src/HOL/Multivariate_Analysis/Integration.thy
author haftmann
Mon Apr 26 11:34:19 2010 +0200 (2010-04-26)
changeset 36350 bc7982c54e37
parent 36334 068a01b4bc56
child 36365 18bf20d0c2df
permissions -rw-r--r--
dropped group_simps, ring_simps, field_eq_simps
     1 
     2 header {* Kurzweil-Henstock gauge integration in many dimensions. *}
     3 (*  Author:                     John Harrison
     4     Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     5 
     6 theory Integration
     7   imports Derivative SMT
     8 begin
     9 
    10 declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]
    11 declare [[smt_fixed=true]]
    12 declare [[z3_proofs=true]]
    13 
    14 subsection {* Sundries *}
    15 
    16 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
    17 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
    18 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
    19 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
    20 
    21 declare smult_conv_scaleR[simp]
    22 
    23 lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast
    24 
    25 lemma linear_simps:  assumes "bounded_linear f"
    26   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)"
    27   apply(rule_tac[!] additive.add additive.minus additive.diff additive.zero bounded_linear.scaleR)
    28   using assms unfolding bounded_linear_def additive_def by auto
    29 
    30 lemma bounded_linearI:assumes "\<And>x y. f (x + y) = f x + f y"
    31   "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K"
    32   shows "bounded_linear f"
    33   unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto
    34  
    35 lemma real_le_inf_subset:
    36   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s" shows "Inf s <= Inf (t::real set)"
    37   apply(rule isGlb_le_isLb) apply(rule Inf[OF assms(1)])
    38   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
    39   unfolding isLb_def setge_def by auto
    40 
    41 lemma real_ge_sup_subset:
    42   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b" shows "Sup s >= Sup (t::real set)"
    43   apply(rule isLub_le_isUb) apply(rule Sup[OF assms(1)])
    44   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
    45   unfolding isUb_def setle_def by auto
    46 
    47 lemma dist_trans[simp]:"dist (vec1 x) (vec1 y) = dist x (y::real)"
    48   unfolding dist_real_def dist_vec1 ..
    49 
    50 lemma Lim_trans[simp]: fixes f::"'a \<Rightarrow> real"
    51   shows "((\<lambda>x. vec1 (f x)) ---> vec1 l) net \<longleftrightarrow> (f ---> l) net"
    52   using assms unfolding Lim dist_trans ..
    53 
    54 lemma bounded_linear_component[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
    55   apply(rule bounded_linearI[where K=1]) 
    56   using component_le_norm[of _ k] unfolding real_norm_def by auto
    57 
    58 lemma bounded_vec1[intro]:  "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"
    59   unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI) by auto
    60 
    61 lemma transitive_stepwise_lt_eq:
    62   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
    63   shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r")
    64 proof(safe) assume ?r fix n m::nat assume "m < n" thus "R m n" apply-
    65   proof(induct n arbitrary: m) case (Suc n) show ?case 
    66     proof(cases "m < n") case True
    67       show ?thesis apply(rule assms[OF Suc(1)[OF True]]) using `?r` by auto
    68     next case False hence "m = n" using Suc(2) by auto
    69       thus ?thesis using `?r` by auto
    70     qed qed auto qed auto
    71 
    72 lemma transitive_stepwise_gt:
    73   assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
    74   shows "\<forall>n>m. R m n"
    75 proof- have "\<forall>m. \<forall>n>m. R m n" apply(subst transitive_stepwise_lt_eq)
    76     apply(rule assms) apply(assumption,assumption) using assms(2) by auto
    77   thus ?thesis by auto qed
    78 
    79 lemma transitive_stepwise_le_eq:
    80   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
    81   shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r")
    82 proof safe assume ?r fix m n::nat assume "m\<le>n" thus "R m n" apply-
    83   proof(induct n arbitrary: m) case (Suc n) show ?case 
    84     proof(cases "m \<le> n") case True show ?thesis apply(rule assms(2))
    85         apply(rule Suc(1)[OF True]) using `?r` by auto
    86     next case False hence "m = Suc n" using Suc(2) by auto
    87       thus ?thesis using assms(1) by auto
    88     qed qed(insert assms(1), auto) qed auto
    89 
    90 lemma transitive_stepwise_le:
    91   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) "
    92   shows "\<forall>n\<ge>m. R m n"
    93 proof- have "\<forall>m. \<forall>n\<ge>m. R m n" apply(subst transitive_stepwise_le_eq)
    94     apply(rule assms) apply(rule assms,assumption,assumption) using assms(3) by auto
    95   thus ?thesis by auto qed
    96 
    97 
    98 subsection {* Some useful lemmas about intervals. *}
    99 
   100 lemma empty_as_interval: "{} = {1..0::real^'n}"
   101   apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval
   102   using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
   103 
   104 lemma interior_subset_union_intervals: 
   105   assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
   106   shows "interior i \<subseteq> interior s" proof-
   107   have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
   108     unfolding assms(1,2) interior_closed_interval by auto
   109   moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
   110     using assms(4) unfolding assms(1,2) by auto
   111   ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
   112     unfolding assms(1,2) interior_closed_interval by auto qed
   113 
   114 lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"
   115   assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
   116   shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
   117   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)
   118     unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
   119   have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
   120   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
   121   thus ?case proof(induct rule:finite_induct) 
   122     case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
   123     case (insert i f) guess x using insert(5) .. note x = this
   124     then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
   125     guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
   126     show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
   127       then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
   128       hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto
   129       hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 by auto hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
   130       hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t" apply-apply(rule insert(3)) using insert(4) by auto thus ?thesis by auto next
   131     case True show ?thesis proof(cases "x\<in>{a<..<b}")
   132       case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
   133       thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
   134 	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
   135     case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) 
   136     hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
   137     hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
   138       let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
   139 	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
   140 	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
   141 	hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
   142 	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed
   143       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
   144 	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"
   145 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
   146 	  unfolding norm_scaleR norm_basis by auto
   147 	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
   148 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
   149       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
   150     next let ?z = "x + (e/2) *\<^sub>R basis k" assume as:"x$k = b$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
   151 	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
   152 	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto
   153 	hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)
   154 	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed
   155       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
   156 	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"
   157 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
   158 	  unfolding norm_scaleR norm_basis by auto
   159 	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)
   160 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed
   161       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 
   162     then guess x .. hence "x \<in> s \<inter> interior (\<Union>f)" unfolding lem1[where U="\<Union>f",THEN sym] using centre_in_ball e[THEN conjunct1] by auto
   163     thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
   164   guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
   165   hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
   166   thus False using `t\<in>f` assms(4) by auto qed
   167 subsection {* Bounds on intervals where they exist. *}
   168 
   169 definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"
   170 
   171 definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"
   172 
   173 lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"
   174   using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   175   apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   176   apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
   177   unfolding mem_interval using assms by auto
   178 
   179 lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"
   180   using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)
   181   apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   182   apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
   183   unfolding mem_interval using assms by auto
   184 
   185 lemmas interval_bounds = interval_upperbound interval_lowerbound
   186 
   187 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
   188   using assms unfolding interval_ne_empty by auto
   189 
   190 lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"
   191   apply(rule interval_upperbound) by auto
   192 
   193 lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"
   194   apply(rule interval_lowerbound) by auto
   195 
   196 lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1
   197 
   198 subsection {* Content (length, area, volume...) of an interval. *}
   199 
   200 definition "content (s::(real^'n) set) =
   201        (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"
   202 
   203 lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"
   204   unfolding interval_eq_empty unfolding not_ex not_less by assumption
   205 
   206 lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"
   207   shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   208   using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
   209 
   210 lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"
   211   apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
   212 
   213 lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"
   214   using content_closed_interval[of a b] by auto
   215 
   216 lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto
   217 
   218 lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-
   219   have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto
   220   have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto
   221   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
   222 
   223 lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")
   224   case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption
   225   have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"
   226     apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
   227   thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
   228 
   229 lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"
   230 proof- have help_lemma1: "\<forall>i. a$i < b$i \<Longrightarrow> \<forall>i. a$i \<le> ((b$i)::real)" apply(rule,erule_tac x=i in allE) by auto
   231   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
   232     using assms apply(erule_tac x=x in allE) by auto qed
   233 
   234 lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"
   235   apply(rule content_pos_lt) by auto
   236 
   237 lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")
   238   case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
   239     apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
   240   guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]
   241   show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]
   242     apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
   243     apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
   244 
   245 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
   246 
   247 lemma content_closed_interval_cases:
   248   "content {a..b} = (if \<forall>i. a$i \<le> b$i then setprod (\<lambda>i. b$i - a$i) UNIV else 0)" apply(rule cond_cases) 
   249   apply(rule content_closed_interval) unfolding content_eq_0 not_all not_le defer apply(erule exE,rule_tac x=x in exI) by auto
   250 
   251 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   252   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
   253 
   254 lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
   255   unfolding content_eq_0 by auto
   256 
   257 lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
   258   apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
   259   hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i. a$i < b$i" unfolding content_eq_0 not_ex not_le by auto qed
   260 
   261 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
   262 
   263 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")
   264   case True thus ?thesis using content_pos_le[of c d] by auto next
   265   case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto
   266   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   267   have "{c..d} \<noteq> {}" using assms False by auto
   268   hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto
   269   show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
   270     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n
   271     show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto
   272     show "b $ i - a $ i \<le> d $ i - c $ i"
   273       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
   274       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed
   275 
   276 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
   277   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto
   278 
   279 subsection {* The notion of a gauge --- simply an open set containing the point. *}
   280 
   281 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
   282 
   283 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
   284   using assms unfolding gauge_def by auto
   285 
   286 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
   287 
   288 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   289   unfolding gauge_def by auto 
   290 
   291 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
   292 
   293 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
   294 
   295 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
   296   unfolding gauge_def by auto 
   297 
   298 lemma gauge_inters: assumes "finite s" "\<forall>d\<in>s. gauge (f d)" shows "gauge(\<lambda>x. \<Inter> {f d x | d. d \<in> s})" proof-
   299   have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
   300   unfolding gauge_def unfolding * 
   301   using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
   302 
   303 lemma gauge_existence_lemma: "(\<forall>x. \<exists>d::real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)" by(meson zero_less_one)
   304 
   305 subsection {* Divisions. *}
   306 
   307 definition division_of (infixl "division'_of" 40) where
   308   "s division_of i \<equiv>
   309         finite s \<and>
   310         (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
   311         (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
   312         (\<Union>s = i)"
   313 
   314 lemma division_ofD[dest]: assumes  "s division_of i"
   315   shows"finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   316   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" using assms unfolding division_of_def by auto
   317 
   318 lemma division_ofI:
   319   assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   320   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
   321   shows "s division_of i" using assms unfolding division_of_def by auto
   322 
   323 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   324   unfolding division_of_def by auto
   325 
   326 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   327   unfolding division_of_def by auto
   328 
   329 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
   330 
   331 lemma division_of_sing[simp]: "s division_of {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
   332   assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 
   333     ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }
   334   ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next
   335   assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]
   336   { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
   337   moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] by auto qed
   338 
   339 lemma elementary_empty: obtains p where "p division_of {}"
   340   unfolding division_of_trivial by auto
   341 
   342 lemma elementary_interval: obtains p where  "p division_of {a..b}"
   343   by(metis division_of_trivial division_of_self)
   344 
   345 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   346   unfolding division_of_def by auto
   347 
   348 lemma forall_in_division:
   349  "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
   350   unfolding division_of_def by fastsimp
   351 
   352 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
   353   apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
   354   show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
   355   { fix k assume "k \<in> q" hence kp:"k\<in>p" using assms(2) by auto show "k\<subseteq>\<Union>q" using `k \<in> q` by auto
   356   show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
   357   fix k1 k2 assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2" hence *:"k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto
   358   show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
   359 
   360 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
   361 
   362 lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
   363   unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
   364   apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
   365 
   366 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"
   367   shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)" (is "?A' division_of _") proof-
   368 let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
   369 show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
   370   moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
   371   have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto show "\<Union>?A = s1 \<inter> s2" apply(rule set_ext) unfolding * and Union_image_eq UN_iff
   372     using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
   373   { fix k assume "k\<in>?A" then obtain k1 k2 where k:"k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto thus "k \<noteq> {}" by auto
   374   show "k \<subseteq> s1 \<inter> s2" using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)] unfolding k by auto
   375   guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
   376   guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
   377   show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
   378   assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
   379   assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
   380   assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
   381   have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
   382       interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
   383       interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
   384       \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
   385   show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
   386     using division_ofD(5)[OF assms(1) k1(2) k2(2)]
   387     using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
   388 
   389 lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"
   390   shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
   391   case True show ?thesis unfolding True and division_of_trivial by auto next
   392   have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 
   393   case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
   394 
   395 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"
   396   shows "\<exists>p. p division_of (s \<inter> t)"
   397   by(rule,rule division_inter[OF assms])
   398 
   399 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"
   400   shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
   401 case (insert x f) show ?case proof(cases "f={}")
   402   case True thus ?thesis unfolding True using insert by auto next
   403   case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
   404   moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
   405   show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
   406 
   407 lemma division_disjoint_union:
   408   assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
   409   shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) 
   410   note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
   411   show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
   412   show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
   413   { fix k1 k2 assume as:"k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2" moreover let ?g="interior k1 \<inter> interior k2 = {}"
   414   { assume as:"k1\<in>p1" "k2\<in>p2" have ?g using subset_interior[OF d1(2)[OF as(1)]] subset_interior[OF d2(2)[OF as(2)]]
   415       using assms(3) by blast } moreover
   416   { assume as:"k1\<in>p2" "k2\<in>p1" have ?g using subset_interior[OF d1(2)[OF as(2)]] subset_interior[OF d2(2)[OF as(1)]]
   417       using assms(3) by blast} ultimately
   418   show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
   419   fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
   420   show "k \<noteq> {}" using k d1(3) d2(3) by auto show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto qed
   421 
   422 lemma partial_division_extend_1:
   423   assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"
   424   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
   425 proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto
   426   guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this
   427   def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
   428   have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
   429   hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
   430   have \<pi>\<pi>'[simp]:"\<And>i. \<pi> (\<pi>' i) = i" unfolding \<pi>'_def apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
   431   have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq) using \<pi> unfolding n_def bij_betw_def by auto
   432   have "{c..d} \<noteq> {}" using assms by auto
   433   let ?p1 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else if \<pi>' i = l then c$\<pi> l else b$i)}"
   434   let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else if \<pi>' i = l then d$\<pi> l else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else b$i)}"
   435   let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
   436   have abcd:"\<And>i. a $ i \<le> c $ i \<and> c$i \<le> d$i \<and> d $ i \<le> b $ i" using assms unfolding subset_interval interval_eq_empty by(auto simp add:not_le not_less)
   437   show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
   438   proof- have "\<And>i. \<pi>' i < Suc n"
   439     proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"
   440       hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto
   441     qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"
   442         "d = (\<chi> i. if \<pi>' i < Suc n then d $ i else if \<pi>' i = n + 1 then c $ \<pi> (n + 1) else b $ i)"
   443       unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto
   444     thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
   445     have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}"  "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"
   446       unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
   447     proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
   448       then guess i unfolding mem_interval not_all .. note i=this
   449       show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
   450         apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
   451     qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
   452     proof- fix x assume x:"x\<in>{a..b}"
   453       { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
   454       let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $ \<pi> i \<le> x $ \<pi> i \<and> x $ \<pi> i \<le> d $ \<pi> i)}"
   455       assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..
   456       hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
   457       hence M:"finite ?M" "?M \<noteq> {}" by auto
   458       def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]
   459         Min_gr_iff[OF M,unfolded l_def[symmetric]]
   460       have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
   461         apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
   462       proof- assume as:"x $ \<pi> l < c $ \<pi> l"
   463         show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta
   464         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   465           thus ?case using as x[unfolded mem_interval,rule_format,of i]
   466             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   467         qed
   468       next assume as:"x $ \<pi> l > d $ \<pi> l"
   469         show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta
   470         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto
   471           thus ?case using as x[unfolded mem_interval,rule_format,of i]
   472             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])
   473         qed qed
   474       thus "x \<in> \<Union>?p" using l(2) by blast 
   475     qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
   476     
   477     show "finite ?p" by auto
   478     fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto
   479     show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
   480     proof- fix i::'n and x assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
   481       ultimately show "a$i \<le> x$i" "x$i \<le> b$i" using abcd[of i] using l by(auto elim:disjE elim!:allE[where x=i] simp add:vector_le_def)
   482     qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
   483     proof- case goal1 thus ?case using abcd[of x] by auto
   484     next   case goal2 thus ?case using abcd[of x] by auto
   485     qed thus "k \<noteq> {}" using k by auto
   486     show "\<exists>a b. k = {a..b}" using k by auto
   487     fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto
   488     { fix k k' l l'
   489       assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
   490       assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
   491       assume "l \<le> l'" fix x
   492       have "x \<notin> interior k \<inter> interior k'" 
   493       proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
   494         case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   495         hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   496         have ln:"l < n + 1" 
   497         proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
   498           hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   499           hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)
   500           thus False using `k\<noteq>k'` k' by auto
   501         qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
   502         have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-
   503         proof(erule disjE)
   504           assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   505           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   506         next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   507           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto
   508         qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
   509           by(auto elim!:allE[where x="\<pi> l"])
   510       next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
   511         hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
   512         note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
   513         assume x:"x \<in> interior k \<inter> interior k'"
   514         show False using l(1) l'(1) apply-
   515         proof(erule_tac[!] disjE)+
   516           assume as:"k = ?p1 l" "k' = ?p1 l'"
   517           note * = x[unfolded as Int_iff interior_closed_interval mem_interval]
   518           have "l \<noteq> l'" using k'(2)[unfolded as] by auto
   519           thus False using * by(smt Cart_lambda_beta \<pi>l)
   520         next assume as:"k = ?p2 l" "k' = ?p2 l'"
   521           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   522           have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
   523           thus False using *[of "\<pi> l"] *[of "\<pi> l'"]
   524             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto
   525         next assume as:"k = ?p1 l" "k' = ?p2 l'"
   526           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   527           show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   528             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt 
   529         next assume as:"k = ?p2 l" "k' = ?p1 l'"
   530           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   531           show False using *[of "\<pi> l"] *[of "\<pi> l'"]
   532             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt
   533         qed qed } 
   534     from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
   535       apply - apply(cases "l' \<le> l") using k'(2) by auto            
   536     thus "interior k \<inter> interior k' = {}" by auto        
   537 qed qed
   538 
   539 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   540   obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")
   541   case True guess q apply(rule elementary_interval[of a b]) .
   542   thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
   543   case False note p = division_ofD[OF assms(1)]
   544   have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
   545     guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
   546     have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
   547     guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
   548   guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
   549   have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" apply(rule,rule_tac p="q x" in division_of_subset) proof-
   550     fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
   551       using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
   552   hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
   553     apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
   554   then guess d .. note d = this
   555   show ?thesis apply(rule that[of "d \<union> p"]) proof-
   556     have *:"\<And>s f t. s \<noteq> {} \<Longrightarrow> (\<forall>i\<in>s. f i \<union> i = t) \<Longrightarrow> t = \<Inter> (f ` s) \<union> (\<Union>s)" by auto
   557     have *:"{a..b} = \<Inter> (\<lambda>i. \<Union>(q i - {i})) ` p \<union> \<Union>p" apply(rule *[OF False]) proof fix i assume i:"i\<in>p"
   558       show "\<Union>(q i - {i}) \<union> i = {a..b}" using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto qed
   559     show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
   560       apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
   561       fix k assume k:"k\<in>p" have *:"\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto
   562       show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
   563 	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
   564 	show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
   565 	show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
   566 	have *:"\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<subseteq> interior (\<Union>(q k - {k}))"
   567 	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
   568 
   569 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"
   570   unfolding division_of_def by(metis bounded_Union bounded_interval) 
   571 
   572 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"
   573   by(meson elementary_bounded bounded_subset_closed_interval)
   574 
   575 lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"
   576   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
   577   case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
   578   case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
   579   have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
   580   case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
   581     using false True assms using interior_subset by auto next
   582   case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
   583   have *:"{u..v} \<subseteq> {c..d}" using uv by auto
   584   guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
   585   have *:"{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s" using p(8) unfolding uv[THEN sym] by auto
   586   show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
   587     apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
   588     unfolding interior_inter[THEN sym] proof-
   589     have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
   590     have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
   591       apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
   592     also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
   593     finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
   594 
   595 lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
   596   "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   597   shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
   598   apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
   599   using division_ofD[OF assms(2)] by auto
   600   
   601 lemma elementary_union_interval: assumes "p division_of \<Union>p"
   602   obtains q where "q division_of ({a..b::real^'n} \<union> \<Union>p)" proof-
   603   note assm=division_ofD[OF assms]
   604   have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
   605   have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
   606 { presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
   607     "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
   608   thus thesis by auto
   609 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
   610   thus thesis apply(rule_tac that[of p]) unfolding as by auto 
   611 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
   612 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
   613   show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
   614     unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
   615     using assm(2-4) as apply- by(fastsimp dest: assm(5))+
   616 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
   617   have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
   618     from assm(4)[OF this] guess c .. then guess d ..
   619     thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
   620   qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
   621   let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
   622   show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
   623     have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
   624     show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
   625     show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
   626       using q(6) by auto
   627     fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
   628     show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
   629     fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
   630     obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
   631     obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
   632     show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
   633       case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
   634     next case False 
   635       { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
   636         "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
   637         thus ?thesis by auto }
   638       { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
   639       { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
   640       assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
   641       guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
   642       have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
   643       hence "interior k \<subseteq> interior x" apply-
   644         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
   645       guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
   646       have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
   647       hence "interior k' \<subseteq> interior x'" apply-
   648         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
   649       ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
   650     qed qed } qed
   651 
   652 lemma elementary_unions_intervals:
   653   assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"
   654   obtains p where "p division_of (\<Union>f)" proof-
   655   have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
   656     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   657     fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
   658     from this(3) guess p .. note p=this
   659     from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
   660     have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
   661     show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
   662       unfolding Union_insert ab * by auto
   663   qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
   664 
   665 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"
   666   obtains p where "p division_of (s \<union> t)"
   667 proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
   668   hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
   669   show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
   670     unfolding * prefer 3 apply(rule_tac p=p in that)
   671     using assms[unfolded division_of_def] by auto qed
   672 
   673 lemma partial_division_extend: fixes t::"(real^'n) set"
   674   assumes "p division_of s" "q division_of t" "s \<subseteq> t"
   675   obtains r where "p \<subseteq> r" "r division_of t" proof-
   676   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   677   obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
   678   guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
   679     apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
   680   guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
   681   then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
   682     apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
   683   { fix x assume x:"x\<in>t" "x\<notin>s"
   684     hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
   685     then guess r unfolding Union_iff .. note r=this moreover
   686     have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
   687       thus False using x by auto qed
   688     ultimately have "x\<in>\<Union>(r1 - p)" by auto }
   689   hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
   690   show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
   691     unfolding divp(6) apply(rule assms r2)+
   692   proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
   693     proof(rule inter_interior_unions_intervals)
   694       show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
   695       have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
   696       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
   697         fix m x assume as:"m\<in>r1-p"
   698         have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
   699           show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
   700           show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
   701         qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
   702       qed qed        
   703     thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
   704   qed auto qed
   705 
   706 subsection {* Tagged (partial) divisions. *}
   707 
   708 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
   709   "(s tagged_partial_division_of i) \<equiv>
   710         finite s \<and>
   711         (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   712         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
   713                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
   714 
   715 lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
   716   shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
   717   "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   718   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> (x1,k1) \<noteq> (x2,k2) \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
   719   using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
   720 
   721 definition tagged_division_of (infixr "tagged'_division'_of" 40) where
   722   "(s tagged_division_of i) \<equiv>
   723         (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   724 
   725 lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
   726   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   727 
   728 lemma tagged_division_of:
   729  "(s tagged_division_of i) \<longleftrightarrow>
   730         finite s \<and>
   731         (\<forall>x k. (x,k) \<in> s
   732                \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   733         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
   734                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
   735         (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   736   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   737 
   738 lemma tagged_division_ofI: assumes
   739   "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   740   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   741   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   742   shows "s tagged_division_of i"
   743   unfolding tagged_division_of apply(rule) defer apply rule
   744   apply(rule allI impI conjI assms)+ apply assumption
   745   apply(rule, rule assms, assumption) apply(rule assms, assumption)
   746   using assms(1,5-) apply- by blast+
   747 
   748 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
   749   shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   750   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   751   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
   752 
   753 lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
   754 proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
   755   show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
   756   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   757   thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
   758   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   759   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   760 qed
   761 
   762 lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
   763   shows "(snd ` s) division_of \<Union>(snd ` s)"
   764 proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
   765   show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
   766   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   767   thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
   768   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   769   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   770 qed
   771 
   772 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
   773   shows "t tagged_partial_division_of i"
   774   using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
   775 
   776 lemma setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"
   777   assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   778   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
   779 proof- note assm=tagged_division_ofD[OF assms(1)]
   780   have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
   781   show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
   782     show "finite p" using assm by auto
   783     fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
   784     obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
   785     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
   786     hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
   787     hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
   788     hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto
   789     thus "d (snd x) = 0" unfolding ab by auto qed qed
   790 
   791 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
   792 
   793 lemma tagged_division_of_empty: "{} tagged_division_of {}"
   794   unfolding tagged_division_of by auto
   795 
   796 lemma tagged_partial_division_of_trivial[simp]:
   797  "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   798   unfolding tagged_partial_division_of_def by auto
   799 
   800 lemma tagged_division_of_trivial[simp]:
   801  "p tagged_division_of {} \<longleftrightarrow> p = {}"
   802   unfolding tagged_division_of by auto
   803 
   804 lemma tagged_division_of_self:
   805  "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
   806   apply(rule tagged_division_ofI) by auto
   807 
   808 lemma tagged_division_union:
   809   assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
   810   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
   811 proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
   812   show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
   813   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
   814   fix x k assume xk:"(x,k)\<in>p1\<union>p2" show "x\<in>k" "\<exists>a b. k = {a..b}" using xk p1(2,4) p2(2,4) by auto
   815   show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
   816   fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
   817   have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
   818   show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
   819     apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
   820     using p1(3) p2(3) using xk xk' by auto qed 
   821 
   822 lemma tagged_division_unions:
   823   assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
   824   "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
   825   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
   826 proof(rule tagged_division_ofI)
   827   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
   828   show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
   829   have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>(\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset" by blast 
   830   also have "\<dots> = \<Union>iset" using assm(6) by auto
   831   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . 
