src/HOL/Multivariate_Analysis/Integration.thy
author huffman
Tue Aug 09 10:30:00 2011 -0700 (2011-08-09)
changeset 44125 230a8665c919
parent 42871 1c0b99f950d9
child 44140 2c10c35dd4be
permissions -rw-r--r--
mark some redundant theorems as legacy
     1 header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
     2 (*  Author:                     John Harrison
     3     Translation from HOL light: Robert Himmelmann, TU Muenchen *)
     4 
     5 theory Integration
     6 imports
     7   Derivative
     8   "~~/src/HOL/Library/Indicator_Function"
     9 begin
    10 
    11 declare [[smt_certificates="Integration.certs"]]
    12 declare [[smt_fixed=true]]
    13 declare [[smt_oracle=false]]
    14 
    15 (*declare not_less[simp] not_le[simp]*)
    16 
    17 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
    18   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
    19   scaleR_cancel_left scaleR_cancel_right scaleR.add_right scaleR.add_left real_vector_class.scaleR_one
    20 
    21 lemma real_arch_invD:
    22   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
    23   by(subst(asm) real_arch_inv)
    24 subsection {* Sundries *}
    25 
    26 (*declare basis_component[simp]*)
    27 
    28 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
    29 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
    30 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
    31 lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
    32 
    33 declare norm_triangle_ineq4[intro] 
    34 
    35 lemma simple_image: "{f x |x . x \<in> s} = f ` s" by blast
    36 
    37 lemma linear_simps:  assumes "bounded_linear f"
    38   shows "f (a + b) = f a + f b" "f (a - b) = f a - f b" "f 0 = 0" "f (- a) = - f a" "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
    39   apply(rule_tac[!] additive.add additive.minus additive.diff additive.zero bounded_linear.scaleR)
    40   using assms unfolding bounded_linear_def additive_def by auto
    41 
    42 lemma bounded_linearI:assumes "\<And>x y. f (x + y) = f x + f y"
    43   "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x" "\<And>x. norm (f x) \<le> norm x * K"
    44   shows "bounded_linear f"
    45   unfolding bounded_linear_def additive_def bounded_linear_axioms_def using assms by auto
    46  
    47 lemma real_le_inf_subset:
    48   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. b <=* s" shows "Inf s <= Inf (t::real set)"
    49   apply(rule isGlb_le_isLb) apply(rule Inf[OF assms(1)])
    50   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
    51   unfolding isLb_def setge_def by auto
    52 
    53 lemma real_ge_sup_subset:
    54   assumes "t \<noteq> {}" "t \<subseteq> s" "\<exists>b. s *<= b" shows "Sup s >= Sup (t::real set)"
    55   apply(rule isLub_le_isUb) apply(rule Sup[OF assms(1)])
    56   using assms apply-apply(erule exE) apply(rule_tac x=b in exI)
    57   unfolding isUb_def setle_def by auto
    58 
    59 lemma bounded_linear_component[intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x $$ k)"
    60   apply(rule bounded_linearI[where K=1]) 
    61   using component_le_norm[of _ k] unfolding real_norm_def by auto
    62 
    63 lemma transitive_stepwise_lt_eq:
    64   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
    65   shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))" (is "?l = ?r")
    66 proof(safe) assume ?r fix n m::nat assume "m < n" thus "R m n" apply-
    67   proof(induct n arbitrary: m) case (Suc n) show ?case 
    68     proof(cases "m < n") case True
    69       show ?thesis apply(rule assms[OF Suc(1)[OF True]]) using `?r` by auto
    70     next case False hence "m = n" using Suc(2) by auto
    71       thus ?thesis using `?r` by auto
    72     qed qed auto qed auto
    73 
    74 lemma transitive_stepwise_gt:
    75   assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
    76   shows "\<forall>n>m. R m n"
    77 proof- have "\<forall>m. \<forall>n>m. R m n" apply(subst transitive_stepwise_lt_eq)
    78     apply(rule assms) apply(assumption,assumption) using assms(2) by auto
    79   thus ?thesis by auto qed
    80 
    81 lemma transitive_stepwise_le_eq:
    82   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
    83   shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))" (is "?l = ?r")
    84 proof safe assume ?r fix m n::nat assume "m\<le>n" thus "R m n" apply-
    85   proof(induct n arbitrary: m) case (Suc n) show ?case 
    86     proof(cases "m \<le> n") case True show ?thesis apply(rule assms(2))
    87         apply(rule Suc(1)[OF True]) using `?r` by auto
    88     next case False hence "m = Suc n" using Suc(2) by auto
    89       thus ?thesis using assms(1) by auto
    90     qed qed(insert assms(1), auto) qed auto
    91 
    92 lemma transitive_stepwise_le:
    93   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n) "
    94   shows "\<forall>n\<ge>m. R m n"
    95 proof- have "\<forall>m. \<forall>n\<ge>m. R m n" apply(subst transitive_stepwise_le_eq)
    96     apply(rule assms) apply(rule assms,assumption,assumption) using assms(3) by auto
    97   thus ?thesis by auto qed
    98 
    99 subsection {* Some useful lemmas about intervals. *}
   100 
   101 abbreviation One  where "One \<equiv> ((\<chi>\<chi> i. 1)::_::ordered_euclidean_space)"
   102 
   103 lemma empty_as_interval: "{} = {One..0}"
   104   apply(rule set_eqI,rule) defer unfolding mem_interval
   105   using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto
   106 
   107 lemma interior_subset_union_intervals: 
   108   assumes "i = {a..b::'a::ordered_euclidean_space}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"
   109   shows "interior i \<subseteq> interior s" proof-
   110   have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
   111     unfolding assms(1,2) interior_closed_interval by auto
   112   moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)
   113     using assms(4) unfolding assms(1,2) by auto
   114   ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)
   115     unfolding assms(1,2) interior_closed_interval by auto qed
   116 
   117 lemma inter_interior_unions_intervals: fixes f::"('a::ordered_euclidean_space) set set"
   118   assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"
   119   shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1
   120   have lem1:"\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U" apply rule  defer apply(rule_tac Int_greatest)
   121     unfolding open_subset_interior[OF open_ball]  using interior_subset by auto
   122   have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
   123   have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow> (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)" proof- case goal1
   124   thus ?case proof(induct rule:finite_induct) 
   125     case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next
   126     case (insert i f) guess x using insert(5) .. note x = this
   127     then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this
   128     guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this
   129     show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto
   130       then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..
   131       hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" unfolding ab ball_min_Int by auto
   132       hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 unfolding  ball_min_Int by auto
   133       hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
   134       hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t" apply-apply(rule insert(3)) using insert(4) by auto thus ?thesis by auto next
   135     case True show ?thesis proof(cases "x\<in>{a<..<b}")
   136       case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..
   137       thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)
   138         unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next
   139     case False then obtain k where "x$$k \<le> a$$k \<or> x$$k \<ge> b$$k" and k:"k<DIM('a)" unfolding mem_interval by(auto simp add:not_less) 
   140     hence "x$$k = a$$k \<or> x$$k = b$$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto
   141     hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)
   142       let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$$k = a$$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
   143         fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
   144         hence "\<bar>(?z - y) $$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
   145         hence "y$$k < a$$k" using e[THEN conjunct1] k by(auto simp add:field_simps basis_component as)
   146         hence "y \<notin> i" unfolding ab mem_interval not_all apply(rule_tac x=k in exI) using k by auto thus False using yi by auto qed
   147       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
   148         fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"
   149            apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])
   150           unfolding norm_scaleR norm_basis by auto
   151         also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
   152         finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed
   153       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto
   154     next let ?z = "x + (e/2) *\<^sub>R basis k" assume as:"x$$k = b$$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)
   155         fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto
   156         hence "\<bar>(?z - y) $$ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding dist_norm by auto
   157         hence "y$$k > b$$k" using e[THEN conjunct1] k by(auto simp add:field_simps as)
   158         hence "y \<notin> i" unfolding ab mem_interval not_all using k by(rule_tac x=k in exI,auto) thus False using yi by auto qed
   159       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof
   160         fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"
   161            apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])
   162           unfolding norm_scaleR by auto
   163         also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball dist_norm using e by(auto simp add:field_simps)
   164         finally show "y\<in>ball x e" unfolding mem_ball dist_norm using e by(auto simp add:field_simps) qed
   165       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 
   166     then guess x .. hence "x \<in> s \<inter> interior (\<Union>f)" unfolding lem1[where U="\<Union>f",THEN sym] using centre_in_ball e[THEN conjunct1] by auto
   167     thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this
   168   guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..
   169   hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto
   170   thus False using `t\<in>f` assms(4) by auto qed
   171 
   172 subsection {* Bounds on intervals where they exist. *}
   173 
   174 definition "interval_upperbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Sup {a. \<exists>x\<in>s. x$$i = a})::'a)"
   175 
   176 definition "interval_lowerbound (s::('a::ordered_euclidean_space) set) = ((\<chi>\<chi> i. Inf {a. \<exists>x\<in>s. x$$i = a})::'a)"
   177 
   178 lemma interval_upperbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i" shows "interval_upperbound {a..b} = b"
   179   using assms unfolding interval_upperbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
   180   unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
   181   apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   182   apply(rule,rule) apply(rule_tac x="b$$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)
   183   unfolding mem_interval using assms by auto
   184 
   185 lemma interval_lowerbound[simp]: assumes "\<forall>i<DIM('a::ordered_euclidean_space). a$$i \<le> (b::'a)$$i" shows "interval_lowerbound {a..b} = a"
   186   using assms unfolding interval_lowerbound_def apply(subst euclidean_eq[where 'a='a]) apply safe
   187   unfolding euclidean_lambda_beta' apply(erule_tac x=i in allE)
   188   apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer
   189   apply(rule,rule) apply(rule_tac x="a$$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)
   190   unfolding mem_interval using assms by auto 
   191 
   192 lemmas interval_bounds = interval_upperbound interval_lowerbound
   193 
   194 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
   195   using assms unfolding interval_ne_empty by auto
   196 
   197 subsection {* Content (length, area, volume...) of an interval. *}
   198 
   199 definition "content (s::('a::ordered_euclidean_space) set) =
   200        (if s = {} then 0 else (\<Prod>i<DIM('a). (interval_upperbound s)$$i - (interval_lowerbound s)$$i))"
   201 
   202 lemma interval_not_empty:"\<forall>i<DIM('a). a$$i \<le> b$$i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
   203   unfolding interval_eq_empty unfolding not_ex not_less by auto
   204 
   205 lemma content_closed_interval: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a$$i \<le> b$$i"
   206   shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
   207   using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto
   208 
   209 lemma content_closed_interval': fixes a::"'a::ordered_euclidean_space" assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i<DIM('a). b$$i - a$$i)"
   210   apply(rule content_closed_interval) using assms unfolding interval_ne_empty .
   211 
   212 lemma content_real:assumes "a\<le>b" shows "content {a..b} = b-a"
   213 proof- have *:"{..<Suc 0} = {0}" by auto
   214   show ?thesis unfolding content_def using assms by(auto simp: *)
   215 qed
   216 
   217 lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1" proof-
   218   have *:"\<forall>i<DIM('a). (0::'a)$$i \<le> (One::'a)$$i" by auto
   219   have "0 \<in> {0..One::'a}" unfolding mem_interval by auto
   220   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed
   221 
   222 lemma content_pos_le[intro]: fixes a::"'a::ordered_euclidean_space" shows "0 \<le> content {a..b}" proof(cases "{a..b}={}")
   223   case False hence *:"\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty by assumption
   224   have "(\<Prod>i<DIM('a). interval_upperbound {a..b} $$ i - interval_lowerbound {a..b} $$ i) \<ge> 0"
   225     apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto
   226   thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)
   227 
   228 lemma content_pos_lt: fixes a::"'a::ordered_euclidean_space" assumes "\<forall>i<DIM('a). a$$i < b$$i" shows "0 < content {a..b}"
   229 proof- have help_lemma1: "\<forall>i<DIM('a). a$$i < b$$i \<Longrightarrow> \<forall>i<DIM('a). a$$i \<le> ((b$$i)::real)" apply(rule,erule_tac x=i in allE) by auto
   230   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)
   231     using assms apply(erule_tac x=x in allE) by auto qed
   232 
   233 lemma content_eq_0: "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i<DIM('a). b$$i \<le> a$$i)" proof(cases "{a..b} = {}")
   234   case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-
   235     apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next
   236   case False note this[unfolded interval_eq_empty not_ex not_less]
   237   hence as:"\<forall>i<DIM('a). b $$ i \<ge> a $$ i" by fastsimp
   238   show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_lessThan]
   239     apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer
   240     apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed
   241 
   242 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto
   243 
   244 lemma content_closed_interval_cases:
   245   "content {a..b::'a::ordered_euclidean_space} = (if \<forall>i<DIM('a). a$$i \<le> b$$i then setprod (\<lambda>i. b$$i - a$$i) {..<DIM('a)} else 0)" apply(rule cond_cases) 
   246   apply(rule content_closed_interval) unfolding content_eq_0 not_all not_le defer apply(erule exE,rule_tac x=x in exI) by auto
   247 
   248 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
   249   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto
   250 
   251 (*lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
   252   unfolding content_eq_0 by auto*)
   253 
   254 lemma content_pos_lt_eq: "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i<DIM('a). a$$i < b$$i)"
   255   apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"
   256   hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i<DIM('a). a$$i < b$$i" unfolding content_eq_0 not_ex not_le by fastsimp qed
   257 
   258 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto
   259 
   260 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}" proof(cases "{a..b}={}")
   261   case True thus ?thesis using content_pos_le[of c d] by auto next
   262   case False hence ab_ne:"\<forall>i<DIM('a). a $$ i \<le> b $$ i" unfolding interval_ne_empty by auto
   263   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto
   264   have "{c..d} \<noteq> {}" using assms False by auto
   265   hence cd_ne:"\<forall>i<DIM('a). c $$ i \<le> d $$ i" using assms unfolding interval_ne_empty by auto
   266   show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
   267     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof
   268     fix i assume i:"i\<in>{..<DIM('a)}"
   269     show "0 \<le> b $$ i - a $$ i" using ab_ne[THEN spec[where x=i]] i by auto
   270     show "b $$ i - a $$ i \<le> d $$ i - c $$ i"
   271       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
   272       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] using i by auto qed qed
   273 
   274 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
   275   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastsimp
   276 
   277 subsection {* The notion of a gauge --- simply an open set containing the point. *}
   278 
   279 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"
   280 
   281 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"
   282   using assms unfolding gauge_def by auto
   283 
   284 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "open (d x)" using assms unfolding gauge_def by auto
   285 
   286 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   287   unfolding gauge_def by auto 
   288 
   289 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)" unfolding gauge_def by auto 
   290 
   291 lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)" apply(rule gauge_ball) by auto
   292 
   293 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. (d1 x) \<inter> (d2 x))"
   294   unfolding gauge_def by auto 
   295 
   296 lemma gauge_inters: assumes "finite s" "\<forall>d\<in>s. gauge (f d)" shows "gauge(\<lambda>x. \<Inter> {f d x | d. d \<in> s})" proof-
   297   have *:"\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s" by auto show ?thesis
   298   unfolding gauge_def unfolding * 
   299   using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto qed
   300 
   301 lemma gauge_existence_lemma: "(\<forall>x. \<exists>d::real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)" by(meson zero_less_one)
   302 
   303 subsection {* Divisions. *}
   304 
   305 definition division_of (infixl "division'_of" 40) where
   306   "s division_of i \<equiv>
   307         finite s \<and>
   308         (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
   309         (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
   310         (\<Union>s = i)"
   311 
   312 lemma division_ofD[dest]: assumes  "s division_of i"
   313   shows"finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   314   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" using assms unfolding division_of_def by auto
   315 
   316 lemma division_ofI:
   317   assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"
   318   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i"
   319   shows "s division_of i" using assms unfolding division_of_def by auto
   320 
   321 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   322   unfolding division_of_def by auto
   323 
   324 lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
   325   unfolding division_of_def by auto
   326 
   327 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}" unfolding division_of_def by auto 
   328 
   329 lemma division_of_sing[simp]: "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof
   330   assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 
   331     ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing by auto }
   332   ultimately show ?l unfolding division_of_def interval_sing by auto next
   333   assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing]]
   334   { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }
   335   moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing by auto qed
   336 
   337 lemma elementary_empty: obtains p where "p division_of {}"
   338   unfolding division_of_trivial by auto
   339 
   340 lemma elementary_interval: obtains p where  "p division_of {a..b}"
   341   by(metis division_of_trivial division_of_self)
   342 
   343 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   344   unfolding division_of_def by auto
   345 
   346 lemma forall_in_division:
   347  "d division_of i \<Longrightarrow> ((\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b}))"
   348   unfolding division_of_def by fastsimp
   349 
   350 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"
   351   apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]
   352   show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto
   353   { fix k assume "k \<in> q" hence kp:"k\<in>p" using assms(2) by auto show "k\<subseteq>\<Union>q" using `k \<in> q` by auto
   354   show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }
   355   fix k1 k2 assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2" hence *:"k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto
   356   show "interior k1 \<inter> interior k2 = {}" using as(5)[OF *] by auto qed auto
   357 
   358 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)" unfolding division_of_def by auto
   359 
   360 lemma division_of_content_0: assumes "content {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"
   361   unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])
   362   apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed
   363 
   364 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::('a::ordered_euclidean_space) set)"
   365   shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)" (is "?A' division_of _") proof-
   366 let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}" have *:"?A' = ?A" by auto
   367 show ?thesis unfolding * proof(rule division_ofI) have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)" by auto
   368   moreover have "finite (p1 \<times> p2)" using assms unfolding division_of_def by auto ultimately show "finite ?A" by auto
   369   have *:"\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s" by auto show "\<Union>?A = s1 \<inter> s2" apply(rule set_eqI) unfolding * and Union_image_eq UN_iff
   370     using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
   371   { fix k assume "k\<in>?A" then obtain k1 k2 where k:"k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto thus "k \<noteq> {}" by auto
   372   show "k \<subseteq> s1 \<inter> s2" using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)] unfolding k by auto
   373   guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this
   374   guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this
   375   show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_interval by auto } fix k1 k2
   376   assume "k1\<in>?A" then obtain x1 y1 where k1:"k1 = x1 \<inter> y1" "x1\<in>p1" "y1\<in>p2" "k1\<noteq>{}" by auto
   377   assume "k2\<in>?A" then obtain x2 y2 where k2:"k2 = x2 \<inter> y2" "x2\<in>p1" "y2\<in>p2" "k2\<noteq>{}" by auto
   378   assume "k1 \<noteq> k2" hence th:"x1\<noteq>x2 \<or> y1\<noteq>y2" unfolding k1 k2 by auto
   379   have *:"(interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {}) \<Longrightarrow>
   380       interior(x1 \<inter> y1) \<subseteq> interior(x1) \<Longrightarrow> interior(x1 \<inter> y1) \<subseteq> interior(y1) \<Longrightarrow>
   381       interior(x2 \<inter> y2) \<subseteq> interior(x2) \<Longrightarrow> interior(x2 \<inter> y2) \<subseteq> interior(y2)
   382       \<Longrightarrow> interior(x1 \<inter> y1) \<inter> interior(x2 \<inter> y2) = {}" by auto
   383   show "interior k1 \<inter> interior k2 = {}" unfolding k1 k2 apply(rule *) defer apply(rule_tac[1-4] subset_interior)
   384     using division_ofD(5)[OF assms(1) k1(2) k2(2)]
   385     using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed
   386 
   387 lemma division_inter_1: assumes "d division_of i" "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
   388   shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")
   389   case True show ?thesis unfolding True and division_of_trivial by auto next
   390   have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 
   391   case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed
   392 
   393 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::('a::ordered_euclidean_space) set)"
   394   shows "\<exists>p. p division_of (s \<inter> t)"
   395   by(rule,rule division_inter[OF assms])
   396 
   397 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
   398   shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)
   399 case (insert x f) show ?case proof(cases "f={}")
   400   case True thus ?thesis unfolding True using insert by auto next
   401   case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
   402   moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately
   403   show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto
   404 
   405 lemma division_disjoint_union:
   406   assumes "p1 division_of s1" "p2 division_of s2" "interior s1 \<inter> interior s2 = {}"
   407   shows "(p1 \<union> p2) division_of (s1 \<union> s2)" proof(rule division_ofI) 
   408   note d1 = division_ofD[OF assms(1)] and d2 = division_ofD[OF assms(2)]
   409   show "finite (p1 \<union> p2)" using d1(1) d2(1) by auto
   410   show "\<Union>(p1 \<union> p2) = s1 \<union> s2" using d1(6) d2(6) by auto
   411   { fix k1 k2 assume as:"k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2" moreover let ?g="interior k1 \<inter> interior k2 = {}"
   412   { assume as:"k1\<in>p1" "k2\<in>p2" have ?g using subset_interior[OF d1(2)[OF as(1)]] subset_interior[OF d2(2)[OF as(2)]]
   413       using assms(3) by blast } moreover
   414   { assume as:"k1\<in>p2" "k2\<in>p1" have ?g using subset_interior[OF d1(2)[OF as(2)]] subset_interior[OF d2(2)[OF as(1)]]
   415       using assms(3) by blast} ultimately
   416   show ?g using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto }
   417   fix k assume k:"k \<in> p1 \<union> p2"  show "k \<subseteq> s1 \<union> s2" using k d1(2) d2(2) by auto
   418   show "k \<noteq> {}" using k d1(3) d2(3) by auto show "\<exists>a b. k = {a..b}" using k d1(4) d2(4) by auto qed
   419 
   420 (* move *)
   421 lemma Eucl_nth_inverse[simp]: fixes x::"'a::euclidean_space" shows "(\<chi>\<chi> i. x $$ i) = x"
   422   apply(subst euclidean_eq) by auto
   423 
   424 lemma partial_division_extend_1:
   425   assumes "{c..d} \<subseteq> {a..b::'a::ordered_euclidean_space}" "{c..d} \<noteq> {}"
   426   obtains p where "p division_of {a..b}" "{c..d} \<in> p"
   427 proof- def n \<equiv> "DIM('a)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def using DIM_positive[where 'a='a] by auto
   428   guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_lessThan[of "DIM('a)"]] .. note \<pi>=this
   429   def \<pi>' \<equiv> "inv_into {1..n} \<pi>"
   430   have \<pi>':"bij_betw \<pi>' {..<DIM('a)} {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto
   431   hence \<pi>'i:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 
   432   have \<pi>i:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi> i <DIM('a)" using \<pi> unfolding bij_betw_def n_def by auto 
   433   have \<pi>\<pi>'[simp]:"\<And>i. i<DIM('a) \<Longrightarrow> \<pi> (\<pi>' i) = i" unfolding \<pi>'_def
   434     apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto
   435   have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq)
   436     using \<pi> unfolding n_def bij_betw_def by auto
   437   have "{c..d} \<noteq> {}" using assms by auto
   438   let ?p1 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else if \<pi>' i = l then c$$\<pi> l else b$$i)}"
   439   let ?p2 = "\<lambda>l. {(\<chi>\<chi> i. if \<pi>' i < l then c$$i else if \<pi>' i = l then d$$\<pi> l else a$$i)::'a .. (\<chi>\<chi> i. if \<pi>' i < l then d$$i else b$$i)}"
   440   let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"
   441   have abcd:"\<And>i. i<DIM('a) \<Longrightarrow> a $$ i \<le> c $$ i \<and> c$$i \<le> d$$i \<and> d $$ i \<le> b $$ i" using assms
   442     unfolding subset_interval interval_eq_empty by auto
   443   show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)
   444   proof- have "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < Suc n"
   445     proof(rule ccontr,unfold not_less) fix i assume i:"i<DIM('a)" and "Suc n \<le> \<pi>' i"
   446       hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' i unfolding bij_betw_def by auto
   447     qed hence "c = (\<chi>\<chi> i. if \<pi>' i < Suc n then c $$ i else a $$ i)"
   448         "d = (\<chi>\<chi> i. if \<pi>' i < Suc n then d $$ i else if \<pi>' i = n + 1 then c $$ \<pi> (n + 1) else b $$ i)"
   449       unfolding euclidean_eq[where 'a='a] using \<pi>' unfolding bij_betw_def by auto
   450     thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto
   451     have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}"  "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"
   452       unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)
   453     proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"
   454       then guess i unfolding mem_interval not_all not_imp .. note i=conjunctD2[OF this]
   455       show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)
   456         apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 
   457     qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"
   458     proof- fix x assume x:"x\<in>{a..b}"
   459       { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }
   460       let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $$ \<pi> i \<le> x $$ \<pi> i \<and> x $$ \<pi> i \<le> d $$ \<pi> i)}"
   461       assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all not_imp ..
   462       hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)
   463       hence M:"finite ?M" "?M \<noteq> {}" by auto
   464       def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]
   465         Min_gr_iff[OF M,unfolded l_def[symmetric]]
   466       have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le
   467         apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)
   468       proof- assume as:"x $$ \<pi> l < c $$ \<pi> l"
   469         show "x \<in> ?p1 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
   470         proof- case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal1 by auto
   471           thus ?case using as x[unfolded mem_interval,rule_format,of i]
   472             apply auto using l(3)[of "\<pi>' i"] using goal1 by(auto elim!:ballE[where x="\<pi>' i"])
   473         next case goal2 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using goal2 by auto
   474           thus ?case using as x[unfolded mem_interval,rule_format,of i]
   475             apply auto using l(3)[of "\<pi>' i"] using goal2 by(auto elim!:ballE[where x="\<pi>' i"])
   476         qed
   477       next assume as:"x $$ \<pi> l > d $$ \<pi> l"
   478         show "x \<in> ?p2 l" unfolding mem_interval apply safe unfolding euclidean_lambda_beta'
   479         proof- fix i assume i:"i<DIM('a)"
   480           have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le using i by auto
   481           thus "(if \<pi>' i < l then c $$ i else if \<pi>' i = l then d $$ \<pi> l else a $$ i) \<le> x $$ i"
   482             "x $$ i \<le> (if \<pi>' i < l then d $$ i else b $$ i)"
   483             using as x[unfolded mem_interval,rule_format,of i]
   484             apply auto using l(3)[of "\<pi>' i"] i by(auto elim!:ballE[where x="\<pi>' i"])
   485         qed qed
   486       thus "x \<in> \<Union>?p" using l(2) by blast 
   487     qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)
   488     
   489     show "finite ?p" by auto
   490     fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto
   491     show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 
   492     proof fix i x assume i:"i<DIM('a)" assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto
   493       ultimately show "a$$i \<le> x$$i" "x$$i \<le> b$$i" using abcd[of i] using l using i
   494         by(auto elim:disjE elim!:allE[where x=i] simp add:eucl_le[where 'a='a])
   495     qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)
   496     proof- case goal1 thus ?case using abcd[of x] by auto
   497     next   case goal2 thus ?case using abcd[of x] by auto
   498     qed thus "k \<noteq> {}" using k by auto
   499     show "\<exists>a b. k = {a..b}" using k by auto
   500     fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto
   501     { fix k k' l l'
   502       assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 
   503       assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 
   504       assume "l \<le> l'" fix x
   505       have "x \<notin> interior k \<inter> interior k'" 
   506       proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"
   507         case True hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l'" using \<pi>'i using l' by(auto simp add:less_Suc_eq_le)
   508         hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l' then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
   509         hence k':"k' = {c..d}" using l'(1) unfolding * by auto
   510         have ln:"l < n + 1" 
   511         proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto
   512           hence "\<And>i. i<DIM('a) \<Longrightarrow> \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)
   513           hence *:"\<And> P Q. (\<chi>\<chi> i. if \<pi>' i < l then P i else Q i) = ((\<chi>\<chi> i. P i)::'a)" apply-apply(subst euclidean_eq) by auto
   514           hence "k = {c..d}" using l(1) \<pi>'i unfolding * by(auto)
   515           thus False using `k\<noteq>k'` k' by auto
   516         qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto
   517         have "x $$ \<pi> l < c $$ \<pi> l \<or> d $$ \<pi> l < x $$ \<pi> l" using l(1) apply-
   518         proof(erule disjE)
   519           assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   520           show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] by(auto simp add:** not_less)
   521         next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   522           show ?thesis using *[of "\<pi> l"] using ln l(2) using \<pi>i[of l] unfolding ** by auto
   523         qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval
   524           by(auto elim!:allE[where x="\<pi> l"])
   525       next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto
   526         hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto
   527         note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]
   528         assume x:"x \<in> interior k \<inter> interior k'"
   529         show False using l(1) l'(1) apply-
   530         proof(erule_tac[!] disjE)+
   531           assume as:"k = ?p1 l" "k' = ?p1 l'"
   532           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   533           have "l \<noteq> l'" using k'(2)[unfolded as] by auto
   534           thus False using *[of "\<pi> l'"] *[of "\<pi> l"] ln using \<pi>i[OF ln(1)] \<pi>i[OF ln(2)] apply(cases "l<l'")
   535             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
   536         next assume as:"k = ?p2 l" "k' = ?p2 l'"
   537           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   538           have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto
   539           thus False using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
   540         next assume as:"k = ?p1 l" "k' = ?p2 l'"
   541           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   542           show False using abcd[of "\<pi> l'"] using *[of "\<pi> l"] *[of "\<pi> l'"]  `l \<le> l'` ln apply(cases "l=l'")
   543             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
   544         next assume as:"k = ?p2 l" "k' = ?p1 l'"
   545           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]
   546           show False using *[of "\<pi> l"] *[of "\<pi> l'"] ln `l \<le> l'` apply(cases "l=l'") using abcd[of "\<pi> l'"] 
   547             by(auto simp add:euclidean_lambda_beta' \<pi>l \<pi>i n_def)
   548         qed qed } 
   549     from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"
   550       apply - apply(cases "l' \<le> l") using k'(2) by auto            
   551     thus "interior k \<inter> interior k' = {}" by auto        
   552 qed qed
   553 
   554 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
   555   obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}" proof(cases "p = {}")
   556   case True guess q apply(rule elementary_interval[of a b]) .
   557   thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next
   558   case False note p = division_ofD[OF assms(1)]
   559   have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1
   560     guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this
   561     have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto
   562     guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed
   563   guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]
   564   have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" apply(rule,rule_tac p="q x" in division_of_subset) proof-
   565     fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)
   566       using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed
   567   hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)
   568     apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto
   569   then guess d .. note d = this
   570   show ?thesis apply(rule that[of "d \<union> p"]) proof-
   571     have *:"\<And>s f t. s \<noteq> {} \<Longrightarrow> (\<forall>i\<in>s. f i \<union> i = t) \<Longrightarrow> t = \<Inter> (f ` s) \<union> (\<Union>s)" by auto
   572     have *:"{a..b} = \<Inter> (\<lambda>i. \<Union>(q i - {i})) ` p \<union> \<Union>p" apply(rule *[OF False]) proof fix i assume i:"i\<in>p"
   573       show "\<Union>(q i - {i}) \<union> i = {a..b}" using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto qed
   574     show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])
   575       apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)
   576       fix k assume k:"k\<in>p" have *:"\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto
   577       show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])
   578         defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- note qk=division_ofD[OF q(1)[OF k]]
   579         show "finite (q k - {k})" "open (interior k)"  "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}" using qk by auto
   580         show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}" using qk(5) using q(2)[OF k] by auto
   581         have *:"\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<subseteq> interior (\<Union>(q k - {k}))"
   582           apply(rule subset_interior *)+ using k by auto qed qed qed auto qed
   583 
   584 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::('a::ordered_euclidean_space) set)"
   585   unfolding division_of_def by(metis bounded_Union bounded_interval) 
   586 
   587 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
   588   by(meson elementary_bounded bounded_subset_closed_interval)
   589 
   590 lemma division_union_intervals_exists: assumes "{a..b::'a::ordered_euclidean_space} \<noteq> {}"
   591   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")
   592   case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next
   593   case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")
   594   have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto
   595   case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)
   596     using false True assms using interior_subset by auto next
   597   case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto
   598   have *:"{u..v} \<subseteq> {c..d}" using uv by auto
   599   guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]
   600   have *:"{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s" using p(8) unfolding uv[THEN sym] by auto
   601   show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)
   602     apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer
   603     unfolding interior_inter[THEN sym] proof-
   604     have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
   605     have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 
   606       apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto
   607     also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto
   608     finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed
   609 
   610 lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"
   611   "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   612   shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+
   613   apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])
   614   using division_ofD[OF assms(2)] by auto
   615   
   616 lemma elementary_union_interval: assumes "p division_of \<Union>p"
   617   obtains q where "q division_of ({a..b::'a::ordered_euclidean_space} \<union> \<Union>p)" proof-
   618   note assm=division_ofD[OF assms]
   619   have lem1:"\<And>f s. \<Union>\<Union> (f ` s) = \<Union>(\<lambda>x.\<Union>(f x)) ` s" by auto
   620   have lem2:"\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f" by auto
   621 { presume "p={} \<Longrightarrow> thesis" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
   622     "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
   623   thus thesis by auto
   624 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .
   625   thus thesis apply(rule_tac that[of p]) unfolding as by auto 
   626 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto
   627 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"
   628   show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)
   629     unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  
   630     using assm(2-4) as apply- by(fastsimp dest: assm(5))+
   631 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"
   632   have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1
   633     from assm(4)[OF this] guess c .. then guess d ..
