src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
 author haftmann Wed Aug 10 18:57:20 2016 +0200 (2016-08-10) changeset 63659 abe0c3872d8a parent 63627 6ddb43c6b711 child 63680 6e1e8b5abbfa permissions -rw-r--r--
keeping lifting rules local
```     1 (*  Author:     John Harrison
```
```     2     Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP
```
```     3 *)
```
```     4
```
```     5 section \<open>Henstock-Kurzweil gauge integration in many dimensions.\<close>
```
```     6
```
```     7 theory Henstock_Kurzweil_Integration
```
```     8 imports
```
```     9   Derivative
```
```    10   Uniform_Limit
```
```    11   "~~/src/HOL/Library/Indicator_Function"
```
```    12 begin
```
```    13
```
```    14 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
```
```    15   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
```
```    16   scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
```
```    17
```
```    18
```
```    19 subsection \<open>Sundries\<close>
```
```    20
```
```    21 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
```
```    22 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
```
```    23 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
```
```    24
```
```    25 declare norm_triangle_ineq4[intro]
```
```    26
```
```    27 lemma simple_image: "{f x |x . x \<in> s} = f ` s"
```
```    28   by blast
```
```    29
```
```    30 lemma linear_simps:
```
```    31   assumes "bounded_linear f"
```
```    32   shows
```
```    33     "f (a + b) = f a + f b"
```
```    34     "f (a - b) = f a - f b"
```
```    35     "f 0 = 0"
```
```    36     "f (- a) = - f a"
```
```    37     "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
```
```    38 proof -
```
```    39   interpret f: bounded_linear f by fact
```
```    40   show "f (a + b) = f a + f b" by (rule f.add)
```
```    41   show "f (a - b) = f a - f b" by (rule f.diff)
```
```    42   show "f 0 = 0" by (rule f.zero)
```
```    43   show "f (- a) = - f a" by (rule f.minus)
```
```    44   show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
```
```    45 qed
```
```    46
```
```    47 lemma bounded_linearI:
```
```    48   assumes "\<And>x y. f (x + y) = f x + f y"
```
```    49     and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
```
```    50     and "\<And>x. norm (f x) \<le> norm x * K"
```
```    51   shows "bounded_linear f"
```
```    52   using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
```
```    53
```
```    54 lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \<bullet> k)"
```
```    55   by (rule bounded_linear_inner_left)
```
```    56
```
```    57 lemma transitive_stepwise_lt_eq:
```
```    58   assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
```
```    59   shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))"
```
```    60   (is "?l = ?r")
```
```    61 proof safe
```
```    62   assume ?r
```
```    63   fix n m :: nat
```
```    64   assume "m < n"
```
```    65   then show "R m n"
```
```    66   proof (induct n arbitrary: m)
```
```    67     case 0
```
```    68     then show ?case by auto
```
```    69   next
```
```    70     case (Suc n)
```
```    71     show ?case
```
```    72     proof (cases "m < n")
```
```    73       case True
```
```    74       show ?thesis
```
```    75         apply (rule assms[OF Suc(1)[OF True]])
```
```    76         using \<open>?r\<close>
```
```    77         apply auto
```
```    78         done
```
```    79     next
```
```    80       case False
```
```    81       then have "m = n"
```
```    82         using Suc(2) by auto
```
```    83       then show ?thesis
```
```    84         using \<open>?r\<close> by auto
```
```    85     qed
```
```    86   qed
```
```    87 qed auto
```
```    88
```
```    89 lemma transitive_stepwise_gt:
```
```    90   assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n)"
```
```    91   shows "\<forall>n>m. R m n"
```
```    92 proof -
```
```    93   have "\<forall>m. \<forall>n>m. R m n"
```
```    94     apply (subst transitive_stepwise_lt_eq)
```
```    95     apply (blast intro: assms)+
```
```    96     done
```
```    97   then show ?thesis by auto
```
```    98 qed
```
```    99
```
```   100 lemma transitive_stepwise_le_eq:
```
```   101   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
```
```   102   shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))"
```
```   103   (is "?l = ?r")
```
```   104 proof safe
```
```   105   assume ?r
```
```   106   fix m n :: nat
```
```   107   assume "m \<le> n"
```
```   108   then show "R m n"
```
```   109   proof (induct n arbitrary: m)
```
```   110     case 0
```
```   111     with assms show ?case by auto
```
```   112   next
```
```   113     case (Suc n)
```
```   114     show ?case
```
```   115     proof (cases "m \<le> n")
```
```   116       case True
```
```   117       with Suc.hyps \<open>\<forall>n. R n (Suc n)\<close> assms show ?thesis
```
```   118         by blast
```
```   119     next
```
```   120       case False
```
```   121       then have "m = Suc n"
```
```   122         using Suc(2) by auto
```
```   123       then show ?thesis
```
```   124         using assms(1) by auto
```
```   125     qed
```
```   126   qed
```
```   127 qed auto
```
```   128
```
```   129 lemma transitive_stepwise_le:
```
```   130   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
```
```   131     and "\<And>n. R n (Suc n)"
```
```   132   shows "\<forall>n\<ge>m. R m n"
```
```   133 proof -
```
```   134   have "\<forall>m. \<forall>n\<ge>m. R m n"
```
```   135     apply (subst transitive_stepwise_le_eq)
```
```   136     apply (blast intro: assms)+
```
```   137     done
```
```   138   then show ?thesis by auto
```
```   139 qed
```
```   140
```
```   141
```
```   142 subsection \<open>Some useful lemmas about intervals.\<close>
```
```   143
```
```   144 lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
```
```   145   using nonempty_Basis
```
```   146   by (fastforce simp add: set_eq_iff mem_box)
```
```   147
```
```   148 lemma interior_subset_union_intervals:
```
```   149   assumes "i = cbox a b"
```
```   150     and "j = cbox c d"
```
```   151     and "interior j \<noteq> {}"
```
```   152     and "i \<subseteq> j \<union> s"
```
```   153     and "interior i \<inter> interior j = {}"
```
```   154   shows "interior i \<subseteq> interior s"
```
```   155 proof -
```
```   156   have "box a b \<inter> cbox c d = {}"
```
```   157      using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
```
```   158      unfolding assms(1,2) interior_cbox by auto
```
```   159   moreover
```
```   160   have "box a b \<subseteq> cbox c d \<union> s"
```
```   161     apply (rule order_trans,rule box_subset_cbox)
```
```   162     using assms(4) unfolding assms(1,2)
```
```   163     apply auto
```
```   164     done
```
```   165   ultimately
```
```   166   show ?thesis
```
```   167     unfolding assms interior_cbox
```
```   168       by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
```
```   169 qed
```
```   170
```
```   171 lemma interior_Union_subset_cbox:
```
```   172   assumes "finite f"
```
```   173   assumes f: "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a b" "\<And>s. s \<in> f \<Longrightarrow> interior s \<subseteq> t"
```
```   174     and t: "closed t"
```
```   175   shows "interior (\<Union>f) \<subseteq> t"
```
```   176 proof -
```
```   177   have [simp]: "s \<in> f \<Longrightarrow> closed s" for s
```
```   178     using f by auto
```
```   179   define E where "E = {s\<in>f. interior s = {}}"
```
```   180   then have "finite E" "E \<subseteq> {s\<in>f. interior s = {}}"
```
```   181     using \<open>finite f\<close> by auto
```
```   182   then have "interior (\<Union>f) = interior (\<Union>(f - E))"
```
```   183   proof (induction E rule: finite_subset_induct')
```
```   184     case (insert s f')
```
```   185     have "interior (\<Union>(f - insert s f') \<union> s) = interior (\<Union>(f - insert s f'))"
```
```   186       using insert.hyps \<open>finite f\<close> by (intro interior_closed_Un_empty_interior) auto
```
```   187     also have "\<Union>(f - insert s f') \<union> s = \<Union>(f - f')"
```
```   188       using insert.hyps by auto
```
```   189     finally show ?case
```
```   190       by (simp add: insert.IH)
```
```   191   qed simp
```
```   192   also have "\<dots> \<subseteq> \<Union>(f - E)"
```
```   193     by (rule interior_subset)
```
```   194   also have "\<dots> \<subseteq> t"
```
```   195   proof (rule Union_least)
```
```   196     fix s assume "s \<in> f - E"
```
```   197     with f[of s] obtain a b where s: "s \<in> f" "s = cbox a b" "box a b \<noteq> {}"
```
```   198       by (fastforce simp: E_def)
```
```   199     have "closure (interior s) \<subseteq> closure t"
```
```   200       by (intro closure_mono f \<open>s \<in> f\<close>)
```
```   201     with s \<open>closed t\<close> show "s \<subseteq> t"
```
```   202       by (simp add: closure_box)
```
```   203   qed
```
```   204   finally show ?thesis .
```
```   205 qed
```
```   206
```
```   207 lemma inter_interior_unions_intervals:
```
```   208     "finite f \<Longrightarrow> open s \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow> \<forall>t\<in>f. s \<inter> (interior t) = {} \<Longrightarrow> s \<inter> interior (\<Union>f) = {}"
```
```   209   using interior_Union_subset_cbox[of f "UNIV - s"] by auto
```
```   210
```
```   211 subsection \<open>Bounds on intervals where they exist.\<close>
```
```   212
```
```   213 definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
```
```   214   where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
```
```   215
```
```   216 definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
```
```   217    where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
```
```   218
```
```   219 lemma interval_upperbound[simp]:
```
```   220   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
```
```   221     interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
```
```   222   unfolding interval_upperbound_def euclidean_representation_setsum cbox_def
```
```   223   by (safe intro!: cSup_eq) auto
```
```   224
```
```   225 lemma interval_lowerbound[simp]:
```
```   226   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
```
```   227     interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
```
```   228   unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def
```
```   229   by (safe intro!: cInf_eq) auto
```
```   230
```
```   231 lemmas interval_bounds = interval_upperbound interval_lowerbound
```
```   232
```
```   233 lemma
```
```   234   fixes X::"real set"
```
```   235   shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
```
```   236     and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
```
```   237   by (auto simp: interval_upperbound_def interval_lowerbound_def)
```
```   238
```
```   239 lemma interval_bounds'[simp]:
```
```   240   assumes "cbox a b \<noteq> {}"
```
```   241   shows "interval_upperbound (cbox a b) = b"
```
```   242     and "interval_lowerbound (cbox a b) = a"
```
```   243   using assms unfolding box_ne_empty by auto
```
```   244
```
```   245
```
```   246 lemma interval_upperbound_Times:
```
```   247   assumes "A \<noteq> {}" and "B \<noteq> {}"
```
```   248   shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
```
```   249 proof-
```
```   250   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
```
```   251   have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:A. x \<bullet> i) *\<^sub>R i)"
```
```   252       by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
```
```   253   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
```
```   254   have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)"
```
```   255       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
```
```   256   ultimately show ?thesis unfolding interval_upperbound_def
```
```   257       by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
```
```   258 qed
```
```   259
```
```   260 lemma interval_lowerbound_Times:
```
```   261   assumes "A \<noteq> {}" and "B \<noteq> {}"
```
```   262   shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
```
```   263 proof-
```
```   264   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
```
```   265   have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:A. x \<bullet> i) *\<^sub>R i)"
```
```   266       by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
```
```   267   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
```
```   268   have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)"
```
```   269       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
```
```   270   ultimately show ?thesis unfolding interval_lowerbound_def
```
```   271       by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
```
```   272 qed
```
```   273
```
```   274 subsection \<open>Content (length, area, volume...) of an interval.\<close>
```
```   275
```
```   276 definition "content (s::('a::euclidean_space) set) =
```
```   277   (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
```
```   278
```
```   279 lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
```
```   280   unfolding box_eq_empty unfolding not_ex not_less by auto
```
```   281
```
```   282 lemma content_cbox:
```
```   283   fixes a :: "'a::euclidean_space"
```
```   284   assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```   285   shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
```
```   286   using interval_not_empty[OF assms]
```
```   287   unfolding content_def
```
```   288   by auto
```
```   289
```
```   290 lemma content_cbox':
```
```   291   fixes a :: "'a::euclidean_space"
```
```   292   assumes "cbox a b \<noteq> {}"
```
```   293   shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
```
```   294     using assms box_ne_empty(1) content_cbox by blast
```
```   295
```
```   296 lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
```
```   297   by (auto simp: interval_upperbound_def interval_lowerbound_def content_def)
```
```   298
```
```   299 lemma abs_eq_content: "\<bar>y - x\<bar> = (if x\<le>y then content {x .. y} else content {y..x})"
```
```   300   by (auto simp: content_real)
```
```   301
```
```   302 lemma content_singleton[simp]: "content {a} = 0"
```
```   303 proof -
```
```   304   have "content (cbox a a) = 0"
```
```   305     by (subst content_cbox) (auto simp: ex_in_conv)
```
```   306   then show ?thesis by (simp add: cbox_sing)
```
```   307 qed
```
```   308
```
```   309 lemma content_unit[iff]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
```
```   310  proof -
```
```   311    have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
```
```   312     by auto
```
```   313   have "0 \<in> cbox 0 (One::'a)"
```
```   314     unfolding mem_box by auto
```
```   315   then show ?thesis
```
```   316      unfolding content_def interval_bounds[OF *] using setprod.neutral_const by auto
```
```   317  qed
```
```   318
```
```   319 lemma content_pos_le[intro]:
```
```   320   fixes a::"'a::euclidean_space"
```
```   321   shows "0 \<le> content (cbox a b)"
```
```   322 proof (cases "cbox a b = {}")
```
```   323   case False
```
```   324   then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```   325     unfolding box_ne_empty .
```
```   326   have "0 \<le> (\<Prod>i\<in>Basis. interval_upperbound (cbox a b) \<bullet> i - interval_lowerbound (cbox a b) \<bullet> i)"
```
```   327     apply (rule setprod_nonneg)
```
```   328     unfolding interval_bounds[OF *]
```
```   329     using *
```
```   330     apply auto
```
```   331     done
```
```   332   also have "\<dots> = content (cbox a b)" using False by (simp add: content_def)
```
```   333   finally show ?thesis .
```
```   334 qed (simp add: content_def)
```
```   335
```
```   336 corollary content_nonneg [simp]:
```
```   337   fixes a::"'a::euclidean_space"
```
```   338   shows "~ content (cbox a b) < 0"
```
```   339 using not_le by blast
```
```   340
```
```   341 lemma content_pos_lt:
```
```   342   fixes a :: "'a::euclidean_space"
```
```   343   assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
```
```   344   shows "0 < content (cbox a b)"
```
```   345   using assms
```
```   346   by (auto simp: content_def box_eq_empty intro!: setprod_pos)
```
```   347
```
```   348 lemma content_eq_0:
```
```   349   "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
```
```   350   by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)
```
```   351
```
```   352 lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
```
```   353   by auto
```
```   354
```
```   355 lemma content_cbox_cases:
```
```   356   "content (cbox a (b::'a::euclidean_space)) =
```
```   357     (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
```
```   358   by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)
```
```   359
```
```   360 lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
```
```   361   unfolding content_eq_0 interior_cbox box_eq_empty
```
```   362   by auto
```
```   363
```
```   364 lemma content_pos_lt_eq:
```
```   365   "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
```
```   366 proof (rule iffI)
```
```   367   assume "0 < content (cbox a b)"
```
```   368   then have "content (cbox a b) \<noteq> 0" by auto
```
```   369   then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
```
```   370     unfolding content_eq_0 not_ex not_le by fastforce
```
```   371 next
```
```   372   assume "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i"
```
```   373   then show "0 < content (cbox a b)"
```
```   374     by (metis content_pos_lt)
```
```   375 qed
```
```   376
```
```   377 lemma content_empty [simp]: "content {} = 0"
```
```   378   unfolding content_def by auto
```
```   379
```
```   380 lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
```
```   381   by (simp add: content_real)
```
```   382
```
```   383 lemma content_subset:
```
```   384   assumes "cbox a b \<subseteq> cbox c d"
```
```   385   shows "content (cbox a b) \<le> content (cbox c d)"
```
```   386 proof (cases "cbox a b = {}")
```
```   387   case True
```
```   388   then show ?thesis
```
```   389     using content_pos_le[of c d] by auto
```
```   390 next
```
```   391   case False
```
```   392   then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```   393     unfolding box_ne_empty by auto
```
```   394   then have ab_ab: "a\<in>cbox a b" "b\<in>cbox a b"
```
```   395     unfolding mem_box by auto
```
```   396   have "cbox c d \<noteq> {}" using assms False by auto
```
```   397   then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
```
```   398     using assms unfolding box_ne_empty by auto
```
```   399   have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> b \<bullet> i - a \<bullet> i"
```
```   400     using ab_ne by auto
```
```   401   moreover
```
```   402   have "\<And>i. i \<in> Basis \<Longrightarrow> b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
```
```   403     using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
```
```   404           assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
```
```   405       by (metis diff_mono)
```
```   406   ultimately show ?thesis
```
```   407     unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
```
```   408     by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF \<open>cbox c d \<noteq> {}\<close>])
```
```   409 qed
```
```   410
```
```   411 lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
```
```   412   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
```
```   413
```
```   414 lemma content_times[simp]: "content (A \<times> B) = content A * content B"
```
```   415 proof (cases "A \<times> B = {}")
```
```   416   let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
```
```   417   let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
```
```   418   assume nonempty: "A \<times> B \<noteq> {}"
```
```   419   hence "content (A \<times> B) = (\<Prod>i\<in>Basis. (?ub1 A, ?ub2 B) \<bullet> i - (?lb1 A, ?lb2 B) \<bullet> i)"
```
```   420       unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
```
```   421   also have "... = content A * content B" unfolding content_def using nonempty
```
```   422     apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
```
```   423     apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
```
```   424     done
```
```   425   finally show ?thesis .
```
```   426 qed (auto simp: content_def)
```
```   427
```
```   428 lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
```
```   429   by (simp add: cbox_Pair_eq)
```
```   430
```
```   431 lemma content_cbox_pair_eq0_D:
```
```   432    "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
```
```   433   by (simp add: content_Pair)
```
```   434
```
```   435 lemma content_eq_0_gen:
```
```   436   fixes s :: "'a::euclidean_space set"
```
```   437   assumes "bounded s"
```
```   438   shows "content s = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. \<exists>v. \<forall>x \<in> s. x \<bullet> i = v)"  (is "_ = ?rhs")
```
```   439 proof safe
```
```   440   assume "content s = 0" then show ?rhs
```
```   441     apply (clarsimp simp: ex_in_conv content_def split: if_split_asm)
```
```   442     apply (rule_tac x=a in bexI)
```
```   443     apply (rule_tac x="interval_lowerbound s \<bullet> a" in exI)
```
```   444     apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
```
```   445     apply (drule cSUP_eq_cINF_D)
```
```   446     apply (auto simp: bounded_inner_imp_bdd_above [OF assms]  bounded_inner_imp_bdd_below [OF assms])
```
```   447     done
```
```   448 next
```
```   449   fix i a
```
```   450   assume "i \<in> Basis" "\<forall>x\<in>s. x \<bullet> i = a"
```
```   451   then show "content s = 0"
```
```   452     apply (clarsimp simp: content_def)
```
```   453     apply (rule_tac x=i in bexI)
```
```   454     apply (auto simp: interval_upperbound_def interval_lowerbound_def)
```
```   455     done
```
```   456 qed
```
```   457
```
```   458 lemma content_0_subset_gen:
```
```   459   fixes a :: "'a::euclidean_space"
```
```   460   assumes "content t = 0" "s \<subseteq> t" "bounded t" shows "content s = 0"
```
```   461 proof -
```
```   462   have "bounded s"
```
```   463     using assms by (metis bounded_subset)
```
```   464   then show ?thesis
```
```   465     using assms
```
```   466     by (auto simp: content_eq_0_gen)
```
```   467 qed
```
```   468
```
```   469 lemma content_0_subset: "\<lbrakk>content(cbox a b) = 0; s \<subseteq> cbox a b\<rbrakk> \<Longrightarrow> content s = 0"
```
```   470   by (simp add: content_0_subset_gen bounded_cbox)
```
```   471
```
```   472
```
```   473 lemma interval_split:
```
```   474   fixes a :: "'a::euclidean_space"
```
```   475   assumes "k \<in> Basis"
```
```   476   shows
```
```   477     "cbox a b \<inter> {x. x\<bullet>k \<le> c} = cbox a (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i) *\<^sub>R i)"
```
```   478     "cbox a b \<inter> {x. x\<bullet>k \<ge> c} = cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i) *\<^sub>R i) b"
```
```   479   apply (rule_tac[!] set_eqI)
```
```   480   unfolding Int_iff mem_box mem_Collect_eq
```
```   481   using assms
```
```   482   apply auto
```
```   483   done
```
```   484
```
```   485 lemma content_split:
```
```   486   fixes a :: "'a::euclidean_space"
```
```   487   assumes "k \<in> Basis"
```
```   488   shows "content (cbox a b) = content(cbox a b \<inter> {x. x\<bullet>k \<le> c}) + content(cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```   489 proof cases
```
```   490   note simps = interval_split[OF assms] content_cbox_cases
```
```   491   have *: "Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
```
```   492     using assms by auto
```
```   493   have *: "\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
```
```   494     "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
```
```   495     apply (subst *(1))
```
```   496     defer
```
```   497     apply (subst *(1))
```
```   498     unfolding setprod.insert[OF *(2-)]
```
```   499     apply auto
```
```   500     done
```
```   501   assume as: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```   502   moreover
```
```   503   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
```
```   504     x * (b\<bullet>k - a\<bullet>k) = x * (max (a \<bullet> k) c - a \<bullet> k) + x * (b \<bullet> k - max (a \<bullet> k) c)"
```
```   505     by  (auto simp add: field_simps)
```
```   506   moreover
```
```   507   have **: "(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
```
```   508       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
```
```   509     "(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
```
```   510       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
```
```   511     by (auto intro!: setprod.cong)
```
```   512   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
```
```   513     unfolding not_le
```
```   514     using as[unfolded ,rule_format,of k] assms
```
```   515     by auto
```
```   516   ultimately show ?thesis
```
```   517     using assms
```
```   518     unfolding simps **
```
```   519     unfolding *(1)[of "\<lambda>i x. b\<bullet>i - x"] *(1)[of "\<lambda>i x. x - a\<bullet>i"]
```
```   520     unfolding *(2)
```
```   521     by auto
```
```   522 next
```
```   523   assume "\<not> (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
```
```   524   then have "cbox a b = {}"
```
```   525     unfolding box_eq_empty by (auto simp: not_le)
```
```   526   then show ?thesis
```
```   527     by (auto simp: not_le)
```
```   528 qed
```
```   529
```
```   530 subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
```
```   531
```
```   532 definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
```
```   533
```
```   534 lemma gaugeI:
```
```   535   assumes "\<And>x. x \<in> g x"
```
```   536     and "\<And>x. open (g x)"
```
```   537   shows "gauge g"
```
```   538   using assms unfolding gauge_def by auto
```
```   539
```
```   540 lemma gaugeD[dest]:
```
```   541   assumes "gauge d"
```
```   542   shows "x \<in> d x"
```
```   543     and "open (d x)"
```
```   544   using assms unfolding gauge_def by auto
```
```   545
```
```   546 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
```
```   547   unfolding gauge_def by auto
```
```   548
```
```   549 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
```
```   550   unfolding gauge_def by auto
```
```   551
```
```   552 lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
```
```   553   by (rule gauge_ball) auto
```
```   554
```
```   555 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
```
```   556   unfolding gauge_def by auto
```
```   557
```
```   558 lemma gauge_inters:
```
```   559   assumes "finite s"
```
```   560     and "\<forall>d\<in>s. gauge (f d)"
```
```   561   shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
```
```   562 proof -
```
```   563   have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
```
```   564     by auto
```
```   565   show ?thesis
```
```   566     unfolding gauge_def unfolding *
```
```   567     using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
```
```   568 qed
```
```   569
```
```   570 lemma gauge_existence_lemma:
```
```   571   "(\<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)"
```
```   572   by (metis zero_less_one)
```
```   573
```
```   574
```
```   575 subsection \<open>Divisions.\<close>
```
```   576
```
```   577 definition division_of (infixl "division'_of" 40)
```
```   578 where
```
```   579   "s division_of i \<longleftrightarrow>
```
```   580     finite s \<and>
```
```   581     (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```   582     (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
```
```   583     (\<Union>s = i)"
```
```   584
```
```   585 lemma division_ofD[dest]:
```
```   586   assumes "s division_of i"
```
```   587   shows "finite s"
```
```   588     and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
```
```   589     and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
```
```   590     and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```   591     and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
```
```   592     and "\<Union>s = i"
```
```   593   using assms unfolding division_of_def by auto
```
```   594
```
```   595 lemma division_ofI:
```
```   596   assumes "finite s"
```
```   597     and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
```
```   598     and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
```
```   599     and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```   600     and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```   601     and "\<Union>s = i"
```
```   602   shows "s division_of i"
```
```   603   using assms unfolding division_of_def by auto
```
```   604
```
```   605 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
```
```   606   unfolding division_of_def by auto
```
```   607
```
```   608 lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
```
```   609   unfolding division_of_def by auto
```
```   610
```
```   611 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
```
```   612   unfolding division_of_def by auto
```
```   613
```
```   614 lemma division_of_sing[simp]:
```
```   615   "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
```
```   616   (is "?l = ?r")
```
```   617 proof
```
```   618   assume ?r
```
```   619   moreover
```
```   620   { fix k
```
```   621     assume "s = {{a}}" "k\<in>s"
```
```   622     then have "\<exists>x y. k = cbox x y"
```
```   623       apply (rule_tac x=a in exI)+
```
```   624       apply (force simp: cbox_sing)
```
```   625       done
```
```   626   }
```
```   627   ultimately show ?l
```
```   628     unfolding division_of_def cbox_sing by auto
```
```   629 next
```
```   630   assume ?l
```
```   631   note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
```
```   632   {
```
```   633     fix x
```
```   634     assume x: "x \<in> s" have "x = {a}"
```
```   635       using *(2)[rule_format,OF x] by auto
```
```   636   }
```
```   637   moreover have "s \<noteq> {}"
```
```   638     using *(4) by auto
```
```   639   ultimately show ?r
```
```   640     unfolding cbox_sing by auto
```
```   641 qed
```
```   642
```
```   643 lemma elementary_empty: obtains p where "p division_of {}"
```
```   644   unfolding division_of_trivial by auto
```
```   645
```
```   646 lemma elementary_interval: obtains p where "p division_of (cbox a b)"
```
```   647   by (metis division_of_trivial division_of_self)
```
```   648
```
```   649 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
```
```   650   unfolding division_of_def by auto
```
```   651
```
```   652 lemma forall_in_division:
```
```   653   "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))"
```
```   654   unfolding division_of_def by fastforce
```
```   655
```
```   656 lemma division_of_subset:
```
```   657   assumes "p division_of (\<Union>p)"
```
```   658     and "q \<subseteq> p"
```
```   659   shows "q division_of (\<Union>q)"
```
```   660 proof (rule division_ofI)
```
```   661   note * = division_ofD[OF assms(1)]
```
```   662   show "finite q"
```
```   663     using "*"(1) assms(2) infinite_super by auto
```
```   664   {
```
```   665     fix k
```
```   666     assume "k \<in> q"
```
```   667     then have kp: "k \<in> p"
```
```   668       using assms(2) by auto
```
```   669     show "k \<subseteq> \<Union>q"
```
```   670       using \<open>k \<in> q\<close> by auto
```
```   671     show "\<exists>a b. k = cbox a b"
```
```   672       using *(4)[OF kp] by auto
```
```   673     show "k \<noteq> {}"
```
```   674       using *(3)[OF kp] by auto
```
```   675   }
```
```   676   fix k1 k2
```
```   677   assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
```
```   678   then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
```
```   679     using assms(2) by auto
```
```   680   show "interior k1 \<inter> interior k2 = {}"
```
```   681     using *(5)[OF **] by auto
```
```   682 qed auto
```
```   683
```
```   684 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
```
```   685   unfolding division_of_def by auto
```
```   686
```
```   687 lemma division_of_content_0:
```
```   688   assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
```
```   689   shows "\<forall>k\<in>d. content k = 0"
```
```   690   unfolding forall_in_division[OF assms(2)]
```
```   691   by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
```
```   692
```
```   693 lemma division_inter:
```
```   694   fixes s1 s2 :: "'a::euclidean_space set"
```
```   695   assumes "p1 division_of s1"
```
```   696     and "p2 division_of s2"
```
```   697   shows "{k1 \<inter> k2 | k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
```
```   698   (is "?A' division_of _")
```
```   699 proof -
```
```   700   let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
```
```   701   have *: "?A' = ?A" by auto
```
```   702   show ?thesis
```
```   703     unfolding *
```
```   704   proof (rule division_ofI)
```
```   705     have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
```
```   706       by auto
```
```   707     moreover have "finite (p1 \<times> p2)"
```
```   708       using assms unfolding division_of_def by auto
```
```   709     ultimately show "finite ?A" by auto
```
```   710     have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
```
```   711       by auto
```
```   712     show "\<Union>?A = s1 \<inter> s2"
```
```   713       apply (rule set_eqI)
```
```   714       unfolding * and UN_iff
```
```   715       using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
```
```   716       apply auto
```
```   717       done
```
```   718     {
```
```   719       fix k
```
```   720       assume "k \<in> ?A"
```
```   721       then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
```
```   722         by auto
```
```   723       then show "k \<noteq> {}"
```
```   724         by auto
```
```   725       show "k \<subseteq> s1 \<inter> s2"
```
```   726         using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
```
```   727         unfolding k by auto
```
```   728       obtain a1 b1 where k1: "k1 = cbox a1 b1"
```
```   729         using division_ofD(4)[OF assms(1) k(2)] by blast
```
```   730       obtain a2 b2 where k2: "k2 = cbox a2 b2"
```
```   731         using division_ofD(4)[OF assms(2) k(3)] by blast
```
```   732       show "\<exists>a b. k = cbox a b"
```
```   733         unfolding k k1 k2 unfolding inter_interval by auto
```
```   734     }
```
```   735     fix k1 k2
```
```   736     assume "k1 \<in> ?A"
```
```   737     then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
```
```   738       by auto
```
```   739     assume "k2 \<in> ?A"
```
```   740     then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
```
```   741       by auto
```
```   742     assume "k1 \<noteq> k2"
```
```   743     then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
```
```   744       unfolding k1 k2 by auto
```
```   745     have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
```
```   746       interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
```
```   747       interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
```
```   748       interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
```
```   749     show "interior k1 \<inter> interior k2 = {}"
```
```   750       unfolding k1 k2
```
```   751       apply (rule *)
```
```   752       using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
```
```   753       done
```
```   754   qed
```
```   755 qed
```
```   756
```
```   757 lemma division_inter_1:
```
```   758   assumes "d division_of i"
```
```   759     and "cbox a (b::'a::euclidean_space) \<subseteq> i"
```
```   760   shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
```
```   761 proof (cases "cbox a b = {}")
```
```   762   case True
```
```   763   show ?thesis
```
```   764     unfolding True and division_of_trivial by auto
```
```   765 next
```
```   766   case False
```
```   767   have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
```
```   768   show ?thesis
```
```   769     using division_inter[OF division_of_self[OF False] assms(1)]
```
```   770     unfolding * by auto
```
```   771 qed
```
```   772
```
```   773 lemma elementary_inter:
```
```   774   fixes s t :: "'a::euclidean_space set"
```
```   775   assumes "p1 division_of s"
```
```   776     and "p2 division_of t"
```
```   777   shows "\<exists>p. p division_of (s \<inter> t)"
```
```   778 using assms division_inter by blast
```
```   779
```
```   780 lemma elementary_inters:
```
```   781   assumes "finite f"
```
```   782     and "f \<noteq> {}"
```
```   783     and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
```
```   784   shows "\<exists>p. p division_of (\<Inter>f)"
```
```   785   using assms
```
```   786 proof (induct f rule: finite_induct)
```
```   787   case (insert x f)
```
```   788   show ?case
```
```   789   proof (cases "f = {}")
```
```   790     case True
```
```   791     then show ?thesis
```
```   792       unfolding True using insert by auto
```
```   793   next
```
```   794     case False
```
```   795     obtain p where "p division_of \<Inter>f"
```
```   796       using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
```
```   797     moreover obtain px where "px division_of x"
```
```   798       using insert(5)[rule_format,OF insertI1] ..