   832   fix x k assume xk:"(x,k)\<in>\<Union>pfn ` iset" then obtain i where i:"i \<in> iset" "(x, k) \<in> pfn i" by auto
   833   show "x\<in>k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset" using assm(2-4)[OF i] using i(1) by auto
   834   fix x' k' assume xk':"(x',k')\<in>\<Union>pfn ` iset" "(x, k) \<noteq> (x', k')" then obtain i' where i':"i' \<in> iset" "(x', k') \<in> pfn i'" by auto
   835   have *:"\<And>a b. i\<noteq>i' \<Longrightarrow> a\<subseteq> i \<Longrightarrow> b\<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" using i(1) i'(1)
   836     using assms(3)[rule_format] subset_interior by blast
   837   show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
   838     using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
   839 qed
   840 
   841 lemma tagged_partial_division_of_union_self:
   842   assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
   843   apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
   844 
   845 lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
   846   shows "p tagged_division_of (\<Union>(snd ` p))"
   847   apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
   848 
   849 subsection {* Fine-ness of a partition w.r.t. a gauge. *}
   850 
   851 definition fine (infixr "fine" 46) where
   852   "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
   853 
   854 lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   855   shows "d fine s" using assms unfolding fine_def by auto
   856 
   857 lemma fineD[dest]: assumes "d fine s"
   858   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
   859 
   860 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   861   unfolding fine_def by auto
   862 
   863 lemma fine_inters:
   864  "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   865   unfolding fine_def by blast
   866 
   867 lemma fine_union:
   868   "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   869   unfolding fine_def by blast
   870 
   871 lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   872   unfolding fine_def by auto
   873 
   874 lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   875   unfolding fine_def by blast
   876 
   877 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
   878 
   879 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
   880   "(f has_integral_compact_interval y) i \<equiv>
   881         (\<forall>e>0. \<exists>d. gauge d \<and>
   882           (\<forall>p. p tagged_division_of i \<and> d fine p
   883                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
   884 
   885 definition has_integral (infixr "has'_integral" 46) where 
   886 "((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
   887         if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
   888         else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   889               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
   890                                        norm(z - y) < e))"
   891 
   892 lemma has_integral:
   893  "(f has_integral y) ({a..b}) \<longleftrightarrow>
   894         (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
   895                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   896   unfolding has_integral_def has_integral_compact_interval_def by auto
   897 
   898 lemma has_integralD[dest]: assumes
   899  "(f has_integral y) ({a..b})" "e>0"
   900   obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
   901                         \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   902   using assms unfolding has_integral by auto
   903 
   904 lemma has_integral_alt:
   905  "(f has_integral y) i \<longleftrightarrow>
   906       (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
   907        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   908                                \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
   909                                         has_integral z) ({a..b}) \<and>
   910                                        norm(z - y) < e)))"
   911   unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   912 
   913 lemma has_integral_altD:
   914   assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
   915   obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
   916   using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   917 
   918 definition integrable_on (infixr "integrable'_on" 46) where
   919   "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
   920 
   921 definition "integral i f \<equiv> SOME y. (f has_integral y) i"
   922 
   923 lemma integrable_integral[dest]:
   924  "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   925   unfolding integrable_on_def integral_def by(rule someI_ex)
   926 
   927 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   928   unfolding integrable_on_def by auto
   929 
   930 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   931   by auto
   932 
   933 lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
   934   shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
   935 proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
   936     unfolding vec_sub Cart_eq by(auto simp add:vec1_dest_vec1_simps split_beta)
   937   show ?thesis using assms unfolding has_integral apply safe
   938     apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
   939     apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
   940 
   941 lemma setsum_content_null:
   942   assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
   943   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
   944 proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
   945   obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
   946   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   947   from this(2) guess c .. then guess d .. note c_d=this
   948   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
   949   also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
   950     unfolding assms(1) c_d by auto
   951   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
   952 qed
   953 
   954 subsection {* Some basic combining lemmas. *}
   955 
   956 lemma tagged_division_unions_exists:
   957   assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
   958   "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
   959    obtains p where "p tagged_division_of i" "d fine p"
   960 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
   961   show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
   962     apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
   963     apply(rule fine_unions) using pfn by auto
   964 qed
   965 
   966 subsection {* The set we're concerned with must be closed. *}
   967 
   968 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"
   969   unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
   970 
   971 subsection {* General bisection principle for intervals; might be useful elsewhere. *}
   972 
   973 lemma interval_bisection_step:
   974   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::real^'n})"
   975   obtains c d where "~(P{c..d})"
   976   "\<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"
   977 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
   978   note ab=this[unfolded interval_eq_empty not_ex not_less]
   979   { fix f have "finite f \<Longrightarrow>
   980         (\<forall>s\<in>f. P s) \<Longrightarrow>
   981         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
   982         (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
   983     proof(induct f rule:finite_induct)
   984       case empty show ?case using assms(1) by auto
   985     next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
   986         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
   987         using insert by auto
   988     qed } note * = this
   989   let ?A = "{{c..d} | c d. \<forall>i. (c$i = a$i) \<and> (d$i = (a$i + b$i) / 2) \<or> (c$i = (a$i + b$i) / 2) \<and> (d$i = b$i)}"
   990   let ?PP = "\<lambda>c d. \<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"
   991   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
   992     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
   993   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
   994   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
   995     let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..
   996       (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"
   997     have "?A \<subseteq> ?B" proof case goal1
   998       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
   999       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
  1000       show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)
  1001         unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta
  1002       proof(rule_tac[1-2] allI) fix i show "c $ i = (if i \<in> {i. c $ i = a $ i} then a $ i else (a $ i + b $ i) / 2)"
  1003           "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"
  1004           using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
  1005       qed auto qed
  1006     thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto
  1007     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
  1008     note c_d=this[rule_format]
  1009     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case 
  1010         using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
  1011     show "\<exists>a b. s = {a..b}" unfolding c_d by auto
  1012     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
  1013     note e_f=this[rule_format]
  1014     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
  1015     then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto
  1016     hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)
  1017     proof- assume "c$i \<noteq> e$i" thus "d$i \<noteq> f$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
  1018     next   assume "d$i \<noteq> f$i" thus "c$i \<noteq> e$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
  1019     qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
  1020     show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
  1021       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
  1022       hence x:"c$i < d$i" "e$i < f$i" "c$i < f$i" "e$i < d$i" unfolding mem_interval apply-apply(erule_tac[!] x=i in allE)+ by auto
  1023       show False using c_d(2)[of i] apply- apply(erule_tac disjE)
  1024       proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"
  1025         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
  1026       next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"
  1027         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
  1028       qed qed qed
  1029   also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)
  1030     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
  1031     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
  1032     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
  1033     show "x\<in>{a..b}" unfolding mem_interval proof 
  1034       fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
  1035         using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
  1036   next fix x assume x:"x\<in>{a..b}"
  1037     have "\<forall>i. \<exists>c d. (c = a$i \<and> d = (a$i + b$i) / 2 \<or> c = (a$i + b$i) / 2 \<and> d = b$i) \<and> c\<le>x$i \<and> x$i \<le> d"
  1038       (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i
  1039       have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"
  1040         using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
  1041     qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq
  1042       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
  1043   qed finally show False using assms by auto qed
  1044 
  1045 lemma interval_bisection:
  1046   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"
  1047   obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
  1048 proof-
  1049   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>
  1050                            2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-
  1051       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
  1052       thus ?thesis apply(cases "P {fst x..snd x}") by auto
  1053     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
  1054       thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
  1055     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
  1056   def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
  1057   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
  1058     (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> 
  1059     2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")
  1060   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
  1061     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
  1062     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
  1063     next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
  1064     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
  1065 
  1066   have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
  1067   proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this
  1068     show ?case apply(rule_tac x=n in exI) proof(rule,rule)
  1069       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
  1070       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)
  1071       also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"
  1072       proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"
  1073           using xy[unfolded mem_interval,THEN spec[where x=i]]
  1074           unfolding vector_minus_component by auto qed
  1075       also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib
  1076       proof(rule setsum_mono) case goal1 thus ?case
  1077         proof(induct n) case 0 thus ?case unfolding AB by auto
  1078         next case (Suc n) have "B (Suc n) $ i - A (Suc n) $ i \<le> (B n $ i - A n $ i) / 2" using AB(4)[of n i] by auto
  1079           also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
  1080         qed qed
  1081       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
  1082     qed qed
  1083   { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
  1084     have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
  1085     proof(induct d) case 0 thus ?case by auto
  1086     next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
  1087         apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
  1088       proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)
  1089       qed qed } note ABsubset = this 
  1090   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
  1091   proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
  1092   then guess x0 .. note x0=this[rule_format]
  1093   show thesis proof(rule that[rule_format,of x0])
  1094     show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
  1095     fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
  1096     show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
  1097       apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
  1098     proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto
  1099       show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
  1100       show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
  1101     qed qed qed 
  1102 
  1103 subsection {* Cousin's lemma. *}
  1104 
  1105 lemma fine_division_exists: assumes "gauge g" 
  1106   obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"
  1107 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
  1108   then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
  1109 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
  1110   guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
  1111     apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
  1112   proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
  1113     fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"
  1114     thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule
  1115       apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
  1116   qed note x=this
  1117   obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
  1118   from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
  1119   have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
  1120   thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
  1121 
  1122 subsection {* Basic theorems about integrals. *}
  1123 
  1124 lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  1125   assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
  1126 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
  1127   have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.
  1128     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
  1129   proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
  1130     guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
  1131     guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
  1132     guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
  1133     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
  1134       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
  1135     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
  1136       apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
  1137     finally show False by auto
  1138   qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
  1139     thus False apply-apply(cases "\<exists>a b. i = {a..b}")
  1140       using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
  1141   assume as:"\<not> (\<exists>a b. i = {a..b})"
  1142   guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
  1143   guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
  1144   have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
  1145     using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
  1146   note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
  1147   guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
  1148   guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
  1149   have "z = w" using lem[OF w(1) z(1)] by auto
  1150   hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
  1151     using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
  1152   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
  1153   finally show False by auto qed
  1154 
  1155 lemma integral_unique[intro]:
  1156   "(f has_integral y) k \<Longrightarrow> integral k f = y"
  1157   unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
  1158 
  1159 lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  1160   assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
  1161 proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.
  1162     (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
  1163   proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"
  1164     assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
  1165     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
  1166       apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
  1167     proof(rule,rule,erule conjE) case goal1
  1168       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
  1169         fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
  1170         thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
  1171       qed thus ?case using as by auto
  1172     qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1173     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
  1174       using assms by(auto simp add:has_integral intro:lem) }
  1175   have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
  1176   assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
  1177   apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
  1178   proof- fix e::real and a b assume "e>0"
  1179     thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
  1180       apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
  1181   qed auto qed
  1182 
  1183 lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"
  1184   apply(rule has_integral_is_0) by auto 
  1185 
  1186 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
  1187   using has_integral_unique[OF has_integral_0] by auto
  1188 
  1189 lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  1190   assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
  1191 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  1192   have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.
  1193     (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
  1194   proof(subst has_integral,rule,rule) case goal1
  1195     from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  1196     have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
  1197     guess g using has_integralD[OF goal1(1) *] . note g=this
  1198     show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
  1199     proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
  1200       have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto
  1201       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
  1202         unfolding o_def unfolding scaleR[THEN sym] * by simp
  1203       also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
  1204       finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
  1205       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
  1206         apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
  1207     qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1208     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  1209   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  1210   proof(rule,rule) fix e::real  assume e:"0<e"
  1211     have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
  1212     guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
  1213     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
  1214       apply(rule_tac x=M in exI) apply(rule,rule M(1))
  1215     proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
  1216       have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
  1217         unfolding o_def apply(rule ext) using zero by auto
  1218       show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
  1219         apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
  1220     qed qed qed
  1221 
  1222 lemma has_integral_cmul:
  1223   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
  1224   unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
  1225   by(rule scaleR.bounded_linear_right)
  1226 
  1227 lemma has_integral_neg:
  1228   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
  1229   apply(drule_tac c="-1" in has_integral_cmul) by auto
  1230 
  1231 lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 
  1232   assumes "(f has_integral k) s" "(g has_integral l) s"
  1233   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
  1234 proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.
  1235     (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
  1236      ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
  1237     show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
  1238       guess d1 using has_integralD[OF goal1(1) *] . note d1=this
  1239       guess d2 using has_integralD[OF goal1(2) *] . note d2=this
  1240       show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
  1241         apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
  1242       proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
  1243         have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
  1244           unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
  1245           by(rule setsum_cong2,auto)
  1246         have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
  1247           unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
  1248         from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
  1249         have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
  1250           apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
  1251         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
  1252       qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1253     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  1254   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  1255   proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
  1256     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
  1257     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
  1258     show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
  1259     proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"
  1260       hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto
  1261       guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
  1262       guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
  1263       have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto
  1264       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"
  1265         apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
  1266         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
  1267     qed qed qed
  1268 
  1269 lemma has_integral_sub:
  1270   shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
  1271   using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
  1272 
  1273 lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"
  1274   by(rule integral_unique has_integral_0)+
  1275 
  1276 lemma integral_add:
  1277   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
  1278    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
  1279   apply(rule integral_unique) apply(drule integrable_integral)+
  1280   apply(rule has_integral_add) by assumption+
  1281 
  1282 lemma integral_cmul:
  1283   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
  1284   apply(rule integral_unique) apply(drule integrable_integral)+
  1285   apply(rule has_integral_cmul) by assumption+
  1286 
  1287 lemma integral_neg:
  1288   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
  1289   apply(rule integral_unique) apply(drule integrable_integral)+
  1290   apply(rule has_integral_neg) by assumption+
  1291 
  1292 lemma integral_sub:
  1293   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
  1294   apply(rule integral_unique) apply(drule integrable_integral)+
  1295   apply(rule has_integral_sub) by assumption+
  1296 
  1297 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
  1298   unfolding integrable_on_def using has_integral_0 by auto
  1299 
  1300 lemma integrable_add:
  1301   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
  1302   unfolding integrable_on_def by(auto intro: has_integral_add)
  1303 
  1304 lemma integrable_cmul:
  1305   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
  1306   unfolding integrable_on_def by(auto intro: has_integral_cmul)
  1307 
  1308 lemma integrable_neg:
  1309   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
  1310   unfolding integrable_on_def by(auto intro: has_integral_neg)
  1311 
  1312 lemma integrable_sub:
  1313   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
  1314   unfolding integrable_on_def by(auto intro: has_integral_sub)
  1315 
  1316 lemma integrable_linear:
  1317   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
  1318   unfolding integrable_on_def by(auto intro: has_integral_linear)
  1319 
  1320 lemma integral_linear:
  1321   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
  1322   apply(rule has_integral_unique) defer unfolding has_integral_integral 
  1323   apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
  1324   apply(rule integrable_linear) by assumption+
  1325 
  1326 lemma integral_component_eq[simp]: fixes f::"real^'n \<Rightarrow> real^'m"
  1327   assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
  1328   using integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] .
  1329 
  1330 lemma has_integral_setsum:
  1331   assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
  1332   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
  1333 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
  1334   case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
  1335     apply(rule has_integral_add) using insert assms by auto
  1336 qed auto
  1337 
  1338 lemma integral_setsum:
  1339   shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
  1340   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
  1341   apply(rule integral_unique) apply(rule has_integral_setsum)
  1342   using integrable_integral by auto
  1343 
  1344 lemma integrable_setsum:
  1345   shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
  1346   unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
  1347 
  1348 lemma has_integral_eq:
  1349   assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
  1350   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
  1351   using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
  1352 
  1353 lemma integrable_eq:
  1354   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
  1355   unfolding integrable_on_def using has_integral_eq[of s f g] by auto
  1356 
  1357 lemma has_integral_eq_eq:
  1358   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
  1359   using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 
  1360 
  1361 lemma has_integral_null[dest]:
  1362   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
  1363   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
  1364 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
  1365   fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
  1366   have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
  1367     using setsum_content_null[OF assms(1) p, of f] . 
  1368   thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
  1369 
  1370 lemma has_integral_null_eq[simp]:
  1371   shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
  1372   apply rule apply(rule has_integral_unique,assumption) 
  1373   apply(drule has_integral_null,assumption)
  1374   apply(drule has_integral_null) by auto
  1375 
  1376 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
  1377   by(rule integral_unique,drule has_integral_null)
  1378 
  1379 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
  1380   unfolding integrable_on_def apply(drule has_integral_null) by auto
  1381 
  1382 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
  1383   unfolding empty_as_interval apply(rule has_integral_null) 
  1384   using content_empty unfolding empty_as_interval .
  1385 
  1386 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
  1387   apply(rule,rule has_integral_unique,assumption) by auto
  1388 
  1389 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
  1390 
  1391 lemma integral_empty[simp]: shows "integral {} f = 0"
  1392   apply(rule integral_unique) using has_integral_empty .
