$\$ i else (a \$\$ i + b \$\$ i) / 2)"
```
```  1010           "d \$\$ i = (if i < DIM('a) \<and> c \$\$ i = a \$\$ i then (a \$\$ i + b \$\$ i) / 2 else b \$\$ i)"
```
```  1011           using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
```
```  1012       qed qed
```
```  1013     thus "finite ?A" apply(rule finite_subset) by auto
```
```  1014     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
```
```  1015     note c_d=this[rule_format]
```
```  1016     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 thus ?case
```
```  1017         using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
```
```  1018     show "\<exists>a b. s = {a..b}" unfolding c_d by auto
```
```  1019     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
```
```  1020     note e_f=this[rule_format]
```
```  1021     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
```
```  1022     then obtain i where "c\$\$i \<noteq> e\$\$i \<or> d\$\$i \<noteq> f\$\$i" and i':"i<DIM('a)" unfolding de_Morgan_conj euclidean_eq[where 'a='a] by auto
```
```  1023     hence i:"c\$\$i \<noteq> e\$\$i" "d\$\$i \<noteq> f\$\$i" apply- apply(erule_tac[!] disjE)
```
```  1024     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
```
```  1025     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
```
```  1026     qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
```
```  1027     show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
```
```  1028       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
```
```  1029       hence x:"c\$\$i < d\$\$i" "e\$\$i < f\$\$i" "c\$\$i < f\$\$i" "e\$\$i < d\$\$i" unfolding mem_interval using i'
```
```  1030         apply-apply(erule_tac[!] x=i in allE)+ by auto
```
```  1031       show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
```
```  1032       proof(erule_tac[!] conjE) assume as:"c \$\$ i = a \$\$ i" "d \$\$ i = (a \$\$ i + b \$\$ i) / 2"
```
```  1033         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
```
```  1034       next assume as:"c \$\$ i = (a \$\$ i + b \$\$ i) / 2" "d \$\$ i = b \$\$ i"
```
```  1035         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
```
```  1036       qed qed qed
```
```  1037   also have "\<Union> ?A = {a..b}" proof(rule set_eqI,rule)
```
```  1038     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
```
```  1039     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
```
```  1040     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
```
```  1041     show "x\<in>{a..b}" unfolding mem_interval proof safe
```
```  1042       fix i assume "i<DIM('a)" thus "a \$\$ i \<le> x \$\$ i" "x \$\$ i \<le> b \$\$ i"
```
```  1043         using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
```
```  1044   next fix x assume x:"x\<in>{a..b}"
```
```  1045     have "\<forall>i<DIM('a). \<exists>c d. (c = a\$\$i \<and> d = (a\$\$i + b\$\$i) / 2 \<or> c = (a\$\$i + b\$\$i) / 2 \<and> d = b\$\$i) \<and> c\<le>x\$\$i \<and> x\$\$i \<le> d"
```
```  1046       (is "\<forall>i<DIM('a). \<exists>c d. ?P i c d") unfolding mem_interval proof(rule,rule) fix i
```
```  1047       have "?P i (a\$\$i) ((a \$\$ i + b \$\$ i) / 2) \<or> ?P i ((a \$\$ i + b \$\$ i) / 2) (b\$\$i)"
```
```  1048         using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
```
```  1049     qed thus "x\<in>\<Union>?A" unfolding Union_iff unfolding lambda_skolem' unfolding Bex_def mem_Collect_eq
```
```  1050       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
```
```  1051   qed finally show False using assms by auto qed
```
```  1052
```
```  1053 lemma interval_bisection: fixes type::"'a::ordered_euclidean_space"
```
```  1054   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::'a}"
```
```  1055   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})"
```
```  1056 proof-
```
```  1057   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
```
```  1058     (\<forall>i<DIM('a). fst x\$\$i \<le> fst y\$\$i \<and> fst y\$\$i \<le> snd y\$\$i \<and> snd y\$\$i \<le> snd x\$\$i \<and>
```
```  1059                            2 * (snd y\$\$i - fst y\$\$i) \<le> snd x\$\$i - fst x\$\$i))" proof case goal1 thus ?case proof-
```
```  1060       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
```
```  1061       thus ?thesis apply(cases "P {fst x..snd x}") by auto
```
```  1062     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d .
```
```  1063       thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
```
```  1064     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
```
```  1065   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
```
```  1066   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
```
```  1067     (\<forall>i<DIM('a). A(n)\$\$i \<le> A(Suc n)\$\$i \<and> A(Suc n)\$\$i \<le> B(Suc n)\$\$i \<and> B(Suc n)\$\$i \<le> B(n)\$\$i \<and>
```
```  1068     2 * (B(Suc n)\$\$i - A(Suc n)\$\$i) \<le> B(n)\$\$i - A(n)\$\$i)" (is "\<And>n. ?P n")
```
```  1069   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
```
```  1070     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
```
```  1071     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
```
```  1072     next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
```
```  1073     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
```
```  1074
```
```  1075   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"
```
```  1076   proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b\$\$i - a\$\$i) {..<DIM('a)}) / e"] .. note n=this
```
```  1077     show ?case apply(rule_tac x=n in exI) proof(rule,rule)
```
```  1078       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
```
```  1079       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\$\$i)) {..<DIM('a)}" unfolding dist_norm by(rule norm_le_l1)
```
```  1080       also have "\<dots> \<le> setsum (\<lambda>i. B n\$\$i - A n\$\$i) {..<DIM('a)}"
```
```  1081       proof(rule setsum_mono) fix i show "\<bar>(x - y) \$\$ i\<bar> \<le> B n \$\$ i - A n \$\$ i"
```
```  1082           using xy[unfolded mem_interval,THEN spec[where x=i]] by auto qed
```
```  1083       also have "\<dots> \<le> setsum (\<lambda>i. b\$\$i - a\$\$i) {..<DIM('a)} / 2^n" unfolding setsum_divide_distrib
```
```  1084       proof(rule setsum_mono) case goal1 thus ?case
```
```  1085         proof(induct n) case 0 thus ?case unfolding AB by auto
```
```  1086         next case (Suc n) have "B (Suc n) \$\$ i - A (Suc n) \$\$ i \<le> (B n \$\$ i - A n \$\$ i) / 2"
```
```  1087             using AB(4)[of i n] using goal1 by auto
```
```  1088           also have "\<dots> \<le> (b \$\$ i - a \$\$ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
```
```  1089         qed qed
```
```  1090       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
```
```  1091     qed qed
```
```  1092   { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
```
```  1093     have "{A n..B n} \<subseteq> {A m..B m}" unfolding d
```
```  1094     proof(induct d) case 0 thus ?case by auto
```
```  1095     next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
```
```  1096         apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
```
```  1097       proof- case goal1 thus ?case using AB(4)[of i "m + d"] by(auto simp add:field_simps)
```
```  1098       qed qed } note ABsubset = this
```
```  1099   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
```
```  1100   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
```
```  1101   then guess x0 .. note x0=this[rule_format]
```
```  1102   show thesis proof(rule that[rule_format,of x0])
```
```  1103     show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
```
```  1104     fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
```
```  1105     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}"
```
```  1106       apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer
```
```  1107     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
```
```  1108       show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
```
```  1109       show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
```
```  1110     qed qed qed
```
```  1111
```
```  1112 subsection {* Cousin's lemma. *}
```
```  1113
```
```  1114 lemma fine_division_exists: assumes "gauge g"
```
```  1115   obtains p where "p tagged_division_of {a..b::'a::ordered_euclidean_space}" "g fine p"
```
```  1116 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
```
```  1117   then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
```
```  1118 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
```
```  1119   guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
```
```  1120     apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
```
```  1121   proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
```
```  1122     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 = {}"
```
```  1123     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
```
```  1124       apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
```
```  1125   qed note x=this
```
```  1126   obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
```
```  1127   from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
```
```  1128   have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
```
```  1129   thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
```
```  1130
```
```  1131 subsection {* Basic theorems about integrals. *}
```
```  1132
```
```  1133 lemma has_integral_unique: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  1134   assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
```
```  1135 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
```
```  1136   have lem:"\<And>f::'n \<Rightarrow> 'a.  \<And> a b k1 k2.
```
```  1137     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
```
```  1138   proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
```
```  1139     guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
```
```  1140     guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
```
```  1141     guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
```
```  1142     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
```
```  1143       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
```
```  1144     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  1145       apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
```
```  1146     finally show False by auto
```
```  1147   qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
```
```  1148     thus False apply-apply(cases "\<exists>a b. i = {a..b}")
```
```  1149       using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
```
```  1150   assume as:"\<not> (\<exists>a b. i = {a..b})"
```
```  1151   guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
```
```  1152   guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
```
```  1153   have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
```
```  1154     using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
```
```  1155   note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  1156   guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
```
```  1157   guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
```
```  1158   have "z = w" using lem[OF w(1) z(1)] by auto
```
```  1159   hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
```
```  1160     using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute)
```
```  1161   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
```
```  1162   finally show False by auto qed
```
```  1163
```
```  1164 lemma integral_unique[intro]:
```
```  1165   "(f has_integral y) k \<Longrightarrow> integral k f = y"
```
```  1166   unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique)
```
```  1167
```
```  1168 lemma has_integral_is_0: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  1169   assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
```
```  1170 proof- have lem:"\<And>a b. \<And>f::'n \<Rightarrow> 'a.
```
```  1171     (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
```
```  1172   proof(rule,rule) fix a b e and f::"'n \<Rightarrow> 'a"
```
```  1173     assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
```
```  1174     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)"
```
```  1175       apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
```
```  1176     proof(rule,rule,erule conjE) case goal1
```
```  1177       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
```
```  1178         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
```
```  1179         thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
```
```  1180       qed thus ?case using as by auto
```
```  1181     qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1182     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
```
```  1183       using assms by(auto simp add:has_integral intro:lem) }
```
```  1184   have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
```
```  1185   assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
```
```  1186   apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
```
```  1187   proof- fix e::real and a b assume "e>0"
```
```  1188     thus "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
```
```  1189       apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
```
```  1190   qed auto qed
```
```  1191
```
```  1192 lemma has_integral_0[simp]: "((\<lambda>x::'n::ordered_euclidean_space. 0) has_integral 0) s"
```
```  1193   apply(rule has_integral_is_0) by auto
```
```  1194
```
```  1195 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
```
```  1196   using has_integral_unique[OF has_integral_0] by auto
```
```  1197
```
```  1198 lemma has_integral_linear: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  1199   assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
```
```  1200 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
```
```  1201   have lem:"\<And>f::'n \<Rightarrow> 'a. \<And> y a b.
```
```  1202     (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
```
```  1203   proof(subst has_integral,rule,rule) case goal1
```
```  1204     from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
```
```  1205     have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
```
```  1206     guess g using has_integralD[OF goal1(1) *] . note g=this
```
```  1207     show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
```
```  1208     proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p"
```
```  1209       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
```
```  1210       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"
```
```  1211         unfolding o_def unfolding scaleR[THEN sym] * by simp
```
```  1212       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
```
```  1213       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)" .
```
```  1214       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
```
```  1215         apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
```
```  1216     qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1217     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
```
```  1218   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
```
```  1219   proof(rule,rule) fix e::real  assume e:"0<e"
```
```  1220     have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
```
```  1221     guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
```
```  1222     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)"
```
```  1223       apply(rule_tac x=M in exI) apply(rule,rule M(1))
```
```  1224     proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
```
```  1225       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)"
```
```  1226         unfolding o_def apply(rule ext) using zero by auto
```
```  1227       show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
```
```  1228         apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
```
```  1229     qed qed qed
```
```  1230
```
```  1231 lemma has_integral_cmul:
```
```  1232   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
```
```  1233   unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
```
```  1234   by(rule scaleR.bounded_linear_right)
```
```  1235
```
```  1236 lemma has_integral_neg:
```
```  1237   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
```
```  1238   apply(drule_tac c="-1" in has_integral_cmul) by auto
```
```  1239
```
```  1240 lemma has_integral_add: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  1241   assumes "(f has_integral k) s" "(g has_integral l) s"
```
```  1242   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
```
```  1243 proof- have lem:"\<And>f g::'n \<Rightarrow> 'a. \<And>a b k l.
```
```  1244     (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
```
```  1245      ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
```
```  1246     show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
```
```  1247       guess d1 using has_integralD[OF goal1(1) *] . note d1=this
```
```  1248       guess d2 using has_integralD[OF goal1(2) *] . note d2=this
```
```  1249       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)"
```
```  1250         apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
```
```  1251       proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  1252         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)"
```
```  1253           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]
```
```  1254           by(rule setsum_cong2,auto)
```
```  1255         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))"
```
```  1256           unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
```
```  1257         from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
```
```  1258         have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
```
```  1259           apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
```
```  1260         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
```
```  1261       qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
```
```  1262     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
```
```  1263   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
```
```  1264   proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
```
```  1265     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
```
```  1266     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
```
```  1267     show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
```
```  1268     proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
```
```  1269       hence *:"ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}" by auto
```
```  1270       guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
```
```  1271       guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
```
```  1272       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
```
```  1273       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"
```
```  1274         apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
```
```  1275         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
```
```  1276     qed qed qed
```
```  1277
```
```  1278 lemma has_integral_sub:
```
```  1279   shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
```
```  1280   using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
```
```  1281
```
```  1282 lemma integral_0: "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
```
```  1283   by(rule integral_unique has_integral_0)+
```
```  1284
```
```  1285 lemma integral_add:
```
```  1286   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  1287    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
```
```  1288   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1289   apply(rule has_integral_add) by assumption+
```
```  1290
```
```  1291 lemma integral_cmul:
```
```  1292   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
```
```  1293   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1294   apply(rule has_integral_cmul) by assumption+
```
```  1295
```
```  1296 lemma integral_neg:
```
```  1297   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
```
```  1298   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1299   apply(rule has_integral_neg) by assumption+
```
```  1300
```
```  1301 lemma integral_sub:
```
```  1302   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"
```
```  1303   apply(rule integral_unique) apply(drule integrable_integral)+
```
```  1304   apply(rule has_integral_sub) by assumption+
```
```  1305
```
```  1306 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
```
```  1307   unfolding integrable_on_def using has_integral_0 by auto
```
```  1308
```
```  1309 lemma integrable_add:
```
```  1310   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
```
```  1311   unfolding integrable_on_def by(auto intro: has_integral_add)
```
```  1312
```
```  1313 lemma integrable_cmul:
```
```  1314   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
```
```  1315   unfolding integrable_on_def by(auto intro: has_integral_cmul)
```
```  1316
```
```  1317 lemma integrable_neg:
```
```  1318   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
```
```  1319   unfolding integrable_on_def by(auto intro: has_integral_neg)
```
```  1320
```
```  1321 lemma integrable_sub:
```
```  1322   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
```
```  1323   unfolding integrable_on_def by(auto intro: has_integral_sub)
```
```  1324
```
```  1325 lemma integrable_linear:
```
```  1326   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
```
```  1327   unfolding integrable_on_def by(auto intro: has_integral_linear)
```
```  1328
```
```  1329 lemma integral_linear:
```
```  1330   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
```
```  1331   apply(rule has_integral_unique) defer unfolding has_integral_integral
```
```  1332   apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
```
```  1333   apply(rule integrable_linear) by assumption+
```
```  1334
```
```  1335 lemma integral_component_eq[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
```
```  1336   assumes "f integrable_on s" shows "integral s (\<lambda>x. f x \$\$ k) = integral s f \$\$ k"
```
```  1337   unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
```
```  1338
```
```  1339 lemma has_integral_setsum:
```
```  1340   assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
```
```  1341   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
```
```  1342 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
```
```  1343   case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
```
```  1344     apply(rule has_integral_add) using insert assms by auto
```
```  1345 qed auto
```
```  1346
```
```  1347 lemma integral_setsum:
```
```  1348   shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
```
```  1349   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
```
```  1350   apply(rule integral_unique) apply(rule has_integral_setsum)
```
```  1351   using integrable_integral by auto
```
```  1352
```
```  1353 lemma integrable_setsum:
```
```  1354   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"
```
```  1355   unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
```
```  1356
```
```  1357 lemma has_integral_eq:
```
```  1358   assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
```
```  1359   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
```
```  1360   using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
```
```  1361
```
```  1362 lemma integrable_eq:
```
```  1363   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
```
```  1364   unfolding integrable_on_def using has_integral_eq[of s f g] by auto
```
```  1365
```
```  1366 lemma has_integral_eq_eq:
```
```  1367   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
```
```  1368   using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
```
```  1369
```
```  1370 lemma has_integral_null[dest]:
```
```  1371   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
```
```  1372   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
```
```  1373 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
```
```  1374   fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
```
```  1375   have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
```
```  1376     using setsum_content_null[OF assms(1) p, of f] .
```
```  1377   thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
```
```  1378
```
```  1379 lemma has_integral_null_eq[simp]:
```
```  1380   shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
```
```  1381   apply rule apply(rule has_integral_unique,assumption)
```
```  1382   apply(drule has_integral_null,assumption)
```
```  1383   apply(drule has_integral_null) by auto
```
```  1384
```
```  1385 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
```
```  1386   by(rule integral_unique,drule has_integral_null)
```
```  1387
```
```  1388 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
```
```  1389   unfolding integrable_on_def apply(drule has_integral_null) by auto
```
```  1390
```
```  1391 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
```
```  1392   unfolding empty_as_interval apply(rule has_integral_null)
```
```  1393   using content_empty unfolding empty_as_interval .
```
```  1394
```
```  1395 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
```
```  1396   apply(rule,rule has_integral_unique,assumption) by auto
```
```  1397
```
```  1398 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
```
```  1399
```
```  1400 lemma integral_empty[simp]: shows "integral {} f = 0"
```
```  1401   apply(rule integral_unique) using has_integral_empty .