```
```   799     ultimately show ?thesis
```
```   800       by (simp add: elementary_inter Inter_insert)
```
```   801   qed
```
```   802 qed auto
```
```   803
```
```   804 lemma division_disjoint_union:
```
```   805   assumes "p1 division_of s1"
```
```   806     and "p2 division_of s2"
```
```   807     and "interior s1 \<inter> interior s2 = {}"
```
```   808   shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
```
```   809 proof (rule division_ofI)
```
```   810   note d1 = division_ofD[OF assms(1)]
```
```   811   note d2 = division_ofD[OF assms(2)]
```
```   812   show "finite (p1 \<union> p2)"
```
```   813     using d1(1) d2(1) by auto
```
```   814   show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
```
```   815     using d1(6) d2(6) by auto
```
```   816   {
```
```   817     fix k1 k2
```
```   818     assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
```
```   819     moreover
```
```   820     let ?g="interior k1 \<inter> interior k2 = {}"
```
```   821     {
```
```   822       assume as: "k1\<in>p1" "k2\<in>p2"
```
```   823       have ?g
```
```   824         using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
```
```   825         using assms(3) by blast
```
```   826     }
```
```   827     moreover
```
```   828     {
```
```   829       assume as: "k1\<in>p2" "k2\<in>p1"
```
```   830       have ?g
```
```   831         using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
```
```   832         using assms(3) by blast
```
```   833     }
```
```   834     ultimately show ?g
```
```   835       using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
```
```   836   }
```
```   837   fix k
```
```   838   assume k: "k \<in> p1 \<union> p2"
```
```   839   show "k \<subseteq> s1 \<union> s2"
```
```   840     using k d1(2) d2(2) by auto
```
```   841   show "k \<noteq> {}"
```
```   842     using k d1(3) d2(3) by auto
```
```   843   show "\<exists>a b. k = cbox a b"
```
```   844     using k d1(4) d2(4) by auto
```
```   845 qed
```
```   846
```
```   847 lemma partial_division_extend_1:
```
```   848   fixes a b c d :: "'a::euclidean_space"
```
```   849   assumes incl: "cbox c d \<subseteq> cbox a b"
```
```   850     and nonempty: "cbox c d \<noteq> {}"
```
```   851   obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
```
```   852 proof
```
```   853   let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
```
```   854     cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)"
```
```   855   define p where "p = ?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
```
```   856
```
```   857   show "cbox c d \<in> p"
```
```   858     unfolding p_def
```
```   859     by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
```
```   860   {
```
```   861     fix i :: 'a
```
```   862     assume "i \<in> Basis"
```
```   863     with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
```
```   864       unfolding box_eq_empty subset_box by (auto simp: not_le)
```
```   865   }
```
```   866   note ord = this
```
```   867
```
```   868   show "p division_of (cbox a b)"
```
```   869   proof (rule division_ofI)
```
```   870     show "finite p"
```
```   871       unfolding p_def by (auto intro!: finite_PiE)
```
```   872     {
```
```   873       fix k
```
```   874       assume "k \<in> p"
```
```   875       then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
```
```   876         by (auto simp: p_def)
```
```   877       then show "\<exists>a b. k = cbox a b"
```
```   878         by auto
```
```   879       have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
```
```   880       proof (simp add: k box_eq_empty subset_box not_less, safe)
```
```   881         fix i :: 'a
```
```   882         assume i: "i \<in> Basis"
```
```   883         with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
```
```   884           by (auto simp: PiE_iff)
```
```   885         with i ord[of i]
```
```   886         show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
```
```   887           by auto
```
```   888       qed
```
```   889       then show "k \<noteq> {}" "k \<subseteq> cbox a b"
```
```   890         by auto
```
```   891       {
```
```   892         fix l
```
```   893         assume "l \<in> p"
```
```   894         then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
```
```   895           by (auto simp: p_def)
```
```   896         assume "l \<noteq> k"
```
```   897         have "\<exists>i\<in>Basis. f i \<noteq> g i"
```
```   898         proof (rule ccontr)
```
```   899           assume "\<not> ?thesis"
```
```   900           with f g have "f = g"
```
```   901             by (auto simp: PiE_iff extensional_def intro!: ext)
```
```   902           with \<open>l \<noteq> k\<close> show False
```
```   903             by (simp add: l k)
```
```   904         qed
```
```   905         then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
```
```   906         then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
```
```   907                   "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
```
```   908           using f g by (auto simp: PiE_iff)
```
```   909         with * ord[of i] show "interior l \<inter> interior k = {}"
```
```   910           by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
```
```   911       }
```
```   912       note \<open>k \<subseteq> cbox a b\<close>
```
```   913     }
```
```   914     moreover
```
```   915     {
```
```   916       fix x assume x: "x \<in> cbox a b"
```
```   917       have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
```
```   918       proof
```
```   919         fix i :: 'a
```
```   920         assume "i \<in> Basis"
```
```   921         with x ord[of i]
```
```   922         have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
```
```   923             (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
```
```   924           by (auto simp: cbox_def)
```
```   925         then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
```
```   926           by auto
```
```   927       qed
```
```   928       then obtain f where
```
```   929         f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}"
```
```   930         unfolding bchoice_iff ..
```
```   931       moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
```
```   932         by auto
```
```   933       moreover from f have "x \<in> ?B (restrict f Basis)"
```
```   934         by (auto simp: mem_box)
```
```   935       ultimately have "\<exists>k\<in>p. x \<in> k"
```
```   936         unfolding p_def by blast
```
```   937     }
```
```   938     ultimately show "\<Union>p = cbox a b"
```
```   939       by auto
```
```   940   qed
```
```   941 qed
```
```   942
```
```   943 lemma partial_division_extend_interval:
```
```   944   assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
```
```   945   obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
```
```   946 proof (cases "p = {}")
```
```   947   case True
```
```   948   obtain q where "q division_of (cbox a b)"
```
```   949     by (rule elementary_interval)
```
```   950   then show ?thesis
```
```   951     using True that by blast
```
```   952 next
```
```   953   case False
```
```   954   note p = division_ofD[OF assms(1)]
```
```   955   have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
```
```   956   proof
```
```   957     fix k
```
```   958     assume kp: "k \<in> p"
```
```   959     obtain c d where k: "k = cbox c d"
```
```   960       using p(4)[OF kp] by blast
```
```   961     have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
```
```   962       using p(2,3)[OF kp, unfolded k] using assms(2)
```
```   963       by (blast intro: order.trans)+
```
```   964     obtain q where "q division_of cbox a b" "cbox c d \<in> q"
```
```   965       by (rule partial_division_extend_1[OF *])
```
```   966     then show "\<exists>q. q division_of cbox a b \<and> k \<in> q"
```
```   967       unfolding k by auto
```
```   968   qed
```
```   969   obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of cbox a b" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
```
```   970     using bchoice[OF div_cbox] by blast
```
```   971   { fix x
```
```   972     assume x: "x \<in> p"
```
```   973     have "q x division_of \<Union>q x"
```
```   974       apply (rule division_ofI)
```
```   975       using division_ofD[OF q(1)[OF x]]
```
```   976       apply auto
```
```   977       done }
```
```   978   then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
```
```   979     by (meson Diff_subset division_of_subset)
```
```   980   then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
```
```   981     apply -
```
```   982     apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
```
```   983     apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
```
```   984     done
```
```   985   then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
```
```   986   have "d \<union> p division_of cbox a b"
```
```   987   proof -
```
```   988     have te: "\<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
```
```   989     have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
```
```   990     proof (rule te[OF False], clarify)
```
```   991       fix i
```
```   992       assume i: "i \<in> p"
```
```   993       show "\<Union>(q i - {i}) \<union> i = cbox a b"
```
```   994         using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
```
```   995     qed
```
```   996     { fix k
```
```   997       assume k: "k \<in> p"
```
```   998       have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}"
```
```   999         by auto
```
```  1000       have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}"
```
```  1001       proof (rule *[OF inter_interior_unions_intervals])
```
```  1002         note qk=division_ofD[OF q(1)[OF k]]
```
```  1003         show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
```
```  1004           using qk by auto
```
```  1005         show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
```
```  1006           using qk(5) using q(2)[OF k] by auto
```
```  1007         show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))"
```
```  1008           apply (rule interior_mono)+
```
```  1009           using k
```
```  1010           apply auto
```
```  1011           done
```
```  1012       qed } note [simp] = this
```
```  1013     show "d \<union> p division_of (cbox a b)"
```
```  1014       unfolding cbox_eq
```
```  1015       apply (rule division_disjoint_union[OF d assms(1)])
```
```  1016       apply (rule inter_interior_unions_intervals)
```
```  1017       apply (rule p open_interior ballI)+
```
```  1018       apply simp_all
```
```  1019       done
```
```  1020   qed
```
```  1021   then show ?thesis
```
```  1022     by (meson Un_upper2 that)
```
```  1023 qed
```
```  1024
```
```  1025 lemma elementary_bounded[dest]:
```
```  1026   fixes s :: "'a::euclidean_space set"
```
```  1027   shows "p division_of s \<Longrightarrow> bounded s"
```
```  1028   unfolding division_of_def by (metis bounded_Union bounded_cbox)
```
```  1029
```
```  1030 lemma elementary_subset_cbox:
```
```  1031   "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
```
```  1032   by (meson elementary_bounded bounded_subset_cbox)
```
```  1033
```
```  1034 lemma division_union_intervals_exists:
```
```  1035   fixes a b :: "'a::euclidean_space"
```
```  1036   assumes "cbox a b \<noteq> {}"
```
```  1037   obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
```
```  1038 proof (cases "cbox c d = {}")
```
```  1039   case True
```
```  1040   show ?thesis
```
```  1041     apply (rule that[of "{}"])
```
```  1042     unfolding True
```
```  1043     using assms
```
```  1044     apply auto
```
```  1045     done
```
```  1046 next
```
```  1047   case False
```
```  1048   show ?thesis
```
```  1049   proof (cases "cbox a b \<inter> cbox c d = {}")
```
```  1050     case True
```
```  1051     then show ?thesis
```
```  1052       by (metis that False assms division_disjoint_union division_of_self insert_is_Un interior_Int interior_empty)
```
```  1053   next
```
```  1054     case False
```
```  1055     obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
```
```  1056       unfolding inter_interval by auto
```
```  1057     have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
```
```  1058     obtain p where "p division_of cbox c d" "cbox u v \<in> p"
```
```  1059       by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
```
```  1060     note p = this division_ofD[OF this(1)]
```
```  1061     have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
```
```  1062       apply (rule arg_cong[of _ _ interior])
```
```  1063       using p(8) uv by auto
```
```  1064     also have "\<dots> = {}"
```
```  1065       unfolding interior_Int
```
```  1066       apply (rule inter_interior_unions_intervals)
```
```  1067       using p(6) p(7)[OF p(2)] p(3)
```
```  1068       apply auto
```
```  1069       done
```
```  1070     finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
```
```  1071     have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
```
```  1072       using p(8) unfolding uv[symmetric] by auto
```
```  1073     have "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
```
```  1074     proof -
```
```  1075       have "{cbox a b} division_of cbox a b"
```
```  1076         by (simp add: assms division_of_self)
```
```  1077       then show "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
```
```  1078         by (metis (no_types) Diff_subset \<open>interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}\<close> division_disjoint_union division_of_subset insert_is_Un p(1) p(8))
```
```  1079     qed
```
```  1080     with that[of "p - {cbox u v}"] show ?thesis by (simp add: cbe)
```
```  1081   qed
```
```  1082 qed
```
```  1083
```
```  1084 lemma division_of_unions:
```
```  1085   assumes "finite f"
```
```  1086     and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
```
```  1087     and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```  1088   shows "\<Union>f division_of \<Union>\<Union>f"
```
```  1089   using assms
```
```  1090   by (auto intro!: division_ofI)
```
```  1091
```
```  1092 lemma elementary_union_interval:
```
```  1093   fixes a b :: "'a::euclidean_space"
```
```  1094   assumes "p division_of \<Union>p"
```
```  1095   obtains q where "q division_of (cbox a b \<union> \<Union>p)"
```
```  1096 proof -
```
```  1097   note assm = division_ofD[OF assms]
```
```  1098   have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
```
```  1099     by auto
```
```  1100   have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
```
```  1101     by auto
```
```  1102   {
```
```  1103     presume "p = {} \<Longrightarrow> thesis"
```
```  1104       "cbox a b = {} \<Longrightarrow> thesis"
```
```  1105       "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
```
```  1106       "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
```
```  1107     then show thesis by auto
```
```  1108   next
```
```  1109     assume as: "p = {}"
```
```  1110     obtain p where "p division_of (cbox a b)"
```
```  1111       by (rule elementary_interval)
```
```  1112     then show thesis
```
```  1113       using as that by auto
```
```  1114   next
```
```  1115     assume as: "cbox a b = {}"
```
```  1116     show thesis
```
```  1117       using as assms that by auto
```
```  1118   next
```
```  1119     assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
```
```  1120     show thesis
```
```  1121       apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
```
```  1122       unfolding finite_insert
```
```  1123       apply (rule assm(1)) unfolding Union_insert
```
```  1124       using assm(2-4) as
```
```  1125       apply -
```
```  1126       apply (fast dest: assm(5))+
```
```  1127       done
```
```  1128   next
```
```  1129     assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
```
```  1130     have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
```
```  1131     proof
```
```  1132       fix k
```
```  1133       assume kp: "k \<in> p"
```
```  1134       from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast
```
```  1135       then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
```
```  1136         by (meson as(3) division_union_intervals_exists)
```
```  1137     qed
```
```  1138     from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" ..
```
```  1139     note q = division_ofD[OF this[rule_format]]
```
```  1140     let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
```
```  1141     show thesis
```
```  1142     proof (rule that[OF division_ofI])
```
```  1143       have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
```
```  1144         by auto
```
```  1145       show "finite ?D"
```
```  1146         using "*" assm(1) q(1) by auto
```
```  1147       show "\<Union>?D = cbox a b \<union> \<Union>p"
```
```  1148         unfolding * lem1
```
```  1149         unfolding lem2[OF as(1), of "cbox a b", symmetric]
```
```  1150         using q(6)
```
```  1151         by auto
```
```  1152       fix k
```
```  1153       assume k: "k \<in> ?D"
```
```  1154       then show "k \<subseteq> cbox a b \<union> \<Union>p"
```
```  1155         using q(2) by auto
```
```  1156       show "k \<noteq> {}"
```
```  1157         using q(3) k by auto
```
```  1158       show "\<exists>a b. k = cbox a b"
```
```  1159         using q(4) k by auto
```
```  1160       fix k'
```
```  1161       assume k': "k' \<in> ?D" "k \<noteq> k'"
```
```  1162       obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
```
```  1163         using k by auto
```
```  1164       obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
```
```  1165         using k' by auto
```
```  1166       show "interior k \<inter> interior k' = {}"
```
```  1167       proof (cases "x = x'")
```
```  1168         case True
```
```  1169         show ?thesis
```
```  1170           using True k' q(5) x' x by auto
```
```  1171       next
```
```  1172         case False
```
```  1173         {
```
```  1174           presume "k = cbox a b \<Longrightarrow> ?thesis"
```
```  1175             and "k' = cbox a b \<Longrightarrow> ?thesis"
```
```  1176             and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
```
```  1177           then show ?thesis by linarith
```
```  1178         next
```
```  1179           assume as': "k  = cbox a b"
```
```  1180           show ?thesis
```
```  1181             using as' k' q(5) x' by blast
```
```  1182         next
```
```  1183           assume as': "k' = cbox a b"
```
```  1184           show ?thesis
```
```  1185             using as' k'(2) q(5) x by blast
```
```  1186         }
```
```  1187         assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
```
```  1188         obtain c d where k: "k = cbox c d"
```
```  1189           using q(4)[OF x(2,1)] by blast
```
```  1190         have "interior k \<inter> interior (cbox a b) = {}"
```
```  1191           using as' k'(2) q(5) x by blast
```
```  1192         then have "interior k \<subseteq> interior x"
```
```  1193         using interior_subset_union_intervals
```
```  1194           by (metis as(2) k q(2) x interior_subset_union_intervals)
```
```  1195         moreover
```
```  1196         obtain c d where c_d: "k' = cbox c d"
```
```  1197           using q(4)[OF x'(2,1)] by blast
```
```  1198         have "interior k' \<inter> interior (cbox a b) = {}"
```
```  1199           using as'(2) q(5) x' by blast
```
```  1200         then have "interior k' \<subseteq> interior x'"
```
```  1201           by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
```
```  1202         ultimately show ?thesis
```
```  1203           using assm(5)[OF x(2) x'(2) False] by auto
```
```  1204       qed
```
```  1205     qed
```
```  1206   }
```
```  1207 qed
```
```  1208
```
```  1209 lemma elementary_unions_intervals:
```
```  1210   assumes fin: "finite f"
```
```  1211     and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
```
```  1212   obtains p where "p division_of (\<Union>f)"
```
```  1213 proof -
```
```  1214   have "\<exists>p. p division_of (\<Union>f)"
```
```  1215   proof (induct_tac f rule:finite_subset_induct)
```
```  1216     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
```
```  1217   next
```
```  1218     fix x F
```
```  1219     assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
```
```  1220     from this(3) obtain p where p: "p division_of \<Union>F" ..
```
```  1221     from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
```
```  1222     have *: "\<Union>F = \<Union>p"
```
```  1223       using division_ofD[OF p] by auto
```
```  1224     show "\<exists>p. p division_of \<Union>insert x F"
```
```  1225       using elementary_union_interval[OF p[unfolded *], of a b]
```
```  1226       unfolding Union_insert x * by metis
```
```  1227   qed (insert assms, auto)
```
```  1228   then show ?thesis
```
```  1229     using that by auto
```
```  1230 qed
```
```  1231
```
```  1232 lemma elementary_union:
```
```  1233   fixes s t :: "'a::euclidean_space set"
```
```  1234   assumes "ps division_of s" "pt division_of t"
```
```  1235   obtains p where "p division_of (s \<union> t)"
```
```  1236 proof -
```
```  1237   have *: "s \<union> t = \<Union>ps \<union> \<Union>pt"
```
```  1238     using assms unfolding division_of_def by auto
```
```  1239   show ?thesis
```
```  1240     apply (rule elementary_unions_intervals[of "ps \<union> pt"])
```
```  1241     using assms apply auto
```
```  1242     by (simp add: * that)
```
```  1243 qed
```
```  1244
```
```  1245 lemma partial_division_extend:
```
```  1246   fixes t :: "'a::euclidean_space set"
```
```  1247   assumes "p division_of s"
```
```  1248     and "q division_of t"
```
```  1249     and "s \<subseteq> t"
```
```  1250   obtains r where "p \<subseteq> r" and "r division_of t"
```
```  1251 proof -
```
```  1252   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
```
```  1253   obtain a b where ab: "t \<subseteq> cbox a b"
```
```  1254     using elementary_subset_cbox[OF assms(2)] by auto
```
```  1255   obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
```
```  1256     using assms
```
```  1257     by (metis ab dual_order.trans partial_division_extend_interval divp(6))
```
```  1258   note r1 = this division_ofD[OF this(2)]
```
```  1259   obtain p' where "p' division_of \<Union>(r1 - p)"
```
```  1260     apply (rule elementary_unions_intervals[of "r1 - p"])
```
```  1261     using r1(3,6)
```
```  1262     apply auto
```
```  1263     done
```
```  1264   then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
```
```  1265     by (metis assms(2) divq(6) elementary_inter)
```
```  1266   {
```
```  1267     fix x
```
```  1268     assume x: "x \<in> t" "x \<notin> s"
```
```  1269     then have "x\<in>\<Union>r1"
```
```  1270       unfolding r1 using ab by auto
```
```  1271     then obtain r where r: "r \<in> r1" "x \<in> r"
```
```  1272       unfolding Union_iff ..