  1393 
  1394 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a}"
  1395 proof- have *:"{a} = {a..a}" apply(rule set_ext) unfolding mem_interval singleton_iff Cart_eq
  1396     apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps)
  1397   show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
  1398     apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
  1399     unfolding interior_closed_interval using interval_sing by auto qed
  1400 
  1401 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
  1402 
  1403 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
  1404 
  1405 subsection {* Cauchy-type criterion for integrability. *}
  1406 
  1407 lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
  1408   shows "f integrable_on {a..b} \<longleftrightarrow>
  1409   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
  1410                             p2 tagged_division_of {a..b} \<and> d fine p2
  1411                             \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
  1412                                      setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
  1413 proof assume ?l
  1414   then guess y unfolding integrable_on_def has_integral .. note y=this
  1415   show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
  1416     then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
  1417     show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
  1418     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
  1419       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1420         apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])
  1421         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
  1422     qed qed
  1423 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
  1424   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  1425   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
  1426   hence "\<forall>n. \<exists>p. p tagged_division_of {a..b} \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p" apply-
  1427   proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
  1428   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
  1429   have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
  1430   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
  1431   proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
  1432     show ?case apply(rule_tac x=N in exI)
  1433     proof(rule,rule,rule,rule) fix m n assume mn:"N \<le> m" "N \<le> n" have *:"N = (N - 1) + 1" using N by auto
  1434       show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
  1435         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
  1436         using dp p(1) using mn by auto 
  1437     qed qed
  1438   then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
  1439   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
  1440   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
  1441     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
  1442     guess N2 using y[OF *] .. note N2=this
  1443     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
  1444       apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
  1445     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
  1446       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
  1447       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
  1448       show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
  1449         apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
  1450         using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]
  1451         using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
  1452 
  1453 subsection {* Additivity of integral on abutting intervals. *}
  1454 
  1455 lemma interval_split:
  1456   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
  1457   "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
  1458   apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq
  1459   unfolding Cart_lambda_beta by auto
  1460 
  1461 lemma content_split:
  1462   "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
  1463 proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def
  1464   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
  1465   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
  1466   have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
  1467     "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 
  1468     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  1469   assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c
  1470     \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
  1471     by  (auto simp add:field_simps)
  1472   moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
  1473     unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
  1474   ultimately show ?thesis 
  1475     unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
  1476 qed
  1477 
  1478 lemma division_split_left_inj:
  1479   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  1480   "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"
  1481   shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  1482 proof- note d=division_ofD[OF assms(1)]
  1483   have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"
  1484     unfolding interval_split content_eq_0_interior by auto
  1485   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  1486   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  1487   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  1488   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  1489     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  1490 
  1491 lemma division_split_right_inj:
  1492   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  1493   "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"
  1494   shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  1495 proof- note d=division_ofD[OF assms(1)]
  1496   have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k >= c}) = {})"
  1497     unfolding interval_split content_eq_0_interior by auto
  1498   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  1499   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  1500   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  1501   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  1502     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  1503 
  1504 lemma tagged_division_split_left_inj:
  1505   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}" 
  1506   shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"
  1507 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  1508   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
  1509     apply(rule_tac[1-2] *) using assms(2-) by auto qed
  1510 
  1511 lemma tagged_division_split_right_inj:
  1512   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" 
  1513   shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"
  1514 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  1515   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
  1516     apply(rule_tac[1-2] *) using assms(2-) by auto qed
  1517 
  1518 lemma division_split:
  1519   assumes "p division_of {a..b::real^'n}"
  1520   shows "{l \<inter> {x. x$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<le> c} = {})} division_of ({a..b} \<inter> {x. x$k \<le> c})" (is "?p1 division_of ?I1") and 
  1521         "{l \<inter> {x. x$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$k \<ge> c})" (is "?p2 division_of ?I2")
  1522 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]
  1523   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
  1524   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  1525     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  1526     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  1527       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  1528     fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  1529     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  1530   { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  1531     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  1532     show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  1533       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto
  1534     fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  1535     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  1536 qed
  1537 
  1538 lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  1539   assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"
  1540   shows "(f has_integral (i + j)) ({a..b})"
  1541 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
  1542   guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]
  1543   guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]
  1544   let ?d = "\<lambda>x. if x$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$k - c)) \<inter> d1 x \<inter> d2 x"
  1545   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
  1546   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
  1547     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
  1548     have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  1549          "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  1550     proof- fix x kk assume as:"(x,kk)\<in>p"
  1551       show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"
  1552       proof(rule ccontr) case goal1
  1553         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  1554           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  1555         hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast 
  1556         then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)
  1557           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  1558         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  1559       qed
  1560       show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"
  1561       proof(rule ccontr) case goal1
  1562         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"
  1563           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  1564         hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast 
  1565         then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)
  1566           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)
  1567         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  1568       qed
  1569     qed
  1570 
  1571     have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
  1572     have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
  1573     proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
  1574     have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>
  1575       setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
  1576                = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
  1577       apply(rule setsum_mono_zero_left) prefer 3
  1578     proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"
  1579       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
  1580       then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
  1581       have "content (g k) = 0" using xk using content_empty by auto
  1582       thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
  1583     qed auto
  1584     have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
  1585 
  1586     let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"
  1587     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
  1588       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  1589     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto
  1590       fix x l assume xl:"(x,l)\<in>?M1"
  1591       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  1592       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  1593       thus "l \<subseteq> d1 x" unfolding xl' by auto
  1594       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  1595         using lem0(1)[OF xl'(3-4)] by auto
  1596       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])
  1597       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
  1598       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  1599       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  1600       proof(cases "l' = r' \<longrightarrow> x' = y'")
  1601         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1602       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  1603         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1604       qed qed moreover
  1605 
  1606     let ?M2 = "{(x,kk \<inter> {x. x$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<ge> c} \<noteq> {}}" 
  1607     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
  1608       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  1609     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x$k \<ge> c}" unfolding p(8)[THEN sym] by auto
  1610       fix x l assume xl:"(x,l)\<in>?M2"
  1611       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  1612       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  1613       thus "l \<subseteq> d2 x" unfolding xl' by auto
  1614       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  1615         using lem0(2)[OF xl'(3-4)] by auto
  1616       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])
  1617       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
  1618       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  1619       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  1620       proof(cases "l' = r' \<longrightarrow> x' = y'")
  1621         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1622       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  1623         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1624       qed qed ultimately
  1625 
  1626     have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"
  1627       apply- apply(rule norm_triangle_lt) by auto
  1628     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto
  1629       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)
  1630        = (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto
  1631       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) + (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) - (i + j)"
  1632         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
  1633         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
  1634       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto
  1635       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto
  1636       qed also note setsum_addf[THEN sym]
  1637       also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) x
  1638         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
  1639       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
  1640         thus "content (b \<inter> {x. x $ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $ k}) *\<^sub>R f a = content b *\<^sub>R f a"
  1641           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto
  1642       qed note setsum_cong2[OF this]
  1643       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
  1644         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
  1645         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
  1646     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
  1647 
  1648 subsection {* A sort of converse, integrability on subintervals. *}
  1649 
  1650 lemma tagged_division_union_interval:
  1651   assumes "p1 tagged_division_of ({a..b} \<inter> {x::real^'n. x$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c})"
  1652   shows "(p1 \<union> p2) tagged_division_of ({a..b})"
  1653 proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto
  1654   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])
  1655     unfolding interval_split interior_closed_interval
  1656     by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed
  1657 
  1658 lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"
  1659   assumes "(f has_integral i) ({a..b})" "e>0"
  1660   obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>
  1661                                 p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2
  1662                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
  1663                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
  1664 proof- guess d using has_integralD[OF assms] . note d=this
  1665   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
  1666   proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
  1667                    assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
  1668     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
  1669     have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
  1670       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
  1671     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
  1672       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
  1673       have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
  1674       moreover have "interior {x. x $ k = c} = {}" 
  1675       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto
  1676         then guess e unfolding mem_interior .. note e=this
  1677         have x:"x$k = c" using x interior_subset by fastsimp
  1678         have *:"\<And>i. \<bar>(x - (x + (\<chi> i. if i = k then e / 2 else 0))) $ i\<bar> = (if i = k then e/2 else 0)" using e by auto
  1679         have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm 
  1680           apply(rule le_less_trans[OF norm_le_l1]) unfolding * 
  1681           unfolding setsum_delta[OF finite_UNIV] using e by auto 
  1682         hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto
  1683         thus False unfolding mem_Collect_eq using e x by auto
  1684       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
  1685       thus "content b *\<^sub>R f a = 0" by auto
  1686     qed auto
  1687     also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+
  1688     finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
  1689 
  1690 lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"
  1691   shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) 
  1692 proof- guess y using assms unfolding integrable_on_def .. note y=this
  1693   def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"
  1694   and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"
  1695   show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]
  1696   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
  1697     from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]
  1698     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
  1699                               norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
  1700     show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  1701     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p2"
  1702       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1703       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
  1704         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
  1705           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  1706           using p using assms by(auto simp add:algebra_simps)
  1707       qed qed  
  1708     show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  1709     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p2"
  1710       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1711       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
  1712         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
  1713           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]
  1714           using p using assms by(auto simp add:algebra_simps) qed qed qed qed
  1715 
  1716 subsection {* Generalized notion of additivity. *}
  1717 
  1718 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
  1719 
  1720 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where
  1721   "operative opp f \<equiv> 
  1722     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
  1723     (\<forall>a b c k. f({a..b}) =
  1724                    opp (f({a..b} \<inter> {x. x$k \<le> c}))
  1725                        (f({a..b} \<inter> {x. x$k \<ge> c})))"
  1726 
  1727 lemma operativeD[dest]: assumes "operative opp f"
  1728   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"
  1729   "\<And>a b c k. f({a..b}) = opp (f({a..b} \<inter> {x. x$k \<le> c})) (f({a..b} \<inter> {x. x$k \<ge> c}))"
  1730   using assms unfolding operative_def by auto
  1731 
  1732 lemma operative_trivial:
  1733  "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
  1734   unfolding operative_def by auto
  1735 
  1736 lemma property_empty_interval:
  1737  "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
  1738   using content_empty unfolding empty_as_interval by auto
  1739 
  1740 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
  1741   unfolding operative_def apply(rule property_empty_interval) by auto
  1742 
  1743 subsection {* Using additivity of lifted function to encode definedness. *}
  1744 
  1745 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
  1746   by (metis map_of.simps option.nchotomy)
  1747 
  1748 lemma exists_option:
  1749  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
  1750   by (metis map_of.simps option.nchotomy)
  1751 
  1752 fun lifted where 
  1753   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
  1754   "lifted opp None _ = (None::'b option)" |
  1755   "lifted opp _ None = None"
  1756 
  1757 lemma lifted_simp_1[simp]: "lifted opp v None = None"
  1758   apply(induct v) by auto
  1759 
  1760 definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
  1761                    (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
  1762                    (\<forall>x. opp (neutral opp) x = x)"
  1763 
  1764 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
  1765   "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
  1766   "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
  1767   unfolding monoidal_def using assms by fastsimp
  1768 
  1769 lemma monoidal_ac: assumes "monoidal opp"
  1770   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
  1771   "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
  1772   using assms unfolding monoidal_def apply- by metis+
  1773 
  1774 lemma monoidal_simps[simp]: assumes "monoidal opp"
  1775   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
  1776   using monoidal_ac[OF assms] by auto
  1777 
  1778 lemma neutral_lifted[cong]: assumes "monoidal opp"
  1779   shows "neutral (lifted opp) = Some(neutral opp)"
  1780   apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
  1781 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
  1782   thus "x = Some (neutral opp)" apply(induct x) defer
  1783     apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
  1784     apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
  1785 qed(auto simp add:monoidal_ac[OF assms])
  1786 
  1787 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
  1788   unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
  1789 
  1790 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
  1791 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
  1792 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
  1793 
  1794 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
  1795 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
  1796 
  1797 lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"
  1798   unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto
  1799 
  1800 lemma support_clauses:
  1801   "\<And>f g s. support opp f {} = {}"
  1802   "\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
  1803   "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
  1804   "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
  1805   "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
  1806   "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
  1807   "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
  1808 unfolding support_def by auto
  1809 
  1810 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
  1811   unfolding support_def by auto
  1812 
  1813 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
  1814   unfolding iterate_def fold'_def by auto 
  1815 
  1816 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
  1817   shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
  1818 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
  1819   show ?thesis unfolding iterate_def if_P[OF True] * by auto
  1820 next case False note x=this
  1821   note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]
  1822   show ?thesis proof(cases "f x = neutral opp")
  1823     case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
  1824       unfolding True monoidal_simps[OF assms(1)] by auto
  1825   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
  1826       apply(subst fun_left_comm.fold_insert[OF * finite_support])
  1827       using `finite s` unfolding support_def using False x by auto qed qed 
  1828 
  1829 lemma iterate_some:
  1830   assumes "monoidal opp"  "finite s"
  1831   shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
  1832 proof(induct s) case empty thus ?case using assms by auto
  1833 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
  1834     defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
  1835 
  1836 subsection {* Two key instances of additivity. *}
  1837 
  1838 lemma neutral_add[simp]:
  1839   "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
  1840   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
  1841 
  1842 lemma operative_content[intro]: "operative (op +) content"
  1843   unfolding operative_def content_split[THEN sym] neutral_add by auto
  1844 
  1845 lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
  1846   unfolding neutral_def apply(rule some_equality) defer
  1847   apply(erule_tac x=0 in allE) by auto
  1848 
  1849 lemma monoidal_monoid[intro]:
  1850   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
  1851   unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps) 
  1852 
  1853 lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
  1854   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
  1855   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
  1856   apply(rule,rule,rule,rule) defer apply(rule allI)+
  1857 proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
  1858               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)
  1859                (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)"
  1860   proof(cases "f integrable_on {a..b}") 
  1861     case True show ?thesis unfolding if_P[OF True]
  1862       unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]
  1863       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) 
  1864       apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+
  1865   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}))"
  1866     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
  1867         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)
  1868         apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto
  1869       thus False using False by auto
  1870     qed thus ?thesis using False by auto 
  1871   qed next 
  1872   fix a b assume as:"content {a..b::real^'n} = 0"
  1873   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
  1874     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
  1875 
  1876 subsection {* Points of division of a partition. *}
  1877 
  1878 definition "division_points (k::(real^'n) set) d = 
  1879     {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>
  1880            (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  1881 
  1882 lemma division_points_finite: assumes "d division_of i"
  1883   shows "finite (division_points i d)"
  1884 proof- note assm = division_ofD[OF assms]
  1885   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$j < x \<and> x < (interval_upperbound i)$j \<and>
  1886            (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"
  1887   have *:"division_points i d = \<Union>(?M ` UNIV)"
  1888     unfolding division_points_def by auto
  1889   show ?thesis unfolding * using assm by auto qed
  1890 
  1891 lemma division_points_subset:
  1892   assumes "d division_of {a..b}" "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  1893   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$k \<le> c} = {})}
  1894                   \<subseteq> division_points ({a..b}) d" (is ?t1) and
  1895         "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}
  1896                   \<subseteq> division_points ({a..b}) d" (is ?t2)
  1897 proof- note assm = division_ofD[OF assms(1)]
  1898   have *:"\<forall>i. a$i \<le> b$i"   "\<forall>i. a$i \<le> (\<chi> i. if i = k then min (b $ k) c else b $ i) $ i"
  1899     "\<forall>i. (\<chi> i. if i = k then max (a $ k) c else a $ i) $ i \<le> b$i"  "min (b $ k) c = c" "max (a $ k) c = c"
  1900     using assms using less_imp_le by auto
  1901   show ?t1 unfolding division_points_def interval_split[of a b]
  1902     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  1903     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  1904   proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"
  1905       "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"  "i = l \<inter> {x. x $ k \<le> c}" "l \<in> d" "l \<inter> {x. x $ k \<le> c} \<noteq> {}"
  1906     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  1907     have *:"\<forall>i. u $ i \<le> (\<chi> i. if i = k then min (v $ k) c else v $ i) $ i" using as(6) unfolding l interval_split interval_ne_empty as .
  1908     have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  1909     show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"
  1910       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  1911       apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  1912       apply(case_tac[!] "fst x = k") using assms by auto
  1913   qed
  1914   show ?t2 unfolding division_points_def interval_split[of a b]
  1915     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *
  1916     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+
  1917   proof- fix i l x assume as:"(if fst x = k then c else a $ fst x) < snd x" "snd x < b $ fst x" "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"
  1918       "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"
  1919     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  1920     have *:"\<forall>i. (\<chi> i. if i = k then max (u $ k) c else u $ i) $ i \<le> v $ i" using as(6) unfolding l interval_split interval_ne_empty as .
  1921     have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  1922     show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"
  1923       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-
  1924       apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  1925       apply(case_tac[!] "fst x = k") using assms by auto qed qed
  1926 
  1927 lemma division_points_psubset:
  1928   assumes "d division_of {a..b}"  "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"
  1929   "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"
  1930   shows "division_points ({a..b} \<inter> {x. x$k \<le> c}) {l \<inter> {x. x$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
  1931         "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 
  1932 proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)
  1933   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
  1934   have uv:"\<forall>i. u$i \<le> v$i" "\<forall>i. a$i \<le> u$i \<and> v$i \<le> b$i" using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
  1935     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
  1936   have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  1937          "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  1938     unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  1939     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  1940   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
  1941     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  1942     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  1943     unfolding division_points_def unfolding interval_bounds[OF ab]
  1944     apply (auto simp add:interval_bounds) unfolding * by auto
  1945   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto
  1946 
  1947   have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"
  1948          "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"
  1949     unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  1950     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto
  1951   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
  1952     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer
  1953     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)
  1954     unfolding division_points_def unfolding interval_bounds[OF ab]
  1955     apply (auto simp add:interval_bounds) unfolding * by auto
  1956   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed
  1957 
  1958 subsection {* Preservation by divisions and tagged divisions. *}
  1959 
  1960 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
  1961   unfolding support_def by auto
  1962 
  1963 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
  1964   unfolding iterate_def support_support by auto
  1965 
  1966 lemma iterate_expand_cases:
  1967   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
  1968   apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
  1969 
  1970 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
  1971   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  1972 proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
  1973      iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  1974   proof- case goal1 show ?case using goal1
  1975     proof(induct s) case empty thus ?case using assms(1) by auto
  1976     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
  1977         unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
  1978         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
  1979         apply(rule finite_imageI insert)+ apply(subst if_not_P)
  1980         unfolding image_iff o_def using insert(2,4) by auto
  1981     qed qed
  1982   show ?thesis 
  1983     apply(cases "finite (support opp g (f ` s))")
  1984     apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
  1985     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
  1986     apply(rule subset_inj_on[OF assms(2) support_subset])+
  1987     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
  1988     apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
  1989 
  1990 
  1991 (* This lemma about iterations comes up in a few places.                     *)
  1992 lemma iterate_nonzero_image_lemma:
  1993   assumes "monoidal opp" "finite s" "g(a) = neutral opp"
  1994   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
  1995   shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
  1996 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
  1997   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
  1998     unfolding support_def using assms(3) by auto
  1999   show ?thesis unfolding *
  2000     apply(subst iterate_support[THEN sym]) unfolding support_clauses
  2001     apply(subst iterate_image[OF assms(1)]) defer
  2002     apply(subst(2) iterate_support[THEN sym]) apply(subst **)
  2003     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
  2004 
  2005 lemma iterate_eq_neutral:
  2006   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
  2007   shows "(iterate opp s f = neutral opp)"
  2008 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
  2009   show ?thesis apply(subst iterate_support[THEN sym]) 
  2010     unfolding * using assms(1) by auto qed
  2011 
  2012 lemma iterate_op: assumes "monoidal opp" "finite s"
  2013   shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
  2014 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
  2015 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
  2016     unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
  2017 
  2018 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
  2019   shows "iterate opp s f = iterate opp s g"
  2020 proof- have *:"support opp g s = support opp f s"
  2021     unfolding support_def using assms(2) by auto
  2022   show ?thesis
  2023   proof(cases "finite (support opp f s)")
  2024     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
  2025       unfolding * by auto
  2026   next def su \<equiv> "support opp f s"
  2027     case True note support_subset[of opp f s] 
  2028     thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
  2029       unfolding su_def[symmetric]
  2030     proof(induct su) case empty show ?case by auto
  2031     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
  2032         unfolding if_not_P[OF insert(2)] apply(subst insert(3))
  2033         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
  2034 
  2035 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
  2036 
  2037 lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"
  2038   assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
  2039   shows "iterate opp d f = f {a..b}"
  2040 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
  2041   proof(induct C arbitrary:a b d rule:full_nat_induct)
  2042     case goal1
  2043     { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
  2044       thus ?case apply-apply(cases) defer apply assumption
  2045       proof- assume as:"content {a..b} = 0"
  2046         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
  2047         proof fix x assume x:"x\<in>d"
  2048           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
  2049           thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
  2050             using operativeD(1)[OF assms(2)] x by auto
  2051         qed qed }
  2052     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
  2053     hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case 
  2054     proof(cases "division_points {a..b} d = {}")
  2055       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
  2056         (\<forall>j. u$j = a$j \<and> v$j = a$j \<or> u$j = b$j \<and> v$j = b$j \<or> u$j = a$j \<and> v$j = b$j)"
  2057         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
  2058         apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)
  2059       proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
  2060         hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto
  2061         have *:"\<And>p r Q. p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as by auto
  2062         have "(j, u$j) \<notin> division_points {a..b} d"
  2063           "(j, v$j) \<notin> division_points {a..b} d" using True by auto
  2064         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
  2065         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
  2066         moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] 
  2067           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
  2068           unfolding interval_ne_empty mem_interval by auto
  2069         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"
  2070           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto
  2071       qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
  2072       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
  2073       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
  2074       have "{a..b} \<in> d"
  2075       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
  2076         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
  2077         show "u = a" "v = b" unfolding Cart_eq
  2078         proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]
  2079           thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto
  2080         qed qed
  2081       hence *:"d = insert {a..b} (d - {{a..b}})" by auto
  2082       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
  2083       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
  2084         then guess u v apply-by(erule exE conjE)+ note uv=this
  2085         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
  2086         then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto
  2087         hence "u$j = v$j" using uv(2)[rule_format,of j] by auto
  2088         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto
  2089         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
  2090       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
  2091         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
  2092     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
  2093       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
  2094         by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]
  2095       from this(3) guess j .. note j=this
  2096       def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"
  2097       def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"
  2098       def cb \<equiv> "(\<chi> i. if i = k then c else b$i)" and ca \<equiv> "(\<chi> i. if i = k then c else a$i)"
  2099       note division_points_psubset[OF goal1(4) ab kc(1-2) j]
  2100       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
  2101       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})"
  2102         apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])
  2103         using division_split[OF goal1(4), where k=k and c=c]
  2104         unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
  2105         using goal1(2-3) using division_points_finite[OF goal1(4)] by auto
  2106       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
  2107         unfolding * apply(rule operativeD(2)) using goal1(3) .
  2108       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"
  2109         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2110         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2111         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2112       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" 
  2113         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2114         show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split
  2115           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)
  2116           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  2117       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"
  2118         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2119         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2120         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2121       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" 
  2122         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2123         show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split
  2124           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)
  2125           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+
  2126       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}))"
  2127         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .
  2128       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})))
  2129         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
  2130         apply(rule iterate_op[THEN sym]) using goal1 by auto
  2131       finally show ?thesis by auto
  2132     qed qed qed 
  2133 
  2134 lemma iterate_image_nonzero: assumes "monoidal opp"
  2135   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
  2136   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
  2137 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
  2138   case goal1 show ?case using assms(1) by auto
  2139 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
  2140   show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
  2141     apply(rule finite_imageI goal2)+
  2142     apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
  2143     apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
  2144     apply(subst iterate_insert[OF assms(1) goal2(1)])
  2145     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
  2146     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
  2147     using goal2 unfolding o_def by auto qed 
  2148 
  2149 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
  2150   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
  2151 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
  2152   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
  2153     apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
  2154     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
  2155   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
  2156     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
  2157     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
  2158       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
  2159       unfolding as(4)[THEN sym] uv by auto
  2160   qed also have "\<dots> = f {a..b}" 
  2161     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
  2162   finally show ?thesis . qed
  2163 
  2164 subsection {* Additivity of content. *}
  2165 
  2166 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
  2167 proof- have *:"setsum f s = setsum f (support op + f s)"
  2168     apply(rule setsum_mono_zero_right)
  2169     unfolding support_def neutral_monoid using assms by auto
  2170   thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
  2171     unfolding neutral_monoid . qed
  2172 
  2173 lemma additive_content_division: assumes "d division_of {a..b}"
  2174   shows "setsum content d = content({a..b})"
  2175   unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
  2176   apply(subst setsum_iterate) using assms by auto
  2177 
  2178 lemma additive_content_tagged_division:
  2179   assumes "d tagged_division_of {a..b}"
  2180   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
  2181   unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
  2182   apply(subst setsum_iterate) using assms by auto
  2183 
  2184 subsection {* Finally, the integral of a constant *}
  2185 
  2186 lemma has_integral_const[intro]:
  2187   "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"
  2188   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
  2189   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
  2190   unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
  2191   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
  2192 
  2193 subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
  2194 
  2195 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
  2196   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
  2197   apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
  2198   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
  2199   apply(subst real_mult_commute) apply(rule mult_left_mono)
  2200   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
  2201   apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
  2202 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
  2203   fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
  2204   thus "0 \<le> content x" using content_pos_le by auto
  2205 qed(insert assms,auto)
  2206 
  2207 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
  2208   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
  2209 proof(cases "{a..b} = {}") case True
  2210   show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
  2211 next case False show ?thesis
  2212     apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
  2213     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
  2214     unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)
  2215     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
  2216     apply(subst o_def, rule abs_of_nonneg)
  2217   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
  2218       unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
  2219     guess w using nonempty_witness[OF False] .
  2220     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
  2221     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
  2222     from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
  2223     show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
  2224     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
  2225   qed(insert assms,auto) qed
  2226 
  2227 lemma rsum_diff_bound:
  2228   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
  2229   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})"
  2230   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2231   unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
  2232 
  2233 lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2234   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
  2235   shows "norm i \<le> B * content {a..b}"
  2236 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
  2237     thus ?thesis proof(cases ?P) case False
  2238       hence *:"content {a..b} = 0" using content_lt_nz by auto
  2239       hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
  2240       show ?thesis unfolding * ** using assms(1) by auto
  2241     qed auto } assume ab:?P
  2242   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2243   assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
  2244   from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  2245   from fine_division_exists[OF this(1), of a b] guess p . note p=this
  2246   have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
  2247   proof- case goal1 thus ?case unfolding not_less
  2248     using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
  2249   qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
  2250 
  2251 subsection {* Similar theorems about relationship among components. *}
  2252 
  2253 lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2254   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"
  2255   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"
  2256   unfolding setsum_component apply(rule setsum_mono)
  2257   apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv
  2258 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
  2259   from this(3) guess u v apply-by(erule exE)+ note b=this
  2260   show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b
  2261     unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)
  2262     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
  2263 
  2264 lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2265   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  2266   shows "i$k \<le> j$k"
  2267 proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> 
  2268     (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"
  2269   proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto
  2270     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
  2271     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
  2272     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
  2273     note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]
  2274     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
  2275     thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt
  2276   qed let ?P = "\<exists>a b. s = {a..b}"
  2277   { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
  2278       case True then guess a b apply-by(erule exE)+ note s=this
  2279       show ?thesis apply(rule lem) using assms[unfolded s] by auto
  2280     qed auto } assume as:"\<not> ?P"
  2281   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2282   assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto
  2283   note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
  2284   have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
  2285   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
  2286   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
  2287   guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
  2288   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
  2289   have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" by smt(*SMTSMT*)
  2290   note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
  2291   have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
  2292   show False unfolding Cart_nth.diff by(rule *) qed
  2293 
  2294 lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  2295   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"
  2296   shows "(integral s f)$k \<le> (integral s g)$k"
  2297   apply(rule has_integral_component_le) using integrable_integral assms by auto
  2298 
  2299 lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  2300   assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
  2301   shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
  2302   using assms(3) unfolding vector_le_def by auto
  2303 
  2304 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  2305   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  2306   shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
  2307   apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto
  2308 
  2309 lemma has_integral_component_nonneg: fixes f::"real^'n \<Rightarrow> real^'m"
  2310   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"
  2311   using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto
  2312 
  2313 lemma integral_component_nonneg: fixes f::"real^'n \<Rightarrow> real^'m"
  2314   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"
  2315   apply(rule has_integral_component_nonneg) using assms by auto
  2316 
  2317 lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
  2318   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
  2319   using has_integral_component_nonneg[OF assms(1), of 1]
  2320   using assms(2) unfolding vector_le_def by auto
  2321 
  2322 lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
  2323   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
  2324   apply(rule has_integral_dest_vec1_nonneg) using assms by auto
  2325 
  2326 lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"
  2327   assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"
  2328   using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto
  2329 
  2330 lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
  2331   assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
  2332   using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto
  2333 
  2334 lemma has_integral_component_lbound:
  2335   assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)$k" shows "B * content {a..b} \<le> i$k"
  2336   using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)
  2337   unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)
  2338 
  2339 lemma has_integral_component_ubound: 
  2340   assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"
  2341   shows "i$k \<le> B * content({a..b})"
  2342   using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]
  2343   unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)
  2344 
  2345 lemma integral_component_lbound:
  2346   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"
  2347   shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"
  2348   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
  2349 
  2350 lemma integral_component_ubound:
  2351   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"
  2352   shows "(integral({a..b}) f)$k \<le> B * content({a..b})"
  2353   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
  2354 
  2355 subsection {* Uniform limit of integrable functions is integrable. *}
  2356 
  2357 lemma real_arch_invD:
  2358   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
  2359   by(subst(asm) real_arch_inv)
  2360 
  2361 lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2362   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
  2363   shows "f integrable_on {a..b}"
  2364 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
  2365     show ?thesis apply cases apply(rule *,assumption)
  2366       unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
  2367   assume as:"content {a..b} > 0"
  2368   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
  2369   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
  2370   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
  2371   
  2372   have "Cauchy i" unfolding Cauchy_def
  2373   proof(rule,rule) fix e::real assume "e>0"
  2374     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
  2375     then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
  2376     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)
  2377     proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
  2378       from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
  2379       from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
  2380       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
  2381       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"
  2382       proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
  2383           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
  2384           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps)
  2385         also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
  2386         finally show ?case .