   634     thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto
   635   qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]
   636   let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
   637   show thesis apply(rule that[of "?D"]) proof(rule division_ofI)
   638     have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto
   639     show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto
   640     show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]
   641       using q(6) by auto
   642     fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto
   643     show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {a..b}" using q(4) k by auto
   644     fix k' assume k':"k'\<in>?D" "k\<noteq>k'"
   645     obtain x  where x: "k \<in>insert {a..b} (q x)"  "x\<in>p"  using k  by auto
   646     obtain x' where x':"k'\<in>insert {a..b} (q x')" "x'\<in>p" using k' by auto
   647     show "interior k \<inter> interior k' = {}" proof(cases "x=x'")
   648       case True show ?thesis apply(rule q(5)) using x x' k' unfolding True by auto
   649     next case False 
   650       { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 
   651         "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
   652         thus ?thesis by auto }
   653       { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }
   654       { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }
   655       assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"
   656       guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this
   657       have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto
   658       hence "interior k \<subseteq> interior x" apply-
   659         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover
   660       guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this
   661       have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto
   662       hence "interior k' \<subseteq> interior x'" apply-
   663         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto
   664       ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto
   665     qed qed } qed
   666 
   667 lemma elementary_unions_intervals:
   668   assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
   669   obtains p where "p division_of (\<Union>f)" proof-
   670   have "\<exists>p. p division_of (\<Union>f)" proof(induct_tac f rule:finite_subset_induct) 
   671     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   672     fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
   673     from this(3) guess p .. note p=this
   674     from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this
   675     have *:"\<Union>F = \<Union>p" using division_ofD[OF p] by auto
   676     show "\<exists>p. p division_of \<Union>insert x F" using elementary_union_interval[OF p[unfolded *], of a b]
   677       unfolding Union_insert ab * by auto
   678   qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed
   679 
   680 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::('a::ordered_euclidean_space) set)"
   681   obtains p where "p division_of (s \<union> t)"
   682 proof- have "s \<union> t = \<Union>ps \<union> \<Union>pt" using assms unfolding division_of_def by auto
   683   hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto
   684   show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])
   685     unfolding * prefer 3 apply(rule_tac p=p in that)
   686     using assms[unfolded division_of_def] by auto qed
   687 
   688 lemma partial_division_extend: fixes t::"('a::ordered_euclidean_space) set"
   689   assumes "p division_of s" "q division_of t" "s \<subseteq> t"
   690   obtains r where "p \<subseteq> r" "r division_of t" proof-
   691   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   692   obtain a b where ab:"t\<subseteq>{a..b}" using elementary_subset_interval[OF assms(2)] by auto
   693   guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])
   694     apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]
   695   guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 
   696   then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 
   697     apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto
   698   { fix x assume x:"x\<in>t" "x\<notin>s"
   699     hence "x\<in>\<Union>r1" unfolding r1 using ab by auto
   700     then guess r unfolding Union_iff .. note r=this moreover
   701     have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto
   702       thus False using x by auto qed
   703     ultimately have "x\<in>\<Union>(r1 - p)" by auto }
   704   hence *:"t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)" unfolding divp divq using assms(3) by auto
   705   show ?thesis apply(rule that[of "p \<union> r2"]) unfolding * defer apply(rule division_disjoint_union)
   706     unfolding divp(6) apply(rule assms r2)+
   707   proof- have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
   708     proof(rule inter_interior_unions_intervals)
   709       show "finite (r1 - p)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}" using r1 by auto
   710       have *:"\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}" by auto
   711       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}" proof(rule)
   712         fix m x assume as:"m\<in>r1-p"
   713         have "interior m \<inter> interior (\<Union>p) = {}" proof(rule inter_interior_unions_intervals)
   714           show "finite p" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto
   715           show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto
   716         qed thus "interior s \<inter> interior m = {}" unfolding divp by auto
   717       qed qed        
   718     thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto
   719   qed auto qed
   720 
   721 subsection {* Tagged (partial) divisions. *}
   722 
   723 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where
   724   "(s tagged_partial_division_of i) \<equiv>
   725         finite s \<and>
   726         (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   727         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ((x1,k1) \<noteq> (x2,k2))
   728                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {}))"
   729 
   730 lemma tagged_partial_division_ofD[dest]: assumes "s tagged_partial_division_of i"
   731   shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
   732   "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   733   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> (x1,k1) \<noteq> (x2,k2) \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
   734   using assms unfolding tagged_partial_division_of_def  apply- by blast+ 
   735 
   736 definition tagged_division_of (infixr "tagged'_division'_of" 40) where
   737   "(s tagged_division_of i) \<equiv>
   738         (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   739 
   740 lemma tagged_division_of_finite[dest]: "s tagged_division_of i \<Longrightarrow> finite s"
   741   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   742 
   743 lemma tagged_division_of:
   744  "(s tagged_division_of i) \<longleftrightarrow>
   745         finite s \<and>
   746         (\<forall>x k. (x,k) \<in> s
   747                \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
   748         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (x2,k2))
   749                        \<longrightarrow> (interior(k1) \<inter> interior(k2) = {})) \<and>
   750         (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   751   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
   752 
   753 lemma tagged_division_ofI: assumes
   754   "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   755   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   756   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
   757   shows "s tagged_division_of i"
   758   unfolding tagged_division_of apply(rule) defer apply rule
   759   apply(rule allI impI conjI assms)+ apply assumption
   760   apply(rule, rule assms, assumption) apply(rule assms, assumption)
   761   using assms(1,5-) apply- by blast+
   762 
   763 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"
   764   shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
   765   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"
   766   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- by blast+
   767 
   768 lemma division_of_tagged_division: assumes"s tagged_division_of i"  shows "(snd ` s) division_of i"
   769 proof(rule division_ofI) note assm=tagged_division_ofD[OF assms]
   770   show "\<Union>snd ` s = i" "finite (snd ` s)" using assm by auto
   771   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   772   thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+
   773   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   774   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   775 qed
   776 
   777 lemma partial_division_of_tagged_division: assumes "s tagged_partial_division_of i"
   778   shows "(snd ` s) division_of \<Union>(snd ` s)"
   779 proof(rule division_ofI) note assm=tagged_partial_division_ofD[OF assms]
   780   show "finite (snd ` s)" "\<Union>snd ` s = \<Union>snd ` s" using assm by auto
   781   fix k assume k:"k \<in> snd ` s" then obtain xk where xk:"(xk, k) \<in> s" by auto
   782   thus "k\<noteq>{}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>snd ` s" using assm by auto
   783   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto
   784   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto
   785 qed
   786 
   787 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "t \<subseteq> s"
   788   shows "t tagged_partial_division_of i"
   789   using assms unfolding tagged_partial_division_of_def using finite_subset[OF assms(2)] by blast
   790 
   791 lemma setsum_over_tagged_division_lemma: fixes d::"('m::ordered_euclidean_space) set \<Rightarrow> 'a::real_normed_vector"
   792   assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
   793   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
   794 proof- note assm=tagged_division_ofD[OF assms(1)]
   795   have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto
   796   show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)
   797     show "finite p" using assm by auto
   798     fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 
   799     obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto
   800     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto
   801     hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 
   802     hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto
   803     hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto
   804     thus "d (snd x) = 0" unfolding ab by auto qed qed
   805 
   806 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x,k) \<in> p \<Longrightarrow> x \<in> i" by auto
   807 
   808 lemma tagged_division_of_empty: "{} tagged_division_of {}"
   809   unfolding tagged_division_of by auto
   810 
   811 lemma tagged_partial_division_of_trivial[simp]:
   812  "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   813   unfolding tagged_partial_division_of_def by auto
   814 
   815 lemma tagged_division_of_trivial[simp]:
   816  "p tagged_division_of {} \<longleftrightarrow> p = {}"
   817   unfolding tagged_division_of by auto
   818 
   819 lemma tagged_division_of_self:
   820  "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
   821   apply(rule tagged_division_ofI) by auto
   822 
   823 lemma tagged_division_union:
   824   assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "interior s1 \<inter> interior s2 = {}"
   825   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
   826 proof(rule tagged_division_ofI) note p1=tagged_division_ofD[OF assms(1)] and p2=tagged_division_ofD[OF assms(2)]
   827   show "finite (p1 \<union> p2)" using p1(1) p2(1) by auto
   828   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2" using p1(6) p2(6) by blast
   829   fix x k assume xk:"(x,k)\<in>p1\<union>p2" show "x\<in>k" "\<exists>a b. k = {a..b}" using xk p1(2,4) p2(2,4) by auto
   830   show "k\<subseteq>s1\<union>s2" using xk p1(3) p2(3) by blast
   831   fix x' k' assume xk':"(x',k')\<in>p1\<union>p2" "(x,k) \<noteq> (x',k')"
   832   have *:"\<And>a b. a\<subseteq> s1 \<Longrightarrow> b\<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}" using assms(3) subset_interior by blast
   833   show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")
   834     apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))
   835     using p1(3) p2(3) using xk xk' by auto qed 
   836 
   837 lemma tagged_division_unions:
   838   assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"
   839   "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})"
   840   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
   841 proof(rule tagged_division_ofI)
   842   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
   843   show "finite (\<Union>pfn ` iset)" apply(rule finite_Union) using assms by auto
   844   have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>(\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset" by blast 
   845   also have "\<dots> = \<Union>iset" using assm(6) by auto
   846   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>pfn ` iset} = \<Union>iset" . 
   847   fix x k assume xk:"(x,k)\<in>\<Union>pfn ` iset" then obtain i where i:"i \<in> iset" "(x, k) \<in> pfn i" by auto
   848   show "x\<in>k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset" using assm(2-4)[OF i] using i(1) by auto
   849   fix x' k' assume xk':"(x',k')\<in>\<Union>pfn ` iset" "(x, k) \<noteq> (x', k')" then obtain i' where i':"i' \<in> iset" "(x', k') \<in> pfn i'" by auto
   850   have *:"\<And>a b. i\<noteq>i' \<Longrightarrow> a\<subseteq> i \<Longrightarrow> b\<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}" using i(1) i'(1)
   851     using assms(3)[rule_format] subset_interior by blast
   852   show "interior k \<inter> interior k' = {}" apply(cases "i=i'")
   853     using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) by auto
   854 qed
   855 
   856 lemma tagged_partial_division_of_union_self:
   857   assumes "p tagged_partial_division_of s" shows "p tagged_division_of (\<Union>(snd ` p))"
   858   apply(rule tagged_division_ofI) using tagged_partial_division_ofD[OF assms] by auto
   859 
   860 lemma tagged_division_of_union_self: assumes "p tagged_division_of s"
   861   shows "p tagged_division_of (\<Union>(snd ` p))"
   862   apply(rule tagged_division_ofI) using tagged_division_ofD[OF assms] by auto
   863 
   864 subsection {* Fine-ness of a partition w.r.t. a gauge. *}
   865 
   866 definition fine (infixr "fine" 46) where
   867   "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d(x))"
   868 
   869 lemma fineI: assumes "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   870   shows "d fine s" using assms unfolding fine_def by auto
   871 
   872 lemma fineD[dest]: assumes "d fine s"
   873   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x" using assms unfolding fine_def by auto
   874 
   875 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   876   unfolding fine_def by auto
   877 
   878 lemma fine_inters:
   879  "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   880   unfolding fine_def by blast
   881 
   882 lemma fine_union:
   883   "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   884   unfolding fine_def by blast
   885 
   886 lemma fine_unions:"(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   887   unfolding fine_def by auto
   888 
   889 lemma fine_subset:  "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   890   unfolding fine_def by blast
   891 
   892 subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
   893 
   894 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where
   895   "(f has_integral_compact_interval y) i \<equiv>
   896         (\<forall>e>0. \<exists>d. gauge d \<and>
   897           (\<forall>p. p tagged_division_of i \<and> d fine p
   898                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
   899 
   900 definition has_integral (infixr "has'_integral" 46) where 
   901 "((f::('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>
   902         if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i
   903         else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   904               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
   905                                        norm(z - y) < e))"
   906 
   907 lemma has_integral:
   908  "(f has_integral y) ({a..b}) \<longleftrightarrow>
   909         (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
   910                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
   911   unfolding has_integral_def has_integral_compact_interval_def by auto
   912 
   913 lemma has_integralD[dest]: assumes
   914  "(f has_integral y) ({a..b})" "e>0"
   915   obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p
   916                         \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
   917   using assms unfolding has_integral by auto
   918 
   919 lemma has_integral_alt:
   920  "(f has_integral y) i \<longleftrightarrow>
   921       (if (\<exists>a b. i = {a..b}) then (f has_integral y) i
   922        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
   923                                \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)
   924                                         has_integral z) ({a..b}) \<and>
   925                                        norm(z - y) < e)))"
   926   unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   927 
   928 lemma has_integral_altD:
   929   assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"
   930   obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
   931   using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto
   932 
   933 definition integrable_on (infixr "integrable'_on" 46) where
   934   "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"
   935 
   936 definition "integral i f \<equiv> SOME y. (f has_integral y) i"
   937 
   938 lemma integrable_integral[dest]:
   939  "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   940   unfolding integrable_on_def integral_def by(rule someI_ex)
   941 
   942 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   943   unfolding integrable_on_def by auto
   944 
   945 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   946   by auto
   947 
   948 lemma setsum_content_null:
   949   assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"
   950   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
   951 proof(rule setsum_0',rule) fix y assume y:"y\<in>p"
   952   obtain x k where xk:"y = (x,k)" using surj_pair[of y] by blast
   953   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
   954   from this(2) guess c .. then guess d .. note c_d=this
   955   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x" unfolding xk by auto
   956   also have "\<dots> = 0" using content_subset[OF assm(1)[unfolded c_d]] content_pos_le[of c d]
   957     unfolding assms(1) c_d by auto
   958   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
   959 qed
   960 
   961 subsection {* Some basic combining lemmas. *}
   962 
   963 lemma tagged_division_unions_exists:
   964   assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"
   965   "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"
   966    obtains p where "p tagged_division_of i" "d fine p"
   967 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]
   968   show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]
   969     apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 
   970     apply(rule fine_unions) using pfn by auto
   971 qed
   972 
   973 subsection {* The set we're concerned with must be closed. *}
   974 
   975 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::('n::ordered_euclidean_space) set)"
   976   unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)
   977 
   978 subsection {* General bisection principle for intervals; might be useful elsewhere. *}
   979 
   980 lemma interval_bisection_step:  fixes type::"'a::ordered_euclidean_space"
   981   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::'a})"
   982   obtains c d where "~(P{c..d})"
   983   "\<forall>i<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$i"
   984 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto
   985   note ab=this[unfolded interval_eq_empty not_ex not_less]
   986   { fix f have "finite f \<Longrightarrow>
   987         (\<forall>s\<in>f. P s) \<Longrightarrow>
   988         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>
   989         (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<Longrightarrow> P(\<Union>f)"
   990     proof(induct f rule:finite_induct)
   991       case empty show ?case using assms(1) by auto
   992     next case (insert x f) show ?case unfolding Union_insert apply(rule assms(2)[rule_format])
   993         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)
   994         using insert by auto
   995     qed } note * = this
   996   let ?A = "{{c..d} | c d::'a. \<forall>i<DIM('a). (c$$i = a$$i) \<and> (d$$i = (a$$i + b$$i) / 2) \<or> (c$$i = (a$$i + b$$i) / 2) \<and> (d$$i = b$$i)}"
   997   let ?PP = "\<lambda>c d. \<forall>i<DIM('a). a$$i \<le> c$$i \<and> c$$i \<le> d$$i \<and> d$$i \<le> b$$i \<and> 2 * (d$$i - c$$i) \<le> b$$i - a$$i"
   998   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
   999     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }
  1000   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
  1001   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 
  1002     let ?B = "(\<lambda>s.{(\<chi>\<chi> i. if i \<in> s then a$$i else (a$$i + b$$i) / 2)::'a ..
  1003       (\<chi>\<chi> i. if i \<in> s then (a$$i + b$$i) / 2 else b$$i)}) ` {s. s \<subseteq> {..<DIM('a)}}"
  1004     have "?A \<subseteq> ?B" proof case goal1
  1005       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]
  1006       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto
  1007       show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. i<DIM('a) \<and> c$$i = a$$i}" in bexI)
  1008         unfolding c_d apply(rule * ) unfolding euclidean_eq[where 'a='a] apply safe unfolding euclidean_lambda_beta' mem_Collect_eq
  1009       proof- fix i assume "i<DIM('a)" thus " c $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then a $$ i else (a $$ i + b $$ i) / 2)"
  1010           "d $$ i = (if i < DIM('a) \<and> c $$ i = a $$ i then (a $$ i + b $$ i) / 2 else b $$ i)"
  1011           using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)
  1012       qed qed
  1013     thus "finite ?A" apply(rule finite_subset) by auto
  1014     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)
  1015     note c_d=this[rule_format]
  1016     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 thus ?case 
  1017         using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed
  1018     show "\<exists>a b. s = {a..b}" unfolding c_d by auto
  1019     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)
  1020     note e_f=this[rule_format]
  1021     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto
  1022     then obtain i where "c$$i \<noteq> e$$i \<or> d$$i \<noteq> f$$i" and i':"i<DIM('a)" unfolding de_Morgan_conj euclidean_eq[where 'a='a] by auto
  1023     hence i:"c$$i \<noteq> e$$i" "d$$i \<noteq> f$$i" apply- apply(erule_tac[!] disjE)
  1024     proof- assume "c$$i \<noteq> e$$i" thus "d$$i \<noteq> f$$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
  1025     next   assume "d$$i \<noteq> f$$i" thus "c$$i \<noteq> e$$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp
  1026     qed have *:"\<And>s t. (\<And>a. a\<in>s \<Longrightarrow> a\<in>t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}" by auto
  1027     show "interior s \<inter> interior t = {}" unfolding e_f c_d interior_closed_interval proof(rule *)
  1028       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"
  1029       hence x:"c$$i < d$$i" "e$$i < f$$i" "c$$i < f$$i" "e$$i < d$$i" unfolding mem_interval using i'
  1030         apply-apply(erule_tac[!] x=i in allE)+ by auto
  1031       show False using c_d(2)[OF i'] apply- apply(erule_tac disjE)
  1032       proof(erule_tac[!] conjE) assume as:"c $$ i = a $$ i" "d $$ i = (a $$ i + b $$ i) / 2"
  1033         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
  1034       next assume as:"c $$ i = (a $$ i + b $$ i) / 2" "d $$ i = b $$ i"
  1035         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)
  1036       qed qed qed
  1037   also have "\<Union> ?A = {a..b}" proof(rule set_eqI,rule)
  1038     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..
  1039     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..
  1040     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]
  1041     show "x\<in>{a..b}" unfolding mem_interval proof safe
  1042       fix i assume "i<DIM('a)" thus "a $$ i \<le> x $$ i" "x $$ i \<le> b $$ i"
  1043         using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed
  1044   next fix x assume x:"x\<in>{a..b}"
  1045     have "\<forall>i<DIM('a). \<exists>c d. (c = a$$i \<and> d = (a$$i + b$$i) / 2 \<or> c = (a$$i + b$$i) / 2 \<and> d = b$$i) \<and> c\<le>x$$i \<and> x$$i \<le> d"
  1046       (is "\<forall>i<DIM('a). \<exists>c d. ?P i c d") unfolding mem_interval proof(rule,rule) fix i
  1047       have "?P i (a$$i) ((a $$ i + b $$ i) / 2) \<or> ?P i ((a $$ i + b $$ i) / 2) (b$$i)"
  1048         using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast
  1049     qed thus "x\<in>\<Union>?A" unfolding Union_iff unfolding lambda_skolem' unfolding Bex_def mem_Collect_eq
  1050       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto
  1051   qed finally show False using assms by auto qed
  1052 
  1053 lemma interval_bisection: fixes type::"'a::ordered_euclidean_space"
  1054   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::'a}"
  1055   obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"
  1056 proof-
  1057   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
  1058     (\<forall>i<DIM('a). fst x$$i \<le> fst y$$i \<and> fst y$$i \<le> snd y$$i \<and> snd y$$i \<le> snd x$$i \<and>
  1059                            2 * (snd y$$i - fst y$$i) \<le> snd x$$i - fst x$$i))" proof case goal1 thus ?case proof-
  1060       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
  1061       thus ?thesis apply(cases "P {fst x..snd x}") by auto
  1062     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 
  1063       thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto
  1064     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this
  1065   def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def
  1066   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
  1067     (\<forall>i<DIM('a). A(n)$$i \<le> A(Suc n)$$i \<and> A(Suc n)$$i \<le> B(Suc n)$$i \<and> B(Suc n)$$i \<le> B(n)$$i \<and> 
  1068     2 * (B(Suc n)$$i - A(Suc n)$$i) \<le> B(n)$$i - A(n)$$i)" (is "\<And>n. ?P n")
  1069   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto
  1070     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case
  1071     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto
  1072     next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto
  1073     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]
  1074 
  1075   have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
  1076   proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$$i - a$$i) {..<DIM('a)}) / e"] .. note n=this
  1077     show ?case apply(rule_tac x=n in exI) proof(rule,rule)
  1078       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"
  1079       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$$i)) {..<DIM('a)}" unfolding dist_norm by(rule norm_le_l1)
  1080       also have "\<dots> \<le> setsum (\<lambda>i. B n$$i - A n$$i) {..<DIM('a)}"
  1081       proof(rule setsum_mono) fix i show "\<bar>(x - y) $$ i\<bar> \<le> B n $$ i - A n $$ i"
  1082           using xy[unfolded mem_interval,THEN spec[where x=i]] by auto qed
  1083       also have "\<dots> \<le> setsum (\<lambda>i. b$$i - a$$i) {..<DIM('a)} / 2^n" unfolding setsum_divide_distrib
  1084       proof(rule setsum_mono) case goal1 thus ?case
  1085         proof(induct n) case 0 thus ?case unfolding AB by auto
  1086         next case (Suc n) have "B (Suc n) $$ i - A (Suc n) $$ i \<le> (B n $$ i - A n $$ i) / 2"
  1087             using AB(4)[of i n] using goal1 by auto
  1088           also have "\<dots> \<le> (b $$ i - a $$ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .
  1089         qed qed
  1090       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .
  1091     qed qed
  1092   { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this
  1093     have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 
  1094     proof(induct d) case 0 thus ?case by auto
  1095     next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])
  1096         apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)
  1097       proof- case goal1 thus ?case using AB(4)[of i "m + d"] by(auto simp add:field_simps)
  1098       qed qed } note ABsubset = this 
  1099   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
  1100   proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto
  1101   then guess x0 .. note x0=this[rule_format]
  1102   show thesis proof(rule that[rule_format,of x0])
  1103     show "x0\<in>{a..b}" using x0[of 0] unfolding AB .
  1104     fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this
  1105     show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
  1106       apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 
  1107     proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto
  1108       show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto
  1109       show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto
  1110     qed qed qed 
  1111 
  1112 subsection {* Cousin's lemma. *}
  1113 
  1114 lemma fine_division_exists: assumes "gauge g" 
  1115   obtains p where "p tagged_division_of {a..b::'a::ordered_euclidean_space}" "g fine p"
  1116 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
  1117   then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto
  1118 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
  1119   guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
  1120     apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+
  1121   proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto
  1122     fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"
  1123     thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule
  1124       apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto
  1125   qed note x=this
  1126   obtain e where e:"e>0" "ball x e \<subseteq> g x" using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
  1127   from x(2)[OF e(1)] guess c d apply-apply(erule exE conjE)+ . note c_d = this
  1128   have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto
  1129   thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed
  1130 
  1131 subsection {* Basic theorems about integrals. *}
  1132 
  1133 lemma has_integral_unique: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
  1134   assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"
  1135 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto
  1136   have lem:"\<And>f::'n \<Rightarrow> 'a.  \<And> a b k1 k2.
  1137     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
  1138   proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto
  1139     guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this
  1140     guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this
  1141     guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
  1142     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
  1143       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:algebra_simps norm_minus_commute)
  1144     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
  1145       apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto
  1146     finally show False by auto
  1147   qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
  1148     thus False apply-apply(cases "\<exists>a b. i = {a..b}")
  1149       using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }
  1150   assume as:"\<not> (\<exists>a b. i = {a..b})"
  1151   guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
  1152   guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
  1153   have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)
  1154     using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+
  1155   note ab=conjunctD2[OF this[unfolded Un_subset_iff]]
  1156   guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]
  1157   guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]
  1158   have "z = w" using lem[OF w(1) z(1)] by auto
  1159   hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
  1160     using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 
  1161   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))
  1162   finally show False by auto qed
  1163 
  1164 lemma integral_unique[intro]:
  1165   "(f has_integral y) k \<Longrightarrow> integral k f = y"
  1166   unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 
  1167 
  1168 lemma has_integral_is_0: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector" 
  1169   assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"
  1170 proof- have lem:"\<And>a b. \<And>f::'n \<Rightarrow> 'a.
  1171     (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral
  1172   proof(rule,rule) fix a b e and f::"'n \<Rightarrow> 'a"
  1173     assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
  1174     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
  1175       apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)
  1176     proof(rule,rule,erule conjE) case goal1
  1177       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)
  1178         fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
  1179         thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto
  1180       qed thus ?case using as by auto
  1181     qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1182     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")
  1183       using assms by(auto simp add:has_integral intro:lem) }
  1184   have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto
  1185   assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *
  1186   apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)
  1187   proof- fix e::real and a b assume "e>0"
  1188     thus "\<exists>z. ((\<lambda>x::'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"
  1189       apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto
  1190   qed auto qed
  1191 
  1192 lemma has_integral_0[simp]: "((\<lambda>x::'n::ordered_euclidean_space. 0) has_integral 0) s"
  1193   apply(rule has_integral_is_0) by auto 
  1194 
  1195 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
  1196   using has_integral_unique[OF has_integral_0] by auto
  1197 
  1198 lemma has_integral_linear: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
  1199   assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"
  1200 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  1201   have lem:"\<And>f::'n \<Rightarrow> 'a. \<And> y a b.
  1202     (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"
  1203   proof(subst has_integral,rule,rule) case goal1
  1204     from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]
  1205     have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto
  1206     guess g using has_integralD[OF goal1(1) *] . note g=this
  1207     show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))
  1208     proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 
  1209       have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto
  1210       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
  1211         unfolding o_def unfolding scaleR[THEN sym] * by simp
  1212       also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
  1213       finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
  1214       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]
  1215         apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)
  1216     qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1217     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  1218   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  1219   proof(rule,rule) fix e::real  assume e:"0<e"
  1220     have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))
  1221     guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this
  1222     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"
  1223       apply(rule_tac x=M in exI) apply(rule,rule M(1))
  1224     proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]
  1225       have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
  1226         unfolding o_def apply(rule ext) using zero by auto
  1227       show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]
  1228         apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)
  1229     qed qed qed
  1230 
  1231 lemma has_integral_cmul:
  1232   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
  1233   unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)
  1234   by(rule scaleR.bounded_linear_right)
  1235 
  1236 lemma has_integral_neg:
  1237   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"
  1238   apply(drule_tac c="-1" in has_integral_cmul) by auto
  1239 
  1240 lemma has_integral_add: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector" 
  1241   assumes "(f has_integral k) s" "(g has_integral l) s"
  1242   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
  1243 proof- have lem:"\<And>f g::'n \<Rightarrow> 'a. \<And>a b k l.
  1244     (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>
  1245      ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1
  1246     show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto
  1247       guess d1 using has_integralD[OF goal1(1) *] . note d1=this
  1248       guess d2 using has_integralD[OF goal1(2) *] . note d2=this
  1249       show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
  1250         apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])
  1251       proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
  1252         have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
  1253           unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]
  1254           by(rule setsum_cong2,auto)
  1255         have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"
  1256           unfolding * by(auto simp add:algebra_simps) also let ?res = "\<dots>"
  1257         from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto
  1258         have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])
  1259           apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto
  1260         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto
  1261       qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"
  1262     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }
  1263   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P
  1264   proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto
  1265     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]
  1266     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]
  1267     show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)
  1268     proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::'n}"
  1269       hence *:"ball 0 B1 \<subseteq> {a..b::'n}" "ball 0 B2 \<subseteq> {a..b::'n}" by auto
  1270       guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]
  1271       guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]
  1272       have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto
  1273       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"
  1274         apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])
  1275         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)
  1276     qed qed qed
  1277 
  1278 lemma has_integral_sub:
  1279   shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"
  1280   using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding algebra_simps by auto
  1281 
  1282 lemma integral_0: "integral s (\<lambda>x::'n::ordered_euclidean_space. 0::'m::real_normed_vector) = 0"
  1283   by(rule integral_unique has_integral_0)+
  1284 
  1285 lemma integral_add:
  1286   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
  1287    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
  1288   apply(rule integral_unique) apply(drule integrable_integral)+
  1289   apply(rule has_integral_add) by assumption+
  1290 
  1291 lemma integral_cmul:
  1292   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
  1293   apply(rule integral_unique) apply(drule integrable_integral)+
  1294   apply(rule has_integral_cmul) by assumption+
  1295 
  1296 lemma integral_neg:
  1297   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"
  1298   apply(rule integral_unique) apply(drule integrable_integral)+
  1299   apply(rule has_integral_neg) by assumption+
  1300 
  1301 lemma integral_sub:
  1302   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
  1303   apply(rule integral_unique) apply(drule integrable_integral)+
  1304   apply(rule has_integral_sub) by assumption+
  1305 
  1306 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
  1307   unfolding integrable_on_def using has_integral_0 by auto
  1308 
  1309 lemma integrable_add:
  1310   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
  1311   unfolding integrable_on_def by(auto intro: has_integral_add)
  1312 
  1313 lemma integrable_cmul:
  1314   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
  1315   unfolding integrable_on_def by(auto intro: has_integral_cmul)
  1316 
  1317 lemma integrable_neg:
  1318   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
  1319   unfolding integrable_on_def by(auto intro: has_integral_neg)
  1320 
  1321 lemma integrable_sub:
  1322   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
  1323   unfolding integrable_on_def by(auto intro: has_integral_sub)
  1324 
  1325 lemma integrable_linear:
  1326   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"
  1327   unfolding integrable_on_def by(auto intro: has_integral_linear)
  1328 
  1329 lemma integral_linear:
  1330   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"
  1331   apply(rule has_integral_unique) defer unfolding has_integral_integral 
  1332   apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]
  1333   apply(rule integrable_linear) by assumption+
  1334 
  1335 lemma integral_component_eq[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
  1336   assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $$ k) = integral s f $$ k"
  1337   unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
  1338 
  1339 lemma has_integral_setsum:
  1340   assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
  1341   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
  1342 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)
  1343   case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]
  1344     apply(rule has_integral_add) using insert assms by auto
  1345 qed auto
  1346 
  1347 lemma integral_setsum:
  1348   shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>
  1349   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
  1350   apply(rule integral_unique) apply(rule has_integral_setsum)
  1351   using integrable_integral by auto
  1352 
  1353 lemma integrable_setsum:
  1354   shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
  1355   unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto
  1356 
  1357 lemma has_integral_eq:
  1358   assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"
  1359   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
  1360   using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto
  1361 
  1362 lemma integrable_eq:
  1363   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
  1364   unfolding integrable_on_def using has_integral_eq[of s f g] by auto
  1365 
  1366 lemma has_integral_eq_eq:
  1367   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"
  1368   using has_integral_eq[of s f g] has_integral_eq[of s g f] by rule auto
  1369 
  1370 lemma has_integral_null[dest]:
  1371   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"
  1372   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer
  1373 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto
  1374   fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)
  1375   have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right
  1376     using setsum_content_null[OF assms(1) p, of f] . 
  1377   thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed
  1378 
  1379 lemma has_integral_null_eq[simp]:
  1380   shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"
  1381   apply rule apply(rule has_integral_unique,assumption) 
  1382   apply(drule has_integral_null,assumption)
  1383   apply(drule has_integral_null) by auto
  1384 
  1385 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"
  1386   by(rule integral_unique,drule has_integral_null)
  1387 
  1388 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"
  1389   unfolding integrable_on_def apply(drule has_integral_null) by auto
  1390 
  1391 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"
  1392   unfolding empty_as_interval apply(rule has_integral_null) 
  1393   using content_empty unfolding empty_as_interval .
  1394 
  1395 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"
  1396   apply(rule,rule has_integral_unique,assumption) by auto
  1397 
  1398 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto
  1399 
  1400 lemma integral_empty[simp]: shows "integral {} f = 0"
  1401   apply(rule integral_unique) using has_integral_empty .