```
```  1402
```
```  1403 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a::'a::ordered_euclidean_space}"
```
```  1404 proof- have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq[where 'a='a]
```
```  1405     apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps)
```
```  1406   show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
```
```  1407     apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
```
```  1408     unfolding interior_closed_interval using interval_sing by auto qed
```
```  1409
```
```  1410 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
```
```  1411
```
```  1412 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
```
```  1413
```
```  1414 subsection {* Cauchy-type criterion for integrability. *}
```
```  1415
```
```  1416 (* XXXXXXX *)
```
```  1417 lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
```
```  1418   shows "f integrable_on {a..b} \<longleftrightarrow>
```
```  1419   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
```
```  1420                             p2 tagged_division_of {a..b} \<and> d fine p2
```
```  1421                             \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
```
```  1422                                      setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
```
```  1423 proof assume ?l
```
```  1424   then guess y unfolding integrable_on_def has_integral .. note y=this
```
```  1425   show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
```
```  1426     then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
```
```  1427     show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
```
```  1428     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
```
```  1429       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"
```
```  1430         apply(rule dist_triangle_half_l[where y=y,unfolded dist_norm])
```
```  1431         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
```
```  1432     qed qed
```
```  1433 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
```
```  1434   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  1435   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
```
```  1436   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-
```
```  1437   proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
```
```  1438   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
```
```  1439   have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
```
```  1440   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
```
```  1441   proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
```
```  1442     show ?case apply(rule_tac x=N in exI)
```
```  1443     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
```
```  1444       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"
```
```  1445         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
```
```  1446         using dp p(1) using mn by auto
```
```  1447     qed qed
```
```  1448   then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
```
```  1449   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
```
```  1450   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
```
```  1451     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
```
```  1452     guess N2 using y[OF *] .. note N2=this
```
```  1453     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)"
```
```  1454       apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer
```
```  1455     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
```
```  1456       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
```
```  1457       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
```
```  1458       show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
```
```  1459         apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
```
```  1460         using N2[rule_format,unfolded dist_norm,of "N1+N2"]
```
```  1461         using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
```
```  1462
```
```  1463 subsection {* Additivity of integral on abutting intervals. *}
```
```  1464
```
```  1465 lemma interval_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
```
```  1466   "{a..b} \<inter> {x. x\$\$k \<le> c} = {a .. (\<chi>\<chi> i. if i = k then min (b\$\$k) c else b\$\$i)}"
```
```  1467   "{a..b} \<inter> {x. x\$\$k \<ge> c} = {(\<chi>\<chi> i. if i = k then max (a\$\$k) c else a\$\$i) .. b}"
```
```  1468   apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval mem_Collect_eq using assms by auto
```
```  1469
```
```  1470 lemma content_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
```
```  1471   "content {a..b} = content({a..b} \<inter> {x. x\$\$k \<le> c}) + content({a..b} \<inter> {x. x\$\$k >= c})"
```
```  1472 proof- note simps = interval_split[OF assms] content_closed_interval_cases eucl_le[where 'a='a]
```
```  1473   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps using assms by auto }
```
```  1474   have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" "\<And>x. finite ({..<DIM('a)}-{x})" "\<And>x. x\<notin>{..<DIM('a)}-{x}"
```
```  1475     using assms by auto
```
```  1476   have *:"\<And>X Y Z. (\<Prod>i\<in>{..<DIM('a)}. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>{..<DIM('a)}-{k}. Z i (Y i))"
```
```  1477     "(\<Prod>i\<in>{..<DIM('a)}. b\$\$i - a\$\$i) = (\<Prod>i\<in>{..<DIM('a)}-{k}. b\$\$i - a\$\$i) * (b\$\$k - a\$\$k)"
```
```  1478     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
```
```  1479   assume as:"a\<le>b" moreover have "\<And>x. min (b \$\$ k) c = max (a \$\$ k) c
```
```  1480     \<Longrightarrow> x* (b\$\$k - a\$\$k) = x*(max (a \$\$ k) c - a \$\$ k) + x*(b \$\$ k - max (a \$\$ k) c)"
```
```  1481     by  (auto simp add:field_simps)
```
```  1482   moreover have **:"(\<Prod>i<DIM('a). ((\<chi>\<chi> i. if i = k then min (b \$\$ k) c else b \$\$ i)::'a) \$\$ i - a \$\$ i) =
```
```  1483     (\<Prod>i<DIM('a). (if i = k then min (b \$\$ k) c else b \$\$ i) - a \$\$ i)"
```
```  1484     "(\<Prod>i<DIM('a). b \$\$ i - ((\<chi>\<chi> i. if i = k then max (a \$\$ k) c else a \$\$ i)::'a) \$\$ i) =
```
```  1485     (\<Prod>i<DIM('a). b \$\$ i - (if i = k then max (a \$\$ k) c else a \$\$ i))"
```
```  1486     apply(rule_tac[!] setprod.cong) by auto
```
```  1487   have "\<not> a \$\$ k \<le> c \<Longrightarrow> \<not> c \<le> b \$\$ k \<Longrightarrow> False"
```
```  1488     unfolding not_le using as[unfolded eucl_le[where 'a='a],rule_format,of k] assms by auto
```
```  1489   ultimately show ?thesis using assms unfolding simps **
```
```  1490     unfolding *(1)[of "\<lambda>i x. b\$\$i - x"] *(1)[of "\<lambda>i x. x - a\$\$i"] unfolding  *(2)
```
```  1491     apply(subst(2) euclidean_lambda_beta''[where 'a='a])
```
```  1492     apply(subst(3) euclidean_lambda_beta''[where 'a='a]) by auto
```
```  1493 qed
```
```  1494
```
```  1495 lemma division_split_left_inj: fixes type::"'a::ordered_euclidean_space"
```
```  1496   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
```
```  1497   "k1 \<inter> {x::'a. x\$\$k \<le> c} = k2 \<inter> {x. x\$\$k \<le> c}"and k:"k<DIM('a)"
```
```  1498   shows "content(k1 \<inter> {x. x\$\$k \<le> c}) = 0"
```
```  1499 proof- note d=division_ofD[OF assms(1)]
```
```  1500   have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x\$\$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x\$\$k \<le> c}) = {})"
```
```  1501     unfolding  interval_split[OF k] content_eq_0_interior by auto
```
```  1502   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
```
```  1503   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
```
```  1504   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
```
```  1505   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
```
```  1506     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
```
```  1507
```
```  1508 lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
```
```  1509   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
```
```  1510   "k1 \<inter> {x::'a. x\$\$k \<ge> c} = k2 \<inter> {x. x\$\$k \<ge> c}" and k:"k<DIM('a)"
```
```  1511   shows "content(k1 \<inter> {x. x\$\$k \<ge> c}) = 0"
```
```  1512 proof- note d=division_ofD[OF assms(1)]
```
```  1513   have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x\$\$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x\$\$k >= c}) = {})"
```
```  1514     unfolding interval_split[OF k] content_eq_0_interior by auto
```
```  1515   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
```
```  1516   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
```
```  1517   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
```
```  1518   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
```
```  1519     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
```
```  1520
```
```  1521 lemma tagged_division_split_left_inj: fixes x1::"'a::ordered_euclidean_space"
```
```  1522   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}"
```
```  1523   and k:"k<DIM('a)"
```
```  1524   shows "content(k1 \<inter> {x. x\$\$k \<le> c}) = 0"
```
```  1525 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
```
```  1526   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
```
```  1527     apply(rule_tac[1-2] *) using assms(2-) by auto qed
```
```  1528
```
```  1529 lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
```
```  1530   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}"
```
```  1531   and k:"k<DIM('a)"
```
```  1532   shows "content(k1 \<inter> {x. x\$\$k \<ge> c}) = 0"
```
```  1533 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
```
```  1534   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
```
```  1535     apply(rule_tac[1-2] *) using assms(2-) by auto qed
```
```  1536
```
```  1537 lemma division_split: fixes a::"'a::ordered_euclidean_space"
```
```  1538   assumes "p division_of {a..b}" and k:"k<DIM('a)"
```
```  1539   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
```
```  1540         "{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")
```
```  1541 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms(1)]
```
```  1542   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
```
```  1543   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
```
```  1544     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
```
```  1545     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
```
```  1546       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
```
```  1547     fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
```
```  1548     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
```
```  1549   { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
```
```  1550     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
```
```  1551     show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
```
```  1552       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
```
```  1553     fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
```
```  1554     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
```
```  1555 qed
```
```  1556
```
```  1557 lemma has_integral_split: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1558   assumes "(f has_integral i) ({a..b} \<inter> {x. x\$\$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x\$\$k \<ge> c})" and k:"k<DIM('a)"
```
```  1559   shows "(f has_integral (i + j)) ({a..b})"
```
```  1560 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
```
```  1561   guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[THEN sym,OF k]]
```
```  1562   guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[THEN sym,OF k]]
```
```  1563   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"
```
```  1564   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
```
```  1565   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
```
```  1566     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
```
```  1567     have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\$\$k \<le> c} = {}) \<Longrightarrow> x\$\$k \<le> c"
```
```  1568          "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x\$\$k \<ge> c} = {}) \<Longrightarrow> x\$\$k \<ge> c"
```
```  1569     proof- fix x kk assume as:"(x,kk)\<in>p"
```
```  1570       show "~(kk \<inter> {x. x\$\$k \<le> c} = {}) \<Longrightarrow> x\$\$k \<le> c"
```
```  1571       proof(rule ccontr) case goal1
```
```  1572         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \$\$ k - c\<bar>"
```
```  1573           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  1574         hence "\<exists>y. y \<in> ball x \<bar>x \$\$ k - c\<bar> \<inter> {x. x \$\$ k \<le> c}" using goal1(1) by blast
```
```  1575         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)
```
```  1576           using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
```
```  1577         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
```
```  1578       qed
```
```  1579       show "~(kk \<inter> {x. x\$\$k \<ge> c} = {}) \<Longrightarrow> x\$\$k \<ge> c"
```
```  1580       proof(rule ccontr) case goal1
```
```  1581         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \$\$ k - c\<bar>"
```
```  1582           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  1583         hence "\<exists>y. y \<in> ball x \<bar>x \$\$ k - c\<bar> \<inter> {x. x \$\$ k \<ge> c}" using goal1(1) by blast
```
```  1584         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)
```
```  1585           using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
```
```  1586         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
```
```  1587       qed
```
```  1588     qed
```
```  1589
```
```  1590     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
```
```  1591     have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
```
```  1592     proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
```
```  1593     have lem3: "\<And>g::('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool. finite p \<Longrightarrow>
```
```  1594       setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
```
```  1595                = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
```
```  1596       apply(rule setsum_mono_zero_left) prefer 3
```
```  1597     proof fix g::"('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" and i::"('a) \<times> (('a) set)"
```
```  1598       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  1599       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
```
```  1600       have "content (g k) = 0" using xk using content_empty by auto
```
```  1601       thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
```
```  1602     qed auto
```
```  1603     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
```
```  1604
```
```  1605     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> {}}"
```
```  1606     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
```
```  1607       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
```
```  1608     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
```
```  1609       fix x l assume xl:"(x,l)\<in>?M1"
```
```  1610       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
```
```  1611       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
```
```  1612       thus "l \<subseteq> d1 x" unfolding xl' by auto
```
```  1613       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)
```
```  1614         using lem0(1)[OF xl'(3-4)] by auto
```
```  1615       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k,where c=c])
```
```  1616       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
```
```  1617       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
```
```  1618       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
```
```  1619       proof(cases "l' = r' \<longrightarrow> x' = y'")
```
```  1620         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1621       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
```
```  1622         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1623       qed qed moreover
```
```  1624
```
```  1625     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> {}}"
```
```  1626     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
```
```  1627       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
```
```  1628     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
```
```  1629       fix x l assume xl:"(x,l)\<in>?M2"
```
```  1630       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
```
```  1631       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
```
```  1632       thus "l \<subseteq> d2 x" unfolding xl' by auto
```
```  1633       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)
```
```  1634         using lem0(2)[OF xl'(3-4)] by auto
```
```  1635       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k, where c=c])
```
```  1636       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
```
```  1637       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
```
```  1638       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
```
```  1639       proof(cases "l' = r' \<longrightarrow> x' = y'")
```
```  1640         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1641       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
```
```  1642         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1643       qed qed ultimately
```
```  1644
```
```  1645     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"
```
```  1646       apply- apply(rule norm_triangle_lt) by auto
```
```  1647     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0" using scaleR_zero_left by auto
```
```  1648       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)
```
```  1649        = (\<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
```
```  1650       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \$\$ k \<le> c}) *\<^sub>R f x) +
```
```  1651         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \$\$ k}) *\<^sub>R f x) - (i + j)"
```
```  1652         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
```
```  1653         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
```
```  1654       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) using k by auto
```
```  1655       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) using k by auto
```
```  1656       qed also note setsum_addf[THEN sym]
```
```  1657       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
```
```  1658         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
```
```  1659       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  1660         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"
```
```  1661           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[OF k,of u v c] by auto
```
```  1662       qed note setsum_cong2[OF this]
```
```  1663       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 +
```
```  1664         ((\<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) =
```
```  1665         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
```
```  1666     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
```
```  1667
```
```  1668 (*lemma has_integral_split_cart: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
```
```  1669   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})"
```
```  1670   shows "(f has_integral (i + j)) ({a..b})" *)
```
```  1671
```
```  1672 subsection {* A sort of converse, integrability on subintervals. *}
```
```  1673
```
```  1674 lemma tagged_division_union_interval: fixes a::"'a::ordered_euclidean_space"
```
```  1675   assumes "p1 tagged_division_of ({a..b} \<inter> {x. x\$\$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x\$\$k \<ge> c})"
```
```  1676   and k:"k<DIM('a)"
```
```  1677   shows "(p1 \<union> p2) tagged_division_of ({a..b})"
```
```  1678 proof- have *:"{a..b} = ({a..b} \<inter> {x. x\$\$k \<le> c}) \<union> ({a..b} \<inter> {x. x\$\$k \<ge> c})" by auto
```
```  1679   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms(1-2)])
```
```  1680     unfolding interval_split[OF k] interior_closed_interval using k
```
```  1681     by(auto simp add: eucl_less[where 'a='a] elim!:allE[where x=k]) qed
```
```  1682
```
```  1683 lemma has_integral_separate_sides: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1684   assumes "(f has_integral i) ({a..b})" "e>0" and k:"k<DIM('a)"
```
```  1685   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>
```
```  1686                                 p2 tagged_division_of ({a..b} \<inter> {x. x\$\$k \<ge> c}) \<and> d fine p2
```
```  1687                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
```
```  1688                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
```
```  1689 proof- guess d using has_integralD[OF assms(1-2)] . note d=this
```
```  1690   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
```
```  1691   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
```
```  1692                    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
```
```  1693     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
```
```  1694     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)"
```
```  1695       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
```
```  1696     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
```
```  1697       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
```
```  1698       have "b \<subseteq> {x. x\$\$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
```
```  1699       moreover have "interior {x::'a. x \$\$ k = c} = {}"
```
```  1700       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x\$\$k = c}" by auto
```
```  1701         then guess e unfolding mem_interior .. note e=this
```
```  1702         have x:"x\$\$k = c" using x interior_subset by fastsimp
```
```  1703         have *:"\<And>i. i<DIM('a) \<Longrightarrow> \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) \$\$ i\<bar>
```
```  1704           = (if i = k then e/2 else 0)" using e by auto
```
```  1705         have "(\<Sum>i<DIM('a). \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) \$\$ i\<bar>) =
```
```  1706           (\<Sum>i<DIM('a). (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
```
```  1707         also have "... < e" apply(subst setsum_delta) using e by auto
```
```  1708         finally have "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball dist_norm
```
```  1709           by(rule le_less_trans[OF norm_le_l1])
```
```  1710         hence "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> {x. x\$\$k = c}" using e by auto
```
```  1711         thus False unfolding mem_Collect_eq using e x k by auto
```
```  1712       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
```
```  1713       thus "content b *\<^sub>R f a = 0" by auto
```
```  1714     qed auto
```
```  1715     also have "\<dots> < e" by(rule k d(2) p12 fine_union p1 p2)+
```
```  1716     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
```
```  1717
```
```  1718 lemma integrable_split[intro]: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
```
```  1719   assumes "f integrable_on {a..b}" and k:"k<DIM('a)"
```
```  1720   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)
```
```  1721 proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
```
```  1722   def b' \<equiv> "(\<chi>\<chi> i. if i = k then min (b\$\$k) c else b\$\$i)::'a"
```
```  1723   and a' \<equiv> "(\<chi>\<chi> i. if i = k then max (a\$\$k) c else a\$\$i)::'a"
```
```  1724   show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
```
```  1725   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
```
```  1726     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
```
```  1727     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
```
```  1728       \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
```
```  1729       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
```
```  1730     show "?P {x. x \$\$ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
```
```  1731     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \$\$ k \<le> c} \<and> d fine p1
```
```  1732         \<and> p2 tagged_division_of {a..b} \<inter> {x. x \$\$ k \<le> c} \<and> d fine p2"
```
```  1733       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"
```
```  1734       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
```
```  1735         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
```
```  1736           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  1737           using p using assms by(auto simp add:algebra_simps)
```
```  1738       qed qed
```
```  1739     show "?P {x. x \$\$ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
```
```  1740     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x \$\$ k \<ge> c} \<and> d fine p1
```
```  1741         \<and> p2 tagged_division_of {a..b} \<inter> {x. x \$\$ k \<ge> c} \<and> d fine p2"
```
```  1742       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"
```
```  1743       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
```
```  1744         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
```
```  1745           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  1746           using p using assms by(auto simp add:algebra_simps) qed qed qed qed
```
```  1747
```
```  1748 subsection {* Generalized notion of additivity. *}
```
```  1749
```
```  1750 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
```
```  1751
```
```  1752 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```  1753   "operative opp f \<equiv>
```
```  1754     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
```
```  1755     (\<forall>a b c. \<forall>k<DIM('b). f({a..b}) =
```
```  1756                    opp (f({a..b} \<inter> {x. x\$\$k \<le> c}))
```
```  1757                        (f({a..b} \<inter> {x. x\$\$k \<ge> c})))"
```
```  1758
```
```  1759 lemma operativeD[dest]: fixes type::"'a::ordered_euclidean_space"  assumes "operative opp f"
```
```  1760   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b::'a} = neutral(opp)"
```
```  1761   "\<And>a b c k. k<DIM('a) \<Longrightarrow> f({a..b}) = opp (f({a..b} \<inter> {x. x\$\$k \<le> c})) (f({a..b} \<inter> {x. x\$\$k \<ge> c}))"
```
```  1762   using assms unfolding operative_def by auto
```
```  1763
```
```  1764 lemma operative_trivial:
```
```  1765  "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
```
```  1766   unfolding operative_def by auto
```
```  1767
```
```  1768 lemma property_empty_interval:
```
```  1769  "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}"
```
```  1770   using content_empty unfolding empty_as_interval by auto
```
```  1771
```
```  1772 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
```
```  1773   unfolding operative_def apply(rule property_empty_interval) by auto
```
```  1774
```
```  1775 subsection {* Using additivity of lifted function to encode definedness. *}
```
```  1776
```
```  1777 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
```
```  1778   by (metis option.nchotomy)
```
```  1779
```
```  1780 lemma exists_option:
```
```  1781  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))"
```
```  1782   by (metis option.nchotomy)
```
```  1783
```
```  1784 fun lifted where
```
```  1785   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
```
```  1786   "lifted opp None _ = (None::'b option)" |
```
```  1787   "lifted opp _ None = None"
```
```  1788
```
```  1789 lemma lifted_simp_1[simp]: "lifted opp v None = None"
```
```  1790   apply(induct v) by auto
```
```  1791
```
```  1792 definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
```
```  1793                    (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
```
```  1794                    (\<forall>x. opp (neutral opp) x = x)"
```
```  1795
```
```  1796 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
```
```  1797   "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
```
```  1798   "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
```
```  1799   unfolding monoidal_def using assms by fastsimp
```
```  1800
```
```  1801 lemma monoidal_ac: assumes "monoidal opp"
```
```  1802   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
```
```  1803   "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
```
```  1804   using assms unfolding monoidal_def apply- by metis+
```
```  1805
```
```  1806 lemma monoidal_simps[simp]: assumes "monoidal opp"
```
```  1807   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
```
```  1808   using monoidal_ac[OF assms] by auto
```
```  1809
```
```  1810 lemma neutral_lifted[cong]: assumes "monoidal opp"
```
```  1811   shows "neutral (lifted opp) = Some(neutral opp)"
```
```  1812   apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
```
```  1813 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
```
```  1814   thus "x = Some (neutral opp)" apply(induct x) defer
```
```  1815     apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
```
```  1816     apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
```
```  1817 qed(auto simp add:monoidal_ac[OF assms])
```
```  1818
```
```  1819 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
```
```  1820   unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
```
```  1821
```
```  1822 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
```
```  1823 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
```
```  1824 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
```
```  1825
```
```  1826 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
```
```  1827 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
```
```  1828
```
```  1829 lemma comp_fun_commute_monoidal[intro]: assumes "monoidal opp" shows "comp_fun_commute opp"
```
```  1830   unfolding comp_fun_commute_def using monoidal_ac[OF assms] by auto
```
```  1831
```
```  1832 lemma support_clauses:
```
```  1833   "\<And>f g s. support opp f {} = {}"
```
```  1834   "\<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))"
```
```  1835   "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
```
```  1836   "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
```
```  1837   "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
```
```  1838   "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
```
```  1839   "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
```
```  1840 unfolding support_def by auto
```
```  1841
```
```  1842 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
```
```  1843   unfolding support_def by auto
```
```  1844
```
```  1845 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
```
```  1846   unfolding iterate_def fold'_def by auto
```
```  1847
```
```  1848 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
```
```  1849   shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))"
```
```  1850 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
```
```  1851   show ?thesis unfolding iterate_def if_P[OF True] * by auto
```
```  1852 next case False note x=this
```
```  1853   note * = comp_fun_commute.comp_comp_fun_commute [OF comp_fun_commute_monoidal[OF assms(1)]]
```
```  1854   show ?thesis proof(cases "f x = neutral opp")
```
```  1855     case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
```
```  1856       unfolding True monoidal_simps[OF assms(1)] by auto
```
```  1857   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
```
```  1858       apply(subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
```
```  1859       using `finite s` unfolding support_def using False x by auto qed qed
```
```  1860
```
```  1861 lemma iterate_some:
```
```  1862   assumes "monoidal opp"  "finite s"
```
```  1863   shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
```
```  1864 proof(induct s) case empty thus ?case using assms by auto
```
```  1865 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
```
```  1866     defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
```
```  1867 subsection {* Two key instances of additivity. *}
```
```  1868
```
```  1869 lemma neutral_add[simp]:
```
```  1870   "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def
```
```  1871   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
```
```  1872
```
```  1873 lemma operative_content[intro]: "operative (op +) content"
```
```  1874   unfolding operative_def neutral_add apply safe
```
```  1875   unfolding content_split[THEN sym] ..
```
```  1876
```
```  1877 lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
```
```  1878   by (rule neutral_add) (* FIXME: duplicate *)
```
```  1879
```
```  1880 lemma monoidal_monoid[intro]:
```
```  1881   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
```
```  1882   unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps)
```
```  1883
```
```  1884 lemma operative_integral: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  1885   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
```
```  1886   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
```
```  1887   apply(rule,rule,rule,rule) defer apply(rule allI impI)+
```
```  1888 proof- fix a b c k assume k:"k<DIM('a)" show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
```
```  1889     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)
```
```  1890     (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)"
```
```  1891   proof(cases "f integrable_on {a..b}")
```
```  1892     case True show ?thesis unfolding if_P[OF True] using k apply-
```
```  1893       unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
```
```  1894       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k])
```
```  1895       apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
```
```  1896   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}))"
```
```  1897     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
```
```  1898         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)
```
```  1899         apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
```
```  1900       thus False using False by auto
```
```  1901     qed thus ?thesis using False by auto
```
```  1902   qed next
```
```  1903   fix a b assume as:"content {a..b::'a} = 0"
```
```  1904   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
```
```  1905     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
```
```  1906
```
```  1907 subsection {* Points of division of a partition. *}
```
```  1908
```
```  1909 definition "division_points (k::('a::ordered_euclidean_space) set) d =
```
```  1910     {(j,x). j<DIM('a) \<and> (interval_lowerbound k)\$\$j < x \<and> x < (interval_upperbound k)\$\$j \<and>
```
```  1911            (\<exists>i\<in>d. (interval_lowerbound i)\$\$j = x \<or> (interval_upperbound i)\$\$j = x)}"
```
```  1912
```
```  1913 lemma division_points_finite: fixes i::"('a::ordered_euclidean_space) set"
```
```  1914   assumes "d division_of i" shows "finite (division_points i d)"
```
```  1915 proof- note assm = division_ofD[OF assms]
```
```  1916   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\$\$j < x \<and> x < (interval_upperbound i)\$\$j \<and>
```
```  1917            (\<exists>i\<in>d. (interval_lowerbound i)\$\$j = x \<or> (interval_upperbound i)\$\$j = x)}"
```
```  1918   have *:"division_points i d = \<Union>(?M ` {..<DIM('a)})"
```
```  1919     unfolding division_points_def by auto
```
```  1920   show ?thesis unfolding * using assm by auto qed
```
```  1921
```
```  1922 lemma division_points_subset: fixes a::"'a::ordered_euclidean_space"
```
```  1923   assumes "d division_of {a..b}" "\<forall>i<DIM('a). a\$\$i < b\$\$i"  "a\$\$k < c" "c < b\$\$k" and k:"k<DIM('a)"
```
```  1924   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} = {})}
```
```  1925                   \<subseteq> division_points ({a..b}) d" (is ?t1) and
```
```  1926         "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} = {})}
```
```  1927                   \<subseteq> division_points ({a..b}) d" (is ?t2)
```
```  1928 proof- note assm = division_ofD[OF assms(1)]
```
```  1929   have *:"\<forall>i<DIM('a). a\$\$i \<le> b\$\$i"   "\<forall>i<DIM('a). a\$\$i \<le> ((\<chi>\<chi> i. if i = k then min (b \$\$ k) c else b \$\$ i)::'a) \$\$ i"
```
```  1930     "\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (a \$\$ k) c else a \$\$ i)::'a) \$\$ i \<le> b\$\$i"  "min (b \$\$ k) c = c" "max (a \$\$ k) c = c"
```
```  1931     using assms using less_imp_le by auto
```
```  1932   show ?t1 unfolding division_points_def interval_split[OF k, of a b]
```
```  1933     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
```
```  1934     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
```
```  1935     unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta'
```
```  1936   proof- fix i l x assume as:"a \$\$ fst x < snd x" "snd x < (if fst x = k then c else b \$\$ fst x)"
```
```  1937       "interval_lowerbound i \$\$ fst x = snd x \<or> interval_upperbound i \$\$ fst x = snd x"
```
```  1938       "i = l \<inter> {x. x \$\$ k \<le> c}" "l \<in> d" "l \<inter> {x. x \$\$ k \<le> c} \<noteq> {}" and fstx:"fst x <DIM('a)"
```
```  1939     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1940     have *:"\<forall>i<DIM('a). u \$\$ i \<le> ((\<chi>\<chi> i. if i = k then min (v \$\$ k) c else v \$\$ i)::'a) \$\$ i"
```
```  1941       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
```
```  1942     have **:"\<forall>i<DIM('a). u\$\$i \<le> v\$\$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
```
```  1943     show "fst x <DIM('a) \<and> a \$\$ fst x < snd x \<and> snd x < b \$\$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i \$\$ fst x = snd x
```
```  1944       \<or> interval_upperbound i \$\$ fst x = snd x)" apply(rule,rule fstx)
```
```  1945       using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
```
```  1946       apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
```
```  1947       apply(case_tac[!] "fst x = k") using assms fstx apply- unfolding euclidean_lambda_beta by auto
```
```  1948   qed
```
```  1949   show ?t2 unfolding division_points_def interval_split[OF k, of a b]
```
```  1950     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
```
```  1951     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
```
```  1952     unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta' apply(rule,assumption)
```
```  1953   proof- fix i l x assume as:"(if fst x = k then c else a \$\$ fst x) < snd x" "snd x < b \$\$ fst x"
```
```  1954       "interval_lowerbound i \$\$ fst x = snd x \<or> interval_upperbound i \$\$ fst x = snd x"
```
```  1955       "i = l \<inter> {x. c \<le> x \$\$ k}" "l \<in> d" "l \<inter> {x. c \<le> x \$\$ k} \<noteq> {}" and fstx:"fst x < DIM('a)"
```
```  1956     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1957     have *:"\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (u \$\$ k) c else u \$\$ i)::'a) \$\$ i \<le> v \$\$ i"
```
```  1958       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
```
```  1959     have **:"\<forall>i<DIM('a). u\$\$i \<le> v\$\$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
```
```  1960     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>
```
```  1961       interval_upperbound i \$\$ fst x = snd x)"
```
```  1962       using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
```
```  1963       apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
```
```  1964       apply(case_tac[!] "fst x = k") using assms fstx apply-  by(auto simp add:euclidean_lambda_beta'[OF k]) qed qed
```
```  1965
```
```  1966 lemma division_points_psubset: fixes a::"'a::ordered_euclidean_space"
```
```  1967   assumes "d division_of {a..b}"  "\<forall>i<DIM('a). a\$\$i < b\$\$i"  "a\$\$k < c" "c < b\$\$k"
```
```  1968   "l \<in> d" "interval_lowerbound l\$\$k = c \<or> interval_upperbound l\$\$k = c" and k:"k<DIM('a)"
```
```  1969   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> {}}
```
```  1970               \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D")
```
```  1971         "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> {}}
```
```  1972               \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D")
```
```  1973 proof- have ab:"\<forall>i<DIM('a). a\$\$i \<le> b\$\$i" using assms(2) by(auto intro!:less_imp_le)
```
```  1974   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
```
```  1975   have uv:"\<forall>i<DIM('a). u\$\$i \<le> v\$\$i" "\<forall>i<DIM('a). a\$\$i \<le> u\$\$i \<and> v\$\$i \<le> b\$\$i"
```
```  1976     using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
```
```  1977     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
```
```  1978   have *:"interval_upperbound ({a..b} \<inter> {x. x \$\$ k \<le> interval_upperbound l \$\$ k}) \$\$ k = interval_upperbound l \$\$ k"
```
```  1979          "interval_upperbound ({a..b} \<inter> {x. x \$\$ k \<le> interval_lowerbound l \$\$ k}) \$\$ k = interval_lowerbound l \$\$ k"
```
```  1980     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
```
```  1981     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
```
```  1982   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
```
```  1983     apply(rule_tac x="(k,(interval_lowerbound l)\$\$k)" in exI) defer
```