```
```  1273     moreover
```
```  1274     have "r \<notin> p"
```
```  1275     proof
```
```  1276       assume "r \<in> p"
```
```  1277       then have "x \<in> s" using divp(2) r by auto
```
```  1278       then show False using x by auto
```
```  1279     qed
```
```  1280     ultimately have "x\<in>\<Union>(r1 - p)" by auto
```
```  1281   }
```
```  1282   then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
```
```  1283     unfolding divp divq using assms(3) by auto
```
```  1284   show ?thesis
```
```  1285     apply (rule that[of "p \<union> r2"])
```
```  1286     unfolding *
```
```  1287     defer
```
```  1288     apply (rule division_disjoint_union)
```
```  1289     unfolding divp(6)
```
```  1290     apply(rule assms r2)+
```
```  1291   proof -
```
```  1292     have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
```
```  1293     proof (rule inter_interior_unions_intervals)
```
```  1294       show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
```
```  1295         using r1 by auto
```
```  1296       have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
```
```  1297         by auto
```
```  1298       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
```
```  1299       proof
```
```  1300         fix m x
```
```  1301         assume as: "m \<in> r1 - p"
```
```  1302         have "interior m \<inter> interior (\<Union>p) = {}"
```
```  1303         proof (rule inter_interior_unions_intervals)
```
```  1304           show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
```
```  1305             using divp by auto
```
```  1306           show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
```
```  1307             by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
```
```  1308         qed
```
```  1309         then show "interior s \<inter> interior m = {}"
```
```  1310           unfolding divp by auto
```
```  1311       qed
```
```  1312     qed
```
```  1313     then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
```
```  1314       using interior_subset by auto
```
```  1315   qed auto
```
```  1316 qed
```
```  1317
```
```  1318 lemma division_split_left_inj:
```
```  1319   fixes type :: "'a::euclidean_space"
```
```  1320   assumes "d division_of i"
```
```  1321     and "k1 \<in> d"
```
```  1322     and "k2 \<in> d"
```
```  1323     and "k1 \<noteq> k2"
```
```  1324     and "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
```
```  1325     and k: "k\<in>Basis"
```
```  1326   shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
```
```  1327 proof -
```
```  1328   note d=division_ofD[OF assms(1)]
```
```  1329   have *: "\<And>(a::'a) b c. content (cbox a b \<inter> {x. x\<bullet>k \<le> c}) = 0 \<longleftrightarrow>
```
```  1330     interior(cbox a b \<inter> {x. x\<bullet>k \<le> c}) = {}"
```
```  1331     unfolding  interval_split[OF k] content_eq_0_interior by auto
```
```  1332   guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
```
```  1333   guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
```
```  1334   have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
```
```  1335     by auto
```
```  1336   show ?thesis
```
```  1337     unfolding uv1 uv2 *
```
```  1338     apply (rule **[OF d(5)[OF assms(2-4)]])
```
```  1339     apply (simp add: uv1)
```
```  1340     using assms(5) uv1 by auto
```
```  1341 qed
```
```  1342
```
```  1343 lemma division_split_right_inj:
```
```  1344   fixes type :: "'a::euclidean_space"
```
```  1345   assumes "d division_of i"
```
```  1346     and "k1 \<in> d"
```
```  1347     and "k2 \<in> d"
```
```  1348     and "k1 \<noteq> k2"
```
```  1349     and "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
```
```  1350     and k: "k \<in> Basis"
```
```  1351   shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
```
```  1352 proof -
```
```  1353   note d=division_ofD[OF assms(1)]
```
```  1354   have *: "\<And>a b::'a. \<And>c. content(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = 0 \<longleftrightarrow>
```
```  1355     interior(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = {}"
```
```  1356     unfolding interval_split[OF k] content_eq_0_interior by auto
```
```  1357   guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
```
```  1358   guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
```
```  1359   have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
```
```  1360     by auto
```
```  1361   show ?thesis
```
```  1362     unfolding uv1 uv2 *
```
```  1363     apply (rule **[OF d(5)[OF assms(2-4)]])
```
```  1364     apply (simp add: uv1)
```
```  1365     using assms(5) uv1 by auto
```
```  1366 qed
```
```  1367
```
```  1368
```
```  1369 lemma division_split:
```
```  1370   fixes a :: "'a::euclidean_space"
```
```  1371   assumes "p division_of (cbox a b)"
```
```  1372     and k: "k\<in>Basis"
```
```  1373   shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} division_of(cbox a b \<inter> {x. x\<bullet>k \<le> c})"
```
```  1374       (is "?p1 division_of ?I1")
```
```  1375     and "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  1376       (is "?p2 division_of ?I2")
```
```  1377 proof (rule_tac[!] division_ofI)
```
```  1378   note p = division_ofD[OF assms(1)]
```
```  1379   show "finite ?p1" "finite ?p2"
```
```  1380     using p(1) by auto
```
```  1381   show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2"
```
```  1382     unfolding p(6)[symmetric] by auto
```
```  1383   {
```
```  1384     fix k
```
```  1385     assume "k \<in> ?p1"
```
```  1386     then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
```
```  1387     guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
```
```  1388     show "k \<subseteq> ?I1"
```
```  1389       using l p(2) uv by force
```
```  1390     show  "k \<noteq> {}"
```
```  1391       by (simp add: l)
```
```  1392     show  "\<exists>a b. k = cbox a b"
```
```  1393       apply (simp add: l uv p(2-3)[OF l(2)])
```
```  1394       apply (subst interval_split[OF k])
```
```  1395       apply (auto intro: order.trans)
```
```  1396       done
```
```  1397     fix k'
```
```  1398     assume "k' \<in> ?p1"
```
```  1399     then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
```
```  1400     assume "k \<noteq> k'"
```
```  1401     then show "interior k \<inter> interior k' = {}"
```
```  1402       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
```
```  1403   }
```
```  1404   {
```
```  1405     fix k
```
```  1406     assume "k \<in> ?p2"
```
```  1407     then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
```
```  1408     guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
```
```  1409     show "k \<subseteq> ?I2"
```
```  1410       using l p(2) uv by force
```
```  1411     show  "k \<noteq> {}"
```
```  1412       by (simp add: l)
```
```  1413     show  "\<exists>a b. k = cbox a b"
```
```  1414       apply (simp add: l uv p(2-3)[OF l(2)])
```
```  1415       apply (subst interval_split[OF k])
```
```  1416       apply (auto intro: order.trans)
```
```  1417       done
```
```  1418     fix k'
```
```  1419     assume "k' \<in> ?p2"
```
```  1420     then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
```
```  1421     assume "k \<noteq> k'"
```
```  1422     then show "interior k \<inter> interior k' = {}"
```
```  1423       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
```
```  1424   }
```
```  1425 qed
```
```  1426
```
```  1427 subsection \<open>Tagged (partial) divisions.\<close>
```
```  1428
```
```  1429 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
```
```  1430   where "s tagged_partial_division_of i \<longleftrightarrow>
```
```  1431     finite s \<and>
```
```  1432     (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```  1433     (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
```
```  1434       interior k1 \<inter> interior k2 = {})"
```
```  1435
```
```  1436 lemma tagged_partial_division_ofD[dest]:
```
```  1437   assumes "s tagged_partial_division_of i"
```
```  1438   shows "finite s"
```
```  1439     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1440     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1441     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1442     and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
```
```  1443       (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```  1444   using assms unfolding tagged_partial_division_of_def by blast+
```
```  1445
```
```  1446 definition tagged_division_of (infixr "tagged'_division'_of" 40)
```
```  1447   where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1448
```
```  1449 lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
```
```  1450   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```  1451
```
```  1452 lemma tagged_division_of:
```
```  1453   "s tagged_division_of i \<longleftrightarrow>
```
```  1454     finite s \<and>
```
```  1455     (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```  1456     (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
```
```  1457       interior k1 \<inter> interior k2 = {}) \<and>
```
```  1458     (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1459   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```  1460
```
```  1461 lemma tagged_division_ofI:
```
```  1462   assumes "finite s"
```
```  1463     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1464     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1465     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1466     and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
```
```  1467       interior k1 \<inter> interior k2 = {}"
```
```  1468     and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1469   shows "s tagged_division_of i"
```
```  1470   unfolding tagged_division_of
```
```  1471   using assms
```
```  1472   apply auto
```
```  1473   apply fastforce+
```
```  1474   done
```
```  1475
```
```  1476 lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
```
```  1477   assumes "s tagged_division_of i"
```
```  1478   shows "finite s"
```
```  1479     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1480     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1481     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1482     and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
```
```  1483       interior k1 \<inter> interior k2 = {}"
```
```  1484     and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1485   using assms unfolding tagged_division_of by blast+
```
```  1486
```
```  1487 lemma division_of_tagged_division:
```
```  1488   assumes "s tagged_division_of i"
```
```  1489   shows "(snd ` s) division_of i"
```
```  1490 proof (rule division_ofI)
```
```  1491   note assm = tagged_division_ofD[OF assms]
```
```  1492   show "\<Union>(snd ` s) = i" "finite (snd ` s)"
```
```  1493     using assm by auto
```
```  1494   fix k
```
```  1495   assume k: "k \<in> snd ` s"
```
```  1496   then obtain xk where xk: "(xk, k) \<in> s"
```
```  1497     by auto
```
```  1498   then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
```
```  1499     using assm by fastforce+
```
```  1500   fix k'
```
```  1501   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
```
```  1502   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
```
```  1503     by auto
```
```  1504   then show "interior k \<inter> interior k' = {}"
```
```  1505     using assm(5) k'(2) xk by blast
```
```  1506 qed
```
```  1507
```
```  1508 lemma partial_division_of_tagged_division:
```
```  1509   assumes "s tagged_partial_division_of i"
```
```  1510   shows "(snd ` s) division_of \<Union>(snd ` s)"
```
```  1511 proof (rule division_ofI)
```
```  1512   note assm = tagged_partial_division_ofD[OF assms]
```
```  1513   show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
```
```  1514     using assm by auto
```
```  1515   fix k
```
```  1516   assume k: "k \<in> snd ` s"
```
```  1517   then obtain xk where xk: "(xk, k) \<in> s"
```
```  1518     by auto
```
```  1519   then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
```
```  1520     using assm by auto
```
```  1521   fix k'
```
```  1522   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
```
```  1523   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
```
```  1524     by auto
```
```  1525   then show "interior k \<inter> interior k' = {}"
```
```  1526     using assm(5) k'(2) xk by auto
```
```  1527 qed
```
```  1528
```
```  1529 lemma tagged_partial_division_subset:
```
```  1530   assumes "s tagged_partial_division_of i"
```
```  1531     and "t \<subseteq> s"
```
```  1532   shows "t tagged_partial_division_of i"
```
```  1533   using assms
```
```  1534   unfolding tagged_partial_division_of_def
```
```  1535   using finite_subset[OF assms(2)]
```
```  1536   by blast
```
```  1537
```
```  1538 lemma (in comm_monoid_set) over_tagged_division_lemma:
```
```  1539   assumes "p tagged_division_of i"
```
```  1540     and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = \<^bold>1"
```
```  1541   shows "F (\<lambda>(x,k). d k) p = F d (snd ` p)"
```
```  1542 proof -
```
```  1543   have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
```
```  1544     unfolding o_def by (rule ext) auto
```
```  1545   note assm = tagged_division_ofD[OF assms(1)]
```
```  1546   show ?thesis
```
```  1547     unfolding *
```
```  1548   proof (rule reindex_nontrivial[symmetric])
```
```  1549     show "finite p"
```
```  1550       using assm by auto
```
```  1551     fix x y
```
```  1552     assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
```
```  1553     obtain a b where ab: "snd x = cbox a b"
```
```  1554       using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
```
```  1555     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
```
```  1556       by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
```
```  1557     with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
```
```  1558       by (intro assm(5)[of "fst x" _ "fst y"]) auto
```
```  1559     then have "content (cbox a b) = 0"
```
```  1560       unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior by auto
```
```  1561     then have "d (cbox a b) = \<^bold>1"
```
```  1562       using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
```
```  1563     then show "d (snd x) = \<^bold>1"
```
```  1564       unfolding ab by auto
```
```  1565   qed
```
```  1566 qed
```
```  1567
```
```  1568 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
```
```  1569   by auto
```
```  1570
```
```  1571 lemma tagged_division_of_empty: "{} tagged_division_of {}"
```
```  1572   unfolding tagged_division_of by auto
```
```  1573
```
```  1574 lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
```
```  1575   unfolding tagged_partial_division_of_def by auto
```
```  1576
```
```  1577 lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
```
```  1578   unfolding tagged_division_of by auto
```
```  1579
```
```  1580 lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
```
```  1581   by (rule tagged_division_ofI) auto
```
```  1582
```
```  1583 lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
```
```  1584   unfolding box_real[symmetric]
```
```  1585   by (rule tagged_division_of_self)
```
```  1586
```
```  1587 lemma tagged_division_union:
```
```  1588   assumes "p1 tagged_division_of s1"
```
```  1589     and "p2 tagged_division_of s2"
```
```  1590     and "interior s1 \<inter> interior s2 = {}"
```
```  1591   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
```
```  1592 proof (rule tagged_division_ofI)
```
```  1593   note p1 = tagged_division_ofD[OF assms(1)]
```
```  1594   note p2 = tagged_division_ofD[OF assms(2)]
```
```  1595   show "finite (p1 \<union> p2)"
```
```  1596     using p1(1) p2(1) by auto
```
```  1597   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
```
```  1598     using p1(6) p2(6) by blast
```
```  1599   fix x k
```
```  1600   assume xk: "(x, k) \<in> p1 \<union> p2"
```
```  1601   show "x \<in> k" "\<exists>a b. k = cbox a b"
```
```  1602     using xk p1(2,4) p2(2,4) by auto
```
```  1603   show "k \<subseteq> s1 \<union> s2"
```
```  1604     using xk p1(3) p2(3) by blast
```
```  1605   fix x' k'
```
```  1606   assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
```
```  1607   have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
```
```  1608     using assms(3) interior_mono by blast
```
```  1609   show "interior k \<inter> interior k' = {}"
```
```  1610     apply (cases "(x, k) \<in> p1")
```
```  1611     apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
```
```  1612     by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
```
```  1613 qed
```
```  1614
```
```  1615 lemma tagged_division_unions:
```
```  1616   assumes "finite iset"
```
```  1617     and "\<forall>i\<in>iset. pfn i tagged_division_of i"
```
```  1618     and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
```
```  1619   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
```
```  1620 proof (rule tagged_division_ofI)
```
```  1621   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
```
```  1622   show "finite (\<Union>(pfn ` iset))"
```
```  1623     apply (rule finite_Union)
```
```  1624     using assms
```
```  1625     apply auto
```
```  1626     done
```
```  1627   have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
```
```  1628     by blast
```
```  1629   also have "\<dots> = \<Union>iset"
```
```  1630     using assm(6) by auto
```
```  1631   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
```
```  1632   fix x k
```
```  1633   assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
```
```  1634   then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
```
```  1635     by auto
```
```  1636   show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
```
```  1637     using assm(2-4)[OF i] using i(1) by auto
```
```  1638   fix x' k'
```
```  1639   assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
```
```  1640   then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
```
```  1641     by auto
```
```  1642   have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
```
```  1643     using i(1) i'(1)
```
```  1644     using assms(3)[rule_format] interior_mono
```
```  1645     by blast
```
```  1646   show "interior k \<inter> interior k' = {}"
```
```  1647     apply (cases "i = i'")
```
```  1648     using assm(5) i' i(2) xk'(2) apply blast
```
```  1649     using "*" assm(3) i' i by auto
```
```  1650 qed
```
```  1651
```
```  1652 lemma tagged_partial_division_of_union_self:
```
```  1653   assumes "p tagged_partial_division_of s"
```
```  1654   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```  1655   apply (rule tagged_division_ofI)
```
```  1656   using tagged_partial_division_ofD[OF assms]
```
```  1657   apply auto
```
```  1658   done
```
```  1659
```
```  1660 lemma tagged_division_of_union_self:
```
```  1661   assumes "p tagged_division_of s"
```
```  1662   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```  1663   apply (rule tagged_division_ofI)
```
```  1664   using tagged_division_ofD[OF assms]
```
```  1665   apply auto
```
```  1666   done
```
```  1667
```
```  1668 subsection \<open>Functions closed on boxes: morphisms from boxes to monoids\<close>
```
```  1669
```
```  1670 text \<open>This auxiliary structure is used to sum up over the elements of a division. Main theorem is
```
```  1671   @{text operative_division}. Instances for the monoid are @{typ "'a option"}, @{typ real}, and
```
```  1672   @{typ bool}.\<close>
```
```  1673
```
```  1674 lemma property_empty_interval: "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b) \<Longrightarrow> P {}"
```
```  1675   using content_empty unfolding empty_as_interval by auto
```
```  1676
```
```  1677 paragraph \<open>Using additivity of lifted function to encode definedness.\<close>
```
```  1678
```
```  1679 definition lift_option :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> 'c option"
```
```  1680 where
```
```  1681   "lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))"
```
```  1682
```
```  1683 lemma lift_option_simps[simp]:
```
```  1684   "lift_option f (Some a) (Some b) = Some (f a b)"
```
```  1685   "lift_option f None b' = None"
```
```  1686   "lift_option f a' None = None"
```
```  1687   by (auto simp: lift_option_def)
```
```  1688
```
```  1689 lemma comm_monoid_lift_option:
```
```  1690   assumes "comm_monoid f z"
```
```  1691   shows "comm_monoid (lift_option f) (Some z)"
```
```  1692 proof -
```
```  1693   from assms interpret comm_monoid f z .
```
```  1694   show ?thesis
```
```  1695     by standard (auto simp: lift_option_def ac_simps split: bind_split)
```
```  1696 qed
```
```  1697
```
```  1698 lemma comm_monoid_and: "comm_monoid HOL.conj True"
```
```  1699   by standard auto
```
```  1700
```
```  1701 lemma comm_monoid_set_and: "comm_monoid_set HOL.conj True"
```
```  1702   by (rule comm_monoid_set.intro) (fact comm_monoid_and)
```
```  1703
```
```  1704 paragraph \<open>Operative\<close>
```
```  1705
```
```  1706 definition (in comm_monoid) operative :: "('b::euclidean_space set \<Rightarrow> 'a) \<Rightarrow> bool"
```
```  1707   where "operative g \<longleftrightarrow>
```
```  1708     (\<forall>a b. content (cbox a b) = 0 \<longrightarrow> g (cbox a b) = \<^bold>1) \<and>
```
```  1709     (\<forall>a b c. \<forall>k\<in>Basis. g (cbox a b) = g (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. x\<bullet>k \<ge> c}))"
```
```  1710
```
```  1711 lemma (in comm_monoid) operativeD[dest]:
```
```  1712   assumes "operative g"
```
```  1713   shows "\<And>a b. content (cbox a b) = 0 \<Longrightarrow> g (cbox a b) = \<^bold>1"
```
```  1714     and "\<And>a b c k. k \<in> Basis \<Longrightarrow> g (cbox a b) = g (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  1715   using assms unfolding operative_def by auto
```
```  1716
```
```  1717 lemma (in comm_monoid) operative_empty: "operative g \<Longrightarrow> g {} = \<^bold>1"
```
```  1718   unfolding operative_def by (rule property_empty_interval) auto
```
```  1719
```
```  1720 lemma operative_content[intro]: "add.operative content"
```
```  1721   by (force simp add: add.operative_def content_split[symmetric])
```
```  1722
```
```  1723 definition "division_points (k::('a::euclidean_space) set) d =
```
```  1724    {(j,x). j \<in> Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
```
```  1725      (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
```
```  1726
```
```  1727 lemma division_points_finite:
```
```  1728   fixes i :: "'a::euclidean_space set"
```
```  1729   assumes "d division_of i"
```
```  1730   shows "finite (division_points i d)"
```
```  1731 proof -
```
```  1732   note assm = division_ofD[OF assms]
```
```  1733   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and>
```
```  1734     (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
```
```  1735   have *: "division_points i d = \<Union>(?M ` Basis)"
```
```  1736     unfolding division_points_def by auto
```
```  1737   show ?thesis
```
```  1738     unfolding * using assm by auto
```
```  1739 qed
```
```  1740
```
```  1741 lemma division_points_subset:
```
```  1742   fixes a :: "'a::euclidean_space"
```
```  1743   assumes "d division_of (cbox a b)"
```
```  1744     and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
```
```  1745     and k: "k \<in> Basis"
```
```  1746   shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l . l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subseteq>
```
```  1747       division_points (cbox a b) d" (is ?t1)
```
```  1748     and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x\<bullet>k \<ge> c} = {})} \<subseteq>
```
```  1749       division_points (cbox a b) d" (is ?t2)
```
```  1750 proof -
```
```  1751   note assm = division_ofD[OF assms(1)]
```
```  1752   have *: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1753     "\<forall>i\<in>Basis. a\<bullet>i \<le> (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else  b \<bullet> i) *\<^sub>R i) \<bullet> i"
```
```  1754     "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i \<le> b\<bullet>i"
```
```  1755     "min (b \<bullet> k) c = c" "max (a \<bullet> k) c = c"
```
```  1756     using assms using less_imp_le by auto
```
```  1757   show ?t1 (*FIXME a horrible mess*)
```
```  1758     unfolding division_points_def interval_split[OF k, of a b]
```
```  1759     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
```
```  1760     unfolding *
```
```  1761     apply (rule subsetI)
```
```  1762     unfolding mem_Collect_eq split_beta
```
```  1763     apply (erule bexE conjE)+
```
```  1764     apply (simp add: )
```
```  1765     apply (erule exE conjE)+
```
```  1766   proof
```
```  1767     fix i l x
```
```  1768     assume as:
```
```  1769       "a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)"
```
```  1770       "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1771       "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
```
```  1772       and fstx: "fst x \<in> Basis"
```
```  1773     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1774     have *: "\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i"
```
```  1775       using as(6) unfolding l interval_split[OF k] box_ne_empty as .
```
```  1776     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
```
```  1777       using l using as(6) unfolding box_ne_empty[symmetric] by auto
```
```  1778     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1779       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
```
```  1780       using as(1-3,5) fstx
```
```  1781       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
```
```  1782       apply (auto split: if_split_asm)
```
```  1783       done
```
```  1784     show "snd x < b \<bullet> fst x"
```
```  1785       using as(2) \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm)
```
```  1786   qed
```
```  1787   show ?t2
```
```  1788     unfolding division_points_def interval_split[OF k, of a b]
```
```  1789     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
```
```  1790     unfolding *
```
```  1791     unfolding subset_eq
```
```  1792     apply rule
```
```  1793     unfolding mem_Collect_eq split_beta
```
```  1794     apply (erule bexE conjE)+
```
```  1795     apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
```
```  1796     apply (erule exE conjE)+
```
```  1797   proof
```
```  1798     fix i l x
```
```  1799     assume as:
```
```  1800       "(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
```
```  1801       "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1802       "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
```
```  1803       and fstx: "fst x \<in> Basis"
```
```  1804     from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
```
```  1805     have *: "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i"
```
```  1806       using as(6) unfolding l interval_split[OF k] box_ne_empty as .
```
```  1807     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
```
```  1808       using l using as(6) unfolding box_ne_empty[symmetric] by auto
```
```  1809     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1810       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
```
```  1811       using as(1-3,5) fstx
```
```  1812       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
```
```  1813       apply (auto split: if_split_asm)
```
```  1814       done
```
```  1815     show "a \<bullet> fst x < snd x"
```
```  1816       using as(1) \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm)
```
```  1817    qed
```
```  1818 qed
```
```  1819
```
```  1820 lemma division_points_psubset:
```
```  1821   fixes a :: "'a::euclidean_space"
```
```  1822   assumes "d division_of (cbox a b)"
```
```  1823       and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
```
```  1824       and "l \<in> d"
```
```  1825       and "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c"
```
```  1826       and k: "k \<in> Basis"
```
```  1827   shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subset>
```
```  1828          division_points (cbox a b) d" (is "?D1 \<subset> ?D")
```
```  1829     and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} \<subset>
```
```  1830          division_points (cbox a b) d" (is "?D2 \<subset> ?D")
```
```  1831 proof -
```
```  1832   have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1833     using assms(2) by (auto intro!:less_imp_le)
```
```  1834   guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
```
```  1835   have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i"
```
```  1836     using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
```
```  1837     using subset_box(1)
```
```  1838     apply auto
```
```  1839     apply blast+
```
```  1840     done
```
```  1841   have *: "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
```
```  1842           "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
```
```  1843     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
```
```  1844     using uv[rule_format, of k] ab k
```
```  1845     by auto
```
```  1846   have "\<exists>x. x \<in> ?D - ?D1"
```
```  1847     using assms(3-)
```
```  1848     unfolding division_points_def interval_bounds[OF ab]
```
```  1849     apply -
```
```  1850     apply (erule disjE)
```
```  1851     apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1852     apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1853     done
```
```  1854   moreover have "?D1 \<subseteq> ?D"
```
```  1855     by (auto simp add: assms division_points_subset)
```
```  1856   ultimately show "?D1 \<subset> ?D"
```
```  1857     by blast
```
```  1858   have *: "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
```
```  1859     "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
```
```  1860     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
```
```  1861     using uv[rule_format, of k] ab k
```
```  1862     by auto
```
```  1863   have "\<exists>x. x \<in> ?D - ?D2"
```
```  1864     using assms(3-)
```
```  1865     unfolding division_points_def interval_bounds[OF ab]
```
```  1866     apply -
```
```  1867     apply (erule disjE)
```
```  1868     apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1869     apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1870     done
```
```  1871   moreover have "?D2 \<subseteq> ?D"
```
```  1872     by (auto simp add: assms division_points_subset)
```
```  1873   ultimately show "?D2 \<subset> ?D"
```
```  1874     by blast
```
```  1875 qed
```
```  1876
```
```  1877 lemma (in comm_monoid_set) operative_division:
```
```  1878   fixes g :: "'b::euclidean_space set \<Rightarrow> 'a"
```
```  1879   assumes g: "operative g" and d: "d division_of (cbox a b)" shows "F g d = g (cbox a b)"
```
```  1880 proof -
```
```  1881   define C where [abs_def]: "C = card (division_points (cbox a b) d)"
```
```  1882   then show ?thesis
```
```  1883     using d
```
```  1884   proof (induction C arbitrary: a b d rule: less_induct)
```
```  1885     case (less a b d)
```
```  1886     show ?case
```
```  1887     proof cases
```
```  1888       show "content (cbox a b) = 0 \<Longrightarrow> F g d = g (cbox a b)"
```
```  1889         using division_of_content_0[OF _ less.prems] operativeD(1)[OF  g] division_ofD(4)[OF less.prems]
```
```  1890         by (fastforce intro!: neutral)
```
```  1891     next
```
```  1892       assume "content (cbox a b) \<noteq> 0"
```
```  1893       note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
```
```  1894       then have ab': "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1895         by (auto intro!: less_imp_le)
```
```  1896       show "F g d = g (cbox a b)"
```
```  1897       proof (cases "division_points (cbox a b) d = {}")
```
```  1898         case True
```
```  1899         { fix u v and j :: 'b
```
```  1900           assume j: "j \<in> Basis" and as: "cbox u v \<in> d"
```
```  1901           then have "cbox u v \<noteq> {}"
```
```  1902             using less.prems by blast
```
```  1903           then have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "u\<bullet>j \<le> v\<bullet>j"
```
```  1904             using j unfolding box_ne_empty by auto
```
```  1905           have *: "\<And>p r Q. \<not> j\<in>Basis \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> Q (cbox u v)"
```
```  1906             using as j by auto
```
```  1907           have "(j, u\<bullet>j) \<notin> division_points (cbox a b) d"
```
```  1908                "(j, v\<bullet>j) \<notin> division_points (cbox a b) d" using True by auto
```
```  1909           note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
```
```  1910           note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
```
```  1911           moreover
```
```  1912           have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j"
```
```  1913             using division_ofD(2,2,3)[OF \<open>d division_of cbox a b\<close> as]
```
```  1914             apply (metis j subset_box(1) uv(1))
```
```  1915             by (metis \<open>cbox u v \<subseteq> cbox a b\<close> j subset_box(1) uv(1))
```
```  1916           ultimately have "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j"
```
```  1917             unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force }
```
```  1918         then have d': "\<forall>i\<in>d. \<exists>u v. i = cbox u v \<and>
```
```  1919           (\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)"
```
```  1920           unfolding forall_in_division[OF less.prems] by blast
```
```  1921         have "(1/2) *\<^sub>R (a+b) \<in> cbox a b"
```
```  1922           unfolding mem_box using ab by(auto intro!: less_imp_le simp: inner_simps)
```
```  1923         note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff]
```
```  1924         then guess i .. note i=this
```
```  1925         guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
```
```  1926         have "cbox a b \<in> d"
```
```  1927         proof -
```
```  1928           have "u = a" "v = b"
```
```  1929             unfolding euclidean_eq_iff[where 'a='b]
```
```  1930           proof safe
```
```  1931             fix j :: 'b
```
```  1932             assume j: "j \<in> Basis"
```
```  1933             note i(2)[unfolded uv mem_box,rule_format,of j]
```
```  1934             then show "u \<bullet> j = a \<bullet> j" and "v \<bullet> j = b \<bullet> j"
```
```  1935               using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
```
```  1936           qed
```
```  1937           then have "i = cbox a b" using uv by auto
```
```  1938           then show ?thesis using i by auto
```
```  1939         qed
```
```  1940         then have deq: "d = insert (cbox a b) (d - {cbox a b})"
```
```  1941           by auto
```
```  1942         have "F g (d - {cbox a b}) = \<^bold>1"
```
```  1943         proof (intro neutral ballI)
```
```  1944           fix x
```
```  1945           assume x: "x \<in> d - {cbox a b}"
```
```  1946           then have "x\<in>d"
```
```  1947             by auto note d'[rule_format,OF this]
```
```  1948           then guess u v by (elim exE conjE) note uv=this
```
```  1949           have "u \<noteq> a \<or> v \<noteq> b"
```
```  1950             using x[unfolded uv] by auto
```
```  1951           then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j: "j \<in> Basis"
```
```  1952             unfolding euclidean_eq_iff[where 'a='b] by auto
```
```  1953           then have "u\<bullet>j = v\<bullet>j"
```
```  1954             using uv(2)[rule_format,OF j] by auto
```
```  1955           then have "content (cbox u v) = 0"
```
```  1956             unfolding content_eq_0 using j
```
```  1957             by force
```
```  1958           then show "g x = \<^bold>1"
```
```  1959             unfolding uv(1) by (rule operativeD(1)[OF g])
```
```  1960         qed
```
```  1961         then show "F g d = g (cbox a b)"
```
```  1962           using division_ofD[OF less.prems]
```
```  1963           apply (subst deq)
```
```  1964           apply (subst insert)
```
```  1965           apply auto
```
```  1966           done
```
```  1967       next
```
```  1968         case False
```
```  1969         then have "\<exists>x. x \<in> division_points (cbox a b) d"
```
```  1970           by auto
```
```  1971         then guess k c
```
```  1972           unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
```
```  1973           apply (elim exE conjE)
```
```  1974           done
```
```  1975         note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
```
```  1976         from this(3) guess j .. note j=this
```
```  1977         define d1 where "d1 = {l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
```
```  1978         define d2 where "d2 = {l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
```
```  1979         define cb where "cb = (\<Sum>i\<in>Basis. (if i = k then c else b\<bullet>i) *\<^sub>R i)"
```
```  1980         define ca where "ca = (\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)"
```
```  1981         note division_points_psubset[OF \<open>d division_of cbox a b\<close> ab kc(1-2) j]
```
```  1982         note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
```
```  1983         then have *: "F g d1 = g (cbox a b \<inter> {x. x\<bullet>k \<le> c})" "F g d2 = g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  1984           unfolding interval_split[OF kc(4)]
```
```  1985           apply (rule_tac[!] "less.hyps"[rule_format])
```
```  1986           using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c]
```
```  1987           apply (simp_all add: interval_split kc d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>])
```
```  1988           done
```
```  1989         { fix l y
```
```  1990           assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
```
```  1991           from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
```
```  1992           have "g (l \<inter> {x. x \<bullet> k \<le> c}) = \<^bold>1"
```
```  1993             unfolding leq interval_split[OF kc(4)]
```
```  1994             apply (rule operativeD[OF g])
```
```  1995             unfolding interval_split[symmetric, OF kc(4)]
```
```  1996             using division_split_left_inj less as kc leq by blast
```
```  1997         } note fxk_le = this
```
```  1998         { fix l y
```
```  1999           assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
```
```  2000           from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
```
```  2001           have "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1"
```
```  2002             unfolding leq interval_split[OF kc(4)]
```
```  2003             apply (rule operativeD(1)[OF g])
```
```  2004             unfolding interval_split[symmetric,OF kc(4)]
```
```  2005             using division_split_right_inj less leq as kc by blast
```
```  2006         } note fxk_ge = this
```
```  2007         have d1_alt: "d1 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<le> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
```
```  2008           using d1_def by auto
```
```  2009         have d2_alt: "d2 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<ge> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
```
```  2010           using d2_def by auto
```
```  2011         have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev")
```
```  2012           unfolding * using g kc(4) by blast
```
```  2013         also have "F g d1 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d"
```
```  2014           unfolding d1_alt using division_of_finite[OF less.prems] fxk_le
```
```  2015           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
```
```  2016         also have "F g d2 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d"
```
```  2017           unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge
```
```  2018           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
```
```  2019         also have *: "\<forall>x\<in>d. g x = g (x \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (x \<inter> {x. c \<le> x \<bullet> k})"
```
```  2020           unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>]
```
```  2021           using g kc(4) by blast
```