  1402 
  1403 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}" "(f has_integral 0) {a::'a::ordered_euclidean_space}"
  1404 proof- have *:"{a} = {a..a}" apply(rule set_eqI) unfolding mem_interval singleton_iff euclidean_eq[where 'a='a]
  1405     apply safe prefer 3 apply(erule_tac x=i in allE) by(auto simp add: field_simps)
  1406   show "(f has_integral 0) {a..a}" "(f has_integral 0) {a}" unfolding *
  1407     apply(rule_tac[!] has_integral_null) unfolding content_eq_0_interior
  1408     unfolding interior_closed_interval using interval_sing by auto qed
  1409 
  1410 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto
  1411 
  1412 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto
  1413 
  1414 subsection {* Cauchy-type criterion for integrability. *}
  1415 
  1416 (* XXXXXXX *)
  1417 lemma integrable_cauchy: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}" 
  1418   shows "f integrable_on {a..b} \<longleftrightarrow>
  1419   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>
  1420                             p2 tagged_division_of {a..b} \<and> d fine p2
  1421                             \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
  1422                                      setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
  1423 proof assume ?l
  1424   then guess y unfolding integrable_on_def has_integral .. note y=this
  1425   show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto
  1426     then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]
  1427     show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)
  1428     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"
  1429       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1430         apply(rule dist_triangle_half_l[where y=y,unfolded dist_norm])
  1431         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
  1432     qed qed
  1433 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto
  1434   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  1435   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto
  1436   hence "\<forall>n. \<exists>p. p tagged_division_of {a..b} \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p" apply-
  1437   proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed
  1438   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
  1439   have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto
  1440   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
  1441   proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this
  1442     show ?case apply(rule_tac x=N in exI)
  1443     proof(rule,rule,rule,rule) fix m n assume mn:"N \<le> m" "N \<le> n" have *:"N = (N - 1) + 1" using N by auto
  1444       show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"
  1445         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))
  1446         using dp p(1) using mn by auto 
  1447     qed qed
  1448   then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]
  1449   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)
  1450   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto
  1451     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto
  1452     guess N2 using y[OF *] .. note N2=this
  1453     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
  1454       apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 
  1455     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto
  1456       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"
  1457       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto
  1458       show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)
  1459         apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer
  1460         using N2[rule_format,unfolded dist_norm,of "N1+N2"]
  1461         using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed
  1462 
  1463 subsection {* Additivity of integral on abutting intervals. *}
  1464 
  1465 lemma interval_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
  1466   "{a..b} \<inter> {x. x$$k \<le> c} = {a .. (\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)}"
  1467   "{a..b} \<inter> {x. x$$k \<ge> c} = {(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i) .. b}"
  1468   apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval mem_Collect_eq using assms by auto
  1469 
  1470 lemma content_split: fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)" shows
  1471   "content {a..b} = content({a..b} \<inter> {x. x$$k \<le> c}) + content({a..b} \<inter> {x. x$$k >= c})"
  1472 proof- note simps = interval_split[OF assms] content_closed_interval_cases eucl_le[where 'a='a]
  1473   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps using assms by auto }
  1474   have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" "\<And>x. finite ({..<DIM('a)}-{x})" "\<And>x. x\<notin>{..<DIM('a)}-{x}"
  1475     using assms by auto
  1476   have *:"\<And>X Y Z. (\<Prod>i\<in>{..<DIM('a)}. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>{..<DIM('a)}-{k}. Z i (Y i))"
  1477     "(\<Prod>i\<in>{..<DIM('a)}. b$$i - a$$i) = (\<Prod>i\<in>{..<DIM('a)}-{k}. b$$i - a$$i) * (b$$k - a$$k)" 
  1478     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto
  1479   assume as:"a\<le>b" moreover have "\<And>x. min (b $$ k) c = max (a $$ k) c
  1480     \<Longrightarrow> x* (b$$k - a$$k) = x*(max (a $$ k) c - a $$ k) + x*(b $$ k - max (a $$ k) c)"
  1481     by  (auto simp add:field_simps)
  1482   moreover have **:"(\<Prod>i<DIM('a). ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i - a $$ i) = 
  1483     (\<Prod>i<DIM('a). (if i = k then min (b $$ k) c else b $$ i) - a $$ i)"
  1484     "(\<Prod>i<DIM('a). b $$ i - ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i) =
  1485     (\<Prod>i<DIM('a). b $$ i - (if i = k then max (a $$ k) c else a $$ i))"
  1486     apply(rule_tac[!] setprod.cong) by auto
  1487   have "\<not> a $$ k \<le> c \<Longrightarrow> \<not> c \<le> b $$ k \<Longrightarrow> False"
  1488     unfolding not_le using as[unfolded eucl_le[where 'a='a],rule_format,of k] assms by auto
  1489   ultimately show ?thesis using assms unfolding simps **
  1490     unfolding *(1)[of "\<lambda>i x. b$$i - x"] *(1)[of "\<lambda>i x. x - a$$i"] unfolding  *(2) 
  1491     apply(subst(2) euclidean_lambda_beta''[where 'a='a])
  1492     apply(subst(3) euclidean_lambda_beta''[where 'a='a]) by auto
  1493 qed
  1494 
  1495 lemma division_split_left_inj: fixes type::"'a::ordered_euclidean_space"
  1496   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2" 
  1497   "k1 \<inter> {x::'a. x$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}"and k:"k<DIM('a)"
  1498   shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
  1499 proof- note d=division_ofD[OF assms(1)]
  1500   have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k \<le> c}) = {})"
  1501     unfolding  interval_split[OF k] content_eq_0_interior by auto
  1502   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  1503   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  1504   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  1505   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  1506     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  1507  
  1508 lemma division_split_right_inj: fixes type::"'a::ordered_euclidean_space"
  1509   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"
  1510   "k1 \<inter> {x::'a. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}" and k:"k<DIM('a)"
  1511   shows "content(k1 \<inter> {x. x$$k \<ge> c}) = 0"
  1512 proof- note d=division_ofD[OF assms(1)]
  1513   have *:"\<And>a b::'a. \<And> c. (content({a..b} \<inter> {x. x$$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$$k >= c}) = {})"
  1514     unfolding interval_split[OF k] content_eq_0_interior by auto
  1515   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this
  1516   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule exE)+ note uv2=this
  1517   have **:"\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}" by auto
  1518   show ?thesis unfolding uv1 uv2 * apply(rule **[OF d(5)[OF assms(2-4)]])
  1519     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed
  1520 
  1521 lemma tagged_division_split_left_inj: fixes x1::"'a::ordered_euclidean_space"
  1522   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$$k \<le> c} = k2 \<inter> {x. x$$k \<le> c}" 
  1523   and k:"k<DIM('a)"
  1524   shows "content(k1 \<inter> {x. x$$k \<le> c}) = 0"
  1525 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  1526   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
  1527     apply(rule_tac[1-2] *) using assms(2-) by auto qed
  1528 
  1529 lemma tagged_division_split_right_inj: fixes x1::"'a::ordered_euclidean_space"
  1530   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$$k \<ge> c} = k2 \<inter> {x. x$$k \<ge> c}" 
  1531   and k:"k<DIM('a)"
  1532   shows "content(k1 \<inter> {x. x$$k \<ge> c}) = 0"
  1533 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto
  1534   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
  1535     apply(rule_tac[1-2] *) using assms(2-) by auto qed
  1536 
  1537 lemma division_split: fixes a::"'a::ordered_euclidean_space"
  1538   assumes "p division_of {a..b}" and k:"k<DIM('a)"
  1539   shows "{l \<inter> {x. x$$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<le> c} = {})} division_of({a..b} \<inter> {x. x$$k \<le> c})" (is "?p1 division_of ?I1") and 
  1540         "{l \<inter> {x. x$$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$$k \<ge> c})" (is "?p2 division_of ?I2")
  1541 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms(1)]
  1542   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto
  1543   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  1544     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  1545     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  1546       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
  1547     fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  1548     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  1549   { fix k assume "k\<in>?p2" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this
  1550     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this
  1551     show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l
  1552       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split[OF k]) by auto
  1553     fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this
  1554     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }
  1555 qed
  1556 
  1557 lemma has_integral_split: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  1558   assumes "(f has_integral i) ({a..b} \<inter> {x. x$$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$$k \<ge> c})" and k:"k<DIM('a)"
  1559   shows "(f has_integral (i + j)) ({a..b})"
  1560 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto
  1561   guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] . note d1=this[unfolded interval_split[THEN sym,OF k]]
  1562   guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] . note d2=this[unfolded interval_split[THEN sym,OF k]]
  1563   let ?d = "\<lambda>x. if x$$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$$k - c)) \<inter> d1 x \<inter> d2 x"
  1564   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)
  1565   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto
  1566     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]
  1567     have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
  1568          "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
  1569     proof- fix x kk assume as:"(x,kk)\<in>p"
  1570       show "~(kk \<inter> {x. x$$k \<le> c} = {}) \<Longrightarrow> x$$k \<le> c"
  1571       proof(rule ccontr) case goal1
  1572         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ k - c\<bar>"
  1573           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  1574         hence "\<exists>y. y \<in> ball x \<bar>x $$ k - c\<bar> \<inter> {x. x $$ k \<le> c}" using goal1(1) by blast 
  1575         then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<le> c" apply-apply(rule le_less_trans)
  1576           using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
  1577         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  1578       qed
  1579       show "~(kk \<inter> {x. x$$k \<ge> c} = {}) \<Longrightarrow> x$$k \<ge> c"
  1580       proof(rule ccontr) case goal1
  1581         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $$ k - c\<bar>"
  1582           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
  1583         hence "\<exists>y. y \<in> ball x \<bar>x $$ k - c\<bar> \<inter> {x. x $$ k \<ge> c}" using goal1(1) by blast 
  1584         then guess y .. hence "\<bar>x $$ k - y $$ k\<bar> < \<bar>x $$ k - c\<bar>" "y$$k \<ge> c" apply-apply(rule le_less_trans)
  1585           using component_le_norm[of "x - y" k] by(auto simp add:dist_norm)
  1586         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)
  1587       qed
  1588     qed
  1589 
  1590     have lem1: "\<And>f P Q. (\<forall>x k. (x,k) \<in> {(x,f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow> (\<forall>x k. P x k \<longrightarrow> Q x (f k))" by auto
  1591     have lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
  1592     proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed
  1593     have lem3: "\<And>g::('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool. finite p \<Longrightarrow>
  1594       setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}
  1595                = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"
  1596       apply(rule setsum_mono_zero_left) prefer 3
  1597     proof fix g::"('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool" and i::"('a) \<times> (('a) set)"
  1598       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
  1599       then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto
  1600       have "content (g k) = 0" using xk using content_empty by auto
  1601       thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto
  1602     qed auto
  1603     have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto
  1604 
  1605     let ?M1 = "{(x,kk \<inter> {x. x$$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$k \<le> c} \<noteq> {}}"
  1606     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)
  1607       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  1608     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$$k \<le> c}" unfolding p(8)[THEN sym] by auto
  1609       fix x l assume xl:"(x,l)\<in>?M1"
  1610       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  1611       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  1612       thus "l \<subseteq> d1 x" unfolding xl' by auto
  1613       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  1614         using lem0(1)[OF xl'(3-4)] by auto
  1615       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k,where c=c])
  1616       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"
  1617       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  1618       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  1619       proof(cases "l' = r' \<longrightarrow> x' = y'")
  1620         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1621       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  1622         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1623       qed qed moreover
  1624 
  1625     let ?M2 = "{(x,kk \<inter> {x. x$$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$$k \<ge> c} \<noteq> {}}" 
  1626     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)
  1627       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)
  1628     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x$$k \<ge> c}" unfolding p(8)[THEN sym] by auto
  1629       fix x l assume xl:"(x,l)\<in>?M2"
  1630       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this
  1631       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto
  1632       thus "l \<subseteq> d2 x" unfolding xl' by auto
  1633       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $$ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)
  1634         using lem0(2)[OF xl'(3-4)] by auto
  1635       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[OF k, where c=c])
  1636       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"
  1637       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this
  1638       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"
  1639       proof(cases "l' = r' \<longrightarrow> x' = y'")
  1640         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1641       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto
  1642         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
  1643       qed qed ultimately
  1644 
  1645     have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"
  1646       apply- apply(rule norm_triangle_lt) by auto
  1647     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0" using scaleR_zero_left by auto
  1648       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)
  1649        = (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto
  1650       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) +
  1651         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $$ k}) *\<^sub>R f x) - (i + j)"
  1652         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])
  1653         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)
  1654       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) using k by auto
  1655       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) using k by auto
  1656       qed also note setsum_addf[THEN sym]
  1657       also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $$ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $$ k}) *\<^sub>R f x) x
  1658         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv
  1659       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this
  1660         thus "content (b \<inter> {x. x $$ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $$ k}) *\<^sub>R f a = content b *\<^sub>R f a"
  1661           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[OF k,of u v c] by auto
  1662       qed note setsum_cong2[OF this]
  1663       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $$ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $$ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
  1664         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $$ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $$ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
  1665         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }
  1666     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed
  1667 
  1668 (*lemma has_integral_split_cart: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"
  1669   assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"
  1670   shows "(f has_integral (i + j)) ({a..b})" *)
  1671 
  1672 subsection {* A sort of converse, integrability on subintervals. *}
  1673 
  1674 lemma tagged_division_union_interval: fixes a::"'a::ordered_euclidean_space"
  1675   assumes "p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c})"
  1676   and k:"k<DIM('a)"
  1677   shows "(p1 \<union> p2) tagged_division_of ({a..b})"
  1678 proof- have *:"{a..b} = ({a..b} \<inter> {x. x$$k \<le> c}) \<union> ({a..b} \<inter> {x. x$$k \<ge> c})" by auto
  1679   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms(1-2)])
  1680     unfolding interval_split[OF k] interior_closed_interval using k
  1681     by(auto simp add: eucl_less[where 'a='a] elim!:allE[where x=k]) qed
  1682 
  1683 lemma has_integral_separate_sides: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  1684   assumes "(f has_integral i) ({a..b})" "e>0" and k:"k<DIM('a)"
  1685   obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$$k \<le> c}) \<and> d fine p1 \<and>
  1686                                 p2 tagged_division_of ({a..b} \<inter> {x. x$$k \<ge> c}) \<and> d fine p2
  1687                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +
  1688                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"
  1689 proof- guess d using has_integralD[OF assms(1-2)] . note d=this
  1690   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)
  1691   proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this
  1692                    assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $$ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this
  1693     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
  1694     have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
  1695       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv
  1696     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"
  1697       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this
  1698       have "b \<subseteq> {x. x$$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp
  1699       moreover have "interior {x::'a. x $$ k = c} = {}" 
  1700       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x::'a. x$$k = c}" by auto
  1701         then guess e unfolding mem_interior .. note e=this
  1702         have x:"x$$k = c" using x interior_subset by fastsimp
  1703         have *:"\<And>i. i<DIM('a) \<Longrightarrow> \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>
  1704           = (if i = k then e/2 else 0)" using e by auto
  1705         have "(\<Sum>i<DIM('a). \<bar>(x - (x + (\<chi>\<chi> i. if i = k then e / 2 else 0))) $$ i\<bar>) =
  1706           (\<Sum>i<DIM('a). (if i = k then e / 2 else 0))" apply(rule setsum_cong2) apply(subst *) by auto
  1707         also have "... < e" apply(subst setsum_delta) using e by auto 
  1708         finally have "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball dist_norm
  1709           by(rule le_less_trans[OF norm_le_l1])
  1710         hence "x + (\<chi>\<chi> i. if i = k then e/2 else 0) \<in> {x. x$$k = c}" using e by auto
  1711         thus False unfolding mem_Collect_eq using e x k by auto
  1712       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto
  1713       thus "content b *\<^sub>R f a = 0" by auto
  1714     qed auto
  1715     also have "\<dots> < e" by(rule k d(2) p12 fine_union p1 p2)+
  1716     finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed
  1717 
  1718 lemma integrable_split[intro]: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
  1719   assumes "f integrable_on {a..b}" and k:"k<DIM('a)"
  1720   shows "f integrable_on ({a..b} \<inter> {x. x$$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$$k \<ge> c})" (is ?t2) 
  1721 proof- guess y using assms(1) unfolding integrable_on_def .. note y=this
  1722   def b' \<equiv> "(\<chi>\<chi> i. if i = k then min (b$$k) c else b$$i)::'a"
  1723   and a' \<equiv> "(\<chi>\<chi> i. if i = k then max (a$$k) c else a$$i)::'a"
  1724   show ?t1 ?t2 unfolding interval_split[OF k] integrable_cauchy unfolding interval_split[THEN sym,OF k]
  1725   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto
  1726     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
  1727     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1
  1728       \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>
  1729       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
  1730     show "?P {x. x $$ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  1731     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c} \<and> d fine p1
  1732         \<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<le> c} \<and> d fine p2"
  1733       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1734       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this
  1735         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
  1736           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
  1737           using p using assms by(auto simp add:algebra_simps)
  1738       qed qed  
  1739     show "?P {x. x $$ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)
  1740     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p1
  1741         \<and> p2 tagged_division_of {a..b} \<inter> {x. x $$ k \<ge> c} \<and> d fine p2"
  1742       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
  1743       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this
  1744         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
  1745           using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
  1746           using p using assms by(auto simp add:algebra_simps) qed qed qed qed
  1747 
  1748 subsection {* Generalized notion of additivity. *}
  1749 
  1750 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"
  1751 
  1752 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (('b::ordered_euclidean_space) set \<Rightarrow> 'a) \<Rightarrow> bool" where
  1753   "operative opp f \<equiv> 
  1754     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>
  1755     (\<forall>a b c. \<forall>k<DIM('b). f({a..b}) =
  1756                    opp (f({a..b} \<inter> {x. x$$k \<le> c}))
  1757                        (f({a..b} \<inter> {x. x$$k \<ge> c})))"
  1758 
  1759 lemma operativeD[dest]: fixes type::"'a::ordered_euclidean_space"  assumes "operative opp f"
  1760   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b::'a} = neutral(opp)"
  1761   "\<And>a b c k. k<DIM('a) \<Longrightarrow> f({a..b}) = opp (f({a..b} \<inter> {x. x$$k \<le> c})) (f({a..b} \<inter> {x. x$$k \<ge> c}))"
  1762   using assms unfolding operative_def by auto
  1763 
  1764 lemma operative_trivial:
  1765  "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"
  1766   unfolding operative_def by auto
  1767 
  1768 lemma property_empty_interval:
  1769  "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 
  1770   using content_empty unfolding empty_as_interval by auto
  1771 
  1772 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"
  1773   unfolding operative_def apply(rule property_empty_interval) by auto
  1774 
  1775 subsection {* Using additivity of lifted function to encode definedness. *}
  1776 
  1777 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"
  1778   by (metis option.nchotomy)
  1779 
  1780 lemma exists_option:
  1781  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 
  1782   by (metis option.nchotomy)
  1783 
  1784 fun lifted where 
  1785   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |
  1786   "lifted opp None _ = (None::'b option)" |
  1787   "lifted opp _ None = None"
  1788 
  1789 lemma lifted_simp_1[simp]: "lifted opp v None = None"
  1790   apply(induct v) by auto
  1791 
  1792 definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>
  1793                    (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>
  1794                    (\<forall>x. opp (neutral opp) x = x)"
  1795 
  1796 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"
  1797   "\<And>x y z. opp x (opp y z) = opp (opp x y) z"
  1798   "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"
  1799   unfolding monoidal_def using assms by fastsimp
  1800 
  1801 lemma monoidal_ac: assumes "monoidal opp"
  1802   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"
  1803   "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"
  1804   using assms unfolding monoidal_def apply- by metis+
  1805 
  1806 lemma monoidal_simps[simp]: assumes "monoidal opp"
  1807   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"
  1808   using monoidal_ac[OF assms] by auto
  1809 
  1810 lemma neutral_lifted[cong]: assumes "monoidal opp"
  1811   shows "neutral (lifted opp) = Some(neutral opp)"
  1812   apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3
  1813 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"
  1814   thus "x = Some (neutral opp)" apply(induct x) defer
  1815     apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)
  1816     apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto
  1817 qed(auto simp add:monoidal_ac[OF assms])
  1818 
  1819 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"
  1820   unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto
  1821 
  1822 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"
  1823 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"
  1824 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"
  1825 
  1826 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto
  1827 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto
  1828 
  1829 lemma comp_fun_commute_monoidal[intro]: assumes "monoidal opp" shows "comp_fun_commute opp"
  1830   unfolding comp_fun_commute_def using monoidal_ac[OF assms] by auto
  1831 
  1832 lemma support_clauses:
  1833   "\<And>f g s. support opp f {} = {}"
  1834   "\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
  1835   "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"
  1836   "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"
  1837   "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"
  1838   "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
  1839   "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"
  1840 unfolding support_def by auto
  1841 
  1842 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"
  1843   unfolding support_def by auto
  1844 
  1845 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"
  1846   unfolding iterate_def fold'_def by auto 
  1847 
  1848 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"
  1849   shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 
  1850 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto
  1851   show ?thesis unfolding iterate_def if_P[OF True] * by auto
  1852 next case False note x=this
  1853   note * = comp_fun_commute.comp_comp_fun_commute [OF comp_fun_commute_monoidal[OF assms(1)]]
  1854   show ?thesis proof(cases "f x = neutral opp")
  1855     case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
  1856       unfolding True monoidal_simps[OF assms(1)] by auto
  1857   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
  1858       apply(subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
  1859       using `finite s` unfolding support_def using False x by auto qed qed 
  1860 
  1861 lemma iterate_some:
  1862   assumes "monoidal opp"  "finite s"
  1863   shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)
  1864 proof(induct s) case empty thus ?case using assms by auto
  1865 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)
  1866     defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed
  1867 subsection {* Two key instances of additivity. *}
  1868 
  1869 lemma neutral_add[simp]:
  1870   "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 
  1871   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto
  1872 
  1873 lemma operative_content[intro]: "operative (op +) content" 
  1874   unfolding operative_def neutral_add apply safe 
  1875   unfolding content_split[THEN sym] ..
  1876 
  1877 lemma neutral_monoid: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"
  1878   by (rule neutral_add) (* FIXME: duplicate *)
  1879 
  1880 lemma monoidal_monoid[intro]:
  1881   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"
  1882   unfolding monoidal_def neutral_monoid by(auto simp add: algebra_simps) 
  1883 
  1884 lemma operative_integral: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  1885   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"
  1886   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add
  1887   apply(rule,rule,rule,rule) defer apply(rule allI impI)+
  1888 proof- fix a b c k assume k:"k<DIM('a)" show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =
  1889     lifted op + (if f integrable_on {a..b} \<inter> {x. x $$ k \<le> c} then Some (integral ({a..b} \<inter> {x. x $$ k \<le> c}) f) else None)
  1890     (if f integrable_on {a..b} \<inter> {x. c \<le> x $$ k} then Some (integral ({a..b} \<inter> {x. c \<le> x $$ k}) f) else None)"
  1891   proof(cases "f integrable_on {a..b}") 
  1892     case True show ?thesis unfolding if_P[OF True] using k apply-
  1893       unfolding if_P[OF integrable_split(1)[OF True]] unfolding if_P[OF integrable_split(2)[OF True]]
  1894       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split[OF _ _ k]) 
  1895       apply(rule_tac[!] integrable_integral integrable_split)+ using True k by auto
  1896   next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x $$ k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x $$ k}))"
  1897     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def
  1898         apply(rule_tac x="integral ({a..b} \<inter> {x. x $$ k \<le> c}) f + integral ({a..b} \<inter> {x. x $$ k \<ge> c}) f" in exI)
  1899         apply(rule has_integral_split[OF _ _ k]) apply(rule_tac[!] integrable_integral) by auto
  1900       thus False using False by auto
  1901     qed thus ?thesis using False by auto 
  1902   qed next 
  1903   fix a b assume as:"content {a..b::'a} = 0"
  1904   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"
  1905     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed
  1906 
  1907 subsection {* Points of division of a partition. *}
  1908 
  1909 definition "division_points (k::('a::ordered_euclidean_space) set) d = 
  1910     {(j,x). j<DIM('a) \<and> (interval_lowerbound k)$$j < x \<and> x < (interval_upperbound k)$$j \<and>
  1911            (\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
  1912 
  1913 lemma division_points_finite: fixes i::"('a::ordered_euclidean_space) set"
  1914   assumes "d division_of i" shows "finite (division_points i d)"
  1915 proof- note assm = division_ofD[OF assms]
  1916   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)$$j < x \<and> x < (interval_upperbound i)$$j \<and>
  1917            (\<exists>i\<in>d. (interval_lowerbound i)$$j = x \<or> (interval_upperbound i)$$j = x)}"
  1918   have *:"division_points i d = \<Union>(?M ` {..<DIM('a)})"
  1919     unfolding division_points_def by auto
  1920   show ?thesis unfolding * using assm by auto qed
  1921 
  1922 lemma division_points_subset: fixes a::"'a::ordered_euclidean_space"
  1923   assumes "d division_of {a..b}" "\<forall>i<DIM('a). a$$i < b$$i"  "a$$k < c" "c < b$$k" and k:"k<DIM('a)"
  1924   shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<le> c} = {})}
  1925                   \<subseteq> division_points ({a..b}) d" (is ?t1) and
  1926         "division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$$k \<ge> c} = {})}
  1927                   \<subseteq> division_points ({a..b}) d" (is ?t2)
  1928 proof- note assm = division_ofD[OF assms(1)]
  1929   have *:"\<forall>i<DIM('a). a$$i \<le> b$$i"   "\<forall>i<DIM('a). a$$i \<le> ((\<chi>\<chi> i. if i = k then min (b $$ k) c else b $$ i)::'a) $$ i"
  1930     "\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (a $$ k) c else a $$ i)::'a) $$ i \<le> b$$i"  "min (b $$ k) c = c" "max (a $$ k) c = c"
  1931     using assms using less_imp_le by auto
  1932   show ?t1 unfolding division_points_def interval_split[OF k, of a b]
  1933     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
  1934     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
  1935     unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta'
  1936   proof- fix i l x assume as:"a $$ fst x < snd x" "snd x < (if fst x = k then c else b $$ fst x)"
  1937       "interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x"
  1938       "i = l \<inter> {x. x $$ k \<le> c}" "l \<in> d" "l \<inter> {x. x $$ k \<le> c} \<noteq> {}" and fstx:"fst x <DIM('a)"
  1939     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  1940     have *:"\<forall>i<DIM('a). u $$ i \<le> ((\<chi>\<chi> i. if i = k then min (v $$ k) c else v $$ i)::'a) $$ i"
  1941       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
  1942     have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  1943     show "fst x <DIM('a) \<and> a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x
  1944       \<or> interval_upperbound i $$ fst x = snd x)" apply(rule,rule fstx)
  1945       using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
  1946       apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  1947       apply(case_tac[!] "fst x = k") using assms fstx apply- unfolding euclidean_lambda_beta by auto
  1948   qed
  1949   show ?t2 unfolding division_points_def interval_split[OF k, of a b]
  1950     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] unfolding *
  1951     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+
  1952     unfolding mem_Collect_eq apply(erule exE conjE)+ unfolding euclidean_lambda_beta' apply(rule,assumption)
  1953   proof- fix i l x assume as:"(if fst x = k then c else a $$ fst x) < snd x" "snd x < b $$ fst x"
  1954       "interval_lowerbound i $$ fst x = snd x \<or> interval_upperbound i $$ fst x = snd x" 
  1955       "i = l \<inter> {x. c \<le> x $$ k}" "l \<in> d" "l \<inter> {x. c \<le> x $$ k} \<noteq> {}" and fstx:"fst x < DIM('a)"
  1956     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
  1957     have *:"\<forall>i<DIM('a). ((\<chi>\<chi> i. if i = k then max (u $$ k) c else u $$ i)::'a) $$ i \<le> v $$ i"
  1958       using as(6) unfolding l interval_split[OF k] interval_ne_empty as .
  1959     have **:"\<forall>i<DIM('a). u$$i \<le> v$$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto
  1960     show "a $$ fst x < snd x \<and> snd x < b $$ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $$ fst x = snd x \<or>
  1961       interval_upperbound i $$ fst x = snd x)"
  1962       using as(1-3,5) unfolding l interval_split[OF k] interval_ne_empty as interval_bounds[OF *] apply-
  1963       apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]
  1964       apply(case_tac[!] "fst x = k") using assms fstx apply-  by(auto simp add:euclidean_lambda_beta'[OF k]) qed qed
  1965 
  1966 lemma division_points_psubset: fixes a::"'a::ordered_euclidean_space"
  1967   assumes "d division_of {a..b}"  "\<forall>i<DIM('a). a$$i < b$$i"  "a$$k < c" "c < b$$k"
  1968   "l \<in> d" "interval_lowerbound l$$k = c \<or> interval_upperbound l$$k = c" and k:"k<DIM('a)"
  1969   shows "division_points ({a..b} \<inter> {x. x$$k \<le> c}) {l \<inter> {x. x$$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}
  1970               \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 
  1971         "division_points ({a..b} \<inter> {x. x$$k \<ge> c}) {l \<inter> {x. x$$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}
  1972               \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 
  1973 proof- have ab:"\<forall>i<DIM('a). a$$i \<le> b$$i" using assms(2) by(auto intro!:less_imp_le)
  1974   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this
  1975   have uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "\<forall>i<DIM('a). a$$i \<le> u$$i \<and> v$$i \<le> b$$i"
  1976     using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty
  1977     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto
  1978   have *:"interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
  1979          "interval_upperbound ({a..b} \<inter> {x. x $$ k \<le> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
  1980     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  1981     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
  1982   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)
  1983     apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
  1984     apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
  1985     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
  1986   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) using k by auto
  1987 
  1988   have *:"interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_lowerbound l $$ k}) $$ k = interval_lowerbound l $$ k"
  1989          "interval_lowerbound ({a..b} \<inter> {x. x $$ k \<ge> interval_upperbound l $$ k}) $$ k = interval_upperbound l $$ k"
  1990     unfolding interval_split[OF k] apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)
  1991     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab k by auto
  1992   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)
  1993     apply(rule_tac x="(k,(interval_lowerbound l)$$k)" in exI) defer
  1994     apply(rule_tac x="(k,(interval_upperbound l)$$k)" in exI)
  1995     unfolding division_points_def unfolding interval_bounds[OF ab] by(auto simp add:*) 
  1996   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4) k]) by auto qed
  1997 
  1998 subsection {* Preservation by divisions and tagged divisions. *}
  1999 
  2000 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
  2001   unfolding support_def by auto
  2002 
  2003 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
  2004   unfolding iterate_def support_support by auto
  2005 
  2006 lemma iterate_expand_cases:
  2007   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
  2008   apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 
  2009 
  2010 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"
  2011   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  2012 proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>
  2013      iterate opp (f ` s) g = iterate opp s (g \<circ> f)"
  2014   proof- case goal1 show ?case using goal1
  2015     proof(induct s) case empty thus ?case using assms(1) by auto
  2016     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]
  2017         unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])
  2018         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])
  2019         apply(rule finite_imageI insert)+ apply(subst if_not_P)
  2020         unfolding image_iff o_def using insert(2,4) by auto
  2021     qed qed
  2022   show ?thesis 
  2023     apply(cases "finite (support opp g (f ` s))")
  2024     apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])
  2025     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]
  2026     apply(rule subset_inj_on[OF assms(2) support_subset])+
  2027     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)
  2028     apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed
  2029 
  2030 
  2031 (* This lemma about iterations comes up in a few places.                     *)
  2032 lemma iterate_nonzero_image_lemma:
  2033   assumes "monoidal opp" "finite s" "g(a) = neutral opp"
  2034   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"
  2035   shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"
  2036 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto
  2037   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"
  2038     unfolding support_def using assms(3) by auto
  2039   show ?thesis unfolding *
  2040     apply(subst iterate_support[THEN sym]) unfolding support_clauses
  2041     apply(subst iterate_image[OF assms(1)]) defer
  2042     apply(subst(2) iterate_support[THEN sym]) apply(subst **)
  2043     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed
  2044 
  2045 lemma iterate_eq_neutral:
  2046   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"
  2047   shows "(iterate opp s f = neutral opp)"
  2048 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto
  2049   show ?thesis apply(subst iterate_support[THEN sym]) 
  2050     unfolding * using assms(1) by auto qed
  2051 
  2052 lemma iterate_op: assumes "monoidal opp" "finite s"
  2053   shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)
  2054 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto
  2055 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
  2056     unfolding monoidal_ac[OF assms(1)] by(rule refl) qed
  2057 
  2058 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
  2059   shows "iterate opp s f = iterate opp s g"
  2060 proof- have *:"support opp g s = support opp f s"
  2061     unfolding support_def using assms(2) by auto
  2062   show ?thesis
  2063   proof(cases "finite (support opp f s)")
  2064     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)
  2065       unfolding * by auto
  2066   next def su \<equiv> "support opp f s"
  2067     case True note support_subset[of opp f s] 
  2068     thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True
  2069       unfolding su_def[symmetric]
  2070     proof(induct su) case empty show ?case by auto
  2071     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 
  2072         unfolding if_not_P[OF insert(2)] apply(subst insert(3))
  2073         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed
  2074 
  2075 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto
  2076 
  2077 lemma operative_division: fixes f::"('a::ordered_euclidean_space) set \<Rightarrow> 'b"
  2078   assumes "monoidal opp" "operative opp f" "d division_of {a..b}"
  2079   shows "iterate opp d f = f {a..b}"
  2080 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms
  2081   proof(induct C arbitrary:a b d rule:full_nat_induct)
  2082     case goal1
  2083     { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"
  2084       thus ?case apply-apply(cases) defer apply assumption
  2085       proof- assume as:"content {a..b} = 0"
  2086         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])
  2087         proof fix x assume x:"x\<in>d"
  2088           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+
  2089           thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 
  2090             using operativeD(1)[OF assms(2)] x by auto
  2091         qed qed }
  2092     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]
  2093     hence ab':"\<forall>i<DIM('a). a$$i \<le> b$$i" by (auto intro!: less_imp_le) show ?case 
  2094     proof(cases "division_points {a..b} d = {}")
  2095       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>
  2096         (\<forall>j<DIM('a). u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j)"
  2097         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)
  2098         apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule,rule)
  2099       proof- fix u v j assume j:"j<DIM('a)" assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]
  2100         hence uv:"\<forall>i<DIM('a). u$$i \<le> v$$i" "u$$j \<le> v$$j" using j unfolding interval_ne_empty by auto
  2101         have *:"\<And>p r Q. \<not> j<DIM('a) \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as j by auto
  2102         have "(j, u$$j) \<notin> division_points {a..b} d"
  2103           "(j, v$$j) \<notin> division_points {a..b} d" using True by auto
  2104         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
  2105         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
  2106         moreover have "a$$j \<le> u$$j" "v$$j \<le> b$$j" using division_ofD(2,2,3)[OF goal1(4) as] 
  2107           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
  2108           unfolding interval_ne_empty mem_interval using j by auto
  2109         ultimately show "u$$j = a$$j \<and> v$$j = a$$j \<or> u$$j = b$$j \<and> v$$j = b$$j \<or> u$$j = a$$j \<and> v$$j = b$$j"
  2110           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by auto
  2111       qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)
  2112       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]
  2113       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this
  2114       have "{a..b} \<in> d"
  2115       proof- { presume "i = {a..b}" thus ?thesis using i by auto }
  2116         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }
  2117         show "u = a" "v = b" unfolding euclidean_eq[where 'a='a]
  2118         proof(safe) fix j assume j:"j<DIM('a)" note i(2)[unfolded uv mem_interval,rule_format,of j]
  2119           thus "u $$ j = a $$ j" "v $$ j = b $$ j" using uv(2)[rule_format,of j] j by auto
  2120         qed qed
  2121       hence *:"d = insert {a..b} (d - {{a..b}})" by auto
  2122       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])
  2123       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]
  2124         then guess u v apply-by(erule exE conjE)+ note uv=this
  2125         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  
  2126         then obtain j where "u$$j \<noteq> a$$j \<or> v$$j \<noteq> b$$j" and j:"j<DIM('a)" unfolding euclidean_eq[where 'a='a] by auto
  2127         hence "u$$j = v$$j" using uv(2)[rule_format,OF j] by auto
  2128         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) using j by auto
  2129         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])
  2130       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 
  2131         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto
  2132     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto
  2133       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv
  2134         by(erule exE conjE)+ note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
  2135       from this(3) guess j .. note j=this
  2136       def d1 \<equiv> "{l \<inter> {x. x$$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<le> c} \<noteq> {}}"
  2137       def d2 \<equiv> "{l \<inter> {x. x$$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$$k \<ge> c} \<noteq> {}}"
  2138       def cb \<equiv> "(\<chi>\<chi> i. if i = k then c else b$$i)::'a" and ca \<equiv> "(\<chi>\<chi> i. if i = k then c else a$$i)::'a"
  2139       note division_points_psubset[OF goal1(4) ab kc(1-2) j]
  2140       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
  2141       hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x$$k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x$$k \<ge> c})"
  2142         apply- unfolding interval_split[OF kc(4)] apply(rule_tac[!] goal1(1)[rule_format])
  2143         using division_split[OF goal1(4), where k=k and c=c]
  2144         unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono
  2145         using goal1(2-3) using division_points_finite[OF goal1(4)] using kc(4) by auto
  2146       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
  2147         unfolding * apply(rule operativeD(2)) using goal1(3) using kc(4) by auto 
  2148       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$k \<le> c}))"
  2149         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2150         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2151         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2152       proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x $$ k \<le> c} = y \<inter> {x. x $$ k \<le> c}" "l \<noteq> y" 
  2153         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2154         show "f (l \<inter> {x. x $$ k \<le> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
  2155           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_left_inj)
  2156           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule kc(4) as)+
  2157       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$$k \<ge> c}))"
  2158         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])
  2159         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+
  2160         unfolding empty_as_interval[THEN sym] apply(rule content_empty)
  2161       proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x $$ k} = y \<inter> {x. c \<le> x $$ k}" "l \<noteq> y" 
  2162         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this
  2163         show "f (l \<inter> {x. x $$ k \<ge> c}) = neutral opp" unfolding l interval_split[OF kc(4)] 
  2164           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym,OF kc(4)] apply(rule division_split_right_inj)
  2165           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as kc(4))+
  2166       qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x $$ k \<le> c})) (f (x \<inter> {x. c \<le> x $$ k}))"
  2167         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) kc by auto 
  2168       have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x $$ k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x $$ k})))
  2169         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3
  2170         apply(rule iterate_op[THEN sym]) using goal1 by auto
  2171       finally show ?thesis by auto
  2172     qed qed qed 
  2173 
  2174 lemma iterate_image_nonzero: assumes "monoidal opp"
  2175   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"
  2176   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms
  2177 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])
  2178   case goal1 show ?case using assms(1) by auto
  2179 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto
  2180   show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])
  2181     apply(rule finite_imageI goal2)+
  2182     apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer
  2183     apply(subst iterate_insert[OF assms(1) goal2(1)]) defer
  2184     apply(subst iterate_insert[OF assms(1) goal2(1)])
  2185     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)
  2186     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])
  2187     using goal2 unfolding o_def by auto qed 
  2188 
  2189 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"
  2190   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"
  2191 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]
  2192   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *
  2193     apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 
  2194     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)
  2195   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"
  2196     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this
  2197     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])
  2198       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]
  2199       unfolding as(4)[THEN sym] uv by auto
  2200   qed also have "\<dots> = f {a..b}" 
  2201     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
  2202   finally show ?thesis . qed
  2203 
  2204 subsection {* Additivity of content. *}
  2205 
  2206 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"
  2207 proof- have *:"setsum f s = setsum f (support op + f s)"
  2208     apply(rule setsum_mono_zero_right)
  2209     unfolding support_def neutral_monoid using assms by auto
  2210   thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def
  2211     unfolding neutral_monoid . qed
  2212 
  2213 lemma additive_content_division: assumes "d division_of {a..b}"
  2214   shows "setsum content d = content({a..b})"
  2215   unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]
  2216   apply(subst setsum_iterate) using assms by auto
  2217 
  2218 lemma additive_content_tagged_division:
  2219   assumes "d tagged_division_of {a..b}"
  2220   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"
  2221   unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]
  2222   apply(subst setsum_iterate) using assms by auto
  2223 
  2224 subsection {* Finally, the integral of a constant *}
  2225 
  2226 lemma has_integral_const[intro]:
  2227   "((\<lambda>x. c) has_integral (content({a..b::'a::ordered_euclidean_space}) *\<^sub>R c)) ({a..b})"
  2228   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)
  2229   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)
  2230   unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])
  2231   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto
  2232 
  2233 subsection {* Bounds on the norm of Riemann sums and the integral itself. *}
  2234 
  2235 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"
  2236   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")
  2237   apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]
  2238   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)
  2239   apply(subst mult_commute) apply(rule mult_left_mono)
  2240   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)
  2241   apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]
  2242 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .
  2243   fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+
  2244   thus "0 \<le> content x" using content_pos_le by auto
  2245 qed(insert assms,auto)
  2246 
  2247 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"
  2248   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"
  2249 proof(cases "{a..b} = {}") case True
  2250   show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto
  2251 next case False show ?thesis
  2252     apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR
  2253     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer
  2254     unfolding setsum_left_distrib[THEN sym] apply(subst mult_commute) apply(rule mult_left_mono)
  2255     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)
  2256     apply(subst o_def, rule abs_of_nonneg)
  2257   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)
  2258       unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto
  2259     guess w using nonempty_witness[OF False] .