  2387       qed
  2388       show ?case unfolding vector_dist_norm apply(rule lem2) defer
  2389         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
  2390         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
  2391         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
  2392       proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
  2393           using M as by(auto simp add:field_simps)
  2394         fix x assume x:"x \<in> {a..b}"
  2395         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
  2396             using g(1)[OF x, of n] g(1)[OF x, of m] by auto
  2397         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
  2398           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
  2399         also have "\<dots> = 2 / real M" unfolding real_divide_def by auto
  2400         finally show "norm (g n x - g m x) \<le> 2 / real M"
  2401           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
  2402           by(auto simp add:algebra_simps simp add:norm_minus_commute)
  2403       qed qed qed
  2404   from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
  2405 
  2406   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
  2407   proof(rule,rule)  
  2408     case goal1 hence *:"e/3 > 0" by auto
  2409     from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
  2410     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
  2411     from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
  2412     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
  2413     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"
  2414     proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
  2415         using norm_triangle_ineq[of "sf - sg" "sg - s"]
  2416         using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:algebra_simps)
  2417       also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
  2418       finally show ?case .
  2419     qed
  2420     show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
  2421     proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
  2422       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
  2423         apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
  2424       proof- have "content {a..b} < e / 3 * (real N2)"
  2425           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
  2426         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
  2427           apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
  2428         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
  2429           unfolding inverse_eq_divide by(auto simp add:field_simps)
  2430         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)
  2431       qed qed qed qed
  2432 
  2433 subsection {* Negligible sets. *}
  2434 
  2435 definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"
  2436 
  2437 lemma dest_vec1_indicator:
  2438  "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto
  2439 
  2440 lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto
  2441 
  2442 lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto
  2443 
  2444 lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"
  2445   unfolding indicator_def by auto
  2446 
  2447 definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"
  2448 
  2449 lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"
  2450   unfolding indicator_def by auto
  2451 
  2452 subsection {* Negligibility of hyperplane. *}
  2453 
  2454 lemma vsum_nonzero_image_lemma: 
  2455   assumes "finite s" "g(a) = 0"
  2456   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
  2457   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
  2458   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
  2459   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
  2460   unfolding assms using neutral_add unfolding neutral_add using assms by auto 
  2461 
  2462 lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =
  2463   {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"
  2464 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2465   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
  2466   show ?thesis unfolding * ** interval_split by(rule refl) qed
  2467 
  2468 lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" 
  2469   shows "{l \<inter> {x. abs(x$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})"
  2470 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2471   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
  2472   note division_split(1)[OF assms, where c="c+e" and k=k,unfolded interval_split]
  2473   note division_split(2)[OF this, where c="c-e" and k=k] 
  2474   thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
  2475     apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
  2476     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule
  2477     apply(rule_tac x=l in exI) by blast+ qed
  2478 
  2479 lemma content_doublesplit: assumes "0 < e"
  2480   obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"
  2481 proof(cases "content {a..b} = 0")
  2482   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit
  2483     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
  2484     unfolding interval_doublesplit[THEN sym] using assms by auto 
  2485 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"
  2486   note False[unfolded content_eq_0 not_ex not_le, rule_format]
  2487   hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt
  2488   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
  2489   proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto
  2490     have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
  2491       (\<Prod>i\<in>UNIV - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i - interval_lowerbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i)
  2492       = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)
  2493       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto
  2494     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)
  2495       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
  2496       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]
  2497     proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto
  2498       also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
  2499       finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"
  2500         unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
  2501 
  2502 lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" 
  2503   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
  2504 proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"
  2505   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
  2506   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
  2507     have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"
  2508       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
  2509       apply(cases,rule disjI1,assumption,rule disjI2)
  2510     proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
  2511       show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
  2512         apply(rule set_ext,rule,rule) unfolding mem_Collect_eq
  2513       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
  2514         note this[unfolded subset_eq mem_ball vector_dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
  2515         thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto
  2516       qed auto qed
  2517     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
  2518     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
  2519       apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
  2520       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
  2521       prefer 2 apply(subst(asm) eq_commute) apply assumption
  2522       apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)
  2523     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}))"
  2524         apply(rule setsum_mono) unfolding split_paired_all split_conv 
  2525         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)
  2526       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
  2527       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"
  2528           unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto
  2529         thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt
  2530       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"
  2531           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
  2532         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)"
  2533           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
  2534           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)
  2535         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]
  2536         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]
  2537         note this[unfolded real_scaleR_def real_norm_def class_semiring.semiring_rules, of k c d] note le_less_trans[OF this d(2)]
  2538         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"
  2539           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
  2540           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
  2541         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
  2542           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}"
  2543           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
  2544           note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
  2545           hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
  2546           thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .
  2547         qed qed
  2548       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .
  2549     qed qed qed
  2550 
  2551 subsection {* A technical lemma about "refinement" of division. *}
  2552 
  2553 lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"
  2554   assumes "p tagged_division_of {a..b}" "gauge d"
  2555   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"
  2556 proof-
  2557   let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
  2558     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
  2559                    (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
  2560   { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
  2561     presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
  2562     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
  2563   } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"
  2564   show "?P p" apply(rule,rule) using as proof(induct p) 
  2565     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
  2566   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
  2567     note tagged_partial_division_subset[OF insert(4) subset_insertI]
  2568     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
  2569     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
  2570     note p = tagged_partial_division_ofD[OF insert(4)]
  2571     from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
  2572 
  2573     have "finite {k. \<exists>x. (x, k) \<in> p}" 
  2574       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
  2575       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
  2576     hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
  2577       apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
  2578       unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
  2579       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
  2580       using insert(2) unfolding uv xk by auto
  2581 
  2582     show ?case proof(cases "{u..v} \<subseteq> d x")
  2583       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
  2584         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
  2585         apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
  2586         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
  2587         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
  2588         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
  2589     next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
  2590       show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
  2591         apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
  2592         unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
  2593         apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
  2594         apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
  2595     qed qed qed
  2596 
  2597 subsection {* Hence the main theorem about negligible sets. *}
  2598 
  2599 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
  2600   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
  2601 proof(induct) case (insert x s) 
  2602   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
  2603   show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
  2604 
  2605 lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
  2606   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
  2607 proof(induct) case (insert a s)
  2608   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
  2609   show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
  2610     prefer 4 apply(subst insert(3)) unfolding add_right_cancel
  2611   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
  2612     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
  2613   qed(insert insert, auto) qed auto
  2614 
  2615 lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2616   assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
  2617   shows "(f has_integral 0) t"
  2618 proof- presume P:"\<And>f::real^'n \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
  2619   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
  2620   show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
  2621     apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
  2622   proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
  2623     show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
  2624   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)"
  2625       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
  2626       apply(rule,rule P) using assms(2) by auto
  2627   qed
  2628 next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
  2629   show "(f has_integral 0) {a..b}" unfolding has_integral
  2630   proof(safe) case goal1
  2631     hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
  2632       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
  2633     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
  2634     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
  2635     show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
  2636     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
  2637       fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
  2638       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
  2639       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
  2640       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
  2641       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
  2642       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)"
  2643         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
  2644       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
  2645       have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) 
  2646         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
  2647       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"
  2648       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
  2649           apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
  2650       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
  2651                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"
  2652         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
  2653         apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 
  2654       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
  2655         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
  2656           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
  2657           using tagged_division_ofD(4)[OF q(1) as''] by auto
  2658       next fix i::nat show "finite (q i)" using q by auto
  2659       next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
  2660         have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
  2661         have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
  2662         hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
  2663         moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
  2664         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
  2665         moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
  2666         proof(cases "x\<in>s") case False thus ?thesis using assm by auto
  2667         next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
  2668           moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
  2669           ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
  2670         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)"
  2671           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
  2672       qed(insert as, auto)
  2673       also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
  2674       proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
  2675           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
  2676       qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]
  2677         apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
  2678         apply(subst sumr_geometric) using goal1 by auto
  2679       finally show "?goal" by auto qed qed qed
  2680 
  2681 lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  2682   assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
  2683   shows "(g has_integral y) t"
  2684 proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a
  2685     assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
  2686     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
  2687       apply(rule has_integral_negligible[OF assms(1)]) using as by auto
  2688     hence "(g has_integral y) {a..b}" by auto } note * = this
  2689   show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
  2690     apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
  2691     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)
  2692     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
  2693 
  2694 lemma has_integral_spike_eq:
  2695   assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2696   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2697   apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
  2698 
  2699 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
  2700   shows "g integrable_on  t"
  2701   using assms unfolding integrable_on_def apply-apply(erule exE)
  2702   apply(rule,rule has_integral_spike) by fastsimp+
  2703 
  2704 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2705   shows "integral t f = integral t g"
  2706   unfolding integral_def using has_integral_spike_eq[OF assms] by auto
  2707 
  2708 subsection {* Some other trivialities about negligible sets. *}
  2709 
  2710 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
  2711 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
  2712     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
  2713     using assms(2) unfolding indicator_def by auto qed
  2714 
  2715 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
  2716 
  2717 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
  2718 
  2719 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
  2720 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
  2721   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
  2722     defer apply assumption unfolding indicator_def by auto qed
  2723 
  2724 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
  2725   using negligible_union by auto
  2726 
  2727 lemma negligible_sing[intro]: "negligible {a::real^'n}" 
  2728 proof- guess x using UNIV_witness[where 'a='n] ..
  2729   show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed
  2730 
  2731 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
  2732   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
  2733 
  2734 lemma negligible_empty[intro]: "negligible {}" by auto
  2735 
  2736 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
  2737   using assms apply(induct s) by auto
  2738 
  2739 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
  2740   using assms by(induct,auto) 
  2741 
  2742 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"
  2743   apply safe defer apply(subst negligible_def)
  2744 proof- fix t::"(real^'n) set" assume as:"negligible s"
  2745   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)  
  2746   show "(indicator s has_integral 0) t" apply(subst has_integral_alt)
  2747     apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
  2748     apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
  2749     using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def unfolding * by auto qed auto
  2750 
  2751 subsection {* Finite case of the spike theorem is quite commonly needed. *}
  2752 
  2753 lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
  2754   "(f has_integral y) t" shows "(g has_integral y) t"
  2755   apply(rule has_integral_spike) using assms by auto
  2756 
  2757 lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
  2758   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2759   apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
  2760 
  2761 lemma integrable_spike_finite:
  2762   assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
  2763   using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
  2764   apply(rule has_integral_spike_finite) by auto
  2765 
  2766 subsection {* In particular, the boundary of an interval is negligible. *}
  2767 
  2768 lemma negligible_frontier_interval: "negligible({a..b} - {a<..<b})"
  2769 proof- let ?A = "\<Union>((\<lambda>k. {x. x$k = a$k} \<union> {x. x$k = b$k}) ` UNIV)"
  2770   have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
  2771     apply(erule conjE exE)+ apply(rule_tac X="{x. x $ xa = a $ xa} \<union> {x. x $ xa = b $ xa}" in UnionI)
  2772     apply(erule_tac[!] x=xa in allE) by auto
  2773   thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
  2774 
  2775 lemma has_integral_spike_interior:
  2776   assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
  2777   apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
  2778 
  2779 lemma has_integral_spike_interior_eq:
  2780   assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
  2781   apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
  2782 
  2783 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}"
  2784   using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
  2785 
  2786 subsection {* Integrability of continuous functions. *}
  2787 
  2788 lemma neutral_and[simp]: "neutral op \<and> = True"
  2789   unfolding neutral_def apply(rule some_equality) by auto
  2790 
  2791 lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
  2792 
  2793 lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
  2794 apply induct unfolding iterate_insert[OF monoidal_and] by auto
  2795 
  2796 lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
  2797   shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
  2798   using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
  2799 
  2800 lemma operative_approximable: assumes "0 \<le> e" fixes f::"real^'n \<Rightarrow> 'a::banach"
  2801   shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::real^'n)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
  2802 proof safe fix a b::"real^'n" { assume "content {a..b} = 0"
  2803     thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
  2804       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
  2805   { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  2806     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}"
  2807       "\<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}"
  2808       apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2)] by auto }
  2809   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}"
  2810                           "\<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}"
  2811   let ?g = "\<lambda>x. if x$k = c then f x else if x$k \<le> c then g1 x else g2 x"
  2812   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)
  2813   proof safe case goal1 thus ?case apply- apply(cases "x$k=c", case_tac "x$k < c") using as assms by auto
  2814   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}"
  2815     then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this]
  2816     show ?case unfolding integrable_on_def by auto
  2817   next show "?g integrable_on {a..b} \<inter> {x. x $ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $ k \<ge> c}"
  2818       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using as(2,4) by auto qed qed
  2819 
  2820 lemma approximable_on_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2821   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"
  2822   obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  2823 proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
  2824   note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
  2825   guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
  2826 
  2827 lemma integrable_continuous: fixes f::"real^'n \<Rightarrow> 'a::banach"
  2828   assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
  2829 proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
  2830   from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
  2831   note d=conjunctD2[OF this,rule_format]
  2832   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
  2833   note p' = tagged_division_ofD[OF p(1)]
  2834   have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  2835   proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
  2836     from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
  2837     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)
  2838     proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
  2839       fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
  2840       note d(2)[OF _ _ this[unfolded mem_ball]]
  2841       thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding vector_dist_norm l norm_minus_commute by fastsimp qed qed
  2842   from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
  2843   thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
  2844 
  2845 subsection {* Specialization of additivity to one dimension. *}
  2846 
  2847 lemma operative_1_lt: assumes "monoidal opp"
  2848   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  2849                 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2850   unfolding operative_def content_eq_0_1 forall_1 vector_le_def vector_less_def
  2851 proof safe fix a b c::"real^1" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))" "a $ 1 < c $ 1" "c $ 1 < b $ 1"
  2852     from this(2-) have "{a..b} \<inter> {x. x $ 1 \<le> c $ 1} = {a..c}" "{a..b} \<inter> {x. x $ 1 \<ge> c $ 1} = {c..b}" by auto
  2853     thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c$1"] by auto
  2854 next fix a b::"real^1" and c::real
  2855   assume as:"\<forall>a b. b $ 1 \<le> a $ 1 \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a $ 1 < c $ 1 \<and> c $ 1 < b $ 1 \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}"
  2856   show "f {a..b} = opp (f ({a..b} \<inter> {x. x $ 1 \<le> c})) (f ({a..b} \<inter> {x. c \<le> x $ 1}))"
  2857   proof(cases "c \<in> {a$1 .. b$1}")
  2858     case False hence "c<a$1 \<or> c>b$1" by auto
  2859     thus ?thesis apply-apply(erule disjE)
  2860     proof- assume "c<a$1" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {1..0}"  "{a..b} \<inter> {x. c \<le> x $ 1} = {a..b}" by auto
  2861       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2862     next   assume "b$1<c" hence *:"{a..b} \<inter> {x. x $ 1 \<le> c} = {a..b}"  "{a..b} \<inter> {x. c \<le> x $ 1} = {1..0}" by auto
  2863       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2864     qed
  2865   next case True hence *:"min (b $ 1) c = c" "max (a $ 1) c = c" by auto
  2866     show ?thesis unfolding interval_split num1_eq_iff if_True * vec_def[THEN sym]
  2867     proof(cases "c = a$1 \<or> c = b$1")
  2868       case False thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})"
  2869         apply-apply(subst as(2)[rule_format]) using True by auto
  2870     next case True thus "f {a..b} = opp (f {a..vec1 c}) (f {vec1 c..b})" apply-
  2871       proof(erule disjE) assume "c=a$1" hence *:"a = vec1 c" unfolding Cart_eq by auto 
  2872         hence "f {a..vec1 c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2873         thus ?thesis using assms unfolding * by auto
  2874       next assume "c=b$1" hence *:"b = vec1 c" unfolding Cart_eq by auto 
  2875         hence "f {vec1 c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2876         thus ?thesis using assms unfolding * by auto qed qed qed qed
  2877 
  2878 lemma operative_1_le: assumes "monoidal opp"
  2879   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real^1} = neutral opp) \<and>
  2880                 (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2881 unfolding operative_1_lt[OF assms]
  2882 proof safe fix a b c::"real^1" assume as:"\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a < c" "c < b"
  2883   show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) unfolding vector_le_def vector_less_def by auto
  2884 next fix a b c ::"real^1"
  2885   assume "\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a \<le> c" "c \<le> b"
  2886   note as = this[rule_format]
  2887   show "opp (f {a..c}) (f {c..b}) = f {a..b}"
  2888   proof(cases "c = a \<or> c = b")
  2889     case False thus ?thesis apply-apply(subst as(2)) using as(3-) unfolding vector_le_def vector_less_def Cart_eq by(auto simp del:dest_vec1_eq)
  2890     next case True thus ?thesis apply-
  2891       proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2892         thus ?thesis using assms unfolding * by auto
  2893       next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2894         thus ?thesis using assms unfolding * by auto qed qed qed 
  2895 
  2896 subsection {* Special case of additivity we need for the FCT. *}
  2897 
  2898 lemma interval_bound_sing[simp]: "interval_upperbound {a} = a"  "interval_lowerbound {a} = a"
  2899   unfolding interval_upperbound_def interval_lowerbound_def unfolding Cart_eq by auto
  2900 
  2901 lemma additive_tagged_division_1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
  2902   assumes "dest_vec1 a \<le> dest_vec1 b" "p tagged_division_of {a..b}"
  2903   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  2904 proof- let ?f = "(\<lambda>k::(real^1) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
  2905   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty_1
  2906     by(auto simp add:not_less interval_bound_1 vector_less_def)
  2907   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
  2908   note * = this[unfolded if_not_P[OF **] interval_bound_1[OF assms(1)],THEN sym ]
  2909   show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
  2910     apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
  2911 
  2912 subsection {* A useful lemma allowing us to factor out the content size. *}
  2913 
  2914 lemma has_integral_factor_content:
  2915   "(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
  2916     \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
  2917 proof(cases "content {a..b} = 0")
  2918   case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
  2919     apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
  2920     apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
  2921     apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
  2922 next case False note F = this[unfolded content_lt_nz[THEN sym]]
  2923   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)"
  2924   show ?thesis apply(subst has_integral)
  2925   proof safe fix e::real assume e:"e>0"
  2926     { 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)
  2927         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2928         using F e by(auto simp add:field_simps intro:mult_pos_pos) }
  2929     {  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)
  2930         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2931         using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
  2932 
  2933 subsection {* Fundamental theorem of calculus. *}
  2934 
  2935 lemma fundamental_theorem_of_calculus: fixes f::"real^1 \<Rightarrow> 'a::banach"
  2936   assumes "a \<le> b"  "\<forall>x\<in>{a..b}. ((f o vec1) has_vector_derivative f'(vec1 x)) (at x within {a..b})"
  2937   shows "(f' has_integral (f(vec1 b) - f(vec1 a))) ({vec1 a..vec1 b})"
  2938 unfolding has_integral_factor_content
  2939 proof safe fix e::real assume e:"e>0" have ab:"dest_vec1 (vec1 a) \<le> dest_vec1 (vec1 b)" using assms(1) by auto
  2940   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
  2941   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
  2942   note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
  2943   guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  2944   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  2945                  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b})"
  2946     apply(rule_tac x="\<lambda>x. ball x (d (dest_vec1 x))" in exI,safe)
  2947     apply(rule gauge_ball_dependent,rule,rule d(1))
  2948   proof- fix p assume as:"p tagged_division_of {vec1 a..vec1 b}" "(\<lambda>x. ball x (d (dest_vec1 x))) fine p"
  2949     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f (vec1 b) - f (vec1 a))) \<le> e * content {vec1 a..vec1 b}" 
  2950       unfolding content_1[OF ab] additive_tagged_division_1[OF ab as(1),of f,THEN sym]
  2951       unfolding vector_minus_component[THEN sym] additive_tagged_division_1[OF ab as(1),of "\<lambda>x. x",THEN sym]
  2952       apply(subst dest_vec1_setsum) unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
  2953     proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
  2954       note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
  2955       have *:"dest_vec1 u \<le> dest_vec1 v" using xk unfolding k by auto
  2956       have ball:"\<forall>xa\<in>k. xa \<in> ball x (d (dest_vec1 x))" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,unfolded split_conv subset_eq] .
  2957       have "norm ((v$1 - u$1) *\<^sub>R f' x - (f v - f u)) \<le> norm (f u - f x - (u$1 - x$1) *\<^sub>R f' x) + norm (f v - f x - (v$1 - x$1) *\<^sub>R f' x)"
  2958         apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2959         unfolding scaleR.diff_left by(auto simp add:algebra_simps)
  2960       also have "... \<le> e * norm (dest_vec1 u - dest_vec1 x) + e * norm (dest_vec1 v - dest_vec1 x)"
  2961         apply(rule add_mono) apply(rule d(2)[of "x$1" "u$1",unfolded o_def vec1_dest_vec1]) prefer 4
  2962         apply(rule d(2)[of "x$1" "v$1",unfolded o_def vec1_dest_vec1])
  2963         using ball[rule_format,of u] ball[rule_format,of v] 
  2964         using xk(1-2) unfolding k subset_eq by(auto simp add:vector_dist_norm norm_real)
  2965       also have "... \<le> e * dest_vec1 (interval_upperbound k - interval_lowerbound k)"
  2966         unfolding k interval_bound_1[OF *] using xk(1) unfolding k by(auto simp add:vector_dist_norm norm_real field_simps)
  2967       finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
  2968         e * dest_vec1 (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bound_1[OF *] content_1[OF *] .
  2969     qed(insert as, auto) qed qed
  2970 
  2971 subsection {* Attempt a systematic general set of "offset" results for components. *}
  2972 
  2973 lemma gauge_modify:
  2974   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
  2975   shows "gauge (\<lambda>x y. d (f x) (f y))"
  2976   using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
  2977   apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
  2978 
  2979 subsection {* Only need trivial subintervals if the interval itself is trivial. *}
  2980 
  2981 lemma division_of_nontrivial: fixes s::"(real^'n) set set"
  2982   assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
  2983   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
  2984 proof(induct "card s" arbitrary:s rule:nat_less_induct)
  2985   fix s::"(real^'n) set set" assume assm:"s division_of {a..b}"
  2986     "\<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}" 
  2987   note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
  2988   { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
  2989     show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
  2990   assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
  2991   then obtain k where k:"k\<in>s" "content k = 0" by auto
  2992   from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
  2993   from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
  2994   hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
  2995   have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
  2996     apply safe apply(rule closed_interval) using assm(1) by auto
  2997   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
  2998   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
  2999     from k(2)[unfolded k content_eq_0] guess i .. 