```  2022         have "F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d \<^bold>* F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d = F g d"
```
```  2023           using * by (simp add: distrib)
```
```  2024         finally show ?thesis by auto
```
```  2025       qed
```
```  2026     qed
```
```  2027   qed
```
```  2028 qed
```
```  2029
```
```  2030 lemma (in comm_monoid_set) operative_tagged_division:
```
```  2031   assumes f: "operative g" and d: "d tagged_division_of (cbox a b)"
```
```  2032   shows "F (\<lambda>(x, l). g l) d = g (cbox a b)"
```
```  2033   unfolding d[THEN division_of_tagged_division, THEN operative_division[OF f], symmetric]
```
```  2034   by (simp add: f[THEN operativeD(1)] over_tagged_division_lemma[OF d])
```
```  2035
```
```  2036 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> setsum content d = content (cbox a b)"
```
```  2037   by (metis operative_content setsum.operative_division)
```
```  2038
```
```  2039 lemma additive_content_tagged_division:
```
```  2040   "d tagged_division_of (cbox a b) \<Longrightarrow> setsum (\<lambda>(x,l). content l) d = content (cbox a b)"
```
```  2041   unfolding setsum.operative_tagged_division[OF operative_content, symmetric] by blast
```
```  2042
```
```  2043 lemma
```
```  2044   shows real_inner_1_left: "inner 1 x = x"
```
```  2045   and real_inner_1_right: "inner x 1 = x"
```
```  2046   by simp_all
```
```  2047
```
```  2048 lemma content_real_eq_0: "content {a .. b::real} = 0 \<longleftrightarrow> a \<ge> b"
```
```  2049   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
```
```  2050
```
```  2051 lemma interval_real_split:
```
```  2052   "{a .. b::real} \<inter> {x. x \<le> c} = {a .. min b c}"
```
```  2053   "{a .. b} \<inter> {x. c \<le> x} = {max a c .. b}"
```
```  2054   apply (metis Int_atLeastAtMostL1 atMost_def)
```
```  2055   apply (metis Int_atLeastAtMostL2 atLeast_def)
```
```  2056   done
```
```  2057
```
```  2058 lemma (in comm_monoid) operative_1_lt:
```
```  2059   "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
```
```  2060     ((\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1) \<and> (\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
```
```  2061   apply (simp add: operative_def content_real_eq_0 atMost_def[symmetric] atLeast_def[symmetric]
```
```  2062               del: content_real_if)
```
```  2063 proof safe
```
```  2064   fix a b c :: real
```
```  2065   assume *: "\<forall>a b c. g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
```
```  2066   assume "a < c" "c < b"
```
```  2067   with *[rule_format, of a b c] show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2068     by (simp add: less_imp_le min.absorb2 max.absorb2)
```
```  2069 next
```
```  2070   fix a b c :: real
```
```  2071   assume as: "\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1"
```
```  2072     "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2073   from as(1)[rule_format, of 0 1] as(1)[rule_format, of a a for a] as(2)
```
```  2074   have [simp]: "g {} = \<^bold>1" "\<And>a. g {a} = \<^bold>1"
```
```  2075     "\<And>a b c. a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2076     by auto
```
```  2077   show "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
```
```  2078     by (auto simp: min_def max_def le_less)
```
```  2079 qed
```
```  2080
```
```  2081 lemma (in comm_monoid) operative_1_le:
```
```  2082   "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
```
```  2083     ((\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1) \<and> (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
```
```  2084   unfolding operative_1_lt
```
```  2085 proof safe
```
```  2086   fix a b c :: real
```
```  2087   assume as: "\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" "a < c" "c < b"
```
```  2088   show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2089     apply (rule as(1)[rule_format])
```
```  2090     using as(2-)
```
```  2091     apply auto
```
```  2092     done
```
```  2093 next
```
```  2094   fix a b c :: real
```
```  2095   assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
```
```  2096     and "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2097     and "a \<le> c"
```
```  2098     and "c \<le> b"
```
```  2099   note as = this[rule_format]
```
```  2100   show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  2101   proof (cases "c = a \<or> c = b")
```
```  2102     case False
```
```  2103     then show ?thesis
```
```  2104       apply -
```
```  2105       apply (subst as(2))
```
```  2106       using as(3-)
```
```  2107       apply auto
```
```  2108       done
```
```  2109   next
```
```  2110     case True
```
```  2111     then show ?thesis
```
```  2112     proof
```
```  2113       assume *: "c = a"
```
```  2114       then have "g {a .. c} = \<^bold>1"
```
```  2115         apply -
```
```  2116         apply (rule as(1)[rule_format])
```
```  2117         apply auto
```
```  2118         done
```
```  2119       then show ?thesis
```
```  2120         unfolding * by auto
```
```  2121     next
```
```  2122       assume *: "c = b"
```
```  2123       then have "g {c .. b} = \<^bold>1"
```
```  2124         apply -
```
```  2125         apply (rule as(1)[rule_format])
```
```  2126         apply auto
```
```  2127         done
```
```  2128       then show ?thesis
```
```  2129         unfolding * by auto
```
```  2130     qed
```
```  2131   qed
```
```  2132 qed
```
```  2133
```
```  2134 subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
```
```  2135
```
```  2136 definition fine  (infixr "fine" 46)
```
```  2137   where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
```
```  2138
```
```  2139 lemma fineI:
```
```  2140   assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```  2141   shows "d fine s"
```
```  2142   using assms unfolding fine_def by auto
```
```  2143
```
```  2144 lemma fineD[dest]:
```
```  2145   assumes "d fine s"
```
```  2146   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```  2147   using assms unfolding fine_def by auto
```
```  2148
```
```  2149 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
```
```  2150   unfolding fine_def by auto
```
```  2151
```
```  2152 lemma fine_inters:
```
```  2153  "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
```
```  2154   unfolding fine_def by blast
```
```  2155
```
```  2156 lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
```
```  2157   unfolding fine_def by blast
```
```  2158
```
```  2159 lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
```
```  2160   unfolding fine_def by auto
```
```  2161
```
```  2162 lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
```
```  2163   unfolding fine_def by blast
```
```  2164
```
```  2165
```
```  2166 subsection \<open>Gauge integral. Define on compact intervals first, then use a limit.\<close>
```
```  2167
```
```  2168 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
```
```  2169   where "(f has_integral_compact_interval y) i \<longleftrightarrow>
```
```  2170     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  2171       (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
```
```  2172         norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
```
```  2173
```
```  2174 definition has_integral ::
```
```  2175     "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
```
```  2176   (infixr "has'_integral" 46)
```
```  2177   where "(f has_integral y) i \<longleftrightarrow>
```
```  2178     (if \<exists>a b. i = cbox a b
```
```  2179      then (f has_integral_compact_interval y) i
```
```  2180      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2181       (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
```
```  2182         norm (z - y) < e)))"
```
```  2183
```
```  2184 lemma has_integral:
```
```  2185   "(f has_integral y) (cbox a b) \<longleftrightarrow>
```
```  2186     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  2187       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  2188         norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```  2189   unfolding has_integral_def has_integral_compact_interval_def
```
```  2190   by auto
```
```  2191
```
```  2192 lemma has_integral_real:
```
```  2193   "(f has_integral y) {a .. b::real} \<longleftrightarrow>
```
```  2194     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  2195       (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
```
```  2196         norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```  2197   unfolding box_real[symmetric]
```
```  2198   by (rule has_integral)
```
```  2199
```
```  2200 lemma has_integralD[dest]:
```
```  2201   assumes "(f has_integral y) (cbox a b)"
```
```  2202     and "e > 0"
```
```  2203   obtains d where "gauge d"
```
```  2204     and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
```
```  2205       norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
```
```  2206   using assms unfolding has_integral by auto
```
```  2207
```
```  2208 lemma has_integral_alt:
```
```  2209   "(f has_integral y) i \<longleftrightarrow>
```
```  2210     (if \<exists>a b. i = cbox a b
```
```  2211      then (f has_integral y) i
```
```  2212      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2213       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
```
```  2214   unfolding has_integral
```
```  2215   unfolding has_integral_compact_interval_def has_integral_def
```
```  2216   by auto
```
```  2217
```
```  2218 lemma has_integral_altD:
```
```  2219   assumes "(f has_integral y) i"
```
```  2220     and "\<not> (\<exists>a b. i = cbox a b)"
```
```  2221     and "e>0"
```
```  2222   obtains B where "B > 0"
```
```  2223     and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2224       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
```
```  2225   using assms
```
```  2226   unfolding has_integral
```
```  2227   unfolding has_integral_compact_interval_def has_integral_def
```
```  2228   by auto
```
```  2229
```
```  2230 definition integrable_on (infixr "integrable'_on" 46)
```
```  2231   where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
```
```  2232
```
```  2233 definition "integral i f = (SOME y. (f has_integral y) i \<or> ~ f integrable_on i \<and> y=0)"
```
```  2234
```
```  2235 lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
```
```  2236   unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
```
```  2237
```
```  2238 lemma not_integrable_integral: "~ f integrable_on i \<Longrightarrow> integral i f = 0"
```
```  2239   unfolding integrable_on_def integral_def by blast
```
```  2240
```
```  2241 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
```
```  2242   unfolding integrable_on_def by auto
```
```  2243
```
```  2244 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
```
```  2245   by auto
```
```  2246
```
```  2247 lemma setsum_content_null:
```
```  2248   assumes "content (cbox a b) = 0"
```
```  2249     and "p tagged_division_of (cbox a b)"
```
```  2250   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
```
```  2251 proof (rule setsum.neutral, rule)
```
```  2252   fix y
```
```  2253   assume y: "y \<in> p"
```
```  2254   obtain x k where xk: "y = (x, k)"
```
```  2255     using surj_pair[of y] by blast
```
```  2256   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
```
```  2257   from this(2) obtain c d where k: "k = cbox c d" by blast
```
```  2258   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
```
```  2259     unfolding xk by auto
```
```  2260   also have "\<dots> = 0"
```
```  2261     using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
```
```  2262     unfolding assms(1) k
```
```  2263     by auto
```
```  2264   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
```
```  2265 qed
```
```  2266
```
```  2267
```
```  2268 subsection \<open>Some basic combining lemmas.\<close>
```
```  2269
```
```  2270 lemma tagged_division_unions_exists:
```
```  2271   assumes "finite iset"
```
```  2272     and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
```
```  2273     and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
```
```  2274     and "\<Union>iset = i"
```
```  2275    obtains p where "p tagged_division_of i" and "d fine p"
```
```  2276 proof -
```
```  2277   obtain pfn where pfn:
```
```  2278     "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
```
```  2279     "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
```
```  2280     using bchoice[OF assms(2)] by auto
```
```  2281   show thesis
```
```  2282     apply (rule_tac p="\<Union>(pfn ` iset)" in that)
```
```  2283     using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
```
```  2284     by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
```
```  2285 qed
```
```  2286
```
```  2287
```
```  2288 subsection \<open>The set we're concerned with must be closed.\<close>
```
```  2289
```
```  2290 lemma division_of_closed:
```
```  2291   fixes i :: "'n::euclidean_space set"
```
```  2292   shows "s division_of i \<Longrightarrow> closed i"
```
```  2293   unfolding division_of_def by fastforce
```
```  2294
```
```  2295 subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
```
```  2296
```
```  2297 lemma interval_bisection_step:
```
```  2298   fixes type :: "'a::euclidean_space"
```
```  2299   assumes "P {}"
```
```  2300     and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
```
```  2301     and "\<not> P (cbox a (b::'a))"
```
```  2302   obtains c d where "\<not> P (cbox c d)"
```
```  2303     and "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
```
```  2304 proof -
```
```  2305   have "cbox a b \<noteq> {}"
```
```  2306     using assms(1,3) by metis
```
```  2307   then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
```
```  2308     by (force simp: mem_box)
```
```  2309   { fix f
```
```  2310     have "\<lbrakk>finite f;
```
```  2311            \<And>s. s\<in>f \<Longrightarrow> P s;
```
```  2312            \<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
```
```  2313            \<And>s t. s\<in>f \<Longrightarrow> t\<in>f \<Longrightarrow> s \<noteq> t \<Longrightarrow> interior s \<inter> interior t = {}\<rbrakk> \<Longrightarrow> P (\<Union>f)"
```
```  2314     proof (induct f rule: finite_induct)
```
```  2315       case empty
```
```  2316       show ?case
```
```  2317         using assms(1) by auto
```
```  2318     next
```
```  2319       case (insert x f)
```
```  2320       show ?case
```
```  2321         unfolding Union_insert
```
```  2322         apply (rule assms(2)[rule_format])
```
```  2323         using inter_interior_unions_intervals [of f "interior x"]
```
```  2324         apply (auto simp: insert)
```
```  2325         by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
```
```  2326     qed
```
```  2327   } note UN_cases = this
```
```  2328   let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
```
```  2329     (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
```
```  2330   let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
```
```  2331   {
```
```  2332     presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
```
```  2333     then show thesis
```
```  2334       unfolding atomize_not not_all
```
```  2335       by (blast intro: that)
```
```  2336   }
```
```  2337   assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
```
```  2338   have "P (\<Union>?A)"
```
```  2339   proof (rule UN_cases)
```
```  2340     let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a)
```
```  2341       (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)) ` {s. s \<subseteq> Basis}"
```
```  2342     have "?A \<subseteq> ?B"
```
```  2343     proof
```
```  2344       fix x
```
```  2345       assume "x \<in> ?A"
```
```  2346       then obtain c d
```
```  2347         where x:  "x = cbox c d"
```
```  2348                   "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2349                         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2350                         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
```
```  2351       show "x \<in> ?B"
```
```  2352         unfolding image_iff x
```
```  2353         apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
```
```  2354         apply (rule arg_cong2 [where f = cbox])
```
```  2355         using x(2) ab
```
```  2356         apply (auto simp add: euclidean_eq_iff[where 'a='a])
```
```  2357         by fastforce
```
```  2358     qed
```
```  2359     then show "finite ?A"
```
```  2360       by (rule finite_subset) auto
```
```  2361   next
```
```  2362     fix s
```
```  2363     assume "s \<in> ?A"
```
```  2364     then obtain c d
```
```  2365       where s: "s = cbox c d"
```
```  2366                "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2367                      c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2368                      c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
```
```  2369       by blast
```
```  2370     show "P s"
```
```  2371       unfolding s
```
```  2372       apply (rule as[rule_format])
```
```  2373       using ab s(2) by force
```
```  2374     show "\<exists>a b. s = cbox a b"
```
```  2375       unfolding s by auto
```
```  2376     fix t
```
```  2377     assume "t \<in> ?A"
```
```  2378     then obtain e f where t:
```
```  2379       "t = cbox e f"
```
```  2380       "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2381         e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2382         e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
```
```  2383       by blast
```
```  2384     assume "s \<noteq> t"
```
```  2385     then have "\<not> (c = e \<and> d = f)"
```
```  2386       unfolding s t by auto
```
```  2387     then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
```
```  2388       unfolding euclidean_eq_iff[where 'a='a] by auto
```
```  2389     then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
```
```  2390       using s(2) t(2) apply fastforce
```
```  2391       using t(2)[OF i'] \<open>c \<bullet> i \<noteq> e \<bullet> i \<or> d \<bullet> i \<noteq> f \<bullet> i\<close> i' s(2) t(2) by fastforce
```
```  2392     have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
```
```  2393       by auto
```
```  2394     show "interior s \<inter> interior t = {}"
```
```  2395       unfolding s t interior_cbox
```
```  2396     proof (rule *)
```
```  2397       fix x
```
```  2398       assume "x \<in> box c d" "x \<in> box e f"
```
```  2399       then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
```
```  2400         unfolding mem_box using i'
```
```  2401         by force+
```
```  2402       show False  using s(2)[OF i']
```
```  2403       proof safe
```
```  2404         assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
```
```  2405         show False
```
```  2406           using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
```
```  2407       next
```
```  2408         assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
```
```  2409         show False
```
```  2410           using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
```
```  2411       qed
```
```  2412     qed
```
```  2413   qed
```
```  2414   also have "\<Union>?A = cbox a b"
```
```  2415   proof (rule set_eqI,rule)
```
```  2416     fix x
```
```  2417     assume "x \<in> \<Union>?A"
```
```  2418     then obtain c d where x:
```
```  2419       "x \<in> cbox c d"
```
```  2420       "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2421         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2422         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
```
```  2423       by blast
```
```  2424     show "x\<in>cbox a b"
```
```  2425       unfolding mem_box
```
```  2426     proof safe
```
```  2427       fix i :: 'a
```
```  2428       assume i: "i \<in> Basis"
```
```  2429       then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
```
```  2430         using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
```
```  2431     qed
```
```  2432   next
```
```  2433     fix x
```
```  2434     assume x: "x \<in> cbox a b"
```
```  2435     have "\<forall>i\<in>Basis.
```
```  2436       \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
```
```  2437       (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
```
```  2438       unfolding mem_box
```
```  2439     proof
```
```  2440       fix i :: 'a
```
```  2441       assume i: "i \<in> Basis"
```
```  2442       have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
```
```  2443         using x[unfolded mem_box,THEN bspec, OF i] by auto
```
```  2444       then show "\<exists>c d. ?P i c d"
```
```  2445         by blast
```
```  2446     qed
```
```  2447     then show "x\<in>\<Union>?A"
```
```  2448       unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
```
```  2449       apply auto
```
```  2450       apply (rule_tac x="cbox xa xaa" in exI)
```
```  2451       unfolding mem_box
```
```  2452       apply auto
```
```  2453       done
```
```  2454   qed
```
```  2455   finally show False
```
```  2456     using assms by auto
```
```  2457 qed
```
```  2458
```
```  2459 lemma interval_bisection:
```
```  2460   fixes type :: "'a::euclidean_space"
```
```  2461   assumes "P {}"
```
```  2462     and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
```
```  2463     and "\<not> P (cbox a (b::'a))"
```
```  2464   obtains x where "x \<in> cbox a b"
```
```  2465     and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
```
```  2466 proof -
```
```  2467   have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
```
```  2468     (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
```
```  2469        2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x")
```
```  2470   proof
```
```  2471     show "?P x" for x
```
```  2472     proof (cases "P (cbox (fst x) (snd x))")
```
```  2473       case True
```
```  2474       then show ?thesis by auto
```
```  2475     next
```
```  2476       case as: False
```
```  2477       obtain c d where "\<not> P (cbox c d)"
```
```  2478         "\<forall>i\<in>Basis.
```
```  2479            fst x \<bullet> i \<le> c \<bullet> i \<and>
```
```  2480            c \<bullet> i \<le> d \<bullet> i \<and>
```
```  2481            d \<bullet> i \<le> snd x \<bullet> i \<and>
```
```  2482            2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
```
```  2483         by (rule interval_bisection_step[of P, OF assms(1-2) as])
```
```  2484       then show ?thesis
```
```  2485         apply -
```
```  2486         apply (rule_tac x="(c,d)" in exI)
```
```  2487         apply auto
```
```  2488         done
```
```  2489     qed
```
```  2490   qed
```
```  2491   then obtain f where f:
```
```  2492     "\<forall>x.
```
```  2493       \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
```
```  2494       \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
```
```  2495         (\<forall>i\<in>Basis.
```
```  2496             fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
```
```  2497             fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
```
```  2498             snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
```
```  2499             2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
```
```  2500     apply -
```
```  2501     apply (drule choice)
```
```  2502     apply blast
```
```  2503     done
```
```  2504   define AB A B where ab_def: "AB n = (f ^^ n) (a,b)" "A n = fst(AB n)" "B n = snd(AB n)" for n
```
```  2505   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
```
```  2506     (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and>
```
```  2507     2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
```
```  2508   proof -
```
```  2509     show "A 0 = a" "B 0 = b"
```
```  2510       unfolding ab_def by auto
```
```  2511     note S = ab_def funpow.simps o_def id_apply
```
```  2512     show "?P n" for n
```
```  2513     proof (induct n)
```
```  2514       case 0
```
```  2515       then show ?case
```
```  2516         unfolding S
```
```  2517         apply (rule f[rule_format]) using assms(3)
```
```  2518         apply auto
```
```  2519         done
```
```  2520     next
```
```  2521       case (Suc n)
```
```  2522       show ?case
```
```  2523         unfolding S
```
```  2524         apply (rule f[rule_format])
```
```  2525         using Suc
```
```  2526         unfolding S
```
```  2527         apply auto
```
```  2528         done
```
```  2529     qed
```
```  2530   qed
```
```  2531   note AB = this(1-2) conjunctD2[OF this(3),rule_format]
```
```  2532
```
```  2533   have interv: "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
```
```  2534     if e: "0 < e" for e
```
```  2535   proof -
```
```  2536     obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
```
```  2537       using real_arch_pow[of 2 "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto
```
```  2538     show ?thesis
```
```  2539     proof (rule exI [where x=n], clarify)
```
```  2540       fix x y
```
```  2541       assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
```
```  2542       have "dist x y \<le> setsum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
```
```  2543         unfolding dist_norm by(rule norm_le_l1)
```
```  2544       also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
```
```  2545       proof (rule setsum_mono)
```
```  2546         fix i :: 'a
```
```  2547         assume i: "i \<in> Basis"
```
```  2548         show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
```
```  2549           using xy[unfolded mem_box,THEN bspec, OF i]
```
```  2550           by (auto simp: inner_diff_left)
```
```  2551       qed
```
```  2552       also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
```
```  2553         unfolding setsum_divide_distrib
```
```  2554       proof (rule setsum_mono)
```
```  2555         show "B n \<bullet> i - A n \<bullet> i \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ n" if i: "i \<in> Basis" for i
```
```  2556         proof (induct n)
```
```  2557           case 0
```
```  2558           then show ?case
```
```  2559             unfolding AB by auto
```
```  2560         next
```
```  2561           case (Suc n)
```
```  2562           have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
```
```  2563             using AB(4)[of i n] using i by auto
```
```  2564           also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
```
```  2565             using Suc by (auto simp add: field_simps)
```
```  2566           finally show ?case .
```
```  2567         qed
```
```  2568       qed
```
```  2569       also have "\<dots> < e"
```
```  2570         using n using e by (auto simp add: field_simps)
```
```  2571       finally show "dist x y < e" .
```
```  2572     qed
```
```  2573   qed
```
```  2574   {
```
```  2575     fix n m :: nat
```
```  2576     assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
```
```  2577     proof (induction rule: inc_induct)
```
```  2578       case (step i)
```
```  2579       show ?case
```
```  2580         using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
```
```  2581     qed simp
```
```  2582   } note ABsubset = this
```
```  2583   have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
```
```  2584     by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
```
```  2585       (metis nat.exhaust AB(1-3) assms(1,3))
```
```  2586   then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
```
```  2587     by blast
```
```  2588   show thesis
```
```  2589   proof (rule that[rule_format, of x0])
```
```  2590     show "x0\<in>cbox a b"
```
```  2591       using x0[of 0] unfolding AB .
```
```  2592     fix e :: real
```
```  2593     assume "e > 0"
```
```  2594     from interv[OF this] obtain n
```
```  2595       where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
```
```  2596     have "\<not> P (cbox (A n) (B n))"
```
```  2597       apply (cases "0 < n")
```
```  2598       using AB(3)[of "n - 1"] assms(3) AB(1-2)
```
```  2599       apply auto
```
```  2600       done
```
```  2601     moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
```
```  2602       using n using x0[of n] by auto
```
```  2603     moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
```
```  2604       unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
```
```  2605     ultimately show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
```
```  2606       apply (rule_tac x="A n" in exI)
```
```  2607       apply (rule_tac x="B n" in exI)
```
```  2608       apply (auto simp: x0)
```
```  2609       done
```
```  2610   qed
```
```  2611 qed
```
```  2612
```
```  2613
```
```  2614 subsection \<open>Cousin's lemma.\<close>
```
```  2615
```
```  2616 lemma fine_division_exists:
```
```  2617   fixes a b :: "'a::euclidean_space"
```
```  2618   assumes "gauge g"
```
```  2619   obtains p where "p tagged_division_of (cbox a b)" "g fine p"
```
```  2620 proof -
```
```  2621   presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
```
```  2622   then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
```
```  2623     by blast
```
```  2624   then show thesis ..
```
```  2625 next
```
```  2626   assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
```
```  2627   obtain x where x:
```
```  2628       "x \<in> (cbox a b)"
```
```  2629       "\<And>e. 0 < e \<Longrightarrow>
```
```  2630         \<exists>c d.
```
```  2631           x \<in> cbox c d \<and>
```
```  2632           cbox c d \<subseteq> ball x e \<and>
```
```  2633           cbox c d \<subseteq> (cbox a b) \<and>
```
```  2634           \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
```
```  2635     apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
```
```  2636     apply (simp add: fine_def)
```
```  2637     apply (metis tagged_division_union fine_union)
```
```  2638     apply (auto simp: )
```
```  2639     done
```
```  2640   obtain e where e: "e > 0" "ball x e \<subseteq> g x"
```
```  2641     using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
```
```  2642   from x(2)[OF e(1)]
```
```  2643   obtain c d where c_d: "x \<in> cbox c d"
```
```  2644                         "cbox c d \<subseteq> ball x e"
```
```  2645                         "cbox c d \<subseteq> cbox a b"
```
```  2646                         "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
```
```  2647     by blast
```
```  2648   have "g fine {(x, cbox c d)}"
```
```  2649     unfolding fine_def using e using c_d(2) by auto
```
```  2650   then show False
```
```  2651     using tagged_division_of_self[OF c_d(1)] using c_d by auto
```
```  2652 qed
```
```  2653
```
```  2654 lemma fine_division_exists_real:
```
```  2655   fixes a b :: real
```
```  2656   assumes "gauge g"
```
```  2657   obtains p where "p tagged_division_of {a .. b}" "g fine p"
```
```  2658   by (metis assms box_real(2) fine_division_exists)
```
```  2659
```
```  2660 subsection \<open>Basic theorems about integrals.\<close>
```
```  2661
```
```  2662 lemma has_integral_unique:
```
```  2663   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2664   assumes "(f has_integral k1) i"
```
```  2665     and "(f has_integral k2) i"
```
```  2666   shows "k1 = k2"
```
```  2667 proof (rule ccontr)
```
```  2668   let ?e = "norm (k1 - k2) / 2"
```
```  2669   assume as: "k1 \<noteq> k2"
```
```  2670   then have e: "?e > 0"
```
```  2671     by auto
```
```  2672   have lem: False
```
```  2673     if f_k1: "(f has_integral k1) (cbox a b)"
```
```  2674     and f_k2: "(f has_integral k2) (cbox a b)"
```
```  2675     and "k1 \<noteq> k2"
```
```  2676     for f :: "'n \<Rightarrow> 'a" and a b k1 k2
```
```  2677   proof -
```
```  2678     let ?e = "norm (k1 - k2) / 2"
```
```  2679     from \<open>k1 \<noteq> k2\<close> have e: "?e > 0" by auto
```
```  2680     obtain d1 where d1:
```
```  2681         "gauge d1"
```
```  2682         "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
```
```  2683           d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
```
```  2684       by (rule has_integralD[OF f_k1 e]) blast
```
```  2685     obtain d2 where d2:
```
```  2686         "gauge d2"
```
```  2687         "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
```
```  2688           d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
```
```  2689       by (rule has_integralD[OF f_k2 e]) blast
```
```  2690     obtain p where p:
```
```  2691         "p tagged_division_of cbox a b"
```
```  2692         "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  2693       by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
```
```  2694     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  2695     have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
```
```  2696       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
```
```  2697       by (auto simp add:algebra_simps norm_minus_commute)
```
```  2698     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  2699       apply (rule add_strict_mono)
```
```  2700       apply (rule_tac[!] d2(2) d1(2))
```
```  2701       using p unfolding fine_def
```
```  2702       apply auto
```
```  2703       done
```
```  2704     finally show False by auto
```
```  2705   qed
```
```  2706   {
```
```  2707     presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
```
```  2708     then show False
```
```  2709       using as assms lem by blast
```
```  2710   }
```
```  2711   assume as: "\<not> (\<exists>a b. i = cbox a b)"
```
```  2712   obtain B1 where B1:
```
```  2713       "0 < B1"
```
```  2714       "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```  2715         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```  2716           norm (z - k1) < norm (k1 - k2) / 2"
```
```  2717     by (rule has_integral_altD[OF assms(1) as,OF e]) blast
```
```  2718   obtain B2 where B2:
```
```  2719       "0 < B2"
```
```  2720       "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
```
```  2721         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```  2722           norm (z - k2) < norm (k1 - k2) / 2"
```
```  2723     by (rule has_integral_altD[OF assms(2) as,OF e]) blast
```
```  2724   have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
```
```  2725     apply (rule bounded_subset_cbox)
```
```  2726     using bounded_Un bounded_ball
```
```  2727     apply auto
```
```  2728     done
```
```  2729   then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
```
```  2730     by blast
```
```  2731   obtain w where w:
```
```  2732     "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
```
```  2733     "norm (w - k1) < norm (k1 - k2) / 2"
```
```  2734     using B1(2)[OF ab(1)] by blast
```
```  2735   obtain z where z:
```
```  2736     "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
```
```  2737     "norm (z - k2) < norm (k1 - k2) / 2"
```
```  2738     using B2(2)[OF ab(2)] by blast
```
```  2739   have "z = w"
```
```  2740     using lem[OF w(1) z(1)] by auto
```
```  2741   then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
```
```  2742     using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
```
```  2743     by (auto simp add: norm_minus_commute)
```
```  2744   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  2745     apply (rule add_strict_mono)
```
```  2746     apply (rule_tac[!] z(2) w(2))
```
```  2747     done
```
```  2748   finally show False by auto
```
```  2749 qed
```
```  2750
```
```  2751 lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
```
```  2752   unfolding integral_def
```
```  2753   by (rule some_equality) (auto intro: has_integral_unique)
```
```  2754
```
```  2755 lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> ~ f integrable_on k \<and> y=0"
```
```  2756   unfolding integral_def integrable_on_def
```
```  2757   apply (erule subst)
```
```  2758   apply (rule someI_ex)
```
```  2759   by blast
```
```  2760
```
```  2761 lemma has_integral_is_0:
```
```  2762   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2763   assumes "\<forall>x\<in>s. f x = 0"
```
```  2764   shows "(f has_integral 0) s"
```
```  2765 proof -
```
```  2766   have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
```
```  2767     (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```  2768     unfolding has_integral
```
```  2769   proof clarify
```
```  2770     fix a b e
```
```  2771     fix f :: "'n \<Rightarrow> 'a"
```
```  2772     assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
```
```  2773     have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  2774       if p: "p tagged_division_of cbox a b" for p
```
```  2775     proof -
```
```  2776       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
```
```  2777       proof (rule setsum.neutral, rule)
```
```  2778         fix x
```
```  2779         assume x: "x \<in> p"
```
```  2780         have "f (fst x) = 0"
```
```  2781           using tagged_division_ofD(2-3)[OF p, of "fst x" "snd x"] using as x by auto
```
```  2782         then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
```
```  2783           apply (subst surjective_pairing[of x])
```
```  2784           unfolding split_conv
```
```  2785           apply auto
```
```  2786           done
```
```  2787       qed
```
```  2788       then show ?thesis
```
```  2789         using as by auto
```
```  2790     qed
```
```  2791     then show "\<exists>d. gauge d \<and>
```
```  2792         (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
```
```  2793       by auto
```
```  2794   qed
```
```  2795   {
```
```  2796     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  2797     with assms lem show ?thesis
```
```  2798       by blast
```
```  2799   }
```
```  2800   have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
```
```  2801     apply (rule ext)
```
```  2802     using assms
```
```  2803     apply auto
```
```  2804     done
```
```  2805   assume "\<not> (\<exists>a b. s = cbox a b)"
```
```  2806   then show ?thesis
```
```  2807     using lem
```
```  2808     by (subst has_integral_alt) (force simp add: *)
```
```  2809 qed
```
```  2810
```
```  2811 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
```
```  2812   by (rule has_integral_is_0) auto
```
```  2813
```
```  2814 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
```
```  2815   using has_integral_unique[OF has_integral_0] by auto
```
```  2816
```
```  2817 lemma has_integral_linear:
```
```  2818   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2819   assumes "(f has_integral y) s"
```
```  2820     and "bounded_linear h"
```
```  2821   shows "((h \<circ> f) has_integral ((h y))) s"
```
```  2822 proof -
```
```  2823   interpret bounded_linear h
```
```  2824     using assms(2) .
```
```  2825   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
```
```  2826     by blast
```
```  2827   have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
```
```  2828     (f has_integral y) (cbox a b) \<Longrightarrow> ((h \<circ> f) has_integral h y) (cbox a b)"
```
```  2829     unfolding has_integral
```
```  2830   proof (clarify, goal_cases)
```
```  2831     case prems: (1 f y a b e)
```
```  2832     from pos_bounded
```
```  2833     obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
```
```  2834       by blast
```
```  2835     have "e / B > 0" using prems(2) B by simp
```
```  2836     then obtain g
```
```  2837       where g: "gauge g"
```
```  2838                "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
```
```  2839                     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
```
```  2840         using prems(1) by auto
```
```  2841     {
```
```  2842       fix p
```
```  2843       assume as: "p tagged_division_of (cbox a b)" "g fine p"
```
```  2844       have hc: "\<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"
```
```  2845         by auto
```
```  2846       then 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"
```
```  2847         unfolding o_def unfolding scaleR[symmetric] hc by simp
```
```  2848       also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  2849         using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
```
```  2850       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)" .