  2260     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto
  2261     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]
  2262     from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this
  2263     show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)
  2264     show "norm (f (fst xk)) \<le> e" unfolding xk fst_conv using tagged_division_ofD(2,3)[OF assms(1) xk(2)] assms(2) by auto
  2265   qed(insert assms,auto) qed
  2266 
  2267 lemma rsum_diff_bound:
  2268   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"
  2269   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le> e * content({a..b})"
  2270   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2271   unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto
  2272 
  2273 lemma has_integral_bound: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  2274   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"
  2275   shows "norm i \<le> B * content {a..b}"
  2276 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"
  2277     thus ?thesis proof(cases ?P) case False
  2278       hence *:"content {a..b} = 0" using content_lt_nz by auto
  2279       hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto
  2280       show ?thesis unfolding * ** using assms(1) by auto
  2281     qed auto } assume ab:?P
  2282   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2283   assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto
  2284   from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  2285   from fine_division_exists[OF this(1), of a b] guess p . note p=this
  2286   have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"
  2287   proof- case goal1 thus ?case unfolding not_less
  2288     using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto
  2289   qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed
  2290 
  2291 subsection {* Similar theorems about relationship among components. *}
  2292 
  2293 lemma rsum_component_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
  2294   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$$i \<le> (g x)$$i"
  2295   shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$$i"
  2296   unfolding  euclidean_component.setsum apply(rule setsum_mono) apply safe
  2297 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
  2298   from this(3) guess u v apply-by(erule exE)+ note b=this
  2299   show "(content b *\<^sub>R f a) $$ i \<le> (content b *\<^sub>R g a) $$ i" unfolding b
  2300     unfolding euclidean_simps real_scaleR_def apply(rule mult_left_mono)
  2301     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed
  2302 
  2303 lemma has_integral_component_le: fixes f g::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
  2304   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
  2305   shows "i$$k \<le> j$$k"
  2306 proof- have lem:"\<And>a b i (j::'b). \<And>g f::'a \<Rightarrow> 'b. (f has_integral i) ({a..b}) \<Longrightarrow> 
  2307     (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$$k \<le> (g x)$$k \<Longrightarrow> i$$k \<le> j$$k"
  2308   proof(rule ccontr) case goal1 hence *:"0 < (i$$k - j$$k) / 3" by auto
  2309     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]
  2310     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]
  2311     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
  2312     note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k] term g
  2313     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
  2314     thus False unfolding euclidean_simps using rsum_component_le[OF p(1) goal1(3)] apply simp by smt
  2315   qed let ?P = "\<exists>a b. s = {a..b}"
  2316   { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)
  2317       case True then guess a b apply-by(erule exE)+ note s=this
  2318       show ?thesis apply(rule lem) using assms[unfolded s] by auto
  2319     qed auto } assume as:"\<not> ?P"
  2320   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }
  2321   assume "\<not> i$$k \<le> j$$k" hence ij:"(i$$k - j$$k) / 3 > 0" by auto
  2322   note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
  2323   have "bounded (ball 0 B1 \<union> ball (0::'a) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+
  2324   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+
  2325   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
  2326   guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
  2327   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
  2328   have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" by smt
  2329   note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover
  2330   have "w1$$k \<le> w2$$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately
  2331   show False unfolding euclidean_simps by(rule *) qed
  2332 
  2333 lemma integral_component_le: fixes g f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
  2334   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$$k \<le> (g x)$$k"
  2335   shows "(integral s f)$$k \<le> (integral s g)$$k"
  2336   apply(rule has_integral_component_le) using integrable_integral assms by auto
  2337 
  2338 (*lemma has_integral_dest_vec1_le: fixes f::"'a::ordered_euclidean_space \<Rightarrow> real^1"
  2339   assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"
  2340   shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])
  2341   using assms(3) unfolding vector_le_def by auto
  2342 
  2343 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"
  2344   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  2345   shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"
  2346   apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto*)
  2347 
  2348 lemma has_integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
  2349   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> i$$k" 
  2350   using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2-) by auto
  2351 
  2352 lemma integral_component_nonneg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::euclidean_space"
  2353   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$$k" shows "0 \<le> (integral s f)$$k"
  2354   apply(rule has_integral_component_nonneg) using assms by auto
  2355 
  2356 (*lemma has_integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
  2357   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"
  2358   using has_integral_component_nonneg[OF assms(1), of 1]
  2359   using assms(2) unfolding vector_le_def by auto
  2360 
  2361 lemma integral_dest_vec1_nonneg: fixes f::"real^'n \<Rightarrow> real^1"
  2362   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"
  2363   apply(rule has_integral_dest_vec1_nonneg) using assms by auto*)
  2364 
  2365 lemma has_integral_component_neg: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
  2366   assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$$k \<le> 0"shows "i$$k \<le> 0" 
  2367   using has_integral_component_le[OF assms(1) has_integral_0] assms(2-) by auto
  2368 
  2369 (*lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"
  2370   assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"
  2371   using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto*)
  2372 
  2373 lemma has_integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
  2374   assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)" shows "B * content {a..b} \<le> i$$k"
  2375   using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi>\<chi> i. B)::'b" k] assms(2-)
  2376   unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] by(auto simp add:field_simps)
  2377 
  2378 lemma has_integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
  2379   assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$$k \<le> B" "k<DIM('b)"
  2380   shows "i$$k \<le> B * content({a..b})"
  2381   using has_integral_component_le[OF assms(1) has_integral_const, of k "\<chi>\<chi> i. B"]
  2382   unfolding euclidean_simps euclidean_lambda_beta'[OF assms(3)] using assms(2) by(auto simp add:field_simps)
  2383 
  2384 lemma integral_component_lbound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
  2385   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$$k" "k<DIM('b)"
  2386   shows "B * content({a..b}) \<le> (integral({a..b}) f)$$k"
  2387   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto
  2388 
  2389 lemma integral_component_ubound: fixes f::"'a::ordered_euclidean_space => 'b::ordered_euclidean_space"
  2390   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$$k \<le> B" "k<DIM('b)" 
  2391   shows "(integral({a..b}) f)$$k \<le> B * content({a..b})"
  2392   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto
  2393 
  2394 subsection {* Uniform limit of integrable functions is integrable. *}
  2395 
  2396 lemma integrable_uniform_limit: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  2397   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"
  2398   shows "f integrable_on {a..b}"
  2399 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"
  2400     show ?thesis apply cases apply(rule *,assumption)
  2401       unfolding content_lt_nz integrable_on_def using has_integral_null by auto }
  2402   assume as:"content {a..b} > 0"
  2403   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto
  2404   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
  2405   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]
  2406   
  2407   have "Cauchy i" unfolding Cauchy_def
  2408   proof(rule,rule) fix e::real assume "e>0"
  2409     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)
  2410     then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this
  2411     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e" apply(rule_tac x=M in exI,rule,rule,rule,rule)
  2412     proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]
  2413       from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]
  2414       from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]
  2415       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
  2416       have lem2:"\<And>s1 s2 i1 i2. norm(s2 - s1) \<le> e/2 \<Longrightarrow> norm(s1 - i1) < e / 4 \<Longrightarrow> norm(s2 - i2) < e / 4 \<Longrightarrow>norm(i1 - i2) < e"
  2417       proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
  2418           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
  2419           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:algebra_simps)
  2420         also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
  2421         finally show ?case .
  2422       qed
  2423       show ?case unfolding dist_norm apply(rule lem2) defer
  2424         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
  2425         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)
  2426         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])
  2427       proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 
  2428           using M as by(auto simp add:field_simps)
  2429         fix x assume x:"x \<in> {a..b}"
  2430         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 
  2431             using g(1)[OF x, of n] g(1)[OF x, of m] by auto
  2432         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)
  2433           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto
  2434         also have "\<dots> = 2 / real M" unfolding divide_inverse by auto
  2435         finally show "norm (g n x - g m x) \<le> 2 / real M"
  2436           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
  2437           by(auto simp add:algebra_simps simp add:norm_minus_commute)
  2438       qed qed qed
  2439   from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this
  2440 
  2441   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral
  2442   proof(rule,rule)  
  2443     case goal1 hence *:"e/3 > 0" by auto
  2444     from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this
  2445     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)
  2446     from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this
  2447     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
  2448     have lem:"\<And>sf sg i. norm(sf - sg) \<le> e / 3 \<Longrightarrow> norm(i - s) < e / 3 \<Longrightarrow> norm(sg - i) < e / 3 \<Longrightarrow> norm(sf - s) < e"
  2449     proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
  2450         using norm_triangle_ineq[of "sf - sg" "sg - s"]
  2451         using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:algebra_simps)
  2452       also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:algebra_simps)
  2453       finally show ?case .
  2454     qed
  2455     show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')
  2456     proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]
  2457       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])
  2458         apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)
  2459       proof- have "content {a..b} < e / 3 * (real N2)"
  2460           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)
  2461         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"
  2462           apply-apply(rule less_le_trans,assumption) using `e>0` by auto 
  2463         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"
  2464           unfolding inverse_eq_divide by(auto simp add:field_simps)
  2465         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded dist_norm],auto)
  2466       qed qed qed qed
  2467 
  2468 subsection {* Negligible sets. *}
  2469 
  2470 definition "negligible (s::('a::ordered_euclidean_space) set) \<equiv> (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) {a..b})"
  2471 
  2472 subsection {* Negligibility of hyperplane. *}
  2473 
  2474 lemma vsum_nonzero_image_lemma: 
  2475   assumes "finite s" "g(a) = 0"
  2476   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"
  2477   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"
  2478   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer
  2479   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+
  2480   unfolding assms using neutral_add unfolding neutral_add using assms by auto 
  2481 
  2482 lemma interval_doublesplit:  fixes a::"'a::ordered_euclidean_space" assumes "k<DIM('a)"
  2483   shows "{a..b} \<inter> {x . abs(x$$k - c) \<le> (e::real)} = 
  2484   {(\<chi>\<chi> i. if i = k then max (a$$k) (c - e) else a$$i) .. (\<chi>\<chi> i. if i = k then min (b$$k) (c + e) else b$$i)}"
  2485 proof- have *:"\<And>x c e::real. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2486   have **:"\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}" by blast
  2487   show ?thesis unfolding * ** interval_split[OF assms] by(rule refl) qed
  2488 
  2489 lemma division_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "p division_of {a..b}" and k:"k<DIM('a)"
  2490   shows "{l \<inter> {x. abs(x$$k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x$$k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x$$k - c) \<le> e})"
  2491 proof- have *:"\<And>x c. abs(x - c) \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e" by auto
  2492   have **:"\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'" by auto
  2493   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
  2494   note division_split(2)[OF this, where c="c-e" and k=k,OF k] 
  2495   thus ?thesis apply(rule **) using k apply- unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit
  2496     apply(rule set_eqI) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer
  2497     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $$ k}" in exI) apply rule defer apply rule
  2498     apply(rule_tac x=l in exI) by blast+ qed
  2499 
  2500 lemma content_doublesplit: fixes a::"'a::ordered_euclidean_space" assumes "0 < e" and k:"k<DIM('a)"
  2501   obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$$k - c) \<le> d}) < e"
  2502 proof(cases "content {a..b} = 0")
  2503   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit[OF k]
  2504     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)
  2505     unfolding interval_doublesplit[THEN sym,OF k] using assms by auto 
  2506 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})"
  2507   note False[unfolded content_eq_0 not_ex not_le, rule_format]
  2508   hence "\<And>x. x<DIM('a) \<Longrightarrow> b$$x > a$$x" by(auto simp add:not_le)
  2509   hence prod0:"0 < setprod (\<lambda>i. b$$i - a$$i) ({..<DIM('a)} - {k})" apply-apply(rule setprod_pos) by(auto simp add:field_simps)
  2510   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis
  2511   proof(rule that[of d]) have *:"{..<DIM('a)} = insert k ({..<DIM('a)} - {k})" using k by auto
  2512     have **:"{a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 
  2513       (\<Prod>i\<in>{..<DIM('a)} - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i
  2514       - interval_lowerbound ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) $$ i)
  2515       = (\<Prod>i\<in>{..<DIM('a)} - {k}. b$$i - a$$i)" apply(rule setprod_cong,rule refl) 
  2516       unfolding interval_doublesplit[OF k] apply(subst interval_bounds) defer apply(subst interval_bounds)
  2517       unfolding interval_eq_empty not_ex not_less by auto
  2518     show "content ({a..b} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)
  2519       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **
  2520       unfolding interval_doublesplit[OF k] interval_eq_empty not_ex not_less prefer 3
  2521       apply(subst interval_bounds) defer apply(subst interval_bounds) unfolding euclidean_lambda_beta'[OF k] if_P[OF refl]
  2522     proof- have "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) \<le> 2 * d" by auto
  2523       also have "... < e / (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)
  2524       finally show "(min (b $$ k) (c + d) - max (a $$ k) (c - d)) * (\<Prod>i\<in>{..<DIM('a)} - {k}. b $$ i - a $$ i) < e"
  2525         unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed
  2526 
  2527 lemma negligible_standard_hyperplane[intro]: fixes type::"'a::ordered_euclidean_space" assumes k:"k<DIM('a)"
  2528   shows "negligible {x::'a. x$$k = (c::real)}" 
  2529   unfolding negligible_def has_integral apply(rule,rule,rule,rule)
  2530 proof-
  2531   case goal1 from content_doublesplit[OF this k,of a b c] guess d . note d=this
  2532   let ?i = "indicator {x::'a. x$$k = c} :: 'a\<Rightarrow>real"
  2533   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)
  2534   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"
  2535     have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$$k - c) \<le> d}) *\<^sub>R ?i x)"
  2536       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
  2537       apply(cases,rule disjI1,assumption,rule disjI2)
  2538     proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$$k = c" unfolding indicator_def apply-by(rule ccontr,auto)
  2539       show "content l = content (l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])
  2540         apply(rule set_eqI,rule,rule) unfolding mem_Collect_eq
  2541       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
  2542         note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]
  2543         thus "\<bar>y $$ k - c\<bar> \<le> d" unfolding euclidean_simps xk by auto
  2544       qed auto qed
  2545     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
  2546     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e" unfolding diff_0_right * unfolding real_scaleR_def real_norm_def
  2547       apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv
  2548       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)
  2549       prefer 2 apply(subst(asm) eq_commute) apply assumption
  2550       apply(subst interval_doublesplit[OF k]) apply(rule content_pos_le) apply(rule indicator_pos_le)
  2551     proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}))"
  2552         apply(rule setsum_mono) unfolding split_paired_all split_conv 
  2553         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit[OF k] intro!:content_pos_le)
  2554       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
  2555       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<le> content {u..v}"
  2556           unfolding interval_doublesplit[OF k] apply(rule content_subset) unfolding interval_doublesplit[THEN sym,OF k] by auto
  2557         thus ?case unfolding goal1 unfolding interval_doublesplit[OF k] using content_pos_le by smt
  2558       next have *:"setsum content {l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"
  2559           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 
  2560         proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"
  2561           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this
  2562           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit[OF k] by(rule content_pos_le)
  2563         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
  2564         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym,OF k]]
  2565         note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d] note le_less_trans[OF this d(2)]
  2566         from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d})) < e"
  2567           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])
  2568           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']
  2569         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v
  2570           assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}"
  2571           have "({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast
  2572           note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]
  2573           hence "interior ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto
  2574           thus "content ({m..n} \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit[OF k] content_eq_0_interior[THEN sym] .
  2575         qed qed
  2576       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $$ k - c\<bar> \<le> d}) * ?i x) < e" .
  2577     qed qed qed
  2578 
  2579 subsection {* A technical lemma about "refinement" of division. *}
  2580 
  2581 lemma tagged_division_finer: fixes p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set"
  2582   assumes "p tagged_division_of {a..b}" "gauge d"
  2583   obtains q where "q tagged_division_of {a..b}" "d fine q" "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
  2584 proof-
  2585   let ?P = "\<lambda>p. p tagged_partial_division_of {a..b} \<longrightarrow> gauge d \<longrightarrow>
  2586     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
  2587                    (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
  2588   { have *:"finite p" "p tagged_partial_division_of {a..b}" using assms(1) unfolding tagged_division_of_def by auto
  2589     presume "\<And>p. finite p \<Longrightarrow> ?P p" from this[rule_format,OF * assms(2)] guess q .. note q=this
  2590     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto
  2591   } fix p::"(('a::ordered_euclidean_space) \<times> (('a::ordered_euclidean_space) set)) set" assume as:"finite p"
  2592   show "?P p" apply(rule,rule) using as proof(induct p) 
  2593     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto
  2594   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule exE)+ note xk=this
  2595     note tagged_partial_division_subset[OF insert(4) subset_insertI]
  2596     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
  2597     have *:"\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}" unfolding xk by auto
  2598     note p = tagged_partial_division_ofD[OF insert(4)]
  2599     from p(4)[unfolded xk, OF insertI1] guess u v apply-by(erule exE)+ note uv=this
  2600 
  2601     have "finite {k. \<exists>x. (x, k) \<in> p}" 
  2602       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq
  2603       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto
  2604     hence int:"interior {u..v} \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
  2605       apply(rule inter_interior_unions_intervals) apply(rule open_interior) apply(rule_tac[!] ballI)
  2606       unfolding mem_Collect_eq apply(erule_tac[!] exE) apply(drule p(4)[OF insertI2],assumption)      
  2607       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption
  2608       using insert(2) unfolding uv xk by auto
  2609 
  2610     show ?case proof(cases "{u..v} \<subseteq> d x")
  2611       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule
  2612         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)
  2613         apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 
  2614         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)
  2615         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)
  2616         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto
  2617     next case False from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
  2618       show ?thesis apply(rule_tac x="q2 \<union> q1" in exI)
  2619         apply rule unfolding * uv apply(rule tagged_division_union q2 q1 int fine_union)+
  2620         unfolding Ball_def split_paired_All split_conv apply rule apply(rule fine_union)
  2621         apply(rule q1 q2)+ apply(rule,rule,rule,rule) apply(erule insertE)
  2622         apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto
  2623     qed qed qed
  2624 
  2625 subsection {* Hence the main theorem about negligible sets. *}
  2626 
  2627 lemma finite_product_dependent: assumes "finite s" "\<And>x. x\<in>s\<Longrightarrow> finite (t x)"
  2628   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" using assms
  2629 proof(induct) case (insert x s) 
  2630   have *:"{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} = (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
  2631   show ?case unfolding * apply(rule finite_UnI) using insert by auto qed auto
  2632 
  2633 lemma sum_sum_product: assumes "finite s" "\<forall>i\<in>s. finite (t i)"
  2634   shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s = setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}" using assms
  2635 proof(induct) case (insert a s)
  2636   have *:"{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} = (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
  2637   show ?case unfolding * apply(subst setsum_Un_disjoint) unfolding setsum_insert[OF insert(1-2)]
  2638     prefer 4 apply(subst insert(3)) unfolding add_right_cancel
  2639   proof- show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in>Pair a ` t a. x xa y)" apply(subst setsum_reindex) unfolding inj_on_def by auto
  2640     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto
  2641   qed(insert insert, auto) qed auto
  2642 
  2643 lemma has_integral_negligible: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
  2644   assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"
  2645   shows "(f has_integral 0) t"
  2646 proof- presume P:"\<And>f::'b::ordered_euclidean_space \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
  2647   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
  2648   show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)
  2649     apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P
  2650   proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this
  2651     show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto
  2652   next show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) {a..b} \<and> norm (z - 0) < e)"
  2653       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)
  2654       apply(rule,rule P) using assms(2) by auto
  2655   qed
  2656 next fix f::"'b \<Rightarrow> 'a" and a b::"'b" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 
  2657   show "(f has_integral 0) {a..b}" unfolding has_integral
  2658   proof(safe) case goal1
  2659     hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 
  2660       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)
  2661     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 
  2662     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
  2663     show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 
  2664     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto
  2665       fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 
  2666       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
  2667       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }
  2668       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
  2669       hence N:"\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N" apply(subst(asm) Sup_finite_le_iff) using as as' by auto
  2670       have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
  2671         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto
  2672       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
  2673       have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)" apply(rule setsum_nonneg,safe) 
  2674         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto
  2675       have **:"\<And>f g s t. finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow> (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t"
  2676       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4
  2677           apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed
  2678       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
  2679                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {0..N+1}"
  2680         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
  2681         apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 
  2682       proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto
  2683         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
  2684           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)
  2685           using tagged_division_ofD(4)[OF q(1) as''] by auto
  2686       next fix i::nat show "finite (q i)" using q by auto
  2687       next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"
  2688         have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto
  2689         have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto
  2690         hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto
  2691         moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
  2692         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]
  2693         moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
  2694         proof(cases "x\<in>s") case False thus ?thesis using assm by auto
  2695         next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto
  2696           moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto
  2697           ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)
  2698         qed ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le> (real y + 1) * (content k *\<^sub>R indicator s x)"
  2699           apply(rule_tac x=n in exI,safe) apply(rule_tac x=n in exI,rule_tac x="(x,k)" in exI,safe) by auto
  2700       qed(insert as, auto)
  2701       also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 
  2702       proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])
  2703           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)
  2704       qed also have "... < e * inverse 2 * 2" unfolding divide_inverse setsum_right_distrib[THEN sym]
  2705         apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]
  2706         apply(subst sumr_geometric) using goal1 by auto
  2707       finally show "?goal" by auto qed qed qed
  2708 
  2709 lemma has_integral_spike: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
  2710   assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"
  2711   shows "(g has_integral y) t"
  2712 proof- { fix a b::"'b" and f g ::"'b \<Rightarrow> 'a" and y::'a
  2713     assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"
  2714     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])
  2715       apply(rule has_integral_negligible[OF assms(1)]) using as by auto
  2716     hence "(g has_integral y) {a..b}" by auto } note * = this
  2717   show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)
  2718     apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P
  2719     apply(erule_tac x=e in allE,safe,rule_tac x=B in exI,safe) apply(erule_tac x=a in allE,erule_tac x=b in allE,safe)
  2720     apply(rule_tac x=z in exI,safe) apply(rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"]) by auto qed
  2721 
  2722 lemma has_integral_spike_eq:
  2723   assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2724   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2725   apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto
  2726 
  2727 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"
  2728   shows "g integrable_on  t"
  2729   using assms unfolding integrable_on_def apply-apply(erule exE)
  2730   apply(rule,rule has_integral_spike) by fastsimp+
  2731 
  2732 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"
  2733   shows "integral t f = integral t g"
  2734   unfolding integral_def using has_integral_spike_eq[OF assms] by auto
  2735 
  2736 subsection {* Some other trivialities about negligible sets. *}
  2737 
  2738 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 
  2739 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]
  2740     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption
  2741     using assms(2) unfolding indicator_def by auto qed
  2742 
  2743 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto
  2744 
  2745 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto
  2746 
  2747 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 
  2748 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]
  2749   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])
  2750     defer apply assumption unfolding indicator_def by auto qed
  2751 
  2752 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"
  2753   using negligible_union by auto
  2754 
  2755 lemma negligible_sing[intro]: "negligible {a::_::ordered_euclidean_space}" 
  2756   using negligible_standard_hyperplane[of 0 "a$$0"] by auto 
  2757 
  2758 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"
  2759   apply(subst insert_is_Un) unfolding negligible_union_eq by auto
  2760 
  2761 lemma negligible_empty[intro]: "negligible {}" by auto
  2762 
  2763 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"
  2764   using assms apply(induct s) by auto
  2765 
  2766 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"
  2767   using assms by(induct,auto) 
  2768 
  2769 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::('a::ordered_euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
  2770   apply safe defer apply(subst negligible_def)
  2771 proof- fix t::"'a set" assume as:"negligible s"
  2772   have *:"(\<lambda>x. if x \<in> s \<inter> t then 1 else 0) = (\<lambda>x. if x\<in>t then if x\<in>s then 1 else 0 else 0)" by(rule ext,auto)  
  2773   show "((indicator s::'a\<Rightarrow>real) has_integral 0) t" apply(subst has_integral_alt)
  2774     apply(cases,subst if_P,assumption) unfolding if_not_P apply(safe,rule as[unfolded negligible_def,rule_format])
  2775     apply(rule_tac x=1 in exI) apply(safe,rule zero_less_one) apply(rule_tac x=0 in exI)
  2776     using negligible_subset[OF as,of "s \<inter> t"] unfolding negligible_def indicator_def_raw unfolding * by auto qed auto
  2777 
  2778 subsection {* Finite case of the spike theorem is quite commonly needed. *}
  2779 
  2780 lemma has_integral_spike_finite: assumes "finite s" "\<forall>x\<in>t-s. g x = f x" 
  2781   "(f has_integral y) t" shows "(g has_integral y) t"
  2782   apply(rule has_integral_spike) using assms by auto
  2783 
  2784 lemma has_integral_spike_finite_eq: assumes "finite s" "\<forall>x\<in>t-s. g x = f x"
  2785   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
  2786   apply rule apply(rule_tac[!] has_integral_spike_finite) using assms by auto
  2787 
  2788 lemma integrable_spike_finite:
  2789   assumes "finite s" "\<forall>x\<in>t-s. g x = f x" "f integrable_on t" shows "g integrable_on  t"
  2790   using assms unfolding integrable_on_def apply safe apply(rule_tac x=y in exI)
  2791   apply(rule has_integral_spike_finite) by auto
  2792 
  2793 subsection {* In particular, the boundary of an interval is negligible. *}
  2794 
  2795 lemma negligible_frontier_interval: "negligible({a::'a::ordered_euclidean_space..b} - {a<..<b})"
  2796 proof- let ?A = "\<Union>((\<lambda>k. {x. x$$k = a$$k} \<union> {x::'a. x$$k = b$$k}) ` {..<DIM('a)})"
  2797   have "{a..b} - {a<..<b} \<subseteq> ?A" apply rule unfolding Diff_iff mem_interval not_all
  2798     apply(erule conjE exE)+ apply(rule_tac X="{x. x $$ xa = a $$ xa} \<union> {x. x $$ xa = b $$ xa}" in UnionI)
  2799     apply(erule_tac[!] x=xa in allE) by auto
  2800   thus ?thesis apply-apply(rule negligible_subset[of ?A]) apply(rule negligible_unions[OF finite_imageI]) by auto qed
  2801 
  2802 lemma has_integral_spike_interior:
  2803   assumes "\<forall>x\<in>{a<..<b}. g x = f x" "(f has_integral y) ({a..b})" shows "(g has_integral y) ({a..b})"
  2804   apply(rule has_integral_spike[OF negligible_frontier_interval _ assms(2)]) using assms(1) by auto
  2805 
  2806 lemma has_integral_spike_interior_eq:
  2807   assumes "\<forall>x\<in>{a<..<b}. g x = f x" shows "((f has_integral y) ({a..b}) \<longleftrightarrow> (g has_integral y) ({a..b}))"
  2808   apply rule apply(rule_tac[!] has_integral_spike_interior) using assms by auto
  2809 
  2810 lemma integrable_spike_interior: assumes "\<forall>x\<in>{a<..<b}. g x = f x" "f integrable_on {a..b}" shows "g integrable_on {a..b}"
  2811   using  assms unfolding integrable_on_def using has_integral_spike_interior[OF assms(1)] by auto
  2812 
  2813 subsection {* Integrability of continuous functions. *}
  2814 
  2815 lemma neutral_and[simp]: "neutral op \<and> = True"
  2816   unfolding neutral_def apply(rule some_equality) by auto
  2817 
  2818 lemma monoidal_and[intro]: "monoidal op \<and>" unfolding monoidal_def by auto
  2819 
  2820 lemma iterate_and[simp]: assumes "finite s" shows "(iterate op \<and>) s p \<longleftrightarrow> (\<forall>x\<in>s. p x)" using assms
  2821 apply induct unfolding iterate_insert[OF monoidal_and] by auto
  2822 
  2823 lemma operative_division_and: assumes "operative op \<and> P" "d division_of {a..b}"
  2824   shows "(\<forall>i\<in>d. P i) \<longleftrightarrow> P {a..b}"
  2825   using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)] by auto
  2826 
  2827 lemma operative_approximable: assumes "0 \<le> e" fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
  2828   shows "operative op \<and> (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)" unfolding operative_def neutral_and
  2829 proof safe fix a b::"'b" { assume "content {a..b} = 0"
  2830     thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" 
  2831       apply(rule_tac x=f in exI) using assms by(auto intro!:integrable_on_null) }
  2832   { fix c k g assume as:"\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}" and k:"k<DIM('b)"
  2833     show "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
  2834       "\<exists>g. (\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
  2835       apply(rule_tac[!] x=g in exI) using as(1) integrable_split[OF as(2) k] by auto }
  2836   fix c k g1 g2 assume as:"\<forall>x\<in>{a..b} \<inter> {x. x $$ k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on {a..b} \<inter> {x. x $$ k \<le> c}"
  2837                           "\<forall>x\<in>{a..b} \<inter> {x. c \<le> x $$ k}. norm (f x - g2 x) \<le> e" "g2 integrable_on {a..b} \<inter> {x. c \<le> x $$ k}"
  2838   assume k:"k<DIM('b)"
  2839   let ?g = "\<lambda>x. if x$$k = c then f x else if x$$k \<le> c then g1 x else g2 x"
  2840   show "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" apply(rule_tac x="?g" in exI)
  2841   proof safe case goal1 thus ?case apply- apply(cases "x$$k=c", case_tac "x$$k < c") using as assms by auto
  2842   next case goal2 presume "?g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
  2843     then guess h1 h2 unfolding integrable_on_def by auto from has_integral_split[OF this k] 
  2844     show ?case unfolding integrable_on_def by auto
  2845   next show "?g integrable_on {a..b} \<inter> {x. x $$ k \<le> c}" "?g integrable_on {a..b} \<inter> {x. x $$ k \<ge> c}"
  2846       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]]) using k as(2,4) by auto qed qed
  2847 
  2848 lemma approximable_on_division: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
  2849   assumes "0 \<le> e" "d division_of {a..b}" "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  2850   obtains g where "\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e" "g integrable_on {a..b}"
  2851 proof- note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
  2852   note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]] from assms(3)[unfolded this[of f]]
  2853   guess g .. thus thesis apply-apply(rule that[of g]) by auto qed
  2854 
  2855 lemma integrable_continuous: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
  2856   assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
  2857 proof(rule integrable_uniform_limit,safe) fix e::real assume e:"0 < e"
  2858   from compact_uniformly_continuous[OF assms compact_interval,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
  2859   note d=conjunctD2[OF this,rule_format]
  2860   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
  2861   note p' = tagged_division_ofD[OF p(1)]
  2862   have *:"\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
  2863   proof(safe,unfold snd_conv) fix x l assume as:"(x,l) \<in> p" 
  2864     from p'(4)[OF this] guess a b apply-by(erule exE)+ note l=this
  2865     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l" apply(rule_tac x="\<lambda>y. f x" in exI)
  2866     proof safe show "(\<lambda>y. f x) integrable_on l" unfolding integrable_on_def l by(rule,rule has_integral_const)
  2867       fix y assume y:"y\<in>l" note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
  2868       note d(2)[OF _ _ this[unfolded mem_ball]]
  2869       thus "norm (f y - f x) \<le> e" using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastsimp qed qed
  2870   from e have "0 \<le> e" by auto from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
  2871   thus "\<exists>g. (\<forall>x\<in>{a..b}. norm (f x - g x) \<le> e) \<and> g integrable_on {a..b}" by auto qed 
  2872 
  2873 subsection {* Specialization of additivity to one dimension. *}
  2874 
  2875 lemma operative_1_lt: assumes "monoidal opp"
  2876   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
  2877                 (\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2878   unfolding operative_def content_eq_0 DIM_real less_one simp_thms(39,41) Eucl_real_simps
  2879     (* The dnf_simps simplify "\<exists> x. x= _ \<and> _" and "\<forall>k. k = _ \<longrightarrow> _" *)
  2880 proof safe fix a b c::"real" assume as:"\<forall>a b c. f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c}))
  2881     (f ({a..b} \<inter> {x. c \<le> x}))" "a < c" "c < b"
  2882     from this(2-) have "{a..b} \<inter> {x. x \<le> c} = {a..c}" "{a..b} \<inter> {x. x \<ge> c} = {c..b}" by auto
  2883     thus "opp (f {a..c}) (f {c..b}) = f {a..b}" unfolding as(1)[rule_format,of a b "c"] by auto
  2884 next fix a b c::real
  2885   assume as:"\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp" "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}"
  2886   show "f {a..b} = opp (f ({a..b} \<inter> {x. x \<le> c})) (f ({a..b} \<inter> {x. c \<le> x}))"
  2887   proof(cases "c \<in> {a .. b}")
  2888     case False hence "c<a \<or> c>b" by auto
  2889     thus ?thesis apply-apply(erule disjE)
  2890     proof- assume "c<a" hence *:"{a..b} \<inter> {x. x \<le> c} = {1..0}"  "{a..b} \<inter> {x. c \<le> x} = {a..b}" by auto
  2891       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2892     next   assume "b<c" hence *:"{a..b} \<inter> {x. x \<le> c} = {a..b}"  "{a..b} \<inter> {x. c \<le> x} = {1..0}" by auto
  2893       show ?thesis unfolding * apply(subst as(1)[rule_format,of 0 1]) using assms by auto
  2894     qed
  2895   next case True hence *:"min (b) c = c" "max a c = c" by auto
  2896     have **:"0 < DIM(real)" by auto
  2897     have ***:"\<And>P Q. (\<chi>\<chi> i. if i = 0 then P i else Q i) = (P 0::real)" apply(subst euclidean_eq)
  2898       apply safe unfolding euclidean_lambda_beta' by auto
  2899     show ?thesis unfolding interval_split[OF **,unfolded Eucl_real_simps(1,3)] unfolding *** *
  2900     proof(cases "c = a \<or> c = b")
  2901       case False thus "f {a..b} = opp (f {a..c}) (f {c..b})"
  2902         apply-apply(subst as(2)[rule_format]) using True by auto
  2903     next case True thus "f {a..b} = opp (f {a..c}) (f {c..b})" apply-
  2904       proof(erule disjE) assume *:"c=a"
  2905         hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2906         thus ?thesis using assms unfolding * by auto
  2907       next assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2908         thus ?thesis using assms unfolding * by auto qed qed qed qed
  2909 
  2910 lemma operative_1_le: assumes "monoidal opp"
  2911   shows "operative opp f \<longleftrightarrow> ((\<forall>a b. b \<le> a \<longrightarrow> f {a..b::real} = neutral opp) \<and>
  2912                 (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f{a..c})(f{c..b}) = f {a..b}))"
  2913 unfolding operative_1_lt[OF assms]
  2914 proof safe fix a b c::"real" assume as:"\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a < c" "c < b"
  2915   show "opp (f {a..c}) (f {c..b}) = f {a..b}" apply(rule as(1)[rule_format]) using as(2-) by auto
  2916 next fix a b c ::"real" assume "\<forall>a b. b \<le> a \<longrightarrow> f {a..b} = neutral opp"
  2917     "\<forall>a b c. a < c \<and> c < b \<longrightarrow> opp (f {a..c}) (f {c..b}) = f {a..b}" "a \<le> c" "c \<le> b"
  2918   note as = this[rule_format]
  2919   show "opp (f {a..c}) (f {c..b}) = f {a..b}"
  2920   proof(cases "c = a \<or> c = b")
  2921     case False thus ?thesis apply-apply(subst as(2)) using as(3-) by(auto)
  2922     next case True thus ?thesis apply-
  2923       proof(erule disjE) assume *:"c=a" hence "f {a..c} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2924         thus ?thesis using assms unfolding * by auto
  2925       next               assume *:"c=b" hence "f {c..b} = neutral opp" apply-apply(rule as(1)[rule_format]) by auto
  2926         thus ?thesis using assms unfolding * by auto qed qed qed 
  2927 
  2928 subsection {* Special case of additivity we need for the FCT. *}
  2929 
  2930 lemma interval_bound_sing[simp]: "interval_upperbound {a} = a"  "interval_lowerbound {a} = a"
  2931   unfolding interval_upperbound_def interval_lowerbound_def  by auto
  2932 
  2933 lemma additive_tagged_division_1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  2934   assumes "a \<le> b" "p tagged_division_of {a..b}"
  2935   shows "setsum (\<lambda>(x,k). f(interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  2936 proof- let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
  2937   have ***:"\<forall>i<DIM(real). a $$ i \<le> b $$ i" using assms by auto 
  2938   have *:"operative op + ?f" unfolding operative_1_lt[OF monoidal_monoid] interval_eq_empty by auto
  2939   have **:"{a..b} \<noteq> {}" using assms(1) by auto note operative_tagged_division[OF monoidal_monoid * assms(2)]
  2940   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],THEN sym]
  2941   show ?thesis unfolding * apply(subst setsum_iterate[THEN sym]) defer
  2942     apply(rule setsum_cong2) unfolding split_paired_all split_conv using assms(2) by auto qed
  2943 
  2944 subsection {* A useful lemma allowing us to factor out the content size. *}
  2945 
  2946 lemma has_integral_factor_content:
  2947   "(f has_integral i) {a..b} \<longleftrightarrow> (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p
  2948     \<longrightarrow> norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b}))"
  2949 proof(cases "content {a..b} = 0")
  2950   case True show ?thesis unfolding has_integral_null_eq[OF True] apply safe
  2951     apply(rule,rule,rule gauge_trivial,safe) unfolding setsum_content_null[OF True] True defer 
  2952     apply(erule_tac x=1 in allE,safe) defer apply(rule fine_division_exists[of _ a b],assumption)
  2953     apply(erule_tac x=p in allE) unfolding setsum_content_null[OF True] by auto
  2954 next case False note F = this[unfolded content_lt_nz[THEN sym]]
  2955   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
  2956   show ?thesis apply(subst has_integral)
  2957   proof safe fix e::real assume e:"e>0"
  2958     { assume "\<forall>e>0. ?P e op <" thus "?P (e * content {a..b}) op \<le>" apply(erule_tac x="e * content {a..b}" in allE)
  2959         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2960         using F e by(auto simp add:field_simps intro:mult_pos_pos) }
  2961     {  assume "\<forall>e>0. ?P (e * content {a..b}) op \<le>" thus "?P e op <" apply(erule_tac x="e / 2 / content {a..b}" in allE)
  2962         apply(erule impE) defer apply(erule exE,rule_tac x=d in exI)
  2963         using F e by(auto simp add:field_simps intro:mult_pos_pos) } qed qed
  2964 
  2965 subsection {* Fundamental theorem of calculus. *}
  2966 
  2967 lemma interval_bounds_real: assumes "a\<le>(b::real)"
  2968   shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"
  2969   apply(rule_tac[!] interval_bounds) using assms by auto
  2970 
  2971 lemma fundamental_theorem_of_calculus: fixes f::"real \<Rightarrow> 'a::banach"
  2972   assumes "a \<le> b"  "\<forall>x\<in>{a..b}. (f has_vector_derivative f' x) (at x within {a..b})"
  2973   shows "(f' has_integral (f b - f a)) ({a..b})"
  2974 unfolding has_integral_factor_content
  2975 proof safe fix e::real assume e:"e>0"
  2976   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
  2977   have *:"\<And>P Q. \<forall>x\<in>{a..b}. P x \<and> (\<forall>e>0. \<exists>d>0. Q x e d) \<Longrightarrow> \<forall>x. \<exists>(d::real)>0. x\<in>{a..b} \<longrightarrow> Q x e d" using e by blast
  2978   note this[OF assm,unfolded gauge_existence_lemma] from choice[OF this,unfolded Ball_def[symmetric]]
  2979   guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
  2980   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
  2981                  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
  2982     apply(rule_tac x="\<lambda>x. ball x (d x)" in exI,safe)
  2983     apply(rule gauge_ball_dependent,rule,rule d(1))
  2984   proof- fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. ball x (d x)) fine p"
  2985     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}" 
  2986       unfolding content_real[OF assms(1)] additive_tagged_division_1[OF assms(1) as(1),of f,THEN sym]
  2987       unfolding additive_tagged_division_1[OF assms(1) as(1),of "\<lambda>x. x",THEN sym]
  2988       unfolding setsum_right_distrib defer unfolding setsum_subtractf[THEN sym] 
  2989     proof(rule setsum_norm_le,safe) fix x k assume "(x,k)\<in>p"
  2990       note xk = tagged_division_ofD(2-4)[OF as(1) this] from this(3) guess u v apply-by(erule exE)+ note k=this
  2991       have *:"u \<le> v" using xk unfolding k by auto
  2992       have ball:"\<forall>xa\<in>k. xa \<in> ball x (d x)" using as(2)[unfolded fine_def,rule_format,OF `(x,k)\<in>p`,
  2993         unfolded split_conv subset_eq] .