```  1984     apply(rule_tac x="(k,(interval_upperbound l)\$\$k)" in exI)
```
```  1985     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
```
```  1986   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
```
```  1987
```
```  1988   have *:"interval_lowerbound ({a..b} \<inter> {x. x \$\$ k \<ge> interval_lowerbound l \$\$ k}) \$\$ k = interval_lowerbound l \$\$ k"
```
```  1989          "interval_lowerbound ({a..b} \<inter> {x. x \$\$ k \<ge> interval_upperbound l \$\$ k}) \$\$ k = interval_upperbound l \$\$ k"
```
```  1990     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
```
```  1991     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
```
```  1992   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
```
```  1993     apply(rule_tac x="(k,(interval_lowerbound l)\$\$k)" in exI) defer
```
```  1994     apply(rule_tac x="(k,(interval_upperbound l)\$\$k)" in exI)
```
```  1995     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*)
```
```  1996   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4) k]) by auto qed
```
```  1997
```
```  1998 subsection {* Preservation by divisions and tagged divisions. *}
```
```  1999
```
```  2000 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
```
```  2001   unfolding support_def by auto
```
```  2002
```
```  2003 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
```
```  2004   unfolding iterate_def support_support by auto
```
```  2005
```
```  2006 lemma iterate_expand_cases:
```
```  2007   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
```
```  2008   apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto
```
```  2009
```
```  2010 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
```
```  2011   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
```
```  2012 proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
```
```  2013      iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
```
```  2014   proof- case goal1 show ?case using goal1
```
```  2015     proof(induct s) case empty thus ?case using assms(1) by auto
```
```  2016     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
```
```  2017         unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
```
```  2018         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
```
```  2019         apply(rule finite_imageI insert)+ apply(subst if_not_P)
```
```  2020         unfolding image_iff o_def using insert(2,4) by auto
```
```  2021     qed qed
```
```  2022   show ?thesis
```
```  2023     apply(cases "finite (support opp g (f ` s))")
```
```  2024     apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
```
```  2025     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
```
```  2026     apply(rule subset_inj_on[OF assms(2) support_subset])+
```
```  2027     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
```
```  2028     apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
```
```  2029
```
```  2030
```
```  2031 (* This lemma about iterations comes up in a few places.                     *)
```
```  2032 lemma iterate_nonzero_image_lemma:
```
```  2033   assumes "monoidal opp" "finite s" "g(a) = neutral opp"
```
```  2034   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
```
```  2035   shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
```
```  2036 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
```
```  2037   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
```
```  2038     unfolding support_def using assms(3) by auto
```
```  2039   show ?thesis unfolding *
```
```  2040     apply(subst iterate_support[THEN sym]) unfolding support_clauses
```
```  2041     apply(subst iterate_image[OF assms(1)]) defer
```
```  2042     apply(subst(2) iterate_support[THEN sym]) apply(subst **)
```
```  2043     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
```
```  2044
```
```  2045 lemma iterate_eq_neutral:
```
```  2046   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
```
```  2047   shows "(iterate opp s f = neutral opp)"
```
```  2048 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
```
```  2049   show ?thesis apply(subst iterate_support[THEN sym])
```
```  2050     unfolding * using assms(1) by auto qed
```
```  2051
```
```  2052 lemma iterate_op: assumes "monoidal opp" "finite s"
```
```  2053   shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
```
```  2054 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
```
```  2055 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
```
```  2056     unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
```
```  2057
```
```  2058 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  2059   shows "iterate opp s f = iterate opp s g"
```
```  2060 proof- have *:"support opp g s = support opp f s"
```
```  2061     unfolding support_def using assms(2) by auto
```
```  2062   show ?thesis
```
```  2063   proof(cases "finite (support opp f s)")
```
```  2064     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
```
```  2065       unfolding * by auto
```
```  2066   next def su \<equiv> "support opp f s"
```
```  2067     case True note support_subset[of opp f s]
```
```  2068     thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
```
```  2069       unfolding su_def[symmetric]
```
```  2070     proof(induct su) case empty show ?case by auto
```
```  2071     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
```
```  2072         unfolding if_not_P[OF insert(2)] apply(subst insert(3))
```
```  2073         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
```
```  2074
```
```  2075 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
```
```  2076
```
```  2077 lemma operative_division: fixes f::"('a::ordered_euclidean_space) set \<Rightarrow> 'b"
```
```  2078   assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
```
```  2079   shows "iterate opp d f = f {a..b}"
```
```  2080 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
```
```  2081   proof(induct C arbitrary:a b d rule:full_nat_induct)
```
```  2082     case goal1
```
```  2083     { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
```
```  2084       thus ?case apply-apply(cases) defer apply assumption
```
```  2085       proof- assume as:"content {a..b} = 0"
```
```  2086         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
```
```  2087         proof fix x assume x:"x\<in>d"
```
```  2088           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
```
```  2089           thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)]
```
```  2090             using operativeD(1)[OF assms(2)] x by auto
```
```  2091         qed qed }
```
```  2092     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
```
```  2093     hence ab':"\<forall>i<DIM('a). a\$\$i \<le> b\$\$i" by (auto intro!: less_imp_le) show ?case
```
```  2094     proof(cases "division_points {a..b} d = {}")
```
```  2095       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
```
```  2096         (\<forall>j<DIM('a). u\$\$j = a\$\$j \<and> v\$\$j = a\$\$j \<or> u\$\$j = b\$\$j \<and> v\$\$j = b\$\$j \<or> u\$\$j = a\$\$j \<and> v\$\$j = b\$\$j)"
```
```  2097         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
```
```  2098         apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule,rule)
```
```  2099       proof- fix u v j assume j:"j<DIM('a)" assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
```
```  2100         hence uv:"\<forall>i<DIM('a). u\$\$i \<le> v\$\$i" "u\$\$j \<le> v\$\$j" using j unfolding interval_ne_empty by auto
```
```  2101         have *:"\<And>p r Q. \<not> j<DIM('a) \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as j by auto
```
```  2102         have "(j, u\$\$j) \<notin> division_points {a..b} d"
```
```  2103           "(j, v\$\$j) \<notin> division_points {a..b} d" using True by auto
```
```  2104         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
```
```  2105         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
```
```  2106         moreover have "a\$\$j \<le> u\$\$j" "v\$\$j \<le> b\$\$j" using division_ofD(2,2,3)[OF goal1(4) as]
```
```  2107           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
```
```  2108           unfolding interval_ne_empty mem_interval using j by auto
```
```  2109         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"
```
```  2110           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by auto
```
```  2111       qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
```
```  2112       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
```
```  2113       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
```
```  2114       have "{a..b} \<in> d"
```
```  2115       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
```
```  2116         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
```
```  2117         show "u = a" "v = b" unfolding euclidean_eq[where 'a='a]
```
```  2118         proof(safe) fix j assume j:"j<DIM('a)" note i(2)[unfolded uv mem_interval,rule_format,of j]
```
```  2119           thus "u \$\$ j = a \$\$ j" "v \$\$ j = b \$\$ j" using uv(2)[rule_format,of j] j by auto
```
```  2120         qed qed
```
```  2121       hence *:"d = insert {a..b} (d - {{a..b}})" by auto
```
```  2122       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
```
```  2123       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
```
```  2124         then guess u v apply-by(erule exE conjE)+ note uv=this
```
```  2125         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto
```
```  2126         then obtain j where "u\$\$j \<noteq> a\$\$j \<or> v\$\$j \<noteq> b\$\$j" and j:"j<DIM('a)" unfolding euclidean_eq[where 'a='a] by auto
```
```  2127         hence "u\$\$j = v\$\$j" using uv(2)[rule_format,OF j] by auto
```
```  2128         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) using j by auto
```
```  2129         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
```
```  2130       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *)
```
```  2131         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
```
```  2132     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
```
```  2133       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
```
```  2134         by(erule exE conjE)+ note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
```
```  2135       from this(3) guess j .. note j=this
```
```  2136       def d1 \<equiv> "{l \<inter> {x. x\$\$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\$\$k \<le> c} \<noteq> {}}"
```
```  2137       def d2 \<equiv> "{l \<inter> {x. x\$\$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\$\$k \<ge> c} \<noteq> {}}"
```
```  2138       def cb \<equiv> "(\<chi>\<chi> i. if i = k then c else b\$\$i)::'a" and ca \<equiv> "(\<chi>\<chi> i. if i = k then c else a\$\$i)::'a"
```
```  2139       note division_points_psubset[OF goal1(4) ab kc(1-2) j]
```
```  2140       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
```
```  2141       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})"
```
```  2142         apply- unfolding interval_split[OF kc(4)] apply(rule_tac[!] goal1(1)[rule_format])
```
```  2143         using division_split[OF goal1(4), where k=k and c=c]
```
```  2144         unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
```
```  2145         using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
```
```  2146       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
```
```  2147         unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto
```
```  2148       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\$\$k \<le> c}))"
```
```  2149         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
```
```  2150         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
```
```  2151         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
```
```  2152       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"
```
```  2153         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
```
```  2154         show "f (l \<inter> {x. x \$\$ k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
```
```  2155           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
```
```  2156           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
```
```  2157       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x\$\$k \<ge> c}))"
```
```  2158         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
```
```  2159         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
```
```  2160         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
```
```  2161       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"
```
```  2162         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
```
```  2163         show "f (l \<inter> {x. x \$\$ k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)]
```
```  2164           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
```
```  2165           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
```
```  2166       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}))"
```
```  2167         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto
```
```  2168       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})))
```
```  2169         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
```
```  2170         apply(rule iterate_op[THEN sym]) using goal1 by auto
```
```  2171       finally show ?thesis by auto
```
```  2172     qed qed qed
```
```  2173
```
```  2174 lemma iterate_image_nonzero: assumes "monoidal opp"
```
```  2175   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
```
```  2176   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
```
```  2177 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
```
```  2178   case goal1 show ?case using assms(1) by auto
```
```  2179 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
```
```  2180   show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
```
```  2181     apply(rule finite_imageI goal2)+
```
```  2182     apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
```
```  2183     apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
```
```  2184     apply(subst iterate_insert[OF assms(1) goal2(1)])
```
```  2185     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
```
```  2186     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
```
```  2187     using goal2 unfolding o_def by auto qed
```
```  2188
```
```  2189 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
```
```  2190   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
```
```  2191 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
```
```  2192   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
```
```  2193     apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+
```
```  2194     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
```
```  2195   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
```
```  2196     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
```
```  2197     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
```
```  2198       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
```
```  2199       unfolding as(4)[THEN sym] uv by auto
```
```  2200   qed also have "\<dots> = f {a..b}"
```
```  2201     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
```
```  2202   finally show ?thesis . qed
```
```  2203
```
```  2204 subsection {* Additivity of content. *}
```
```  2205
```
```  2206 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
```
```  2207 proof- have *:"setsum f s = setsum f (support op + f s)"
```
```  2208     apply(rule setsum_mono_zero_right)
```
```  2209     unfolding support_def neutral_monoid using assms by auto
```
```  2210   thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
```
```  2211     unfolding neutral_monoid . qed
```
```  2212
```
```  2213 lemma additive_content_division: assumes "d division_of {a..b}"
```
```  2214   shows "setsum content d = content({a..b})"
```
```  2215   unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
```
```  2216   apply(subst setsum_iterate) using assms by auto
```
```  2217
```
```  2218 lemma additive_content_tagged_division:
```
```  2219   assumes "d tagged_division_of {a..b}"
```
```  2220   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
```
```  2221   unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
```
```  2222   apply(subst setsum_iterate) using assms by auto
```
```  2223
```
```  2224 subsection {* Finally, the integral of a constant *}
```
```  2225
```
```  2226 lemma has_integral_const[intro]:
```
```  2227   "((\<lambda>x. c) has_integral (content({a..b::'a::ordered_euclidean_space}) *\<^sub>R c)) ({a..b})"
```
```  2228   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
```
```  2229   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
```
```  2230   unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
```
```  2231   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
```
```  2232
```
```  2233 subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
```
```  2234
```
```  2235 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
```
```  2236   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
```
```  2237   apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
```
```  2238   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
```
```  2239   apply(subst mult_commute) apply(rule mult_left_mono)
```
```  2240   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
```
```  2241   apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
```
```  2242 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
```
```  2243   fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
```
```  2244   thus "0 \<le> content x" using content_pos_le by auto
```
```  2245 qed(insert assms,auto)
```
```  2246
```
```  2247 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
```
```  2248   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
```
```  2249 proof(cases "{a..b} = {}") case True
```
```  2250   show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
```
```  2251 next case False show ?thesis
```
```  2252     apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
```
```  2253     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
```
```  2254     unfolding setsum_left_distrib[THEN sym] apply(subst mult_commute) apply(rule mult_left_mono)
```
```  2255     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
```
```  2256     apply(subst o_def, rule abs_of_nonneg)
```
```  2257   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
```
```  2258       unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
```
```  2259     guess w using nonempty_witness[OF False] .
```
```  2260     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
```
```  2261     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
```
```  2262     from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
```
```  2263     show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
```
```  2264     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
```
```  2265   qed(insert assms,auto) qed
```
```  2266
```
```  2267 lemma rsum_diff_bound:
```
```  2268   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
```
```  2269   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})"
```
```  2270   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
```
```  2271   unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
```
```  2272
```
```  2273 lemma has_integral_bound: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  2274   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
```
```  2275   shows "norm i \<le> B * content {a..b}"
```
```  2276 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
```
```  2277     thus ?thesis proof(cases ?P) case False
```
```  2278       hence *:"content {a..b} = 0" using content_lt_nz by auto
```
```  2279       hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
```
```  2280       show ?thesis unfolding * ** using assms(1) by auto
```
```  2281     qed auto } assume ab:?P
```
```  2282   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
```
```  2283   assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
```
```  2284   from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
```
```  2285   from fine_division_exists[OF this(1), of a b] guess p . note p=this
```
```  2286   have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
```
```  2287   proof- case goal1 thus ?case unfolding not_less
```
```  2288     using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
```
```  2289   qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
```
```  2290
```
```  2291 subsection {* Similar theorems about relationship among components. *}
```
```  2292
```
```  2293 lemma rsum_component_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2294   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)\$\$i \<le> (g x)\$\$i"
```
```  2295   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"
```
```  2296   unfolding  euclidean_component.setsum apply(rule setsum_mono) apply safe
```
```  2297 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
```
```  2298   from this(3) guess u v apply-by(erule exE)+ note b=this
```
```  2299   show "(content b *\<^sub>R f a) \$\$ i \<le> (content b *\<^sub>R g a) \$\$ i" unfolding b
```
```  2300     unfolding euclidean_simps real_scaleR_def apply(rule mult_left_mono)
```
```  2301     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
```
```  2302
```
```  2303 lemma has_integral_component_le: fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2304   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)\$\$k \<le> (g x)\$\$k"
```
```  2305   shows "i\$\$k \<le> j\$\$k"
```
```  2306 proof- have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow>
```
```  2307     (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"
```
```  2308   proof(rule ccontr) case goal1 hence *:"0 < (i\$\$k - j\$\$k) / 3" by auto
```
```  2309     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
```
```  2310     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
```
```  2311     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
```
```  2312     note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k] term g
```
```  2313     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
```
```  2314     thus False unfolding euclidean_simps using rsum_component_le[OF p(1) goal1(3)] apply simp by smt
```
```  2315   qed let ?P = "\<exists>a b. s = {a..b}"
```
```  2316   { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
```
```  2317       case True then guess a b apply-by(erule exE)+ note s=this
```
```  2318       show ?thesis apply(rule lem) using assms[unfolded s] by auto
```
```  2319     qed auto } assume as:"\<not> ?P"
```
```  2320   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
```
```  2321   assume "\<not> i\$\$k \<le> j\$\$k" hence ij:"(i\$\$k - j\$\$k) / 3 > 0" by auto
```
```  2322   note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
```
```  2323   have "bounded (ball 0 B1 \<union> ball (0::'a) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
```
```  2324   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
```
```  2325   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  2326   guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
```
```  2327   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
```
```  2328   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
```
```  2329   note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
```
```  2330   have "w1\$\$k \<le> w2\$\$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
```
```  2331   show False unfolding euclidean_simps by(rule *) qed
```
```  2332
```
```  2333 lemma integral_component_le: fixes g f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2334   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)\$\$k \<le> (g x)\$\$k"
```
```  2335   shows "(integral s f)\$\$k \<le> (integral s g)\$\$k"
```
```  2336   apply(rule has_integral_component_le) using integrable_integral assms by auto
```
```  2337
```
```  2338 (*lemma has_integral_dest_vec1_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> real^1"
```
```  2339   assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
```
```  2340   shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
```
```  2341   using assms(3) unfolding vector_le_def by auto
```
```  2342
```
```  2343 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2344   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
```
```  2345   shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
```
```  2346   apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto*)
```
```  2347
```
```  2348 lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2349   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)\$\$k" shows "0 \<le> i\$\$k"
```
```  2350   using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2-) by auto
```
```  2351
```
```  2352 lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  2353   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)\$\$k" shows "0 \<le> (integral s f)\$\$k"
```
```  2354   apply(rule has_integral_component_nonneg) using assms by auto
```
```  2355
```
```  2356 (*lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2357   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
```
```  2358   using has_integral_component_nonneg[OF assms(1), of 1]
```
```  2359   using assms(2) unfolding vector_le_def by auto
```
```  2360
```
```  2361 lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2362   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
```
```  2363   apply(rule has_integral_dest_vec1_nonneg) using assms by auto*)
```
```  2364
```
```  2365 lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
```
```  2366   assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)\$\$k \<le> 0"shows "i\$\$k \<le> 0"
```
```  2367   using has_integral_component_le[OF assms(1) has_integral_0] assms(2-) by auto
```
```  2368
```
```  2369 (*lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
```
```  2370   assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
```
```  2371   using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto*)
```
```  2372
```
```  2373 lemma has_integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
```
```  2374   assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)\$\$k" "k<DIM('b)" shows "B * content {a..b} \<le> i\$\$k"
```
```  2375   using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi>\<chi> i. B)::'b" k] assms(2-)
```
```  2376   unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] by(auto simp add:field_simps)
```
```  2377
```
```  2378 lemma has_integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
```
```  2379   assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x\$\$k \<le> B" "k<DIM('b)"
```
```  2380   shows "i\$\$k \<le> B * content({a..b})"
```
```  2381   using has_integral_component_le[OF assms(1) has_integral_const, of k "\<chi>\<chi> i. B"]
```
```  2382   unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] using assms(2) by(auto simp add:field_simps)
```
```  2383
```
```  2384 lemma integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
```
```  2385   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)\$\$k" "k<DIM('b)"
```
```  2386   shows "B * content({a..b}) \<le> (integral({a..b}) f)\$\$k"
```
```  2387   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
```
```  2388
```
```  2389 lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
```
```  2390   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)\$\$k \<le> B" "k<DIM('b)"
```
```  2391   shows "(integral({a..b}) f)\$\$k \<le> B * content({a..b})"
```
```  2392   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
```
```  2393
```
```  2394 subsection {* Uniform limit of integrable functions is integrable. *}
```
```  2395
```
```  2396 lemma integrable_uniform_limit: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  2397   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
```
```  2398   shows "f integrable_on {a..b}"
```
```  2399 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
```
```  2400     show ?thesis apply cases apply(rule *,assumption)
```
```  2401       unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
```
```  2402   assume as:"content {a..b} > 0"
```
```  2403   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
```
```  2404   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
```
```  2405   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
```
```  2406
```
```  2407   have "Cauchy i" unfolding Cauchy_def
```
```  2408   proof(rule,rule) fix e::real assume "e>0"
```
```  2409     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
```
```  2410     then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
```
```  2411     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)
```
```  2412     proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
```
```  2413       from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
```
```  2414       from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
```
```  2415       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
```
```  2416       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"
```
```  2417       proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
```
```  2418           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
```
```  2419           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps)
```
```  2420         also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
```
```  2421         finally show ?case .
```
```  2422       qed
```
```  2423       show ?case unfolding dist_norm apply(rule lem2) defer
```
```  2424         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
```
```  2425         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
```
```  2426         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
```
```  2427       proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse
```
```  2428           using M as by(auto simp add:field_simps)
```
```  2429         fix x assume x:"x \<in> {a..b}"
```
```  2430         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
```
```  2431             using g(1)[OF x, of n] g(1)[OF x, of m] by auto
```
```  2432         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
```
```  2433           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
```
```  2434         also have "\<dots> = 2 / real M" unfolding divide_inverse by auto
```
```  2435         finally show "norm (g n x - g m x) \<le> 2 / real M"
```
```  2436           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
```
```  2437           by(auto simp add:algebra_simps simp add:norm_minus_commute)
```
```  2438       qed qed qed
```
```  2439   from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
```
```  2440
```
```  2441   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
```
```  2442   proof(rule,rule)
```
```  2443     case goal1 hence *:"e/3 > 0" by auto
```
```  2444     from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
```
```  2445     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
```
```  2446     from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
```
```  2447     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
```
```  2448     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"
```
```  2449     proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
```
```  2450         using norm_triangle_ineq[of "sf - sg" "sg - s"]
```
```  2451         using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:algebra_simps)
```
```  2452       also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
```
```  2453       finally show ?case .
```
```  2454     qed
```
```  2455     show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
```
```  2456     proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
```
```  2457       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
```
```  2458         apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
```
```  2459       proof- have "content {a..b} < e / 3 * (real N2)"
```
```  2460           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
```
```  2461         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
```
```  2462           apply-apply(rule less_le_trans,assumption) using `e>0` by auto
```
```  2463         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
```
```  2464           unfolding inverse_eq_divide by(auto simp add:field_simps)
```
```  2465         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded dist_norm],auto)
```
```  2466       qed qed qed qed
```
```  2467
```
```  2468 subsection {* Negligible sets. *}
```
```  2469
```
```  2470 definition "negligible (s::('a::ordered_euclidean_space) set) \<equiv> (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) {a..b})"
```
```  2471
```
```  2472 subsection {* Negligibility of hyperplane. *}
```
```  2473
```
```  2474 lemma vsum_nonzero_image_lemma:
```
```  2475   assumes "finite s" "g(a) = 0"
```
```  2476   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
```
```  2477   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
```
```  2478   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
```
```  2479   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
```
```  2480   unfolding assms using neutral_add unfolding neutral_add using assms by auto
```
```  2481
```
```  2482 lemma interval_doublesplit:  fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)"
```
```  2483   shows "{a..b} \<inter> {x . abs(x\$\$k - c) \<le> (e::real)} =
```
```  2484   {(\<chi>\<chi> i. if i = k then max (a\$\$k) (c - e) else a\$\$i) .. (\<chi>\<chi> i. if i = k then min (b\$\$k) (c + e) else b\$\$i)}"
```
```  2485 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
```
```  2486   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
```
```  2487   show ?thesis unfolding * ** interval_split[OF assms] by(rule refl) qed
```
```  2488
```
```  2489 lemma division_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "p division_of {a..b}" and k:"k<DIM('a)"
```
```  2490   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})"
```
```  2491 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
```
```  2492   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
```
```  2493   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
```
```  2494   note division_split(2)[OF this, where c="c-e" and k=k,OF k]
```
```  2495   thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
```
```  2496     apply(rule set_eqI) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
```
```  2497     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x \$\$ k}" in exI) apply rule defer apply rule
```
```  2498     apply(rule_tac x=l in exI) by blast+ qed
```
```  2499
```
```  2500 lemma content_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "0 < e" and k:"k<DIM('a)"
```
```  2501   obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x\$\$k - c) \<le> d}) < e"
```
```  2502 proof(cases "content {a..b} = 0")
```
```  2503   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
```
```  2504     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
```
```  2505     unfolding interval_doublesplit[THEN sym,OF k] using assms by auto
```
```  2506 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b\$\$i - a\$\$i) ({..<DIM('a)} - {k})"
```
```  2507   note False[unfolded content_eq_0 not_ex not_le, rule_format]
```
```  2508   hence "\<And>x. x<DIM('a) \<Longrightarrow> b\$\$x > a\$\$x" by(auto simp add:not_le)
```
```  2509   hence prod0:"0 < setprod (\<lambda>i. b\$\$i - a\$\$i) ({..<DIM('a)} - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
```
```  2510   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
```
```  2511   proof(rule that[of d]) have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" using k by auto
```
```  2512     have **:"{a..b} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow>
```
```  2513       (\<Prod>i\<in>{..<DIM('a)} - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) \$\$ i
```
```  2514       - interval_lowerbound ({a..b} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) \$\$ i)
```
```  2515       = (\<Prod>i\<in>{..<DIM('a)} - {k}. b\$\$i - a\$\$i)" apply(rule setprod_cong,rule refl)
```
```  2516       unfolding interval_doublesplit[OF k] apply(subst interval_bounds) defer apply(subst interval_bounds)
```
```  2517       unfolding interval_eq_empty not_ex not_less by auto
```
```  2518     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)
```
```  2519       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
```
```  2520       unfolding interval_doublesplit[OF k] interval_eq_empty not_ex not_less prefer 3
```
```  2521       apply(subst interval_bounds) defer apply(subst interval_bounds) unfolding euclidean_lambda_beta'[OF k] if_P[OF refl]
```
```  2522     proof- have "(min (b \$\$ k) (c + d) - max (a \$\$ k) (c - d)) \<le> 2 * d" by auto
```
```  2523       also have "... < e / (\<Prod>i\<in>{..<DIM('a)} - {k}. b \$\$ i - a \$\$ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
```
```  2524       finally show "(min (b \$\$ k) (c + d) - max (a \$\$ k) (c - d)) * (\<Prod>i\<in>{..<DIM('a)} - {k}. b \$\$ i - a \$\$ i) < e"
```
```  2525         unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
```
```  2526
```
```  2527 lemma negligible_standard_hyperplane[intro]: fixes type::"'a::ordered_euclidean_space" assumes k:"k<DIM('a)"
```
```  2528   shows "negligible {x::'a. x\$\$k = (c::real)}"
```
```  2529   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
```
```  2530 proof-
```
```  2531   case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
```
```  2532   let ?i = "indicator {x::'a. x\$\$k = c} :: 'a\<Rightarrow>real"
```
```  2533   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
```
```  2534   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
```
```  2535     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)"
```
```  2536       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
```
```  2537       apply(cases,rule disjI1,assumption,rule disjI2)
```
```  2538     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)
```
```  2539       show "content l = content (l \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
```
```  2540         apply(rule set_eqI,rule,rule) unfolding mem_Collect_eq
```
```  2541       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
```
```  2542         note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
```
```  2543         thus "\<bar>y \$\$ k - c\<bar> \<le> d" unfolding euclidean_simps xk by auto
```
```  2544       qed auto qed
```
```  2545     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
```
```  2546     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
```
```  2547       apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
```
```  2548       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
```
```  2549       prefer 2 apply(subst(asm) eq_commute) apply assumption
```
```  2550       apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
```
```  2551     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}))"
```
```  2552         apply(rule setsum_mono) unfolding split_paired_all split_conv
```
```  2553         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k] intro!:content_pos_le)
```
```  2554       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
```
```  2555       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) \<le> content {u..v}"
```
```  2556           unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
```
```  2557         thus ?case unfolding goal1 unfolding interval_doublesplit[OF k] using content_pos_le by smt
```
```  2558       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"
```
```  2559           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all
```
```  2560         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)"
```
```  2561           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
```
```  2562           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
```
```  2563         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
```
```  2564         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
```
```  2565         note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
```
```  2566         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"
```
```  2567           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
```
```  2568           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
```
```  2569         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
```
```  2570           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}"
```
```  2571           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
```
```  2572           note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
```
```  2573           hence "interior ({m..n} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
```
```  2574           thus "content ({m..n} \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
```
```  2575         qed qed
```
```  2576       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \$\$ k - c\<bar> \<le> d}) * ?i x) < e" .