```
```  2851       then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
```
```  2852         apply (simp add: diff[symmetric])
```
```  2853         apply (rule le_less_trans[OF B(2)])
```
```  2854         using g(2)[OF as] B(1)
```
```  2855         apply (auto simp add: field_simps)
```
```  2856         done
```
```  2857     }
```
```  2858     with g show ?case
```
```  2859       by (rule_tac x=g in exI) auto
```
```  2860   qed
```
```  2861   {
```
```  2862     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  2863     then show ?thesis
```
```  2864       using assms(1) lem by blast
```
```  2865   }
```
```  2866   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```  2867   then show ?thesis
```
```  2868   proof (subst has_integral_alt, clarsimp)
```
```  2869     fix e :: real
```
```  2870     assume e: "e > 0"
```
```  2871     have *: "0 < e/B" using e B(1) by simp
```
```  2872     obtain M where M:
```
```  2873       "M > 0"
```
```  2874       "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
```
```  2875         \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e / B"
```
```  2876       using has_integral_altD[OF assms(1) as *] by blast
```
```  2877     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2878       (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) (cbox a b) \<and> norm (z - h y) < e)"
```
```  2879     proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
```
```  2880       case prems: (1 a b)
```
```  2881       obtain z where z:
```
```  2882         "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
```
```  2883         "norm (z - y) < e / B"
```
```  2884         using M(2)[OF prems(1)] by blast
```
```  2885       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)"
```
```  2886         using zero by auto
```
```  2887       show ?case
```
```  2888         apply (rule_tac x="h z" in exI)
```
```  2889         apply (simp add: * lem z(1))
```
```  2890         apply (metis B diff le_less_trans pos_less_divide_eq z(2))
```
```  2891         done
```
```  2892     qed
```
```  2893   qed
```
```  2894 qed
```
```  2895
```
```  2896 lemma has_integral_scaleR_left:
```
```  2897   "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
```
```  2898   using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
```
```  2899
```
```  2900 lemma has_integral_mult_left:
```
```  2901   fixes c :: "_ :: real_normed_algebra"
```
```  2902   shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
```
```  2903   using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
```
```  2904
```
```  2905 text\<open>The case analysis eliminates the condition @{term "f integrable_on s"} at the cost
```
```  2906      of the type class constraint \<open>division_ring\<close>\<close>
```
```  2907 corollary integral_mult_left [simp]:
```
```  2908   fixes c:: "'a::{real_normed_algebra,division_ring}"
```
```  2909   shows "integral s (\<lambda>x. f x * c) = integral s f * c"
```
```  2910 proof (cases "f integrable_on s \<or> c = 0")
```
```  2911   case True then show ?thesis
```
```  2912     by (force intro: has_integral_mult_left)
```
```  2913 next
```
```  2914   case False then have "~ (\<lambda>x. f x * c) integrable_on s"
```
```  2915     using has_integral_mult_left [of "(\<lambda>x. f x * c)" _ s "inverse c"]
```
```  2916     by (force simp add: mult.assoc)
```
```  2917   with False show ?thesis by (simp add: not_integrable_integral)
```
```  2918 qed
```
```  2919
```
```  2920 corollary integral_mult_right [simp]:
```
```  2921   fixes c:: "'a::{real_normed_field}"
```
```  2922   shows "integral s (\<lambda>x. c * f x) = c * integral s f"
```
```  2923 by (simp add: mult.commute [of c])
```
```  2924
```
```  2925 corollary integral_divide [simp]:
```
```  2926   fixes z :: "'a::real_normed_field"
```
```  2927   shows "integral S (\<lambda>x. f x / z) = integral S (\<lambda>x. f x) / z"
```
```  2928 using integral_mult_left [of S f "inverse z"]
```
```  2929   by (simp add: divide_inverse_commute)
```
```  2930
```
```  2931 lemma has_integral_mult_right:
```
```  2932   fixes c :: "'a :: real_normed_algebra"
```
```  2933   shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
```
```  2934   using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
```
```  2935
```
```  2936 lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
```
```  2937   unfolding o_def[symmetric]
```
```  2938   by (metis has_integral_linear bounded_linear_scaleR_right)
```
```  2939
```
```  2940 lemma has_integral_cmult_real:
```
```  2941   fixes c :: real
```
```  2942   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
```
```  2943   shows "((\<lambda>x. c * f x) has_integral c * x) A"
```
```  2944 proof (cases "c = 0")
```
```  2945   case True
```
```  2946   then show ?thesis by simp
```
```  2947 next
```
```  2948   case False
```
```  2949   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
```
```  2950     unfolding real_scaleR_def .
```
```  2951 qed
```
```  2952
```
```  2953 lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral -k) s"
```
```  2954   by (drule_tac c="-1" in has_integral_cmul) auto
```
```  2955
```
```  2956 lemma has_integral_add:
```
```  2957   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2958   assumes "(f has_integral k) s"
```
```  2959     and "(g has_integral l) s"
```
```  2960   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
```
```  2961 proof -
```
```  2962   have lem: "((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
```
```  2963     if f_k: "(f has_integral k) (cbox a b)"
```
```  2964     and g_l: "(g has_integral l) (cbox a b)"
```
```  2965     for f :: "'n \<Rightarrow> 'a" and g a b k l
```
```  2966     unfolding has_integral
```
```  2967   proof clarify
```
```  2968     fix e :: real
```
```  2969     assume e: "e > 0"
```
```  2970     then have *: "e / 2 > 0"
```
```  2971       by auto
```
```  2972     obtain d1 where d1:
```
```  2973       "gauge d1"
```
```  2974       "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d1 fine p \<Longrightarrow>
```
```  2975         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) < e / 2"
```
```  2976       using has_integralD[OF f_k *] by blast
```
```  2977     obtain d2 where d2:
```
```  2978       "gauge d2"
```
```  2979       "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d2 fine p \<Longrightarrow>
```
```  2980         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l) < e / 2"
```
```  2981       using has_integralD[OF g_l *] by blast
```
```  2982     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  2983               norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
```
```  2984     proof (rule exI [where x="\<lambda>x. (d1 x) \<inter> (d2 x)"], clarsimp simp add: gauge_inter[OF d1(1) d2(1)])
```
```  2985       fix p
```
```  2986       assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  2987       have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) =
```
```  2988         (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
```
```  2989         unfolding scaleR_right_distrib setsum.distrib[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,symmetric]
```
```  2990         by (rule setsum.cong) auto
```
```  2991       from as have fine: "d1 fine p" "d2 fine p"
```
```  2992         unfolding fine_inter by auto
```
```  2993       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) =
```
```  2994             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))"
```
```  2995         unfolding * by (auto simp add: algebra_simps)
```
```  2996       also have "\<dots> < e/2 + e/2"
```
```  2997         apply (rule le_less_trans[OF norm_triangle_ineq])
```
```  2998         using as d1 d2 fine
```
```  2999         apply (blast intro: add_strict_mono)
```
```  3000         done
```
```  3001       finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e"
```
```  3002         by auto
```
```  3003     qed
```
```  3004   qed
```
```  3005   {
```
```  3006     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  3007     then show ?thesis
```
```  3008       using assms lem by force
```
```  3009   }
```
```  3010   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```  3011   then show ?thesis
```
```  3012   proof (subst has_integral_alt, clarsimp, goal_cases)
```
```  3013     case (1 e)
```
```  3014     then have *: "e / 2 > 0"
```
```  3015       by auto
```
```  3016     from has_integral_altD[OF assms(1) as *]
```
```  3017     obtain B1 where B1:
```
```  3018         "0 < B1"
```
```  3019         "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```  3020           \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - k) < e / 2"
```
```  3021       by blast
```
```  3022     from has_integral_altD[OF assms(2) as *]
```
```  3023     obtain B2 where B2:
```
```  3024         "0 < B2"
```
```  3025         "\<And>a b. ball 0 B2 \<subseteq> (cbox a b) \<Longrightarrow>
```
```  3026           \<exists>z. ((\<lambda>x. if x \<in> s then g x else 0) has_integral z) (cbox a b) \<and> norm (z - l) < e / 2"
```
```  3027       by blast
```
```  3028     show ?case
```
```  3029     proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
```
```  3030       fix a b
```
```  3031       assume "ball 0 (max B1 B2) \<subseteq> cbox a (b::'n)"
```
```  3032       then have *: "ball 0 B1 \<subseteq> cbox a (b::'n)" "ball 0 B2 \<subseteq> cbox a (b::'n)"
```
```  3033         by auto
```
```  3034       obtain w where w:
```
```  3035         "((\<lambda>x. if x \<in> s then f x else 0) has_integral w) (cbox a b)"
```
```  3036         "norm (w - k) < e / 2"
```
```  3037         using B1(2)[OF *(1)] by blast
```
```  3038       obtain z where z:
```
```  3039         "((\<lambda>x. if x \<in> s then g x else 0) has_integral z) (cbox a b)"
```
```  3040         "norm (z - l) < e / 2"
```
```  3041         using B2(2)[OF *(2)] by blast
```
```  3042       have *: "\<And>x. (if x \<in> s then f x + g x else 0) =
```
```  3043         (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)"
```
```  3044         by auto
```
```  3045       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) (cbox a b) \<and> norm (z - (k + l)) < e"
```
```  3046         apply (rule_tac x="w + z" in exI)
```
```  3047         apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
```
```  3048         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
```
```  3049         apply (auto simp add: field_simps)
```
```  3050         done
```
```  3051     qed
```
```  3052   qed
```
```  3053 qed
```
```  3054
```
```  3055 lemma has_integral_sub:
```
```  3056   "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow>
```
```  3057     ((\<lambda>x. f x - g x) has_integral (k - l)) s"
```
```  3058   using has_integral_add[OF _ has_integral_neg, of f k s g l]
```
```  3059   by (auto simp: algebra_simps)
```
```  3060
```
```  3061 lemma integral_0 [simp]:
```
```  3062   "integral s (\<lambda>x::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
```
```  3063   by (rule integral_unique has_integral_0)+
```
```  3064
```
```  3065 lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  3066     integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
```
```  3067   by (rule integral_unique) (metis integrable_integral has_integral_add)
```
```  3068
```
```  3069 lemma integral_cmul [simp]: "integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
```
```  3070 proof (cases "f integrable_on s \<or> c = 0")
```
```  3071   case True with has_integral_cmul show ?thesis by force
```
```  3072 next
```
```  3073   case False then have "~ (\<lambda>x. c *\<^sub>R f x) integrable_on s"
```
```  3074     using has_integral_cmul [of "(\<lambda>x. c *\<^sub>R f x)" _ s "inverse c"]
```
```  3075     by force
```
```  3076   with False show ?thesis by (simp add: not_integrable_integral)
```
```  3077 qed
```
```  3078
```
```  3079 lemma integral_neg [simp]: "integral s (\<lambda>x. - f x) = - integral s f"
```
```  3080 proof (cases "f integrable_on s")
```
```  3081   case True then show ?thesis
```
```  3082     by (simp add: has_integral_neg integrable_integral integral_unique)
```
```  3083 next
```
```  3084   case False then have "~ (\<lambda>x. - f x) integrable_on s"
```
```  3085     using has_integral_neg [of "(\<lambda>x. - f x)" _ s ]
```
```  3086     by force
```
```  3087   with False show ?thesis by (simp add: not_integrable_integral)
```
```  3088 qed
```
```  3089
```
```  3090 lemma integral_diff: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  3091     integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
```
```  3092   by (rule integral_unique) (metis integrable_integral has_integral_sub)
```
```  3093
```
```  3094 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
```
```  3095   unfolding integrable_on_def using has_integral_0 by auto
```
```  3096
```
```  3097 lemma integrable_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
```
```  3098   unfolding integrable_on_def by(auto intro: has_integral_add)
```
```  3099
```
```  3100 lemma integrable_cmul: "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
```
```  3101   unfolding integrable_on_def by(auto intro: has_integral_cmul)
```
```  3102
```
```  3103 lemma integrable_on_cmult_iff:
```
```  3104   fixes c :: real
```
```  3105   assumes "c \<noteq> 0"
```
```  3106   shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  3107   using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] \<open>c \<noteq> 0\<close>
```
```  3108   by auto
```
```  3109
```
```  3110 lemma integrable_on_cmult_left:
```
```  3111   assumes "f integrable_on s"
```
```  3112   shows "(\<lambda>x. of_real c * f x) integrable_on s"
```
```  3113     using integrable_cmul[of f s "of_real c"] assms
```
```  3114     by (simp add: scaleR_conv_of_real)
```
```  3115
```
```  3116 lemma integrable_neg: "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
```
```  3117   unfolding integrable_on_def by(auto intro: has_integral_neg)
```
```  3118
```
```  3119 lemma integrable_diff:
```
```  3120   "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
```
```  3121   unfolding integrable_on_def by(auto intro: has_integral_sub)
```
```  3122
```
```  3123 lemma integrable_linear:
```
```  3124   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on s"
```
```  3125   unfolding integrable_on_def by(auto intro: has_integral_linear)
```
```  3126
```
```  3127 lemma integral_linear:
```
```  3128   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
```
```  3129   apply (rule has_integral_unique [where i=s and f = "h \<circ> f"])
```
```  3130   apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
```
```  3131   done
```
```  3132
```
```  3133 lemma integral_component_eq[simp]:
```
```  3134   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  3135   assumes "f integrable_on s"
```
```  3136   shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
```
```  3137   unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
```
```  3138
```
```  3139 lemma has_integral_setsum:
```
```  3140   assumes "finite t"
```
```  3141     and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
```
```  3142   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
```
```  3143   using assms(1) subset_refl[of t]
```
```  3144 proof (induct rule: finite_subset_induct)
```
```  3145   case empty
```
```  3146   then show ?case by auto
```
```  3147 next
```
```  3148   case (insert x F)
```
```  3149   with assms show ?case
```
```  3150     by (simp add: has_integral_add)
```
```  3151 qed
```
```  3152
```
```  3153 lemma integral_setsum:
```
```  3154   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow>
```
```  3155    integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
```
```  3156   by (auto intro: has_integral_setsum integrable_integral)
```
```  3157
```
```  3158 lemma integrable_setsum:
```
```  3159   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"
```
```  3160   unfolding integrable_on_def
```
```  3161   apply (drule bchoice)
```
```  3162   using has_integral_setsum[of t]
```
```  3163   apply auto
```
```  3164   done
```
```  3165
```
```  3166 lemma has_integral_eq:
```
```  3167   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  3168     and "(f has_integral k) s"
```
```  3169   shows "(g has_integral k) s"
```
```  3170   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
```
```  3171   using has_integral_is_0[of s "\<lambda>x. f x - g x"]
```
```  3172   using assms(1)
```
```  3173   by auto
```
```  3174
```
```  3175 lemma integrable_eq: "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
```
```  3176   unfolding integrable_on_def
```
```  3177   using has_integral_eq[of s f g] has_integral_eq by blast
```
```  3178
```
```  3179 lemma has_integral_cong:
```
```  3180   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  3181   shows "(f has_integral i) s = (g has_integral i) s"
```
```  3182   using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
```
```  3183   by auto
```
```  3184
```
```  3185 lemma integral_cong:
```
```  3186   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  3187   shows "integral s f = integral s g"
```
```  3188   unfolding integral_def
```
```  3189 by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
```
```  3190
```
```  3191 lemma integrable_on_cmult_left_iff [simp]:
```
```  3192   assumes "c \<noteq> 0"
```
```  3193   shows "(\<lambda>x. of_real c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  3194         (is "?lhs = ?rhs")
```
```  3195 proof
```
```  3196   assume ?lhs
```
```  3197   then have "(\<lambda>x. of_real (1 / c) * (of_real c * f x)) integrable_on s"
```
```  3198     using integrable_cmul[of "\<lambda>x. of_real c * f x" s "1 / of_real c"]
```
```  3199     by (simp add: scaleR_conv_of_real)
```
```  3200   then have "(\<lambda>x. (of_real (1 / c) * of_real c * f x)) integrable_on s"
```
```  3201     by (simp add: algebra_simps)
```
```  3202   with \<open>c \<noteq> 0\<close> show ?rhs
```
```  3203     by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
```
```  3204 qed (blast intro: integrable_on_cmult_left)
```
```  3205
```
```  3206 lemma integrable_on_cmult_right:
```
```  3207   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```  3208   assumes "f integrable_on s"
```
```  3209   shows "(\<lambda>x. f x * of_real c) integrable_on s"
```
```  3210 using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
```
```  3211
```
```  3212 lemma integrable_on_cmult_right_iff [simp]:
```
```  3213   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```  3214   assumes "c \<noteq> 0"
```
```  3215   shows "(\<lambda>x. f x * of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  3216 using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
```
```  3217
```
```  3218 lemma integrable_on_cdivide:
```
```  3219   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```  3220   assumes "f integrable_on s"
```
```  3221   shows "(\<lambda>x. f x / of_real c) integrable_on s"
```
```  3222 by (simp add: integrable_on_cmult_right divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```  3223
```
```  3224 lemma integrable_on_cdivide_iff [simp]:
```
```  3225   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```  3226   assumes "c \<noteq> 0"
```
```  3227   shows "(\<lambda>x. f x / of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  3228 by (simp add: divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```  3229
```
```  3230 lemma has_integral_null [intro]:
```
```  3231   assumes "content(cbox a b) = 0"
```
```  3232   shows "(f has_integral 0) (cbox a b)"
```
```  3233 proof -
```
```  3234   have "gauge (\<lambda>x. ball x 1)"
```
```  3235     by auto
```
```  3236   moreover
```
```  3237   {
```
```  3238     fix e :: real
```
```  3239     fix p
```
```  3240     assume e: "e > 0"
```
```  3241     assume p: "p tagged_division_of (cbox a b)"
```
```  3242     have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0"
```
```  3243       unfolding norm_eq_zero diff_0_right
```
```  3244       using setsum_content_null[OF assms(1) p, of f] .
```
```  3245     then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  3246       using e by auto
```
```  3247   }
```
```  3248   ultimately show ?thesis
```
```  3249     by (auto simp: has_integral)
```
```  3250 qed
```
```  3251
```
```  3252 lemma has_integral_null_real [intro]:
```
```  3253   assumes "content {a .. b::real} = 0"
```
```  3254   shows "(f has_integral 0) {a .. b}"
```
```  3255   by (metis assms box_real(2) has_integral_null)
```
```  3256
```
```  3257 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
```
```  3258   by (auto simp add: has_integral_null dest!: integral_unique)
```
```  3259
```
```  3260 lemma integral_null [simp]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
```
```  3261   by (metis has_integral_null integral_unique)
```
```  3262
```
```  3263 lemma integrable_on_null [intro]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
```
```  3264   by (simp add: has_integral_integrable)
```
```  3265
```
```  3266 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
```
```  3267   by (simp add: has_integral_is_0)
```
```  3268
```
```  3269 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
```
```  3270   by (auto simp add: has_integral_empty has_integral_unique)
```
```  3271
```
```  3272 lemma integrable_on_empty[intro]: "f integrable_on {}"
```
```  3273   unfolding integrable_on_def by auto
```
```  3274
```
```  3275 lemma integral_empty[simp]: "integral {} f = 0"
```
```  3276   by (rule integral_unique) (rule has_integral_empty)
```
```  3277
```
```  3278 lemma has_integral_refl[intro]:
```
```  3279   fixes a :: "'a::euclidean_space"
```
```  3280   shows "(f has_integral 0) (cbox a a)"
```
```  3281     and "(f has_integral 0) {a}"
```
```  3282 proof -
```
```  3283   have *: "{a} = cbox a a"
```
```  3284     apply (rule set_eqI)
```
```  3285     unfolding mem_box singleton_iff euclidean_eq_iff[where 'a='a]
```
```  3286     apply safe
```
```  3287     prefer 3
```
```  3288     apply (erule_tac x=b in ballE)
```
```  3289     apply (auto simp add: field_simps)
```
```  3290     done
```
```  3291   show "(f has_integral 0) (cbox a a)" "(f has_integral 0) {a}"
```
```  3292     unfolding *
```
```  3293     apply (rule_tac[!] has_integral_null)
```
```  3294     unfolding content_eq_0_interior
```
```  3295     unfolding interior_cbox
```
```  3296     using box_sing
```
```  3297     apply auto
```
```  3298     done
```
```  3299 qed
```
```  3300
```
```  3301 lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
```
```  3302   unfolding integrable_on_def by auto
```
```  3303
```
```  3304 lemma integral_refl [simp]: "integral (cbox a a) f = 0"
```
```  3305   by (rule integral_unique) auto
```
```  3306
```
```  3307 lemma integral_singleton [simp]: "integral {a} f = 0"
```
```  3308   by auto
```
```  3309
```
```  3310 lemma integral_blinfun_apply:
```
```  3311   assumes "f integrable_on s"
```
```  3312   shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
```
```  3313   by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
```
```  3314
```
```  3315 lemma blinfun_apply_integral:
```
```  3316   assumes "f integrable_on s"
```
```  3317   shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
```
```  3318   by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
```
```  3319
```
```  3320 lemma has_integral_componentwise_iff:
```
```  3321   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3322   shows "(f has_integral y) A \<longleftrightarrow> (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3323 proof safe
```
```  3324   fix b :: 'b assume "(f has_integral y) A"
```
```  3325   from has_integral_linear[OF this(1) bounded_linear_component, of b]
```
```  3326     show "((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A" by (simp add: o_def)
```
```  3327 next
```
```  3328   assume "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3329   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral ((y \<bullet> b) *\<^sub>R b)) A"
```
```  3330     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```  3331   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. (y \<bullet> b) *\<^sub>R b)) A"
```
```  3332     by (intro has_integral_setsum) (simp_all add: o_def)
```
```  3333   thus "(f has_integral y) A" by (simp add: euclidean_representation)
```
```  3334 qed
```
```  3335
```
```  3336 lemma has_integral_componentwise:
```
```  3337   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3338   shows "(\<And>b. b \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A) \<Longrightarrow> (f has_integral y) A"
```
```  3339   by (subst has_integral_componentwise_iff) blast
```
```  3340
```
```  3341 lemma integrable_componentwise_iff:
```
```  3342   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3343   shows "f integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```  3344 proof
```
```  3345   assume "f integrable_on A"
```
```  3346   then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def)
```
```  3347   hence "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3348     by (subst (asm) has_integral_componentwise_iff)
```
```  3349   thus "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)" by (auto simp: integrable_on_def)
```
```  3350 next
```
```  3351   assume "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```  3352   then obtain y where "\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y b) A"
```
```  3353     unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
```
```  3354   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral (y b *\<^sub>R b)) A"
```
```  3355     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```  3356   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. y b *\<^sub>R b)) A"
```
```  3357     by (intro has_integral_setsum) (simp_all add: o_def)
```
```  3358   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
```
```  3359 qed
```
```  3360
```
```  3361 lemma integrable_componentwise:
```
```  3362   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3363   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A) \<Longrightarrow> f integrable_on A"
```
```  3364   by (subst integrable_componentwise_iff) blast
```
```  3365
```
```  3366 lemma integral_componentwise:
```
```  3367   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3368   assumes "f integrable_on A"
```
```  3369   shows "integral A f = (\<Sum>b\<in>Basis. integral A (\<lambda>x. (f x \<bullet> b) *\<^sub>R b))"
```
```  3370 proof -
```
```  3371   from assms have integrable: "\<forall>b\<in>Basis. (\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. (f x \<bullet> b)) integrable_on A"
```
```  3372     by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI)
```
```  3373        (simp_all add: bounded_linear_scaleR_left)
```
```  3374   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
```
```  3375     by (simp add: euclidean_representation)
```
```  3376   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
```
```  3377     by (subst integral_setsum) (simp_all add: o_def)
```
```  3378   finally show ?thesis .