  2994       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
  2995         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
  2996         apply(rule order_trans[OF _ norm_triangle_ineq4]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])
  2997         unfolding scaleR.diff_left by(auto simp add:algebra_simps)
  2998       also have "... \<le> e * norm (u - x) + e * norm (v - x)"
  2999         apply(rule add_mono) apply(rule d(2)[of "x" "u",unfolded o_def]) prefer 4
  3000         apply(rule d(2)[of "x" "v",unfolded o_def])
  3001         using ball[rule_format,of u] ball[rule_format,of v] 
  3002         using xk(1-2) unfolding k subset_eq by(auto simp add:dist_real_def) 
  3003       also have "... \<le> e * (interval_upperbound k - interval_lowerbound k)"
  3004         unfolding k interval_bounds_real[OF *] using xk(1) unfolding k by(auto simp add:dist_real_def field_simps)
  3005       finally show "norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k))) \<le>
  3006         e * (interval_upperbound k - interval_lowerbound k)" unfolding k interval_bounds_real[OF *] content_real[OF *] .
  3007     qed(insert as, auto) qed qed
  3008 
  3009 subsection {* Attempt a systematic general set of "offset" results for components. *}
  3010 
  3011 lemma gauge_modify:
  3012   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
  3013   shows "gauge (\<lambda>x y. d (f x) (f y))"
  3014   using assms unfolding gauge_def apply safe defer apply(erule_tac x="f x" in allE)
  3015   apply(erule_tac x="d (f x)" in allE) unfolding mem_def Collect_def by auto
  3016 
  3017 subsection {* Only need trivial subintervals if the interval itself is trivial. *}
  3018 
  3019 lemma division_of_nontrivial: fixes s::"('a::ordered_euclidean_space) set set"
  3020   assumes "s division_of {a..b}" "content({a..b}) \<noteq> 0"
  3021   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of {a..b}" using assms(1) apply-
  3022 proof(induct "card s" arbitrary:s rule:nat_less_induct)
  3023   fix s::"'a set set" assume assm:"s division_of {a..b}"
  3024     "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow> x division_of {a..b} \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of {a..b}" 
  3025   note s = division_ofD[OF assm(1)] let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of {a..b}"
  3026   { presume *:"{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
  3027     show ?thesis apply cases defer apply(rule *,assumption) using assm(1) by auto }
  3028   assume noteq:"{k \<in> s. content k \<noteq> 0} \<noteq> s"
  3029   then obtain k where k:"k\<in>s" "content k = 0" by auto
  3030   from s(4)[OF k(1)] guess c d apply-by(erule exE)+ note k=k this
  3031   from k have "card s > 0" unfolding card_gt_0_iff using assm(1) by auto
  3032   hence card:"card (s - {k}) < card s" using assm(1) k(1) apply(subst card_Diff_singleton_if) by auto
  3033   have *:"closed (\<Union>(s - {k}))" apply(rule closed_Union) defer apply rule apply(drule DiffD1,drule s(4))
  3034     apply safe apply(rule closed_interval) using assm(1) by auto
  3035   have "k \<subseteq> \<Union>(s - {k})" apply safe apply(rule *[unfolded closed_limpt,rule_format]) unfolding islimpt_approachable
  3036   proof safe fix x and e::real assume as:"x\<in>k" "e>0"
  3037     from k(2)[unfolded k content_eq_0] guess i .. 
  3038     hence i:"c$$i = d$$i" "i<DIM('a)" using s(3)[OF k(1),unfolded k] unfolding interval_ne_empty by auto
  3039     hence xi:"x$$i = d$$i" using as unfolding k mem_interval by smt
  3040     def y \<equiv> "(\<chi>\<chi> j. if j = i then if c$$i \<le> (a$$i + b$$i) / 2 then c$$i +
  3041       min e (b$$i - c$$i) / 2 else c$$i - min e (c$$i - a$$i) / 2 else x$$j)::'a"
  3042     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x=y in bexI) 
  3043     proof have "d \<in> {c..d}" using s(3)[OF k(1)] unfolding k interval_eq_empty mem_interval by(fastsimp simp add: not_less)
  3044       hence "d \<in> {a..b}" using s(2)[OF k(1)] unfolding k by auto note di = this[unfolded mem_interval,THEN spec[where x=i]]
  3045       hence xyi:"y$$i \<noteq> x$$i" unfolding y_def unfolding i xi euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
  3046         apply(cases) apply(subst if_P,assumption) unfolding if_not_P not_le using as(2)
  3047         using assms(2)[unfolded content_eq_0] using i(2) by smt+ 
  3048       thus "y \<noteq> x" unfolding euclidean_eq[where 'a='a] using i by auto
  3049       have *:"{..<DIM('a)} = insert i ({..<DIM('a)} - {i})" using i by auto
  3050       have "norm (y - x) < e + setsum (\<lambda>i. 0) {..<DIM('a)}" apply(rule le_less_trans[OF norm_le_l1])
  3051         apply(subst *,subst setsum_insert) prefer 3 apply(rule add_less_le_mono)
  3052       proof- show "\<bar>(y - x) $$ i\<bar> < e" unfolding y_def euclidean_simps euclidean_lambda_beta'[OF i(2)] if_P[OF refl]
  3053           apply(cases) apply(subst if_P,assumption) unfolding if_not_P unfolding i xi using di as(2) by auto
  3054         show "(\<Sum>i\<in>{..<DIM('a)} - {i}. \<bar>(y - x) $$ i\<bar>) \<le> (\<Sum>i\<in>{..<DIM('a)}. 0)" unfolding y_def by auto 
  3055       qed auto thus "dist y x < e" unfolding dist_norm by auto
  3056       have "y\<notin>k" unfolding k mem_interval apply rule apply(erule_tac x=i in allE) using xyi unfolding k i xi by auto
  3057       moreover have "y \<in> \<Union>s" unfolding s mem_interval
  3058       proof(rule,rule) note simps = y_def euclidean_lambda_beta' if_not_P
  3059         fix j assume j:"j<DIM('a)" show "a $$ j \<le> y $$ j \<and> y $$ j \<le> b $$ j" 
  3060         proof(cases "j = i") case False have "x \<in> {a..b}" using s(2)[OF k(1)] as(1) by auto
  3061           thus ?thesis using j apply- unfolding simps if_not_P[OF False] unfolding mem_interval by auto
  3062         next case True note T = this show ?thesis
  3063           proof(cases "c $$ i \<le> (a $$ i + b $$ i) / 2")
  3064             case True show ?thesis unfolding simps if_P[OF T] if_P[OF True] unfolding i
  3065               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps) 
  3066           next case False thus ?thesis unfolding simps if_P[OF T] if_not_P[OF False] unfolding i
  3067               using True as(2) di apply-apply rule unfolding T by (auto simp add:field_simps)
  3068           qed qed qed
  3069       ultimately show "y \<in> \<Union>(s - {k})" by auto
  3070     qed qed hence "\<Union>(s - {k}) = {a..b}" unfolding s(6)[THEN sym] by auto
  3071   hence  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of {a..b}" apply-apply(rule assm(2)[rule_format,OF card refl])
  3072     apply(rule division_ofI) defer apply(rule_tac[1-4] s) using assm(1) by auto
  3073   moreover have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}" using k by auto ultimately show ?thesis by auto qed
  3074 
  3075 subsection {* Integrabibility on subintervals. *}
  3076 
  3077 lemma operative_integrable: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
  3078   "operative op \<and> (\<lambda>i. f integrable_on i)"
  3079   unfolding operative_def neutral_and apply safe apply(subst integrable_on_def)
  3080   unfolding has_integral_null_eq apply(rule,rule refl) apply(rule,assumption,assumption)+
  3081   unfolding integrable_on_def by(auto intro!: has_integral_split)
  3082 
  3083 lemma integrable_subinterval: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach" 
  3084   assumes "f integrable_on {a..b}" "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}" 
  3085   apply(cases "{c..d} = {}") defer apply(rule partial_division_extend_1[OF assms(2)],assumption)
  3086   using operative_division_and[OF operative_integrable,THEN sym,of _ _ _ f] assms(1) by auto
  3087 
  3088 subsection {* Combining adjacent intervals in 1 dimension. *}
  3089 
  3090 lemma has_integral_combine: assumes "(a::real) \<le> c" "c \<le> b"
  3091   "(f has_integral i) {a..c}" "(f has_integral (j::'a::banach)) {c..b}"
  3092   shows "(f has_integral (i + j)) {a..b}"
  3093 proof- note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
  3094   note conjunctD2[OF this,rule_format] note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
  3095   hence "f integrable_on {a..b}" apply- apply(rule ccontr) apply(subst(asm) if_P) defer
  3096     apply(subst(asm) if_P) using assms(3-) by auto
  3097   with * show ?thesis apply-apply(subst(asm) if_P) defer apply(subst(asm) if_P) defer apply(subst(asm) if_P)
  3098     unfolding lifted.simps using assms(3-) by(auto simp add: integrable_on_def integral_unique) qed
  3099 
  3100 lemma integral_combine: fixes f::"real \<Rightarrow> 'a::banach"
  3101   assumes "a \<le> c" "c \<le> b" "f integrable_on ({a..b})"
  3102   shows "integral {a..c} f + integral {c..b} f = integral({a..b}) f"
  3103   apply(rule integral_unique[THEN sym]) apply(rule has_integral_combine[OF assms(1-2)])
  3104   apply(rule_tac[!] integrable_integral integrable_subinterval[OF assms(3)])+ using assms(1-2) by auto
  3105 
  3106 lemma integrable_combine: fixes f::"real \<Rightarrow> 'a::banach"
  3107   assumes "a \<le> c" "c \<le> b" "f integrable_on {a..c}" "f integrable_on {c..b}"
  3108   shows "f integrable_on {a..b}" using assms unfolding integrable_on_def by(fastsimp intro!:has_integral_combine)
  3109 
  3110 subsection {* Reduce integrability to "local" integrability. *}
  3111 
  3112 lemma integrable_on_little_subintervals: fixes f::"'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
  3113   assumes "\<forall>x\<in>{a..b}. \<exists>d>0. \<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v}"
  3114   shows "f integrable_on {a..b}"
  3115 proof- have "\<forall>x. \<exists>d. x\<in>{a..b} \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> {u..v} \<and> {u..v} \<subseteq> ball x d \<and> {u..v} \<subseteq> {a..b} \<longrightarrow> f integrable_on {u..v})"
  3116     using assms by auto note this[unfolded gauge_existence_lemma] from choice[OF this] guess d .. note d=this[rule_format]
  3117   guess p apply(rule fine_division_exists[OF gauge_ball_dependent,of d a b]) using d by auto note p=this(1-2)
  3118   note division_of_tagged_division[OF this(1)] note * = operative_division_and[OF operative_integrable,OF this,THEN sym,of f]
  3119   show ?thesis unfolding * apply safe unfolding snd_conv
  3120   proof- fix x k assume "(x,k) \<in> p" note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
  3121     thus "f integrable_on k" apply safe apply(rule d[THEN conjunct2,rule_format,of x]) by auto qed qed
  3122 
  3123 subsection {* Second FCT or existence of antiderivative. *}
  3124 
  3125 lemma integrable_const[intro]:"(\<lambda>x. c) integrable_on {a..b}"
  3126   unfolding integrable_on_def by(rule,rule has_integral_const)
  3127 
  3128 lemma integral_has_vector_derivative: fixes f::"real \<Rightarrow> 'a::banach"
  3129   assumes "continuous_on {a..b} f" "x \<in> {a..b}"
  3130   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
  3131   unfolding has_vector_derivative_def has_derivative_within_alt
  3132 apply safe apply(rule scaleR.bounded_linear_left)
  3133 proof- fix e::real assume e:"e>0"
  3134   note compact_uniformly_continuous[OF assms(1) compact_interval,unfolded uniformly_continuous_on_def]
  3135   from this[rule_format,OF e] guess d apply-by(erule conjE exE)+ note d=this[rule_format]
  3136   let ?I = "\<lambda>a b. integral {a..b} f"
  3137   show "\<exists>d>0. \<forall>y\<in>{a..b}. norm (y - x) < d \<longrightarrow> norm (?I a y - ?I a x - (y - x) *\<^sub>R f x) \<le> e * norm (y - x)"
  3138   proof(rule,rule,rule d,safe) case goal1 show ?case proof(cases "y < x")
  3139       case False have "f integrable_on {a..y}" apply(rule integrable_subinterval,rule integrable_continuous)
  3140         apply(rule assms)  unfolding not_less using assms(2) goal1 by auto
  3141       hence *:"?I a y - ?I a x = ?I x y" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
  3142         using False unfolding not_less using assms(2) goal1 by auto
  3143       have **:"norm (y - x) = content {x..y}" apply(subst content_real) using False unfolding not_less by auto
  3144       show ?thesis unfolding ** apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
  3145         defer apply(rule has_integral_sub) apply(rule integrable_integral)
  3146         apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
  3147       proof- show "{x..y} \<subseteq> {a..b}" using goal1 assms(2) by auto
  3148         have *:"y - x = norm(y - x)" using False by auto
  3149         show "((\<lambda>xa. f x) has_integral (y - x) *\<^sub>R f x) {x.. y}" apply(subst *) unfolding ** by auto
  3150         show "\<forall>xa\<in>{x..y}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
  3151           apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
  3152       qed(insert e,auto)
  3153     next case True have "f integrable_on {a..x}" apply(rule integrable_subinterval,rule integrable_continuous)
  3154         apply(rule assms)+  unfolding not_less using assms(2) goal1 by auto
  3155       hence *:"?I a x - ?I a y = ?I y x" unfolding algebra_simps apply(subst eq_commute) apply(rule integral_combine)
  3156         using True using assms(2) goal1 by auto
  3157       have **:"norm (y - x) = content {y..x}" apply(subst content_real) using True unfolding not_less by auto
  3158       have ***:"\<And>fy fx c::'a. fx - fy - (y - x) *\<^sub>R c = -(fy - fx - (x - y) *\<^sub>R c)" unfolding scaleR_left.diff by auto 
  3159       show ?thesis apply(subst ***) unfolding norm_minus_cancel **
  3160         apply(rule has_integral_bound[where f="(\<lambda>u. f u - f x)"]) unfolding * unfolding o_def
  3161         defer apply(rule has_integral_sub) apply(subst minus_minus[THEN sym]) unfolding minus_minus
  3162         apply(rule integrable_integral) apply(rule integrable_subinterval,rule integrable_continuous) apply(rule assms)+
  3163       proof- show "{y..x} \<subseteq> {a..b}" using goal1 assms(2) by auto
  3164         have *:"x - y = norm(y - x)" using True by auto
  3165         show "((\<lambda>xa. f x) has_integral (x - y) *\<^sub>R f x) {y..x}" apply(subst *) unfolding ** by auto
  3166         show "\<forall>xa\<in>{y..x}. norm (f xa - f x) \<le> e" apply safe apply(rule less_imp_le)
  3167           apply(rule d(2)[unfolded dist_norm]) using assms(2) using goal1 by auto
  3168       qed(insert e,auto) qed qed qed
  3169 
  3170 lemma antiderivative_continuous: assumes "continuous_on {a..b::real} f"
  3171   obtains g where "\<forall>x\<in> {a..b}. (g has_vector_derivative (f(x)::_::banach)) (at x within {a..b})"
  3172   apply(rule that,rule) using integral_has_vector_derivative[OF assms] by auto
  3173 
  3174 subsection {* Combined fundamental theorem of calculus. *}
  3175 
  3176 lemma antiderivative_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach" assumes "continuous_on {a..b} f"
  3177   obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u..v}"
  3178 proof- from antiderivative_continuous[OF assms] guess g . note g=this
  3179   show ?thesis apply(rule that[of g])
  3180   proof safe case goal1 have "\<forall>x\<in>{u..v}. (g has_vector_derivative f x) (at x within {u..v})"
  3181       apply(rule,rule has_vector_derivative_within_subset) apply(rule g[rule_format]) using goal1(1-2) by auto
  3182     thus ?case using fundamental_theorem_of_calculus[OF goal1(3),of "g" "f"] by auto qed qed
  3183 
  3184 subsection {* General "twiddling" for interval-to-interval function image. *}
  3185 
  3186 lemma has_integral_twiddle:
  3187   assumes "0 < r" "\<forall>x. h(g x) = x" "\<forall>x. g(h x) = x" "\<forall>x. continuous (at x) g"
  3188   "\<forall>u v. \<exists>w z. g ` {u..v} = {w..z}"
  3189   "\<forall>u v. \<exists>w z. h ` {u..v} = {w..z}"
  3190   "\<forall>u v. content(g ` {u..v}) = r * content {u..v}"
  3191   "(f has_integral i) {a..b}"
  3192   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` {a..b})"
  3193 proof- { presume *:"{a..b} \<noteq> {} \<Longrightarrow> ?thesis"
  3194     show ?thesis apply cases defer apply(rule *,assumption)
  3195     proof- case goal1 thus ?thesis unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed }
  3196   assume "{a..b} \<noteq> {}" from assms(6)[rule_format,of a b] guess w z apply-by(erule exE)+ note wz=this
  3197   have inj:"inj g" "inj h" unfolding inj_on_def apply safe apply(rule_tac[!] ccontr)
  3198     using assms(2) apply(erule_tac x=x in allE) using assms(2) apply(erule_tac x=y in allE) defer
  3199     using assms(3) apply(erule_tac x=x in allE) using assms(3) apply(erule_tac x=y in allE) by auto
  3200   show ?thesis unfolding has_integral_def has_integral_compact_interval_def apply(subst if_P) apply(rule,rule,rule wz)
  3201   proof safe fix e::real assume e:"e>0" hence "e * r > 0" using assms(1) by(rule mult_pos_pos)
  3202     from assms(8)[unfolded has_integral,rule_format,OF this] guess d apply-by(erule exE conjE)+ note d=this[rule_format]
  3203     def d' \<equiv> "\<lambda>x y. d (g x) (g y)" have d':"\<And>x. d' x = {y. g y \<in> (d (g x))}" unfolding d'_def by(auto simp add:mem_def)
  3204     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
  3205     proof(rule_tac x=d' in exI,safe) show "gauge d'" using d(1) unfolding gauge_def d' using continuous_open_preimage_univ[OF assms(4)] by auto
  3206       fix p assume as:"p tagged_division_of h ` {a..b}" "d' fine p" note p = tagged_division_ofD[OF as(1)] 
  3207       have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of {a..b} \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p" unfolding tagged_division_of 
  3208       proof safe show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)" using as by auto
  3209         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p" using as(2) unfolding fine_def d' by auto
  3210         fix x k assume xk[intro]:"(x,k) \<in> p" show "g x \<in> g ` k" using p(2)[OF xk] by auto
  3211         show "\<exists>u v. g ` k = {u..v}" using p(4)[OF xk] using assms(5-6) by auto
  3212         { fix y assume "y \<in> k" thus "g y \<in> {a..b}" "g y \<in> {a..b}" using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
  3213             using assms(2)[rule_format,of y] unfolding inj_image_mem_iff[OF inj(2)] by auto }
  3214         fix x' k' assume xk':"(x',k') \<in> p" fix z assume "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
  3215         hence *:"interior (g ` k) \<inter> interior (g ` k') \<noteq> {}" by auto
  3216         have same:"(x, k) = (x', k')" apply-apply(rule ccontr,drule p(5)[OF xk xk'])
  3217         proof- assume as:"interior k \<inter> interior k' = {}" from nonempty_witness[OF *] guess z .
  3218           hence "z \<in> g ` (interior k \<inter> interior k')" using interior_image_subset[OF assms(4) inj(1)]
  3219             unfolding image_Int[OF inj(1)] by auto thus False using as by blast
  3220         qed thus "g x = g x'" by auto
  3221         { fix z assume "z \<in> k"  thus  "g z \<in> g ` k'" using same by auto }
  3222         { fix z assume "z \<in> k'" thus  "g z \<in> g ` k"  using same by auto }
  3223       next fix x assume "x \<in> {a..b}" hence "h x \<in>  \<Union>{k. \<exists>x. (x, k) \<in> p}" using p(6) by auto
  3224         then guess X unfolding Union_iff .. note X=this from this(1) guess y unfolding mem_Collect_eq ..
  3225         thus "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}" apply-
  3226           apply(rule_tac X="g ` X" in UnionI) defer apply(rule_tac x="h x" in image_eqI)
  3227           using X(2) assms(3)[rule_format,of x] by auto
  3228       qed note ** = d(2)[OF this] have *:"inj_on (\<lambda>(x, k). (g x, g ` k)) p" using inj(1) unfolding inj_on_def by fastsimp
  3229        have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _") unfolding algebra_simps add_left_cancel
  3230         unfolding setsum_reindex[OF *] apply(subst scaleR_right.setsum) defer apply(rule setsum_cong2) unfolding o_def split_paired_all split_conv
  3231         apply(drule p(4)) apply safe unfolding assms(7)[rule_format] using p by auto
  3232       also have "... = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r") unfolding scaleR.diff_right scaleR.scaleR_left[THEN sym]
  3233         unfolding real_scaleR_def using assms(1) by auto finally have *:"?l = ?r" .
  3234       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e" using ** unfolding * unfolding norm_scaleR
  3235         using assms(1) by(auto simp add:field_simps) qed qed qed
  3236 
  3237 subsection {* Special case of a basic affine transformation. *}
  3238 
  3239 lemma interval_image_affinity_interval: shows "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space) + c) ` {a..b} = {u..v}"
  3240   unfolding image_affinity_interval by auto
  3241 
  3242 lemma setprod_cong2: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "setprod f A = setprod g A"
  3243   apply(rule setprod_cong) using assms by auto
  3244 
  3245 lemma content_image_affinity_interval: 
  3246  "content((\<lambda>x::'a::ordered_euclidean_space. m *\<^sub>R x + c) ` {a..b}) = (abs m) ^ DIM('a) * content {a..b}" (is "?l = ?r")
  3247 proof- { presume *:"{a..b}\<noteq>{} \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  3248       unfolding not_not using content_empty by auto }
  3249   have *:"DIM('a) = card {..<DIM('a)}" by auto
  3250   assume as:"{a..b}\<noteq>{}" show ?thesis proof(cases "m \<ge> 0")
  3251     case True show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_P[OF True]
  3252       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
  3253       apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
  3254       apply(rule setprod_cong2) using True as unfolding interval_ne_empty euclidean_simps not_le  
  3255       by(auto simp add:field_simps intro:mult_left_mono)
  3256   next case False show ?thesis unfolding image_affinity_interval if_not_P[OF as] if_not_P[OF False]
  3257       unfolding content_closed_interval'[OF as] apply(subst content_closed_interval') defer apply(subst(2) *)
  3258       apply(subst setprod_constant[THEN sym]) apply(rule finite_lessThan) unfolding setprod_timesf[THEN sym]
  3259       apply(rule setprod_cong2) using False as unfolding interval_ne_empty euclidean_simps not_le 
  3260       by(auto simp add:field_simps mult_le_cancel_left_neg) qed qed
  3261 
  3262 lemma has_integral_affinity: fixes a::"'a::ordered_euclidean_space" assumes "(f has_integral i) {a..b}" "m \<noteq> 0"
  3263   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (abs(m) ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` {a..b})"
  3264   apply(rule has_integral_twiddle,safe) apply(rule zero_less_power) unfolding euclidean_eq[where 'a='a]
  3265   unfolding scaleR_right_distrib euclidean_simps scaleR.scaleR_left[THEN sym]
  3266   defer apply(insert assms(2), simp add:field_simps) apply(insert assms(2), simp add:field_simps)
  3267   apply(rule continuous_intros)+ apply(rule interval_image_affinity_interval)+ apply(rule content_image_affinity_interval) using assms by auto
  3268 
  3269 lemma integrable_affinity: assumes "f integrable_on {a..b}" "m \<noteq> 0"
  3270   shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` {a..b})"
  3271   using assms unfolding integrable_on_def apply safe apply(drule has_integral_affinity) by auto
  3272 
  3273 subsection {* Special case of stretching coordinate axes separately. *}
  3274 
  3275 lemma image_stretch_interval:
  3276   "(\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} =
  3277   (if {a..b} = {} then {} else {(\<chi>\<chi> k. min (m(k) * a$$k) (m(k) * b$$k))::'a ..  (\<chi>\<chi> k. max (m(k) * a$$k) (m(k) * b$$k))})"
  3278   (is "?l = ?r")
  3279 proof(cases "{a..b}={}") case True thus ?thesis unfolding True by auto
  3280 next have *:"\<And>P Q. (\<forall>i<DIM('a). P i) \<and> (\<forall>i<DIM('a). Q i) \<longleftrightarrow> (\<forall>i<DIM('a). P i \<and> Q i)" by auto
  3281   case False note ab = this[unfolded interval_ne_empty]
  3282   show ?thesis apply-apply(rule set_eqI)
  3283   proof- fix x::"'a" have **:"\<And>P Q. (\<forall>i<DIM('a). P i = Q i) \<Longrightarrow> (\<forall>i<DIM('a). P i) = (\<forall>i<DIM('a). Q i)" by auto
  3284     show "x \<in> ?l \<longleftrightarrow> x \<in> ?r" unfolding if_not_P[OF False] 
  3285       unfolding image_iff mem_interval Bex_def euclidean_simps euclidean_eq[where 'a='a] *
  3286       unfolding imp_conjR[THEN sym] apply(subst euclidean_lambda_beta'') apply(subst lambda_skolem'[THEN sym])
  3287       apply(rule **,rule,rule) unfolding euclidean_lambda_beta'
  3288     proof- fix i assume i:"i<DIM('a)" show "(\<exists>xa. (a $$ i \<le> xa \<and> xa \<le> b $$ i) \<and> x $$ i = m i * xa) =
  3289         (min (m i * a $$ i) (m i * b $$ i) \<le> x $$ i \<and> x $$ i \<le> max (m i * a $$ i) (m i * b $$ i))"
  3290       proof(cases "m i = 0") case True thus ?thesis using ab i by auto
  3291       next case False hence "0 < m i \<or> 0 > m i" by auto thus ?thesis apply-
  3292         proof(erule disjE) assume as:"0 < m i" hence *:"min (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
  3293             "max (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab i unfolding min_def max_def by auto
  3294           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
  3295             using as by(auto simp add:field_simps)
  3296         next assume as:"0 > m i" hence *:"max (m i * a $$ i) (m i * b $$ i) = m i * a $$ i"
  3297             "min (m i * a $$ i) (m i * b $$ i) = m i * b $$ i" using ab as i unfolding min_def max_def 
  3298             by(auto simp add:field_simps mult_le_cancel_left_neg intro: order_antisym)
  3299           show ?thesis unfolding * apply rule defer apply(rule_tac x="1 / m i * x$$i" in exI)
  3300             using as by(auto simp add:field_simps) qed qed qed qed qed 
  3301 
  3302 lemma interval_image_stretch_interval: "\<exists>u v. (\<lambda>x. \<chi>\<chi> k. m k * x$$k) ` {a..b::'a::ordered_euclidean_space} = {u..v::'a}"
  3303   unfolding image_stretch_interval by auto 
  3304 
  3305 lemma content_image_stretch_interval:
  3306   "content((\<lambda>x::'a::ordered_euclidean_space. (\<chi>\<chi> k. m k * x$$k)::'a) ` {a..b}) = abs(setprod m {..<DIM('a)}) * content({a..b})"
  3307 proof(cases "{a..b} = {}") case True thus ?thesis
  3308     unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
  3309 next case False hence "(\<lambda>x. (\<chi>\<chi> k. m k * x $$ k)::'a) ` {a..b} \<noteq> {}" by auto
  3310   thus ?thesis using False unfolding content_def image_stretch_interval apply- unfolding interval_bounds' if_not_P
  3311     unfolding abs_setprod setprod_timesf[THEN sym] apply(rule setprod_cong2) unfolding lessThan_iff euclidean_lambda_beta'
  3312   proof- fix i assume i:"i<DIM('a)" have "(m i < 0 \<or> m i > 0) \<or> m i = 0" by auto
  3313     thus "max (m i * a $$ i) (m i * b $$ i) - min (m i * a $$ i) (m i * b $$ i) = \<bar>m i\<bar> * (b $$ i - a $$ i)"
  3314       apply-apply(erule disjE)+ unfolding min_def max_def using False[unfolded interval_ne_empty,rule_format,of i] i 
  3315       by(auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos) qed qed
  3316 
  3317 lemma has_integral_stretch: fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
  3318   assumes "(f has_integral i) {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
  3319   shows "((\<lambda>x. f(\<chi>\<chi> k. m k * x$$k)) has_integral
  3320              ((1/(abs(setprod m {..<DIM('a)}))) *\<^sub>R i)) ((\<lambda>x. (\<chi>\<chi> k. 1/(m k) * x$$k)::'a) ` {a..b})"
  3321   apply(rule has_integral_twiddle[where f=f]) unfolding zero_less_abs_iff content_image_stretch_interval
  3322   unfolding image_stretch_interval empty_as_interval euclidean_eq[where 'a='a] using assms
  3323 proof- show "\<forall>y::'a. continuous (at y) (\<lambda>x. (\<chi>\<chi> k. m k * x $$ k)::'a)"
  3324    apply(rule,rule linear_continuous_at) unfolding linear_linear
  3325    unfolding linear_def euclidean_simps euclidean_eq[where 'a='a] by(auto simp add:field_simps) qed auto
  3326 
  3327 lemma integrable_stretch:  fixes f::"'a::ordered_euclidean_space => 'b::real_normed_vector"
  3328   assumes "f integrable_on {a..b}" "\<forall>k<DIM('a). ~(m k = 0)"
  3329   shows "(\<lambda>x::'a. f(\<chi>\<chi> k. m k * x$$k)) integrable_on ((\<lambda>x. \<chi>\<chi> k. 1/(m k) * x$$k) ` {a..b})"
  3330   using assms unfolding integrable_on_def apply-apply(erule exE) 
  3331   apply(drule has_integral_stretch,assumption) by auto
  3332 
  3333 subsection {* even more special cases. *}
  3334 
  3335 lemma uminus_interval_vector[simp]:"uminus ` {a..b} = {-b .. -a::'a::ordered_euclidean_space}"
  3336   apply(rule set_eqI,rule) defer unfolding image_iff
  3337   apply(rule_tac x="-x" in bexI) by(auto simp add:minus_le_iff le_minus_iff eucl_le[where 'a='a])
  3338 
  3339 lemma has_integral_reflect_lemma[intro]: assumes "(f has_integral i) {a..b}"
  3340   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
  3341   using has_integral_affinity[OF assms, of "-1" 0] by auto
  3342 
  3343 lemma has_integral_reflect[simp]: "((\<lambda>x. f(-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) ({a..b})"
  3344   apply rule apply(drule_tac[!] has_integral_reflect_lemma) by auto
  3345 
  3346 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on {-b..-a} \<longleftrightarrow> f integrable_on {a..b}"
  3347   unfolding integrable_on_def by auto
  3348 
  3349 lemma integral_reflect[simp]: "integral {-b..-a} (\<lambda>x. f(-x)) = integral ({a..b}) f"
  3350   unfolding integral_def by auto
  3351 
  3352 subsection {* Stronger form of FCT; quite a tedious proof. *}
  3353 
  3354 lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)" by(meson zero_less_one)
  3355 
  3356 lemma additive_tagged_division_1': fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  3357   assumes "a \<le> b" "p tagged_division_of {a..b}"
  3358   shows "setsum (\<lambda>(x,k). f (interval_upperbound k) - f(interval_lowerbound k)) p = f b - f a"
  3359   using additive_tagged_division_1[OF _ assms(2), of f] using assms(1) by auto
  3360 
  3361 lemma split_minus[simp]:"(\<lambda>(x, k). f x k) x - (\<lambda>(x, k). g x k) x = (\<lambda>(x, k). f x k - g x k) x"
  3362   unfolding split_def by(rule refl)
  3363 
  3364 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
  3365   apply(subst(asm)(2) norm_minus_cancel[THEN sym])
  3366   apply(drule norm_triangle_le) by(auto simp add:algebra_simps)
  3367 
  3368 lemma fundamental_theorem_of_calculus_interior: fixes f::"real => 'a::real_normed_vector"
  3369   assumes"a \<le> b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_vector_derivative f'(x)) (at x)"
  3370   shows "(f' has_integral (f b - f a)) {a..b}"
  3371 proof- { presume *:"a < b \<Longrightarrow> ?thesis" 
  3372     show ?thesis proof(cases,rule *,assumption)
  3373       assume "\<not> a < b" hence "a = b" using assms(1) by auto
  3374       hence *:"{a .. b} = {b}" "f b - f a = 0" by(auto simp add:  order_antisym)
  3375       show ?thesis unfolding *(2) apply(rule has_integral_null) unfolding content_eq_0 using * `a=b` by auto
  3376     qed } assume ab:"a < b"
  3377   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
  3378                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
  3379   { presume "\<And>e. e>0 \<Longrightarrow> ?P e" thus ?thesis unfolding has_integral_factor_content by auto }
  3380   fix e::real assume e:"e>0"
  3381   note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
  3382   note conjunctD2[OF this] note bounded=this(1) and this(2)
  3383   from this(2) have "\<forall>x\<in>{a<..<b}. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow> norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
  3384     apply-apply safe apply(erule_tac x=x in ballE,erule_tac x="e/2" in allE) using e by auto note this[unfolded bgauge_existence_lemma]
  3385   from choice[OF this] guess d .. note conjunctD2[OF this[rule_format]] note d = this[rule_format]
  3386   have "bounded (f ` {a..b})" apply(rule compact_imp_bounded compact_continuous_image)+ using compact_interval assms by auto
  3387   from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
  3388 
  3389   have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a..c} \<subseteq> {a..b} \<and> {a..c} \<subseteq> ball a da
  3390     \<longrightarrow> norm(content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
  3391   proof- have "a\<in>{a..b}" using ab by auto
  3392     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  3393     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0" using e ab by(auto simp add:field_simps)
  3394     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  3395     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
  3396     proof(cases "f' a = 0") case True
  3397       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  3398     next case False thus ?thesis
  3399         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI) using ab e by(auto simp add:field_simps) 
  3400     qed then guess l .. note l = conjunctD2[OF this]
  3401     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  3402     proof- fix c assume as:"a \<le> c" "{a..c} \<subseteq> {a..b}" "{a..c} \<subseteq> ball a (min k l)" 
  3403       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  3404       have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)" by(rule norm_triangle_ineq4)
  3405       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  3406       proof(rule add_mono) case goal1 have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>" using as' by auto
  3407         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  3408       next case goal2 show ?case apply(rule less_imp_le) apply(cases "a = c") defer
  3409           apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
  3410       qed finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4"
  3411         unfolding content_real[OF as(1)] by auto
  3412     qed qed then guess da .. note da=conjunctD2[OF this,rule_format]
  3413 
  3414   have "\<exists>db>0. \<forall>c\<le>b. {c..b} \<subseteq> {a..b} \<and> {c..b} \<subseteq> ball b db \<longrightarrow>
  3415     norm(content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
  3416   proof- have "b\<in>{a..b}" using ab by auto
  3417     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