  3000     hence i:"c$i = d$i" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by smt
  3001     hence xi:"x$i = d$i" using as unfolding k mem_interval by smt
  3002     def y \<equiv> "(\<chi> j. if j = i then if c$i \<le> (a$i + b$i) / 2 then c$i + min e (b$i - c$i) / 2 else c$i - min e (c$i - a$i) / 2 else x$j)"
  3003     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
  3004     proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
  3005       hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
  3006       hence xyi:"y$i \<noteq> x$i" unfolding y_def unfolding i xi Cart_lambda_beta if_P[OF refl]
  3007         apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2) using assms(2)[unfolded content_eq_0] by smt+ 
  3008       thus "y \<noteq> x" unfolding Cart_eq by auto
  3009       have *:"UNIV = insert i (UNIV - {i})" by auto
  3010       have "norm (y - x) < e + setsum (\<lambda>i. 0) (UNIV::'n set)" apply(rule le_less_trans[OF norm_le_l1])
  3011         apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
  3012       proof- show "\<bar>(y - x) $ i\<bar> < e" unfolding y_def Cart_lambda_beta vector_minus_component if_P[OF refl]
  3013           apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
  3014         show "(\<Sum>i\<in>UNIV - {i}. \<bar>(y - x) $ i\<bar>) \<le> (\<Sum>i\<in>UNIV. 0)" unfolding y_def by auto 
  3015       qed auto thus "dist y x < e" unfolding vector_dist_norm by auto
  3016       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
  3017       moreover have "y \<in> \<Union>s" unfolding s mem_interval
  3018       proof note simps = y_def Cart_lambda_beta if_not_P
  3019         fix j::'n show "a $ j \<le> y $ j \<and> y $ j \<le> b $ j" 
  3020         proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
  3021           thus ?thesis unfolding simps if_not_P[OF False] unfolding mem_interval by auto
  3022         next case True note T = this show ?thesis
  3023           proof(cases "c $ i \<le> (a $ i + b $ i) / 2")
  3024             case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
  3025               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
  3026           next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
  3027               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
  3028           qed qed qed
  3029       ultimately show "y \<in> \<Union>(s - {k})" by auto
  3030     qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
  3031   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
  3032     apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
  3033   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
  3034 
  3035 subsection {* Integrabibility on subintervals. *}
  3036 
  3037 lemma operative_integrable: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  3038   "operative op \<and> (\<lambda>i. f integrable_on i)"
  3039   unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
  3040   unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption)+
  3041   unfolding integrable_on_def by(auto intro: has_integral_split)
  3042 
  3043 lemma integrable_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  3044   assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
  3045   apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
  3046   using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
  3047 
  3048 subsection {* Combining adjacent intervals in 1 dimension. *}
  3049 
  3050 lemma has_integral_combine: assumes "(a::real^1) \<le> c" "c \<le> b"
  3051   "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
  3052   shows "(f has_integral (i + j)) {a..b}"
  3053 proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
  3054   note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
  3055   hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
  3056     apply(subst(asm) if_P) using assms(3-) by auto
  3057   with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
  3058     unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
  3059 
  3060 lemma integral_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  3061   assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
  3062   shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
  3063   apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
  3064   apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
  3065 
  3066 lemma integrable_combine: fixes f::"real^1 \<Rightarrow> 'a::banach"
  3067   assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
  3068   shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
  3069 
  3070 subsection {* Reduce integrability to "local" integrability. *}
  3071 
  3072 lemma integrable_on_little_subintervals: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3073   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}"
  3074   shows "f integrable_on {a..b}"
  3075 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})"
  3076     using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
  3077   guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
  3078   note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
  3079   show ?thesis unfolding * apply safe unfolding snd_conv
  3080   proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
  3081     thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
  3082 
  3083 subsection {* Second FCT or existence of antiderivative. *}
  3084 
  3085 lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
  3086   unfolding integrable_on_def by(rule,rule has_integral_const)
  3087 
  3088 lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
  3089   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  3090   shows "((\<lambda>u. integral {vec a..vec u} (f o dest_vec1)) has_vector_derivative f(x)) (at x within {a..b})"
  3091   unfolding has_vector_derivative_def has_derivative_within_alt
  3092 apply safe apply(rule scaleR.bounded_linear_left)
  3093 proof- fix e::real assume e:"e>0"
  3094   note compact_uniformly_continuous[OF assms(1) compact_real_interval,unfolded uniformly_continuous_on_def]
  3095   from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
  3096   let ?I = "\<lambda>a b. integral {vec1 a..vec1 b} (f \<circ> dest_vec1)"
  3097   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)"
  3098   proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
  3099       case False have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 y}" apply(rule integrable_subinterval,rule integrable_continuous)
  3100         apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  3101       hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
  3102         using False unfolding not_less using assms(2) goal1 by auto
  3103       have **:"norm (y - x) = content {vec1 x..vec1 y}" apply(subst content_1) using False unfolding not_less by auto
  3104       show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  3105         defer apply(rule has_integral_sub) apply(rule integrable_integral)
  3106         apply(rule integrable_subinterval,rule integrable_continuous) apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  3107       proof- show "{vec1 x..vec1 y} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  3108         have *:"y - x = norm(y - x)" using False by auto
  3109         show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {vec1 x..vec1 y}" apply(subst *) unfolding ** by auto
  3110         show "\<forall>xa\<in>{vec1 x..vec1 y}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  3111           apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  3112       qed(insert e,auto)
  3113     next case True have "f \<circ> dest_vec1 integrable_on {vec1 a..vec1 x}" apply(rule integrable_subinterval,rule integrable_continuous)
  3114         apply(rule continuous_on_o_dest_vec1 assms)+  unfolding not_less using assms(2) goal1 by auto
  3115       hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
  3116         using True using assms(2) goal1 by auto
  3117       have **:"norm (y - x) = content {vec1 y..vec1 x}" apply(subst content_1) using True unfolding not_less by auto
  3118       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 
  3119       show ?thesis apply(subst ***) unfolding norm_minus_cancel **
  3120         apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x) o dest_vec1"]) unfolding * unfolding o_def
  3121         defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
  3122         apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous)
  3123         apply(rule continuous_on_o_dest_vec1[unfolded o_def] assms)+
  3124       proof- show "{vec1 y..vec1 x} \<subseteq> {vec1 a..vec1 b}" using goal1 assms(2) by auto
  3125         have *:"x - y = norm(y - x)" using True by auto
  3126         show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {vec1 y..vec1 x}" apply(subst *) unfolding ** by auto
  3127         show "\<forall>xa\<in>{vec1 y..vec1 x}. norm (f (dest_vec1 xa) - f x) \<le> e" apply safe apply(rule less_imp_le)
  3128           apply(rule d(2)[unfolded vector_dist_norm]) using assms(2) using goal1 by auto
  3129       qed(insert e,auto) qed qed qed
  3130 
  3131 lemma integral_has_vector_derivative': fixes f::"real^1 \<Rightarrow> 'a::banach"
  3132   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  3133   shows "((\<lambda>u. (integral {a..vec u} f)) has_vector_derivative f x) (at (x$1) within {a$1..b$1})"
  3134   using integral_has_vector_derivative[OF continuous_on_o_vec1[OF assms(1)], of "x$1"]
  3135   unfolding o_def vec1_dest_vec1 using assms(2) by auto
  3136 
  3137 lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
  3138   obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
  3139   apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
  3140 
  3141 subsection {* Combined fundamental theorem of calculus. *}
  3142 
  3143 lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
  3144   obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> ((f o dest_vec1) has_integral (g v - g u)) {vec u..vec v}"
  3145 proof- from antiderivative_continuous[OF assms] guess g . note g=this
  3146   show ?thesis apply(rule that[of g])
  3147   proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
  3148       apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
  3149     thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g o dest_vec1" "f o dest_vec1"]
  3150       unfolding o_def vec1_dest_vec1 by auto qed qed
  3151 
  3152 subsection {* General "twiddling" for interval-to-interval function image. *}
  3153 
  3154 lemma has_integral_twiddle:
  3155   assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
  3156   "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
  3157   "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
  3158   "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
  3159   "(f has_integral i) {a..b}"
  3160   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
  3161 proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
  3162     show ?thesis apply cases defer apply(rule *,assumption)
  3163     proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
  3164   assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
  3165   have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
  3166     using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
  3167     using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
  3168   show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
  3169   proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
  3170     from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  3171     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)
  3172     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)"
  3173     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
  3174       fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
  3175       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 
  3176       proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
  3177         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
  3178         fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
  3179         show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
  3180         { 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)"]
  3181             using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
  3182         fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
  3183         hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
  3184         have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
  3185         proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
  3186           hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
  3187             unfolding image_Int[OF inj(1)] by auto thus False using as by blast
  3188         qed thus "g x = g x'" by auto
  3189         { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
  3190         { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
  3191       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
  3192         then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
  3193         thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
  3194           apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
  3195           using X(2) assms(3)[rule_format,of x] by auto
  3196       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
  3197        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
  3198         unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
  3199         apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
  3200       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]
  3201         unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
  3202       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
  3203         using assms(1) by(auto simp add:field_simps) qed qed qed
  3204 
  3205 subsection {* Special case of a basic affine transformation. *}
  3206 
  3207 lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::real^'n) + c) ` {a..b} = {u..v}"
  3208   unfolding image_affinity_interval by auto
  3209 
  3210 lemmas Cart_simps = Cart_nth.add Cart_nth.minus Cart_nth.zero Cart_nth.diff Cart_nth.scaleR real_scaleR_def Cart_lambda_beta
  3211    Cart_eq vector_le_def vector_less_def
  3212 
  3213 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
  3214   apply(rule setprod_cong) using assms by auto
  3215 
  3216 lemma content_image_affinity_interval: 
  3217  "content((\<lambda>x::real^'n. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ CARD('n) * content {a..b}" (is "?l = ?r")
  3218 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  3219       unfolding not_not using content_empty by auto }
  3220   assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
  3221     case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
  3222       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  3223       defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  3224       apply(rule setprod_cong2) using True as unfolding interval_ne_empty Cart_simps not_le  
  3225       by(auto simp add:field_simps intro:mult_left_mono)
  3226   next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
  3227       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') 
  3228       defer apply(subst setprod_constant[THEN sym]) apply(rule finite_UNIV) unfolding setprod_timesf[THEN sym]
  3229       apply(rule setprod_cong2) using False as unfolding interval_ne_empty Cart_simps not_le 
  3230       by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
  3231 
  3232 lemma has_integral_affinity: assumes "(f has_integral i) {a..b::real^'n}" "m \<noteq> 0"
  3233   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ CARD('n::finite))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
  3234   apply(rule has_integral_twiddle,safe) unfolding Cart_eq Cart_simps apply(rule zero_less_power)
  3235   defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
  3236   apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
  3237 
  3238 lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
  3239   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})"
  3240   using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
  3241 
  3242 subsection {* Special case of stretching coordinate axes separately. *}
  3243 
  3244 lemma image_stretch_interval:
  3245   "(\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} =
  3246   (if {a..b} = {} then {} else {(\<chi> k. min (m(k) * a$k) (m(k) * b$k)) ..  (\<chi> k. max (m(k) * a$k) (m(k) * b$k))})" (is "?l = ?r")
  3247 proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
  3248 next have *:"\<And>P Q. (\<forall>i. P i) \<and> (\<forall>i. Q i) \<longleftrightarrow> (\<forall>i. P i \<and> Q i)" by auto
  3249   case False note ab = this[unfolded interval_ne_empty]
  3250   show ?thesis apply-apply(rule set_ext)
  3251   proof- fix x::"real^'n" have **:"\<And>P Q. (\<forall>i. P i = Q i) \<Longrightarrow> (\<forall>i. P i) = (\<forall>i. Q i)" by auto
  3252     show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
  3253       unfolding image_iff mem_interval Bex_def Cart_simps Cart_eq *
  3254       unfolding lambda_skolem[THEN sym,of "\<lambda> i xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa"]
  3255     proof(rule **,rule) fix i::'n show "(\<exists>xa. (a $ i \<le> xa \<and> xa \<le> b $ i) \<and> x $ i = m i * xa) =
  3256         (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))"
  3257       proof(cases "m i = 0") case True thus ?thesis using ab by auto
  3258       next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
  3259         proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $ i) (m i * b $ i) = m i * a $ i"
  3260             "max (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab unfolding min_def max_def by auto
  3261           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  3262             using as by(auto simp add:field_simps)
  3263         next assume as:"0 > m i" hence *:"max (m i * a $ i) (m i * b $ i) = m i * a $ i"
  3264             "min (m i * a $ i) (m i * b $ i) = m i * b $ i" using ab as unfolding min_def max_def 
  3265             by(auto simp add:field_simps mult_le_cancel_left_neg intro:real_le_antisym)
  3266           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$i" in exI)
  3267             using as by(auto simp add:field_simps) qed qed qed qed qed 
  3268 
  3269 lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi> k. m k * x$k) ` {a..b::real^'n} = {u..v}"
  3270   unfolding image_stretch_interval by auto 
  3271 
  3272 lemma content_image_stretch_interval:
  3273   "content((\<lambda>x::real^'n. \<chi> k. m k * x$k) ` {a..b}) = abs(setprod m UNIV) * content({a..b})"
  3274 proof(cases "{a..b} = {}") case True thus ?thesis
  3275     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
  3276 next case False hence "(\<lambda>x. \<chi> k. m k * x $ k) ` {a..b} \<noteq> {}" by auto
  3277   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
  3278     unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding Cart_lambda_beta
  3279   proof- fix i::'n have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
  3280     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)"
  3281       apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] 
  3282       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
  3283 
  3284 lemma has_integral_stretch: assumes "(f has_integral i) {a..b}" "\<forall>k. ~(m k = 0)"
  3285   shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral
  3286              ((1/(abs(setprod m UNIV))) *\<^sub>R i)) ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  3287   apply(rule has_integral_twiddle) unfolding zero_less_abs_iff content_image_stretch_interval
  3288   unfolding image_stretch_interval empty_as_interval Cart_eq using assms
  3289 proof- show "\<forall>x. continuous (at x) (\<lambda>x. \<chi> k. m k * x $ k)"
  3290    apply(rule,rule linear_continuous_at) unfolding linear_linear
  3291    unfolding linear_def Cart_simps Cart_eq by(auto simp add:field_simps) qed auto
  3292 
  3293 lemma integrable_stretch: 
  3294   assumes "f integrable_on {a..b}" "\<forall>k. ~(m k = 0)"
  3295   shows "(\<lambda>x. f(\<chi> k. m k * x$k)) integrable_on ((\<lambda>x. \<chi> k. 1/(m k) * x$k) ` {a..b})"
  3296   using assms unfolding integrable_on_def apply-apply(erule exE) apply(drule has_integral_stretch) by auto
  3297 
  3298 subsection {* even more special cases. *}
  3299 
  3300 lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::real^'n}"
  3301   apply(rule set_ext,rule) defer unfolding image_iff
  3302   apply(rule_tac x="-x" in bexI) by(auto simp add:vector_le_def minus_le_iff le_minus_iff)
  3303 
  3304 lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
  3305   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
  3306   using has_integral_affinity[OF assms, of "-1" 0] by auto
  3307 
  3308 lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
  3309   apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
  3310 
  3311 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
  3312   unfolding integrable_on_def by auto
  3313 
  3314 lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
  3315   unfolding integral_def by auto
  3316 
  3317 subsection {* Stronger form of FCT; quite a tedious proof. *}
  3318 
  3319 (** move this **)
  3320 declare norm_triangle_ineq4[intro] 
  3321 
  3322 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)
  3323 
  3324 lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  3325   assumes "a \<le> b" "p tagged_division_of {vec1 a..vec1 b}"
  3326   shows "setsum (\<lambda>(x,k). f (dest_vec1 (interval_upperbound k)) - f(dest_vec1 (interval_lowerbound k))) p = f b - f a"
  3327   using additive_tagged_division_1[OF _ assms(2), of "f o dest_vec1"]
  3328   unfolding o_def vec1_dest_vec1 using assms(1) by auto
  3329 
  3330 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"
  3331   unfolding split_def by(rule refl)
  3332 
  3333 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
  3334   apply(subst(asm)(2) norm_minus_cancel[THEN sym])
  3335   apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
  3336 
  3337 lemma fundamental_theorem_of_calculus_interior:
  3338   assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
  3339   shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}"
  3340 proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
  3341     show ?thesis proof(cases,rule *,assumption)
  3342       assume "\<not> a < b" hence "a = b" using assms(1) by auto
  3343       hence *:"{vec a .. vec b} = {vec b}" "f b - f a = 0" apply(auto simp add: Cart_simps) by smt
  3344       show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0_1 using * `a=b` by auto
  3345     qed } assume ab:"a < b"
  3346   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {vec1 a..vec1 b} \<and> d fine p \<longrightarrow>
  3347                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f' \<circ> dest_vec1) x) - (f b - f a)) \<le> e * content {vec1 a..vec1 b})"
  3348   { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
  3349   fix e::real assume e:"e>0"
  3350   note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
  3351   note conjunctD2[OF this] note bounded=this(1) and this(2)
  3352   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)"
  3353     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]
  3354   from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
  3355   have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_real_interval assms by auto
  3356   from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
  3357 
  3358   have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
  3359     \<longrightarrow> norm(content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
  3360   proof- have "a\<in>{a..b}" using ab by auto
  3361     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  3362     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  3363     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  3364     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
  3365     proof(cases "f' a = 0") case True
  3366       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  3367     next case False thus ?thesis 
  3368         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
  3369         using ab e by(auto simp add:field_simps)
  3370     qed then guess l .. note l = conjunctD2[OF this]
  3371     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  3372     proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
  3373       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  3374       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)
  3375       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  3376       proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
  3377         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  3378       next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
  3379           apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  3380       qed finally show "norm (content {vec1 a..vec1 c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto
  3381     qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
  3382 
  3383   have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow> norm(content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
  3384   proof- have "b\<in>{a..b}" using ab by auto
  3385     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  3386     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  3387     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  3388     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
  3389     proof(cases "f' b = 0") case True
  3390       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  3391     next case False thus ?thesis 
  3392         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
  3393         using ab e by(auto simp add:field_simps)
  3394     qed then guess l .. note l = conjunctD2[OF this]
  3395     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  3396     proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
  3397       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  3398       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)
  3399       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  3400       proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
  3401         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  3402       next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
  3403           apply(rule k(2)[unfolded vector_dist_norm]) using as' e ab by(auto simp add:field_simps)
  3404       qed finally show "norm (content {vec1 c..vec1 b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4" unfolding content_1'[OF as(1)] by auto
  3405     qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
  3406 
  3407   let ?d = "(\<lambda>x. ball x (if x=vec1 a then da else if x=vec b then db else d (dest_vec1 x)))"
  3408   show "?P e" apply(rule_tac x="?d" in exI)
  3409   proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
  3410   next case goal2 note as=this let ?A = "{t. fst t \<in> {vec1 a, vec1 b}}" note p = tagged_division_ofD[OF goal2(1)]
  3411     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
  3412     note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
  3413     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
  3414     show ?case unfolding content_1'[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
  3415       unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
  3416     proof(rule norm_triangle_le,rule **) 
  3417       case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
  3418       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"
  3419           "e * (dest_vec1 (interval_upperbound k) - dest_vec1 (interval_lowerbound k)) / 2
  3420           < norm (content k *\<^sub>R f' (dest_vec1 x) - (f (dest_vec1 (interval_upperbound k)) - f (dest_vec1 (interval_lowerbound k))))"
  3421         from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
  3422         hence "\<forall>i. u$i \<le> v$i" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto note this(1) this(1)[unfolded forall_1]
  3423         note result = as(2)[unfolded k interval_bounds[OF this(1)] content_1[OF this(2)]]
  3424 
  3425         assume as':"x \<noteq> vec1 a" "x \<noteq> vec1 b" hence "x$1 \<in> {a<..<b}" using p(2-3)[OF as(1)] by(auto simp add:Cart_simps) note  * = d(2)[OF this] 
  3426         have "norm ((v$1 - u$1) *\<^sub>R f' (x$1) - (f (v$1) - f (u$1))) =
  3427           norm ((f (u$1) - f (x$1) - (u$1 - x$1) *\<^sub>R f' (x$1)) - (f (v$1) - f (x$1) - (v$1 - x$1) *\<^sub>R f' (x$1)))" 
  3428           apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
  3429         also have "... \<le> e / 2 * norm (u$1 - x$1) + e / 2 * norm (v$1 - x$1)" apply(rule norm_triangle_le_sub)
  3430           apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
  3431           apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp add:dist_real)
  3432         also have "... \<le> e / 2 * norm (v$1 - u$1)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
  3433         finally have "e * (dest_vec1 v - dest_vec1 u) / 2 < e * (dest_vec1 v - dest_vec1 u) / 2"
  3434           apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
  3435 
  3436     next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
  3437       case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
  3438         defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
  3439         apply(subst additive_tagged_division_1[OF _ as(1)]) unfolding vec1_dest_vec1 apply(rule assms)
  3440       proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}}" note xk=IntD1[OF this]
  3441         from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  3442         with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
  3443         thus "0 \<le> e * ((interval_upperbound k)$1 - (interval_lowerbound k)$1)"
  3444           unfolding uv using e by(auto simp add:field_simps)
  3445       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
  3446         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R (f' \<circ> dest_vec1) x -
  3447           (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) \<le> e * (b - a) / 2" 
  3448           apply(rule *[where t="p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0}"])
  3449           apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
  3450         proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {vec1 a, vec1 b}} - p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content (snd t) \<noteq> 0}"
  3451           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
  3452           have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk unfolding uv content_eq_0_1 interval_eq_empty by auto
  3453           thus "content k *\<^sub>R (f' (x$1)) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1)) = 0" using xk unfolding uv by auto
  3454         next have *:"p \<inter> {t. fst t \<in> {vec1 a, vec1 b} \<and> content(snd t) \<noteq> 0} = 
  3455             {t. t\<in>p \<and> fst t = vec1 a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = vec1 b \<and> content(snd t) \<noteq> 0}" by blast
  3456           have **:"\<And>s f. \<And>e::real. (\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y) \<Longrightarrow> (\<forall>x. x \<in> s \<longrightarrow> norm(f x) \<le> e) \<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
  3457           proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
  3458             thus ?case using `x\<in>s` goal2(2) by auto
  3459           qed auto
  3460           case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4 apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"]) 
  3461             apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
  3462           proof- let ?B = "\<lambda>x. {t \<in> p. fst t = vec1 x \<and> content (snd t) \<noteq> 0}"
  3463             have pa:"\<And>k. (vec1 a, k) \<in> p \<Longrightarrow> \<exists>v. k = {vec1 a .. v} \<and> vec1 a \<le> v" 
  3464             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  3465               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  3466               have u:"u = vec1 a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  3467                 have "u \<ge> vec1 a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>vec1 a" ultimately
  3468                 have "u > vec1 a" unfolding Cart_simps by auto
  3469                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  3470               qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
  3471             qed