```
```  2577     qed qed qed
```
```  2578
```
```  2579 subsection {* A technical lemma about "refinement" of division. *}
```
```  2580
```
```  2581 lemma tagged_division_finer: fixes p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set"
```
```  2582   assumes "p tagged_division_of {a..b}" "gauge d"
```
```  2583   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"
```
```  2584 proof-
```
```  2585   let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
```
```  2586     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
```
```  2587                    (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
```
```  2588   { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
```
```  2589     presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
```
```  2590     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
```
```  2591   } fix p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set" assume as:"finite p"
```
```  2592   show "?P p" apply(rule,rule) using as proof(induct p)
```
```  2593     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
```
```  2594   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
```
```  2595     note tagged_partial_division_subset[OF insert(4) subset_insertI]
```
```  2596     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
```
```  2597     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
```
```  2598     note p = tagged_partial_division_ofD[OF insert(4)]
```
```  2599     from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
```
```  2600
```
```  2601     have "finite {k. \<exists>x. (x, k) \<in> p}"
```
```  2602       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
```
```  2603       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
```
```  2604     hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
```
```  2605       apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
```
```  2606       unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)
```
```  2607       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
```
```  2608       using insert(2) unfolding uv xk by auto
```
```  2609
```
```  2610     show ?case proof(cases "{u..v} \<subseteq> d x")
```
```  2611       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
```
```  2612         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
```
```  2613         apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int)
```
```  2614         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
```
```  2615         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
```
```  2616         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
```
```  2617     next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
```
```  2618       show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
```
```  2619         apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
```
```  2620         unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
```
```  2621         apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
```
```  2622         apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
```
```  2623     qed qed qed
```
```  2624
```
```  2625 subsection {* Hence the main theorem about negligible sets. *}
```
```  2626
```
```  2627 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
```
```  2628   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
```
```  2629 proof(induct) case (insert x s)
```
```  2630   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
```
```  2631   show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
```
```  2632
```
```  2633 lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
```
```  2634   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
```
```  2635 proof(induct) case (insert a s)
```
```  2636   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
```
```  2637   show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
```
```  2638     prefer 4 apply(subst insert(3)) unfolding add_right_cancel
```
```  2639   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
```
```  2640     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
```
```  2641   qed(insert insert, auto) qed auto
```
```  2642
```
```  2643 lemma has_integral_negligible: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2644   assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
```
```  2645   shows "(f has_integral 0) t"
```
```  2646 proof- presume P:"\<And>f::'b::ordered_euclidean_space \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
```
```  2647   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
```
```  2648   show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
```
```  2649     apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
```
```  2650   proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
```
```  2651     show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
```
```  2652   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)"
```
```  2653       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
```
```  2654       apply(rule,rule P) using assms(2) by auto
```
```  2655   qed
```
```  2656 next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  2657   show "(f has_integral 0) {a..b}" unfolding has_integral
```
```  2658   proof(safe) case goal1
```
```  2659     hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
```
```  2660       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
```
```  2661     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"]
```
```  2662     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
```
```  2663     show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
```
```  2664     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
```
```  2665       fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
```
```  2666       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  2667       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
```
```  2668       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
```
```  2669       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
```
```  2670       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)"
```
```  2671         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
```
```  2672       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
```
```  2673       have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe)
```
```  2674         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
```
```  2675       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"
```
```  2676       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
```
```  2677           apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
```
```  2678       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
```
```  2679                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {0..N+1}"
```
```  2680         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
```
```  2681         apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3
```
```  2682       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
```
```  2683         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
```
```  2684           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
```
```  2685           using tagged_division_ofD(4)[OF q(1) as''] by auto
```
```  2686       next fix i::nat show "finite (q i)" using q by auto
```
```  2687       next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
```
```  2688         have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
```
```  2689         have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
```
```  2690         hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
```
```  2691         moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
```
```  2692         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
```
```  2693         moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
```
```  2694         proof(cases "x\<in>s") case False thus ?thesis using assm by auto
```
```  2695         next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
```
```  2696           moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
```
```  2697           ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
```
```  2698         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)"
```
```  2699           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
```
```  2700       qed(insert as, auto)
```
```  2701       also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono)
```
```  2702       proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
```
```  2703           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
```
```  2704       qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[THEN sym]
```
```  2705         apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
```
```  2706         apply(subst sumr_geometric) using goal1 by auto
```
```  2707       finally show "?goal" by auto qed qed qed
```
```  2708
```
```  2709 lemma has_integral_spike: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2710   assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
```
```  2711   shows "(g has_integral y) t"
```
```  2712 proof- { fix a b::"'b" and f g ::"'b \<Rightarrow> 'a" and y::'a
```
```  2713     assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
```
```  2714     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
```
```  2715       apply(rule has_integral_negligible[OF assms(1)]) using as by auto
```
```  2716     hence "(g has_integral y) {a..b}" by auto } note * = this
```
```  2717   show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
```
```  2718     apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
```
```  2719     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)
```
```  2720     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
```
```  2721
```
```  2722 lemma has_integral_spike_eq:
```
```  2723   assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
```
```  2724   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2725   apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
```
```  2726
```
```  2727 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
```
```  2728   shows "g integrable_on  t"
```
```  2729   using assms unfolding integrable_on_def apply-apply(erule exE)
```
```  2730   apply(rule,rule has_integral_spike) by fastsimp+
```
```  2731
```
```  2732 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
```
```  2733   shows "integral t f = integral t g"
```
```  2734   unfolding integral_def using has_integral_spike_eq[OF assms] by auto
```
```  2735
```
```  2736 subsection {* Some other trivialities about negligible sets. *}
```
```  2737
```
```  2738 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def
```
```  2739 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
```
```  2740     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
```
```  2741     using assms(2) unfolding indicator_def by auto qed
```
```  2742
```
```  2743 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
```
```  2744
```
```  2745 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
```
```  2746
```
```  2747 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def
```
```  2748 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
```
```  2749   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
```
```  2750     defer apply assumption unfolding indicator_def by auto qed
```
```  2751
```
```  2752 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
```
```  2753   using negligible_union by auto
```
```  2754
```
```  2755 lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}"
```
```  2756   using negligible_standard_hyperplane[of 0 "a\$\$0"] by auto
```
```  2757
```
```  2758 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
```
```  2759   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
```
```  2760
```
```  2761 lemma negligible_empty[intro]: "negligible {}" by auto
```
```  2762
```
```  2763 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
```
```  2764   using assms apply(induct s) by auto
```
```  2765
```
```  2766 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
```
```  2767   using assms by(induct,auto)
```
```  2768
```
```  2769 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
```
```  2770   apply safe defer apply(subst negligible_def)
```
```  2771 proof- fix t::"'a set" assume as:"negligible s"
```
```  2772   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)
```
```  2773   show "((indicator s::'a\<Rightarrow>real) has_integral 0) t" apply(subst has_integral_alt)
```
```  2774     apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
```
```  2775     apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
```
```  2776     using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def_raw unfolding * by auto qed auto
```
```  2777
```
```  2778 subsection {* Finite case of the spike theorem is quite commonly needed. *}
```
```  2779
```
```  2780 lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
```
```  2781   "(f has_integral y) t" shows "(g has_integral y) t"
```
```  2782   apply(rule has_integral_spike) using assms by auto
```
```  2783
```
```  2784 lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
```
```  2785   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2786   apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
```
```  2787
```
```  2788 lemma integrable_spike_finite:
```
```  2789   assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
```
```  2790   using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
```
```  2791   apply(rule has_integral_spike_finite) by auto
```
```  2792
```
```  2793 subsection {* In particular, the boundary of an interval is negligible. *}
```
```  2794
```
```  2795 lemma negligible_frontier_interval: "negligible({a::'a::ordered_euclidean_space..b} - {a<..<b})"
```
```  2796 proof- let ?A = "\<Union>((\<lambda>k. {x. x\$\$k = a\$\$k} \<union> {x::'a. x\$\$k = b\$\$k}) ` {..<DIM('a)})"
```
```  2797   have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
```
```  2798     apply(erule conjE exE)+ apply(rule_tac X="{x. x \$\$ xa = a \$\$ xa} \<union> {x. x \$\$ xa = b \$\$ xa}" in UnionI)
```
```  2799     apply(erule_tac[!] x=xa in allE) by auto
```
```  2800   thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
```
```  2801
```
```  2802 lemma has_integral_spike_interior:
```
```  2803   assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
```
```  2804   apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
```
```  2805
```
```  2806 lemma has_integral_spike_interior_eq:
```
```  2807   assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
```
```  2808   apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
```
```  2809
```
```  2810 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}"
```
```  2811   using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
```
```  2812
```
```  2813 subsection {* Integrability of continuous functions. *}
```
```  2814
```
```  2815 lemma neutral_and[simp]: "neutral op \<and> = True"
```
```  2816   unfolding neutral_def apply(rule some_equality) by auto
```
```  2817
```
```  2818 lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
```
```  2819
```
```  2820 lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
```
```  2821 apply induct unfolding iterate_insert[OF monoidal_and] by auto
```
```  2822
```
```  2823 lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
```
```  2824   shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
```
```  2825   using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
```
```  2826
```
```  2827 lemma operative_approximable: assumes "0 \<le> e" fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  2828   shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
```
```  2829 proof safe fix a b::"'b" { assume "content {a..b} = 0"
```
```  2830     thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}"
```
```  2831       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
```
```  2832   { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k<DIM('b)"
```
```  2833     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}"
```
```  2834       "\<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}"
```
```  2835       apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2) k] by auto }
```
```  2836   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}"
```
```  2837                           "\<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}"
```
```  2838   assume k:"k<DIM('b)"
```
```  2839   let ?g = "\<lambda>x. if x\$\$k = c then f x else if x\$\$k \<le> c then g1 x else g2 x"
```
```  2840   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)
```
```  2841   proof safe case goal1 thus ?case apply- apply(cases "x\$\$k=c", case_tac "x\$\$k < c") using as assms by auto
```
```  2842   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}"
```
```  2843     then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k]
```
```  2844     show ?case unfolding integrable_on_def by auto
```
```  2845   next show "?g integrable_on {a..b} \<inter> {x. x \$\$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x \$\$ k \<ge> c}"
```
```  2846       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
```
```  2847
```
```  2848 lemma approximable_on_division: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  2849   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"
```
```  2850   obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
```
```  2851 proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
```
```  2852   note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
```
```  2853   guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
```
```  2854
```
```  2855 lemma integrable_continuous: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  2856   assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
```
```  2857 proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
```
```  2858   from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
```
```  2859   note d=conjunctD2[OF this,rule_format]
```
```  2860   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
```
```  2861   note p' = tagged_division_ofD[OF p(1)]
```
```  2862   have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  2863   proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p"
```
```  2864     from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
```
```  2865     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)
```
```  2866     proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
```
```  2867       fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
```
```  2868       note d(2)[OF _ _ this[unfolded mem_ball]]
```
```  2869       thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastsimp qed qed
```
```  2870   from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
```
```  2871   thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed
```
```  2872
```
```  2873 subsection {* Specialization of additivity to one dimension. *}
```
```  2874
```
```  2875 lemma operative_1_lt: assumes "monoidal opp"
```
```  2876   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
```
```  2877                 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
```
```  2878   unfolding operative_def content_eq_0 DIM_real less_one simp_thms(39,41) Eucl_real_simps
```
```  2879     (* The dnf_simps simplify "\<exists> x. x= _ \<and> _" and "\<forall>k. k = _ \<longrightarrow> _" *)
```
```  2880 proof safe fix a b c::"real" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c}))
```
```  2881     (f ({a..b} \<inter> {x. c \<le> x}))" "a < c" "c < b"
```
```  2882     from this(2-) have "{a..b} \<inter> {x. x \<le> c} = {a..c}" "{a..b} \<inter> {x. x \<ge> c} = {c..b}" by auto
```
```  2883     thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c"] by auto
```
```  2884 next fix a b c::real
```
```  2885   assume as:"\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}"
```
```  2886   show "f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c})) (f ({a..b} \<inter> {x. c \<le> x}))"
```
```  2887   proof(cases "c \<in> {a .. b}")
```
```  2888     case False hence "c<a \<or> c>b" by auto
```
```  2889     thus ?thesis apply-apply(erule disjE)
```
```  2890     proof- assume "c<a" hence *:"{a..b} \<inter> {x. x \<le> c} = {1..0}"  "{a..b} \<inter> {x. c \<le> x} = {a..b}" by auto
```
```  2891       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
```
```  2892     next   assume "b<c" hence *:"{a..b} \<inter> {x. x \<le> c} = {a..b}"  "{a..b} \<inter> {x. c \<le> x} = {1..0}" by auto
```
```  2893       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
```
```  2894     qed
```
```  2895   next case True hence *:"min (b) c = c" "max a c = c" by auto
```
```  2896     have **:"0 < DIM(real)" by auto
```
```  2897     have ***:"\<And>P Q. (\<chi>\<chi> i. if i = 0 then P i else Q i) = (P 0::real)" apply(subst euclidean_eq)
```
```  2898       apply safe unfolding euclidean_lambda_beta' by auto
```
```  2899     show ?thesis unfolding interval_split[OF **,unfolded Eucl_real_simps(1,3)] unfolding *** *
```
```  2900     proof(cases "c = a \<or> c = b")
```
```  2901       case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
```
```  2902         apply-apply(subst as(2)[rule_format]) using True by auto
```
```  2903     next case True thus "f {a..b} = opp (f {a..c}) (f {c..b})" apply-
```
```  2904       proof(erule disjE) assume *:"c=a"
```
```  2905         hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2906         thus ?thesis using assms unfolding * by auto
```
```  2907       next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2908         thus ?thesis using assms unfolding * by auto qed qed qed qed
```
```  2909
```
```  2910 lemma operative_1_le: assumes "monoidal opp"
```
```  2911   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
```
```  2912                 (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
```
```  2913 unfolding operative_1_lt[OF assms]
```
```  2914 proof safe fix a b c::"real" assume as:"\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a < c" "c < b"
```
```  2915   show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) by auto
```
```  2916 next fix a b c ::"real" assume "\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp"
```
```  2917     "\<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"
```
```  2918   note as = this[rule_format]
```
```  2919   show "opp (f {a..c}) (f {c..b}) = f {a..b}"
```
```  2920   proof(cases "c = a \<or> c = b")
```
```  2921     case False thus ?thesis apply-apply(subst as(2)) using as(3-) by(auto)
```
```  2922     next case True thus ?thesis apply-
```
```  2923       proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2924         thus ?thesis using assms unfolding * by auto
```
```  2925       next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
```
```  2926         thus ?thesis using assms unfolding * by auto qed qed qed
```
```  2927
```
```  2928 subsection {* Special case of additivity we need for the FCT. *}
```
```  2929
```
```  2930 lemma interval_bound_sing[simp]: "interval_upperbound {a} = a"  "interval_lowerbound {a} = a"
```
```  2931   unfolding interval_upperbound_def interval_lowerbound_def  by auto
```
```  2932
```
```  2933 lemma additive_tagged_division_1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
```
```  2934   assumes "a \<le> b" "p tagged_division_of {a..b}"
```
```  2935   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
```
```  2936 proof- let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
```
```  2937   have ***:"\<forall>i<DIM(real). a \$\$ i \<le> b \$\$ i" using assms by auto
```
```  2938   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty by auto
```
```  2939   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
```
```  2940   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],THEN sym]
```
```  2941   show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
```
```  2942     apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
```
```  2943
```
```  2944 subsection {* A useful lemma allowing us to factor out the content size. *}
```
```  2945
```
```  2946 lemma has_integral_factor_content:
```
```  2947   "(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
```
```  2948     \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
```
```  2949 proof(cases "content {a..b} = 0")
```
```  2950   case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
```
```  2951     apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer
```
```  2952     apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
```
```  2953     apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
```
```  2954 next case False note F = this[unfolded content_lt_nz[THEN sym]]
```
```  2955   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)"
```
```  2956   show ?thesis apply(subst has_integral)
```
```  2957   proof safe fix e::real assume e:"e>0"
```
```  2958     { 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)
```
```  2959         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
```
```  2960         using F e by(auto simp add:field_simps intro:mult_pos_pos) }
```
```  2961     {  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)
```
```  2962         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
```
```  2963         using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
```
```  2964
```
```  2965 subsection {* Fundamental theorem of calculus. *}
```
```  2966
```
```  2967 lemma interval_bounds_real: assumes "a\<le>(b::real)"
```
```  2968   shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
```
```  2969   apply(rule_tac[!] interval_bounds) using assms by auto
```
```  2970
```
```  2971 lemma fundamental_theorem_of_calculus: fixes f::"real \<Rightarrow> 'a::banach"
```
```  2972   assumes "a \<le> b"  "\<forall>x\<in>{a..b}. (f has_vector_derivative f' x) (at x within {a..b})"
```
```  2973   shows "(f' has_integral (f b - f a)) ({a..b})"
```
```  2974 unfolding has_integral_factor_content
```
```  2975 proof safe fix e::real assume e:"e>0"
```
```  2976   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
```
```  2977   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
```
```  2978   note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
```
```  2979   guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  2980   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
```
```  2981                  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
```
```  2982     apply(rule_tac x="\<lambda>x. ball x (d x)" in exI,safe)
```
```  2983     apply(rule gauge_ball_dependent,rule,rule d(1))
```
```  2984   proof- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. ball x (d x)) fine p"
```
```  2985     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
```
```  2986       unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,THEN sym]
```
```  2987       unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",THEN sym]
```
```  2988       unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym]
```
```  2989     proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
```
```  2990       note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
```
```  2991       have *:"u \<le> v" using xk unfolding k by auto
```
```  2992       have ball:"\<forall>xa\<in>k. xa \<in> ball x (d x)" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,
```
```  2993         unfolded split_conv subset_eq] .
```
```  2994       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
```
```  2995         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
```
```  2996         apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
```
```  2997         unfolding scaleR.diff_left by(auto simp add:algebra_simps)
```
```  2998       also have "... \<le> e * norm (u - x) + e * norm (v - x)"
```
```  2999         apply(rule add_mono) apply(rule d(2)[of "x" "u",unfolded o_def]) prefer 4
```
```  3000         apply(rule d(2)[of "x" "v",unfolded o_def])
```
```  3001         using ball[rule_format,of u] ball[rule_format,of v]
```
```  3002         using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def)
```
```  3003       also have "... \<le> e * (interval_upperbound k - interval_lowerbound k)"
```
```  3004         unfolding k interval_bounds_real[OF *] using xk(1) unfolding k by(auto simp add:dist_real_def field_simps)
```
```  3005       finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
```
```  3006         e * (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bounds_real[OF *] content_real[OF *] .
```
```  3007     qed(insert as, auto) qed qed
```
```  3008
```
```  3009 subsection {* Attempt a systematic general set of "offset" results for components. *}
```
```  3010
```
```  3011 lemma gauge_modify:
```
```  3012   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
```
```  3013   shows "gauge (\<lambda>x y. d (f x) (f y))"
```
```  3014   using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
```
```  3015   apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
```
```  3016
```
```  3017 subsection {* Only need trivial subintervals if the interval itself is trivial. *}
```
```  3018
```
```  3019 lemma division_of_nontrivial: fixes s::"('a::ordered_euclidean_space) set set"
```
```  3020   assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
```
```  3021   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
```
```  3022 proof(induct "card s" arbitrary:s rule:nat_less_induct)
```
```  3023   fix s::"'a set set" assume assm:"s division_of {a..b}"
```
```  3024     "\<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}"
```
```  3025   note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
```
```  3026   { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
```
```  3027     show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
```
```  3028   assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
```
```  3029   then obtain k where k:"k\<in>s" "content k = 0" by auto
```
```  3030   from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
```
```  3031   from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
```
```  3032   hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
```
```  3033   have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
```
```  3034     apply safe apply(rule closed_interval) using assm(1) by auto
```
```  3035   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
```
```  3036   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
```
```  3037     from k(2)[unfolded k content_eq_0] guess i ..