  3418     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
  3419       using e ab by(auto simp add:field_simps)
  3420     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
  3421     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
  3422     proof(cases "f' b = 0") case True
  3423       thus ?thesis apply(rule_tac x=1 in exI) using ab e by(auto intro!:mult_nonneg_nonneg) 
  3424     next case False thus ?thesis 
  3425         apply(rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
  3426         using ab e by(auto simp add:field_simps)
  3427     qed then guess l .. note l = conjunctD2[OF this]
  3428     show ?thesis apply(rule_tac x="min k l" in exI) apply safe unfolding min_less_iff_conj apply(rule,(rule l k)+)
  3429     proof- fix c assume as:"c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)" 
  3430       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_interval]
  3431       have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)" by(rule norm_triangle_ineq4)
  3432       also have "... \<le> e * (b - a) / 8 + e * (b - a) / 8" 
  3433       proof(rule add_mono) case goal1 have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>" using as' by auto
  3434         thus ?case apply-apply(rule order_trans[OF _ l(2)]) unfolding norm_scaleR apply(rule mult_right_mono) by auto
  3435       next case goal2 show ?case apply(rule less_imp_le) apply(cases "b = c") defer apply(subst norm_minus_commute)
  3436           apply(rule k(2)[unfolded dist_norm]) using as' e ab by(auto simp add:field_simps)
  3437       qed finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4"
  3438         unfolding content_real[OF as(1)] by auto
  3439     qed qed then guess db .. note db=conjunctD2[OF this,rule_format]
  3440 
  3441   let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
  3442   show "?P e" apply(rule_tac x="?d" in exI)
  3443   proof safe case goal1 show ?case apply(rule gauge_ball_dependent) using ab db(1) da(1) d(1) by auto
  3444   next case goal2 note as=this let ?A = "{t. fst t \<in> {a, b}}" note p = tagged_division_ofD[OF goal2(1)]
  3445     have pA:"p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"  using goal2 by auto
  3446     note * = additive_tagged_division_1'[OF assms(1) goal2(1), THEN sym]
  3447     have **:"\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2" by arith
  3448     show ?case unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[THEN sym] split_minus
  3449       unfolding setsum_right_distrib apply(subst(2) pA,subst pA) unfolding setsum_Un_disjoint[OF pA(2-)]
  3450     proof(rule norm_triangle_le,rule **) 
  3451       case goal1 show ?case apply(rule order_trans,rule setsum_norm_le) apply(rule pA) defer apply(subst divide.setsum)
  3452       proof(rule order_refl,safe,unfold not_le o_def split_conv fst_conv,rule ccontr) fix x k assume as:"(x,k) \<in> p"
  3453           "e * (interval_upperbound k -  interval_lowerbound k) / 2
  3454           < norm (content k *\<^sub>R f' x - (f (interval_upperbound k) - f (interval_lowerbound k)))"
  3455         from p(4)[OF this(1)] guess u v apply-by(erule exE)+ note k=this
  3456         hence "u \<le> v" and uv:"{u,v}\<subseteq>{u..v}" using p(2)[OF as(1)] by auto
  3457         note result = as(2)[unfolded k interval_bounds_real[OF this(1)] content_real[OF this(1)]]
  3458 
  3459         assume as':"x \<noteq> a" "x \<noteq> b" hence "x \<in> {a<..<b}" using p(2-3)[OF as(1)] by auto
  3460         note  * = d(2)[OF this]
  3461         have "norm ((v - u) *\<^sub>R f' (x) - (f (v) - f (u))) =
  3462           norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))" 
  3463           apply(rule arg_cong[of _ _ norm]) unfolding scaleR_left.diff by auto 
  3464         also have "... \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)" apply(rule norm_triangle_le_sub)
  3465           apply(rule add_mono) apply(rule_tac[!] *) using fineD[OF goal2(2) as(1)] as' unfolding k subset_eq
  3466           apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE) using uv by(auto simp:dist_real_def)
  3467         also have "... \<le> e / 2 * norm (v - u)" using p(2)[OF as(1)] unfolding k by(auto simp add:field_simps)
  3468         finally have "e * (v - u) / 2 < e * (v - u) / 2"
  3469           apply- apply(rule less_le_trans[OF result]) using uv by auto thus False by auto qed
  3470 
  3471     next have *:"\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2" by auto
  3472       case goal2 show ?case apply(rule *) apply(rule setsum_nonneg) apply(rule,unfold split_paired_all split_conv)
  3473         defer unfolding setsum_Un_disjoint[OF pA(2-),THEN sym] pA(1)[THEN sym] unfolding setsum_right_distrib[THEN sym] 
  3474         apply(subst additive_tagged_division_1[OF _ as(1)]) apply(rule assms)
  3475       proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}}" note xk=IntD1[OF this]
  3476         from p(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  3477         with p(2)[OF xk] have "{u..v} \<noteq> {}" by auto
  3478         thus "0 \<le> e * ((interval_upperbound k) - (interval_lowerbound k))"
  3479           unfolding uv using e by(auto simp add:field_simps)
  3480       next have *:"\<And>s f t e. setsum f s = setsum f t \<Longrightarrow> norm(setsum f t) \<le> e \<Longrightarrow> norm(setsum f s) \<le> e" by auto
  3481         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
  3482           (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) \<le> e * (b - a) / 2" 
  3483           apply(rule *[where t="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
  3484           apply(rule setsum_mono_zero_right[OF pA(2)]) defer apply(rule) unfolding split_paired_all split_conv o_def
  3485         proof- fix x k assume "(x,k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
  3486           hence xk:"(x,k)\<in>p" "content k = 0" by auto from p(4)[OF xk(1)] guess u v apply-by(erule exE)+ note uv=this
  3487           have "k\<noteq>{}" using p(2)[OF xk(1)] by auto hence *:"u = v" using xk
  3488             unfolding uv content_eq_0 interval_eq_empty by auto
  3489           thus "content k *\<^sub>R (f' (x)) - (f ((interval_upperbound k)) - f ((interval_lowerbound k))) = 0" using xk unfolding uv by auto
  3490         next have *:"p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} = 
  3491             {t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}" by blast
  3492           have **:"\<And>s f. \<And>e::real. (\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y) \<Longrightarrow> (\<forall>x. x \<in> s \<longrightarrow> norm(f x) \<le> e)
  3493             \<Longrightarrow> e>0 \<Longrightarrow> norm(setsum f s) \<le> e"
  3494           proof(case_tac "s={}") case goal2 then obtain x where "x\<in>s" by auto hence *:"s = {x}" using goal2(1) by auto
  3495             thus ?case using `x\<in>s` goal2(2) by auto
  3496           qed auto
  3497           case goal2 show ?case apply(subst *, subst setsum_Un_disjoint) prefer 4
  3498             apply(rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"]) 
  3499             apply(rule norm_triangle_le,rule add_mono) apply(rule_tac[1-2] **)
  3500           proof- let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
  3501             have pa:"\<And>k. (a, k) \<in> p \<Longrightarrow> \<exists>v. k = {a .. v} \<and> a \<le> v" 
  3502             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  3503               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  3504               have u:"u = a" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  3505                 have "u \<ge> a" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "u\<noteq>a" ultimately
  3506                 have "u > a" by auto
  3507                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
  3508               qed thus ?case apply(rule_tac x=v in exI) unfolding uv using * by auto
  3509             qed
  3510             have pb:"\<And>k. (b, k) \<in> p \<Longrightarrow> \<exists>v. k = {v .. b} \<and> b \<ge> v" 
  3511             proof- case goal1 guess u v using p(4)[OF goal1] apply-by(erule exE)+ note uv=this
  3512               have *:"u \<le> v" using p(2)[OF goal1] unfolding uv by auto
  3513               have u:"v =  b" proof(rule ccontr)  have "u \<in> {u..v}" using p(2-3)[OF goal1(1)] unfolding uv by auto 
  3514                 have "v \<le>  b" using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto moreover assume "v\<noteq> b" ultimately
  3515                 have "v <  b" by auto
  3516                 thus False using p(2)[OF goal1(1)] unfolding uv by(auto simp add:)
  3517               qed thus ?case apply(rule_tac x=u in exI) unfolding uv using * by auto
  3518             qed
  3519 
  3520             show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  3521               unfolding mem_Collect_eq fst_conv snd_conv apply safe
  3522             proof- fix x k k' assume k:"( a, k) \<in> p" "( a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  3523               guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
  3524               guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (min (v) (v'))"
  3525               have "{ a <..< ?v} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
  3526               moreover have " ((a + ?v)/2) \<in> { a <..< ?v}" using k(3-)
  3527                 unfolding v v' content_eq_0 not_le by(auto simp add:not_le)
  3528               ultimately have " ((a + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  3529               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  3530               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  3531             qed 
  3532             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y" apply(rule,rule,rule,unfold split_paired_all)
  3533               unfolding mem_Collect_eq fst_conv snd_conv apply safe
  3534             proof- fix x k k' assume k:"( b, k) \<in> p" "( b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
  3535               guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
  3536               guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = " (max (v) (v'))"
  3537               have "{?v <..<  b} \<subseteq> k \<inter> k'" unfolding v v' by(auto simp add:) note subset_interior[OF this,unfolded interior_inter]
  3538               moreover have " ((b + ?v)/2) \<in> {?v <..<  b}" using k(3-) unfolding v v' content_eq_0 not_le by auto
  3539               ultimately have " ((b + ?v)/2) \<in> interior k \<inter> interior k'" unfolding interior_open[OF open_interval] by auto
  3540               hence *:"k = k'" apply- apply(rule ccontr) using p(5)[OF k(1-2)] by auto
  3541               { assume "x\<in>k" thus "x\<in>k'" unfolding * . } { assume "x\<in>k'" thus "x\<in>k" unfolding * . }
  3542             qed
  3543 
  3544             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
  3545             show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) - (f ((interval_upperbound k)) -
  3546               f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4" apply(rule,rule) unfolding mem_Collect_eq
  3547               unfolding split_paired_all fst_conv snd_conv 
  3548             proof safe case goal1 guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
  3549               have " ?a\<in>{ ?a..v}" using v(2) by auto hence "v \<le> ?b" using p(3)[OF goal1(1)] unfolding subset_eq v by auto
  3550               moreover have "{?a..v} \<subseteq> ball ?a da" using fineD[OF as(2) goal1(1)]
  3551                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe apply(erule_tac x=" x" in ballE)
  3552                 by(auto simp add:subset_eq dist_real_def v) ultimately
  3553               show ?case unfolding v interval_bounds_real[OF v(2)] apply- apply(rule da(2)[of "v"])
  3554                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
  3555             qed
  3556             show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' (x) -
  3557               (f ((interval_upperbound k)) - f ((interval_lowerbound k)))) x) \<le> e * (b - a) / 4"
  3558               apply(rule,rule) unfolding mem_Collect_eq unfolding split_paired_all fst_conv snd_conv 
  3559             proof safe case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
  3560               have " ?b\<in>{v.. ?b}" using v(2) by auto hence "v \<ge> ?a" using p(3)[OF goal1(1)]
  3561                 unfolding subset_eq v by auto
  3562               moreover have "{v..?b} \<subseteq> ball ?b db" using fineD[OF as(2) goal1(1)]
  3563                 apply-apply(subst(asm) if_P,rule refl) unfolding subset_eq apply safe
  3564                 apply(erule_tac x=" x" in ballE) using ab
  3565                 by(auto simp add:subset_eq v dist_real_def) ultimately
  3566               show ?case unfolding v unfolding interval_bounds_real[OF v(2)] apply- apply(rule db(2)[of "v"])
  3567                 using goal1 fineD[OF as(2) goal1(1)] unfolding v content_eq_0 by auto
  3568             qed
  3569           qed(insert p(1) ab e, auto simp add:field_simps) qed auto qed qed qed qed
  3570 
  3571 subsection {* Stronger form with finite number of exceptional points. *}
  3572 
  3573 lemma fundamental_theorem_of_calculus_interior_strong: fixes f::"real \<Rightarrow> 'a::banach"
  3574   assumes"finite s" "a \<le> b" "continuous_on {a..b} f"
  3575   "\<forall>x\<in>{a<..<b} - s. (f has_vector_derivative f'(x)) (at x)"
  3576   shows "(f' has_integral (f b - f a)) {a..b}" using assms apply- 
  3577 proof(induct "card s" arbitrary:s a b)
  3578   case 0 show ?case apply(rule fundamental_theorem_of_calculus_interior) using 0 by auto
  3579 next case (Suc n) from this(2) guess c s' apply-apply(subst(asm) eq_commute) unfolding card_Suc_eq
  3580     apply(subst(asm)(2) eq_commute) by(erule exE conjE)+ note cs = this[rule_format]
  3581   show ?case proof(cases "c\<in>{a<..<b}")
  3582     case False thus ?thesis apply- apply(rule Suc(1)[OF cs(3) _ Suc(4,5)]) apply safe defer
  3583       apply(rule Suc(6)[rule_format]) using Suc(3) unfolding cs by auto
  3584   next have *:"f b - f a = (f c - f a) + (f b - f c)" by auto
  3585     case True hence "a \<le> c" "c \<le> b" by auto
  3586     thus ?thesis apply(subst *) apply(rule has_integral_combine) apply assumption+
  3587       apply(rule_tac[!] Suc(1)[OF cs(3)]) using Suc(3) unfolding cs
  3588     proof- show "continuous_on {a..c} f" "continuous_on {c..b} f"
  3589         apply(rule_tac[!] continuous_on_subset[OF Suc(5)]) using True by auto
  3590       let ?P = "\<lambda>i j. \<forall>x\<in>{i<..<j} - s'. (f has_vector_derivative f' x) (at x)"
  3591       show "?P a c" "?P c b" apply safe apply(rule_tac[!] Suc(6)[rule_format]) using True unfolding cs by auto
  3592     qed auto qed qed
  3593 
  3594 lemma fundamental_theorem_of_calculus_strong: fixes f::"real \<Rightarrow> 'a::banach"
  3595   assumes "finite s" "a \<le> b" "continuous_on {a..b} f"
  3596   "\<forall>x\<in>{a..b} - s. (f has_vector_derivative f'(x)) (at x)"
  3597   shows "(f' has_integral (f(b) - f(a))) {a..b}"
  3598   apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
  3599   using assms(4) by auto
  3600 
  3601 lemma indefinite_integral_continuous_left: fixes f::"real \<Rightarrow> 'a::banach"
  3602   assumes "f integrable_on {a..b}" "a < c" "c \<le> b" "0 < e"
  3603   obtains d where "0 < d" "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm(integral {a..c} f - integral {a..t} f) < e"
  3604 proof- have "\<exists>w>0. \<forall>t. c - w < t \<and> t < c \<longrightarrow> norm(f c) * norm(c - t) < e / 3"
  3605   proof(cases "f c = 0") case False hence "0 < e / 3 / norm (f c)"
  3606       apply-apply(rule divide_pos_pos) using `e>0` by auto
  3607     thus ?thesis apply-apply(rule,rule,assumption,safe)
  3608     proof- fix t assume as:"t < c" and "c - e / 3 / norm (f c) < t"
  3609       hence "c - t < e / 3 / norm (f c)" by auto
  3610       hence "norm (c - t) < e / 3 / norm (f c)" using as by auto
  3611       thus "norm (f c) * norm (c - t) < e / 3" using False apply-
  3612         apply(subst mult_commute) apply(subst pos_less_divide_eq[THEN sym]) by auto
  3613     qed next case True show ?thesis apply(rule_tac x=1 in exI) unfolding True using `e>0` by auto
  3614   qed then guess w .. note w = conjunctD2[OF this,rule_format]
  3615   
  3616   have *:"e / 3 > 0" using assms by auto
  3617   have "f integrable_on {a..c}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) by auto
  3618   from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d1 ..
  3619   note d1 = conjunctD2[OF this,rule_format] def d \<equiv> "\<lambda>x. ball x w \<inter> d1 x"
  3620   have "gauge d" unfolding d_def using w(1) d1 by auto
  3621   note this[unfolded gauge_def,rule_format,of c] note conjunctD2[OF this]
  3622   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k .. note k=conjunctD2[OF this]
  3623 
  3624   let ?d = "min k (c - a)/2" show ?thesis apply(rule that[of ?d])
  3625   proof safe show "?d > 0" using k(1) using assms(2) by auto
  3626     fix t assume as:"c - ?d < t" "t \<le> c" let ?thesis = "norm (integral {a..c} f - integral {a..t} f) < e"
  3627     { presume *:"t < c \<Longrightarrow> ?thesis"
  3628       show ?thesis apply(cases "t = c") defer apply(rule *)
  3629         apply(subst less_le) using `e>0` as(2) by auto } assume "t < c"
  3630 
  3631     have "f integrable_on {a..t}" apply(rule integrable_subinterval[OF assms(1)]) using assms(2-3) as(2) by auto
  3632     from integrable_integral[OF this,unfolded has_integral,rule_format,OF *] guess d2 ..
  3633     note d2 = conjunctD2[OF this,rule_format]
  3634     def d3 \<equiv> "\<lambda>x. if x \<le> t then d1 x \<inter> d2 x else d1 x"
  3635     have "gauge d3" using d2(1) d1(1) unfolding d3_def gauge_def by auto
  3636     from fine_division_exists[OF this, of a t] guess p . note p=this
  3637     note p'=tagged_division_ofD[OF this(1)]
  3638     have pt:"\<forall>(x,k)\<in>p. x \<le> t" proof safe case goal1 from p'(2,3)[OF this] show ?case by auto qed
  3639     with p(2) have "d2 fine p" unfolding fine_def d3_def apply safe apply(erule_tac x="(a,b)" in ballE)+ by auto
  3640     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
  3641     
  3642     have *:"{a..c} \<inter> {x. x $$0 \<le> t} = {a..t}" "{a..c} \<inter> {x. x$$0 \<ge> t} = {t..c}"
  3643       using assms(2-3) as by(auto simp add:field_simps)
  3644     have "p \<union> {(c, {t..c})} tagged_division_of {a..c} \<and> d1 fine p \<union> {(c, {t..c})}" apply rule
  3645       apply(rule tagged_division_union_interval[of _ _ _ 0 "t"]) unfolding * apply(rule p)
  3646       apply(rule tagged_division_of_self) unfolding fine_def
  3647     proof safe fix x k y assume "(x,k)\<in>p" "y\<in>k" thus "y\<in>d1 x"
  3648         using p(2) pt unfolding fine_def d3_def apply- apply(erule_tac x="(x,k)" in ballE)+ by auto
  3649     next fix x assume "x\<in>{t..c}" hence "dist c x < k" unfolding dist_real_def
  3650         using as(1) by(auto simp add:field_simps) 
  3651       thus "x \<in> d1 c" using k(2) unfolding d_def by auto
  3652     qed(insert as(2), auto) note d1_fin = d1(2)[OF this]
  3653 
  3654     have *:"integral{a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
  3655         integral {a..c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a..t} f) + (c - t) *\<^sub>R f c" 
  3656       "e = (e/3 + e/3) + e/3" by auto
  3657     have **:"(\<Sum>(x, k)\<in>p \<union> {(c, {t..c})}. content k *\<^sub>R f x) = (c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
  3658     proof- have **:"\<And>x F. F \<union> {x} = insert x F" by auto
  3659       have "(c, {t..c}) \<notin> p" proof safe case goal1 from p'(2-3)[OF this]
  3660         have "c \<in> {a..t}" by auto thus False using `t<c` by auto
  3661       qed thus ?thesis unfolding ** apply- apply(subst setsum_insert) apply(rule p')
  3662         unfolding split_conv defer apply(subst content_real) using as(2) by auto qed 
  3663 
  3664     have ***:"c - w < t \<and> t < c"
  3665     proof- have "c - k < t" using `k>0` as(1) by(auto simp add:field_simps)
  3666       moreover have "k \<le> w" apply(rule ccontr) using k(2) 
  3667         unfolding subset_eq apply(erule_tac x="c + ((k + w)/2)" in ballE)
  3668         unfolding d_def using `k>0` `w>0` by(auto simp add:field_simps not_le not_less dist_real_def)
  3669       ultimately show  ?thesis using `t<c` by(auto simp add:field_simps) qed
  3670 
  3671     show ?thesis unfolding *(1) apply(subst *(2)) apply(rule norm_triangle_lt add_strict_mono)+
  3672       unfolding norm_minus_cancel apply(rule d1_fin[unfolded **]) apply(rule d2_fin)
  3673       using w(2)[OF ***] unfolding norm_scaleR by(auto simp add:field_simps) qed qed 
  3674 
  3675 lemma indefinite_integral_continuous_right: fixes f::"real \<Rightarrow> 'a::banach"
  3676   assumes "f integrable_on {a..b}" "a \<le> c" "c < b" "0 < e"
  3677   obtains d where "0 < d" "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm(integral{a..c} f - integral{a..t} f) < e"
  3678 proof- have *:"(\<lambda>x. f (- x)) integrable_on {- b..- a}" "- b < - c" "- c \<le> - a" using assms by auto
  3679   from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this let ?d = "min d (b - c)"
  3680   show ?thesis apply(rule that[of "?d"])
  3681   proof safe show "0 < ?d" using d(1) assms(3) by auto
  3682     fix t::"real" assume as:"c \<le> t" "t < c + ?d"
  3683     have *:"integral{a..c} f = integral{a..b} f - integral{c..b} f"
  3684       "integral{a..t} f = integral{a..b} f - integral{t..b} f" unfolding algebra_simps
  3685       apply(rule_tac[!] integral_combine) using assms as by auto
  3686     have "(- c) - d < (- t) \<and> - t \<le> - c" using as by auto note d(2)[rule_format,OF this]
  3687     thus "norm (integral {a..c} f - integral {a..t} f) < e" unfolding * 
  3688       unfolding integral_reflect apply-apply(subst norm_minus_commute) by(auto simp add:algebra_simps) qed qed
  3689    
  3690 lemma indefinite_integral_continuous: fixes f::"real \<Rightarrow> 'a::banach"
  3691   assumes "f integrable_on {a..b}" shows  "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
  3692 proof(unfold continuous_on_iff, safe)  fix x e assume as:"x\<in>{a..b}" "0<(e::real)"
  3693   let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e"
  3694   { presume *:"a<b \<Longrightarrow> ?thesis"
  3695     show ?thesis apply(cases,rule *,assumption)
  3696     proof- case goal1 hence "{a..b} = {x}" using as(1) apply-apply(rule set_eqI)
  3697         unfolding atLeastAtMost_iff by(auto simp only:field_simps not_less DIM_real)
  3698       thus ?case using `e>0` by auto
  3699     qed } assume "a<b"
  3700   have "(x=a \<or> x=b) \<or> (a<x \<and> x<b)" using as(1) by (auto simp add:)
  3701   thus ?thesis apply-apply(erule disjE)+
  3702   proof- assume "x=a" have "a \<le> a" by auto
  3703     from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
  3704     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
  3705       unfolding `x=a` dist_norm apply(rule d(2)[rule_format]) by auto
  3706   next   assume "x=b" have "b \<le> b" by auto
  3707     from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
  3708     show ?thesis apply(rule,rule,rule d,safe) apply(subst dist_commute)
  3709       unfolding `x=b` dist_norm apply(rule d(2)[rule_format])  by auto
  3710   next assume "a<x \<and> x<b" hence xl:"a<x" "x\<le>b" and xr:"a\<le>x" "x<b" by(auto simp add: )
  3711     from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
  3712     from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
  3713     show ?thesis apply(rule_tac x="min d1 d2" in exI)
  3714     proof safe show "0 < min d1 d2" using d1 d2 by auto
  3715       fix y assume "y\<in>{a..b}" "dist y x < min d1 d2"
  3716       thus "dist (integral {a..y} f) (integral {a..x} f) < e" apply-apply(subst dist_commute)
  3717         apply(cases "y < x") unfolding dist_norm apply(rule d1(2)[rule_format]) defer
  3718         apply(rule d2(2)[rule_format]) unfolding not_less by(auto simp add:field_simps)
  3719     qed qed qed 
  3720 
  3721 subsection {* This doesn't directly involve integration, but that gives an easy proof. *}
  3722 
  3723 lemma has_derivative_zero_unique_strong_interval: fixes f::"real \<Rightarrow> 'a::banach"
  3724   assumes "finite k" "continuous_on {a..b} f" "f a = y"
  3725   "\<forall>x\<in>({a..b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a..b})" "x \<in> {a..b}"
  3726   shows "f x = y"
  3727 proof- have ab:"a\<le>b" using assms by auto
  3728   have *:"a\<le>x" using assms(5) by auto
  3729   have "((\<lambda>x. 0\<Colon>'a) has_integral f x - f a) {a..x}"
  3730     apply(rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
  3731     apply(rule continuous_on_subset[OF assms(2)]) defer
  3732     apply safe unfolding has_vector_derivative_def apply(subst has_derivative_within_open[THEN sym])
  3733     apply assumption apply(rule open_interval) apply(rule has_derivative_within_subset[where s="{a..b}"])
  3734     using assms(4) assms(5) by auto note this[unfolded *]
  3735   note has_integral_unique[OF has_integral_0 this]
  3736   thus ?thesis unfolding assms by auto qed
  3737 
  3738 subsection {* Generalize a bit to any convex set. *}
  3739 
  3740 lemma has_derivative_zero_unique_strong_convex: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  3741   assumes "convex s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
  3742   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x \<in> s"
  3743   shows "f x = y"
  3744 proof- { presume *:"x \<noteq> c \<Longrightarrow> ?thesis" show ?thesis apply(cases,rule *,assumption)
  3745       unfolding assms(5)[THEN sym] by auto } assume "x\<noteq>c"
  3746   note conv = assms(1)[unfolded convex_alt,rule_format]
  3747   have as1:"continuous_on {0..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
  3748     apply(rule continuous_on_intros)+ apply(rule continuous_on_subset[OF assms(3)])
  3749     apply safe apply(rule conv) using assms(4,7) by auto
  3750   have *:"\<And>t xa. (1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x \<Longrightarrow> t = xa"
  3751   proof- case goal1 hence "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c" 
  3752       unfolding scaleR_simps by(auto simp add:algebra_simps)
  3753     thus ?case using `x\<noteq>c` by auto qed
  3754   have as2:"finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}" using assms(2) 
  3755     apply(rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
  3756     apply safe unfolding image_iff apply rule defer apply assumption
  3757     apply(rule sym) apply(rule some_equality) defer apply(drule *) by auto
  3758   have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
  3759     apply(rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
  3760     unfolding o_def using assms(5) defer apply-apply(rule)
  3761   proof- fix t assume as:"t\<in>{0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
  3762     have *:"c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k" apply safe apply(rule conv[unfolded scaleR_simps]) 
  3763       using `x\<in>s` `c\<in>s` as by(auto simp add: algebra_simps)
  3764     have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
  3765       apply(rule diff_chain_within) apply(rule has_derivative_add)
  3766       unfolding scaleR_simps apply(rule has_derivative_sub) apply(rule has_derivative_const)
  3767       apply(rule has_derivative_vmul_within,rule has_derivative_id)+ 
  3768       apply(rule has_derivative_within_subset,rule assms(6)[rule_format])
  3769       apply(rule *) apply safe apply(rule conv[unfolded scaleR_simps]) using `x\<in>s` `c\<in>s` by auto
  3770     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 .
  3771   qed auto thus ?thesis by auto qed
  3772 
  3773 subsection {* Also to any open connected set with finite set of exceptions. Could 
  3774  generalize to locally convex set with limpt-free set of exceptions. *}
  3775 
  3776 lemma has_derivative_zero_unique_strong_connected: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  3777   assumes "connected s" "open s" "finite k" "continuous_on s f" "c \<in> s" "f c = y"
  3778   "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
  3779   shows "f x = y"
  3780 proof- have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
  3781     apply(rule assms(1)[unfolded connected_clopen,rule_format]) apply rule defer
  3782     apply(rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
  3783     apply(rule open_openin_trans[OF assms(2)]) unfolding open_contains_ball
  3784   proof safe fix x assume "x\<in>s" 
  3785     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
  3786     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}" apply(rule,rule,rule e)
  3787     proof safe fix y assume y:"y \<in> ball x e" thus "y\<in>s" using e by auto
  3788       show "f y = f x" apply(rule has_derivative_zero_unique_strong_convex[OF convex_ball])
  3789         apply(rule assms) apply(rule continuous_on_subset,rule assms) apply(rule e)+
  3790         apply(subst centre_in_ball,rule e,rule) apply safe
  3791         apply(rule has_derivative_within_subset) apply(rule assms(7)[rule_format])
  3792         using y e by auto qed qed
  3793   thus ?thesis using `x\<in>s` `f c = y` `c\<in>s` by auto qed
  3794 
  3795 subsection {* Integrating characteristic function of an interval. *}
  3796 
  3797 lemma has_integral_restrict_open_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  3798   assumes "(f has_integral i) {c..d}" "{c..d} \<subseteq> {a..b}"
  3799   shows "((\<lambda>x. if x \<in> {c<..<d} then f x else 0) has_integral i) {a..b}"
  3800 proof- def g \<equiv> "\<lambda>x. if x \<in>{c<..<d} then f x else 0"
  3801   { presume *:"{c..d}\<noteq>{} \<Longrightarrow> ?thesis"
  3802     show ?thesis apply(cases,rule *,assumption)
  3803     proof- case goal1 hence *:"{c<..<d} = {}" using interval_open_subset_closed by auto 
  3804       show ?thesis using assms(1) unfolding * using goal1 by auto
  3805     qed } assume "{c..d}\<noteq>{}"
  3806   from partial_division_extend_1[OF assms(2) this] guess p . note p=this
  3807   note mon = monoidal_lifted[OF monoidal_monoid] 
  3808   note operat = operative_division[OF this operative_integral p(1), THEN sym]
  3809   let ?P = "(if g integrable_on {a..b} then Some (integral {a..b} g) else None) = Some i"
  3810   { presume "?P" hence "g integrable_on {a..b} \<and> integral {a..b} g = i"
  3811       apply- apply(cases,subst(asm) if_P,assumption) by auto
  3812     thus ?thesis using integrable_integral unfolding g_def by auto }
  3813 
  3814   note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
  3815   note * = this[unfolded neutral_monoid]
  3816   have iterate:"iterate (lifted op +) (p - {{c..d}})
  3817       (\<lambda>i. if g integrable_on i then Some (integral i g) else None) = Some 0"
  3818   proof(rule *,rule) case goal1 hence "x\<in>p" by auto note div = division_ofD(2-5)[OF p(1) this]
  3819     from div(3) guess u v apply-by(erule exE)+ note uv=this
  3820     have "interior x \<inter> interior {c..d} = {}" using div(4)[OF p(2)] goal1 by auto
  3821     hence "(g has_integral 0) x" unfolding uv apply-apply(rule has_integral_spike_interior[where f="\<lambda>x. 0"])
  3822       unfolding g_def interior_closed_interval by auto thus ?case by auto
  3823   qed
  3824 
  3825   have *:"p = insert {c..d} (p - {{c..d}})" using p by auto
  3826   have **:"g integrable_on {c..d}" apply(rule integrable_spike_interior[where f=f])
  3827     unfolding g_def defer apply(rule has_integral_integrable) using assms(1) by auto
  3828   moreover have "integral {c..d} g = i" apply(rule has_integral_unique[OF _ assms(1)])
  3829     apply(rule has_integral_spike_interior[where f=g]) defer
  3830     apply(rule integrable_integral[OF **]) unfolding g_def by auto
  3831   ultimately show ?P unfolding operat apply- apply(subst *) apply(subst iterate_insert) apply rule+
  3832     unfolding iterate defer apply(subst if_not_P) defer using p by auto qed
  3833 
  3834 lemma has_integral_restrict_closed_subinterval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach"
  3835   assumes "(f has_integral i) ({c..d})" "{c..d} \<subseteq> {a..b}" 
  3836   shows "((\<lambda>x. if x \<in> {c..d} then f x else 0) has_integral i) {a..b}"
  3837 proof- note has_integral_restrict_open_subinterval[OF assms]
  3838   note * = has_integral_spike[OF negligible_frontier_interval _ this]
  3839   show ?thesis apply(rule *[of c d]) using interval_open_subset_closed[of c d] by auto qed
  3840 
  3841 lemma has_integral_restrict_closed_subintervals_eq: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::banach" assumes "{c..d} \<subseteq> {a..b}" 
  3842   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")
  3843 proof(cases "{c..d} = {}") case False let ?g = "\<lambda>x. if x \<in> {c..d} then f x else 0"
  3844   show ?thesis apply rule defer apply(rule has_integral_restrict_closed_subinterval[OF _ assms])
  3845   proof assumption assume ?l hence "?g integrable_on {c..d}"
  3846       apply-apply(rule integrable_subinterval[OF _ assms]) by auto
  3847     hence *:"f integrable_on {c..d}"apply-apply(rule integrable_eq) by auto
  3848     hence "i = integral {c..d} f" apply-apply(rule has_integral_unique)
  3849       apply(rule `?l`) apply(rule has_integral_restrict_closed_subinterval[OF _ assms]) by auto
  3850     thus ?r using * by auto qed qed auto
  3851 
  3852 subsection {* Hence we can apply the limit process uniformly to all integrals. *}
  3853 
  3854 lemma has_integral': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
  3855  "(f has_integral i) s \<longleftrightarrow> (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}
  3856   \<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)")
  3857 proof- { presume *:"\<exists>a b. s = {a..b} \<Longrightarrow> ?thesis"
  3858     show ?thesis apply(cases,rule *,assumption)
  3859       apply(subst has_integral_alt) by auto }
  3860   assume "\<exists>a b. s = {a..b}" then guess a b apply-by(erule exE)+ note s=this
  3861   from bounded_interval[of a b, THEN conjunct1, unfolded bounded_pos] guess B ..