  3472             have pb:"\<And>k. (vec1 b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. vec1 b} \<and> vec1 b \<ge> v" 
  3473             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  3474               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  3475               have u:"v = vec1 b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  3476                 have "v \<le> vec1 b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq>vec1 b" ultimately
  3477                 have "v < vec1 b" unfolding Cart_simps by auto
  3478                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:Cart_simps)
  3479               qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
  3480             qed
  3481 
  3482             show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  3483               unfolding mem_Collect_eq fst_conv snd_conv apply safe
  3484             proof- fix x k k' assume k:"(vec1 a, k) \<in> p" "(vec1 a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  3485               guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
  3486               guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (min (v$1) (v'$1))"
  3487               have "{vec1 a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
  3488               moreover have "vec1 ((a + ?v$1)/2) \<in> {vec1 a <..< ?v}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
  3489               ultimately have "vec1 ((a + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  3490               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  3491               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  3492             qed 
  3493             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  3494               unfolding mem_Collect_eq fst_conv snd_conv apply safe
  3495             proof- fix x k k' assume k:"(vec1 b, k) \<in> p" "(vec1 b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  3496               guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
  3497               guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "vec1 (max (v$1) (v'$1))"
  3498               have "{?v <..< vec1 b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:Cart_simps) note subset_interior[OF this,unfolded interior_inter]
  3499               moreover have "vec1 ((b + ?v$1)/2) \<in> {?v <..< vec1 b}" using k(3-) unfolding v v' content_eq_0_1 not_le by(auto simp add:Cart_simps)
  3500               ultimately have "vec1 ((b + ?v$1)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  3501               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  3502               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  3503             qed
  3504 
  3505             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
  3506             show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
  3507               \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
  3508             proof- case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
  3509               have "vec1 ?a\<in>{vec1 ?a..v}" using v(2) by auto hence "dest_vec1 v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
  3510               moreover have "{?a..dest_vec1 v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
  3511                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE)
  3512                 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
  3513               show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
  3514                 apply(rule da(2)[of "v$1",unfolded vec1_dest_vec1])
  3515                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
  3516             qed
  3517             show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x$1) - (f ((interval_upperbound k)$1) - f ((interval_lowerbound k)$1))) x)
  3518               \<le> e * (b - a) / 4" apply safe unfolding fst_conv snd_conv apply safe unfolding vec1_dest_vec1
  3519             proof- case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
  3520               have "vec1 ?b\<in>{v..vec1 ?b}" using v(2) by auto hence "dest_vec1 v \<ge> ?a" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
  3521               moreover have "{dest_vec1 v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
  3522                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x="vec1 x" in ballE) using ab
  3523                 by(auto simp add:Cart_simps subset_eq dist_real v dist_real_def) ultimately
  3524               show ?case unfolding v unfolding interval_bounds[OF v(2)[unfolded v vector_le_def]] vec1_dest_vec1 apply-
  3525                 apply(rule db(2)[of "v$1",unfolded vec1_dest_vec1])
  3526                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0_1 by auto
  3527             qed
  3528           qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
  3529 
  3530 subsection {* Stronger form with finite number of exceptional points. *}
  3531 
  3532 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
  3533   assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
  3534   "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
  3535   shows "((f' o dest_vec1) has_integral (f b - f a)) {vec a..vec b}" using assms apply- 
  3536 proof(induct "card s" arbitrary:s a b)
  3537   case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
  3538 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
  3539     apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
  3540   show ?case proof(cases "c\<in>{a<..<b}")
  3541     case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
  3542       apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
  3543   next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
  3544     case True hence "vec1 a \<le> vec1 c" "vec1 c \<le> vec1 b" by auto
  3545     thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
  3546       apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
  3547     proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
  3548         apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
  3549       let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
  3550       show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
  3551     qed auto qed qed
  3552 
  3553 lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
  3554   assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
  3555   "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
  3556   shows "((f' o dest_vec1) has_integral (f(b) - f(a))) {vec1 a..vec1 b}"
  3557   apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
  3558   using assms(4) by auto
  3559 
  3560 lemma indefinite_integral_continuous_left: fixes f::"real^1 \<Rightarrow> 'a::banach"
  3561   assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e"
  3562   obtains d where "0 < d" "\<forall>t. c$1 - d < t$1 \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e"
  3563 proof- have "\<exists>w>0. \<forall>t. c$1 - w < t$1 \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3"
  3564   proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)"
  3565       apply-apply(rule divide_pos_pos) using `e>0` by auto
  3566     thus ?thesis apply-apply(rule,rule,assumption,safe)
  3567     proof- fix t assume as:"t < c" and "c$1 - e / 3 / norm (f c) < t$(1::1)"
  3568       hence "c$1 - t$1 < e / 3 / norm (f c)" by auto
  3569       hence "norm (c - t) < e / 3 / norm (f c)" using as unfolding norm_vector_1 vector_less_def by auto
  3570       thus "norm (f c) * norm (c - t) < e / 3" using False apply-
  3571         apply(subst real_mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
  3572     qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
  3573   qed then guess w .. note w = conjunctD2[OF this,rule_format]
  3574   
  3575   have *:"e / 3 > 0" using assms by auto
  3576   have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
  3577   from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
  3578   note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x"
  3579   have "gauge d" unfolding d_def using w(1) d1 by auto
  3580   note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this]
  3581   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
  3582 
  3583   let ?d = "min k (c$1 - a$1)/2" show ?thesis apply(rule that[of ?d])
  3584   proof safe show "?d > 0" using k(1) using assms(2) unfolding vector_less_def by auto
  3585     fix t assume as:"c$1 - ?d < t$1" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
  3586     { presume *:"t < c \<Longrightarrow> ?thesis"
  3587       show ?thesis apply(cases "t = c") defer apply(rule *)
  3588         unfolding vector_less_def apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
  3589 
  3590     have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
  3591     from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
  3592     note d2 = conjunctD2[OF this,rule_format]
  3593     def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x"
  3594     have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto
  3595     from fine_division_exists[OF this, of a t] guess p . note p=this
  3596     note p'=tagged_division_ofD[OF this(1)]
  3597     have pt:"\<forall>(x,k)\<in>p. x$1 \<le> t$1" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
  3598     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
  3599     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
  3600     
  3601     have *:"{a..c} \<inter> {x. x$1 \<le> t$1} = {a..t}" "{a..c} \<inter> {x. x$1 \<ge> t$1} = {t..c}"
  3602       using assms(2-3) as by(auto simp add:field_simps)
  3603     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
  3604       apply(rule tagged_division_union_interval[of _ _ _ 1 "t$1"]) unfolding * apply(rule p)
  3605       apply(rule tagged_division_of_self) unfolding fine_def
  3606     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
  3607         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
  3608     next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real
  3609         using as(1) by(auto simp add:field_simps) 
  3610       thus "x \<in> d1 c" using k(2) unfolding d_def by auto
  3611     qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
  3612 
  3613     have *:"integral{a..c} f - integral {a..t} f = -(((c$1 - t$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
  3614         integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c$1 - t$1) *\<^sub>R f c" 
  3615       "e = (e/3 + e/3) + e/3" by auto
  3616     have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c$1 - t$1) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
  3617     proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
  3618       have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
  3619         have "c \<in> {a..t}" by auto thus False using `t<c` unfolding vector_less_def by auto
  3620       qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
  3621         unfolding split_conv defer apply(subst content_1) using as(2) by auto qed 
  3622 
  3623     have ***:"c$1 - w < t$1 \<and> t < c"
  3624     proof- have "c$1 - k < t$1" using `k>0` as(1) by(auto simp add:field_simps)
  3625       moreover have "k \<le> w" apply(rule ccontr) using k(2) 
  3626         unfolding subset_eq apply(erule_tac x="c + vec ((k + w)/2)" in ballE)
  3627         unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real)
  3628       ultimately show  ?thesis using `t<c` by(auto simp add:field_simps) qed
  3629 
  3630     show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
  3631       unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
  3632       using w(2)[OF ***] unfolding norm_scaleR norm_real by(auto simp add:field_simps) qed qed 
  3633 
  3634 lemma indefinite_integral_continuous_right: fixes f::"real^1 \<Rightarrow> 'a::banach"
  3635   assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
  3636   obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t$1 < c$1 + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e"
  3637 proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a"
  3638     using assms unfolding Cart_simps by auto
  3639   from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b$1 - c$1)"
  3640   show ?thesis apply(rule that[of "?d"])
  3641   proof safe show "0 < ?d" using d(1) assms(3) unfolding Cart_simps by auto
  3642     fix t::"_^1" assume as:"c \<le> t" "t$1 < c$1 + ?d"
  3643     have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
  3644       "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
  3645       apply(rule_tac[!] integral_combine) using assms as unfolding Cart_simps by auto
  3646     have "(- c)$1 - d < (- t)$1 \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
  3647     thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding * 
  3648       unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
  3649 
  3650 declare dest_vec1_eq[simp del] not_less[simp] not_le[simp]
  3651 
  3652 lemma indefinite_integral_continuous: fixes f::"real^1 \<Rightarrow> 'a::banach"
  3653   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
  3654 proof(unfold continuous_on_def, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
  3655   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"
  3656   { presume *:"a<b \<Longrightarrow> ?thesis"
  3657     show ?thesis apply(cases,rule *,assumption)
  3658     proof- case goal1 hence "{a..b} = {x}" using as(1) unfolding Cart_simps  
  3659         by(auto simp only:field_simps not_less Cart_eq forall_1 mem_interval) 
  3660       thus ?case using `e>0` by auto
  3661     qed } assume "a<b"
  3662   have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add: Cart_simps)
  3663   thus ?thesis apply-apply(erule disjE)+
  3664   proof- assume "x=a" have "a \<le> a" by auto
  3665     from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
  3666     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
  3667       unfolding `x=a` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
  3668   next   assume "x=b" have "b \<le> b" by auto
  3669     from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
  3670     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
  3671       unfolding `x=b` vector_dist_norm apply(rule d(2)[rule_format]) unfolding norm_real by auto
  3672   next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add:Cart_simps)
  3673     from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
  3674     from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
  3675     show ?thesis apply(rule_tac x="min d1 d2" in exI)
  3676     proof safe show "0 < min d1 d2" using d1 d2 by auto
  3677       fix y assume "y\<in>{a..b}" "dist y x < min d1 d2"
  3678       thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute)
  3679         apply(cases "y < x") unfolding vector_dist_norm apply(rule d1(2)[rule_format]) defer
  3680         apply(rule d2(2)[rule_format]) unfolding Cart_simps not_less norm_real by(auto simp add:field_simps)
  3681     qed qed qed 
  3682 
  3683 subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
  3684 
  3685 lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach"
  3686   assumes "finite k" "continuous_on {a..b} f" "f a = y"
  3687   "\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
  3688   shows "f x = y"
  3689 proof- have ab:"a\<le>b" using assms by auto
  3690   have *:"(\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 = (\<lambda>x. 0)" unfolding o_def by auto have **:"a \<le> x" using assms by auto
  3691   have "((\<lambda>x. 0\<Colon>'a) \<circ> dest_vec1 has_integral f x - f a) {vec1 a..vec1 x}"
  3692     apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) ** ])
  3693     apply(rule continuous_on_subset[OF assms(2)]) defer
  3694     apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
  3695     apply assumption apply(rule open_interval_real) apply(rule has_derivative_within_subset[where s="{a..b}"])
  3696     using assms(4) assms(5) by auto note this[unfolded *]
  3697   note has_integral_unique[OF has_integral_0 this]
  3698   thus ?thesis unfolding assms by auto qed
  3699 
  3700 subsection {* Generalize a bit to any convex set. *}
  3701 
  3702 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
  3703   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
  3704   scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one
  3705 
  3706 lemma has_derivative_zero_unique_strong_convex: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3707   assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
  3708   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
  3709   shows "f x = y"
  3710 proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  3711       unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
  3712   note conv = assms(1)[unfolded convex_alt,rule_format]
  3713   have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
  3714     apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
  3715     apply safe apply(rule conv) using assms(4,7) by auto
  3716   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"
  3717   proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c" 
  3718       unfolding scaleR_simps by(auto simp add:algebra_simps)
  3719     thus ?case using `x\<noteq>c` by auto qed
  3720   have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2) 
  3721     apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
  3722     apply safe unfolding image_iff apply rule defer apply assumption
  3723     apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
  3724   have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
  3725     apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
  3726     unfolding o_def using assms(5) defer apply-apply(rule)
  3727   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
  3728     have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps]) 
  3729       using `x\<in>s` `c\<in>s` as by(auto simp add:scaleR_simps)
  3730     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})"
  3731       apply(rule diff_chain_within) apply(rule has_derivative_add)
  3732       unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
  3733       apply(rule has_derivative_vmul_within,rule has_derivative_id)+ 
  3734       apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
  3735       apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
  3736     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 .
  3737   qed auto thus ?thesis by auto qed
  3738 
  3739 subsection {* Also to any open connected set with finite set of exceptions. Could 
  3740  generalize to locally convex set with limpt-free set of exceptions. *}
  3741 
  3742 lemma has_derivative_zero_unique_strong_connected: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3743   assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
  3744   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
  3745   shows "f x = y"
  3746 proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
  3747     apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
  3748     apply(rule continuous_closed_in_preimage[OF assms(4) closed_sing])
  3749     apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
  3750   proof safe fix x assume "x\<in>s" 
  3751     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
  3752     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
  3753     proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
  3754       show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball])
  3755         apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+
  3756         apply(subst centre_in_ball,rule e,rule) apply safe
  3757         apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format])
  3758         using y e by auto qed qed
  3759   thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed
  3760 
  3761 subsection {* Integrating characteristic function of an interval. *}
  3762 
  3763 lemma has_integral_restrict_open_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3764   assumes "(f has_integral i) {c..d}" "{c..d} \<subseteq> {a..b}"
  3765   shows "((\<lambda>x. if x \<in> {c<..<d} then f x else 0) has_integral i) {a..b}"
  3766 proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
  3767   { presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
  3768     show ?thesis apply(cases,rule *,assumption)
  3769     proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto 
  3770       show ?thesis using assms(1) unfolding * using goal1 by auto
  3771     qed } assume "{c..d}\<noteq>{}"
  3772   from partial_division_extend_1[OF assms(2) this] guess p . note p=this
  3773   note mon = monoidal_lifted[OF monoidal_monoid] 
  3774   note operat = operative_division[OF this operative_integral p(1), THEN sym]
  3775   let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
  3776   { presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
  3777       apply- apply(cases,subst(asm) if_P,assumption) by auto
  3778     thus ?thesis using integrable_integral unfolding g_def by auto }
  3779 
  3780   note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
  3781   note * = this[unfolded neutral_monoid]
  3782   have iterate:"iterate (lifted op +) (p - {{c..d}})
  3783       (\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0"
  3784   proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this]
  3785     from div(3) guess u v apply-by(erule exE)+ note uv=this
  3786     have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto
  3787     hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"])
  3788       unfolding g_def interior_closed_interval by auto thus ?case by auto
  3789   qed
  3790 
  3791   have *:"p = insert {c..d} (p - {{c..d}})" using p by auto
  3792   have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f])
  3793     unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto
  3794   moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)])
  3795     apply(rule has_integral_spike_interior[where f=g]) defer
  3796     apply(rule integrable_integral[OF **]) unfolding g_def by auto
  3797   ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+
  3798     unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
  3799 
  3800 lemma has_integral_restrict_closed_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3801   assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}" 
  3802   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
  3803 proof- note has_integral_restrict_open_subinterval[OF assms]
  3804   note * = has_integral_spike[OF negligible_frontier_interval _ this]
  3805   show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
  3806 
  3807 lemma has_integral_restrict_closed_subintervals_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" assumes "{c..d} \<subseteq> {a..b}" 
  3808   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")
  3809 proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
  3810   show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
  3811   proof assumption assume ?l hence "?g integrable_on {c..d}"
  3812       apply-apply(rule integrable_subinterval[OF _ assms]) by auto
  3813     hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto
  3814     hence "i = integral {c..d} f" apply-apply(rule has_integral_unique)
  3815       apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto
  3816     thus ?r using * by auto qed qed auto
  3817 
  3818 subsection {* Hence we can apply the limit process uniformly to all integrals. *}
  3819 
  3820 lemma has_integral': fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  3821  "(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
  3822   \<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)")
  3823 proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis"
  3824     show ?thesis apply(cases,rule *,assumption)
  3825       apply(subst has_integral_alt) by auto }
  3826   assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this
  3827   from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B ..
  3828   note B = conjunctD2[OF this,rule_format] show ?thesis apply safe
  3829   proof- fix e assume ?l "e>(0::real)"
  3830     show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI)
  3831     proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::real^'n}"
  3832       thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s
  3833         apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s])
  3834         apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
  3835         by(auto simp add:vector_dist_norm)
  3836     qed(insert B `e>0`, auto)
  3837   next assume as:"\<forall>e>0. ?r e" 
  3838     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
  3839     def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n"
  3840     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
  3841     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
  3842         by(auto simp add:field_simps) qed
  3843     have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball vector_dist_norm 
  3844     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  3845     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
  3846       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
  3847     then guess y .. note y=this
  3848 
  3849     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
  3850       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
  3851       def c \<equiv> "(\<chi> i. - max B C)::real^'n" and d \<equiv> "(\<chi> i. max B C)::real^'n"
  3852       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
  3853       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
  3854           by(auto simp add:field_simps) qed
  3855       have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball vector_dist_norm 
  3856       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  3857       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
  3858       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
  3859       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
  3860       thus False by auto qed
  3861     thus ?l using y unfolding s by auto qed qed 
  3862 
  3863 lemma has_integral_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows
  3864   "((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s"
  3865   unfolding has_integral'[unfolded has_integral] 
  3866 proof case goal1 thus ?case apply safe
  3867   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
  3868   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
  3869   apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
  3870   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
  3871   apply(subst(asm)(2) norm_vector_1) unfolding split_def
  3872   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
  3873     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
  3874   apply(subst(asm)(2) norm_vector_1) by auto
  3875 next case goal2 thus ?case apply safe
  3876   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
  3877   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
  3878   apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
  3879   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
  3880   apply(subst norm_vector_1) unfolding split_def
  3881   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
  3882     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
  3883   apply(subst norm_vector_1) by auto qed
  3884 
  3885 lemma integral_trans[simp]: assumes "(f::real^'n \<Rightarrow> real) integrable_on s"
  3886   shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)"
  3887   apply(rule integral_unique) using assms by auto
  3888 
  3889 lemma integrable_on_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows
  3890   "(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)"
  3891   unfolding integrable_on_def
  3892   apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans
  3893   apply safe defer apply(rule_tac x="vec1 y" in exI) by auto
  3894 
  3895 lemma has_integral_le: fixes f::"real^'n \<Rightarrow> real"
  3896   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x) \<le> (g x)"
  3897   shows "i \<le> j" using has_integral_component_le[of "vec1 o f" "vec1 i" s "vec1 o g" "vec1 j" 1]
  3898   unfolding o_def using assms by auto 
  3899 
  3900 lemma integral_le: fixes f::"real^'n \<Rightarrow> real"
  3901   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  3902   shows "integral s f \<le> integral s g"
  3903   using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] .
  3904 
  3905 lemma has_integral_nonneg: fixes f::"real^'n \<Rightarrow> real"
  3906   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" 
  3907   using has_integral_component_nonneg[of "vec1 o f" "vec1 i" s 1]
  3908   unfolding o_def using assms by auto
  3909 
  3910 lemma integral_nonneg: fixes f::"real^'n \<Rightarrow> real"
  3911   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" 
  3912   using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
  3913 
  3914 subsection {* Hence a general restriction property. *}
  3915 
  3916 lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows
  3917   "((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
  3918 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
  3919   show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed
  3920 
  3921 lemma has_integral_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  3922   "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
  3923 
  3924 lemma has_integral_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  3925   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
  3926   shows "(f has_integral i) t"
  3927 proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
  3928     apply(rule) using assms(1-2) by auto
  3929   thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
  3930   apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
  3931 
  3932 lemma integrable_on_superset: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  3933   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
  3934   shows "f integrable_on t"
  3935   using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
  3936 
  3937 lemma integral_restrict_univ[intro]: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  3938   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
  3939   apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
  3940 
  3941 lemma integrable_restrict_univ: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  3942  "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
  3943   unfolding integrable_on_def by auto
  3944 
  3945 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r")
  3946 proof assume ?r show ?l unfolding negligible_def
  3947   proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
  3948       unfolding indicator_def by auto qed qed auto
  3949 
  3950 lemma has_integral_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach" 
  3951   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
  3952   unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by auto
  3953 
  3954 lemma has_integral_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3955   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
  3956   shows "(f has_integral y) t"
  3957   using assms has_integral_spike_set_eq by auto
  3958 
  3959 lemma integrable_spike_set[dest]: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3960   assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
  3961   shows "f integrable_on t" using assms(2) unfolding integrable_on_def 
  3962   unfolding has_integral_spike_set_eq[OF assms(1)] .
  3963 
  3964 lemma integrable_spike_set_eq: fixes f::"real^'n \<Rightarrow> 'a::banach"
  3965   assumes "negligible((s - t) \<union> (t - s))"
  3966   shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)"
  3967   apply(rule,rule_tac[!] integrable_spike_set) using assms by auto
  3968 
  3969 (*lemma integral_spike_set:
  3970  "\<forall>f:real^M->real^N g s t.
  3971         negligible(s DIFF t \<union> t DIFF s)
  3972         \<longrightarrow> integral s f = integral t f"
  3973 qed  REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
  3974   AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  3975   ASM_MESON_TAC[]);;
  3976 
  3977 lemma has_integral_interior:
  3978  "\<forall>f:real^M->real^N y s.
  3979         negligible(frontier s)
  3980         \<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
  3981 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  3982   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
  3983     NEGLIGIBLE_SUBSET)) THEN
  3984   REWRITE_TAC[frontier] THEN
  3985   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
  3986   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
  3987   SET_TAC[]);;
  3988 
  3989 lemma has_integral_closure:
  3990  "\<forall>f:real^M->real^N y s.