```
```  3038     hence i:"c\$\$i = d\$\$i" "i<DIM('a)" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
```
```  3039     hence xi:"x\$\$i = d\$\$i" using as unfolding k mem_interval by smt
```
```  3040     def y \<equiv> "(\<chi>\<chi> j. if j = i then if c\$\$i \<le> (a\$\$i + b\$\$i) / 2 then c\$\$i +
```
```  3041       min e (b\$\$i - c\$\$i) / 2 else c\$\$i - min e (c\$\$i - a\$\$i) / 2 else x\$\$j)::'a"
```
```  3042     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI)
```
```  3043     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)
```
```  3044       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]]
```
```  3045       hence xyi:"y\$\$i \<noteq> x\$\$i" unfolding y_def unfolding i xi euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
```
```  3046         apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2)
```
```  3047         using assms(2)[unfolded content_eq_0] using i(2) by smt+
```
```  3048       thus "y \<noteq> x" unfolding euclidean_eq[where 'a='a] using i by auto
```
```  3049       have *:"{..<DIM('a)} = insert i ({..<DIM('a)} - {i})" using i by auto
```
```  3050       have "norm (y - x) < e + setsum (\<lambda>i. 0) {..<DIM('a)}" apply(rule le_less_trans[OF norm_le_l1])
```
```  3051         apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
```
```  3052       proof- show "\<bar>(y - x) \$\$ i\<bar> < e" unfolding y_def euclidean_simps euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
```
```  3053           apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
```
```  3054         show "(\<Sum>i\<in>{..<DIM('a)} - {i}. \<bar>(y - x) \$\$ i\<bar>) \<le> (\<Sum>i\<in>{..<DIM('a)}. 0)" unfolding y_def by auto
```
```  3055       qed auto thus "dist y x < e" unfolding dist_norm by auto
```
```  3056       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
```
```  3057       moreover have "y \<in> \<Union>s" unfolding s mem_interval
```
```  3058       proof(rule,rule) note simps = y_def euclidean_lambda_beta' if_not_P
```
```  3059         fix j assume j:"j<DIM('a)" show "a \$\$ j \<le> y \$\$ j \<and> y \$\$ j \<le> b \$\$ j"
```
```  3060         proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
```
```  3061           thus ?thesis using j apply- unfolding simps if_not_P[OF False] unfolding mem_interval by auto
```
```  3062         next case True note T = this show ?thesis
```
```  3063           proof(cases "c \$\$ i \<le> (a \$\$ i + b \$\$ i) / 2")
```
```  3064             case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
```
```  3065               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
```
```  3066           next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
```
```  3067               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
```
```  3068           qed qed qed
```
```  3069       ultimately show "y \<in> \<Union>(s - {k})" by auto
```
```  3070     qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
```
```  3071   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
```
```  3072     apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
```
```  3073   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
```
```  3074
```
```  3075 subsection {* Integrabibility on subintervals. *}
```
```  3076
```
```  3077 lemma operative_integrable: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  3078   "operative op \<and> (\<lambda>i. f integrable_on i)"
```
```  3079   unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
```
```  3080   unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption,assumption)+
```
```  3081   unfolding integrable_on_def by(auto intro!: has_integral_split)
```
```  3082
```
```  3083 lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3084   assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
```
```  3085   apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
```
```  3086   using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
```
```  3087
```
```  3088 subsection {* Combining adjacent intervals in 1 dimension. *}
```
```  3089
```
```  3090 lemma has_integral_combine: assumes "(a::real) \<le> c" "c \<le> b"
```
```  3091   "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
```
```  3092   shows "(f has_integral (i + j)) {a..b}"
```
```  3093 proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
```
```  3094   note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
```
```  3095   hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
```
```  3096     apply(subst(asm) if_P) using assms(3-) by auto
```
```  3097   with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
```
```  3098     unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
```
```  3099
```
```  3100 lemma integral_combine: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3101   assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
```
```  3102   shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
```
```  3103   apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
```
```  3104   apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
```
```  3105
```
```  3106 lemma integrable_combine: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3107   assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
```
```  3108   shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
```
```  3109
```
```  3110 subsection {* Reduce integrability to "local" integrability. *}
```
```  3111
```
```  3112 lemma integrable_on_little_subintervals: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3113   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}"
```
```  3114   shows "f integrable_on {a..b}"
```
```  3115 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})"
```
```  3116     using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
```
```  3117   guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
```
```  3118   note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
```
```  3119   show ?thesis unfolding * apply safe unfolding snd_conv
```
```  3120   proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
```
```  3121     thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
```
```  3122
```
```  3123 subsection {* Second FCT or existence of antiderivative. *}
```
```  3124
```
```  3125 lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
```
```  3126   unfolding integrable_on_def by(rule,rule has_integral_const)
```
```  3127
```
```  3128 lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3129   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
```
```  3130   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
```
```  3131   unfolding has_vector_derivative_def has_derivative_within_alt
```
```  3132 apply safe apply(rule scaleR.bounded_linear_left)
```
```  3133 proof- fix e::real assume e:"e>0"
```
```  3134   note compact_uniformly_continuous[OF assms(1) compact_interval,unfolded uniformly_continuous_on_def]
```
```  3135   from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
```
```  3136   let ?I = "\<lambda>a b. integral {a..b} f"
```
```  3137   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)"
```
```  3138   proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
```
```  3139       case False have "f integrable_on {a..y}" apply(rule integrable_subinterval,rule integrable_continuous)
```
```  3140         apply(rule assms)  unfolding not_less using assms(2) goal1 by auto
```
```  3141       hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
```
```  3142         using False unfolding not_less using assms(2) goal1 by auto
```
```  3143       have **:"norm (y - x) = content {x..y}" apply(subst content_real) using False unfolding not_less by auto
```
```  3144       show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
```
```  3145         defer apply(rule has_integral_sub) apply(rule integrable_integral)
```
```  3146         apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
```
```  3147       proof- show "{x..y} \<subseteq> {a..b}" using goal1 assms(2) by auto
```
```  3148         have *:"y - x = norm(y - x)" using False by auto
```
```  3149         show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {x.. y}" apply(subst *) unfolding ** by auto
```
```  3150         show "\<forall>xa\<in>{x..y}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
```
```  3151           apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
```
```  3152       qed(insert e,auto)
```
```  3153     next case True have "f integrable_on {a..x}" apply(rule integrable_subinterval,rule integrable_continuous)
```
```  3154         apply(rule assms)+  unfolding not_less using assms(2) goal1 by auto
```
```  3155       hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
```
```  3156         using True using assms(2) goal1 by auto
```
```  3157       have **:"norm (y - x) = content {y..x}" apply(subst content_real) using True unfolding not_less by auto
```
```  3158       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
```
```  3159       show ?thesis apply(subst ***) unfolding norm_minus_cancel **
```
```  3160         apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
```
```  3161         defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
```
```  3162         apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
```
```  3163       proof- show "{y..x} \<subseteq> {a..b}" using goal1 assms(2) by auto
```
```  3164         have *:"x - y = norm(y - x)" using True by auto
```
```  3165         show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {y..x}" apply(subst *) unfolding ** by auto
```
```  3166         show "\<forall>xa\<in>{y..x}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
```
```  3167           apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
```
```  3168       qed(insert e,auto) qed qed qed
```
```  3169
```
```  3170 lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
```
```  3171   obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
```
```  3172   apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
```
```  3173
```
```  3174 subsection {* Combined fundamental theorem of calculus. *}
```
```  3175
```
```  3176 lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
```
```  3177   obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u..v}"
```
```  3178 proof- from antiderivative_continuous[OF assms] guess g . note g=this
```
```  3179   show ?thesis apply(rule that[of g])
```
```  3180   proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
```
```  3181       apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
```
```  3182     thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g" "f"] by auto qed qed
```
```  3183
```
```  3184 subsection {* General "twiddling" for interval-to-interval function image. *}
```
```  3185
```
```  3186 lemma has_integral_twiddle:
```
```  3187   assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
```
```  3188   "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
```
```  3189   "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
```
```  3190   "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
```
```  3191   "(f has_integral i) {a..b}"
```
```  3192   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
```
```  3193 proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
```
```  3194     show ?thesis apply cases defer apply(rule *,assumption)
```
```  3195     proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
```
```  3196   assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
```
```  3197   have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
```
```  3198     using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
```
```  3199     using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
```
```  3200   show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
```
```  3201   proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
```
```  3202     from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
```
```  3203     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)
```
```  3204     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)"
```
```  3205     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
```
```  3206       fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)]
```
```  3207       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
```
```  3208       proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
```
```  3209         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
```
```  3210         fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
```
```  3211         show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
```
```  3212         { 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)"]
```
```  3213             using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
```
```  3214         fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
```
```  3215         hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
```
```  3216         have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
```
```  3217         proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
```
```  3218           hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
```
```  3219             unfolding image_Int[OF inj(1)] by auto thus False using as by blast
```
```  3220         qed thus "g x = g x'" by auto
```
```  3221         { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
```
```  3222         { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
```
```  3223       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
```
```  3224         then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
```
```  3225         thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
```
```  3226           apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
```
```  3227           using X(2) assms(3)[rule_format,of x] by auto
```
```  3228       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
```
```  3229        have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding algebra_simps add_left_cancel
```
```  3230         unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
```
```  3231         apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
```
```  3232       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]
```
```  3233         unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
```
```  3234       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
```
```  3235         using assms(1) by(auto simp add:field_simps) qed qed qed
```
```  3236
```
```  3237 subsection {* Special case of a basic affine transformation. *}
```
```  3238
```
```  3239 lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space) + c) ` {a..b} = {u..v}"
```
```  3240   unfolding image_affinity_interval by auto
```
```  3241
```
```  3242 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
```
```  3243   apply(rule setprod_cong) using assms by auto
```
```  3244
```
```  3245 lemma content_image_affinity_interval:
```
```  3246  "content((\<lambda>x::'a::ordered_euclidean_space. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ DIM('a) * content {a..b}" (is "?l = ?r")
```
```  3247 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
```
```  3248       unfolding not_not using content_empty by auto }
```
```  3249   have *:"DIM('a) = card {..<DIM('a)}" by auto
```
```  3250   assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
```
```  3251     case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
```
```  3252       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
```
```  3253       apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
```
```  3254       apply(rule setprod_cong2) using True as unfolding interval_ne_empty euclidean_simps not_le
```
```  3255       by(auto simp add:field_simps intro:mult_left_mono)
```
```  3256   next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
```
```  3257       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
```
```  3258       apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
```
```  3259       apply(rule setprod_cong2) using False as unfolding interval_ne_empty euclidean_simps not_le
```
```  3260       by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
```
```  3261
```
```  3262 lemma has_integral_affinity: fixes a::"'a::ordered_euclidean_space" assumes "(f has_integral i) {a..b}" "m \<noteq> 0"
```
```  3263   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
```
```  3264   apply(rule has_integral_twiddle,safe) apply(rule zero_less_power) unfolding euclidean_eq[where 'a='a]
```
```  3265   unfolding scaleR_right_distrib euclidean_simps scaleR.scaleR_left[THEN sym]
```
```  3266   defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
```
```  3267   apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
```
```  3268
```
```  3269 lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
```
```  3270   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})"
```
```  3271   using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
```
```  3272
```
```  3273 subsection {* Special case of stretching coordinate axes separately. *}
```
```  3274
```
```  3275 lemma image_stretch_interval:
```
```  3276   "(\<lambda>x. \<chi>\<chi> k. m k * x\$\$k) ` {a..b::'a::ordered_euclidean_space} =
```
```  3277   (if {a..b} = {} then {} else {(\<chi>\<chi> k. min (m(k) * a\$\$k) (m(k) * b\$\$k))::'a ..  (\<chi>\<chi> k. max (m(k) * a\$\$k) (m(k) * b\$\$k))})"
```
```  3278   (is "?l = ?r")
```
```  3279 proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
```
```  3280 next have *:"\<And>P Q. (\<forall>i<DIM('a). P i) \<and> (\<forall>i<DIM('a). Q i) \<longleftrightarrow> (\<forall>i<DIM('a). P i \<and> Q i)" by auto
```
```  3281   case False note ab = this[unfolded interval_ne_empty]
```
```  3282   show ?thesis apply-apply(rule set_eqI)
```
```  3283   proof- fix x::"'a" have **:"\<And>P Q. (\<forall>i<DIM('a). P i = Q i) \<Longrightarrow> (\<forall>i<DIM('a). P i) = (\<forall>i<DIM('a). Q i)" by auto
```
```  3284     show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False]
```
```  3285       unfolding image_iff mem_interval Bex_def euclidean_simps euclidean_eq[where 'a='a] *
```
```  3286       unfolding imp_conjR[THEN sym] apply(subst euclidean_lambda_beta'') apply(subst lambda_skolem'[THEN sym])
```
```  3287       apply(rule **,rule,rule) unfolding euclidean_lambda_beta'
```
```  3288     proof- fix i assume i:"i<DIM('a)" show "(\<exists>xa. (a \$\$ i \<le> xa \<and> xa \<le> b \$\$ i) \<and> x \$\$ i = m i * xa) =
```
```  3289         (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))"
```
```  3290       proof(cases "m i = 0") case True thus ?thesis using ab i by auto
```
```  3291       next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
```
```  3292         proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a \$\$ i) (m i * b \$\$ i) = m i * a \$\$ i"
```
```  3293             "max (m i * a \$\$ i) (m i * b \$\$ i) = m i * b \$\$ i" using ab i unfolding min_def max_def by auto
```
```  3294           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x\$\$i" in exI)
```
```  3295             using as by(auto simp add:field_simps)
```
```  3296         next assume as:"0 > m i" hence *:"max (m i * a \$\$ i) (m i * b \$\$ i) = m i * a \$\$ i"
```
```  3297             "min (m i * a \$\$ i) (m i * b \$\$ i) = m i * b \$\$ i" using ab as i unfolding min_def max_def
```
```  3298             by(auto simp add:field_simps mult_le_cancel_left_neg intro: order_antisym)
```
```  3299           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x\$\$i" in exI)
```
```  3300             using as by(auto simp add:field_simps) qed qed qed qed qed
```
```  3301
```
```  3302 lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi>\<chi> k. m k * x\$\$k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
```
```  3303   unfolding image_stretch_interval by auto
```
```  3304
```
```  3305 lemma content_image_stretch_interval:
```
```  3306   "content((\<lambda>x::'a::ordered_euclidean_space. (\<chi>\<chi> k. m k * x\$\$k)::'a) ` {a..b}) = abs(setprod m {..<DIM('a)}) * content({a..b})"
```
```  3307 proof(cases "{a..b} = {}") case True thus ?thesis
```
```  3308     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
```
```  3309 next case False hence "(\<lambda>x. (\<chi>\<chi> k. m k * x \$\$ k)::'a) ` {a..b} \<noteq> {}" by auto
```
```  3310   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
```
```  3311     unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff euclidean_lambda_beta'
```
```  3312   proof- fix i assume i:"i<DIM('a)" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
```
```  3313     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)"
```
```  3314       apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i
```
```  3315       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
```
```  3316
```
```  3317 lemma has_integral_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
```
```  3318   assumes "(f has_integral i) {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
```
```  3319   shows "((\<lambda>x. f(\<chi>\<chi> k. m k * x\$\$k)) has_integral
```
```  3320              ((1/(abs(setprod m {..<DIM('a)}))) *\<^sub>R i)) ((\<lambda>x. (\<chi>\<chi> k. 1/(m k) * x\$\$k)::'a) ` {a..b})"
```
```  3321   apply(rule has_integral_twiddle[where f=f]) unfolding zero_less_abs_iff content_image_stretch_interval
```
```  3322   unfolding image_stretch_interval empty_as_interval euclidean_eq[where 'a='a] using assms
```
```  3323 proof- show "\<forall>y::'a. continuous (at y) (\<lambda>x. (\<chi>\<chi> k. m k * x \$\$ k)::'a)"
```
```  3324    apply(rule,rule linear_continuous_at) unfolding linear_linear
```
```  3325    unfolding linear_def euclidean_simps euclidean_eq[where 'a='a] by(auto simp add:field_simps) qed auto
```
```  3326
```
```  3327 lemma integrable_stretch:  fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
```
```  3328   assumes "f integrable_on {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
```
```  3329   shows "(\<lambda>x::'a. f(\<chi>\<chi> k. m k * x\$\$k)) integrable_on ((\<lambda>x. \<chi>\<chi> k. 1/(m k) * x\$\$k) ` {a..b})"
```
```  3330   using assms unfolding integrable_on_def apply-apply(erule exE)
```
```  3331   apply(drule has_integral_stretch,assumption) by auto
```
```  3332
```
```  3333 subsection {* even more special cases. *}
```
```  3334
```
```  3335 lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::'a::ordered_euclidean_space}"
```
```  3336   apply(rule set_eqI,rule) defer unfolding image_iff
```
```  3337   apply(rule_tac x="-x" in bexI) by(auto simp add:minus_le_iff le_minus_iff eucl_le[where 'a='a])
```
```  3338
```
```  3339 lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
```
```  3340   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
```
```  3341   using has_integral_affinity[OF assms, of "-1" 0] by auto
```
```  3342
```
```  3343 lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
```
```  3344   apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
```
```  3345
```
```  3346 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
```
```  3347   unfolding integrable_on_def by auto
```
```  3348
```
```  3349 lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
```
```  3350   unfolding integral_def by auto
```
```  3351
```
```  3352 subsection {* Stronger form of FCT; quite a tedious proof. *}
```
```  3353
```
```  3354 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)
```
```  3355
```
```  3356 lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
```
```  3357   assumes "a \<le> b" "p tagged_division_of {a..b}"
```
```  3358   shows "setsum (\<lambda>(x,k). f (interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
```
```  3359   using additive_tagged_division_1[OF _ assms(2), of f] using assms(1) by auto
```
```  3360
```
```  3361 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"
```
```  3362   unfolding split_def by(rule refl)
```
```  3363
```
```  3364 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
```
```  3365   apply(subst(asm)(2) norm_minus_cancel[THEN sym])
```
```  3366   apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
```
```  3367
```
```  3368 lemma fundamental_theorem_of_calculus_interior: fixes f::"real => 'a::real_normed_vector"
```
```  3369   assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
```
```  3370   shows "(f' has_integral (f b - f a)) {a..b}"
```
```  3371 proof- { presume *:"a < b \<Longrightarrow> ?thesis"
```
```  3372     show ?thesis proof(cases,rule *,assumption)
```
```  3373       assume "\<not> a < b" hence "a = b" using assms(1) by auto
```
```  3374       hence *:"{a .. b} = {b}" "f b - f a = 0" by(auto simp add:  order_antisym)
```
```  3375       show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0 using * `a=b` by auto
```
```  3376     qed } assume ab:"a < b"
```
```  3377   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
```
```  3378                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
```
```  3379   { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
```
```  3380   fix e::real assume e:"e>0"
```
```  3381   note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
```
```  3382   note conjunctD2[OF this] note bounded=this(1) and this(2)
```
```  3383   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)"
```
```  3384     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]
```
```  3385   from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
```
```  3386   have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_interval assms by auto
```
```  3387   from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
```
```  3388
```
```  3389   have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
```
```  3390     \<longrightarrow> norm(content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
```
```  3391   proof- have "a\<in>{a..b}" using ab by auto
```
```  3392     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3393     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
```
```  3394     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3395     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
```
```  3396     proof(cases "f' a = 0") case True
```
```  3397       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
```
```  3398     next case False thus ?thesis
```
```  3399         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps)
```
```  3400     qed then guess l .. note l = conjunctD2[OF this]
```
```  3401     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
```
```  3402     proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)"
```
```  3403       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
```
```  3404       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)
```
```  3405       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3406       proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
```
```  3407         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
```
```  3408       next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
```
```  3409           apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
```
```  3410       qed finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4"
```
```  3411         unfolding content_real[OF as(1)] by auto
```
```  3412     qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
```
```  3413
```
```  3414   have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow>
```
```  3415     norm(content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
```
```  3416   proof- have "b\<in>{a..b}" using ab by auto
```
```  3417     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3418     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
```
```  3419       using e ab by(auto simp add:field_simps)
```
```  3420     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3421     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
```
```  3422     proof(cases "f' b = 0") case True
```
```  3423       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg)
```
```  3424     next case False thus ?thesis
```
```  3425         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
```
```  3426         using ab e by(auto simp add:field_simps)
```
```  3427     qed then guess l .. note l = conjunctD2[OF this]
```
```  3428     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
```
```  3429     proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
```
```  3430       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
```
```  3431       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)
```
```  3432       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3433       proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
```
```  3434         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
```
```  3435       next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
```
```  3436           apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
```
```  3437       qed finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4"
```
```  3438         unfolding content_real[OF as(1)] by auto
```
```  3439     qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
```
```  3440
```
```  3441   let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
```
```  3442   show "?P e" apply(rule_tac x="?d" in exI)
```
```  3443   proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
```
```  3444   next case goal2 note as=this let ?A = "{t. fst t \<in> {a, b}}" note p = tagged_division_ofD[OF goal2(1)]
```
```  3445     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
```
```  3446     note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
```
```  3447     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
```
```  3448     show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
```
```  3449       unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
```
```  3450     proof(rule norm_triangle_le,rule **)
```
```  3451       case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
```
```  3452       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"
```
```  3453           "e * (interval_upperbound k -  interval_lowerbound k) / 2
```
```  3454           < norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k)))"
```
```  3455         from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
```
```  3456         hence "u \<le> v" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto
```
```  3457         note result = as(2)[unfolded k interval_bounds_real[OF this(1)] content_real[OF this(1)]]
```
```  3458
```
```  3459         assume as':"x \<noteq> a" "x \<noteq> b" hence "x \<in> {a<..<b}" using p(2-3)[OF as(1)] by auto
```
```  3460         note  * = d(2)[OF this]
```
```  3461         have "norm ((v - u) *\<^sub>R f' (x) - (f (v) - f (u))) =
```
```  3462           norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))"
```
```  3463           apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto
```
```  3464         also have "... \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)" apply(rule norm_triangle_le_sub)
```
```  3465           apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
```
```  3466           apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp:dist_real_def)
```
```  3467         also have "... \<le> e / 2 * norm (v - u)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
```
```  3468         finally have "e * (v - u) / 2 < e * (v - u) / 2"
```
```  3469           apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
```
```  3470
```
```  3471     next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
```
```  3472       case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
```
```  3473         defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym]
```
```  3474         apply(subst additive_tagged_division_1[OF _ as(1)]) apply(rule assms)
```
```  3475       proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}}" note xk=IntD1[OF this]
```
```  3476         from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  3477         with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
```
```  3478         thus "0 \<le> e * ((interval_upperbound k) - (interval_lowerbound k))"
```
```  3479           unfolding uv using e by(auto simp add:field_simps)
```
```  3480       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
```
```  3481         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
```
```  3482           (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2"
```
```  3483           apply(rule *[where t="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
```
```  3484           apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
```
```  3485         proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
```
```  3486           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
```
```  3487           have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk
```
```  3488             unfolding uv content_eq_0 interval_eq_empty by auto
```
```  3489           thus "content k *\<^sub>R (f' (x)) - (f ((interval_upperbound k)) - f ((interval_lowerbound k))) = 0" using xk unfolding uv by auto
```
```  3490         next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
```
```  3491             {t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}" by blast
```
```  3492           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)
```
```  3493             \<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
```
```  3494           proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
```
```  3495             thus ?case using `x\<in>s` goal2(2) by auto
```
```  3496           qed auto
```
```  3497           case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4
```
```  3498             apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
```
```  3499             apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
```
```  3500           proof- let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
```
```  3501             have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v"
```
```  3502             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
```
```  3503               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
```
```  3504               have u:"u = a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
```
```  3505                 have "u \<ge> a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>a" ultimately
```
```  3506                 have "u > a" by auto
```
```  3507                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
```
```  3508               qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
```