  3862   note B = conjunctD2[OF this,rule_format] show ?thesis apply safe
  3863   proof- fix e assume ?l "e>(0::real)"
  3864     show "?r e" apply(rule_tac x="B+1" in exI) apply safe defer apply(rule_tac x=i in exI)
  3865     proof fix c d assume as:"ball 0 (B+1) \<subseteq> {c..d::'n::ordered_euclidean_space}"
  3866       thus "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) {c..d}" unfolding s
  3867         apply-apply(rule has_integral_restrict_closed_subinterval) apply(rule `?l`[unfolded s])
  3868         apply safe apply(drule B(2)[rule_format]) unfolding subset_eq apply(erule_tac x=x in ballE)
  3869         by(auto simp add:dist_norm)
  3870     qed(insert B `e>0`, auto)
  3871   next assume as:"\<forall>e>0. ?r e" 
  3872     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
  3873     def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
  3874     have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
  3875     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
  3876         by(auto simp add:field_simps) qed
  3877     have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
  3878     proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  3879     from C(2)[OF this] have "\<exists>y. (f has_integral y) {a..b}"
  3880       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,THEN sym] unfolding s by auto
  3881     then guess y .. note y=this
  3882 
  3883     have "y = i" proof(rule ccontr) assume "y\<noteq>i" hence "0 < norm (y - i)" by auto
  3884       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
  3885       def c \<equiv> "(\<chi>\<chi> i. - max B C)::'n::ordered_euclidean_space" and d \<equiv> "(\<chi>\<chi> i. max B C)::'n::ordered_euclidean_space"
  3886       have c_d:"{a..b} \<subseteq> {c..d}" apply safe apply(drule B(2)) unfolding mem_interval
  3887       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def
  3888           by(auto simp add:field_simps) qed
  3889       have "ball 0 C \<subseteq> {c..d}" apply safe unfolding mem_interval mem_ball dist_norm 
  3890       proof case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  3891       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
  3892       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
  3893       hence "z = y" "norm (z - i) < norm (y - i)" apply- apply(rule has_integral_unique[OF _ y(1)]) .
  3894       thus False by auto qed
  3895     thus ?l using y unfolding s by auto qed qed 
  3896 
  3897 (*lemma has_integral_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
  3898   "((\<lambda>x. vec1 (f x)) has_integral vec1 i) s \<longleftrightarrow> (f has_integral i) s"
  3899   unfolding has_integral'[unfolded has_integral] 
  3900 proof case goal1 thus ?case apply safe
  3901   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
  3902   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
  3903   apply(rule_tac x="dest_vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
  3904   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
  3905   apply(subst(asm)(2) norm_vector_1) unfolding split_def
  3906   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
  3907     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
  3908   apply(subst(asm)(2) norm_vector_1) by auto
  3909 next case goal2 thus ?case apply safe
  3910   apply(erule_tac x=e in allE,safe) apply(rule_tac x=B in exI,safe)
  3911   apply(erule_tac x=a in allE, erule_tac x=b in allE,safe) 
  3912   apply(rule_tac x="vec1 z" in exI,safe) apply(erule_tac x=ea in allE,safe) 
  3913   apply(rule_tac x=d in exI,safe) apply(erule_tac x=p in allE,safe)
  3914   apply(subst norm_vector_1) unfolding split_def
  3915   unfolding setsum_component Cart_nth.diff cond_value_iff[of dest_vec1]
  3916     Cart_nth.scaleR vec1_dest_vec1 zero_index apply assumption
  3917   apply(subst norm_vector_1) by auto qed
  3918 
  3919 lemma integral_trans[simp]: assumes "(f::'n::ordered_euclidean_space \<Rightarrow> real) integrable_on s"
  3920   shows "integral s (\<lambda>x. vec1 (f x)) = vec1 (integral s f)"
  3921   apply(rule integral_unique) using assms by auto
  3922 
  3923 lemma integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
  3924   "(\<lambda>x. vec1 (f x)) integrable_on s \<longleftrightarrow> (f integrable_on s)"
  3925   unfolding integrable_on_def
  3926   apply(subst(2) vec1_dest_vec1(1)[THEN sym]) unfolding has_integral_trans
  3927   apply safe defer apply(rule_tac x="vec1 y" in exI) by auto *)
  3928 
  3929 lemma has_integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  3930   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x) \<le> (g x)"
  3931   shows "i \<le> j" using has_integral_component_le[OF assms(1-2), of 0] using assms(3) by auto
  3932 
  3933 lemma integral_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  3934   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"
  3935   shows "integral s f \<le> integral s g"
  3936   using has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)] .
  3937 
  3938 lemma has_integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  3939   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i" 
  3940   using has_integral_component_nonneg[of "f" "i" s 0]
  3941   unfolding o_def using assms by auto
  3942 
  3943 lemma integral_nonneg: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  3944   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f" 
  3945   using has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)] .
  3946 
  3947 subsection {* Hence a general restriction property. *}
  3948 
  3949 lemma has_integral_restrict[simp]: assumes "s \<subseteq> t" shows
  3950   "((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
  3951 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
  3952   show ?thesis apply(subst(2) has_integral') apply(subst has_integral') unfolding * by rule qed
  3953 
  3954 lemma has_integral_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
  3955   "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s" by auto
  3956 
  3957 lemma has_integral_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
  3958   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "(f has_integral i) s"
  3959   shows "(f has_integral i) t"
  3960 proof- have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
  3961     apply(rule) using assms(1-2) by auto
  3962   thus ?thesis apply- using assms(3) apply(subst has_integral_restrict_univ[THEN sym])
  3963   apply- apply(subst(asm) has_integral_restrict_univ[THEN sym]) by auto qed
  3964 
  3965 lemma integrable_on_superset: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
  3966   assumes "\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0" "s \<subseteq> t" "f integrable_on s"
  3967   shows "f integrable_on t"
  3968   using assms unfolding integrable_on_def by(auto intro:has_integral_on_superset)
  3969 
  3970 lemma integral_restrict_univ[intro]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
  3971   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
  3972   apply(rule integral_unique) unfolding has_integral_restrict_univ by auto
  3973 
  3974 lemma integrable_restrict_univ: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
  3975  "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
  3976   unfolding integrable_on_def by auto
  3977 
  3978 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> {a..b}))" (is "?l = ?r")
  3979 proof assume ?r show ?l unfolding negligible_def
  3980   proof safe case goal1 show ?case apply(rule has_integral_negligible[OF `?r`[rule_format,of a b]])
  3981       unfolding indicator_def by auto qed qed auto
  3982 
  3983 lemma has_integral_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" 
  3984   assumes "negligible((s - t) \<union> (t - s))" shows "((f has_integral y) s \<longleftrightarrow> (f has_integral y) t)"
  3985   unfolding has_integral_restrict_univ[THEN sym,of f] apply(rule has_integral_spike_eq[OF assms]) by (safe, auto split: split_if_asm)
  3986 
  3987 lemma has_integral_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  3988   assumes "negligible((s - t) \<union> (t - s))" "(f has_integral y) s"
  3989   shows "(f has_integral y) t"
  3990   using assms has_integral_spike_set_eq by auto
  3991 
  3992 lemma integrable_spike_set[dest]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  3993   assumes "negligible((s - t) \<union> (t - s))" "f integrable_on s"
  3994   shows "f integrable_on t" using assms(2) unfolding integrable_on_def 
  3995   unfolding has_integral_spike_set_eq[OF assms(1)] .
  3996 
  3997 lemma integrable_spike_set_eq: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  3998   assumes "negligible((s - t) \<union> (t - s))"
  3999   shows "(f integrable_on s \<longleftrightarrow> f integrable_on t)"
  4000   apply(rule,rule_tac[!] integrable_spike_set) using assms by auto
  4001 
  4002 (*lemma integral_spike_set:
  4003  "\<forall>f:real^M->real^N g s t.
  4004         negligible(s DIFF t \<union> t DIFF s)
  4005         \<longrightarrow> integral s f = integral t f"
  4006 qed  REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
  4007   AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  4008   ASM_MESON_TAC[]);;
  4009 
  4010 lemma has_integral_interior:
  4011  "\<forall>f:real^M->real^N y s.
  4012         negligible(frontier s)
  4013         \<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
  4014 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  4015   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
  4016     NEGLIGIBLE_SUBSET)) THEN
  4017   REWRITE_TAC[frontier] THEN
  4018   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
  4019   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
  4020   SET_TAC[]);;
  4021 
  4022 lemma has_integral_closure:
  4023  "\<forall>f:real^M->real^N y s.
  4024         negligible(frontier s)
  4025         \<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
  4026 qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
  4027   FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
  4028     NEGLIGIBLE_SUBSET)) THEN
  4029   REWRITE_TAC[frontier] THEN
  4030   MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
  4031   MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
  4032   SET_TAC[]);;*)
  4033 
  4034 subsection {* More lemmas that are useful later. *}
  4035 
  4036 lemma has_integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
  4037   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)$$k"
  4038   shows "i$$k \<le> j$$k"
  4039 proof- note has_integral_restrict_univ[THEN sym, of f]
  4040   note assms(2-3)[unfolded this] note * = has_integral_component_le[OF this]
  4041   show ?thesis apply(rule *) using assms(1,4) by auto qed
  4042 
  4043 lemma has_integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  4044   assumes "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)"
  4045   shows "i \<le> j" using has_integral_subset_component_le[OF assms(1), of "f" "i" "j" 0] using assms by auto
  4046 
  4047 lemma integral_subset_component_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space"
  4048   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)$$k"
  4049   shows "(integral s f)$$k \<le> (integral t f)$$k"
  4050   apply(rule has_integral_subset_component_le) using assms by auto
  4051 
  4052 lemma integral_subset_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  4053   assumes "s \<subseteq> t" "f integrable_on s" "f integrable_on t" "\<forall>x \<in> t. 0 \<le> f(x)"
  4054   shows "(integral s f) \<le> (integral t f)"
  4055   apply(rule has_integral_subset_le) using assms by auto
  4056 
  4057 lemma has_integral_alt': fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4058   shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
  4059   (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e)" (is "?l = ?r")
  4060 proof assume ?r
  4061   show ?l apply- apply(subst has_integral')
  4062   proof safe case goal1 from `?r`[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
  4063     show ?case apply(rule,rule,rule B,safe)
  4064       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then f x else 0)" in exI)
  4065       apply(drule B(2)[rule_format]) using integrable_integral[OF `?r`[THEN conjunct1,rule_format]] by auto
  4066   qed next
  4067   assume ?l note as = this[unfolded has_integral'[of f],rule_format]
  4068   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
  4069   show ?r proof safe fix a b::"'n::ordered_euclidean_space"
  4070     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
  4071     let ?a = "(\<chi>\<chi> i. min (a$$i) (-B))::'n::ordered_euclidean_space" and ?b = "(\<chi>\<chi> i. max (b$$i) B)::'n::ordered_euclidean_space"
  4072     show "?f integrable_on {a..b}" apply(rule integrable_subinterval[of _ ?a ?b])
  4073     proof- have "ball 0 B \<subseteq> {?a..?b}" apply safe unfolding mem_ball mem_interval dist_norm
  4074       proof case goal1 thus ?case using component_le_norm[of x i] by(auto simp add:field_simps) qed
  4075       from B(2)[OF this] guess z .. note conjunct1[OF this]
  4076       thus "?f integrable_on {?a..?b}" unfolding integrable_on_def by auto
  4077       show "{a..b} \<subseteq> {?a..?b}" apply safe unfolding mem_interval apply(rule,erule_tac x=i in allE) by auto qed
  4078 
  4079     fix e::real assume "e>0" from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
  4080     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
  4081                     norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
  4082     proof(rule,rule,rule B,safe) case goal1 from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
  4083       from integral_unique[OF this(1)] show ?case using z(2) by auto qed qed qed 
  4084 
  4085 
  4086 subsection {* Continuity of the integral (for a 1-dimensional interval). *}
  4087 
  4088 lemma integrable_alt: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows 
  4089   "f integrable_on s \<longleftrightarrow>
  4090           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}) \<and>
  4091           (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> {a..b} \<and> ball 0 B \<subseteq> {c..d}
  4092   \<longrightarrow> norm(integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) -
  4093           integral {c..d}  (\<lambda>x. if x \<in> s then f x else 0)) < e)" (is "?l = ?r")
  4094 proof assume ?l then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
  4095   note y=conjunctD2[OF this,rule_format] show ?r apply safe apply(rule y)
  4096   proof- case goal1 hence "e/2 > 0" by auto from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
  4097     show ?case apply(rule,rule,rule B)
  4098     proof safe case goal1 show ?case apply(rule norm_triangle_half_l)
  4099         using B(2)[OF goal1(1)] B(2)[OF goal1(2)] by auto qed qed
  4100         
  4101 next assume ?r note as = conjunctD2[OF this,rule_format]
  4102   have "Cauchy (\<lambda>n. integral ({(\<chi>\<chi> i. - real n)::'n .. (\<chi>\<chi> i. real n)}) (\<lambda>x. if x \<in> s then f x else 0))"
  4103   proof(unfold Cauchy_def,safe) case goal1
  4104     from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
  4105     from real_arch_simple[of B] guess N .. note N = this
  4106     { fix n assume n:"n \<ge> N" have "ball 0 B \<subseteq> {(\<chi>\<chi> i. - real n)::'n..\<chi>\<chi> i. real n}" apply safe
  4107         unfolding mem_ball mem_interval dist_norm
  4108       proof case goal1 thus ?case using component_le_norm[of x i]
  4109           using n N by(auto simp add:field_simps) qed }
  4110     thus ?case apply-apply(rule_tac x=N in exI) apply safe unfolding dist_norm apply(rule B(2)) by auto
  4111   qed from this[unfolded convergent_eq_cauchy[THEN sym]] guess i ..
  4112   note i = this[unfolded Lim_sequentially, rule_format]
  4113 
  4114   show ?l unfolding integrable_on_def has_integral_alt'[of f] apply(rule_tac x=i in exI)
  4115     apply safe apply(rule as(1)[unfolded integrable_on_def])
  4116   proof- case goal1 hence *:"e/2 > 0" by auto
  4117     from i[OF this] guess N .. note N =this[rule_format]
  4118     from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format] let ?B = "max (real N) B"
  4119     show ?case apply(rule_tac x="?B" in exI)
  4120     proof safe show "0 < ?B" using B(1) by auto
  4121       fix a b assume ab:"ball 0 ?B \<subseteq> {a..b::'n::ordered_euclidean_space}"
  4122       from real_arch_simple[of ?B] guess n .. note n=this
  4123       show "norm (integral {a..b} (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
  4124         apply(rule norm_triangle_half_l) apply(rule B(2)) defer apply(subst norm_minus_commute)
  4125         apply(rule N[unfolded dist_norm, of n])
  4126       proof safe show "N \<le> n" using n by auto
  4127         fix x::"'n::ordered_euclidean_space" assume x:"x \<in> ball 0 B" hence "x\<in> ball 0 ?B" by auto
  4128         thus "x\<in>{a..b}" using ab by blast 
  4129         show "x\<in>{\<chi>\<chi> i. - real n..\<chi>\<chi> i. real n}" using x unfolding mem_interval mem_ball dist_norm apply-
  4130         proof case goal1 thus ?case using component_le_norm[of x i]
  4131             using n by(auto simp add:field_simps) qed qed qed qed qed
  4132 
  4133 lemma integrable_altD: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4134   assumes "f integrable_on s"
  4135   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
  4136   "\<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}
  4137   \<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"
  4138   using assms[unfolded integrable_alt[of f]] by auto
  4139 
  4140 lemma integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4141   assumes "f integrable_on s" "{a..b} \<subseteq> s" shows "f integrable_on {a..b}"
  4142   apply(rule integrable_eq) defer apply(rule integrable_altD(1)[OF assms(1)])
  4143   using assms(2) by auto
  4144 
  4145 subsection {* A straddling criterion for integrability. *}
  4146 
  4147 lemma integrable_straddle_interval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  4148   assumes "\<forall>e>0. \<exists>g  h i j. (g has_integral i) ({a..b}) \<and> (h has_integral j) ({a..b}) \<and>
  4149   norm(i - j) < e \<and> (\<forall>x\<in>{a..b}. (g x) \<le> (f x) \<and> (f x) \<le>(h x))"
  4150   shows "f integrable_on {a..b}"
  4151 proof(subst integrable_cauchy,safe)
  4152   case goal1 hence e:"e/3 > 0" by auto note assms[rule_format,OF this]
  4153   then guess g h i j apply- by(erule exE conjE)+ note obt = this
  4154   from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
  4155   from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
  4156   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
  4157   proof safe have **:"\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
  4158       abs(i - j) < e / 3 \<Longrightarrow> abs(g2 - i) < e / 3 \<Longrightarrow>  abs(g1 - i) < e / 3 \<Longrightarrow> 
  4159       abs(h2 - j) < e / 3 \<Longrightarrow> abs(h1 - j) < e / 3 \<Longrightarrow> abs(f1 - f2) < e" using `e>0` by arith
  4160     case goal1 note tagged_division_ofD(2-4) note * = this[OF goal1(1)] this[OF goal1(4)]
  4161 
  4162     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"
  4163       "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)" 
  4164       "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
  4165       "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)" 
  4166       unfolding setsum_subtractf[THEN sym] apply- apply(rule_tac[!] setsum_nonneg)
  4167       apply safe unfolding real_scaleR_def mult.diff_right[THEN sym]
  4168       apply(rule_tac[!] mult_nonneg_nonneg)
  4169     proof- fix a b assume ab:"(a,b) \<in> p1"
  4170       show "0 \<le> content b" using *(3)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
  4171       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
  4172     next fix a b assume ab:"(a,b) \<in> p2"
  4173       show "0 \<le> content b" using *(6)[OF ab] apply safe using content_pos_le . thus "0 \<le> content b" .
  4174       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 
  4175 
  4176     thus ?case apply- unfolding real_norm_def apply(rule **) defer defer
  4177       unfolding real_norm_def[THEN sym] apply(rule obt(3))
  4178       apply(rule d1(2)[OF conjI[OF goal1(4,5)]])
  4179       apply(rule d1(2)[OF conjI[OF goal1(1,2)]])
  4180       apply(rule d2(2)[OF conjI[OF goal1(4,6)]])
  4181       apply(rule d2(2)[OF conjI[OF goal1(1,3)]]) by auto qed qed 
  4182      
  4183 lemma integrable_straddle: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real"
  4184   assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
  4185   norm(i - j) < e \<and> (\<forall>x\<in>s. (g x) \<le>(f x) \<and>(f x) \<le>(h x))"
  4186   shows "f integrable_on s"
  4187 proof- have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on {a..b}"
  4188   proof(rule integrable_straddle_interval,safe) case goal1 hence *:"e/4 > 0" by auto
  4189     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
  4190     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
  4191     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
  4192     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
  4193     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
  4194     def c \<equiv> "(\<chi>\<chi> i. min (a$$i) (- (max B1 B2)))::'n" and d \<equiv> "(\<chi>\<chi> i. max (b$$i) (max B1 B2))::'n"
  4195     have *:"ball 0 B1 \<subseteq> {c..d}" "ball 0 B2 \<subseteq> {c..d}" apply safe unfolding mem_ball mem_interval dist_norm
  4196     proof(rule_tac[!] allI)
  4197       case goal1 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto next
  4198       case goal2 thus ?case using component_le_norm[of x i] unfolding c_def d_def by auto qed
  4199     have **:"\<And>ch cg ag ah::real. norm(ah - ag) \<le> norm(ch - cg) \<Longrightarrow> norm(cg - i) < e / 4 \<Longrightarrow>
  4200       norm(ch - j) < e / 4 \<Longrightarrow> norm(ag - ah) < e"
  4201       using obt(3) unfolding real_norm_def by arith 
  4202     show ?case apply(rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
  4203                apply(rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
  4204       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)" in exI)
  4205       apply(rule_tac x="integral {a..b} (\<lambda>x. if x \<in> s then h x else 0)" in exI)
  4206       apply safe apply(rule_tac[1-2] integrable_integral,rule g,rule h)
  4207       apply(rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
  4208     proof- have *:"\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
  4209         (if x \<in> s then f x - g x else (0::real))" by auto
  4210       note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_sub, OF h g]]
  4211       show " norm (integral {a..b} (\<lambda>x. if x \<in> s then h x else 0) -
  4212                    integral {a..b} (\<lambda>x. if x \<in> s then g x else 0))
  4213            \<le> norm (integral {c..d} (\<lambda>x. if x \<in> s then h x else 0) -
  4214                    integral {c..d} (\<lambda>x. if x \<in> s then g x else 0))"
  4215         unfolding integral_sub[OF h g,THEN sym] real_norm_def apply(subst **) defer apply(subst **) defer
  4216         apply(rule has_integral_subset_le) defer apply(rule integrable_integral integrable_sub h g)+
  4217       proof safe fix x assume "x\<in>{a..b}" thus "x\<in>{c..d}" unfolding mem_interval c_def d_def
  4218           apply - apply rule apply(erule_tac x=i in allE) by auto
  4219       qed(insert obt(4), auto) qed(insert obt(4), auto) qed note interv = this
  4220 
  4221   show ?thesis unfolding integrable_alt[of f] apply safe apply(rule interv)
  4222   proof- case goal1 hence *:"e/3 > 0" by auto
  4223     from assms[rule_format,OF this] guess g h i j apply-by(erule exE conjE)+ note obt=this
  4224     note obt(1)[unfolded has_integral_alt'[of g]] note conjunctD2[OF this, rule_format]
  4225     note g = this(1) and this(2)[OF *] from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
  4226     note obt(2)[unfolded has_integral_alt'[of h]] note conjunctD2[OF this, rule_format]
  4227     note h = this(1) and this(2)[OF *] from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
  4228     show ?case apply(rule_tac x="max B1 B2" in exI) apply safe apply(rule min_max.less_supI1,rule B1)
  4229     proof- fix a b c d::"'n::ordered_euclidean_space" assume as:"ball 0 (max B1 B2) \<subseteq> {a..b}" "ball 0 (max B1 B2) \<subseteq> {c..d}"
  4230       have **:"ball 0 B1 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n::ordered_euclidean_space) (max B1 B2)" by auto
  4231       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>
  4232         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
  4233       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"
  4234         unfolding real_norm_def apply(rule *, safe) unfolding real_norm_def[THEN sym]
  4235         apply(rule B1(2),rule order_trans,rule **,rule as(1)) 
  4236         apply(rule B1(2),rule order_trans,rule **,rule as(2)) 
  4237         apply(rule B2(2),rule order_trans,rule **,rule as(1)) 
  4238         apply(rule B2(2),rule order_trans,rule **,rule as(2)) 
  4239         apply(rule obt) apply(rule_tac[!] integral_le) using obt
  4240         by(auto intro!: h g interv) qed qed qed 
  4241 
  4242 subsection {* Adding integrals over several sets. *}
  4243 
  4244 lemma has_integral_union: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4245   assumes "(f has_integral i) s" "(f has_integral j) t" "negligible(s \<inter> t)"
  4246   shows "(f has_integral (i + j)) (s \<union> t)"
  4247 proof- note * = has_integral_restrict_univ[THEN sym, of f]
  4248   show ?thesis unfolding * apply(rule has_integral_spike[OF assms(3)])
  4249     defer apply(rule has_integral_add[OF assms(1-2)[unfolded *]]) by auto qed
  4250 
  4251 lemma has_integral_unions: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4252   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')"
  4253   shows "(f has_integral (setsum i t)) (\<Union>t)"
  4254 proof- note * = has_integral_restrict_univ[THEN sym, of f]
  4255   have **:"negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> ~(a = y)}}))"
  4256     apply(rule negligible_unions) apply(rule finite_imageI) apply(rule finite_subset[of _ "t \<times> t"]) defer 
  4257     apply(rule finite_cartesian_product[OF assms(1,1)]) using assms(3) by auto 
  4258   note assms(2)[unfolded *] note has_integral_setsum[OF assms(1) this]
  4259   thus ?thesis unfolding * apply-apply(rule has_integral_spike[OF **]) defer apply assumption
  4260   proof safe case goal1 thus ?case
  4261     proof(cases "x\<in>\<Union>t") case True then guess s unfolding Union_iff .. note s=this
  4262       hence *:"\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s" using goal1(3) by blast
  4263       show ?thesis unfolding if_P[OF True] apply(rule trans) defer
  4264         apply(rule setsum_cong2) apply(subst *, assumption) apply(rule refl)
  4265         unfolding setsum_delta[OF assms(1)] using s by auto qed auto qed qed
  4266 
  4267 subsection {* In particular adding integrals over a division, maybe not of an interval. *}
  4268 
  4269 lemma has_integral_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4270   assumes "d division_of s" "\<forall>k\<in>d. (f has_integral (i k)) k"
  4271   shows "(f has_integral (setsum i d)) s"
  4272 proof- note d = division_ofD[OF assms(1)]
  4273   show ?thesis unfolding d(6)[THEN sym] apply(rule has_integral_unions)
  4274     apply(rule d assms)+ apply(rule,rule,rule)
  4275   proof- case goal1 from d(4)[OF this(1)] d(4)[OF this(2)]
  4276     guess a c b d apply-by(erule exE)+ note obt=this
  4277     from d(5)[OF goal1] show ?case unfolding obt interior_closed_interval
  4278       apply-apply(rule negligible_subset[of "({a..b}-{a<..<b}) \<union> ({c..d}-{c<..<d})"])
  4279       apply(rule negligible_union negligible_frontier_interval)+ by auto qed qed
  4280 
  4281 lemma integral_combine_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4282   assumes "d division_of s" "\<forall>k\<in>d. f integrable_on k"
  4283   shows "integral s f = setsum (\<lambda>i. integral i f) d"
  4284   apply(rule integral_unique) apply(rule has_integral_combine_division[OF assms(1)])
  4285   using assms(2) unfolding has_integral_integral .
  4286 
  4287 lemma has_integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4288   assumes "f integrable_on s" "d division_of k" "k \<subseteq> s"
  4289   shows "(f has_integral (setsum (\<lambda>i. integral i f) d)) k"
  4290   apply(rule has_integral_combine_division[OF assms(2)])
  4291   apply safe unfolding has_integral_integral[THEN sym]
  4292 proof- case goal1 from division_ofD(2,4)[OF assms(2) this]
  4293   show ?case apply safe apply(rule integrable_on_subinterval)
  4294     apply(rule assms) using assms(3) by auto qed
  4295 
  4296 lemma integral_combine_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4297   assumes "f integrable_on s" "d division_of s"
  4298   shows "integral s f = setsum (\<lambda>i. integral i f) d"
  4299   apply(rule integral_unique,rule has_integral_combine_division_topdown) using assms by auto
  4300 
  4301 lemma integrable_combine_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4302   assumes "d division_of s" "\<forall>i\<in>d. f integrable_on i"
  4303   shows "f integrable_on s"
  4304   using assms(2) unfolding integrable_on_def
  4305   by(metis has_integral_combine_division[OF assms(1)])
  4306 
  4307 lemma integrable_on_subdivision: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4308   assumes "d division_of i" "f integrable_on s" "i \<subseteq> s"
  4309   shows "f integrable_on i"
  4310   apply(rule integrable_combine_division assms)+
  4311 proof safe case goal1 note division_ofD(2,4)[OF assms(1) this]
  4312   thus ?case apply safe apply(rule integrable_on_subinterval[OF assms(2)])
  4313     using assms(3) by auto qed
  4314 
  4315 subsection {* Also tagged divisions. *}
  4316 
  4317 lemma has_integral_combine_tagged_division: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4318   assumes "p tagged_division_of s" "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
  4319   shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
  4320 proof- have *:"(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
  4321     apply(rule has_integral_combine_division) apply(rule division_of_tagged_division[OF assms(1)])
  4322     using assms(2) unfolding has_integral_integral[THEN sym] by(safe,auto)
  4323   thus ?thesis apply- apply(rule subst[where P="\<lambda>i. (f has_integral i) s"]) defer apply assumption
  4324     apply(rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"]) apply(subst eq_commute)
  4325     apply(rule setsum_over_tagged_division_lemma[OF assms(1)]) apply(rule integral_null,assumption)
  4326     apply(rule setsum_cong2) using assms(2) by auto qed
  4327 
  4328 lemma integral_combine_tagged_division_bottomup: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4329   assumes "p tagged_division_of {a..b}" "\<forall>(x,k)\<in>p. f integrable_on k"
  4330   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
  4331   apply(rule integral_unique) apply(rule has_integral_combine_tagged_division[OF assms(1)])
  4332   using assms(2) by auto
  4333 
  4334 lemma has_integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4335   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
  4336   shows "(f has_integral (setsum (\<lambda>(x,k). integral k f) p)) {a..b}"
  4337   apply(rule has_integral_combine_tagged_division[OF assms(2)])
  4338 proof safe case goal1 note tagged_division_ofD(3-4)[OF assms(2) this]
  4339   thus ?case using integrable_subinterval[OF assms(1)] by auto qed
  4340 
  4341 lemma integral_combine_tagged_division_topdown: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4342   assumes "f integrable_on {a..b}" "p tagged_division_of {a..b}"
  4343   shows "integral {a..b} f = setsum (\<lambda>(x,k). integral k f) p"
  4344   apply(rule integral_unique,rule has_integral_combine_tagged_division_topdown) using assms by auto
  4345 
  4346 subsection {* Henstock's lemma. *}
  4347 
  4348 lemma henstock_lemma_part1: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4349   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
  4350   "(\<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)"
  4351   and p:"p tagged_partial_division_of {a..b}" "d fine p"
  4352   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e" (is "?x \<le> e")
  4353 proof-  { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" thus ?thesis by (blast intro: field_le_epsilon) }
  4354   fix k::real assume k:"k>0" note p' = tagged_partial_division_ofD[OF p(1)]
  4355   have "\<Union>snd ` p \<subseteq> {a..b}" using p'(3) by fastsimp
  4356   note partial_division_of_tagged_division[OF p(1)] this
  4357   from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
  4358   def r \<equiv> "q - snd ` p" have "snd ` p \<inter> r = {}" unfolding r_def by auto
  4359   have r:"finite r" using q' unfolding r_def by auto
  4360 
  4361   have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
  4362     norm(setsum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
  4363   proof safe case goal1 hence i:"i \<in> q" unfolding r_def by auto
  4364     from q'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4365     have *:"k / (real (card r) + 1) > 0" apply(rule divide_pos_pos,rule k) by auto
  4366     have "f integrable_on {u..v}" apply(rule integrable_subinterval[OF assms(1)])
  4367       using q'(2)[OF i] unfolding uv by auto
  4368     note integrable_integral[OF this, unfolded has_integral[of f]]
  4369     from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
  4370     note gauge_inter[OF `gauge d` dd(1)] from fine_division_exists[OF this,of u v] guess qq .
  4371     thus ?case apply(rule_tac x=qq in exI) using dd(2)[of qq] unfolding fine_inter uv by auto qed
  4372   from bchoice[OF this] guess qq .. note qq=this[rule_format]
  4373 
  4374   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"
  4375     apply(rule assms(4)[rule_format])
  4376   proof show "d fine ?p" apply(rule fine_union,rule p) apply(rule fine_unions) using qq by auto 
  4377     note * = tagged_partial_division_of_union_self[OF p(1)]
  4378     have "p \<union> \<Union>qq ` r tagged_division_of \<Union>snd ` p \<union> \<Union>r"
  4379     proof(rule tagged_division_union[OF * tagged_division_unions])
  4380       show "finite r" by fact case goal2 thus ?case using qq by auto
  4381     next case goal3 thus ?case apply(rule,rule,rule) apply(rule q'(5)) unfolding r_def by auto
  4382     next case goal4 thus ?case apply(rule inter_interior_unions_intervals) apply(fact,rule)
  4383         apply(rule,rule q') defer apply(rule,subst Int_commute) 
  4384         apply(rule inter_interior_unions_intervals) apply(rule finite_imageI,rule p',rule) defer
  4385         apply(rule,rule q') using q(1) p' unfolding r_def by auto qed
  4386     moreover have "\<Union>snd ` p \<union> \<Union>r = {a..b}" "{qq i |i. i \<in> r} = qq ` r"
  4387       unfolding Union_Un_distrib[THEN sym] r_def using q by auto
  4388     ultimately show "?p tagged_division_of {a..b}" by fastsimp qed
  4389 
  4390   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) -
  4391     integral {a..b} f) < e" apply(subst setsum_Un_zero[THEN sym]) apply(rule p') prefer 3 
  4392     apply assumption apply rule apply(rule finite_imageI,rule r) apply safe apply(drule qq)
  4393   proof- fix x l k assume as:"(x,l)\<in>p" "(x,l)\<in>qq k" "k\<in>r"
  4394     note qq[OF this(3)] note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
  4395     from this(2) guess u v apply-by(erule exE)+ note uv=this
  4396     have "l\<in>snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
  4397     hence "l\<in>q" "k\<in>q" "l\<noteq>k" using as(1,3) q(1) unfolding r_def by auto
  4398     note q'(5)[OF this] hence "interior l = {}" using subset_interior[OF `l \<subseteq> k`] by blast
  4399     thus "content l *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto qed auto
  4400 
  4401   hence "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + setsum (setsum (\<lambda>(x, k). content k *\<^sub>R f x))
  4402     (qq ` r) - integral {a..b} f) < e" apply(subst(asm) setsum_UNION_zero)
  4403     prefer 4 apply assumption apply(rule finite_imageI,fact)
  4404     unfolding split_paired_all split_conv image_iff defer apply(erule bexE)+
  4405   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"
  4406     note as = this(1-5)[unfolded this(6-)] note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
  4407     from this(2)[OF as(4,1)] guess u v apply-by(erule exE)+ note uv=this
  4408     have *:"interior (k \<inter> l) = {}" unfolding interior_inter apply(rule q')
  4409       using as unfolding r_def by auto
  4410     have "interior m = {}" unfolding subset_empty[THEN sym] unfolding *[THEN sym]
  4411       apply(rule subset_interior) using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)] by auto
  4412     thus "content m *\<^sub>R f x = 0" unfolding uv content_eq_0_interior[THEN sym] by auto 
  4413   qed(insert qq, auto)
  4414 
  4415   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 -
  4416     integral {a..b} f) < e" apply(subst(asm) setsum_reindex_nonzero) apply fact
  4417     apply(rule setsum_0',rule) unfolding split_paired_all split_conv defer apply assumption
  4418   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"
  4419     note tagged_division_ofD(6)[OF qq[THEN conjunct1]] from this[OF as(1)] this[OF as(2)] 
  4420     show "content m *\<^sub>R f x = 0"  using as(3) unfolding as by auto qed
  4421   
  4422   have *:"\<And>ir ip i cr cp. norm((cp + cr) - i) < e \<Longrightarrow> norm(cr - ir) < k \<Longrightarrow> 
  4423     ip + ir = i \<Longrightarrow> norm(cp - ip) \<le> e + k" 
  4424   proof- case goal1 thus ?case  using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]  
  4425       unfolding goal1(3)[THEN sym] norm_minus_cancel by(auto simp add:algebra_simps) qed
  4426   
  4427   have "?x =  norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p. integral k f))"
  4428     unfolding split_def setsum_subtractf ..