  3991         negligible(frontier s)
  3992         \<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
  3993 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  3994   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
  3995     NEGLIGIBLE_SUBSET)) THEN
  3996   REWRITE_TAC[frontier] THEN
  3997   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
  3998   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
  3999   SET_TAC[]);;*)
  4000 
  4001 subsection {* More lemmas that are useful later. *}
  4002 
  4003 lemma has_integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  4004   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)$k"
  4005   shows "i$k \<le> j$k"
  4006 proof- note has_integral_restrict_univ[THEN sym, of f]
  4007   note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
  4008   show ?thesis apply(rule *) using assms(1,4) by auto qed
  4009 
  4010 lemma has_integral_subset_le: fixes f::"real^'n \<Rightarrow> real"
  4011   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
  4012   shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "vec1 o f" "vec1 i" "vec1 j" 1]
  4013   unfolding o_def using assms by auto
  4014 
  4015 lemma integral_subset_component_le: fixes f::"real^'n \<Rightarrow> real^'m"
  4016   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)$k"
  4017   shows "(integral s f)$k \<le> (integral t f)$k"
  4018   apply(rule has_integral_subset_component_le) using assms by auto
  4019 
  4020 lemma integral_subset_le: fixes f::"real^'n \<Rightarrow> real"
  4021   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)"
  4022   shows "(integral s f) \<le> (integral t f)"
  4023   apply(rule has_integral_subset_le) using assms by auto
  4024 
  4025 lemma has_integral_alt': fixes f::"real^'n \<Rightarrow> 'a::banach"
  4026   shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
  4027   (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e)" (is "?l = ?r")
  4028 proof assume ?r
  4029   show ?l apply- apply(subst has_integral')
  4030   proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
  4031     show ?case apply(rule,rule,rule B,safe)
  4032       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI)
  4033       apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto
  4034   qed next
  4035   assume ?l note as = this[unfolded has_integral'[of f],rule_format]
  4036   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
  4037   show ?r proof safe fix a b::"real^'n"
  4038     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
  4039     let ?a = "(\<chi> i. min (a$i) (-B))::real^'n" and ?b = "(\<chi> i. max (b$i) B)::real^'n"
  4040     show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
  4041     proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval vector_dist_norm
  4042       proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
  4043       from B(2)[OF this] guess z .. note conjunct1[OF this]
  4044       thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
  4045       show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
  4046 
  4047     fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
  4048     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
  4049                     norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
  4050     proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
  4051       from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed 
  4052 
  4053 
  4054 subsection {* Continuity of the integral (for a 1-dimensional interval). *}
  4055 
  4056 lemma integrable_alt: fixes f::"real^'n \<Rightarrow> 'a::banach" shows 
  4057   "f integrable_on s \<longleftrightarrow>
  4058           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
  4059           (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
  4060   \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) -
  4061           integral {c..d}  (\<lambda>x. if x \<in> s then f x else 0)) < e)" (is "?l = ?r")
  4062 proof assume ?l then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
  4063   note y=conjunctD2[OF this,rule_format] show ?r apply safe apply(rule y)
  4064   proof- case goal1 hence "e/2 > 0" by auto from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
  4065     show ?case apply(rule,rule,rule B)
  4066     proof safe case goal1 show ?case apply(rule norm_triangle_half_l)
  4067         using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
  4068         
  4069 next assume ?r note as = conjunctD2[OF this,rule_format]
  4070   have "Cauchy (\<lambda>n. integral ({(\<chi> i. - real n) .. (\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))"
  4071   proof(unfold Cauchy_def,safe) case goal1
  4072     from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
  4073     from real_arch_simple[of B] guess N .. note N = this
  4074     { fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {\<chi> i. - real n..\<chi> i. real n}" apply safe
  4075         unfolding mem_ball mem_interval vector_dist_norm
  4076       proof case goal1 thus ?case using component_le_norm[of x i]
  4077           using n N by(auto simp add:field_simps) qed }
  4078     thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding vector_dist_norm apply(rule B(2)) by auto
  4079   qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
  4080   note i = this[unfolded Lim_sequentially, rule_format]
  4081 
  4082   show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
  4083     apply safe apply(rule as(1)[unfolded integrable_on_def])
  4084   proof- case goal1 hence *:"e/2 > 0" by auto
  4085     from i[OF this] guess N .. note N =this[rule_format]
  4086     from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format] let ?B = "max (real N) B"
  4087     show ?case apply(rule_tac x="?B" in exI)
  4088     proof safe show "0 < ?B" using B(1) by auto
  4089       fix a b assume ab:"ball 0 ?B \<subseteq> {a..b::real^'n}"
  4090       from real_arch_simple[of ?B] guess n .. note n=this
  4091       show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
  4092         apply(rule norm_triangle_half_l) apply(rule B(2)) defer apply(subst norm_minus_commute)
  4093         apply(rule N[unfolded vector_dist_norm, of n])
  4094       proof safe show "N \<le> n" using n by auto
  4095         fix x::"real^'n" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
  4096         thus "x\<in>{a..b}" using ab by blast 
  4097         show "x\<in>{\<chi> i. - real n..\<chi> i. real n}" using x unfolding mem_interval mem_ball vector_dist_norm apply-
  4098         proof case goal1 thus ?case using component_le_norm[of x i]
  4099             using n by(auto simp add:field_simps) qed qed qed qed qed
  4100 
  4101 lemma integrable_altD: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4102   assumes "f integrable_on s"
  4103   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
  4104   "\<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}
  4105   \<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"
  4106   using assms[unfolded integrable_alt[of f]] by auto
  4107 
  4108 lemma integrable_on_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4109   assumes "f integrable_on s" "{a..b} \<subseteq> s" shows "f integrable_on {a..b}"
  4110   apply(rule integrable_eq) defer apply(rule integrable_altD(1)[OF assms(1)])
  4111   using assms(2) by auto
  4112 
  4113 subsection {* A straddling criterion for integrability. *}
  4114 
  4115 lemma integrable_straddle_interval: fixes f::"real^'n \<Rightarrow> real"
  4116   assumes "\<forall>e>0. \<exists>g  h i j. (g has_integral i) ({a..b}) \<and> (h has_integral j) ({a..b}) \<and>
  4117   norm(i - j) < e \<and> (\<forall>x\<in>{a..b}. (g x) \<le> (f x) \<and> (f x) \<le>(h x))"
  4118   shows "f integrable_on {a..b}"
  4119 proof(subst integrable_cauchy,safe)
  4120   case goal1 hence e:"e/3 > 0" by auto note assms[rule_format,OF this]
  4121   then guess g h i j apply- by(erule exE conjE)+ note obt = this
  4122   from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
  4123   from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
  4124   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
  4125   proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
  4126       abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow>  abs(g1 - i) < e / 3 \<Longrightarrow> 
  4127       abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith
  4128     case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)]
  4129 
  4130     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"
  4131       "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)" 
  4132       "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
  4133       "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)" 
  4134       unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg)
  4135       apply safe unfolding real_scaleR_def mult.diff_right[THEN sym]
  4136       apply(rule_tac[!] mult_nonneg_nonneg)
  4137     proof- fix a b assume ab:"(a,b) \<in> p1"
  4138       show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
  4139       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
  4140     next fix a b assume ab:"(a,b) \<in> p2"
  4141       show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
  4142       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 
  4143 
  4144     thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
  4145       unfolding real_norm_def[THEN sym] apply(rule obt(3))
  4146       apply(rule d1(2)[OF conjI[OF goal1(4,5)]])
  4147       apply(rule d1(2)[OF conjI[OF goal1(1,2)]])
  4148       apply(rule d2(2)[OF conjI[OF goal1(4,6)]])
  4149       apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed 
  4150      
  4151 lemma integrable_straddle: fixes f::"real^'n \<Rightarrow> real"
  4152   assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
  4153   norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))"
  4154   shows "f integrable_on s"
  4155 proof- have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
  4156   proof(rule integrable_straddle_interval,safe) case goal1 hence *:"e/4 > 0" by auto
  4157     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
  4158     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
  4159     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
  4160     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
  4161     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
  4162     def c \<equiv> "\<chi> i. min (a$i) (- (max B1 B2))" and d \<equiv> "\<chi> i. max (b$i) (max B1 B2)"
  4163     have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval vector_dist_norm
  4164     proof(rule_tac[!] allI)
  4165       case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next
  4166       case goal2 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  4167     have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow>
  4168       norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e"
  4169       using obt(3) unfolding real_norm_def by arith 
  4170     show ?case apply(rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
  4171                apply(rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
  4172       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI)
  4173       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then h x else 0)" in exI)
  4174       apply safe apply(rule_tac[1-2] integrable_integral,rule g,rule h)
  4175       apply(rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
  4176     proof- have *:"\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
  4177         (if x \<in> s then f x - g x else (0::real))" by auto
  4178       note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_sub, OF h g]]
  4179       show " norm (integral {a..b} (\<lambda>x. if x \<in> s then h x else 0) -
  4180                    integral {a..b} (\<lambda>x. if x \<in> s then g x else 0))
  4181            \<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) -
  4182                    integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))"
  4183         unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
  4184         apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+
  4185       proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
  4186           apply - apply rule apply(erule_tac x=i in allE) by auto
  4187       qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this
  4188 
  4189   show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv)
  4190   proof- case goal1 hence *:"e/3 > 0" by auto
  4191     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
  4192     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
  4193     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
  4194     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
  4195     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
  4196     show ?case apply(rule_tac x="max B1 B2" in exI) apply safe apply(rule min_max.less_supI1,rule B1)
  4197     proof- fix a b c d::"real^'n" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}"
  4198       have **:"ball 0 B1 \<subseteq> ball (0::real^'n) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::real^'n) (max B1 B2)" by auto
  4199       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>
  4200         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
  4201       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"
  4202         unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
  4203         apply(rule B1(2),rule order_trans,rule **,rule as(1)) 
  4204         apply(rule B1(2),rule order_trans,rule **,rule as(2)) 
  4205         apply(rule B2(2),rule order_trans,rule **,rule as(1)) 
  4206         apply(rule B2(2),rule order_trans,rule **,rule as(2)) 
  4207         apply(rule obt) apply(rule_tac[!] integral_le) using obt
  4208         by(auto intro!: h g interv) qed qed qed 
  4209 
  4210 subsection {* Adding integrals over several sets. *}
  4211 
  4212 lemma has_integral_union: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4213   assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)"
  4214   shows "(f has_integral (i + j)) (s \<union> t)"
  4215 proof- note * = has_integral_restrict_univ[THEN sym, of f]
  4216   show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)])
  4217     defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed
  4218 
  4219 lemma has_integral_unions: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4220   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')"
  4221   shows "(f has_integral (setsum i t)) (\<Union>t)"
  4222 proof- note * = has_integral_restrict_univ[THEN sym, of f]
  4223   have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))"
  4224     apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer 
  4225     apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto 
  4226   note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this]
  4227   thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption
  4228   proof safe case goal1 thus ?case
  4229     proof(cases "x\<in>\<Union>t") case True then guess s unfolding Union_iff .. note s=this
  4230       hence *:"\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s" using goal1(3) by blast
  4231       show ?thesis unfolding if_P[OF True] apply(rule trans) defer
  4232         apply(rule setsum_cong2) apply(subst *, assumption) apply(rule refl)
  4233         unfolding setsum_delta[OF assms(1)] using s by auto qed auto qed qed
  4234 
  4235 subsection {* In particular adding integrals over a division, maybe not of an interval. *}
  4236 
  4237 lemma has_integral_combine_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4238   assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k"
  4239   shows "(f has_integral (setsum i d)) s"
  4240 proof- note d = division_ofD[OF assms(1)]
  4241   show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions)
  4242     apply(rule d assms)+ apply(rule,rule,rule)
  4243   proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)]
  4244     guess a c b d apply-by(erule exE)+ note obt=this
  4245     from d(5)[OF goal1] show ?case unfolding obt interior_closed_interval
  4246       apply-apply(rule negligible_subset[of "({a..b}-{a<..<b}) \<union> ({c..d}-{c<..<d})"])
  4247       apply(rule negligible_union negligible_frontier_interval)+ by auto qed qed
  4248 
  4249 lemma integral_combine_division_bottomup: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4250   assumes "d division_of s" "\<forall>k\<in>d. f integrable_on k"
  4251   shows "integral s f = setsum (\<lambda>i. integral i f) d"
  4252   apply(rule integral_unique) apply(rule has_integral_combine_division[OF assms(1)])
  4253   using assms(2) unfolding has_integral_integral .
  4254 
  4255 lemma has_integral_combine_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4256   assumes "f integrable_on s" "d division_of k" "k \<subseteq> s"
  4257   shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
  4258   apply(rule has_integral_combine_division[OF assms(2)])
  4259   apply safe unfolding has_integral_integral[THEN sym]
  4260 proof- case goal1 from division_ofD(2,4)[OF assms(2) this]
  4261   show ?case apply safe apply(rule integrable_on_subinterval)
  4262     apply(rule assms) using assms(3) by auto qed
  4263 
  4264 lemma integral_combine_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4265   assumes "f integrable_on s" "d division_of s"
  4266   shows "integral s f = setsum (\<lambda>i. integral i f) d"
  4267   apply(rule integral_unique,rule has_integral_combine_division_topdown) using assms by auto
  4268 
  4269 lemma integrable_combine_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4270   assumes "d division_of s" "\<forall>i\<in>d. f integrable_on i"
  4271   shows "f integrable_on s"
  4272   using assms(2) unfolding integrable_on_def
  4273   by(metis has_integral_combine_division[OF assms(1)])
  4274 
  4275 lemma integrable_on_subdivision: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4276   assumes "d division_of i" "f integrable_on s" "i \<subseteq> s"
  4277   shows "f integrable_on i"
  4278   apply(rule integrable_combine_division assms)+
  4279 proof safe case goal1 note division_ofD(2,4)[OF assms(1) this]
  4280   thus ?case apply safe apply(rule integrable_on_subinterval[OF assms(2)])
  4281     using assms(3) by auto qed
  4282 
  4283 subsection {* Also tagged divisions. *}
  4284 
  4285 lemma has_integral_combine_tagged_division: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4286   assumes "p tagged_division_of s" "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
  4287   shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
  4288 proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
  4289     apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)])
  4290     using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto)
  4291   thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption
  4292     apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute)
  4293     apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption)
  4294     apply(rule setsum_cong2) using assms(2) by auto qed
  4295 
  4296 lemma integral_combine_tagged_division_bottomup: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4297   assumes "p tagged_division_of {a..b}" "\<forall>(x,k)\<in>p. f integrable_on k"
  4298   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
  4299   apply(rule integral_unique) apply(rule has_integral_combine_tagged_division[OF assms(1)])
  4300   using assms(2) by auto
  4301 
  4302 lemma has_integral_combine_tagged_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4303   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
  4304   shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) {a..b}"
  4305   apply(rule has_integral_combine_tagged_division[OF assms(2)])
  4306 proof safe case goal1 note tagged_division_ofD(3-4)[OF assms(2) this]
  4307   thus ?case using integrable_subinterval[OF assms(1)] by auto qed
  4308 
  4309 lemma integral_combine_tagged_division_topdown: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4310   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
  4311   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
  4312   apply(rule integral_unique,rule has_integral_combine_tagged_division_topdown) using assms by auto
  4313 
  4314 subsection {* Henstock's lemma. *}
  4315 
  4316 lemma henstock_lemma_part1: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4317   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
  4318   "(\<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)"
  4319   and p:"p tagged_partial_division_of {a..b}" "d fine p"
  4320   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e" (is "?x \<le> e")
  4321 proof-  { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" thus ?thesis by arith }
  4322   fix k::real assume k:"k>0" note p' = tagged_partial_division_ofD[OF p(1)]
  4323   have "\<Union>snd ` p \<subseteq> {a..b}" using p'(3) by fastsimp
  4324   note partial_division_of_tagged_division[OF p(1)] this
  4325   from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
  4326   def r \<equiv> "q - snd ` p" have "snd ` p \<inter> r = {}" unfolding r_def by auto
  4327   have r:"finite r" using q' unfolding r_def by auto
  4328 
  4329   have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
  4330     norm(setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
  4331   proof safe case goal1 hence i:"i \<in> q" unfolding r_def by auto
  4332     from q'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4333     have *:"k / (real (card r) + 1) > 0" apply(rule divide_pos_pos,rule k) by auto
  4334     have "f integrable_on {u..v}" apply(rule integrable_subinterval[OF assms(1)])
  4335       using q'(2)[OF i] unfolding uv by auto
  4336     note integrable_integral[OF this, unfolded has_integral[of f]]
  4337     from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
  4338     note gauge_inter[OF `gauge d` dd(1)] from fine_division_exists[OF this,of u v] guess qq .
  4339     thus ?case apply(rule_tac x=qq in exI) using dd(2)[of qq] unfolding fine_inter uv by auto qed
  4340   from bchoice[OF this] guess qq .. note qq=this[rule_format]
  4341 
  4342   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"
  4343     apply(rule assms(4)[rule_format])
  4344   proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto 
  4345     note * = tagged_partial_division_of_union_self[OF p(1)]
  4346     have "p \<union> \<Union>qq ` r tagged_division_of \<Union>snd ` p \<union> \<Union>r"
  4347     proof(rule tagged_division_union[OF * tagged_division_unions])
  4348       show "finite r" by fact case goal2 thus ?case using qq by auto
  4349     next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto
  4350     next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule)
  4351         apply(rule,rule q') defer apply(rule,subst Int_commute) 
  4352         apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer
  4353         apply(rule,rule q') using q(1) p' unfolding r_def by auto qed
  4354     moreover have "\<Union>snd ` p \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r"
  4355       unfolding Union_Un_distrib[THEN sym] r_def using q by auto
  4356     ultimately show "?p tagged_division_of {a..b}" by fastsimp qed
  4357 
  4358   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) -
  4359     integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3 
  4360     apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq)
  4361   proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r"
  4362     note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
  4363     from this(2) guess u v apply-by(erule exE)+ note uv=this
  4364     have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
  4365     hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto
  4366     note q'(5)[OF this] hence "interior l = {}" using subset_interior[OF `l \<subseteq> k`] by blast
  4367     thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto
  4368 
  4369   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
  4370     (qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero)
  4371     prefer 4 apply assumption apply(rule finite_imageI,fact)
  4372     unfolding split_paired_all split_conv image_iff defer apply(erule bexE)+
  4373   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"
  4374     note as = this(1-5)[unfolded this(6-)] note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
  4375     from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this
  4376     have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q')
  4377       using as unfolding r_def by auto
  4378     have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym]
  4379       apply(rule subset_interior) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto
  4380     thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto 
  4381   qed(insert qq, auto)
  4382 
  4383   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 -
  4384     integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact
  4385     apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption
  4386   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"
  4387     note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)] 
  4388     show "content m *\<^sub>R f x = 0"  using as(3) unfolding as by auto qed
  4389   
  4390   have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow> 
  4391     ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k" 
  4392   proof- case goal1 thus ?case  using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]  
  4393       unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
  4394   
  4395   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
  4396     unfolding split_def setsum_subtractf ..
  4397   also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"])
  4398   proof- case goal2 have *:"(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
  4399       apply(subst setsum_reindex_nonzero) apply fact
  4400       unfolding split_paired_all snd_conv split_def o_def
  4401     proof- fix x l y m assume as:"(x,l)\<in>p" "(y,m)\<in>p" "(x,l)\<noteq>(y,m)" "l = m"
  4402       from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this
  4403       show "integral l f = 0" unfolding uv apply(rule integral_unique)
  4404         apply(rule has_integral_null) unfolding content_eq_0_interior
  4405         using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto
  4406     qed auto 
  4407     show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
  4408       apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto
  4409   next  case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps)
  4410     show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
  4411       unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le,fact)
  4412       apply rule apply(drule qq) defer unfolding real_divide_def setsum_left_distrib[THEN sym]
  4413       unfolding real_divide_def[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat)
  4414   qed finally show "?x \<le> e + k" . qed
  4415 
  4416 lemma henstock_lemma_part2: fixes f::"real^'m \<Rightarrow> real^'n"
  4417   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
  4418   "\<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 -
  4419           integral({a..b}) f) < e"    "p tagged_partial_division_of {a..b}" "d fine p"
  4420   shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (CARD('n)) * e"
  4421   unfolding split_def apply(rule vsum_norm_allsubsets_bound) defer 
  4422   apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
  4423   apply safe apply(rule assms[rule_format,unfolded split_def]) defer
  4424   apply(rule tagged_partial_division_subset,rule assms,assumption)
  4425   apply(rule fine_subset,assumption,rule assms) using assms(5) by auto
  4426   
  4427 lemma henstock_lemma: fixes f::"real^'m \<Rightarrow> real^'n"
  4428   assumes "f integrable_on {a..b}" "e>0"
  4429   obtains d where "gauge d"
  4430   "\<forall>p. p tagged_partial_division_of {a..b} \<and> d fine p
  4431   \<longrightarrow> setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
  4432 proof- have *:"e / (2 * (real CARD('n) + 1)) > 0" apply(rule divide_pos_pos) using assms(2) by auto
  4433   from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
  4434   guess d .. note d = conjunctD2[OF this] show thesis apply(rule that,rule d)
  4435   proof safe case goal1 note * = henstock_lemma_part2[OF assms(1) * d this]
  4436     show ?case apply(rule le_less_trans[OF *]) using `e>0` by(auto simp add:field_simps) qed qed
  4437 
  4438 subsection {* monotone convergence (bounded interval first). *}
  4439 
  4440 lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1"
  4441   assumes "\<forall>k. (f k) integrable_on {a..b}"
  4442   "\<forall>k. \<forall>x\<in>{a..b}. dest_vec1(f k x) \<le> dest_vec1(f (Suc k) x)"
  4443   "\<forall>x\<in>{a..b}. ((\<lambda>k. f k x) ---> g x) sequentially"
  4444   "bounded {integral {a..b} (f k) | k . k \<in> UNIV}"
  4445   shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially"
  4446 proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
  4447   show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using Lim_const by auto
  4448 next assume ab:"content {a..b} \<noteq> 0"
  4449   have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x)$1 \<le> (g x)$1"
  4450   proof safe case goal1 note assms(3)[rule_format,OF this]
  4451     note * = Lim_component_ge[OF this trivial_limit_sequentially]
  4452     show ?case apply(rule *) unfolding eventually_sequentially
  4453       apply(rule_tac x=k in exI) apply- apply(rule transitive_stepwise_le)
  4454       using assms(2)[rule_format,OF goal1] by auto qed
  4455   have "\<exists>i. ((\<lambda>k. integral ({a..b}) (f k)) ---> i) sequentially"
  4456     apply(rule bounded_increasing_convergent) defer
  4457     apply rule apply(rule integral_component_le) apply safe
  4458     apply(rule assms(1-2)[rule_format])+ using assms(4) by auto
  4459   then guess i .. note i=this
  4460   have i':"\<And>k. dest_vec1(integral({a..b}) (f k)) \<le> dest_vec1 i"
  4461     apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially)
  4462     unfolding eventually_sequentially apply(rule_tac x=k in exI)
  4463     apply(rule transitive_stepwise_le) prefer 3 apply(rule integral_dest_vec1_le)
  4464     apply(rule assms(1-2)[rule_format])+ using assms(2) by auto
  4465 
  4466   have "(g has_integral i) {a..b}" unfolding has_integral
  4467   proof safe case goal1 note e=this
  4468     hence "\<forall>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
  4469              norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral {a..b} (f k)) < e / 2 ^ (k + 2)))"
  4470       apply-apply(rule,rule assms(1)[unfolded has_integral_integral has_integral,rule_format])
  4471       apply(rule divide_pos_pos) by auto
  4472     from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
  4473 
  4474     have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i$1 - dest_vec1(integral {a..b} (f k)) \<and>
  4475                    i$1 - dest_vec1(integral {a..b} (f k)) < e / 4"
  4476     proof- case goal1 have "e/4 > 0" using e by auto
  4477       from i[unfolded Lim_sequentially,rule_format,OF this] guess r ..
  4478       thus ?case apply(rule_tac x=r in exI) apply rule
  4479         apply(erule_tac x=k in allE)
  4480       proof- case goal1 thus ?case using i'[of k] unfolding dist_real by auto qed qed
  4481     then guess r .. note r=conjunctD2[OF this[rule_format]]
  4482 
  4483     have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)$1 - (f k x)$1 \<and>
  4484            (g x)$1 - (f k x)$1 < e / (4 * content({a..b}))"
  4485     proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact)
  4486         using ab content_pos_le[of a b] by auto
  4487       from assms(3)[rule_format,OF goal1,unfolded Lim_sequentially,rule_format,OF this]
  4488       guess n .. note n=this
  4489       thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE)
  4490         unfolding dist_real using fg[rule_format,OF goal1] by(auto simp add:field_simps) qed
  4491     from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
  4492     def d \<equiv> "\<lambda>x. c (m x) x" 
  4493 
  4494     show ?case apply(rule_tac x=d in exI)
  4495     proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
  4496     next fix p assume p:"p tagged_division_of {a..b}" "d fine p"
  4497       note p'=tagged_division_ofD[OF p(1)]
  4498       have "\<exists>a. \<forall>x\<in>p. m (fst x) \<le> a" by(rule upper_bound_finite_set,fact)
  4499       then guess s .. note s=this
  4500       have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and>
  4501             norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e" 
  4502       proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
  4503           norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
  4504           by(auto simp add:algebra_simps) qed
  4505       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where
  4506           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))"])
  4507       proof safe case goal1
  4508          show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"])
  4509            unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule setsum_norm[OF p'(1)])
  4510            apply(rule setsum_mono) unfolding split_paired_all split_conv
  4511            unfolding split_def setsum_left_distrib[THEN sym] scaleR.diff_right[THEN sym]
  4512            unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
  4513          proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto
  4514            from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this
  4515            show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))"
  4516              unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le] 
  4517              apply(rule mult_left_mono) unfolding norm_real using m(2)[OF x,of "m x"] by auto
  4518          qed(insert ab,auto)
  4519          
  4520        next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
  4521            \<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
  4522            apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer
  4523            apply(subst split_def)+ unfolding setsum_subtractf apply rule
  4524          proof- show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p.