```  3509             qed
```
```  3510             have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v"
```
```  3511             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
```
```  3512               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
```
```  3513               have u:"v =  b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto
```
```  3514                 have "v \<le>  b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq> b" ultimately
```
```  3515                 have "v <  b" by auto
```
```  3516                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
```
```  3517               qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
```
```  3518             qed
```
```  3519
```
```  3520             show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
```
```  3521               unfolding mem_Collect_eq fst_conv snd_conv apply safe
```
```  3522             proof- fix x k k' assume k:"( a, k) \<in> p" "( a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
```
```  3523               guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
```
```  3524               guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (min (v) (v'))"
```
```  3525               have "{ a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
```
```  3526               moreover have " ((a + ?v)/2) \<in> { a <..< ?v}" using k(3-)
```
```  3527                 unfolding v v' content_eq_0 not_le by(auto simp add:not_le)
```
```  3528               ultimately have " ((a + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
```
```  3529               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
```
```  3530               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
```
```  3531             qed
```
```  3532             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
```
```  3533               unfolding mem_Collect_eq fst_conv snd_conv apply safe
```
```  3534             proof- fix x k k' assume k:"( b, k) \<in> p" "( b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
```
```  3535               guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
```
```  3536               guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (max (v) (v'))"
```
```  3537               have "{?v <..<  b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
```
```  3538               moreover have " ((b + ?v)/2) \<in> {?v <..<  b}" using k(3-) unfolding v v' content_eq_0 not_le by auto
```
```  3539               ultimately have " ((b + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
```
```  3540               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
```
```  3541               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
```
```  3542             qed
```
```  3543
```
```  3544             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
```
```  3545             show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) - (f ((interval_upperbound k)) -
```
```  3546               f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4" apply(rule,rule) unfolding mem_Collect_eq
```
```  3547               unfolding split_paired_all fst_conv snd_conv
```
```  3548             proof safe case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
```
```  3549               have " ?a\<in>{ ?a..v}" using v(2) by auto hence "v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
```
```  3550               moreover have "{?a..v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
```
```  3551                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x=" x" in ballE)
```
```  3552                 by(auto simp add:subset_eq dist_real_def v) ultimately
```
```  3553               show ?case unfolding v interval_bounds_real[OF v(2)] apply- apply(rule da(2)[of "v"])
```
```  3554                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
```
```  3555             qed
```
```  3556             show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) -
```
```  3557               (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4"
```
```  3558               apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv
```
```  3559             proof safe case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
```
```  3560               have " ?b\<in>{v.. ?b}" using v(2) by auto hence "v \<ge> ?a" using p(3)[OF goal1(1)]
```
```  3561                 unfolding subset_eq v by auto
```
```  3562               moreover have "{v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
```
```  3563                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe
```
```  3564                 apply(erule_tac x=" x" in ballE) using ab
```
```  3565                 by(auto simp add:subset_eq v dist_real_def) ultimately
```
```  3566               show ?case unfolding v unfolding interval_bounds_real[OF v(2)] apply- apply(rule db(2)[of "v"])
```
```  3567                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
```
```  3568             qed
```
```  3569           qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
```
```  3570
```
```  3571 subsection {* Stronger form with finite number of exceptional points. *}
```
```  3572
```
```  3573 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3574   assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
```
```  3575   "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  3576   shows "(f' has_integral (f b - f a)) {a..b}" using assms apply-
```
```  3577 proof(induct "card s" arbitrary:s a b)
```
```  3578   case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
```
```  3579 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
```
```  3580     apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
```
```  3581   show ?case proof(cases "c\<in>{a<..<b}")
```
```  3582     case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
```
```  3583       apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
```
```  3584   next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
```
```  3585     case True hence "a \<le> c" "c \<le> b" by auto
```
```  3586     thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
```
```  3587       apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
```
```  3588     proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
```
```  3589         apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
```
```  3590       let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
```
```  3591       show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
```
```  3592     qed auto qed qed
```
```  3593
```
```  3594 lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3595   assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
```
```  3596   "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  3597   shows "(f' has_integral (f(b) - f(a))) {a..b}"
```
```  3598   apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
```
```  3599   using assms(4) by auto
```
```  3600
```
```  3601 lemma indefinite_integral_continuous_left: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3602   assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e"
```
```  3603   obtains d where "0 < d" "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e"
```
```  3604 proof- have "\<exists>w>0. \<forall>t. c - w < t \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3"
```
```  3605   proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)"
```
```  3606       apply-apply(rule divide_pos_pos) using `e>0` by auto
```
```  3607     thus ?thesis apply-apply(rule,rule,assumption,safe)
```
```  3608     proof- fix t assume as:"t < c" and "c - e / 3 / norm (f c) < t"
```
```  3609       hence "c - t < e / 3 / norm (f c)" by auto
```
```  3610       hence "norm (c - t) < e / 3 / norm (f c)" using as by auto
```
```  3611       thus "norm (f c) * norm (c - t) < e / 3" using False apply-
```
```  3612         apply(subst mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
```
```  3613     qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
```
```  3614   qed then guess w .. note w = conjunctD2[OF this,rule_format]
```
```  3615
```
```  3616   have *:"e / 3 > 0" using assms by auto
```
```  3617   have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
```
```  3618   from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
```
```  3619   note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x"
```
```  3620   have "gauge d" unfolding d_def using w(1) d1 by auto
```
```  3621   note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this]
```
```  3622   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
```
```  3623
```
```  3624   let ?d = "min k (c - a)/2" show ?thesis apply(rule that[of ?d])
```
```  3625   proof safe show "?d > 0" using k(1) using assms(2) by auto
```
```  3626     fix t assume as:"c - ?d < t" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
```
```  3627     { presume *:"t < c \<Longrightarrow> ?thesis"
```
```  3628       show ?thesis apply(cases "t = c") defer apply(rule *)
```
```  3629         apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
```
```  3630
```
```  3631     have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
```
```  3632     from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
```
```  3633     note d2 = conjunctD2[OF this,rule_format]
```
```  3634     def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x"
```
```  3635     have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto
```
```  3636     from fine_division_exists[OF this, of a t] guess p . note p=this
```
```  3637     note p'=tagged_division_ofD[OF this(1)]
```
```  3638     have pt:"\<forall>(x,k)\<in>p. x \<le> t" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
```
```  3639     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
```
```  3640     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
```
```  3641
```
```  3642     have *:"{a..c} \<inter> {x. x \$\$0 \<le> t} = {a..t}" "{a..c} \<inter> {x. x\$\$0 \<ge> t} = {t..c}"
```
```  3643       using assms(2-3) as by(auto simp add:field_simps)
```
```  3644     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
```
```  3645       apply(rule tagged_division_union_interval[of _ _ _ 0 "t"]) unfolding * apply(rule p)
```
```  3646       apply(rule tagged_division_of_self) unfolding fine_def
```
```  3647     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
```
```  3648         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
```
```  3649     next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real_def
```
```  3650         using as(1) by(auto simp add:field_simps)
```
```  3651       thus "x \<in> d1 c" using k(2) unfolding d_def by auto
```
```  3652     qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
```
```  3653
```
```  3654     have *:"integral{a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
```
```  3655         integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c"
```
```  3656       "e = (e/3 + e/3) + e/3" by auto
```
```  3657     have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  3658     proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
```
```  3659       have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
```
```  3660         have "c \<in> {a..t}" by auto thus False using `t<c` by auto
```
```  3661       qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
```
```  3662         unfolding split_conv defer apply(subst content_real) using as(2) by auto qed
```
```  3663
```
```  3664     have ***:"c - w < t \<and> t < c"
```
```  3665     proof- have "c - k < t" using `k>0` as(1) by(auto simp add:field_simps)
```
```  3666       moreover have "k \<le> w" apply(rule ccontr) using k(2)
```
```  3667         unfolding subset_eq apply(erule_tac x="c + ((k + w)/2)" in ballE)
```
```  3668         unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real_def)
```
```  3669       ultimately show  ?thesis using `t<c` by(auto simp add:field_simps) qed
```
```  3670
```
```  3671     show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
```
```  3672       unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
```
```  3673       using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed
```
```  3674
```
```  3675 lemma indefinite_integral_continuous_right: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3676   assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
```
```  3677   obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e"
```
```  3678 proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a" using assms by auto
```
```  3679   from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b - c)"
```
```  3680   show ?thesis apply(rule that[of "?d"])
```
```  3681   proof safe show "0 < ?d" using d(1) assms(3) by auto
```
```  3682     fix t::"real" assume as:"c \<le> t" "t < c + ?d"
```
```  3683     have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
```
```  3684       "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
```
```  3685       apply(rule_tac[!] integral_combine) using assms as by auto
```
```  3686     have "(- c) - d < (- t) \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
```
```  3687     thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding *
```
```  3688       unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
```
```  3689
```
```  3690 lemma indefinite_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3691   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
```
```  3692 proof(unfold continuous_on_iff, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
```
```  3693   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"
```
```  3694   { presume *:"a<b \<Longrightarrow> ?thesis"
```
```  3695     show ?thesis apply(cases,rule *,assumption)
```
```  3696     proof- case goal1 hence "{a..b} = {x}" using as(1) apply-apply(rule set_eqI)
```
```  3697         unfolding atLeastAtMost_iff by(auto simp only:field_simps not_less DIM_real)
```
```  3698       thus ?case using `e>0` by auto
```
```  3699     qed } assume "a<b"
```
```  3700   have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add:)
```
```  3701   thus ?thesis apply-apply(erule disjE)+
```
```  3702   proof- assume "x=a" have "a \<le> a" by auto
```
```  3703     from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
```
```  3704     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
```
```  3705       unfolding `x=a` dist_norm apply(rule d(2)[rule_format]) by auto
```
```  3706   next   assume "x=b" have "b \<le> b" by auto
```
```  3707     from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
```
```  3708     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
```
```  3709       unfolding `x=b` dist_norm apply(rule d(2)[rule_format])  by auto
```
```  3710   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: )
```
```  3711     from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
```
```  3712     from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
```
```  3713     show ?thesis apply(rule_tac x="min d1 d2" in exI)
```
```  3714     proof safe show "0 < min d1 d2" using d1 d2 by auto
```
```  3715       fix y assume "y\<in>{a..b}" "dist y x < min d1 d2"
```
```  3716       thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute)
```
```  3717         apply(cases "y < x") unfolding dist_norm apply(rule d1(2)[rule_format]) defer
```
```  3718         apply(rule d2(2)[rule_format]) unfolding not_less by(auto simp add:field_simps)
```
```  3719     qed qed qed
```
```  3720
```
```  3721 subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
```
```  3722
```
```  3723 lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach"
```
```  3724   assumes "finite k" "continuous_on {a..b} f" "f a = y"
```
```  3725   "\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
```
```  3726   shows "f x = y"
```
```  3727 proof- have ab:"a\<le>b" using assms by auto
```
```  3728   have *:"a\<le>x" using assms(5) by auto
```
```  3729   have "((\<lambda>x. 0\<Colon>'a) has_integral f x - f a) {a..x}"
```
```  3730     apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
```
```  3731     apply(rule continuous_on_subset[OF assms(2)]) defer
```
```  3732     apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
```
```  3733     apply assumption apply(rule open_interval) apply(rule has_derivative_within_subset[where s="{a..b}"])
```
```  3734     using assms(4) assms(5) by auto note this[unfolded *]
```
```  3735   note has_integral_unique[OF has_integral_0 this]
```
```  3736   thus ?thesis unfolding assms by auto qed
```
```  3737
```
```  3738 subsection {* Generalize a bit to any convex set. *}
```
```  3739
```
```  3740 lemma has_derivative_zero_unique_strong_convex: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  3741   assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
```
```  3742   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
```
```  3743   shows "f x = y"
```
```  3744 proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
```
```  3745       unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
```
```  3746   note conv = assms(1)[unfolded convex_alt,rule_format]
```
```  3747   have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
```
```  3748     apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
```
```  3749     apply safe apply(rule conv) using assms(4,7) by auto
```
```  3750   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"
```
```  3751   proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
```
```  3752       unfolding scaleR_simps by(auto simp add:algebra_simps)
```
```  3753     thus ?case using `x\<noteq>c` by auto qed
```
```  3754   have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2)
```
```  3755     apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
```
```  3756     apply safe unfolding image_iff apply rule defer apply assumption
```
```  3757     apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
```
```  3758   have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
```
```  3759     apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
```
```  3760     unfolding o_def using assms(5) defer apply-apply(rule)
```
```  3761   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
```
```  3762     have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps])
```
```  3763       using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
```
```  3764     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})"
```
```  3765       apply(rule diff_chain_within) apply(rule has_derivative_add)
```
```  3766       unfolding scaleR_simps
```
```  3767       apply(intro has_derivative_intros)
```
```  3768       apply(intro has_derivative_intros)
```
```  3769       apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
```
```  3770       apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
```
```  3771     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 .
```
```  3772   qed auto thus ?thesis by auto qed
```
```  3773
```
```  3774 subsection {* Also to any open connected set with finite set of exceptions. Could
```
```  3775  generalize to locally convex set with limpt-free set of exceptions. *}
```
```  3776
```
```  3777 lemma has_derivative_zero_unique_strong_connected: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  3778   assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
```
```  3779   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
```
```  3780   shows "f x = y"
```
```  3781 proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
```
```  3782     apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
```
```  3783     apply(rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
```
```  3784     apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
```
```  3785   proof safe fix x assume "x\<in>s"
```
```  3786     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
```
```  3787     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
```
```  3788     proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
```
```  3789       show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball])
```
```  3790         apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+
```
```  3791         apply(subst centre_in_ball,rule e,rule) apply safe
```
```  3792         apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format])
```
```  3793         using y e by auto qed qed
```
```  3794   thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed
```
```  3795
```
```  3796 subsection {* Integrating characteristic function of an interval. *}
```
```  3797
```
```  3798 lemma has_integral_restrict_open_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  3799   assumes "(f has_integral i) {c..d}" "{c..d} \<subseteq> {a..b}"
```
```  3800   shows "((\<lambda>x. if x \<in> {c<..<d} then f x else 0) has_integral i) {a..b}"
```
```  3801 proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
```
```  3802   { presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
```
```  3803     show ?thesis apply(cases,rule *,assumption)
```
```  3804     proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto
```
```  3805       show ?thesis using assms(1) unfolding * using goal1 by auto
```
```  3806     qed } assume "{c..d}\<noteq>{}"
```
```  3807   from partial_division_extend_1[OF assms(2) this] guess p . note p=this
```
```  3808   note mon = monoidal_lifted[OF monoidal_monoid]
```
```  3809   note operat = operative_division[OF this operative_integral p(1), THEN sym]
```
```  3810   let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
```
```  3811   { presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
```
```  3812       apply- apply(cases,subst(asm) if_P,assumption) by auto
```
```  3813     thus ?thesis using integrable_integral unfolding g_def by auto }
```
```  3814
```
```  3815   note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
```
```  3816   note * = this[unfolded neutral_monoid]
```
```  3817   have iterate:"iterate (lifted op +) (p - {{c..d}})
```
```  3818       (\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0"
```
```  3819   proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this]
```
```  3820     from div(3) guess u v apply-by(erule exE)+ note uv=this
```
```  3821     have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto
```
```  3822     hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"])
```
```  3823       unfolding g_def interior_closed_interval by auto thus ?case by auto
```
```  3824   qed
```
```  3825
```
```  3826   have *:"p = insert {c..d} (p - {{c..d}})" using p by auto
```
```  3827   have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f])
```
```  3828     unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto
```
```  3829   moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)])
```
```  3830     apply(rule has_integral_spike_interior[where f=g]) defer
```
```  3831     apply(rule integrable_integral[OF **]) unfolding g_def by auto
```
```  3832   ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+
```
```  3833     unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
```
```  3834
```
```  3835 lemma has_integral_restrict_closed_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
```
```  3836   assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}"
```
```  3837   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
```
```  3838 proof- note has_integral_restrict_open_subinterval[OF assms]
```
```  3839   note * = has_integral_spike[OF negligible_frontier_interval _ this]
```
```  3840   show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
```
```  3841
```
```  3842 lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}"
```
```  3843   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")
```
```  3844 proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
```
```  3845   show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
```
```  3846   proof assumption assume ?l hence "?g integrable_on {c..d}"
```
```  3847       apply-apply(rule integrable_subinterval[OF _ assms]) by auto
```
```  3848     hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto
```
```  3849     hence "i = integral {c..d} f" apply-apply(rule has_integral_unique)
```
```  3850       apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto
```
```  3851     thus ?r using * by auto qed qed auto
```
```  3852
```
```  3853 subsection {* Hence we can apply the limit process uniformly to all integrals. *}
```
```  3854
```
```  3855 lemma has_integral': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  3856  "(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
```
```  3857   \<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)")
```
```  3858 proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis"
```
```  3859     show ?thesis apply(cases,rule *,assumption)
```
```  3860       apply(subst has_integral_alt) by auto }
```
```  3861   assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this
```
```  3862   from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B ..
```
```  3863   note B = conjunctD2[OF this,rule_format] show ?thesis apply safe
```
```  3864   proof- fix e assume ?l "e>(0::real)"
```
```  3865     show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI)
```
```  3866     proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::'n::ordered_euclidean_space}"
```
```  3867       thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s
```
```  3868         apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s])
```
```  3869         apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
```
```  3870         by(auto simp add:dist_norm)
```
```  3871     qed(insert B `e>0`, auto)
```
```  3872   next assume as:"\<forall>e>0. ?r e"
```
```  3873     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
```
```  3874     def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
```
```  3875     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
```
```  3876     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
```
```  3877         by(auto simp add:field_simps) qed
```
```  3878     have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
```
```  3879     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
```
```  3880     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
```
```  3881       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
```
```  3882     then guess y .. note y=this
```
```  3883
```
```  3884     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
```
```  3885       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
```
```  3886       def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
```
```  3887       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
```
```  3888       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
```
```  3889           by(auto simp add:field_simps) qed
```
```  3890       have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm
```
```  3891       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
```
```  3892       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
```
```  3893       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
```
```  3894       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
```
```  3895       thus False by auto qed
```
```  3896     thus ?l using y unfolding s by auto qed qed
```
```  3897
```
```  3898 (*lemma has_integral_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
```
```  3899   "((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s"
```
```  3900   unfolding has_integral'[unfolded has_integral]
```
```  3901 proof case goal1 thus ?case apply safe
```
```  3902   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
```
```  3903   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe)
```
```  3904   apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe)
```
```  3905   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
```
```  3906   apply(subst(asm)(2) norm_vector_1) unfolding split_def
```
```  3907   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
```
```  3908     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
```
```  3909   apply(subst(asm)(2) norm_vector_1) by auto
```
```  3910 next case goal2 thus ?case apply safe
```
```  3911   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
```
```  3912   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe)
```
```  3913   apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe)
```
```  3914   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
```
```  3915   apply(subst norm_vector_1) unfolding split_def
```
```  3916   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
```
```  3917     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
```
```  3918   apply(subst norm_vector_1) by auto qed
```
```  3919
```
```  3920 lemma integral_trans[simp]: assumes "(f::'n::ordered_euclidean_space \<Rightarrow> real) integrable_on s"
```
```  3921   shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)"
```
```  3922   apply(rule integral_unique) using assms by auto
```
```  3923
```
```  3924 lemma integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
```
```  3925   "(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)"
```
```  3926   unfolding integrable_on_def
```
```  3927   apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans
```
```  3928   apply safe defer apply(rule_tac x="vec1 y" in exI) by auto *)
```
```  3929
```
```  3930 lemma has_integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  3931   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x) \<le> (g x)"
```
```  3932   shows "i \<le> j" using has_integral_component_le[OF assms(1-2), of 0] using assms(3) by auto
```
```  3933
```
```  3934 lemma integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  3935   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
```
```  3936   shows "integral s f \<le> integral s g"
```
```  3937   using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] .
```
```  3938
```
```  3939 lemma has_integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  3940   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
```
```  3941   using has_integral_component_nonneg[of "f" "i" s 0]
```
```  3942   unfolding o_def using assms by auto
```
```  3943
```
```  3944 lemma integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  3945   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
```
```  3946   using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
```
```  3947
```
```  3948 subsection {* Hence a general restriction property. *}
```
```  3949
```
```  3950 lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows
```
```  3951   "((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
```
```  3952 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
```
```  3953   show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed
```
```  3954
```
```  3955 lemma has_integral_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  3956   "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
```
```  3957
```
```  3958 lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3959   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
```
```  3960   shows "(f has_integral i) t"
```
```  3961 proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
```
```  3962     apply(rule) using assms(1-2) by auto
```
```  3963   thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
```
```  3964   apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
```
```  3965
```
```  3966 lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3967   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
```
```  3968   shows "f integrable_on t"
```
```  3969   using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
```
```  3970
```
```  3971 lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3972   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
```
```  3973   apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
```
```  3974
```
```  3975 lemma integrable_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  3976  "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
```
```  3977   unfolding integrable_on_def by auto
```
```  3978
```
```  3979 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r")
```
```  3980 proof assume ?r show ?l unfolding negligible_def
```
```  3981   proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
```
```  3982       unfolding indicator_def by auto qed qed auto
```
```  3983
```
```  3984 lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3985   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
```
```  3986   unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (safe, auto split: split_if_asm)
```
```  3987
```
```  3988 lemma has_integral_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3989   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
```
```  3990   shows "(f has_integral y) t"
```
```  3991   using assms has_integral_spike_set_eq by auto
```
```  3992
```
```  3993 lemma integrable_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3994   assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
```
```  3995   shows "f integrable_on t" using assms(2) unfolding integrable_on_def
```
```  3996   unfolding has_integral_spike_set_eq[OF assms(1)] .
```
```  3997
```
```  3998 lemma integrable_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  3999   assumes "negligible((s - t) \<union> (t - s))"
```
```  4000   shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)"
```
```  4001   apply(rule,rule_tac[!] integrable_spike_set) using assms by auto
```
```  4002
```
```  4003 (*lemma integral_spike_set:
```
```  4004  "\<forall>f:real^M->real^N g s t.
```
```  4005         negligible(s DIFF t \<union> t DIFF s)
```
```  4006         \<longrightarrow> integral s f = integral t f"
```
```  4007 qed  REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
```
```  4008   AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  4009   ASM_MESON_TAC[]);;
```
```  4010
```
```  4011 lemma has_integral_interior:
```
```  4012  "\<forall>f:real^M->real^N y s.
```
```  4013         negligible(frontier s)
```
```  4014         \<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
```
```  4015 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  4016   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
```
```  4017     NEGLIGIBLE_SUBSET)) THEN
```
```  4018   REWRITE_TAC[frontier] THEN
```
```  4019   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
```
```  4020   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
```
```  4021   SET_TAC[]);;
```
```  4022
```
```  4023 lemma has_integral_closure:
```
```  4024  "\<forall>f:real^M->real^N y s.