  4429   also have "... \<le> e + k" apply(rule *[OF **, where ir="setsum (\<lambda>k. integral k f) r"])
  4430   proof- case goal2 have *:"(\<Sum>(x, k)\<in>p. integral k f) = (\<Sum>k\<in>snd ` p. integral k f)"
  4431       apply(subst setsum_reindex_nonzero) apply fact
  4432       unfolding split_paired_all snd_conv split_def o_def
  4433     proof- fix x l y m assume as:"(x,l)\<in>p" "(y,m)\<in>p" "(x,l)\<noteq>(y,m)" "l = m"
  4434       from p'(4)[OF as(1)] guess u v apply-by(erule exE)+ note uv=this
  4435       show "integral l f = 0" unfolding uv apply(rule integral_unique)
  4436         apply(rule has_integral_null) unfolding content_eq_0_interior
  4437         using p'(5)[OF as(1-3)] unfolding uv as(4)[THEN sym] by auto
  4438     qed auto 
  4439     show ?case unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
  4440       apply(rule setsum_Un_disjoint'[THEN sym]) using q(1) q'(1) p'(1) by auto
  4441   next  case goal1 have *:"k * real (card r) / (1 + real (card r)) < k" using k by(auto simp add:field_simps)
  4442     show ?case apply(rule le_less_trans[of _ "setsum (\<lambda>x. k / (real (card r) + 1)) r"])
  4443       unfolding setsum_subtractf[THEN sym] apply(rule setsum_norm_le,fact)
  4444       apply rule apply(drule qq) defer unfolding divide_inverse setsum_left_distrib[THEN sym]
  4445       unfolding divide_inverse[THEN sym] using * by(auto simp add:field_simps real_eq_of_nat)
  4446   qed finally show "?x \<le> e + k" . qed
  4447 
  4448 lemma henstock_lemma_part2: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
  4449   assumes "f integrable_on {a..b}" "0 < e" "gauge d"
  4450   "\<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 -
  4451           integral({a..b}) f) < e"    "p tagged_partial_division_of {a..b}" "d fine p"
  4452   shows "setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
  4453   unfolding split_def apply(rule setsum_norm_allsubsets_bound) defer 
  4454   apply(rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
  4455   apply safe apply(rule assms[rule_format,unfolded split_def]) defer
  4456   apply(rule tagged_partial_division_subset,rule assms,assumption)
  4457   apply(rule fine_subset,assumption,rule assms) using assms(5) by auto
  4458   
  4459 lemma henstock_lemma: fixes f::"'m::ordered_euclidean_space \<Rightarrow> 'n::ordered_euclidean_space"
  4460   assumes "f integrable_on {a..b}" "e>0"
  4461   obtains d where "gauge d"
  4462   "\<forall>p. p tagged_partial_division_of {a..b} \<and> d fine p
  4463   \<longrightarrow> setsum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
  4464 proof- have *:"e / (2 * (real DIM('n) + 1)) > 0" apply(rule divide_pos_pos) using assms(2) by auto
  4465   from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
  4466   guess d .. note d = conjunctD2[OF this] show thesis apply(rule that,rule d)
  4467   proof safe case goal1 note * = henstock_lemma_part2[OF assms(1) * d this]
  4468     show ?case apply(rule le_less_trans[OF *]) using `e>0` by(auto simp add:field_simps) qed qed
  4469 
  4470 subsection {* monotone convergence (bounded interval first). *}
  4471 
  4472 lemma monotone_convergence_interval: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
  4473   assumes "\<forall>k. (f k) integrable_on {a..b}"
  4474   "\<forall>k. \<forall>x\<in>{a..b}.(f k x) \<le> (f (Suc k) x)"
  4475   "\<forall>x\<in>{a..b}. ((\<lambda>k. f k x) ---> g x) sequentially"
  4476   "bounded {integral {a..b} (f k) | k . k \<in> UNIV}"
  4477   shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially"
  4478 proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
  4479   show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using tendsto_const by auto
  4480 next assume ab:"content {a..b} \<noteq> 0"
  4481   have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) $$ 0 \<le> (g x) $$ 0"
  4482   proof safe case goal1 note assms(3)[rule_format,OF this]
  4483     note * = Lim_component_ge[OF this trivial_limit_sequentially]
  4484     show ?case apply(rule *) unfolding eventually_sequentially
  4485       apply(rule_tac x=k in exI) apply- apply(rule transitive_stepwise_le)
  4486       using assms(2)[rule_format,OF goal1] by auto qed
  4487   have "\<exists>i. ((\<lambda>k. integral ({a..b}) (f k)) ---> i) sequentially"
  4488     apply(rule bounded_increasing_convergent) defer
  4489     apply rule apply(rule integral_le) apply safe
  4490     apply(rule assms(1-2)[rule_format])+ using assms(4) by auto
  4491   then guess i .. note i=this
  4492   have i':"\<And>k. (integral({a..b}) (f k)) \<le> i$$0"
  4493     apply(rule Lim_component_ge,rule i) apply(rule trivial_limit_sequentially)
  4494     unfolding eventually_sequentially apply(rule_tac x=k in exI)
  4495     apply(rule transitive_stepwise_le) prefer 3 unfolding Eucl_real_simps apply(rule integral_le)
  4496     apply(rule assms(1-2)[rule_format])+ using assms(2) by auto
  4497 
  4498   have "(g has_integral i) {a..b}" unfolding has_integral
  4499   proof safe case goal1 note e=this
  4500     hence "\<forall>k. (\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
  4501              norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f k x) - integral {a..b} (f k)) < e / 2 ^ (k + 2)))"
  4502       apply-apply(rule,rule assms(1)[unfolded has_integral_integral has_integral,rule_format])
  4503       apply(rule divide_pos_pos) by auto
  4504     from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
  4505 
  4506     have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i$$0 - (integral {a..b} (f k)) \<and> i$$0 - (integral {a..b} (f k)) < e / 4"
  4507     proof- case goal1 have "e/4 > 0" using e by auto
  4508       from i[unfolded Lim_sequentially,rule_format,OF this] guess r ..
  4509       thus ?case apply(rule_tac x=r in exI) apply rule
  4510         apply(erule_tac x=k in allE)
  4511       proof- case goal1 thus ?case using i'[of k] unfolding dist_real_def by auto qed qed
  4512     then guess r .. note r=conjunctD2[OF this[rule_format]]
  4513 
  4514     have "\<forall>x\<in>{a..b}. \<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x)$$0 - (f k x)$$0 \<and>
  4515            (g x)$$0 - (f k x)$$0 < e / (4 * content({a..b}))"
  4516     proof case goal1 have "e / (4 * content {a..b}) > 0" apply(rule divide_pos_pos,fact)
  4517         using ab content_pos_le[of a b] by auto
  4518       from assms(3)[rule_format,OF goal1,unfolded Lim_sequentially,rule_format,OF this]
  4519       guess n .. note n=this
  4520       thus ?case apply(rule_tac x="n + r" in exI) apply safe apply(erule_tac[2-3] x=k in allE)
  4521         unfolding dist_real_def using fg[rule_format,OF goal1] by(auto simp add:field_simps) qed
  4522     from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
  4523     def d \<equiv> "\<lambda>x. c (m x) x" 
  4524 
  4525     show ?case apply(rule_tac x=d in exI)
  4526     proof safe show "gauge d" using c(1) unfolding gauge_def d_def by auto
  4527     next fix p assume p:"p tagged_division_of {a..b}" "d fine p"
  4528       note p'=tagged_division_ofD[OF p(1)]
  4529       have "\<exists>a. \<forall>x\<in>p. m (fst x) \<le> a"
  4530         by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
  4531       then guess s .. note s=this
  4532       have *:"\<forall>a b c d. norm(a - b) \<le> e / 4 \<and> norm(b - c) < e / 2 \<and>
  4533             norm(c - d) < e / 4 \<longrightarrow> norm(a - d) < e" 
  4534       proof safe case goal1 thus ?case using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
  4535           norm_triangle_lt[of "a - b + (b - c)" "c - d" e] unfolding norm_minus_cancel
  4536           by(auto simp add:algebra_simps) qed
  4537       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - i) < e" apply(rule *[rule_format,where
  4538           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))"])
  4539       proof safe case goal1
  4540          show ?case apply(rule order_trans[of _ "\<Sum>(x, k)\<in>p. content k * (e / (4 * content {a..b}))"])
  4541            unfolding setsum_subtractf[THEN sym] apply(rule order_trans,rule setsum_norm[OF p'(1)])
  4542            apply(rule setsum_mono) unfolding split_paired_all split_conv
  4543            unfolding split_def setsum_left_distrib[THEN sym] scaleR.diff_right[THEN sym]
  4544            unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
  4545          proof- fix x k assume xk:"(x,k) \<in> p" hence x:"x\<in>{a..b}" using p'(2-3)[OF xk] by auto
  4546            from p'(4)[OF xk] guess u v apply-by(erule exE)+ note uv=this
  4547            show " norm (content k *\<^sub>R (g x - f (m x) x)) \<le> content k * (e / (4 * content {a..b}))"
  4548              unfolding norm_scaleR uv unfolding abs_of_nonneg[OF content_pos_le] 
  4549              apply(rule mult_left_mono) using m(2)[OF x,of "m x"] by auto
  4550          qed(insert ab,auto)
  4551          
  4552        next case goal2 show ?case apply(rule le_less_trans[of _ "norm (\<Sum>j = 0..s.
  4553            \<Sum>(x, k)\<in>{xk\<in>p. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))"])
  4554            apply(subst setsum_group) apply fact apply(rule finite_atLeastAtMost) defer
  4555            apply(subst split_def)+ unfolding setsum_subtractf apply rule
  4556          proof- show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> p.
  4557              m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e / 2"
  4558              apply(rule le_less_trans[of _ "setsum (\<lambda>i. e / 2^(i+2)) {0..s}"])
  4559              apply(rule setsum_norm_le[OF finite_atLeastAtMost])
  4560            proof show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
  4561                unfolding power_add divide_inverse inverse_mult_distrib
  4562                unfolding setsum_right_distrib[THEN sym] setsum_left_distrib[THEN sym]
  4563                unfolding power_inverse sum_gp apply(rule mult_strict_left_mono[OF _ e])
  4564                unfolding power2_eq_square by auto
  4565              fix t assume "t\<in>{0..s}"
  4566              show "norm (\<Sum>(x, k)\<in>{xk \<in> p. m (fst xk) = t}. content k *\<^sub>R f (m x) x -
  4567                integral k (f (m x))) \<le> e / 2 ^ (t + 2)"apply(rule order_trans[of _
  4568                "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f t x - integral k (f t)) {xk \<in> p. m (fst xk) = t})"])
  4569                apply(rule eq_refl) apply(rule arg_cong[where f=norm]) apply(rule setsum_cong2) defer
  4570                apply(rule henstock_lemma_part1) apply(rule assms(1)[rule_format])
  4571                apply(rule divide_pos_pos,rule e) defer  apply safe apply(rule c)+
  4572                apply rule apply assumption+ apply(rule tagged_partial_division_subset[of p])
  4573                apply(rule p(1)[unfolded tagged_division_of_def,THEN conjunct1]) defer
  4574                unfolding fine_def apply safe apply(drule p(2)[unfolded fine_def,rule_format])
  4575                unfolding d_def by auto qed
  4576          qed(insert s, auto)
  4577 
  4578        next case goal3
  4579          note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
  4580          have *:"\<And>sr sx ss ks kr::real. kr = sr \<longrightarrow> ks = ss \<longrightarrow> ks \<le> i \<and> sr \<le> sx \<and> sx \<le> ss \<and> 0 \<le> i$$0 - kr$$0
  4581            \<and> i$$0 - kr$$0 < e/4 \<longrightarrow> abs(sx - i) < e/4" by auto 
  4582          show ?case unfolding real_norm_def apply(rule *[rule_format],safe)
  4583            apply(rule comb[of r],rule comb[of s]) apply(rule i'[unfolded Eucl_real_simps]) 
  4584            apply(rule_tac[1-2] setsum_mono) unfolding split_paired_all split_conv
  4585            apply(rule_tac[1-2] integral_le[OF ])
  4586          proof safe show "0 \<le> i$$0 - (integral {a..b} (f r))$$0" using r(1) by auto
  4587            show "i$$0 - (integral {a..b} (f r))$$0 < e / 4" using r(2) by auto
  4588            fix x k assume xk:"(x,k)\<in>p" from p'(4)[OF this] guess u v apply-by(erule exE)+ note uv=this
  4589            show "f r integrable_on k" "f s integrable_on k" "f (m x) integrable_on k" "f (m x) integrable_on k" 
  4590              unfolding uv apply(rule_tac[!] integrable_on_subinterval[OF assms(1)[rule_format]])
  4591              using p'(3)[OF xk] unfolding uv by auto 
  4592            fix y assume "y\<in>k" hence "y\<in>{a..b}" using p'(3)[OF xk] by auto
  4593            hence *:"\<And>m. \<forall>n\<ge>m. (f m y) \<le> (f n y)" apply-apply(rule transitive_stepwise_le) using assms(2) by auto
  4594            show "(f r y) \<le> (f (m x) y)" "(f (m x) y) \<le> (f s y)"
  4595              apply(rule_tac[!] *[rule_format]) using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk] by auto
  4596          qed qed qed qed note * = this 
  4597 
  4598    have "integral {a..b} g = i" apply(rule integral_unique) using * .
  4599    thus ?thesis using i * by auto qed
  4600 
  4601 lemma monotone_convergence_increasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
  4602   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x)"
  4603   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4604   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4605 proof- have lem:"\<And>f::nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real. \<And> g s. \<forall>k.\<forall>x\<in>s. 0 \<le> (f k x) \<Longrightarrow> \<forall>k. (f k) integrable_on s \<Longrightarrow>
  4606     \<forall>k. \<forall>x\<in>s. (f k x) \<le> (f (Suc k) x) \<Longrightarrow> \<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially  \<Longrightarrow>
  4607     bounded {integral s (f k)| k. True} \<Longrightarrow> g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4608   proof- case goal1 note assms=this[rule_format]
  4609     have "\<forall>x\<in>s. \<forall>k. (f k x)$$0 \<le> (g x)$$0" apply safe apply(rule Lim_component_ge)
  4610       apply(rule goal1(4)[rule_format],assumption) apply(rule trivial_limit_sequentially)
  4611       unfolding eventually_sequentially apply(rule_tac x=k in exI)
  4612       apply(rule transitive_stepwise_le) using goal1(3) by auto note fg=this[rule_format]
  4613 
  4614     have "\<exists>i. ((\<lambda>k. integral s (f k)) ---> i) sequentially" apply(rule bounded_increasing_convergent)
  4615       apply(rule goal1(5)) apply(rule,rule integral_le) apply(rule goal1(2)[rule_format])+
  4616       using goal1(3) by auto then guess i .. note i=this
  4617     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
  4618     hence i':"\<forall>k. (integral s (f k))$$0 \<le> i$$0" apply-apply(rule,rule Lim_component_ge)
  4619       apply(rule i,rule trivial_limit_sequentially) unfolding eventually_sequentially
  4620       apply(rule_tac x=k in exI,safe) apply(rule integral_component_le)
  4621       apply(rule goal1(2)[rule_format])+ by auto 
  4622 
  4623     note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
  4624     have ifif:"\<And>k t. (\<lambda>x. if x \<in> t then if x \<in> s then f k x else 0 else 0) =
  4625       (\<lambda>x. if x \<in> t\<inter>s then f k x else 0)" apply(rule ext) by auto
  4626     have int':"\<And>k a b. f k integrable_on {a..b} \<inter> s" apply(subst integrable_restrict_univ[THEN sym])
  4627       apply(subst ifif[THEN sym]) apply(subst integrable_restrict_univ) using int .
  4628     have "\<And>a b. (\<lambda>x. if x \<in> s then g x else 0) integrable_on {a..b} \<and>
  4629       ((\<lambda>k. integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) --->
  4630       integral {a..b} (\<lambda>x. if x \<in> s then g x else 0)) sequentially"
  4631     proof(rule monotone_convergence_interval,safe)
  4632       case goal1 show ?case using int .
  4633     next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto
  4634     next case goal3 thus ?case apply-apply(cases "x\<in>s")
  4635         using assms(4) by (auto intro: tendsto_const)
  4636     next case goal4 note * = integral_nonneg
  4637       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))"
  4638         unfolding real_norm_def apply(subst abs_of_nonneg) apply(rule *[OF int])
  4639         apply(safe,case_tac "x\<in>s") apply(drule assms(1)) prefer 3
  4640         apply(subst abs_of_nonneg) apply(rule *[OF assms(2) goal1(1)[THEN spec]])
  4641         apply(subst integral_restrict_univ[THEN sym,OF int]) 
  4642         unfolding ifif unfolding integral_restrict_univ[OF int']
  4643         apply(rule integral_subset_le[OF _ int' assms(2)]) using assms(1) by auto
  4644       thus ?case using assms(5) unfolding bounded_iff apply safe
  4645         apply(rule_tac x=aa in exI,safe) apply(erule_tac x="integral s (f k)" in ballE)
  4646         apply(rule order_trans) apply assumption by auto qed note g = conjunctD2[OF this]
  4647 
  4648     have "(g has_integral i) s" unfolding has_integral_alt' apply safe apply(rule g(1))
  4649     proof- case goal1 hence "e/4>0" by auto
  4650       from i[unfolded Lim_sequentially,rule_format,OF this] guess N .. note N=this
  4651       note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]]
  4652       from this[THEN conjunct2,rule_format,OF `e/4>0`] guess B .. note B=conjunctD2[OF this]
  4653       show ?case apply(rule,rule,rule B,safe)
  4654       proof- fix a b::"'n::ordered_euclidean_space" assume ab:"ball 0 B \<subseteq> {a..b}"
  4655         from `e>0` have "e/2>0" by auto
  4656         from g(2)[unfolded Lim_sequentially,of a b,rule_format,OF this] guess M .. note M=this
  4657         have **:"norm (integral {a..b} (\<lambda>x. if x \<in> s then f N x else 0) - i) < e/2"
  4658           apply(rule norm_triangle_half_l) using B(2)[rule_format,OF ab] N[rule_format,of N]
  4659           unfolding dist_norm apply-defer apply(subst norm_minus_commute) by auto
  4660         have *:"\<And>f1 f2 g. abs(f1 - i) < e / 2 \<longrightarrow> abs(f2 - g) < e / 2 \<longrightarrow> f1 \<le> f2 \<longrightarrow> f2 \<le> i
  4661           \<longrightarrow> abs(g - i) < e" unfolding Eucl_real_simps by arith
  4662         show "norm (integral {a..b} (\<lambda>x. if x \<in> s then g x else 0) - i) < e" 
  4663           unfolding real_norm_def apply(rule *[rule_format])
  4664           apply(rule **[unfolded real_norm_def])
  4665           apply(rule M[rule_format,of "M + N",unfolded dist_real_def]) apply(rule le_add1)
  4666           apply(rule integral_le[OF int int]) defer
  4667           apply(rule order_trans[OF _ i'[rule_format,of "M + N",unfolded Eucl_real_simps]])
  4668         proof safe case goal2 have "\<And>m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x)$$0 \<le> (f n x)$$0"
  4669             apply(rule transitive_stepwise_le) using assms(3) by auto thus ?case by auto
  4670         next case goal1 show ?case apply(subst integral_restrict_univ[THEN sym,OF int]) 
  4671             unfolding ifif integral_restrict_univ[OF int']
  4672             apply(rule integral_subset_le[OF _ int']) using assms by auto
  4673         qed qed qed 
  4674     thus ?case apply safe defer apply(drule integral_unique) using i by auto qed
  4675 
  4676   have sub:"\<And>k. integral s (\<lambda>x. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
  4677     apply(subst integral_sub) apply(rule assms(1)[rule_format])+ by rule
  4678   have "\<And>x m. x\<in>s \<Longrightarrow> \<forall>n\<ge>m. (f m x) \<le> (f n x)" apply(rule transitive_stepwise_le)
  4679     using assms(2) by auto note * = this[rule_format]
  4680   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)) --->
  4681       integral s (\<lambda>x. g x - f 0 x)) sequentially" apply(rule lem,safe)
  4682   proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
  4683   next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
  4684   next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
  4685   next case goal4 thus ?case apply-apply(rule tendsto_diff) 
  4686       using seq_offset[OF assms(3)[rule_format],of x 1] by (auto intro: tendsto_const)
  4687   next case goal5 thus ?case using assms(4) unfolding bounded_iff
  4688       apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
  4689       apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub
  4690       apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed
  4691   note conjunctD2[OF this] note tendsto_add[OF this(2) tendsto_const[of "integral s (f 0)"]]
  4692     integrable_add[OF this(1) assms(1)[rule_format,of 0]]
  4693   thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub)
  4694     using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed
  4695 
  4696 lemma monotone_convergence_decreasing: fixes f::"nat \<Rightarrow> 'n::ordered_euclidean_space \<Rightarrow> real"
  4697   assumes "\<forall>k. (f k) integrable_on s"  "\<forall>k. \<forall>x\<in>s. (f (Suc k) x) \<le> (f k x)"
  4698   "\<forall>x\<in>s. ((\<lambda>k. f k x) ---> g x) sequentially" "bounded {integral s (f k)| k. True}"
  4699   shows "g integrable_on s \<and> ((\<lambda>k. integral s (f k)) ---> integral s g) sequentially"
  4700 proof- note assm = assms[rule_format]
  4701   have *:"{integral s (\<lambda>x. - f k x) |k. True} = op *\<^sub>R -1 ` {integral s (f k)| k. True}"
  4702     apply safe unfolding image_iff apply(rule_tac x="integral s (f k)" in bexI) prefer 3
  4703     apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto
  4704   have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x))
  4705     ---> integral s (\<lambda>x. - g x))  sequentially" apply(rule monotone_convergence_increasing)
  4706     apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule tendsto_minus)
  4707     apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto
  4708   note * = conjunctD2[OF this]
  4709   show ?thesis apply rule using integrable_neg[OF *(1)] defer
  4710     using tendsto_minus[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
  4711     unfolding integral_neg[OF *(1),THEN sym] by auto qed
  4712 
  4713 subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
  4714 
  4715 definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46) where
  4716   "f absolutely_integrable_on s \<longleftrightarrow> f integrable_on s \<and> (\<lambda>x. (norm(f x))) integrable_on s"
  4717 
  4718 lemma absolutely_integrable_onI[intro?]:
  4719   "f integrable_on s \<Longrightarrow> (\<lambda>x. (norm(f x))) integrable_on s \<Longrightarrow> f absolutely_integrable_on s"
  4720   unfolding absolutely_integrable_on_def by auto
  4721 
  4722 lemma absolutely_integrable_onD[dest]: assumes "f absolutely_integrable_on s"
  4723   shows "f integrable_on s" "(\<lambda>x. (norm(f x))) integrable_on s"
  4724   using assms unfolding absolutely_integrable_on_def by auto
  4725 
  4726 (*lemma absolutely_integrable_on_trans[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real" shows
  4727   "(vec1 o f) absolutely_integrable_on s \<longleftrightarrow> f absolutely_integrable_on s"
  4728   unfolding absolutely_integrable_on_def o_def by auto*)
  4729 
  4730 lemma integral_norm_bound_integral: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4731   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. norm(f x) \<le> g x"
  4732   shows "norm(integral s f) \<le> (integral s g)"
  4733 proof- have *:"\<And>x y. (\<forall>e::real. 0 < e \<longrightarrow> x < y + e) \<longrightarrow> x \<le> y" apply(safe,rule ccontr)
  4734     apply(erule_tac x="x - y" in allE) by auto
  4735   have "\<And>e sg dsa dia ig. norm(sg) \<le> dsa \<longrightarrow> abs(dsa - dia) < e / 2 \<longrightarrow> norm(sg - ig) < e / 2
  4736     \<longrightarrow> norm(ig) < dia + e" 
  4737   proof safe case goal1 show ?case apply(rule le_less_trans[OF norm_triangle_sub[of ig sg]])
  4738       apply(subst real_sum_of_halves[of e,THEN sym]) unfolding add_assoc[symmetric]
  4739       apply(rule add_le_less_mono) defer apply(subst norm_minus_commute,rule goal1)
  4740       apply(rule order_trans[OF goal1(1)]) using goal1(2) by arith
  4741   qed note norm=this[rule_format]
  4742   have lem:"\<And>f::'n::ordered_euclidean_space \<Rightarrow> 'a. \<And> g a b. f integrable_on {a..b} \<Longrightarrow> g integrable_on {a..b} \<Longrightarrow>
  4743     \<forall>x\<in>{a..b}. norm(f x) \<le> (g x) \<Longrightarrow> norm(integral({a..b}) f) \<le> (integral({a..b}) g)"
  4744   proof(rule *[rule_format]) case goal1 hence *:"e/2>0" by auto
  4745     from integrable_integral[OF goal1(1),unfolded has_integral[of f],rule_format,OF *]
  4746     guess d1 .. note d1 = conjunctD2[OF this,rule_format]
  4747     from integrable_integral[OF goal1(2),unfolded has_integral[of g],rule_format,OF *]
  4748     guess d2 .. note d2 = conjunctD2[OF this,rule_format]
  4749     note gauge_inter[OF d1(1) d2(1)]
  4750     from fine_division_exists[OF this, of a b] guess p . note p=this
  4751     show ?case apply(rule norm) defer
  4752       apply(rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def]) defer
  4753       apply(rule d1(2)[OF conjI[OF p(1)]]) defer apply(rule setsum_norm_le)
  4754     proof safe fix x k assume "(x,k)\<in>p" note as = tagged_division_ofD(2-4)[OF p(1) this]
  4755       from this(3) guess u v apply-by(erule exE)+ note uv=this
  4756       show "norm (content k *\<^sub>R f x) \<le> content k *\<^sub>R g x" unfolding uv norm_scaleR
  4757         unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
  4758         apply(rule mult_left_mono) using goal1(3) as by auto
  4759     qed(insert p[unfolded fine_inter],auto) qed
  4760 
  4761   { presume "\<And>e. 0 < e \<Longrightarrow> norm (integral s f) < integral s g + e" 
  4762     thus ?thesis apply-apply(rule *[rule_format]) by auto }
  4763   fix e::real assume "e>0" hence e:"e/2 > 0" by auto
  4764   note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
  4765   note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format]
  4766   from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e]
  4767   guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format]
  4768   from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e]
  4769   guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format]
  4770   from bounded_subset_closed_interval[OF bounded_ball, of "0::'n::ordered_euclidean_space" "max B1 B2"]
  4771   guess a b apply-by(erule exE)+ note ab=this[unfolded ball_max_Un]
  4772 
  4773   have "ball 0 B1 \<subseteq> {a..b}" using ab by auto
  4774   from B1(2)[OF this] guess z .. note z=conjunctD2[OF this]
  4775   have "ball 0 B2 \<subseteq> {a..b}" using ab by auto
  4776   from B2(2)[OF this] guess w .. note w=conjunctD2[OF this]
  4777 
  4778   show "norm (integral s f) < integral s g + e" apply(rule norm)
  4779     apply(rule lem[OF f g, of a b]) unfolding integral_unique[OF z(1)] integral_unique[OF w(1)]
  4780     defer apply(rule w(2)[unfolded real_norm_def],rule z(2))
  4781     apply safe apply(case_tac "x\<in>s") unfolding if_P apply(rule assms(3)[rule_format]) by auto qed
  4782 
  4783 lemma integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4784   fixes g::"'n => 'b::ordered_euclidean_space"
  4785   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
  4786   shows "norm(integral s f) \<le> (integral s g)$$k"
  4787 proof- have "norm (integral s f) \<le> integral s ((\<lambda>x. x $$ k) o g)"
  4788     apply(rule integral_norm_bound_integral[OF assms(1)])
  4789     apply(rule integrable_linear[OF assms(2)],rule)
  4790     unfolding o_def by(rule assms)
  4791   thus ?thesis unfolding o_def integral_component_eq[OF assms(2)] . qed
  4792 
  4793 lemma has_integral_norm_bound_integral_component: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4794   fixes g::"'n => 'b::ordered_euclidean_space"
  4795   assumes "(f has_integral i) s" "(g has_integral j) s" "\<forall>x\<in>s. norm(f x) \<le> (g x)$$k"
  4796   shows "norm(i) \<le> j$$k" using integral_norm_bound_integral_component[of f s g k]
  4797   unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
  4798   using assms by auto
  4799 
  4800 lemma absolutely_integrable_le: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4801   assumes "f absolutely_integrable_on s"
  4802   shows "norm(integral s f) \<le> integral s (\<lambda>x. norm(f x))"
  4803   apply(rule integral_norm_bound_integral) using assms by auto
  4804 
  4805 lemma absolutely_integrable_0[intro]: "(\<lambda>x. 0) absolutely_integrable_on s"
  4806   unfolding absolutely_integrable_on_def by auto
  4807 
  4808 lemma absolutely_integrable_cmul[intro]:
  4809  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) absolutely_integrable_on s"
  4810   unfolding absolutely_integrable_on_def using integrable_cmul[of f s c]
  4811   using integrable_cmul[of "\<lambda>x. norm (f x)" s "abs c"] by auto
  4812 
  4813 lemma absolutely_integrable_neg[intro]:
  4814  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) absolutely_integrable_on s"
  4815   apply(drule absolutely_integrable_cmul[where c="-1"]) by auto
  4816 
  4817 lemma absolutely_integrable_norm[intro]:
  4818  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. norm(f x)) absolutely_integrable_on s"
  4819   unfolding absolutely_integrable_on_def by auto
  4820 
  4821 lemma absolutely_integrable_abs[intro]:
  4822  "f absolutely_integrable_on s \<Longrightarrow> (\<lambda>x. abs(f x::real)) absolutely_integrable_on s"
  4823   apply(drule absolutely_integrable_norm) unfolding real_norm_def .
  4824 
  4825 lemma absolutely_integrable_on_subinterval: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach" shows
  4826   "f absolutely_integrable_on s \<Longrightarrow> {a..b} \<subseteq> s \<Longrightarrow> f absolutely_integrable_on {a..b}" 
  4827   unfolding absolutely_integrable_on_def by(meson integrable_on_subinterval)
  4828 
  4829 lemma absolutely_integrable_bounded_variation: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
  4830   assumes "f absolutely_integrable_on UNIV"
  4831   obtains B where "\<forall>d. d division_of (\<Union>d) \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
  4832   apply(rule that[of "integral UNIV (\<lambda>x. norm (f x))"])
  4833 proof safe case goal1 note d = division_ofD[OF this(2)]
  4834   have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. integral i (\<lambda>x. norm (f x)))"
  4835     apply(rule setsum_mono,rule absolutely_integrable_le) apply(drule d(4),safe)
  4836     apply(rule absolutely_integrable_on_subinterval[OF assms]) by auto
  4837   also have "... \<le> integral (\<Union>d) (\<lambda>x. norm (f x))"
  4838     apply(subst integral_combine_division_topdown[OF _ goal1(2)])
  4839     using integrable_on_subdivision[OF goal1(2)] using assms by auto
  4840   also have "... \<le> integral UNIV (\<lambda>x. norm (f x))"
  4841     apply(rule integral_subset_le) 
  4842     using integrable_on_subdivision[OF goal1(2)] using assms by auto
  4843   finally show ?case . qed
  4844 
  4845 lemma helplemma:
  4846   assumes "setsum (\<lambda>x. norm(f x - g x)) s < e" "finite s"
  4847   shows "abs(setsum (\<lambda>x. norm(f x)) s - setsum (\<lambda>x. norm(g x)) s) < e"
  4848   unfolding setsum_subtractf[THEN sym] apply(rule le_less_trans[OF setsum_abs])
  4849   apply(rule le_less_trans[OF _ assms(1)]) apply(rule setsum_mono)
  4850   using norm_triangle_ineq3 .
  4851 
  4852 lemma bounded_variation_absolutely_integrable_interval:
  4853   fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'm::ordered_euclidean_space" assumes "f integrable_on {a..b}"
  4854   "\<forall>d. d division_of {a..b} \<longrightarrow> setsum (\<lambda>k. norm(integral k f)) d \<le> B"
  4855   shows "f absolutely_integrable_on {a..b}"
  4856 proof- let ?S = "(\<lambda>d. setsum (\<lambda>k. norm(integral k f)) d) ` {d. d division_of {a..b} }" def i \<equiv> "Sup ?S"
  4857   have i:"isLub UNIV ?S i" unfolding i_def apply(rule Sup)
  4858     apply(rule elementary_interval) defer apply(rule_tac x=B in exI)
  4859     apply(rule setleI) using assms(2) by auto
  4860   show ?thesis apply(rule,rule assms) apply rule apply(subst has_integral[of _ i])
  4861   proof safe case goal1 hence "i - e / 2 \<notin> Collect (isUb UNIV (setsum (\<lambda>k. norm (integral k f)) `
  4862         {d. d division_of {a..b}}))" using isLub_ubs[OF i,rule_format]
  4863       unfolding setge_def ubs_def by auto 
  4864     hence " \<exists>y. y division_of {a..b} \<and> i - e / 2 < (\<Sum>k\<in>y. norm (integral k f))"
  4865       unfolding mem_Collect_eq isUb_def setle_def by(simp add:not_le) then guess d .. note d=conjunctD2[OF this]
  4866     note d' = division_ofD[OF this(1)]
  4867 
  4868     have "\<forall>x. \<exists>e>0. \<forall>i\<in>d. x \<notin> i \<longrightarrow> ball x e \<inter> i = {}"
  4869     proof case goal1 have "\<exists>da>0. \<forall>xa\<in>\<Union>{i \<in> d. x \<notin> i}. da \<le> dist x xa"
  4870         apply(rule separate_point_closed) apply(rule closed_Union)
  4871         apply(rule finite_subset[OF _ d'(1)]) apply safe apply(drule d'(4)) by auto
  4872       thus ?case apply safe apply(rule_tac x=da in exI,safe)
  4873         apply(erule_tac x=xa in ballE) by auto
  4874     qed from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
  4875 
  4876     have "e/2 > 0" using goal1 by auto
  4877     from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
  4878     let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
  4879     show ?case apply(rule_tac x="?g" in exI) apply safe
  4880     proof- show "gauge ?g" using g(1) unfolding gauge_def using k(1) by auto
  4881       fix p assume "p tagged_division_of {a..b}" "?g fine p"
  4882       note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
  4883       note p' = tagged_division_ofD[OF p(1)]
  4884       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}"
  4885       have gp':"g fine p'" using p(2) unfolding p'_def fine_def by auto
  4886       have p'':"p' tagged_division_of {a..b}" apply(rule tagged_division_ofI)
  4887       proof- show "finite p'" apply(rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l))
  4888           ` {(k,xl) | k xl. k \<in> d \<and> xl \<in> p}"]) unfolding p'_def 
  4889           defer apply(rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
  4890           apply safe unfolding image_iff apply(rule_tac x="(i,x,l)" in bexI) by auto
  4891         fix x k assume "(x,k)\<in>p'"
  4892         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
  4893         then guess i l apply-by(erule exE)+ note il=conjunctD4[OF this]
  4894         show "x\<in>k" "k\<subseteq>{a..b}" using p'(2-3)[OF il(3)] il by auto
  4895         show "\<exists>a b. k = {a..b}" unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
  4896           apply safe unfolding inter_interval by auto
  4897       next fix x1 k1 assume "(x1,k1)\<in>p'"
  4898         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
  4899         then guess i1 l1 apply-by(erule exE)+ note il1=conjunctD4[OF this]
  4900         fix x2 k2 assume "(x2,k2)\<in>p'"
  4901         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
  4902         then guess i2 l2 apply-by(erule exE)+ note il2=conjunctD4[OF this]
  4903         assume "(x1, k1) \<noteq> (x2, k2)"
  4904         hence "interior(i1) \<inter> interior(i2) = {} \<or> interior(l1) \<inter> interior(l2) = {}"
  4905           using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] unfolding il1 il2 by auto
  4906         thus "interior k1 \<inter> interior k2 = {}" unfolding il1 il2 by auto
  4907       next have *:"\<forall>(x, X) \<in> p'. X \<subseteq> {a..b}" unfolding p'_def using d' by auto
  4908         show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = {a..b}" apply rule apply(rule Union_least)
  4909           unfolding mem_Collect_eq apply(erule exE) apply(drule *[rule_format]) apply safe
  4910         proof- fix y assume y:"y\<in>{a..b}"
  4911           hence "\<exists>x l. (x, l) \<in> p \<and> y\<in>l" unfolding p'(6)[THEN sym] by auto
  4912           then guess x l apply-by(erule exE)+ note xl=conjunctD2[OF this]
  4913           hence "\<exists>k. k\<in>d \<and> y\<in>k" using y unfolding d'(6)[THEN sym] by auto
  4914           then guess i .. note i = conjunctD2[OF this]
  4915           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
  4916           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)
  4917             defer unfolding mem_Collect_eq apply(rule_tac x=x in exI)+ apply(rule_tac x="i\<inter>l" in exI)
  4918             apply safe apply(rule_tac x=i in exI) apply(rule_tac x=l in exI) using i xl by auto 
  4919         qed qed 
  4920 
  4921       hence "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"