```
```  3379 qed
```
```  3380
```
```  3381 lemma integrable_component:
```
```  3382   "f integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (y :: 'b :: euclidean_space)) integrable_on A"
```
```  3383   by (drule integrable_linear[OF _ bounded_linear_component[of y]]) (simp add: o_def)
```
```  3384
```
```  3385
```
```  3386
```
```  3387 subsection \<open>Cauchy-type criterion for integrability.\<close>
```
```  3388
```
```  3389 (* XXXXXXX *)
```
```  3390 lemma integrable_cauchy:
```
```  3391   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
```
```  3392   shows "f integrable_on cbox a b \<longleftrightarrow>
```
```  3393     (\<forall>e>0.\<exists>d. gauge d \<and>
```
```  3394       (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<and> d fine p1 \<and>
```
```  3395         p2 tagged_division_of (cbox a b) \<and> d fine p2 \<longrightarrow>
```
```  3396         norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
```
```  3397         setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))"
```
```  3398   (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
```
```  3399 proof
```
```  3400   assume ?l
```
```  3401   then guess y unfolding integrable_on_def has_integral .. note y=this
```
```  3402   show "\<forall>e>0. \<exists>d. ?P e d"
```
```  3403   proof (clarify, goal_cases)
```
```  3404     case (1 e)
```
```  3405     then have "e/2 > 0" by auto
```
```  3406     then guess d
```
```  3407       apply -
```
```  3408       apply (drule y[rule_format])
```
```  3409       apply (elim exE conjE)
```
```  3410       done
```
```  3411     note d=this[rule_format]
```
```  3412     show ?case
```
```  3413     proof (rule_tac x=d in exI, clarsimp simp: d)
```
```  3414       fix p1 p2
```
```  3415       assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
```
```  3416                  "p2 tagged_division_of (cbox a b)" "d fine p2"
```
```  3417       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"
```
```  3418         apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
```
```  3419         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
```
```  3420     qed
```
```  3421   qed
```
```  3422 next
```
```  3423   assume "\<forall>e>0. \<exists>d. ?P e d"
```
```  3424   then have "\<forall>n::nat. \<exists>d. ?P (inverse(of_nat (n + 1))) d"
```
```  3425     by auto
```
```  3426   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  3427   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})"
```
```  3428     apply (rule gauge_inters)
```
```  3429     using d(1)
```
```  3430     apply auto
```
```  3431     done
```
```  3432   then have "\<forall>n. \<exists>p. p tagged_division_of (cbox a b) \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p"
```
```  3433     by (meson fine_division_exists)
```
```  3434   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
```
```  3435   have dp: "\<And>i n. i\<le>n \<Longrightarrow> d i fine p n"
```
```  3436     using p(2) unfolding fine_inters by auto
```
```  3437   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
```
```  3438   proof (rule CauchyI, goal_cases)
```
```  3439     case (1 e)
```
```  3440     then guess N unfolding real_arch_inverse[of e] .. note N=this
```
```  3441     show ?case
```
```  3442       apply (rule_tac x=N in exI)
```
```  3443     proof clarify
```
```  3444       fix m n
```
```  3445       assume mn: "N \<le> m" "N \<le> n"
```
```  3446       have *: "N = (N - 1) + 1" using N by auto
```
```  3447       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"
```
```  3448         apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
```
```  3449         apply(subst *)
```
```  3450         using dp p(1) mn d(2) by auto
```
```  3451     qed
```
```  3452   qed
```
```  3453   then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
```
```  3454   show ?l
```
```  3455     unfolding integrable_on_def has_integral
```
```  3456   proof (rule_tac x=y in exI, clarify)
```
```  3457     fix e :: real
```
```  3458     assume "e>0"
```
```  3459     then have *:"e/2 > 0" by auto
```
```  3460     then guess N1 unfolding real_arch_inverse[of "e/2"] .. note N1=this
```
```  3461     then have N1': "N1 = N1 - 1 + 1"
```
```  3462       by auto
```
```  3463     guess N2 using y[OF *] .. note N2=this
```
```  3464     have "gauge (d (N1 + N2))"
```
```  3465       using d by auto
```
```  3466     moreover
```
```  3467     {
```
```  3468       fix q
```
```  3469       assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
```
```  3470       have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
```
```  3471         apply (rule less_trans)
```
```  3472         using N1
```
```  3473         apply auto
```
```  3474         done
```
```  3475       have "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e"
```
```  3476         apply (rule norm_triangle_half_r)
```
```  3477         apply (rule less_trans[OF _ *])
```
```  3478         apply (subst N1', rule d(2)[of "p (N1+N2)"])
```
```  3479         using N1' as(1) as(2) dp
```
```  3480         apply (simp add: \<open>\<forall>x. p x tagged_division_of cbox a b \<and> (\<lambda>xa. \<Inter>{d i xa |i. i \<in> {0..x}}) fine p x\<close>)
```
```  3481         using N2 le_add2 by blast
```
```  3482     }
```
```  3483     ultimately show "\<exists>d. gauge d \<and>
```
```  3484       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  3485         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
```
```  3486       by (rule_tac x="d (N1 + N2)" in exI) auto
```
```  3487   qed
```
```  3488 qed
```
```  3489
```
```  3490
```
```  3491 subsection \<open>Additivity of integral on abutting intervals.\<close>
```
```  3492
```
```  3493 lemma tagged_division_split_left_inj:
```
```  3494   fixes x1 :: "'a::euclidean_space"
```
```  3495   assumes d: "d tagged_division_of i"
```
```  3496     and k12: "(x1, k1) \<in> d"
```
```  3497              "(x2, k2) \<in> d"
```
```  3498              "k1 \<noteq> k2"
```
```  3499              "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
```
```  3500              "k \<in> Basis"
```
```  3501   shows "content (k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
```
```  3502 proof -
```
```  3503   have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
```
```  3504     by force
```
```  3505   show ?thesis
```
```  3506     using k12
```
```  3507     by (fastforce intro!:  division_split_left_inj[OF division_of_tagged_division[OF d]] *)
```
```  3508 qed
```
```  3509
```
```  3510 lemma tagged_division_split_right_inj:
```
```  3511   fixes x1 :: "'a::euclidean_space"
```
```  3512   assumes d: "d tagged_division_of i"
```
```  3513     and k12: "(x1, k1) \<in> d"
```
```  3514              "(x2, k2) \<in> d"
```
```  3515              "k1 \<noteq> k2"
```
```  3516              "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
```
```  3517              "k \<in> Basis"
```
```  3518   shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
```
```  3519 proof -
```
```  3520   have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
```
```  3521     by force
```
```  3522   show ?thesis
```
```  3523     using k12
```
```  3524     by (fastforce intro!:  division_split_right_inj[OF division_of_tagged_division[OF d]] *)
```
```  3525 qed
```
```  3526
```
```  3527 lemma has_integral_split:
```
```  3528   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3529   assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
```
```  3530       and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3531       and k: "k \<in> Basis"
```
```  3532   shows "(f has_integral (i + j)) (cbox a b)"
```
```  3533 proof (unfold has_integral, rule, rule, goal_cases)
```
```  3534   case (1 e)
```
```  3535   then have e: "e/2 > 0"
```
```  3536     by auto
```
```  3537     obtain d1
```
```  3538     where d1: "gauge d1"
```
```  3539       and d1norm:
```
```  3540         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c};
```
```  3541                d1 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - i) < e / 2"
```
```  3542        apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
```
```  3543        apply (simp add: interval_split[symmetric] k)
```
```  3544        done
```
```  3545     obtain d2
```
```  3546     where d2: "gauge d2"
```
```  3547       and d2norm:
```
```  3548         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k};
```
```  3549                d2 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - j) < e / 2"
```
```  3550        apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
```
```  3551        apply (simp add: interval_split[symmetric] k)
```
```  3552        done
```
```  3553   let ?d = "\<lambda>x. if x\<bullet>k = c then (d1 x \<inter> d2 x) else ball x \<bar>x\<bullet>k - c\<bar> \<inter> d1 x \<inter> d2 x"
```
```  3554   have "gauge ?d"
```
```  3555     using d1 d2 unfolding gauge_def by auto
```
```  3556   then show ?case
```
```  3557   proof (rule_tac x="?d" in exI, safe)
```
```  3558     fix p
```
```  3559     assume "p tagged_division_of (cbox a b)" "?d fine p"
```
```  3560     note p = this tagged_division_ofD[OF this(1)]
```
```  3561     have xk_le_c: "\<And>x kk. (x, kk) \<in> p \<Longrightarrow> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {} \<Longrightarrow> x\<bullet>k \<le> c"
```
```  3562     proof -
```
```  3563       fix x kk
```
```  3564       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}"
```
```  3565       show "x\<bullet>k \<le> c"
```
```  3566       proof (rule ccontr)
```
```  3567         assume **: "\<not> ?thesis"
```
```  3568         from this[unfolded not_le]
```
```  3569         have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  3570           using p(2)[unfolded fine_def, rule_format,OF as] by auto
```
```  3571         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
```
```  3572           by blast
```
```  3573         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  3574           using Basis_le_norm[OF k, of "x - y"]
```
```  3575           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  3576         with y show False
```
```  3577           using ** by (auto simp add: field_simps)
```
```  3578       qed
```
```  3579     qed
```
```  3580     have xk_ge_c: "\<And>x kk. (x, kk) \<in> p \<Longrightarrow> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {} \<Longrightarrow> x\<bullet>k \<ge> c"
```
```  3581     proof -
```
```  3582       fix x kk
```
```  3583       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}"
```
```  3584       show "x\<bullet>k \<ge> c"
```
```  3585       proof (rule ccontr)
```
```  3586         assume **: "\<not> ?thesis"
```
```  3587         from this[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  3588           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  3589         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
```
```  3590           by blast
```
```  3591         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  3592           using Basis_le_norm[OF k, of "x - y"]
```
```  3593           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  3594         with y show False
```
```  3595           using ** by (auto simp add: field_simps)
```
```  3596       qed
```
```  3597     qed
```
```  3598
```
```  3599     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>
```
```  3600                          (\<forall>x k. P x k \<longrightarrow> Q x (f k))"
```
```  3601       by auto
```
```  3602     have fin_finite: "finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}" if "finite s" for f s P
```
```  3603     proof -
```
```  3604       from that have "finite ((\<lambda>(x, k). (x, f k)) ` s)"
```
```  3605         by auto
```
```  3606       then show ?thesis
```
```  3607         by (rule rev_finite_subset) auto
```
```  3608     qed
```
```  3609     { fix g :: "'a set \<Rightarrow> 'a set"
```
```  3610       fix i :: "'a \<times> 'a set"
```
```  3611       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  3612       then obtain x k where xk:
```
```  3613               "i = (x, g k)"  "(x, k) \<in> p"
```
```  3614               "(x, g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  3615           by auto
```
```  3616       have "content (g k) = 0"
```
```  3617         using xk using content_empty by auto
```
```  3618       then have "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0"
```
```  3619         unfolding xk split_conv by auto
```
```  3620     } note [simp] = this
```
```  3621     have lem3: "\<And>g :: 'a set \<Rightarrow> 'a set. finite p \<Longrightarrow>
```
```  3622                   setsum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
```
```  3623                   setsum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
```
```  3624       by (rule setsum.mono_neutral_left) auto
```
```  3625     let ?M1 = "{(x, kk \<inter> {x. x\<bullet>k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
```
```  3626     have d1_fine: "d1 fine ?M1"
```
```  3627       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  3628     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
```
```  3629     proof (rule d1norm [OF tagged_division_ofI d1_fine])
```
```  3630       show "finite ?M1"
```
```  3631         by (rule fin_finite p(3))+
```
```  3632       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
```
```  3633         unfolding p(8)[symmetric] by auto
```
```  3634       fix x l
```
```  3635       assume xl: "(x, l) \<in> ?M1"
```
```  3636       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  3637       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  3638         unfolding xl'
```
```  3639         using p(4-6)[OF xl'(3)] using xl'(4)
```
```  3640         using xk_le_c[OF xl'(3-4)] by auto
```
```  3641       show "\<exists>a b. l = cbox a b"
```
```  3642         unfolding xl'
```
```  3643         using p(6)[OF xl'(3)]
```
```  3644         by (fastforce simp add: interval_split[OF k,where c=c])
```
```  3645       fix y r
```
```  3646       let ?goal = "interior l \<inter> interior r = {}"
```
```  3647       assume yr: "(y, r) \<in> ?M1"
```
```  3648       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  3649       assume as: "(x, l) \<noteq> (y, r)"
```
```  3650       show "interior l \<inter> interior r = {}"
```
```  3651       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  3652         case False
```
```  3653         then show ?thesis
```
```  3654           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3655       next
```
```  3656         case True
```
```  3657         then have "l' \<noteq> r'"
```
```  3658           using as unfolding xl' yr' by auto
```
```  3659         then show ?thesis
```
```  3660           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3661       qed
```
```  3662     qed
```
```  3663     moreover
```
```  3664     let ?M2 = "{(x,kk \<inter> {x. x\<bullet>k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
```
```  3665     have d2_fine: "d2 fine ?M2"
```
```  3666       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  3667     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
```
```  3668     proof (rule d2norm [OF tagged_division_ofI d2_fine])
```
```  3669       show "finite ?M2"
```
```  3670         by (rule fin_finite p(3))+
```
```  3671       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
```
```  3672         unfolding p(8)[symmetric] by auto
```
```  3673       fix x l
```
```  3674       assume xl: "(x, l) \<in> ?M2"
```
```  3675       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  3676       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  3677         unfolding xl'
```
```  3678         using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
```
```  3679         by auto
```
```  3680       show "\<exists>a b. l = cbox a b"
```
```  3681         unfolding xl'
```
```  3682         using p(6)[OF xl'(3)]
```
```  3683         by (fastforce simp add: interval_split[OF k, where c=c])
```
```  3684       fix y r
```
```  3685       let ?goal = "interior l \<inter> interior r = {}"
```
```  3686       assume yr: "(y, r) \<in> ?M2"
```
```  3687       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  3688       assume as: "(x, l) \<noteq> (y, r)"
```
```  3689       show "interior l \<inter> interior r = {}"
```
```  3690       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  3691         case False
```
```  3692         then show ?thesis
```
```  3693           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3694       next
```
```  3695         case True
```
```  3696         then have "l' \<noteq> r'"
```
```  3697           using as unfolding xl' yr' by auto
```
```  3698         then show ?thesis
```
```  3699           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3700       qed
```
```  3701     qed
```
```  3702     ultimately
```
```  3703     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"
```
```  3704       using norm_add_less by blast
```
```  3705     also {
```
```  3706       have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
```
```  3707         using scaleR_zero_left by auto
```
```  3708       have cont_eq: "\<And>g. (\<lambda>(x,l). content l *\<^sub>R f x) \<circ> (\<lambda>(x,l). (x,g l)) = (\<lambda>(x,l). content (g l) *\<^sub>R f x)"
```
```  3709         by auto
```
```  3710       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) =
```
```  3711         (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
```
```  3712         by auto
```
```  3713       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
```
```  3714         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
```
```  3715         unfolding lem3[OF p(3)]
```
```  3716         by (subst setsum.reindex_nontrivial[OF p(3)], auto intro!: k eq0 tagged_division_split_left_inj[OF p(1)] tagged_division_split_right_inj[OF p(1)]
```
```  3717               simp: cont_eq)+
```
```  3718       also note setsum.distrib[symmetric]
```
```  3719       also have "\<And>x. x \<in> p \<Longrightarrow>
```
```  3720                     (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
```
```  3721                     (\<lambda>(x,ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x =
```
```  3722                     (\<lambda>(x,ka). content ka *\<^sub>R f x) x"
```
```  3723       proof clarify
```
```  3724         fix a b
```
```  3725         assume "(a, b) \<in> p"
```
```  3726         from p(6)[OF this] guess u v by (elim exE) note uv=this
```
```  3727         then show "content (b \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a =
```
```  3728           content b *\<^sub>R f a"
```
```  3729           unfolding scaleR_left_distrib[symmetric]
```
```  3730           unfolding uv content_split[OF k,of u v c]
```
```  3731           by auto
```
```  3732       qed
```
```  3733       note setsum.cong [OF _ this]
```
```  3734       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x \<bullet> k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +
```
```  3735         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x \<bullet> k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}. content k *\<^sub>R f x) - j) =
```
```  3736         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
```
```  3737         by auto
```
```  3738     }
```
```  3739     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
```
```  3740       by auto
```
```  3741   qed
```
```  3742 qed
```
```  3743
```
```  3744
```
```  3745 subsection \<open>A sort of converse, integrability on subintervals.\<close>
```
```  3746
```
```  3747 lemma tagged_division_union_interval:
```
```  3748   fixes a :: "'a::euclidean_space"
```
```  3749   assumes "p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> (c::real)})"
```
```  3750     and "p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3751     and k: "k \<in> Basis"
```
```  3752   shows "(p1 \<union> p2) tagged_division_of (cbox a b)"
```
```  3753 proof -
```
```  3754   have *: "cbox a b = (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<union> (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3755     by auto
```
```  3756   show ?thesis
```
```  3757     apply (subst *)
```
```  3758     apply (rule tagged_division_union[OF assms(1-2)])
```
```  3759     unfolding interval_split[OF k] interior_cbox
```
```  3760     using k
```
```  3761     apply (auto simp add: box_def elim!: ballE[where x=k])
```
```  3762     done
```
```  3763 qed
```
```  3764
```
```  3765 lemma tagged_division_union_interval_real:
```
```  3766   fixes a :: real
```
```  3767   assumes "p1 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<le> (c::real)})"
```
```  3768     and "p2 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3769     and k: "k \<in> Basis"
```
```  3770   shows "(p1 \<union> p2) tagged_division_of {a .. b}"
```
```  3771   using assms
```
```  3772   unfolding box_real[symmetric]
```
```  3773   by (rule tagged_division_union_interval)
```
```  3774
```
```  3775 lemma has_integral_separate_sides:
```
```  3776   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3777   assumes "(f has_integral i) (cbox a b)"
```
```  3778     and "e > 0"
```
```  3779     and k: "k \<in> Basis"
```
```  3780   obtains d where "gauge d"
```
```  3781     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
```
```  3782         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
```
```  3783         norm ((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
```
```  3784 proof -
```
```  3785   guess d using has_integralD[OF assms(1-2)] . note d=this
```
```  3786   { fix p1 p2
```
```  3787     assume "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  3788     note p1=tagged_division_ofD[OF this(1)] this
```
```  3789     assume "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" "d fine p2"
```
```  3790     note p2=tagged_division_ofD[OF this(1)] this
```
```  3791     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
```
```  3792     { fix a b
```
```  3793       assume ab: "(a, b) \<in> p1 \<inter> p2"
```
```  3794       have "(a, b) \<in> p1"
```
```  3795         using ab by auto
```
```  3796       with p1 obtain u v where uv: "b = cbox u v" by auto
```
```  3797       have "b \<subseteq> {x. x\<bullet>k = c}"
```
```  3798         using ab p1(3)[of a b] p2(3)[of a b] by fastforce
```
```  3799       moreover
```
```  3800       have "interior {x::'a. x \<bullet> k = c} = {}"
```
```  3801       proof (rule ccontr)
```
```  3802         assume "\<not> ?thesis"
```
```  3803         then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
```
```  3804           by auto
```
```  3805         then guess e unfolding mem_interior .. note e=this
```
```  3806         have x: "x\<bullet>k = c"
```
```  3807           using x interior_subset by fastforce
```
```  3808         have *: "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>(x - (x + (e / 2) *\<^sub>R k)) \<bullet> i\<bar> = (if i = k then e/2 else 0)"
```
```  3809           using e k by (auto simp: inner_simps inner_not_same_Basis)
```
```  3810         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
```
```  3811               (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))"
```
```  3812           using "*" by (blast intro: setsum.cong)
```
```  3813         also have "\<dots> < e"
```
```  3814           apply (subst setsum.delta)
```
```  3815           using e
```
```  3816           apply auto
```
```  3817           done
```
```  3818         finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
```
```  3819           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
```
```  3820         then have "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
```
```  3821           using e by auto
```
```  3822         then show False
```
```  3823           unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
```
```  3824       qed
```
```  3825       ultimately have "content b = 0"
```
```  3826         unfolding uv content_eq_0_interior
```
```  3827         using interior_mono by blast
```
```  3828       then have "content b *\<^sub>R f a = 0"
```
```  3829         by auto
```
```  3830     }
```
```  3831     then 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) =
```
```  3832                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
```
```  3833       by (subst setsum.union_inter_neutral) (auto simp: p1 p2)
```
```  3834     also have "\<dots> < e"
```
```  3835       by (rule k d(2) p12 fine_union p1 p2)+
```
```  3836     finally 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) < e" .
```
```  3837    }
```
```  3838   then show ?thesis
```
```  3839     by (auto intro: that[of d] d elim: )
```
```  3840 qed
```
```  3841
```
```  3842 lemma integrable_split[intro]:
```
```  3843   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
```
```  3844   assumes "f integrable_on cbox a b"
```
```  3845     and k: "k \<in> Basis"
```
```  3846   shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})" (is ?t1)
```
```  3847     and "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
```
```  3848 proof -
```
```  3849   guess y using assms(1) unfolding integrable_on_def .. note y=this
```
```  3850   define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
```
```  3851   define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
```
```  3852   show ?t1 ?t2
```
```  3853     unfolding interval_split[OF k] integrable_cauchy
```
```  3854     unfolding interval_split[symmetric,OF k]
```
```  3855   proof (rule_tac[!] allI impI)+
```
```  3856     fix e :: real
```
```  3857     assume "e > 0"
```
```  3858     then have "e/2>0"
```
```  3859       by auto
```
```  3860     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
```
```  3861     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<inter> A \<and> d fine p1 \<and>
```
```  3862       p2 tagged_division_of (cbox a b) \<inter> A \<and> d fine p2 \<longrightarrow>
```
```  3863       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
```
```  3864     show "?P {x. x \<bullet> k \<le> c}"
```
```  3865     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  3866       fix p1 p2
```
```  3867       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  3868                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
```
```  3869       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"
```
```  3870       proof (rule fine_division_exists[OF d(1), of a' b] )
```
```  3871         fix p
```
```  3872         assume "p tagged_division_of cbox a' b" "d fine p"
```
```  3873         then show ?thesis
```
```  3874           using as norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
```
```  3875           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  3876           by (auto simp add: algebra_simps)
```
```  3877       qed
```
```  3878     qed
```
```  3879     show "?P {x. x \<bullet> k \<ge> c}"
```
```  3880     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  3881       fix p1 p2
```
```  3882       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
```
```  3883                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
```
```  3884       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"
```
```  3885       proof (rule fine_division_exists[OF d(1), of a b'] )
```
```  3886         fix p
```
```  3887         assume "p tagged_division_of cbox a b'" "d fine p"
```
```  3888         then show ?thesis
```
```  3889           using as norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
```
```  3890           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  3891           by (auto simp add: algebra_simps)
```
```  3892       qed
```
```  3893     qed
```
```  3894   qed
```
```  3895 qed
```
```  3896
```
```  3897 lemma operative_integral:
```
```  3898   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  3899   shows "comm_monoid.operative (lift_option op +) (Some 0)
```
```  3900     (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
```
```  3901 proof -
```
```  3902   interpret comm_monoid "lift_option plus" "Some (0::'b)"
```
```  3903     by (rule comm_monoid_lift_option)
```
```  3904       (rule add.comm_monoid_axioms)
```
```  3905   show ?thesis
```
```  3906   proof (unfold operative_def, safe)
```
```  3907     fix a b c
```
```  3908     fix k :: 'a
```
```  3909     assume k: "k \<in> Basis"
```
```  3910     show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
```
```  3911           lift_option op + (if f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} then Some (integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f) else None)
```
```  3912           (if f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k} then Some (integral (cbox a b \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
```
```  3913     proof (cases "f integrable_on cbox a b")
```
```  3914       case True
```
```  3915       with k show ?thesis
```
```  3916         apply (simp add: integrable_split)
```
```  3917         apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
```
```  3918         apply (auto intro: integrable_integral)
```
```  3919         done
```
```  3920     next
```
```  3921     case False
```
```  3922       have "\<not> (f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<or> \<not> ( f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k})"
```
```  3923       proof (rule ccontr)
```
```  3924         assume "\<not> ?thesis"
```
```  3925         then have "f integrable_on cbox a b"
```
```  3926           unfolding integrable_on_def
```
```  3927           apply (rule_tac x="integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f + integral (cbox a b \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
```
```  3928           apply (rule has_integral_split[OF _ _ k])
```
```  3929           apply (auto intro: integrable_integral)
```
```  3930           done
```
```  3931         then show False
```
```  3932           using False by auto
```
```  3933       qed
```
```  3934       then show ?thesis
```
```  3935         using False by auto
```
```  3936     qed
```
```  3937   next
```
```  3938     fix a b :: 'a
```
```  3939     assume "content (cbox a b) = 0"
```
```  3940     then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
```
```  3941       using has_integral_null_eq
```
```  3942       by (auto simp: integrable_on_null)
```
```  3943   qed
```
```  3944 qed
```
```  3945
```
```  3946 subsection \<open>Finally, the integral of a constant\<close>
```
```  3947
```
```  3948 lemma has_integral_const [intro]:
```
```  3949   fixes a b :: "'a::euclidean_space"
```
```  3950   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
```
```  3951   apply (auto intro!: exI [where x="\<lambda>x. ball x 1"] simp: split_def has_integral)
```
```  3952   apply (subst scaleR_left.setsum[symmetric, unfolded o_def])
```
```  3953   apply (subst additive_content_tagged_division[unfolded split_def])
```
```  3954   apply auto
```
```  3955   done
```
```  3956
```
```  3957 lemma has_integral_const_real [intro]:
```
```  3958   fixes a b :: real
```
```  3959   shows "((\<lambda>x. c) has_integral (content {a .. b} *\<^sub>R c)) {a .. b}"
```
```  3960   by (metis box_real(2) has_integral_const)
```
```  3961
```
```  3962 lemma integral_const [simp]:
```
```  3963   fixes a b :: "'a::euclidean_space"
```
```  3964   shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
```
```  3965   by (rule integral_unique) (rule has_integral_const)
```
```  3966
```
```  3967 lemma integral_const_real [simp]:
```
```  3968   fixes a b :: real
```
```  3969   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
```
```  3970   by (metis box_real(2) integral_const)
```
```  3971
```
```  3972
```
```  3973 subsection \<open>Bounds on the norm of Riemann sums and the integral itself.\<close>
```
```  3974
```
```  3975 lemma dsum_bound:
```
```  3976   assumes "p division_of (cbox a b)"
```
```  3977     and "norm c \<le> e"
```
```  3978   shows "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
```
```  3979 proof -
```
```  3980   have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = setsum content p"
```
```  3981     apply (rule setsum.cong)
```
```  3982     using assms
```
```  3983     apply simp
```
```  3984     apply (metis abs_of_nonneg assms(1) content_pos_le division_ofD(4))
```
```  3985     done
```
```  3986   have e: "0 \<le> e"
```
```  3987     using assms(2) norm_ge_zero order_trans by blast
```
```  3988   have "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
```
```  3989     using norm_setsum by blast
```
```  3990   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
```
```  3991     apply (simp add: setsum_right_distrib[symmetric] mult.commute)
```
```  3992     using assms(2) mult_right_mono by blast
```
```  3993   also have "... \<le> e * content (cbox a b)"
```
```  3994     apply (rule mult_left_mono [OF _ e])
```
```  3995     apply (simp add: sumeq)
```
```  3996     using additive_content_division assms(1) eq_iff apply blast
```
```  3997     done
```
```  3998   finally show ?thesis .
```
```  3999 qed
```
```  4000
```
```  4001 lemma rsum_bound:
```
```  4002   assumes p: "p tagged_division_of (cbox a b)"
```
```  4003       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
```
```  4004     shows "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
```
```  4005 proof (cases "cbox a b = {}")
```
```  4006   case True show ?thesis
```
```  4007     using p unfolding True tagged_division_of_trivial by auto
```
```  4008 next
```
```  4009   case False
```
```  4010   then have e: "e \<ge> 0"
```
```  4011     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
```
```  4012   have setsum_le: "setsum (content \<circ> snd) p \<le> content (cbox a b)"
```
```  4013     unfolding additive_content_tagged_division[OF p, symmetric] split_def
```
```  4014     by (auto intro: eq_refl)
```
```  4015   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
```
```  4016     using tagged_division_ofD(4) [OF p] content_pos_le
```
```  4017     by force
```
```  4018   have norm: "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
```
```  4019     unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
```
```  4020     by (metis prod.collapse subset_eq)
```
```  4021   have "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> (\<Sum>i\<in>p. norm (case i of (x, k) \<Rightarrow> content k *\<^sub>R f x))"
```
```  4022     by (rule norm_setsum)
```
```  4023   also have "...  \<le> e * content (cbox a b)"
```
```  4024     unfolding split_def norm_scaleR
```
```  4025     apply (rule order_trans[OF setsum_mono])
```
```  4026     apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
```
```  4027     apply (metis norm)
```
```  4028     unfolding setsum_left_distrib[symmetric]
```
```  4029     using con setsum_le
```
```  4030     apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
```
```  4031     done
```
```  4032   finally show ?thesis .
```
```  4033 qed
```
```  4034
```
```  4035 lemma rsum_diff_bound:
```
```  4036   assumes "p tagged_division_of (cbox a b)"
```
```  4037     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
```
```  4038   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>
```
```  4039          e * content (cbox a b)"
```
```  4040   apply (rule order_trans[OF _ rsum_bound[OF assms]])
```
```  4041   apply (simp add: split_def scaleR_diff_right setsum_subtractf eq_refl)
```
```  4042   done
```
```  4043
```
```  4044 lemma has_integral_bound:
```
```  4045   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  4046   assumes "0 \<le> B"
```
```  4047       and "(f has_integral i) (cbox a b)"
```
```  4048       and "\<forall>x\<in>cbox a b. norm (f x) \<le> B"
```
```  4049     shows "norm i \<le> B * content (cbox a b)"
```
```  4050 proof (rule ccontr)
```
```  4051   assume "\<not> ?thesis"
```
```  4052   then have *: "norm i - B * content (cbox a b) > 0"
```
```  4053     by auto
```
```  4054   from assms(2)[unfolded has_integral,rule_format,OF *]
```
```  4055   guess d by (elim exE conjE) note d=this[rule_format]
```
```  4056   from fine_division_exists[OF this(1), of a b] guess p . note p=this
```
```  4057   have *: "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
```
```  4058     unfolding not_less
```
```  4059     by (metis norm_triangle_sub[of i] add.commute le_less_trans less_diff_eq linorder_not_le norm_minus_commute)
```
```  4060   show False
```
```  4061     using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
```
```  4062 qed
```
```  4063
```
```  4064 corollary has_integral_bound_real:
```
```  4065   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
```
```  4066   assumes "0 \<le> B"
```
```  4067       and "(f has_integral i) {a .. b}"
```
```  4068       and "\<forall>x\<in>{a .. b}. norm (f x) \<le> B"
```
```  4069     shows "norm i \<le> B * content {a .. b}"
```
```  4070   by (metis assms box_real(2) has_integral_bound)
```
```  4071
```
```  4072 corollary integrable_bound:
```
```  4073   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  4074   assumes "0 \<le> B"
```
```  4075       and "f integrable_on (cbox a b)"
```
```  4076       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
```
```  4077     shows "norm (integral (cbox a b) f) \<le> B * content (cbox a b)"
```
```  4078 by (metis integrable_integral has_integral_bound assms)
```
```  4079
```
```  4080
```
```  4081 subsection \<open>Similar theorems about relationship among components.\<close>
```
```  4082
```
```  4083 lemma rsum_component_le:
```
```  4084   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4085   assumes "p tagged_division_of (cbox a b)"
```
```  4086       and "\<forall>x\<in>cbox a b. (f x)\<bullet>i \<le> (g x)\<bullet>i"
```
```  4087     shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
```
```  4088 unfolding inner_setsum_left
```
```  4089 proof (rule setsum_mono, clarify)
```
```  4090   fix a b
```
```  4091   assume ab: "(a, b) \<in> p"
```
```  4092   note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
```
```  4093   from this(3) guess u v by (elim exE) note b=this
```
```  4094   show "(content b *\<^sub>R f a) \<bullet> i \<le> (content b *\<^sub>R g a) \<bullet> i"
```
```  4095     unfolding b inner_simps real_scaleR_def
```
```  4096     apply (rule mult_left_mono)
```
```  4097     using assms(2) tagged
```
```  4098     by (auto simp add: content_pos_le)
```
```  4099 qed
```
```  4100
```
```  4101 lemma has_integral_component_le:
```
```  4102   fixes f g :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4103   assumes k: "k \<in> Basis"
```
```  4104   assumes "(f has_integral i) s" "(g has_integral j) s"
```
```  4105     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  4106   shows "i\<bullet>k \<le> j\<bullet>k"
```
```  4107 proof -
```
```  4108   have lem: "i\<bullet>k \<le> j\<bullet>k"
```
```  4109     if f_i: "(f has_integral i) (cbox a b)"
```
```  4110     and g_j: "(g has_integral j) (cbox a b)"
```
```  4111     and le: "\<forall>x\<in>cbox a b. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  4112     for a b i and j :: 'b and f g :: "'a \<Rightarrow> 'b"
```
```  4113   proof (rule ccontr)
```
```  4114     assume "\<not> ?thesis"
```
```  4115     then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3"
```
```  4116       by auto
```
```  4117     guess d1 using f_i[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
```
```  4118     guess d2 using g_j[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
```
```  4119     obtain p where p: "p tagged_division_of cbox a b" "d1 fine p" "d2 fine p"
```
```  4120        using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter
```
```  4121        by metis
```
```  4122     note le_less_trans[OF Basis_le_norm[OF k]]
```
```  4123     then have "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
```
```  4124               "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
```
```  4125       using  k norm_bound_Basis_lt d1 d2 p
```
```  4126       by blast+
```
```  4127     then show False
```
```  4128       unfolding inner_simps
```
```  4129       using rsum_component_le[OF p(1) le]
```
```  4130       by (simp add: abs_real_def split: if_split_asm)
```
```  4131   qed
```
```  4132   show ?thesis
```
```  4133   proof (cases "\<exists>a b. s = cbox a b")
```
```  4134     case True
```
```  4135     with lem assms show ?thesis
```
```  4136       by auto
```
```  4137   next
```
```  4138     case False
```
```  4139     show ?thesis
```
```  4140     proof (rule ccontr)
```
```  4141       assume "\<not> i\<bullet>k \<le> j\<bullet>k"
```
```  4142       then have ij: "(i\<bullet>k - j\<bullet>k) / 3 > 0"
```
```  4143         by auto
```
```  4144       note has_integral_altD[OF _ False this]
```
```  4145       from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
```
```  4146       have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
```
```  4147         unfolding bounded_Un by(rule conjI bounded_ball)+
```
```  4148       from bounded_subset_cbox[OF this] guess a b by (elim exE)
```
```  4149       note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  4150       guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
```
```  4151       guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
```
```  4152       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"
```
```  4153         by (simp add: abs_real_def split: if_split_asm)
```
```  4154       note le_less_trans[OF Basis_le_norm[OF k]]
```
```  4155       note this[OF w1(2)] this[OF w2(2)]
```
```  4156       moreover
```
```  4157       have "w1\<bullet>k \<le> w2\<bullet>k"
```
```  4158         by (rule lem[OF w1(1) w2(1)]) (simp add: assms(4))
```
```  4159       ultimately show False
```
```  4160         unfolding inner_simps by(rule *)
```
```  4161     qed
```
```  4162   qed
```
```  4163 qed
```
```  4164
```
```  4165 lemma integral_component_le:
```
```  4166   fixes g f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4167   assumes "k \<in> Basis"
```
```  4168     and "f integrable_on s" "g integrable_on s"
```
```  4169     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  4170   shows "(integral s f)\<bullet>k \<le> (integral s g)\<bullet>k"
```
```  4171   apply (rule has_integral_component_le)
```
```  4172   using integrable_integral assms
```
```  4173   apply auto
```
```  4174   done
```
```  4175
```
```  4176 lemma has_integral_component_nonneg:
```
```  4177   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4178   assumes "k \<in> Basis"
```
```  4179     and "(f has_integral i) s"
```
```  4180     and "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  4181   shows "0 \<le> i\<bullet>k"
```
```  4182   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
```
```  4183   using assms(3-)
```
```  4184   by auto
```
```  4185
```
```  4186 lemma integral_component_nonneg:
```
```  4187   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4188   assumes "k \<in> Basis"
```
```  4189     and  "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  4190   shows "0 \<le> (integral s f)\<bullet>k"
```
```  4191 proof (cases "f integrable_on s")
```
```  4192   case True show ?thesis
```
```  4193     apply (rule has_integral_component_nonneg)
```
```  4194     using assms True
```
```  4195     apply auto
```
```  4196     done
```
```  4197 next
```
```  4198   case False then show ?thesis by (simp add: not_integrable_integral)
```
```  4199 qed
```
```  4200
```
```  4201 lemma has_integral_component_neg:
```
```  4202   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4203   assumes "k \<in> Basis"
```
```  4204     and "(f has_integral i) s"
```
```  4205     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"
```
```  4206   shows "i\<bullet>k \<le> 0"
```
```  4207   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
```
```  4208   by auto
```
```  4209
```
```  4210 lemma has_integral_component_lbound:
```
```  4211   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4212   assumes "(f has_integral i) (cbox a b)"
```
```  4213     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  4214     and "k \<in> Basis"
```
```  4215   shows "B * content (cbox a b) \<le> i\<bullet>k"
```
```  4216   using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(\<Sum>i\<in>Basis. B *\<^sub>R i)::'b"] assms(2-)
```
```  4217   by (auto simp add: field_simps)
```
```  4218
```
```  4219 lemma has_integral_component_ubound:
```
```  4220   fixes f::"'a::euclidean_space => 'b::euclidean_space"
```
```  4221   assumes "(f has_integral i) (cbox a b)"
```
```  4222     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  4223     and "k \<in> Basis"
```
```  4224   shows "i\<bullet>k \<le> B * content (cbox a b)"
```
```  4225   using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"] assms(2-)
```
```  4226   by (auto simp add: field_simps)
```
```  4227
```
```  4228 lemma integral_component_lbound:
```
```  4229   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4230   assumes "f integrable_on cbox a b"
```
```  4231     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  4232     and "k \<in> Basis"
```
```  4233   shows "B * content (cbox a b) \<le> (integral(cbox a b) f)\<bullet>k"
```
```  4234   apply (rule has_integral_component_lbound)
```
```  4235   using assms
```
```  4236   unfolding has_integral_integral
```
```  4237   apply auto
```
```  4238   done
```
```  4239
```
```  4240 lemma integral_component_lbound_real:
```
```  4241   assumes "f integrable_on {a ::real .. b}"
```
```  4242     and "\<forall>x\<in>{a .. b}. B \<le> f(x)\<bullet>k"
```
```  4243     and "k \<in> Basis"
```
```  4244   shows "B * content {a .. b} \<le> (integral {a .. b} f)\<bullet>k"
```
```  4245   using assms
```
```  4246   by (metis box_real(2) integral_component_lbound)
```
```  4247
```
```  4248 lemma integral_component_ubound:
```
```  4249   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4250   assumes "f integrable_on cbox a b"
```
```  4251     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  4252     and "k \<in> Basis"
```
```  4253   shows "(integral (cbox a b) f)\<bullet>k \<le> B * content (cbox a b)"
```
```  4254   apply (rule has_integral_component_ubound)
```
```  4255   using assms
```
```  4256   unfolding has_integral_integral
```
```  4257   apply auto
```
```  4258   done
```
```  4259
```
```  4260 lemma integral_component_ubound_real:
```
```  4261   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```  4262   assumes "f integrable_on {a .. b}"
```
```  4263     and "\<forall>x\<in>{a .. b}. f x\<bullet>k \<le> B"
```
```  4264     and "k \<in> Basis"
```
```  4265   shows "(integral {a .. b} f)\<bullet>k \<le> B * content {a .. b}"
```
```  4266   using assms
```
```  4267   by (metis box_real(2) integral_component_ubound)
```
```  4268
```
```  4269 subsection \<open>Uniform limit of integrable functions is integrable.\<close>
```
```  4270
```
```  4271 lemma real_arch_invD:
```
```  4272   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```  4273   by (subst(asm) real_arch_inverse)
```
```  4274
```
```  4275 lemma integrable_uniform_limit:
```
```  4276   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4277   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  4278   shows "f integrable_on cbox a b"
```
```  4279 proof (cases "content (cbox a b) > 0")
```
```  4280   case False then show ?thesis
```
```  4281       using has_integral_null
```
```  4282       by (simp add: content_lt_nz integrable_on_def)
```
```  4283 next
```
```  4284   case True
```
```  4285   have *: "\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n + 1))"
```
```  4286     by auto
```
```  4287   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
```
```  4288   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]]
```
```  4289   obtain i where i: "\<And>x. (g x has_integral i x) (cbox a b)"
```
```  4290       by auto
```
```  4291   have "Cauchy i"
```
```  4292     unfolding Cauchy_def
```
```  4293   proof clarify
```
```  4294     fix e :: real
```
```  4295     assume "e>0"
```
```  4296     then have "e / 4 / content (cbox a b) > 0"
```
```  4297       using True by (auto simp add: field_simps)
```
```  4298     then obtain M :: nat
```
```  4299          where M: "M \<noteq> 0" "0 < inverse (real_of_nat M)" "inverse (of_nat M) < e / 4 / content (cbox a b)"
```
```  4300       by (subst (asm) real_arch_inverse) auto
```
```  4301     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e"
```
```  4302     proof (rule exI [where x=M], clarify)
```
```  4303       fix m n
```
```  4304       assume m: "M \<le> m" and n: "M \<le> n"
```
```  4305       have "e/4>0" using \<open>e>0\<close> by auto
```
```  4306       note * = i[unfolded has_integral,rule_format,OF this]
```
```  4307       from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
```
```  4308       from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
```
```  4309       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b]
```
```  4310       obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine p"
```
```  4311         by auto
```
```  4312       { fix s1 s2 i1 and i2::'b
```
```  4313         assume no: "norm(s2 - s1) \<le> e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4"
```
```  4314         have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
```
```  4315           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
```
```  4316           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
```
```  4317           by (auto simp add: algebra_simps)
```
```  4318         also have "\<dots> < e"
```
```  4319           using no
```
```  4320           unfolding norm_minus_commute
```
```  4321           by (auto simp add: algebra_simps)
```
```  4322         finally have "norm (i1 - i2) < e" .