```
```  4025         negligible(frontier s)
```
```  4026         \<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
```
```  4027 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
```
```  4028   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
```
```  4029     NEGLIGIBLE_SUBSET)) THEN
```
```  4030   REWRITE_TAC[frontier] THEN
```
```  4031   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
```
```  4032   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
```
```  4033   SET_TAC[]);;*)
```
```  4034
```
```  4035 subsection {* More lemmas that are useful later. *}
```
```  4036
```
```  4037 lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
```
```  4038   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\$\$k"
```
```  4039   shows "i\$\$k \<le> j\$\$k"
```
```  4040 proof- note has_integral_restrict_univ[THEN sym, of f]
```
```  4041   note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
```
```  4042   show ?thesis apply(rule *) using assms(1,4) by auto qed
```
```  4043
```
```  4044 lemma has_integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4045   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
```
```  4046   shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "f" "i" "j" 0] using assms by auto
```
```  4047
```
```  4048 lemma integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
```
```  4049   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)\$\$k"
```
```  4050   shows "(integral s f)\$\$k \<le> (integral t f)\$\$k"
```
```  4051   apply(rule has_integral_subset_component_le) using assms by auto
```
```  4052
```
```  4053 lemma integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4054   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)"
```
```  4055   shows "(integral s f) \<le> (integral t f)"
```
```  4056   apply(rule has_integral_subset_le) using assms by auto
```
```  4057
```
```  4058 lemma has_integral_alt': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4059   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>
```
```  4060   (\<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")
```
```  4061 proof assume ?r
```
```  4062   show ?l apply- apply(subst has_integral')
```
```  4063   proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
```
```  4064     show ?case apply(rule,rule,rule B,safe)
```
```  4065       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI)
```
```  4066       apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto
```
```  4067   qed next
```
```  4068   assume ?l note as = this[unfolded has_integral'[of f],rule_format]
```
```  4069   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
```
```  4070   show ?r proof safe fix a b::"'n::ordered_euclidean_space"
```
```  4071     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  4072     let ?a = "(\<chi>\<chi> i. min (a\$\$i) (-B))::'n::ordered_euclidean_space" and ?b = "(\<chi>\<chi> i. max (b\$\$i) B)::'n::ordered_euclidean_space"
```
```  4073     show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
```
```  4074     proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval dist_norm
```
```  4075       proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
```
```  4076       from B(2)[OF this] guess z .. note conjunct1[OF this]
```
```  4077       thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
```
```  4078       show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
```
```  4079
```
```  4080     fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  4081     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
```
```  4082                     norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
```
```  4083     proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
```
```  4084       from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed
```
```  4085
```
```  4086
```
```  4087 subsection {* Continuity of the integral (for a 1-dimensional interval). *}
```
```  4088
```
```  4089 lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  4090   "f integrable_on s \<longleftrightarrow>
```
```  4091           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
```
```  4092           (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
```
```  4093   \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) -
```
```  4094           integral {c..d}  (\<lambda>x. if x \<in> s then f x else 0)) < e)" (is "?l = ?r")
```
```  4095 proof assume ?l then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
```
```  4096   note y=conjunctD2[OF this,rule_format] show ?r apply safe apply(rule y)
```
```  4097   proof- case goal1 hence "e/2 > 0" by auto from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  4098     show ?case apply(rule,rule,rule B)
```
```  4099     proof safe case goal1 show ?case apply(rule norm_triangle_half_l)
```
```  4100         using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
```
```  4101
```
```  4102 next assume ?r note as = conjunctD2[OF this,rule_format]
```
```  4103   have "Cauchy (\<lambda>n. integral ({(\<chi>\<chi> i. - real n)::'n .. (\<chi>\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))"
```
```  4104   proof(unfold Cauchy_def,safe) case goal1
```
```  4105     from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
```
```  4106     from real_arch_simple[of B] guess N .. note N = this
```
```  4107     { fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {(\<chi>\<chi> i. - real n)::'n..\<chi>\<chi> i. real n}" apply safe
```
```  4108         unfolding mem_ball mem_interval dist_norm
```
```  4109       proof case goal1 thus ?case using component_le_norm[of x i]
```
```  4110           using n N by(auto simp add:field_simps) qed }
```
```  4111     thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto
```
```  4112   qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
```
```  4113   note i = this[unfolded Lim_sequentially, rule_format]
```
```  4114
```
```  4115   show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
```
```  4116     apply safe apply(rule as(1)[unfolded integrable_on_def])
```
```  4117   proof- case goal1 hence *:"e/2 > 0" by auto
```
```  4118     from i[OF this] guess N .. note N =this[rule_format]
```
```  4119     from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format] let ?B = "max (real N) B"
```
```  4120     show ?case apply(rule_tac x="?B" in exI)
```
```  4121     proof safe show "0 < ?B" using B(1) by auto
```
```  4122       fix a b assume ab:"ball 0 ?B \<subseteq> {a..b::'n::ordered_euclidean_space}"
```
```  4123       from real_arch_simple[of ?B] guess n .. note n=this
```
```  4124       show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
```
```  4125         apply(rule norm_triangle_half_l) apply(rule B(2)) defer apply(subst norm_minus_commute)
```
```  4126         apply(rule N[unfolded dist_norm, of n])
```
```  4127       proof safe show "N \<le> n" using n by auto
```
```  4128         fix x::"'n::ordered_euclidean_space" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
```
```  4129         thus "x\<in>{a..b}" using ab by blast
```
```  4130         show "x\<in>{\<chi>\<chi> i. - real n..\<chi>\<chi> i. real n}" using x unfolding mem_interval mem_ball dist_norm apply-
```
```  4131         proof case goal1 thus ?case using component_le_norm[of x i]
```
```  4132             using n by(auto simp add:field_simps) qed qed qed qed qed
```
```  4133
```
```  4134 lemma integrable_altD: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4135   assumes "f integrable_on s"
```
```  4136   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
```
```  4137   "\<And>e. e>0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
```
```  4138   \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - integral {c..d}  (\<lambda>x. if x \<in> s then f x else 0)) < e"
```
```  4139   using assms[unfolded integrable_alt[of f]] by auto
```
```  4140
```
```  4141 lemma integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4142   assumes "f integrable_on s" "{a..b} \<subseteq> s" shows "f integrable_on {a..b}"
```
```  4143   apply(rule integrable_eq) defer apply(rule integrable_altD(1)[OF assms(1)])
```
```  4144   using assms(2) by auto
```
```  4145
```
```  4146 subsection {* A straddling criterion for integrability. *}
```
```  4147
```
```  4148 lemma integrable_straddle_interval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4149   assumes "\<forall>e>0. \<exists>g  h i j. (g has_integral i) ({a..b}) \<and> (h has_integral j) ({a..b}) \<and>
```
```  4150   norm(i - j) < e \<and> (\<forall>x\<in>{a..b}. (g x) \<le> (f x) \<and> (f x) \<le>(h x))"
```
```  4151   shows "f integrable_on {a..b}"
```
```  4152 proof(subst integrable_cauchy,safe)
```
```  4153   case goal1 hence e:"e/3 > 0" by auto note assms[rule_format,OF this]
```
```  4154   then guess g h i j apply- by(erule exE conjE)+ note obt = this
```
```  4155   from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
```
```  4156   from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
```
```  4157   show ?case apply(rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI) apply(rule conjI gauge_inter d1 d2)+ unfolding fine_inter
```
```  4158   proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
```
```  4159       abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow>  abs(g1 - i) < e / 3 \<Longrightarrow>
```
```  4160       abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith
```
```  4161     case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)]
```
```  4162
```
```  4163     have "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
```
```  4164       "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
```
```  4165       "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
```
```  4166       "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
```
```  4167       unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg)
```
```  4168       apply safe unfolding real_scaleR_def mult.diff_right[THEN sym]
```
```  4169       apply(rule_tac[!] mult_nonneg_nonneg)
```
```  4170     proof- fix a b assume ab:"(a,b) \<in> p1"
```
```  4171       show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
```
```  4172       show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(1-2)[OF ab] using obt(4)[rule_format,of a] by auto
```
```  4173     next fix a b assume ab:"(a,b) \<in> p2"
```
```  4174       show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
```
```  4175       show "0 \<le> f a - g a" "0 \<le> h a - f a" using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto qed
```
```  4176
```
```  4177     thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
```
```  4178       unfolding real_norm_def[THEN sym] apply(rule obt(3))
```
```  4179       apply(rule d1(2)[OF conjI[OF goal1(4,5)]])
```
```  4180       apply(rule d1(2)[OF conjI[OF goal1(1,2)]])
```
```  4181       apply(rule d2(2)[OF conjI[OF goal1(4,6)]])
```
```  4182       apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed
```
```  4183
```
```  4184 lemma integrable_straddle: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4185   assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
```
```  4186   norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))"
```
```  4187   shows "f integrable_on s"
```
```  4188 proof- have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
```
```  4189   proof(rule integrable_straddle_interval,safe) case goal1 hence *:"e/4 > 0" by auto
```
```  4190     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
```
```  4191     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
```
```  4192     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
```
```  4193     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
```
```  4194     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
```
```  4195     def c \<equiv> "(\<chi>\<chi> i. min (a\$\$i) (- (max B1 B2)))::'n" and d \<equiv> "(\<chi>\<chi> i. max (b\$\$i) (max B1 B2))::'n"
```
```  4196     have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval dist_norm
```
```  4197     proof(rule_tac[!] allI)
```
```  4198       case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next
```
```  4199       case goal2 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
```
```  4200     have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow>
```
```  4201       norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e"
```
```  4202       using obt(3) unfolding real_norm_def by arith
```
```  4203     show ?case apply(rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
```
```  4204                apply(rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
```
```  4205       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI)
```
```  4206       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then h x else 0)" in exI)
```
```  4207       apply safe apply(rule_tac[1-2] integrable_integral,rule g,rule h)
```
```  4208       apply(rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
```
```  4209     proof- have *:"\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
```
```  4210         (if x \<in> s then f x - g x else (0::real))" by auto
```
```  4211       note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_sub, OF h g]]
```
```  4212       show " norm (integral {a..b} (\<lambda>x. if x \<in> s then h x else 0) -
```
```  4213                    integral {a..b} (\<lambda>x. if x \<in> s then g x else 0))
```
```  4214            \<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) -
```
```  4215                    integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))"
```
```  4216         unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
```
```  4217         apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+
```
```  4218       proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
```
```  4219           apply - apply rule apply(erule_tac x=i in allE) by auto
```
```  4220       qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this
```
```  4221
```
```  4222   show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv)
```
```  4223   proof- case goal1 hence *:"e/3 > 0" by auto
```
```  4224     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
```
```  4225     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
```
```  4226     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
```
```  4227     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
```
```  4228     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
```
```  4229     show ?case apply(rule_tac x="max B1 B2" in exI) apply safe apply(rule min_max.less_supI1,rule B1)
```
```  4230     proof- fix a b c d::"'n::ordered_euclidean_space" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}"
```
```  4231       have **:"ball 0 B1 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" by auto
```
```  4232       have *:"\<And>ga gc ha hc fa fc::real. abs(ga - i) < e / 3 \<and> abs(gc - i) < e / 3 \<and> abs(ha - j) < e / 3 \<and>
```
```  4233         abs(hc - j) < e / 3 \<and> abs(i - j) < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc\<Longrightarrow> abs(fa - fc) < e" by smt
```
```  4234       show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - integral {c..d} (\<lambda>x. if x \<in> s then f x else 0)) < e"
```
```  4235         unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
```
```  4236         apply(rule B1(2),rule order_trans,rule **,rule as(1))
```
```  4237         apply(rule B1(2),rule order_trans,rule **,rule as(2))
```
```  4238         apply(rule B2(2),rule order_trans,rule **,rule as(1))
```
```  4239         apply(rule B2(2),rule order_trans,rule **,rule as(2))
```
```  4240         apply(rule obt) apply(rule_tac[!] integral_le) using obt
```
```  4241         by(auto intro!: h g interv) qed qed qed
```
```  4242
```
```  4243 subsection {* Adding integrals over several sets. *}
```
```  4244
```
```  4245 lemma has_integral_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4246   assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)"
```
```  4247   shows "(f has_integral (i + j)) (s \<union> t)"
```
```  4248 proof- note * = has_integral_restrict_univ[THEN sym, of f]
```
```  4249   show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)])
```
```  4250     defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed
```
```  4251
```
```  4252 lemma has_integral_unions: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4253   assumes "finite t" "\<forall>s\<in>t. (f has_integral (i s)) s"  "\<forall>s\<in>t. \<forall>s'\<in>t. ~(s = s') \<longrightarrow> negligible(s \<inter> s')"
```
```  4254   shows "(f has_integral (setsum i t)) (\<Union>t)"
```
```  4255 proof- note * = has_integral_restrict_univ[THEN sym, of f]
```
```  4256   have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))"
```
```  4257     apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer
```
```  4258     apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto
```
```  4259   note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this]
```
```  4260   thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption
```
```  4261   proof safe case goal1 thus ?case
```
```  4262     proof(cases "x\<in>\<Union>t") case True then guess s unfolding Union_iff .. note s=this
```
```  4263       hence *:"\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s" using goal1(3) by blast
```
```  4264       show ?thesis unfolding if_P[OF True] apply(rule trans) defer
```
```  4265         apply(rule setsum_cong2) apply(subst *, assumption) apply(rule refl)
```
```  4266         unfolding setsum_delta[OF assms(1)] using s by auto qed auto qed qed
```
```  4267
```
```  4268 subsection {* In particular adding integrals over a division, maybe not of an interval. *}
```
```  4269
```
```  4270 lemma has_integral_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4271   assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k"
```
```  4272   shows "(f has_integral (setsum i d)) s"
```
```  4273 proof- note d = division_ofD[OF assms(1)]
```
```  4274   show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions)
```
```  4275     apply(rule d assms)+ apply(rule,rule,rule)
```
```  4276   proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)]
```
```  4277     guess a c b d apply-by(erule exE)+ note obt=this
```
```  4278     from d(5)[OF goal1] show ?case unfolding obt interior_closed_interval
```
```  4279       apply-apply(rule negligible_subset[of "({a..b}-{a<..<b}) \<union> ({c..d}-{c<..<d})"])
```
```  4280       apply(rule negligible_union negligible_frontier_interval)+ by auto qed qed
```
```  4281
```
```  4282 lemma integral_combine_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4283   assumes "d division_of s" "\<forall>k\<in>d. f integrable_on k"
```
```  4284   shows "integral s f = setsum (\<lambda>i. integral i f) d"
```
```  4285   apply(rule integral_unique) apply(rule has_integral_combine_division[OF assms(1)])
```
```  4286   using assms(2) unfolding has_integral_integral .
```
```  4287
```
```  4288 lemma has_integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4289   assumes "f integrable_on s" "d division_of k" "k \<subseteq> s"
```
```  4290   shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
```
```  4291   apply(rule has_integral_combine_division[OF assms(2)])
```
```  4292   apply safe unfolding has_integral_integral[THEN sym]
```
```  4293 proof- case goal1 from division_ofD(2,4)[OF assms(2) this]
```
```  4294   show ?case apply safe apply(rule integrable_on_subinterval)
```
```  4295     apply(rule assms) using assms(3) by auto qed
```
```  4296
```
```  4297 lemma integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4298   assumes "f integrable_on s" "d division_of s"
```
```  4299   shows "integral s f = setsum (\<lambda>i. integral i f) d"
```
```  4300   apply(rule integral_unique,rule has_integral_combine_division_topdown) using assms by auto
```
```  4301
```
```  4302 lemma integrable_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4303   assumes "d division_of s" "\<forall>i\<in>d. f integrable_on i"
```
```  4304   shows "f integrable_on s"
```
```  4305   using assms(2) unfolding integrable_on_def
```
```  4306   by(metis has_integral_combine_division[OF assms(1)])
```
```  4307
```
```  4308 lemma integrable_on_subdivision: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4309   assumes "d division_of i" "f integrable_on s" "i \<subseteq> s"
```
```  4310   shows "f integrable_on i"
```
```  4311   apply(rule integrable_combine_division assms)+
```
```  4312 proof safe case goal1 note division_ofD(2,4)[OF assms(1) this]
```
```  4313   thus ?case apply safe apply(rule integrable_on_subinterval[OF assms(2)])
```
```  4314     using assms(3) by auto qed
```
```  4315
```
```  4316 subsection {* Also tagged divisions. *}
```
```  4317
```
```  4318 lemma has_integral_combine_tagged_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4319   assumes "p tagged_division_of s" "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
```
```  4320   shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
```
```  4321 proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
```
```  4322     apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)])
```
```  4323     using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto)
```
```  4324   thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption
```
```  4325     apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute)
```
```  4326     apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption)
```
```  4327     apply(rule setsum_cong2) using assms(2) by auto qed
```
```  4328
```
```  4329 lemma integral_combine_tagged_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4330   assumes "p tagged_division_of {a..b}" "\<forall>(x,k)\<in>p. f integrable_on k"
```
```  4331   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
```
```  4332   apply(rule integral_unique) apply(rule has_integral_combine_tagged_division[OF assms(1)])
```
```  4333   using assms(2) by auto
```
```  4334
```
```  4335 lemma has_integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4336   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
```
```  4337   shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) {a..b}"
```
```  4338   apply(rule has_integral_combine_tagged_division[OF assms(2)])
```
```  4339 proof safe case goal1 note tagged_division_ofD(3-4)[OF assms(2) this]
```
```  4340   thus ?case using integrable_subinterval[OF assms(1)] by auto qed
```
```  4341
```
```  4342 lemma integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4343   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
```
```  4344   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
```
```  4345   apply(rule integral_unique,rule has_integral_combine_tagged_division_topdown) using assms by auto
```
```  4346
```
```  4347 subsection {* Henstock's lemma. *}
```
```  4348
```
```  4349 lemma henstock_lemma_part1: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4350   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
```
```  4351   "(\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral({a..b}) f) < e)"
```
```  4352   and p:"p tagged_partial_division_of {a..b}" "d fine p"
```
```  4353   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e" (is "?x \<le> e")
```
```  4354 proof-  { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" thus ?thesis by (blast intro: field_le_epsilon) }
```
```  4355   fix k::real assume k:"k>0" note p' = tagged_partial_division_ofD[OF p(1)]
```
```  4356   have "\<Union>snd ` p \<subseteq> {a..b}" using p'(3) by fastsimp
```
```  4357   note partial_division_of_tagged_division[OF p(1)] this
```
```  4358   from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
```
```  4359   def r \<equiv> "q - snd ` p" have "snd ` p \<inter> r = {}" unfolding r_def by auto
```
```  4360   have r:"finite r" using q' unfolding r_def by auto
```
```  4361
```
```  4362   have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
```
```  4363     norm(setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
```
```  4364   proof safe case goal1 hence i:"i \<in> q" unfolding r_def by auto
```
```  4365     from q'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  4366     have *:"k / (real (card r) + 1) > 0" apply(rule divide_pos_pos,rule k) by auto
```
```  4367     have "f integrable_on {u..v}" apply(rule integrable_subinterval[OF assms(1)])
```
```  4368       using q'(2)[OF i] unfolding uv by auto
```
```  4369     note integrable_integral[OF this, unfolded has_integral[of f]]
```
```  4370     from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
```
```  4371     note gauge_inter[OF `gauge d` dd(1)] from fine_division_exists[OF this,of u v] guess qq .
```
```  4372     thus ?case apply(rule_tac x=qq in exI) using dd(2)[of qq] unfolding fine_inter uv by auto qed
```
```  4373   from bchoice[OF this] guess qq .. note qq=this[rule_format]
```
```  4374
```
```  4375   let ?p = "p \<union> \<Union>qq ` r" have "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - integral {a..b} f) < e"
```
```  4376     apply(rule assms(4)[rule_format])
```
```  4377   proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto
```
```  4378     note * = tagged_partial_division_of_union_self[OF p(1)]
```
```  4379     have "p \<union> \<Union>qq ` r tagged_division_of \<Union>snd ` p \<union> \<Union>r"
```
```  4380     proof(rule tagged_division_union[OF * tagged_division_unions])
```
```  4381       show "finite r" by fact case goal2 thus ?case using qq by auto
```
```  4382     next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto
```
```  4383     next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule)
```
```  4384         apply(rule,rule q') defer apply(rule,subst Int_commute)
```
```  4385         apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer
```
```  4386         apply(rule,rule q') using q(1) p' unfolding r_def by auto qed
```
```  4387     moreover have "\<Union>snd ` p \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r"
```
```  4388       unfolding Union_Un_distrib[THEN sym] r_def using q by auto
```
```  4389     ultimately show "?p tagged_division_of {a..b}" by fastsimp qed
```
```  4390
```
```  4391   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>\<Union>qq ` r. content k *\<^sub>R f x) -
```
```  4392     integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3
```
```  4393     apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq)
```
```  4394   proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r"
```
```  4395     note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
```
```  4396     from this(2) guess u v apply-by(erule exE)+ note uv=this
```
```  4397     have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
```
```  4398     hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto
```
```  4399     note q'(5)[OF this] hence "interior l = {}" using subset_interior[OF `l \<subseteq> k`] by blast
```
```  4400     thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto
```
```  4401
```
```  4402   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
```
```  4403     (qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero)
```
```  4404     prefer 4 apply assumption apply(rule finite_imageI,fact)
```
```  4405     unfolding split_paired_all split_conv image_iff defer apply(erule bexE)+
```
```  4406   proof- fix x m k l T1 T2 assume "(x,m)\<in>T1" "(x,m)\<in>T2" "T1\<noteq>T2" "k\<in>r" "l\<in>r" "T1 = qq k" "T2 = qq l"
```
```  4407     note as = this(1-5)[unfolded this(6-)] note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
```
```  4408     from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this
```
```  4409     have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q')
```
```  4410       using as unfolding r_def by auto
```
```  4411     have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym]
```
```  4412       apply(rule subset_interior) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto
```
```  4413     thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto
```
```  4414   qed(insert qq, auto)
```
```  4415
```
```  4416   hence **:"norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x) \<circ> qq) r -
```
```  4417     integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact
```
```  4418     apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption
```
```  4419   proof- fix k l x m assume as:"k\<in>r" "l\<in>r" "k\<noteq>l" "qq k = qq l" "(x,m)\<in>qq k"
```
```  4420     note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)]
```
```  4421     show "content m *\<^sub>R f x = 0"  using as(3) unfolding as by auto qed
```
```  4422
```
```  4423   have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow>
```
```  4424     ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k"
```
```  4425   proof- case goal1 thus ?case  using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
```
```  4426       unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
```
```  4427
```
```  4428   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
```
```  4429     unfolding split_def setsum_subtractf ..
```
```  4430   also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"])
```
```  4431   proof- case goal2 have *:"(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
```
```  4432       apply(subst setsum_reindex_nonzero) apply fact
```
```  4433       unfolding split_paired_all snd_conv split_def o_def
```
```  4434     proof- fix x l y m assume as:"(x,l)\<in>p" "(y,m)\<in>p" "(x,l)\<noteq>(y,m)" "l = m"
```
```  4435       from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this
```
```  4436       show "integral l f = 0" unfolding uv apply(rule integral_unique)
```
```  4437         apply(rule has_integral_null) unfolding content_eq_0_interior
```
```  4438         using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto
```
```  4439     qed auto
```
```  4440     show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
```
```  4441       apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto
```
```  4442   next  case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps)
```
```  4443     show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
```
```  4444       unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le,fact)
```
```  4445       apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[THEN sym]
```
```  4446       unfolding divide_inverse[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat)
```
```  4447   qed finally show "?x \<le> e + k" . qed
```
```  4448
```
```  4449 lemma henstock_lemma_part2: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
```
```  4450   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
```
```  4451   "\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p -
```
```  4452           integral({a..b}) f) < e"    "p tagged_partial_division_of {a..b}" "d fine p"
```
```  4453   shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
```
```  4454   unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer
```
```  4455   apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
```
```  4456   apply safe apply(rule assms[rule_format,unfolded split_def]) defer
```
```  4457   apply(rule tagged_partial_division_subset,rule assms,assumption)
```
```  4458   apply(rule fine_subset,assumption,rule assms) using assms(5) by auto
```
```  4459
```
```  4460 lemma henstock_lemma: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
```
```  4461   assumes "f integrable_on {a..b}" "e>0"
```
```  4462   obtains d where "gauge d"
```
```  4463   "\<forall>p. p tagged_partial_division_of {a..b} \<and> d fine p
```
```  4464   \<longrightarrow> setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
```
```  4465 proof- have *:"e / (2 * (real DIM('n) + 1)) > 0" apply(rule divide_pos_pos) using assms(2) by auto
```
```  4466   from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
```
```  4467   guess d .. note d = conjunctD2[OF this] show thesis apply(rule that,rule d)
```
```  4468   proof safe case goal1 note * = henstock_lemma_part2[OF assms(1) * d this]
```
```  4469     show ?case apply(rule le_less_trans[OF *]) using `e>0` by(auto simp add:field_simps) qed qed
```
```  4470
```
```  4471 subsection {* monotone convergence (bounded interval first). *}
```
```  4472
```
```  4473 lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4474   assumes "\<forall>k. (f k) integrable_on {a..b}"
```
```  4475   "\<forall>k. \<forall>x\<in>{a..b}.(f k x) \<le> (f (Suc k) x)"
```
```  4476   "\<forall>x\<in>{a..b}. ((\<lambda>k. f k x) ---> g x) sequentially"
```
```  4477   "bounded {integral {a..b} (f k) | k . k \<in> UNIV}"
```
```  4478   shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially"
```
```  4479 proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
```
```  4480   show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using tendsto_const by auto
```
```  4481 next assume ab:"content {a..b} \<noteq> 0"
```
```  4482   have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) \$\$ 0 \<le> (g x) \$\$ 0"
```
```  4483   proof safe case goal1 note assms(3)[rule_format,OF this]
```
```  4484     note * = Lim_component_ge[OF this trivial_limit_sequentially]
```
```  4485     show ?case apply(rule *) unfolding eventually_sequentially
```
```  4486       apply(rule_tac x=k in exI) apply- apply(rule transitive_stepwise_le)
```
```  4487       using assms(2)[rule_format,OF goal1] by auto qed
```
```  4488   have "\<exists>i. ((\<lambda>k. integral ({a..b}) (f k)) ---> i) sequentially"
```
```  4489     apply(rule bounded_increasing_convergent) defer
```
```  4490     apply rule apply(rule integral_le) apply safe
```
```  4491     apply(rule assms(1-2)[rule_format])+ using assms(4) by auto
```
```  4492   then guess i .. note i=this
```
```  4493   have i':"\<And>k. (integral({a..b}) (f k)) \<le> i\$\$0"
```
```  4494     apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially)
```
```  4495     unfolding eventually_sequentially apply(rule_tac x=k in exI)
```
```  4496     apply(rule transitive_stepwise_le) prefer 3 unfolding Eucl_real_simps apply(rule integral_le)
```
```  4497     apply(rule assms(1-2)[rule_format])+ using assms(2) by auto
```
```  4498
```
```  4499   have "(g has_integral i) {a..b}" unfolding has_integral
```
```  4500   proof safe case goal1 note e=this
```
```  4501     hence "\<forall>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
```
```  4502              norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral {a..b} (f k)) < e / 2 ^ (k + 2)))"
```
```  4503       apply-apply(rule,rule assms(1)[unfolded has_integral_integral has_integral,rule_format])
```
```  4504       apply(rule divide_pos_pos) by auto
```
```  4505     from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
```
```  4506
```
```  4507     have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i\$\$0 - (integral {a..b} (f k)) \<and> i\$\$0 - (integral {a..b} (f k)) < e / 4"
```
```  4508     proof- case goal1 have "e/4 > 0" using e by auto
```
```  4509       from i[unfolded Lim_sequentially,rule_format,OF this] guess r ..
```
```  4510       thus ?case apply(rule_tac x=r in exI) apply rule
```
```  4511         apply(erule_tac x=k in allE)
```
```  4512       proof- case goal1 thus ?case using i'[of k] unfolding dist_real_def by auto qed qed
```
```  4513     then guess r .. note r=conjunctD2[OF this[rule_format]]
```
```  4514
```
```  4515     have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)\$\$0 - (f k x)\$\$0 \<and>
```
```  4516            (g x)\$\$0 - (f k x)\$\$0 < e / (4 * content({a..b}))"
```
```  4517     proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact)
```
```  4518         using ab content_pos_le[of a b] by auto
```
```  4519       from assms(3)[rule_format,OF goal1,unfolded Lim_sequentially,rule_format,OF this]
```
```  4520       guess n .. note n=this
```
```  4521       thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE)
```
```  4522         unfolding dist_real_def using fg[rule_format,OF goal1] by(auto simp add:field_simps) qed
```
```  4523     from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
```
```  4524     def d \<equiv> "\<lambda>x. c (m x) x"
```
```  4525
```
```  4526     show ?case apply(rule_tac x=d in exI)
```
```  4527     proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
```
```  4528     next fix p assume p:"p tagged_division_of {a..b}" "d fine p"
```
```  4529       note p'=tagged_division_ofD[OF p(1)]
```
```  4530       have "\<exists>a. \<forall>x\<in>p. m (fst x) \<le> a"
```
```  4531         by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
```
```  4532       then guess s .. note s=this
```
```  4533       have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and>
```
```  4534             norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e"
```
```  4535       proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
```
```  4536           norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
```
```  4537           by(auto simp add:algebra_simps) qed
```
```  4538       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where
```
```  4539           b="\<Sum>(x, k)\<in>p. content k *\<^sub>R f (m x) x" and c="\<Sum>(x, k)\<in>p. integral k (f (m x))"])
```
```  4540       proof safe case goal1
```
```  4541          show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"])
```
```  4542            unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule setsum_norm[OF p'(1)])
```
```  4543            apply(rule setsum_mono) unfolding split_paired_all split_conv
```
```  4544            unfolding split_def setsum_left_distrib[THEN sym] scaleR.diff_right[THEN sym]
```
```  4545            unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
```
```  4546          proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto
```
```  4547            from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this
```
```  4548            show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))"
```
```  4549              unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le]
```
```  4550              apply(rule mult_left_mono) using m(2)[OF x,of "m x"] by auto
```
```  4551          qed(insert ab,auto)
```
```  4552
```
```  4553        next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
```
```  4554            \<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
```
```  4555            apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer
```
```  4556            apply(subst split_def)+ unfolding setsum_subtractf apply rule
```
```  4557          proof- show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p.