  4525              m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e / 2"
  4526              apply(rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"])
  4527              apply(rule setsum_norm_le[OF finite_atLeastAtMost])
  4528            proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
  4529                unfolding power_add real_divide_def inverse_mult_distrib
  4530                unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym]
  4531                unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e])
  4532                unfolding power2_eq_square by auto
  4533              fix t assume "t\<in>{0..s}"
  4534              show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x -
  4535                integral k (f (m x))) \<le> e / 2 ^ (t + 2)"apply(rule order_trans[of _
  4536                "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
  4537                apply(rule eq_refl) apply(rule arg_cong[where f=norm]) apply(rule setsum_cong2) defer
  4538                apply(rule henstock_lemma_part1) apply(rule assms(1)[rule_format])
  4539                apply(rule divide_pos_pos,rule e) defer  apply safe apply(rule c)+
  4540                apply rule apply assumption+ apply(rule tagged_partial_division_subset[of p])
  4541                apply(rule p(1)[unfolded tagged_division_of_def,THEN conjunct1]) defer
  4542                unfolding fine_def apply safe apply(drule p(2)[unfolded fine_def,rule_format])
  4543                unfolding d_def by auto qed
  4544          qed(insert s, auto)
  4545 
  4546        next case goal3
  4547          note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
  4548          have *:"\<And>sr sx ss ks kr::real^1. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i$1 - kr$1
  4549            \<and> i$1 - kr$1 < e/4 \<longrightarrow> abs(sx$1 - i$1) < e/4" unfolding Cart_eq forall_1 vector_le_def by arith
  4550          show ?case unfolding norm_real Cart_nth.diff apply(rule *[rule_format],safe)
  4551            apply(rule comb[of r],rule comb[of s]) unfolding vector_le_def forall_1 setsum_component
  4552            apply(rule i') apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
  4553            apply(rule_tac[1-2] integral_component_le[OF ])
  4554          proof safe show "0 \<le> i$1 - (integral {a..b} (f r))$1" using r(1) by auto
  4555            show "i$1 - (integral {a..b} (f r))$1 < e / 4" using r(2) by auto
  4556            fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4557            show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k" 
  4558              unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
  4559              using p'(3)[OF xk] unfolding uv by auto 
  4560            fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto
  4561            hence *:"\<And>m. \<forall>n\<ge>m. (f m y)$1 \<le> (f n y)$1" apply-apply(rule transitive_stepwise_le) using assms(2) by auto
  4562            show "(f r y)$1 \<le> (f (m x) y)$1" "(f (m x) y)$1 \<le> (f s y)$1"
  4563              apply(rule_tac[!] *[rule_format]) using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk] by auto
  4564          qed qed qed qed note * = this 
  4565 
  4566    have "integral {a..b} g = i" apply(rule integral_unique) using * .
  4567    thus ?thesis using i * by auto qed
  4568 
  4569 lemma monotone_convergence_increasing: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1"
  4570   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f k x)$1 \<le> (f (Suc k) x)$1"
  4571   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4572   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4573 proof- have lem:"\<And>f::nat \<Rightarrow> real^'n \<Rightarrow> real^1. \<And> g s. \<forall>k.\<forall>x\<in>s. 0\<le>dest_vec1 (f k x) \<Longrightarrow> \<forall>k. (f k) integrable_on s \<Longrightarrow>
  4574     \<forall>k. \<forall>x\<in>s. (f k x)$1 \<le> (f (Suc k) x)$1 \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially  \<Longrightarrow>
  4575     bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4576   proof- case goal1 note assms=this[rule_format]
  4577     have "\<forall>x\<in>s. \<forall>k. dest_vec1(f k x) \<le> dest_vec1(g x)" apply safe apply(rule Lim_component_ge)
  4578       apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially)
  4579       unfolding eventually_sequentially apply(rule_tac x=k in exI)
  4580       apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format]
  4581 
  4582     have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially" apply(rule bounded_increasing_convergent)
  4583       apply(rule goal1(5)) apply(rule,rule integral_component_le) apply(rule goal1(2)[rule_format])+
  4584       using goal1(3) by auto then guess i .. note i=this
  4585     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
  4586     hence i':"\<forall>k. (integral s (f k))$1 \<le> i$1" apply-apply(rule,rule Lim_component_ge)
  4587       apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
  4588       apply(rule_tac x=k in exI,safe) apply(rule integral_dest_vec1_le)
  4589       apply(rule goal1(2)[rule_format])+ by auto 
  4590 
  4591     note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
  4592     have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) =
  4593       (\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto
  4594     have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym])
  4595       apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int .
  4596     have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and>
  4597       ((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) --->
  4598       integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
  4599     proof(rule monotone_convergence_interval,safe)
  4600       case goal1 show ?case using int .
  4601     next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto
  4602     next case goal3 thus ?case apply-apply(cases "x\<in>s") using assms(4) by auto
  4603     next case goal4 note * = integral_dest_vec1_nonneg[unfolded vector_le_def forall_1 zero_index]
  4604       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))"
  4605         unfolding norm_real apply(subst abs_of_nonneg) apply(rule *[OF int])
  4606         apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3
  4607         apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]])
  4608         apply(subst integral_restrict_univ[THEN sym,OF int]) 
  4609         unfolding ifif unfolding integral_restrict_univ[OF int']
  4610         apply(rule integral_subset_component_le[OF _ int' assms(2)]) using assms(1) by auto
  4611       thus ?case using assms(5) unfolding bounded_iff apply safe
  4612         apply(rule_tac x=aa in exI,safe) apply(erule_tac x="integral s (f k)" in ballE)
  4613         apply(rule order_trans) apply assumption by auto qed note g = conjunctD2[OF this]
  4614 
  4615     have "(g has_integral i) s" unfolding has_integral_alt' apply safe apply(rule g(1))
  4616     proof- case goal1 hence "e/4>0" by auto
  4617       from i[unfolded Lim_sequentially,rule_format,OF this] guess N .. note N=this
  4618       note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]]
  4619       from this[THEN conjunct2,rule_format,OF `e/4>0`] guess B .. note B=conjunctD2[OF this]
  4620       show ?case apply(rule,rule,rule B,safe)
  4621       proof- fix a b::"real^'n" assume ab:"ball 0 B \<subseteq> {a..b}"
  4622         from `e>0` have "e/2>0" by auto
  4623         from g(2)[unfolded Lim_sequentially,of a b,rule_format,OF this] guess M .. note M=this
  4624         have **:"norm (integral {a..b} (\<lambda>x. if x \<in> s then f N x else 0) - i) < e/2"
  4625           apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N]
  4626           unfolding vector_dist_norm apply-defer apply(subst norm_minus_commute) by auto
  4627         have *:"\<And>f1 f2 g. abs(f1 - i$1) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i$1
  4628           \<longrightarrow> abs(g - i$1) < e" by arith
  4629         show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e" 
  4630           unfolding norm_real Cart_simps apply(rule *[rule_format])
  4631           apply(rule **[unfolded norm_real Cart_simps])
  4632           apply(rule M[rule_format,of "M + N",unfolded dist_real]) apply(rule le_add1)
  4633           apply(rule integral_component_le[OF int int]) defer
  4634           apply(rule order_trans[OF _ i'[rule_format,of "M + N"]])
  4635         proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)$1 \<le> (f n x)$1"
  4636             apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto
  4637         next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int]) 
  4638             unfolding ifif integral_restrict_univ[OF int']
  4639             apply(rule integral_subset_component_le[OF _ int']) using assms by auto
  4640         qed qed qed 
  4641     thus ?case apply safe defer apply(drule integral_unique) using i by auto qed
  4642 
  4643   have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
  4644     apply(subst integral_sub) apply(rule assms(1)[rule_format])+ by rule
  4645   have "\<And>x m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. dest_vec1 (f m x) \<le> dest_vec1 (f n x)" apply(rule transitive_stepwise_le)
  4646     using assms(2) by auto note * = this[rule_format]
  4647   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)) --->
  4648       integral s (\<lambda>x. g x - f 0 x)) sequentially" apply(rule lem,safe)
  4649   proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
  4650   next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
  4651   next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
  4652   next case goal4 thus ?case apply-apply(rule Lim_sub) 
  4653       using seq_offset[OF assms(3)[rule_format],of x 1] by auto
  4654   next case goal5 thus ?case using assms(4) unfolding bounded_iff
  4655       apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
  4656       apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub
  4657       apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed
  4658   note conjunctD2[OF this] note Lim_add[OF this(2) Lim_const[of "integral s (f 0)"]]
  4659     integrable_add[OF this(1) assms(1)[rule_format,of 0]]
  4660   thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub)
  4661     using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed
  4662 
  4663 lemma monotone_convergence_decreasing: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real^1"
  4664   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f (Suc k) x)$1 \<le> (f k x)$1"
  4665   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4666   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4667 proof- note assm = assms[rule_format]
  4668   have *:"{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R -1 ` {integral s (f k)| k. True}"
  4669     apply safe unfolding image_iff apply(rule_tac x="integral s (f k)" in bexI) prefer 3
  4670     apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto
  4671   have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x))
  4672     ---> integral s (\<lambda>x. - g x))  sequentially" apply(rule monotone_convergence_increasing)
  4673     apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule Lim_neg)
  4674     apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto
  4675   note * = conjunctD2[OF this]
  4676   show ?thesis apply rule using integrable_neg[OF *(1)] defer
  4677     using Lim_neg[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
  4678     unfolding integral_neg[OF *(1),THEN sym] by auto qed
  4679 
  4680 lemma monotone_convergence_increasing_real: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real"
  4681   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<ge> (f k x)"
  4682   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4683   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4684 proof- have *:"{integral s (\<lambda>x. vec1 (f k x)) |k. True} =  vec1 ` {integral s (f k) |k. True}"
  4685     unfolding integral_trans[OF assms(1)[rule_format]] by auto
  4686   have "vec1 \<circ> g integrable_on s \<and> ((\<lambda>k. integral s (vec1 \<circ> f k)) ---> integral s (vec1 \<circ> g)) sequentially"
  4687     apply(rule monotone_convergence_increasing) unfolding o_def integrable_on_trans
  4688     unfolding vec1_dest_vec1 apply(rule assms)+ unfolding Lim_trans unfolding * using assms(3,4) by auto
  4689   thus ?thesis unfolding o_def unfolding integral_trans[OF assms(1)[rule_format]] by auto qed
  4690 
  4691 lemma monotone_convergence_decreasing_real: fixes f::"nat \<Rightarrow> real^'n \<Rightarrow> real"
  4692   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<le> (f k x)"
  4693   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4694   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4695 proof- have *:"{integral s (\<lambda>x. vec1 (f k x)) |k. True} =  vec1 ` {integral s (f k) |k. True}"
  4696     unfolding integral_trans[OF assms(1)[rule_format]] by auto
  4697   have "vec1 \<circ> g integrable_on s \<and> ((\<lambda>k. integral s (vec1 \<circ> f k)) ---> integral s (vec1 \<circ> g)) sequentially"
  4698     apply(rule monotone_convergence_decreasing) unfolding o_def integrable_on_trans
  4699     unfolding vec1_dest_vec1 apply(rule assms)+ unfolding Lim_trans unfolding * using assms(3,4) by auto
  4700   thus ?thesis unfolding o_def unfolding integral_trans[OF assms(1)[rule_format]] by auto qed
  4701 
  4702 subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
  4703 
  4704 definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46) where
  4705   "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. (norm(f x))) integrable_on s"
  4706 
  4707 lemma absolutely_integrable_onI[intro?]:
  4708   "f integrable_on s \<Longrightarrow> (\<lambda>x. (norm(f x))) integrable_on s \<Longrightarrow> f absolutely_integrable_on s"
  4709   unfolding absolutely_integrable_on_def by auto
  4710 
  4711 lemma absolutely_integrable_onD[dest]: assumes "f absolutely_integrable_on s"
  4712   shows "f integrable_on s" "(\<lambda>x. (norm(f x))) integrable_on s"
  4713   using assms unfolding absolutely_integrable_on_def by auto
  4714 
  4715 lemma absolutely_integrable_on_trans[simp]: fixes f::"real^'n \<Rightarrow> real" shows
  4716   "(vec1 o f) absolutely_integrable_on s \<longleftrightarrow> f absolutely_integrable_on s"
  4717   unfolding absolutely_integrable_on_def o_def by auto
  4718 
  4719 lemma integral_norm_bound_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4720   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> g x"
  4721   shows "norm(integral s f) \<le> (integral s g)"
  4722 proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr)
  4723     apply(erule_tac x="x - y" in allE) by auto
  4724   have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
  4725     \<longrightarrow> norm(ig) < dia + e" 
  4726   proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
  4727       apply(subst real_sum_of_halves[of e,THEN sym]) unfolding class_semiring.add_a
  4728       apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
  4729       apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
  4730   qed note norm=this[rule_format]
  4731   have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> g a b. f integrable_on {a..b} \<Longrightarrow> g integrable_on {a..b} \<Longrightarrow>
  4732     \<forall>x\<in>{a..b}. norm(f x) \<le> (g x) \<Longrightarrow> norm(integral({a..b}) f) \<le> (integral({a..b}) g)"
  4733   proof(rule *[rule_format]) case goal1 hence *:"e/2>0" by auto
  4734     from integrable_integral[OF goal1(1),unfolded has_integral[of f],rule_format,OF *]
  4735     guess d1 .. note d1 = conjunctD2[OF this,rule_format]
  4736     from integrable_integral[OF goal1(2),unfolded has_integral[of g],rule_format,OF *]
  4737     guess d2 .. note d2 = conjunctD2[OF this,rule_format]
  4738     note gauge_inter[OF d1(1) d2(1)]
  4739     from fine_division_exists[OF this, of a b] guess p . note p=this
  4740     show ?case apply(rule norm) defer
  4741       apply(rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def]) defer
  4742       apply(rule d1(2)[OF conjI[OF p(1)]]) defer apply(rule setsum_norm_le)
  4743     proof safe fix x k assume "(x,k)\<in>p" note as = tagged_division_ofD(2-4)[OF p(1) this]
  4744       from this(3) guess u v apply-by(erule exE)+ note uv=this
  4745       show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x" unfolding uv norm_scaleR
  4746         unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
  4747         apply(rule mult_left_mono) using goal1(3) as by auto
  4748     qed(insert p[unfolded fine_inter],auto) qed
  4749 
  4750   { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e" 
  4751     thus ?thesis apply-apply(rule *[rule_format]) by auto }
  4752   fix e::real assume "e>0" hence e:"e/2 > 0" by auto
  4753   note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
  4754   note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format]
  4755   from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e]
  4756   guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format]
  4757   from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e]
  4758   guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format]
  4759   from bounded_subset_closed_interval[OF bounded_ball, of "0::real^'n" "max B1 B2"]
  4760   guess a b apply-by(erule exE)+ note ab=this[unfolded ball_max_Un]
  4761 
  4762   have "ball 0 B1 \<subseteq> {a..b}" using ab by auto
  4763   from B1(2)[OF this] guess z .. note z=conjunctD2[OF this]
  4764   have "ball 0 B2 \<subseteq> {a..b}" using ab by auto
  4765   from B2(2)[OF this] guess w .. note w=conjunctD2[OF this]
  4766 
  4767   show "norm (integral s f) < integral s g + e" apply(rule norm)
  4768     apply(rule lem[OF f g, of a b]) unfolding integral_unique[OF z(1)] integral_unique[OF w(1)]
  4769     defer apply(rule w(2)[unfolded real_norm_def],rule z(2))
  4770     apply safe apply(case_tac "x\<in>s") unfolding if_P apply(rule assms(3)[rule_format]) by auto qed
  4771 
  4772 lemma integral_norm_bound_integral_component: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4773   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. norm(f x) \<le> (g x)$k"
  4774   shows "norm(integral s f) \<le> (integral s g)$k"
  4775 proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x $ k) o g)"
  4776     apply(rule integral_norm_bound_integral[OF assms(1)])
  4777     apply(rule integrable_linear[OF assms(2)],rule)
  4778     unfolding o_def by(rule assms)
  4779   thus ?thesis unfolding o_def integral_component_eq[OF assms(2)] . qed
  4780 
  4781 lemma has_integral_norm_bound_integral_component: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4782   assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$k"
  4783   shows "norm(i) \<le> j$k" using integral_norm_bound_integral_component[of f s g k]
  4784   unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
  4785   using assms by auto
  4786 
  4787 lemma absolutely_integrable_le: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4788   assumes "f absolutely_integrable_on s"
  4789   shows "norm(integral s f) \<le> integral s (\<lambda>x. norm(f x))"
  4790   apply(rule integral_norm_bound_integral) using assms by auto
  4791 
  4792 lemma absolutely_integrable_0[intro]: "(\<lambda>x. 0) absolutely_integrable_on s"
  4793   unfolding absolutely_integrable_on_def by auto
  4794 
  4795 lemma absolutely_integrable_cmul[intro]:
  4796  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on s"
  4797   unfolding absolutely_integrable_on_def using integrable_cmul[of f s c]
  4798   using integrable_cmul[of "\<lambda>x. norm (f x)" s "abs c"] by auto
  4799 
  4800 lemma absolutely_integrable_neg[intro]:
  4801  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) absolutely_integrable_on s"
  4802   apply(drule absolutely_integrable_cmul[where c="-1"]) by auto
  4803 
  4804 lemma absolutely_integrable_norm[intro]:
  4805  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. norm(f x)) absolutely_integrable_on s"
  4806   unfolding absolutely_integrable_on_def by auto
  4807 
  4808 lemma absolutely_integrable_abs[intro]:
  4809  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. abs(f x::real)) absolutely_integrable_on s"
  4810   apply(drule absolutely_integrable_norm) unfolding real_norm_def .
  4811 
  4812 lemma absolutely_integrable_on_subinterval: fixes f::"real^'n \<Rightarrow> 'a::banach" shows
  4813   "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}" 
  4814   unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
  4815 
  4816 lemma absolutely_integrable_bounded_variation: fixes f::"real^'n \<Rightarrow> 'a::banach"
  4817   assumes "f absolutely_integrable_on UNIV"
  4818   obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
  4819   apply(rule that[of "integral UNIV (\<lambda>x. norm (f x))"])
  4820 proof safe case goal1 note d = division_ofD[OF this(2)]
  4821   have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. integral i (\<lambda>x. norm (f x)))"
  4822     apply(rule setsum_mono,rule absolutely_integrable_le) apply(drule d(4),safe)
  4823     apply(rule absolutely_integrable_on_subinterval[OF assms]) by auto
  4824   also have "... \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
  4825     apply(subst integral_combine_division_topdown[OF _ goal1(2)])
  4826     using integrable_on_subdivision[OF goal1(2)] using assms by auto
  4827   also have "... \<le> integral UNIV (\<lambda>x. norm (f x))"
  4828     apply(rule integral_subset_le) 
  4829     using integrable_on_subdivision[OF goal1(2)] using assms by auto
  4830   finally show ?case . qed
  4831 
  4832 lemma helplemma:
  4833   assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s"
  4834   shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e"
  4835   unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs])
  4836   apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
  4837   using norm_triangle_ineq3 .
  4838 
  4839 lemma bounded_variation_absolutely_integrable_interval:
  4840   fixes f::"real^'n \<Rightarrow> real^'m" assumes "f integrable_on {a..b}"
  4841   "\<forall>d. d division_of {a..b} \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
  4842   shows "f absolutely_integrable_on {a..b}"
  4843 proof- let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of {a..b} }" def i \<equiv> "Sup ?S"
  4844   have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
  4845     apply(rule elementary_interval) defer apply(rule_tac x=B in exI)
  4846     apply(rule setleI) using assms(2) by auto
  4847   show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i])
  4848   proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) `
  4849         {d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format]
  4850       unfolding setge_def ubs_def by auto 
  4851     hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))"
  4852       unfolding mem_Collect_eq isUb_def setle_def by simp then guess d .. note d=conjunctD2[OF this]
  4853     note d' = division_ofD[OF this(1)]
  4854 
  4855     have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
  4856     proof case goal1 have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
  4857         apply(rule separate_point_closed) apply(rule closed_Union)
  4858         apply(rule finite_subset[OF _ d'(1)]) apply safe apply(drule d'(4)) by auto
  4859       thus ?case apply safe apply(rule_tac x=da in exI,safe)
  4860         apply(erule_tac x=xa in ballE) by auto
  4861     qed from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
  4862 
  4863     have "e/2 > 0" using goal1 by auto
  4864     from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
  4865     let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
  4866     show ?case apply(rule_tac x="?g" in exI) apply safe
  4867     proof- show "gauge ?g" using g(1) unfolding gauge_def using k(1) by auto
  4868       fix p assume "p tagged_division_of {a..b}" "?g fine p"
  4869       note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
  4870       note p' = tagged_division_ofD[OF p(1)]
  4871       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}"
  4872       have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto
  4873       have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI)
  4874       proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l))
  4875           ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def 
  4876           defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
  4877           apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto
  4878         fix x k assume "(x,k)\<in>p'"
  4879         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
  4880         then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
  4881         show "x\<in>k" "k\<subseteq>{a..b}" using p'(2-3)[OF il(3)] il by auto
  4882         show "\<exists>a b. k = {a..b}" unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
  4883           apply safe unfolding inter_interval by auto
  4884       next fix x1 k1 assume "(x1,k1)\<in>p'"
  4885         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
  4886         then guess i1 l1 apply-by(erule exE)+ note il1=conjunctD4[OF this]
  4887         fix x2 k2 assume "(x2,k2)\<in>p'"
  4888         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
  4889         then guess i2 l2 apply-by(erule exE)+ note il2=conjunctD4[OF this]
  4890         assume "(x1, k1) \<noteq> (x2, k2)"
  4891         hence "interior(i1) \<inter> interior(i2) = {} \<or> interior(l1) \<inter> interior(l2) = {}"
  4892           using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] unfolding il1 il2 by auto
  4893         thus "interior k1 \<inter> interior k2 = {}" unfolding il1 il2 by auto
  4894       next have *:"\<forall>(x, X) \<in> p'. X \<subseteq> {a..b}" unfolding p'_def using d' by auto
  4895         show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least)
  4896           unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe
  4897         proof- fix y assume y:"y\<in>{a..b}"
  4898           hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto
  4899           then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this]
  4900           hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto
  4901           then guess i .. note i = conjunctD2[OF this]
  4902           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
  4903           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)
  4904             defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI)
  4905             apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto 
  4906         qed qed 
  4907 
  4908       hence "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
  4909         apply-apply(rule g(2)[rule_format]) unfolding tagged_division_of_def apply safe using gp' .
  4910       hence **:" \<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
  4911         unfolding split_def apply(rule helplemma) using p'' by auto
  4912 
  4913       have p'alt:"p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> ~(i \<inter> l = {})}"
  4914       proof safe case goal2
  4915         have "x\<in>i" using fineD[OF p(3) goal2(1)] k(2)[OF goal2(2), of x] goal2(4-) by auto
  4916         hence "(x, i \<inter> l) \<in> p'" unfolding p'_def apply safe
  4917           apply(rule_tac x=x in exI,rule_tac x="i\<inter>l" in exI) apply safe using goal2 by auto
  4918         thus ?case using goal2(3) by auto
  4919       next fix x k assume "(x,k)\<in>p'"
  4920         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
  4921         then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
  4922         thus "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
  4923           apply(rule_tac x=x in exI,rule_tac x=i in exI,rule_tac x=l in exI)
  4924           using p'(2)[OF il(3)] by auto
  4925       qed
  4926       have sum_p':"(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
  4927         apply(subst setsum_over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
  4928         unfolding norm_eq_zero apply(rule integral_null,assumption) ..
  4929       note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
  4930 
  4931       have *:"\<And>sni sni' sf sf'. abs(sf' - sni') < e / 2 \<longrightarrow> i - e / 2 < sni \<and> sni' \<le> i \<and>
  4932         sni \<le> sni' \<and> sf' = sf \<longrightarrow> abs(sf - i) < e" by arith
  4933       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - i) < e" 
  4934         unfolding real_norm_def apply(rule *[rule_format,OF **],safe) apply(rule d(2))
  4935       proof- case goal1 show ?case unfolding sum_p'
  4936           apply(rule isLubD2[OF i]) using division_of_tagged_division[OF p''] by auto
  4937       next case goal2 have *:"{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
  4938           (\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}" by auto
  4939         have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
  4940         proof(rule setsum_mono) case goal1 note k=this
  4941           from d'(4)[OF this] gues