```
```  4323       } note triangle3 = this
```
```  4324       have finep: "gm fine p" "gn fine p"
```
```  4325         using fine_inter p  by auto
```
```  4326       { fix x
```
```  4327         assume x: "x \<in> cbox a b"
```
```  4328         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
```
```  4329           using g(1)[OF x, of n] g(1)[OF x, of m] by auto
```
```  4330         also have "\<dots> \<le> inverse (real M) + inverse (real M)"
```
```  4331           apply (rule add_mono)
```
```  4332           using M(2) m n by auto
```
```  4333         also have "\<dots> = 2 / real M"
```
```  4334           unfolding divide_inverse by auto
```
```  4335         finally have "norm (g n x - g m x) \<le> 2 / real M"
```
```  4336           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
```
```  4337           by (auto simp add: algebra_simps simp add: norm_minus_commute)
```
```  4338       } note norm_le = this
```
```  4339       have le_e2: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g n x) - (\<Sum>(x, k)\<in>p. content k *\<^sub>R g m x)) \<le> e / 2"
```
```  4340         apply (rule order_trans [OF rsum_diff_bound[OF p(1), where e="2 / real M"]])
```
```  4341         apply (blast intro: norm_le)
```
```  4342         using M True
```
```  4343         by (auto simp add: field_simps)
```
```  4344       then show "dist (i m) (i n) < e"
```
```  4345         unfolding dist_norm
```
```  4346         using gm gn p finep
```
```  4347         by (auto intro!: triangle3)
```
```  4348     qed
```
```  4349   qed
```
```  4350   then obtain s where s: "i \<longlonglongrightarrow> s"
```
```  4351     using convergent_eq_cauchy[symmetric] by blast
```
```  4352   show ?thesis
```
```  4353     unfolding integrable_on_def has_integral
```
```  4354   proof (rule_tac x=s in exI, clarify)
```
```  4355     fix e::real
```
```  4356     assume e: "0 < e"
```
```  4357     then have *: "e/3 > 0" by auto
```
```  4358     then obtain N1 where N1: "\<forall>n\<ge>N1. norm (i n - s) < e / 3"
```
```  4359       using LIMSEQ_D [OF s] by metis
```
```  4360     from e True have "e / 3 / content (cbox a b) > 0"
```
```  4361       by (auto simp add: field_simps)
```
```  4362     from real_arch_invD[OF this] guess N2 by (elim exE conjE) note N2=this
```
```  4363     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
```
```  4364     { fix sf sg i
```
```  4365       assume no: "norm (sf - sg) \<le> e / 3"
```
```  4366                  "norm(i - s) < e / 3"
```
```  4367                  "norm (sg - i) < e / 3"
```
```  4368       have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
```
```  4369         using norm_triangle_ineq[of "sf - sg" "sg - s"]
```
```  4370         using norm_triangle_ineq[of "sg -  i" " i - s"]
```
```  4371         by (auto simp add: algebra_simps)
```
```  4372       also have "\<dots> < e"
```
```  4373         using no
```
```  4374         unfolding norm_minus_commute
```
```  4375         by (auto simp add: algebra_simps)
```
```  4376       finally have "norm (sf - s) < e" .
```
```  4377     } note lem = this
```
```  4378     { fix p
```
```  4379       assume p: "p tagged_division_of (cbox a b) \<and> g' fine p"
```
```  4380       then have norm_less: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g (N1 + N2) x) - i (N1 + N2)) < e / 3"
```
```  4381         using g' by blast
```
```  4382       have "content (cbox a b) < e / 3 * (of_nat N2)"
```
```  4383         using N2 unfolding inverse_eq_divide using True by (auto simp add: field_simps)
```
```  4384       moreover have "e / 3 * of_nat N2 \<le> e / 3 * (of_nat (N1 + N2) + 1)"
```
```  4385         using \<open>e>0\<close> by auto
```
```  4386       ultimately have "content (cbox a b) < e / 3 * (of_nat (N1 + N2) + 1)"
```
```  4387         by linarith
```
```  4388       then have le_e3: "inverse (real (N1 + N2) + 1) * content (cbox a b) \<le> e / 3"
```
```  4389         unfolding inverse_eq_divide
```
```  4390         by (auto simp add: field_simps)
```
```  4391       have ne3: "norm (i (N1 + N2) - s) < e / 3"
```
```  4392         using N1 by auto
```
```  4393       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e"
```
```  4394         apply (rule lem[OF order_trans [OF _ le_e3] ne3 norm_less])
```
```  4395         apply (rule rsum_diff_bound[OF p[THEN conjunct1]])
```
```  4396         apply (blast intro: g)
```
```  4397         done }
```
```  4398     then show "\<exists>d. gauge d \<and>
```
```  4399              (\<forall>p. p tagged_division_of cbox a b \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e)"
```
```  4400       by (blast intro: g')
```
```  4401   qed
```
```  4402 qed
```
```  4403
```
```  4404 lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
```
```  4405
```
```  4406
```
```  4407 subsection \<open>Negligible sets.\<close>
```
```  4408
```
```  4409 definition "negligible (s:: 'a::euclidean_space set) \<longleftrightarrow>
```
```  4410   (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) (cbox a b))"
```
```  4411
```
```  4412
```
```  4413 subsection \<open>Negligibility of hyperplane.\<close>
```
```  4414
```
```  4415 lemma interval_doublesplit:
```
```  4416   fixes a :: "'a::euclidean_space"
```
```  4417   assumes "k \<in> Basis"
```
```  4418   shows "cbox a b \<inter> {x . \<bar>x\<bullet>k - c\<bar> \<le> (e::real)} =
```
```  4419     cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i)
```
```  4420      (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)"
```
```  4421 proof -
```
```  4422   have *: "\<And>x c e::real. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
```
```  4423     by auto
```
```  4424   have **: "\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}"
```
```  4425     by blast
```
```  4426   show ?thesis
```
```  4427     unfolding * ** interval_split[OF assms] by (rule refl)
```
```  4428 qed
```
```  4429
```
```  4430 lemma division_doublesplit:
```
```  4431   fixes a :: "'a::euclidean_space"
```
```  4432   assumes "p division_of (cbox a b)"
```
```  4433     and k: "k \<in> Basis"
```
```  4434   shows "(\<lambda>l. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e}) ` {l\<in>p. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e} \<noteq> {}}
```
```  4435          division_of  (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e})"
```
```  4436 proof -
```
```  4437   have *: "\<And>x c. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
```
```  4438     by auto
```
```  4439   have **: "\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'"
```
```  4440     by auto
```
```  4441   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
```
```  4442   note division_split(2)[OF this, where c="c-e" and k=k,OF k]
```
```  4443   then show ?thesis
```
```  4444     apply (rule **)
```
```  4445     subgoal
```
```  4446       apply (simp add: abs_diff_le_iff field_simps Collect_conj_eq setcompr_eq_image[symmetric])
```
```  4447       apply (rule equalityI)
```
```  4448       apply blast
```
```  4449       apply clarsimp
```
```  4450       apply (rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI)
```
```  4451       apply auto
```
```  4452       done
```
```  4453     by (simp add: interval_split k interval_doublesplit)
```
```  4454 qed
```
```  4455
```
```  4456 lemma content_doublesplit:
```
```  4457   fixes a :: "'a::euclidean_space"
```
```  4458   assumes "0 < e"
```
```  4459     and k: "k \<in> Basis"
```
```  4460   obtains d where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) < e"
```
```  4461 proof (cases "content (cbox a b) = 0")
```
```  4462   case True
```
```  4463   then have ce: "content (cbox a b) < e"
```
```  4464     by (metis \<open>0 < e\<close>)
```
```  4465   show ?thesis
```
```  4466     apply (rule that[of 1])
```
```  4467     apply simp
```
```  4468     unfolding interval_doublesplit[OF k]
```
```  4469     apply (rule le_less_trans[OF content_subset ce])
```
```  4470     apply (auto simp: interval_doublesplit[symmetric] k)
```
```  4471     done
```
```  4472 next
```
```  4473   case False
```
```  4474   define d where "d = e / 3 / setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})"
```
```  4475   note False[unfolded content_eq_0 not_ex not_le, rule_format]
```
```  4476   then have "\<And>x. x \<in> Basis \<Longrightarrow> b\<bullet>x > a\<bullet>x"
```
```  4477     by (auto simp add:not_le)
```
```  4478   then have prod0: "0 < setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) (Basis - {k})"
```
```  4479     by (force simp add: setprod_pos field_simps)
```
```  4480   then have "d > 0"
```
```  4481     using assms
```
```  4482     by (auto simp add: d_def field_simps)
```
```  4483   then show ?thesis
```
```  4484   proof (rule that[of d])
```
```  4485     have *: "Basis = insert k (Basis - {k})"
```
```  4486       using k by auto
```
```  4487     have less_e: "(min (b \<bullet> k) (c + d) - max (a \<bullet> k) (c - d)) * (\<Prod>i\<in>Basis - {k}. b \<bullet> i - a \<bullet> i) < e"
```
```  4488     proof -
```
```  4489       have "(min (b \<bullet> k) (c + d) - max (a \<bullet> k) (c - d)) \<le> 2 * d"
```
```  4490         by auto
```
```  4491       also have "\<dots> < e / (\<Prod>i\<in>Basis - {k}. b \<bullet> i - a \<bullet> i)"
```
```  4492         unfolding d_def
```
```  4493         using assms prod0
```
```  4494         by (auto simp add: field_simps)
```
```  4495       finally show "(min (b \<bullet> k) (c + d) - max (a \<bullet> k) (c - d)) * (\<Prod>i\<in>Basis - {k}. b \<bullet> i - a \<bullet> i) < e"
```
```  4496         unfolding pos_less_divide_eq[OF prod0] .
```
```  4497     qed
```
```  4498     show "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
```
```  4499     proof (cases "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = {}")
```
```  4500       case True
```
```  4501       then show ?thesis
```
```  4502         using assms by simp
```
```  4503     next
```
```  4504       case False
```
```  4505       then have
```
```  4506           "(\<Prod>i\<in>Basis - {k}. interval_upperbound (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i -
```
```  4507                 interval_lowerbound (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<bullet> i)
```
```  4508            = (\<Prod>i\<in>Basis - {k}. b\<bullet>i - a\<bullet>i)"
```
```  4509         by (simp add: box_eq_empty interval_doublesplit[OF k])
```
```  4510       then show "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
```
```  4511         unfolding content_def
```
```  4512         using assms False
```
```  4513         apply (subst *)
```
```  4514         apply (subst setprod.insert)
```
```  4515         apply (simp_all add: interval_doublesplit[OF k] box_eq_empty not_less less_e)
```
```  4516         done
```
```  4517     qed
```
```  4518   qed
```
```  4519 qed
```
```  4520
```
```  4521 lemma negligible_standard_hyperplane[intro]:
```
```  4522   fixes k :: "'a::euclidean_space"
```
```  4523   assumes k: "k \<in> Basis"
```
```  4524   shows "negligible {x. x\<bullet>k = c}"
```
```  4525   unfolding negligible_def has_integral
```
```  4526 proof (clarify, goal_cases)
```
```  4527   case (1 a b e)
```
```  4528   from this and k obtain d where d: "0 < d" "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
```
```  4529     by (rule content_doublesplit)
```
```  4530   let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
```
```  4531   show ?case
```
```  4532     apply (rule_tac x="\<lambda>x. ball x d" in exI)
```
```  4533     apply rule
```
```  4534     apply (rule gauge_ball)
```
```  4535     apply (rule d)
```
```  4536   proof (rule, rule)
```
```  4537     fix p
```
```  4538     assume p: "p tagged_division_of (cbox a b) \<and> (\<lambda>x. ball x d) fine p"
```
```  4539     have *: "(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) =
```
```  4540       (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
```
```  4541       apply (rule setsum.cong)
```
```  4542       apply (rule refl)
```
```  4543       unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
```
```  4544       apply cases
```
```  4545       apply (rule disjI1)
```
```  4546       apply assumption
```
```  4547       apply (rule disjI2)
```
```  4548     proof -
```
```  4549       fix x l
```
```  4550       assume as: "(x, l) \<in> p" "?i x \<noteq> 0"
```
```  4551       then have xk: "x\<bullet>k = c"
```
```  4552         unfolding indicator_def
```
```  4553         apply -
```
```  4554         apply (rule ccontr)
```
```  4555         apply auto
```
```  4556         done
```
```  4557       show "content l = content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  4558         apply (rule arg_cong[where f=content])
```
```  4559         apply (rule set_eqI)
```
```  4560         apply rule
```
```  4561         apply rule
```
```  4562         unfolding mem_Collect_eq
```
```  4563       proof -
```
```  4564         fix y
```
```  4565         assume y: "y \<in> l"
```
```  4566         note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
```
```  4567         note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y]
```
```  4568         note le_less_trans[OF Basis_le_norm[OF k] this]
```
```  4569         then show "\<bar>y \<bullet> k - c\<bar> \<le> d"
```
```  4570           unfolding inner_simps xk by auto
```
```  4571       qed auto
```
```  4572     qed
```
```  4573     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
```
```  4574     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e"
```
```  4575       unfolding diff_0_right *
```
```  4576       unfolding real_scaleR_def real_norm_def
```
```  4577       apply (subst abs_of_nonneg)
```
```  4578       apply (rule setsum_nonneg)
```
```  4579       apply rule
```
```  4580       unfolding split_paired_all split_conv
```
```  4581       apply (rule mult_nonneg_nonneg)
```
```  4582       apply (drule p'(4))
```
```  4583       apply (erule exE)+
```
```  4584       apply(rule_tac b=b in back_subst)
```
```  4585       prefer 2
```
```  4586       apply (subst(asm) eq_commute)
```
```  4587       apply assumption
```
```  4588       apply (subst interval_doublesplit[OF k])
```
```  4589       apply (rule content_pos_le)
```
```  4590       apply (rule indicator_pos_le)
```
```  4591     proof -
```
```  4592       have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le>
```
```  4593         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
```
```  4594         apply (rule setsum_mono)
```
```  4595         unfolding split_paired_all split_conv
```
```  4596         apply (rule mult_right_le_one_le)
```
```  4597         apply (drule p'(4))
```
```  4598         apply (auto simp add:interval_doublesplit[OF k])
```
```  4599         done
```
```  4600       also have "\<dots> < e"
```
```  4601       proof (subst setsum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
```
```  4602         case prems: (1 u v)
```
```  4603         have "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content (cbox u v)"
```
```  4604           unfolding interval_doublesplit[OF k]
```
```  4605           apply (rule content_subset)
```
```  4606           unfolding interval_doublesplit[symmetric,OF k]
```
```  4607           apply auto
```
```  4608           done
```
```  4609         then show ?case
```
```  4610           unfolding prems interval_doublesplit[OF k]
```
```  4611           by (blast intro: antisym)
```
```  4612       next
```
```  4613         have "(\<Sum>l\<in>snd ` p. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
```
```  4614           setsum content ((\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}})"
```
```  4615         proof (subst (2) setsum.reindex_nontrivial)
```
```  4616           fix x y assume "x \<in> {l \<in> snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}" "y \<in> {l \<in> snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}"
```
```  4617             "x \<noteq> y" and eq: "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
```
```  4618           then obtain x' y' where "(x', x) \<in> p" "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}" "(y', y) \<in> p" "y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}"
```
```  4619             by (auto)
```
```  4620           from p'(5)[OF \<open>(x', x) \<in> p\<close> \<open>(y', y) \<in> p\<close>] \<open>x \<noteq> y\<close> have "interior (x \<inter> y) = {}"
```
```  4621             by auto
```
```  4622           moreover have "interior ((x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> (y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<subseteq> interior (x \<inter> y)"
```
```  4623             by (auto intro: interior_mono)
```
```  4624           ultimately have "interior (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}"
```
```  4625             by (auto simp: eq)
```
```  4626           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
```
```  4627             using p'(4)[OF \<open>(x', x) \<in> p\<close>] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int)
```
```  4628         qed (insert p'(1), auto intro!: setsum.mono_neutral_right)
```
```  4629         also have "\<dots> \<le> norm (\<Sum>l\<in>(\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` p. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}. content l *\<^sub>R 1::real)"
```
```  4630           by simp
```
```  4631         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  4632           using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
```
```  4633           unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
```
```  4634         also have "\<dots> < e"
```
```  4635           using d(2) by simp
```
```  4636         finally show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e" .
```
```  4637       qed
```
```  4638       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
```
```  4639     qed
```
```  4640   qed
```
```  4641 qed
```
```  4642
```
```  4643
```
```  4644 subsection \<open>A technical lemma about "refinement" of division.\<close>
```
```  4645
```
```  4646 lemma tagged_division_finer:
```
```  4647   fixes p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
```
```  4648   assumes "p tagged_division_of (cbox a b)"
```
```  4649     and "gauge d"
```
```  4650   obtains q where "q tagged_division_of (cbox a b)"
```
```  4651     and "d fine q"
```
```  4652     and "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
```
```  4653 proof -
```
```  4654   let ?P = "\<lambda>p. p tagged_partial_division_of (cbox a b) \<longrightarrow> gauge d \<longrightarrow>
```
```  4655     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
```
```  4656       (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
```
```  4657   {
```
```  4658     have *: "finite p" "p tagged_partial_division_of (cbox a b)"
```
```  4659       using assms(1)
```
```  4660       unfolding tagged_division_of_def
```
```  4661       by auto
```
```  4662     presume "\<And>p. finite p \<Longrightarrow> ?P p"
```
```  4663     from this[rule_format,OF * assms(2)] guess q .. note q=this
```
```  4664     then show ?thesis
```
```  4665       apply -
```
```  4666       apply (rule that[of q])
```
```  4667       unfolding tagged_division_ofD[OF assms(1)]
```
```  4668       apply auto
```
```  4669       done
```
```  4670   }
```
```  4671   fix p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
```
```  4672   assume as: "finite p"
```
```  4673   show "?P p"
```
```  4674     apply rule
```
```  4675     apply rule
```
```  4676     using as
```
```  4677   proof (induct p)
```
```  4678     case empty
```
```  4679     show ?case
```
```  4680       apply (rule_tac x="{}" in exI)
```
```  4681       unfolding fine_def
```
```  4682       apply auto
```
```  4683       done
```
```  4684   next
```
```  4685     case (insert xk p)
```
```  4686     guess x k using surj_pair[of xk] by (elim exE) note xk=this
```
```  4687     note tagged_partial_division_subset[OF insert(4) subset_insertI]
```
```  4688     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
```
```  4689     have *: "\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}"
```
```  4690       unfolding xk by auto
```
```  4691     note p = tagged_partial_division_ofD[OF insert(4)]
```
```  4692     from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
```
```  4693
```
```  4694     have "finite {k. \<exists>x. (x, k) \<in> p}"
```
```  4695       apply (rule finite_subset[of _ "snd ` p"])
```
```  4696       using p
```
```  4697       apply safe
```
```  4698       apply (metis image_iff snd_conv)
```
```  4699       apply auto
```
```  4700       done
```
```  4701     then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
```
```  4702       apply (rule inter_interior_unions_intervals)
```
```  4703       apply (rule open_interior)
```
```  4704       apply (rule_tac[!] ballI)
```
```  4705       unfolding mem_Collect_eq
```
```  4706       apply (erule_tac[!] exE)
```
```  4707       apply (drule p(4)[OF insertI2])
```
```  4708       apply assumption
```
```  4709       apply (rule p(5))
```
```  4710       unfolding uv xk
```
```  4711       apply (rule insertI1)
```
```  4712       apply (rule insertI2)
```
```  4713       apply assumption
```
```  4714       using insert(2)
```
```  4715       unfolding uv xk
```
```  4716       apply auto
```
```  4717       done
```
```  4718     show ?case
```
```  4719     proof (cases "cbox u v \<subseteq> d x")
```
```  4720       case True
```
```  4721       then show ?thesis
```
```  4722         apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
```
```  4723         apply rule
```
```  4724         unfolding * uv
```
```  4725         apply (rule tagged_division_union)
```
```  4726         apply (rule tagged_division_of_self)
```
```  4727         apply (rule p[unfolded xk uv] insertI1)+
```
```  4728         apply (rule q1)
```
```  4729         apply (rule int)
```
```  4730         apply rule
```
```  4731         apply (rule fine_union)
```
```  4732         apply (subst fine_def)
```
```  4733         defer
```
```  4734         apply (rule q1)
```
```  4735         unfolding Ball_def split_paired_All split_conv
```
```  4736         apply rule
```
```  4737         apply rule
```
```  4738         apply rule
```
```  4739         apply rule
```
```  4740         apply (erule insertE)
```
```  4741         apply (simp add: uv xk)
```
```  4742         apply (rule UnI2)
```
```  4743         apply (drule q1(3)[rule_format])
```
```  4744         unfolding xk uv
```
```  4745         apply auto
```
```  4746         done
```
```  4747     next
```
```  4748       case False
```
```  4749       from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
```
```  4750       show ?thesis
```
```  4751         apply (rule_tac x="q2 \<union> q1" in exI)
```
```  4752         apply rule
```
```  4753         unfolding * uv
```
```  4754         apply (rule tagged_division_union q2 q1 int fine_union)+
```
```  4755         unfolding Ball_def split_paired_All split_conv
```
```  4756         apply rule
```
```  4757         apply (rule fine_union)
```
```  4758         apply (rule q1 q2)+
```
```  4759         apply rule
```
```  4760         apply rule
```
```  4761         apply rule
```
```  4762         apply rule
```
```  4763         apply (erule insertE)
```
```  4764         apply (rule UnI2)
```
```  4765         apply (simp add: False uv xk)
```
```  4766         apply (drule q1(3)[rule_format])
```
```  4767         using False
```
```  4768         unfolding xk uv
```
```  4769         apply auto
```
```  4770         done
```
```  4771     qed
```
```  4772   qed
```
```  4773 qed
```
```  4774
```
```  4775
```
```  4776 subsection \<open>Hence the main theorem about negligible sets.\<close>
```
```  4777
```
```  4778 lemma finite_product_dependent:
```
```  4779   assumes "finite s"
```
```  4780     and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
```
```  4781   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
```
```  4782   using assms
```
```  4783 proof induct
```
```  4784   case (insert x s)
```
```  4785   have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
```
```  4786     (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
```
```  4787   show ?case
```
```  4788     unfolding *
```
```  4789     apply (rule finite_UnI)
```
```  4790     using insert
```
```  4791     apply auto
```
```  4792     done
```
```  4793 qed auto
```
```  4794
```
```  4795 lemma sum_sum_product:
```
```  4796   assumes "finite s"
```
```  4797     and "\<forall>i\<in>s. finite (t i)"
```
```  4798   shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s =
```
```  4799     setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
```
```  4800   using assms
```
```  4801 proof induct
```
```  4802   case (insert a s)
```
```  4803   have *: "{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} =
```
```  4804     (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
```
```  4805   show ?case
```
```  4806     unfolding *
```
```  4807     apply (subst setsum.union_disjoint)
```
```  4808     unfolding setsum.insert[OF insert(1-2)]
```
```  4809     prefer 4
```
```  4810     apply (subst insert(3))
```
```  4811     unfolding add_right_cancel
```
```  4812   proof -
```
```  4813     show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
```
```  4814       apply (subst setsum.reindex)
```
```  4815       unfolding inj_on_def
```
```  4816       apply auto
```
```  4817       done
```
```  4818     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
```
```  4819       apply (rule finite_product_dependent)
```
```  4820       using insert
```
```  4821       apply auto
```
```  4822       done
```
```  4823   qed (insert insert, auto)
```
```  4824 qed auto
```
```  4825
```
```  4826 lemma has_integral_negligible:
```
```  4827   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  4828   assumes "negligible s"
```
```  4829     and "\<forall>x\<in>(t - s). f x = 0"
```
```  4830   shows "(f has_integral 0) t"
```
```  4831 proof -
```
```  4832   presume P: "\<And>f::'b::euclidean_space \<Rightarrow> 'a.