```
```  4558              m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e / 2"
```
```  4559              apply(rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"])
```
```  4560              apply(rule setsum_norm_le[OF finite_atLeastAtMost])
```
```  4561            proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
```
```  4562                unfolding power_add divide_inverse inverse_mult_distrib
```
```  4563                unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym]
```
```  4564                unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e])
```
```  4565                unfolding power2_eq_square by auto
```
```  4566              fix t assume "t\<in>{0..s}"
```
```  4567              show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x -
```
```  4568                integral k (f (m x))) \<le> e / 2 ^ (t + 2)"apply(rule order_trans[of _
```
```  4569                "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
```
```  4570                apply(rule eq_refl) apply(rule arg_cong[where f=norm]) apply(rule setsum_cong2) defer
```
```  4571                apply(rule henstock_lemma_part1) apply(rule assms(1)[rule_format])
```
```  4572                apply(rule divide_pos_pos,rule e) defer  apply safe apply(rule c)+
```
```  4573                apply rule apply assumption+ apply(rule tagged_partial_division_subset[of p])
```
```  4574                apply(rule p(1)[unfolded tagged_division_of_def,THEN conjunct1]) defer
```
```  4575                unfolding fine_def apply safe apply(drule p(2)[unfolded fine_def,rule_format])
```
```  4576                unfolding d_def by auto qed
```
```  4577          qed(insert s, auto)
```
```  4578
```
```  4579        next case goal3
```
```  4580          note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
```
```  4581          have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i\$\$0 - kr\$\$0
```
```  4582            \<and> i\$\$0 - kr\$\$0 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto
```
```  4583          show ?case unfolding real_norm_def apply(rule *[rule_format],safe)
```
```  4584            apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded Eucl_real_simps])
```
```  4585            apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
```
```  4586            apply(rule_tac[1-2] integral_le[OF ])
```
```  4587          proof safe show "0 \<le> i\$\$0 - (integral {a..b} (f r))\$\$0" using r(1) by auto
```
```  4588            show "i\$\$0 - (integral {a..b} (f r))\$\$0 < e / 4" using r(2) by auto
```
```  4589            fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
```
```  4590            show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k"
```
```  4591              unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
```
```  4592              using p'(3)[OF xk] unfolding uv by auto
```
```  4593            fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto
```
```  4594            hence *:"\<And>m. \<forall>n\<ge>m. (f m y) \<le> (f n y)" apply-apply(rule transitive_stepwise_le) using assms(2) by auto
```
```  4595            show "(f r y) \<le> (f (m x) y)" "(f (m x) y) \<le> (f s y)"
```
```  4596              apply(rule_tac[!] *[rule_format]) using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk] by auto
```
```  4597          qed qed qed qed note * = this
```
```  4598
```
```  4599    have "integral {a..b} g = i" apply(rule integral_unique) using * .
```
```  4600    thus ?thesis using i * by auto qed
```
```  4601
```
```  4602 lemma monotone_convergence_increasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4603   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x)"
```
```  4604   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
```
```  4605   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
```
```  4606 proof- have lem:"\<And>f::nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real. \<And> g s. \<forall>k.\<forall>x\<in>s. 0 \<le> (f k x) \<Longrightarrow> \<forall>k. (f k) integrable_on s \<Longrightarrow>
```
```  4607     \<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x) \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially  \<Longrightarrow>
```
```  4608     bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
```
```  4609   proof- case goal1 note assms=this[rule_format]
```
```  4610     have "\<forall>x\<in>s. \<forall>k. (f k x)\$\$0 \<le> (g x)\$\$0" apply safe apply(rule Lim_component_ge)
```
```  4611       apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially)
```
```  4612       unfolding eventually_sequentially apply(rule_tac x=k in exI)
```
```  4613       apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format]
```
```  4614
```
```  4615     have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially" apply(rule bounded_increasing_convergent)
```
```  4616       apply(rule goal1(5)) apply(rule,rule integral_le) apply(rule goal1(2)[rule_format])+
```
```  4617       using goal1(3) by auto then guess i .. note i=this
```
```  4618     have "\<And>k. \<forall>x\<in>s. \<forall>n\<ge>k. f k x \<le> f n x" apply(rule,rule transitive_stepwise_le) using goal1(3) by auto
```
```  4619     hence i':"\<forall>k. (integral s (f k))\$\$0 \<le> i\$\$0" apply-apply(rule,rule Lim_component_ge)
```
```  4620       apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
```
```  4621       apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
```
```  4622       apply(rule goal1(2)[rule_format])+ by auto
```
```  4623
```
```  4624     note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
```
```  4625     have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) =
```
```  4626       (\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto
```
```  4627     have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym])
```
```  4628       apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int .
```
```  4629     have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and>
```
```  4630       ((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) --->
```
```  4631       integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
```
```  4632     proof(rule monotone_convergence_interval,safe)
```
```  4633       case goal1 show ?case using int .
```
```  4634     next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto
```
```  4635     next case goal3 thus ?case apply-apply(cases "x\<in>s")
```
```  4636         using assms(4) by (auto intro: tendsto_const)
```
```  4637     next case goal4 note * = integral_nonneg
```
```  4638       have "\<And>k. norm (integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) \<le> norm (integral s (f k))"
```
```  4639         unfolding real_norm_def apply(subst abs_of_nonneg) apply(rule *[OF int])
```
```  4640         apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3
```
```  4641         apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]])
```
```  4642         apply(subst integral_restrict_univ[THEN sym,OF int])
```
```  4643         unfolding ifif unfolding integral_restrict_univ[OF int']
```
```  4644         apply(rule integral_subset_le[OF _ int' assms(2)]) using assms(1) by auto
```
```  4645       thus ?case using assms(5) unfolding bounded_iff apply safe
```
```  4646         apply(rule_tac x=aa in exI,safe) apply(erule_tac x="integral s (f k)" in ballE)
```
```  4647         apply(rule order_trans) apply assumption by auto qed note g = conjunctD2[OF this]
```
```  4648
```
```  4649     have "(g has_integral i) s" unfolding has_integral_alt' apply safe apply(rule g(1))
```
```  4650     proof- case goal1 hence "e/4>0" by auto
```
```  4651       from i[unfolded Lim_sequentially,rule_format,OF this] guess N .. note N=this
```
```  4652       note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]]
```
```  4653       from this[THEN conjunct2,rule_format,OF `e/4>0`] guess B .. note B=conjunctD2[OF this]
```
```  4654       show ?case apply(rule,rule,rule B,safe)
```
```  4655       proof- fix a b::"'n::ordered_euclidean_space" assume ab:"ball 0 B \<subseteq> {a..b}"
```
```  4656         from `e>0` have "e/2>0" by auto
```
```  4657         from g(2)[unfolded Lim_sequentially,of a b,rule_format,OF this] guess M .. note M=this
```
```  4658         have **:"norm (integral {a..b} (\<lambda>x. if x \<in> s then f N x else 0) - i) < e/2"
```
```  4659           apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N]
```
```  4660           unfolding dist_norm apply-defer apply(subst norm_minus_commute) by auto
```
```  4661         have *:"\<And>f1 f2 g. abs(f1 - i) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i
```
```  4662           \<longrightarrow> abs(g - i) < e" unfolding Eucl_real_simps by arith
```
```  4663         show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e"
```
```  4664           unfolding real_norm_def apply(rule *[rule_format])
```
```  4665           apply(rule **[unfolded real_norm_def])
```
```  4666           apply(rule M[rule_format,of "M + N",unfolded dist_real_def]) apply(rule le_add1)
```
```  4667           apply(rule integral_le[OF int int]) defer
```
```  4668           apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded Eucl_real_simps]])
```
```  4669         proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)\$\$0 \<le> (f n x)\$\$0"
```
```  4670             apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto
```
```  4671         next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int])
```
```  4672             unfolding ifif integral_restrict_univ[OF int']
```
```  4673             apply(rule integral_subset_le[OF _ int']) using assms by auto
```
```  4674         qed qed qed
```
```  4675     thus ?case apply safe defer apply(drule integral_unique) using i by auto qed
```
```  4676
```
```  4677   have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
```
```  4678     apply(subst integral_sub) apply(rule assms(1)[rule_format])+ by rule
```
```  4679   have "\<And>x m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x) \<le> (f n x)" apply(rule transitive_stepwise_le)
```
```  4680     using assms(2) by auto note * = this[rule_format]
```
```  4681   have "(\<lambda>x. g x - f 0 x) integrable_on s \<and>((\<lambda>k. integral s (\<lambda>x. f (Suc k) x - f 0 x)) --->
```
```  4682       integral s (\<lambda>x. g x - f 0 x)) sequentially" apply(rule lem,safe)
```
```  4683   proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
```
```  4684   next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
```
```  4685   next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
```
```  4686   next case goal4 thus ?case apply-apply(rule tendsto_diff)
```
```  4687       using seq_offset[OF assms(3)[rule_format],of x 1] by (auto intro: tendsto_const)
```
```  4688   next case goal5 thus ?case using assms(4) unfolding bounded_iff
```
```  4689       apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
```
```  4690       apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub
```
```  4691       apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed
```
```  4692   note conjunctD2[OF this] note tendsto_add[OF this(2) tendsto_const[of "integral s (f 0)"]]
```
```  4693     integrable_add[OF this(1) assms(1)[rule_format,of 0]]
```
```  4694   thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub)
```
```  4695     using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed
```
```  4696
```
```  4697 lemma monotone_convergence_decreasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
```
```  4698   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<le> (f k x)"
```
```  4699   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
```
```  4700   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
```
```  4701 proof- note assm = assms[rule_format]
```
```  4702   have *:"{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R -1 ` {integral s (f k)| k. True}"
```
```  4703     apply safe unfolding image_iff apply(rule_tac x="integral s (f k)" in bexI) prefer 3
```
```  4704     apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto
```
```  4705   have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x))
```
```  4706     ---> integral s (\<lambda>x. - g x))  sequentially" apply(rule monotone_convergence_increasing)
```
```  4707     apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule tendsto_minus)
```
```  4708     apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto
```
```  4709   note * = conjunctD2[OF this]
```
```  4710   show ?thesis apply rule using integrable_neg[OF *(1)] defer
```
```  4711     using tendsto_minus[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
```
```  4712     unfolding integral_neg[OF *(1),THEN sym] by auto qed
```
```  4713
```
```  4714 subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
```
```  4715
```
```  4716 definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46) where
```
```  4717   "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. (norm(f x))) integrable_on s"
```
```  4718
```
```  4719 lemma absolutely_integrable_onI[intro?]:
```
```  4720   "f integrable_on s \<Longrightarrow> (\<lambda>x. (norm(f x))) integrable_on s \<Longrightarrow> f absolutely_integrable_on s"
```
```  4721   unfolding absolutely_integrable_on_def by auto
```
```  4722
```
```  4723 lemma absolutely_integrable_onD[dest]: assumes "f absolutely_integrable_on s"
```
```  4724   shows "f integrable_on s" "(\<lambda>x. (norm(f x))) integrable_on s"
```
```  4725   using assms unfolding absolutely_integrable_on_def by auto
```
```  4726
```
```  4727 (*lemma absolutely_integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
```
```  4728   "(vec1 o f) absolutely_integrable_on s \<longleftrightarrow> f absolutely_integrable_on s"
```
```  4729   unfolding absolutely_integrable_on_def o_def by auto*)
```
```  4730
```
```  4731 lemma integral_norm_bound_integral: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4732   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> g x"
```
```  4733   shows "norm(integral s f) \<le> (integral s g)"
```
```  4734 proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr)
```
```  4735     apply(erule_tac x="x - y" in allE) by auto
```
```  4736   have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
```
```  4737     \<longrightarrow> norm(ig) < dia + e"
```
```  4738   proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
```
```  4739       apply(subst real_sum_of_halves[of e,THEN sym]) unfolding add_assoc[symmetric]
```
```  4740       apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
```
```  4741       apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
```
```  4742   qed note norm=this[rule_format]
```
```  4743   have lem:"\<And>f::'n::ordered_euclidean_space \<Rightarrow> 'a. \<And> g a b. f integrable_on {a..b} \<Longrightarrow> g integrable_on {a..b} \<Longrightarrow>
```
```  4744     \<forall>x\<in>{a..b}. norm(f x) \<le> (g x) \<Longrightarrow> norm(integral({a..b}) f) \<le> (integral({a..b}) g)"
```
```  4745   proof(rule *[rule_format]) case goal1 hence *:"e/2>0" by auto
```
```  4746     from integrable_integral[OF goal1(1),unfolded has_integral[of f],rule_format,OF *]
```
```  4747     guess d1 .. note d1 = conjunctD2[OF this,rule_format]
```
```  4748     from integrable_integral[OF goal1(2),unfolded has_integral[of g],rule_format,OF *]
```
```  4749     guess d2 .. note d2 = conjunctD2[OF this,rule_format]
```
```  4750     note gauge_inter[OF d1(1) d2(1)]
```
```  4751     from fine_division_exists[OF this, of a b] guess p . note p=this
```
```  4752     show ?case apply(rule norm) defer
```
```  4753       apply(rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def]) defer
```
```  4754       apply(rule d1(2)[OF conjI[OF p(1)]]) defer apply(rule setsum_norm_le)
```
```  4755     proof safe fix x k assume "(x,k)\<in>p" note as = tagged_division_ofD(2-4)[OF p(1) this]
```
```  4756       from this(3) guess u v apply-by(erule exE)+ note uv=this
```
```  4757       show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x" unfolding uv norm_scaleR
```
```  4758         unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
```
```  4759         apply(rule mult_left_mono) using goal1(3) as by auto
```
```  4760     qed(insert p[unfolded fine_inter],auto) qed
```
```  4761
```
```  4762   { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e"
```
```  4763     thus ?thesis apply-apply(rule *[rule_format]) by auto }
```
```  4764   fix e::real assume "e>0" hence e:"e/2 > 0" by auto
```
```  4765   note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
```
```  4766   note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format]
```
```  4767   from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e]
```
```  4768   guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format]
```
```  4769   from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e]
```
```  4770   guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format]
```
```  4771   from bounded_subset_closed_interval[OF bounded_ball, of "0::'n::ordered_euclidean_space" "max B1 B2"]
```
```  4772   guess a b apply-by(erule exE)+ note ab=this[unfolded ball_max_Un]
```
```  4773
```
```  4774   have "ball 0 B1 \<subseteq> {a..b}" using ab by auto
```
```  4775   from B1(2)[OF this] guess z .. note z=conjunctD2[OF this]
```
```  4776   have "ball 0 B2 \<subseteq> {a..b}" using ab by auto
```
```  4777   from B2(2)[OF this] guess w .. note w=conjunctD2[OF this]
```
```  4778
```
```  4779   show "norm (integral s f) < integral s g + e" apply(rule norm)
```
```  4780     apply(rule lem[OF f g, of a b]) unfolding integral_unique[OF z(1)] integral_unique[OF w(1)]
```
```  4781     defer apply(rule w(2)[unfolded real_norm_def],rule z(2))
```
```  4782     apply safe apply(case_tac "x\<in>s") unfolding if_P apply(rule assms(3)[rule_format]) by auto qed
```
```  4783
```
```  4784 lemma integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4785   fixes g::"'n => 'b::ordered_euclidean_space"
```
```  4786   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. norm(f x) \<le> (g x)\$\$k"
```
```  4787   shows "norm(integral s f) \<le> (integral s g)\$\$k"
```
```  4788 proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x \$\$ k) o g)"
```
```  4789     apply(rule integral_norm_bound_integral[OF assms(1)])
```
```  4790     apply(rule integrable_linear[OF assms(2)],rule)
```
```  4791     unfolding o_def by(rule assms)
```
```  4792   thus ?thesis unfolding o_def integral_component_eq[OF assms(2)] . qed
```
```  4793
```
```  4794 lemma has_integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4795   fixes g::"'n => 'b::ordered_euclidean_space"
```
```  4796   assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)\$\$k"
```
```  4797   shows "norm(i) \<le> j\$\$k" using integral_norm_bound_integral_component[of f s g k]
```
```  4798   unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
```
```  4799   using assms by auto
```
```  4800
```
```  4801 lemma absolutely_integrable_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4802   assumes "f absolutely_integrable_on s"
```
```  4803   shows "norm(integral s f) \<le> integral s (\<lambda>x. norm(f x))"
```
```  4804   apply(rule integral_norm_bound_integral) using assms by auto
```
```  4805
```
```  4806 lemma absolutely_integrable_0[intro]: "(\<lambda>x. 0) absolutely_integrable_on s"
```
```  4807   unfolding absolutely_integrable_on_def by auto
```
```  4808
```
```  4809 lemma absolutely_integrable_cmul[intro]:
```
```  4810  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on s"
```
```  4811   unfolding absolutely_integrable_on_def using integrable_cmul[of f s c]
```
```  4812   using integrable_cmul[of "\<lambda>x. norm (f x)" s "abs c"] by auto
```
```  4813
```
```  4814 lemma absolutely_integrable_neg[intro]:
```
```  4815  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) absolutely_integrable_on s"
```
```  4816   apply(drule absolutely_integrable_cmul[where c="-1"]) by auto
```
```  4817
```
```  4818 lemma absolutely_integrable_norm[intro]:
```
```  4819  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. norm(f x)) absolutely_integrable_on s"
```
```  4820   unfolding absolutely_integrable_on_def by auto
```
```  4821
```
```  4822 lemma absolutely_integrable_abs[intro]:
```
```  4823  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. abs(f x::real)) absolutely_integrable_on s"
```
```  4824   apply(drule absolutely_integrable_norm) unfolding real_norm_def .
```
```  4825
```
```  4826 lemma absolutely_integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
```
```  4827   "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}"
```
```  4828   unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
```
```  4829
```
```  4830 lemma absolutely_integrable_bounded_variation: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
```
```  4831   assumes "f absolutely_integrable_on UNIV"
```
```  4832   obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
```
```  4833   apply(rule that[of "integral UNIV (\<lambda>x. norm (f x))"])
```
```  4834 proof safe case goal1 note d = division_ofD[OF this(2)]
```
```  4835   have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. integral i (\<lambda>x. norm (f x)))"
```
```  4836     apply(rule setsum_mono,rule absolutely_integrable_le) apply(drule d(4),safe)
```
```  4837     apply(rule absolutely_integrable_on_subinterval[OF assms]) by auto
```
```  4838   also have "... \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
```
```  4839     apply(subst integral_combine_division_topdown[OF _ goal1(2)])
```
```  4840     using integrable_on_subdivision[OF goal1(2)] using assms by auto
```
```  4841   also have "... \<le> integral UNIV (\<lambda>x. norm (f x))"
```
```  4842     apply(rule integral_subset_le)
```
```  4843     using integrable_on_subdivision[OF goal1(2)] using assms by auto
```
```  4844   finally show ?case . qed
```
```  4845
```
```  4846 lemma helplemma:
```
```  4847   assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s"
```
```  4848   shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e"
```
```  4849   unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs])
```
```  4850   apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
```
```  4851   using norm_triangle_ineq3 .
```
```  4852
```
```  4853 lemma bounded_variation_absolutely_integrable_interval:
```
```  4854   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assumes "f integrable_on {a..b}"
```
```  4855   "\<forall>d. d division_of {a..b} \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
```
```  4856   shows "f absolutely_integrable_on {a..b}"
```
```  4857 proof- let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of {a..b} }" def i \<equiv> "Sup ?S"
```
```  4858   have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
```
```  4859     apply(rule elementary_interval) defer apply(rule_tac x=B in exI)
```
```  4860     apply(rule setleI) using assms(2) by auto
```
```  4861   show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i])
```
```  4862   proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) `
```
```  4863         {d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format]
```
```  4864       unfolding setge_def ubs_def by auto
```
```  4865     hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))"
```
```  4866       unfolding mem_Collect_eq isUb_def setle_def by(simp add:not_le) then guess d .. note d=conjunctD2[OF this]
```
```  4867     note d' = division_ofD[OF this(1)]
```
```  4868
```
```  4869     have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
```
```  4870     proof case goal1 have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
```
```  4871         apply(rule separate_point_closed) apply(rule closed_Union)
```
```  4872         apply(rule finite_subset[OF _ d'(1)]) apply safe apply(drule d'(4)) by auto
```
```  4873       thus ?case apply safe apply(rule_tac x=da in exI,safe)
```
```  4874         apply(erule_tac x=xa in ballE) by auto
```
```  4875     qed from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
```
```  4876
```
```  4877     have "e/2 > 0" using goal1 by auto
```
```  4878     from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
```
```  4879     let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
```
```  4880     show ?case apply(rule_tac x="?g" in exI) apply safe
```
```  4881     proof- show "gauge ?g" using g(1) unfolding gauge_def using k(1) by auto
```
```  4882       fix p assume "p tagged_division_of {a..b}" "?g fine p"
```
```  4883       note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
```
```  4884       note p' = tagged_division_ofD[OF p(1)]
```
```  4885       def p' \<equiv> "{(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
```
```  4886       have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto
```
```  4887       have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI)
```
```  4888       proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l))
```
```  4889           ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def
```
```  4890           defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
```
```  4891           apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto
```
```  4892         fix x k assume "(x,k)\<in>p'"
```
```  4893         hence "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l" unfolding p'_def by auto
```
```  4894         then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
```
```  4895         show "x\<in>k" "k\<subseteq>{a..b}" using p'(2-3)[OF il(3)] il by auto
```
```  4896         show "\<exists>a b. k = {a..b}" unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
```
```  4897           apply safe unfolding inter_interval by auto
```
```  4898       next fix x1 k1 assume "(x1,k1)\<in>p'"
```
```  4899         hence "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> k1 = i \<inter> l" unfolding p'_def by auto
```
```  4900         then guess i1 l1 apply-by(erule exE)+ note il1=conjunctD4[OF this]
```
```  4901         fix x2 k2 assume "(x2,k2)\<in>p'"
```
```  4902         hence "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> k2 = i \<inter> l" unfolding p'_def by auto
```
```  4903         then guess i2 l2 apply-by(erule exE)+ note il2=conjunctD4[OF this]
```
```  4904         assume "(x1, k1) \<noteq> (x2, k2)"
```
```  4905         hence "interior(i1) \<inter> interior(i2) = {} \<or> interior(l1) \<inter> interior(l2) = {}"
```
```  4906           using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] unfolding il1 il2 by auto
```
```  4907         thus "interior k1 \<inter> interior k2 = {}" unfolding il1 il2 by auto
```
```  4908       next have *:"\<forall>(x, X) \<in> p'. X \<subseteq> {a..b}" unfolding p'_def using d' by auto
```
```  4909         show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least)
```
```  4910           unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe
```
```  4911         proof- fix y assume y:"y\<in>{a..b}"
```
```  4912           hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto
```
```  4913           then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this]
```
```  4914           hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto
```
```  4915           then guess i .. note i = conjunctD2[OF this]
```
```  4916           have "x\<in>i" using fineD[OF p(3) xl(1)] using k(2)[OF i(1), of x] using i(2) xl(2) by auto
```
```  4917           thus "y\<in>\<Union>{k. \<exists>x. (x, k) \<in> p'}" unfolding p'_def Union_iff apply(rule_tac x="i \<inter> l" in bexI)
```
```  4918             defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI)
```
```  4919             apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto
```
```  4920         qed qed
```
```  4921
```
`  4922       hence "(\<Sum&`