```
```  4833     \<And>a b. \<forall>x. x \<notin> s \<longrightarrow> f x = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```  4834   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
```
```  4835   show ?thesis
```
```  4836     apply (rule_tac f="?f" in has_integral_eq)
```
```  4837     unfolding if_P
```
```  4838     apply (rule refl)
```
```  4839     apply (subst has_integral_alt)
```
```  4840     apply cases
```
```  4841     apply (subst if_P, assumption)
```
```  4842     unfolding if_not_P
```
```  4843   proof -
```
```  4844     assume "\<exists>a b. t = cbox a b"
```
```  4845     then guess a b apply - by (erule exE)+ note t = this
```
```  4846     show "(?f has_integral 0) t"
```
```  4847       unfolding t
```
```  4848       apply (rule P)
```
```  4849       using assms(2)
```
```  4850       unfolding t
```
```  4851       apply auto
```
```  4852       done
```
```  4853   next
```
```  4854     show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  4855       (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) (cbox a b) \<and> norm (z - 0) < e)"
```
```  4856       apply safe
```
```  4857       apply (rule_tac x=1 in exI)
```
```  4858       apply rule
```
```  4859       apply (rule zero_less_one)
```
```  4860       apply safe
```
```  4861       apply (rule_tac x=0 in exI)
```
```  4862       apply rule
```
```  4863       apply (rule P)
```
```  4864       using assms(2)
```
```  4865       apply auto
```
```  4866       done
```
```  4867   qed
```
```  4868 next
```
```  4869   fix f :: "'b \<Rightarrow> 'a"
```
```  4870   fix a b :: 'b
```
```  4871   assume assm: "\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  4872   show "(f has_integral 0) (cbox a b)"
```
```  4873     unfolding has_integral
```
```  4874   proof (safe, goal_cases)
```
```  4875     case prems: (1 e)
```
```  4876     then have "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
```
```  4877       apply -
```
```  4878       apply (rule divide_pos_pos)
```
```  4879       defer
```
```  4880       apply (rule mult_pos_pos)
```
```  4881       apply (auto simp add:field_simps)
```
```  4882       done
```
```  4883     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b]
```
```  4884     note allI[OF this,of "\<lambda>x. x"]
```
```  4885     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
```
```  4886     show ?case
```
```  4887       apply (rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
```
```  4888     proof safe
```
```  4889       show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)"
```
```  4890         using d(1) unfolding gauge_def by auto
```
```  4891       fix p
```
```  4892       assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
```
```  4893       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  4894       {
```
```  4895         presume "p \<noteq> {} \<Longrightarrow> ?goal"
```
```  4896         then show ?goal
```
```  4897           apply (cases "p = {}")
```
```  4898           using prems
```
```  4899           apply auto
```
```  4900           done
```
```  4901       }
```
```  4902       assume as': "p \<noteq> {}"
```
```  4903       from real_arch_simple[of "Max((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
```
```  4904       then have N: "\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N"
```
```  4905         by (meson Max_ge as(1) dual_order.trans finite_imageI tagged_division_of_finite)
```
```  4906       have "\<forall>i. \<exists>q. q tagged_division_of (cbox a b) \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"
```
```  4907         by (auto intro: tagged_division_finer[OF as(1) d(1)])
```
```  4908       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
```
```  4909       have *: "\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)"
```
```  4910         apply (rule setsum_nonneg)
```
```  4911         apply safe
```
```  4912         unfolding real_scaleR_def
```
```  4913         apply (drule tagged_division_ofD(4)[OF q(1)])
```
```  4914         apply (auto intro: mult_nonneg_nonneg)
```
```  4915         done
```
```  4916       have **: "finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow>
```
```  4917         (\<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" for f g s t
```
```  4918         apply (rule setsum_le_included[of s t g snd f])
```
```  4919         prefer 4
```
```  4920         apply safe
```
```  4921         apply (erule_tac x=x in ballE)
```
```  4922         apply (erule exE)
```
```  4923         apply (rule_tac x="(xa,x)" in bexI)
```
```  4924         apply auto
```
```  4925         done
```
```  4926       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
```
```  4927         norm (setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
```
```  4928         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
```
```  4929         apply (rule order_trans)
```
```  4930         apply (rule norm_setsum)
```
```  4931         apply (subst sum_sum_product)
```
```  4932         prefer 3
```
```  4933       proof (rule **, safe)
```
```  4934         show "finite {(i, j) |i j. i \<in> {..N + 1} \<and> j \<in> q i}"
```
```  4935           apply (rule finite_product_dependent)
```
```  4936           using q
```
```  4937           apply auto
```
```  4938           done
```
```  4939         fix i a b
```
```  4940         assume as'': "(a, b) \<in> q i"
```
```  4941         show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
```
```  4942           unfolding real_scaleR_def
```
```  4943           using tagged_division_ofD(4)[OF q(1) as'']
```
```  4944           by (auto intro!: mult_nonneg_nonneg)
```
```  4945       next
```
```  4946         fix i :: nat
```
```  4947         show "finite (q i)"
```
```  4948           using q by auto
```
```  4949       next
```
```  4950         fix x k
```
```  4951         assume xk: "(x, k) \<in> p"
```
```  4952         define n where "n = nat \<lfloor>norm (f x)\<rfloor>"
```
```  4953         have *: "norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p"
```
```  4954           using xk by auto
```
```  4955         have nfx: "real n \<le> norm (f x)" "norm (f x) \<le> real n + 1"
```
```  4956           unfolding n_def by auto
```
```  4957         then have "n \<in> {0..N + 1}"
```
```  4958           using N[rule_format,OF *] by auto
```
```  4959         moreover
```
```  4960         note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
```
```  4961         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this]
```
```  4962         note this[unfolded n_def[symmetric]]
```
```  4963         moreover
```
```  4964         have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
```
```  4965         proof (cases "x \<in> s")
```
```  4966           case False
```
```  4967           then show ?thesis
```
```  4968             using assm by auto
```
```  4969         next
```
```  4970           case True
```
```  4971           have *: "content k \<ge> 0"
```
```  4972             using tagged_division_ofD(4)[OF as(1) xk] by auto
```
```  4973           moreover
```
```  4974           have "content k * norm (f x) \<le> content k * (real n + 1)"
```
```  4975             apply (rule mult_mono)
```
```  4976             using nfx *
```
```  4977             apply auto
```
```  4978             done
```
```  4979           ultimately
```
```  4980           show ?thesis
```
```  4981             unfolding abs_mult
```
```  4982             using nfx True
```
```  4983             by (auto simp add: field_simps)
```
```  4984         qed
```
```  4985         ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le>
```
```  4986           (real y + 1) * (content k *\<^sub>R indicator s x)"
```
```  4987           apply (rule_tac x=n in exI)
```
```  4988           apply safe
```
```  4989           apply (rule_tac x=n in exI)
```
```  4990           apply (rule_tac x="(x,k)" in exI)
```
```  4991           apply safe
```
```  4992           apply auto
```
```  4993           done
```
```  4994       qed (insert as, auto)
```
```  4995       also have "\<dots> \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {..N+1}"
```
```  4996       proof (rule setsum_mono, goal_cases)
```
```  4997         case (1 i)
```
```  4998         then show ?case
```
```  4999           apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
```
```  5000           using d(2)[rule_format, of "q i" i]
```
```  5001           using q[rule_format]
```
```  5002           apply (auto simp add: field_simps)
```
```  5003           done
```
```  5004       qed
```
```  5005       also have "\<dots> < e * inverse 2 * 2"
```
```  5006         unfolding divide_inverse setsum_right_distrib[symmetric]
```
```  5007         apply (rule mult_strict_left_mono)
```
```  5008         unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
```
```  5009         apply (subst geometric_sum)
```
```  5010         using prems
```
```  5011         apply auto
```
```  5012         done
```
```  5013       finally show "?goal" by auto
```
```  5014     qed
```
```  5015   qed
```
```  5016 qed
```
```  5017
```
```  5018 lemma has_integral_spike:
```
```  5019   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  5020   assumes "negligible s"
```
```  5021     and "(\<forall>x\<in>(t - s). g x = f x)"
```
```  5022     and "(f has_integral y) t"
```
```  5023   shows "(g has_integral y) t"
```
```  5024 proof -
```
```  5025   {
```
```  5026     fix a b :: 'b
```
```  5027     fix f g :: "'b \<Rightarrow> 'a"
```
```  5028     fix y :: 'a
```
```  5029     assume as: "\<forall>x \<in> cbox a b - s. g x = f x" "(f has_integral y) (cbox a b)"
```
```  5030     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
```
```  5031       apply (rule has_integral_add[OF as(2)])
```
```  5032       apply (rule has_integral_negligible[OF assms(1)])
```
```  5033       using as
```
```  5034       apply auto
```
```  5035       done
```
```  5036     then have "(g has_integral y) (cbox a b)"
```
```  5037       by auto
```
```  5038   } note * = this
```
```  5039   show ?thesis
```
```  5040     apply (subst has_integral_alt)
```
```  5041     using assms(2-)
```
```  5042     apply -
```
```  5043     apply (rule cond_cases)
```
```  5044     apply safe
```
```  5045     apply (rule *)
```
```  5046     apply assumption+
```
```  5047     apply (subst(asm) has_integral_alt)
```
```  5048     unfolding if_not_P
```
```  5049     apply (erule_tac x=e in allE)
```
```  5050     apply safe
```
```  5051     apply (rule_tac x=B in exI)
```
```  5052     apply safe
```
```  5053     apply (erule_tac x=a in allE)
```
```  5054     apply (erule_tac x=b in allE)
```
```  5055     apply safe
```
```  5056     apply (rule_tac x=z in exI)
```
```  5057     apply safe
```
```  5058     apply (rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"])
```
```  5059     apply auto
```
```  5060     done
```
```  5061 qed
```
```  5062
```
```  5063 lemma has_integral_spike_eq:
```
```  5064   assumes "negligible s"
```
```  5065     and "\<forall>x\<in>(t - s). g x = f x"
```
```  5066   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  5067   apply rule
```
```  5068   apply (rule_tac[!] has_integral_spike[OF assms(1)])
```
```  5069   using assms(2)
```
```  5070   apply auto
```
```  5071   done
```
```  5072
```
```  5073 lemma integrable_spike:
```
```  5074   assumes "negligible s"
```
```  5075     and "\<forall>x\<in>(t - s). g x = f x"
```
```  5076     and "f integrable_on t"
```
```  5077   shows "g integrable_on  t"
```
```  5078   using assms
```
```  5079   unfolding integrable_on_def
```
```  5080   apply -
```
```  5081   apply (erule exE)
```
```  5082   apply rule
```
```  5083   apply (rule has_integral_spike)
```
```  5084   apply fastforce+
```
```  5085   done
```
```  5086
```
```  5087 lemma integral_spike:
```
```  5088   assumes "negligible s"
```
```  5089     and "\<forall>x\<in>(t - s). g x = f x"
```
```  5090   shows "integral t f = integral t g"
```
```  5091   using has_integral_spike_eq[OF assms] by (simp add: integral_def integrable_on_def)
```
```  5092
```
```  5093
```
```  5094 subsection \<open>Some other trivialities about negligible sets.\<close>
```
```  5095
```
```  5096 lemma negligible_subset[intro]:
```
```  5097   assumes "negligible s"
```
```  5098     and "t \<subseteq> s"
```
```  5099   shows "negligible t"
```
```  5100   unfolding negligible_def
```
```  5101 proof (safe, goal_cases)
```
```  5102   case (1 a b)
```
```  5103   show ?case
```
```  5104     using assms(1)[unfolded negligible_def,rule_format,of a b]
```
```  5105     apply -
```
```  5106     apply (rule has_integral_spike[OF assms(1)])
```
```  5107     defer
```
```  5108     apply assumption
```
```  5109     using assms(2)
```
```  5110     unfolding indicator_def
```
```  5111     apply auto
```
```  5112     done
```
```  5113 qed
```
```  5114
```
```  5115 lemma negligible_diff[intro?]:
```
```  5116   assumes "negligible s"
```
```  5117   shows "negligible (s - t)"
```
```  5118   using assms by auto
```
```  5119
```
```  5120 lemma negligible_Int:
```
```  5121   assumes "negligible s \<or> negligible t"
```
```  5122   shows "negligible (s \<inter> t)"
```
```  5123   using assms by auto
```
```  5124
```
```  5125 lemma negligible_Un:
```
```  5126   assumes "negligible s"
```
```  5127     and "negligible t"
```
```  5128   shows "negligible (s \<union> t)"
```
```  5129   unfolding negligible_def
```
```  5130 proof (safe, goal_cases)
```
```  5131   case (1 a b)
```
```  5132   note assm = assms[unfolded negligible_def,rule_format,of a b]
```
```  5133   then show ?case
```
```  5134     apply (subst has_integral_spike_eq[OF assms(2)])
```
```  5135     defer
```
```  5136     apply assumption
```
```  5137     unfolding indicator_def
```
```  5138     apply auto
```
```  5139     done
```
```  5140 qed
```
```  5141
```
```  5142 lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
```
```  5143   using negligible_Un by auto
```
```  5144
```
```  5145 lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
```
```  5146   using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto
```
```  5147
```
```  5148 lemma negligible_insert[simp]: "negligible (insert a s) \<longleftrightarrow> negligible s"
```
```  5149   apply (subst insert_is_Un)
```
```  5150   unfolding negligible_Un_eq
```
```  5151   apply auto
```
```  5152   done
```
```  5153
```
```  5154 lemma negligible_empty[iff]: "negligible {}"
```
```  5155   by auto
```
```  5156
```
```  5157 lemma negligible_finite[intro]:
```
```  5158   assumes "finite s"
```
```  5159   shows "negligible s"
```
```  5160   using assms by (induct s) auto
```
```  5161
```
```  5162 lemma negligible_Union[intro]:
```
```  5163   assumes "finite s"
```
```  5164     and "\<forall>t\<in>s. negligible t"
```
```  5165   shows "negligible(\<Union>s)"
```
```  5166   using assms by induct auto
```
```  5167
```
```  5168 lemma negligible:
```
```  5169   "negligible s \<longleftrightarrow> (\<forall>t::('a::euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
```
```  5170   apply safe
```
```  5171   defer
```
```  5172   apply (subst negligible_def)
```
```  5173 proof -
```
```  5174   fix t :: "'a set"
```
```  5175   assume as: "negligible s"
```
```  5176   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)"
```
```  5177     by auto
```
```  5178   show "((indicator s::'a\<Rightarrow>real) has_integral 0) t"
```
```  5179     apply (subst has_integral_alt)
```
```  5180     apply cases
```
```  5181     apply (subst if_P,assumption)
```
```  5182     unfolding if_not_P
```
```  5183     apply safe
```
```  5184     apply (rule as[unfolded negligible_def,rule_format])
```
```  5185     apply (rule_tac x=1 in exI)
```
```  5186     apply safe
```
```  5187     apply (rule zero_less_one)
```
```  5188     apply (rule_tac x=0 in exI)
```
```  5189     using negligible_subset[OF as,of "s \<inter> t"]
```
```  5190     unfolding negligible_def indicator_def [abs_def]
```
```  5191     unfolding *
```
```  5192     apply auto
```
```  5193     done
```
```  5194 qed auto
```
```  5195
```
```  5196
```
```  5197 subsection \<open>Finite case of the spike theorem is quite commonly needed.\<close>
```
```  5198
```
```  5199 lemma has_integral_spike_finite:
```
```  5200   assumes "finite s"
```
```  5201     and "\<forall>x\<in>t-s. g x = f x"
```
```  5202     and "(f has_integral y) t"
```
```  5203   shows "(g has_integral y) t"
```
```  5204   apply (rule has_integral_spike)
```
```  5205   using assms
```
```  5206   apply auto
```
```  5207   done
```
```  5208
```
```  5209 lemma has_integral_spike_finite_eq:
```
```  5210   assumes "finite s"
```
```  5211     and "\<forall>x\<in>t-s. g x = f x"
```
```  5212   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  5213   apply rule
```
```  5214   apply (rule_tac[!] has_integral_spike_finite)
```
```  5215   using assms
```
```  5216   apply auto
```
```  5217   done
```
```  5218
```
```  5219 lemma integrable_spike_finite:
```
```  5220   assumes "finite s"
```
```  5221     and "\<forall>x\<in>t-s. g x = f x"
```
```  5222     and "f integrable_on t"
```
```  5223   shows "g integrable_on  t"
```
```  5224   using assms
```
```  5225   unfolding integrable_on_def
```
```  5226   apply safe
```
```  5227   apply (rule_tac x=y in exI)
```
```  5228   apply (rule has_integral_spike_finite)
```
```  5229   apply auto
```
```  5230   done
```
```  5231
```
```  5232
```
```  5233 subsection \<open>In particular, the boundary of an interval is negligible.\<close>
```
```  5234
```
```  5235 lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
```
```  5236 proof -
```
```  5237   let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
```
```  5238   have "cbox a b - box a b \<subseteq> ?A"
```
```  5239     apply rule unfolding Diff_iff mem_box
```
```  5240     apply simp
```
```  5241     apply(erule conjE bexE)+
```
```  5242     apply(rule_tac x=i in bexI)
```
```  5243     apply auto
```
```  5244     done
```
```  5245   then show ?thesis
```
```  5246     apply -
```
```  5247     apply (rule negligible_subset[of ?A])
```
```  5248     apply (rule negligible_Union[OF finite_imageI])
```
```  5249     apply auto
```
```  5250     done
```
```  5251 qed
```
```  5252
```
```  5253 lemma has_integral_spike_interior:
```
```  5254   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5255     and "(f has_integral y) (cbox a b)"
```
```  5256   shows "(g has_integral y) (cbox a b)"
```
```  5257   apply (rule has_integral_spike[OF negligible_frontier_interval _ assms(2)])
```
```  5258   using assms(1)
```
```  5259   apply auto
```
```  5260   done
```
```  5261
```
```  5262 lemma has_integral_spike_interior_eq:
```
```  5263   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5264   shows "(f has_integral y) (cbox a b) \<longleftrightarrow> (g has_integral y) (cbox a b)"
```
```  5265   apply rule
```
```  5266   apply (rule_tac[!] has_integral_spike_interior)
```
```  5267   using assms
```
```  5268   apply auto
```
```  5269   done
```
```  5270
```
```  5271 lemma integrable_spike_interior:
```
```  5272   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5273     and "f integrable_on cbox a b"
```
```  5274   shows "g integrable_on cbox a b"
```
```  5275   using assms
```
```  5276   unfolding integrable_on_def
```
```  5277   using has_integral_spike_interior[OF assms(1)]
```
```  5278   by auto
```
```  5279
```
```  5280
```
```  5281 subsection \<open>Integrability of continuous functions.\<close>
```
```  5282
```
```  5283 lemma operative_approximable:
```
```  5284   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5285   assumes "0 \<le> e"
```
```  5286   shows "comm_monoid.operative op \<and> True (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)"
```
```  5287   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  5288 proof safe
```
```  5289   fix a b :: 'b
```
```  5290   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5291     if "content (cbox a b) = 0"
```
```  5292     apply (rule_tac x=f in exI)
```
```  5293     using assms that
```
```  5294     apply (auto intro!: integrable_on_null)
```
```  5295     done
```
```  5296   {
```
```  5297     fix c g
```
```  5298     fix k :: 'b
```
```  5299     assume as: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  5300     assume k: "k \<in> Basis"
```
```  5301     show "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  5302       "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
```
```  5303       apply (rule_tac[!] x=g in exI)
```
```  5304       using as(1) integrable_split[OF as(2) k]
```
```  5305       apply auto
```
```  5306       done
```
```  5307   }
```
```  5308   fix c k g1 g2
```
```  5309   assume as: "\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e" "g1 integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  5310     "\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e" "g2 integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
```
```  5311   assume k: "k \<in> Basis"
```
```  5312   let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
```
```  5313   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5314     apply (rule_tac x="?g" in exI)
```
```  5315     apply safe
```
```  5316   proof goal_cases
```
```  5317     case (1 x)
```
```  5318     then show ?case
```
```  5319       apply -
```
```  5320       apply (cases "x\<bullet>k=c")
```
```  5321       apply (case_tac "x\<bullet>k < c")
```
```  5322       using as assms
```
```  5323       apply auto
```
```  5324       done
```
```  5325   next
```
```  5326     case 2
```
```  5327     presume "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  5328       and "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  5329     then guess h1 h2 unfolding integrable_on_def by auto
```
```  5330     from has_integral_split[OF this k] show ?case
```
```  5331       unfolding integrable_on_def by auto
```
```  5332   next
```
```  5333     show "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  5334       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
```
```  5335       using k as(2,4)
```
```  5336       apply auto
```
```  5337       done
```
```  5338   qed
```
```  5339 qed
```
```  5340
```
```  5341 lemma comm_monoid_set_F_and: "comm_monoid_set.F op \<and> True f s \<longleftrightarrow> (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
```
```  5342 proof -
```
```  5343   interpret bool: comm_monoid_set "op \<and>" True
```
```  5344     proof qed auto
```
```  5345   show ?thesis
```
```  5346     by (induction s rule: infinite_finite_induct) auto
```
```  5347 qed
```
```  5348
```
```  5349 lemma approximable_on_division:
```
```  5350   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5351   assumes "0 \<le> e"
```
```  5352     and "d division_of (cbox a b)"
```
```  5353     and "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  5354   obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  5355 proof -
```
```  5356   note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_approximable[OF assms(1)] assms(2)]
```
```  5357   from assms(3) this[unfolded comm_monoid_set_F_and, of f] division_of_finite[OF assms(2)]
```
```  5358   guess g by auto
```
```  5359   then show thesis
```
```  5360     apply -
```
```  5361     apply (rule that[of g])
```
```  5362     apply auto
```
```  5363     done
```
```  5364 qed
```
```  5365
```
```  5366 lemma integrable_continuous:
```
```  5367   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5368   assumes "continuous_on (cbox a b) f"
```
```  5369   shows "f integrable_on cbox a b"
```
```  5370 proof (rule integrable_uniform_limit, safe)
```
```  5371   fix e :: real
```
```  5372   assume e: "e > 0"
```
```  5373   from compact_uniformly_continuous[OF assms compact_cbox,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
```
```  5374   note d=conjunctD2[OF this,rule_format]
```
```  5375   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
```
```  5376   note p' = tagged_division_ofD[OF p(1)]
```
```  5377   have *: "\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  5378   proof (safe, unfold snd_conv)
```
```  5379     fix x l
```
```  5380     assume as: "(x, l) \<in> p"
```
```  5381     from p'(4)[OF this] guess a b by (elim exE) note l=this
```
```  5382     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l"
```
```  5383       apply (rule_tac x="\<lambda>y. f x" in exI)
```
```  5384     proof safe
```
```  5385       show "(\<lambda>y. f x) integrable_on l"
```
```  5386         unfolding integrable_on_def l
```
```  5387         apply rule
```
```  5388         apply (rule has_integral_const)
```
```  5389         done
```
```  5390       fix y
```
```  5391       assume y: "y \<in> l"
```
```  5392       note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
```
```  5393       note d(2)[OF _ _ this[unfolded mem_ball]]
```
```  5394       then show "norm (f y - f x) \<le> e"
```
```  5395         using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce
```
```  5396     qed
```
```  5397   qed
```
```  5398   from e have "e \<ge> 0"
```
```  5399     by auto
```
```  5400   from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
```
```  5401   then show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5402     by auto
```
```  5403 qed
```
```  5404
```
```  5405 lemma integrable_continuous_real:
```
```  5406   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5407   assumes "continuous_on {a .. b} f"
```
```  5408   shows "f integrable_on {a .. b}"
```
```  5409   by (metis assms box_real(2) integrable_continuous)
```
```  5410
```
```  5411 subsection \<open>Specialization of additivity to one dimension.\<close>
```
```  5412
```
```  5413 subsection \<open>Special case of additivity we need for the FTC.\<close>
```
```  5414
```
```  5415 lemma additive_tagged_division_1:
```
```  5416   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
```
```  5417   assumes "a \<le> b"
```
```  5418     and "p tagged_division_of {a..b}"
```
```  5419   shows "setsum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
```
```  5420 proof -
```
```  5421   let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
```
```  5422   have ***: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```  5423     using assms by auto
```
```  5424   have *: "add.operative ?f"
```
```  5425     unfolding add.operative_1_lt box_eq_empty
```
```  5426     by auto
```
```  5427   have **: "cbox a b \<noteq> {}"
```
```  5428     using assms(1) by auto
```
```  5429   note setsum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
```
```  5430   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
```
```  5431   show ?thesis
```
```  5432     unfolding *
```
```  5433     apply (rule setsum.cong)
```
```  5434     unfolding split_paired_all split_conv
```
```  5435     using assms(2)
```
```  5436     apply auto
```
```  5437     done
```
```  5438 qed
```
```  5439
```
```  5440
```
```  5441 subsection \<open>A useful lemma allowing us to factor out the content size.\<close>
```
```  5442
```
```  5443 lemma has_integral_factor_content:
```
```  5444   "(f has_integral i) (cbox a b) \<longleftrightarrow>
```
```  5445     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  5446       norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
```
```  5447 proof (cases "content (cbox a b) = 0")
```
```  5448   case True
```
```  5449   show ?thesis
```
```  5450     unfolding has_integral_null_eq[OF True]
```
```  5451     apply safe
```
```  5452     apply (rule, rule, rule gauge_trivial, safe)
```
```  5453     unfolding setsum_content_null[OF True] True
```
```  5454     defer
```
```  5455     apply (erule_tac x=1 in allE)
```
```  5456     apply safe
```
```  5457     defer
```
```  5458     apply (rule fine_division_exists[of _ a b])
```
```  5459     apply assumption
```
```  5460     apply (erule_tac x=p in allE)
```
```  5461     unfolding setsum_content_null[OF True]
```
```  5462     apply auto
```
```  5463     done
```
```  5464 next
```
```  5465   case False
```
```  5466   note F = this[unfolded content_lt_nz[symmetric]]
```
```  5467   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
```
```  5468     (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
```
```  5469   show ?thesis
```
```  5470     apply (subst has_integral)
```
```  5471   proof safe
```
```  5472     fix e :: real
```
```  5473     assume e: "e > 0"
```
```  5474     {
```
```  5475       assume "\<forall>e>0. ?P e op <"
```
```  5476       then show "?P (e * content (cbox a b)) op \<le>"
```
```  5477         apply (erule_tac x="e * content (cbox a b)" in allE)
```
```  5478         apply (erule impE)
```
```  5479         defer
```
```  5480         apply (erule exE,rule_tac x=d in exI)
```
```  5481         using F e
```
```  5482         apply (auto simp add:field_simps)
```
```  5483         done
```
```  5484     }
```
```  5485     {
```
```  5486       assume "\<forall>e>0. ?P (e * content (cbox a b)) op \<le>"
```
```  5487       then show "?P e op <"
```
```  5488         apply (erule_tac x="e / 2 / content (cbox a b)" in allE)
```
```  5489         apply (erule impE)
```
```  5490         defer
```
```  5491         apply (erule exE,rule_tac x=d in exI)
```
```  5492         using F e
```
```  5493         apply (auto simp add: field_simps)
```
```  5494         done
```
```  5495     }
```
```  5496   qed
```
```  5497 qed
```
```  5498
```
```  5499 lemma has_integral_factor_content_real:
```
```  5500   "(f has_integral i) {a .. b::real} \<longleftrightarrow>
```
```  5501     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a .. b}  \<and> d fine p \<longrightarrow>
```
```  5502       norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a .. b} ))"
```
```  5503   unfolding box_real[symmetric]
```
```  5504   by (rule has_integral_factor_content)
```
```  5505
```
```  5506
```
```  5507 subsection \<open>Fundamental theorem of calculus.\<close>
```
```  5508
```
```  5509 lemma interval_bounds_real:
```
```  5510   fixes q b :: real
```
```  5511   assumes "a \<le> b"
```
```  5512   shows "Sup {a..b} = b"
```
```  5513     and "Inf {a..b} = a"
```
```  5514   using assms by auto
```
```  5515
```
```  5516 lemma fundamental_theorem_of_calculus:
```
```  5517   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5518   assumes "a \<le> b"
```
```  5519     and "\<forall>x\<in>{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
```
```  5520   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  5521   unfolding has_integral_factor_content box_real[symmetric]
```
```  5522 proof safe
```
```  5523   fix e :: real
```
```  5524   assume e: "e > 0"
```
```  5525   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
```
```  5526   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"
```
```  5527     using e by blast
```
```  5528   note this[OF assm,unfolded gauge_existence_lemma]
```
```  5529   from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
```
```  5530   note d=conjunctD2[OF this[rule_format],rule_format]
```
```  5531   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  5532     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
```
```  5533     apply (rule_tac x="\<lambda>x. ball x (d x)" in exI)
```
```  5534     apply safe
```
```  5535     apply (rule gauge_ball_dependent)
```
```  5536     apply rule
```
```  5537     apply (rule d(1))
```
```  5538   proof -
```
```  5539     fix p
```
```  5540     assume as: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
```
```  5541     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
```
```  5542       unfolding content_real[OF assms(1), simplified box_real[symmetric]] additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of f,symmetric]
```
```  5543       unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "\<lambda>x. x",symmetric]
```
```  5544       unfolding setsum_right_distrib
```
```  5545       defer
```
```  5546       unfolding setsum_subtractf[symmetric]
```
```  5547     proof (rule setsum_norm_le,safe)
```
```  5548       fix x k
```
```  5549       assume "(x, k) \<in> p"
```
```  5550       note xk = tagged_division_ofD(2-4)[OF as(1) this]
```
```  5551       from this(3) guess u v by (elim exE) note k=this
```
```  5552       have *: "u \<le> v"
```
```  5553         using xk unfolding k by auto
```
```  5554       have ball: "\<forall>xa\<in>k. xa \<in> ball x (d x)"
```
```  5555         using as(2)[unfolded fine_def,rule_format,OF \<open>(x,k)\<in>p\<close>,unfolded split_conv subset_eq] .
```
```  5556       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
```
```  5557         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
```
```  5558         apply (rule order_trans[OF _ norm_triangle_ineq4])
```
```  5559         apply (rule eq_refl)
```
```  5560         apply (rule arg_cong[where f=norm])
```
```  5561         unfolding scaleR_diff_left
```
```  5562         apply (auto simp add:algebra_simps)
```
```  5563         done
```
```  5564       also have "\<dots> \<le> e * norm (u - x) + e * norm (v - x)"
```
```  5565         apply (rule add_mono)
```
```  5566         apply (rule d(2)[of "x" "u",unfolded o_def])
```
```  5567         prefer 4
```
```  5568         apply (rule d(2)[of "x" "v",unfolded o_def])
```
```  5569         using ball[rule_format,of u] ball[rule_format,of v]
```
```  5570         using xk(1-2)
```
```  5571         unfolding k subset_eq
```
```  5572         apply (auto simp add:dist_real_def)
```
```  5573         done
```
```  5574       also have "\<dots> \<le> e * (Sup k - Inf k)"
```
```  5575         unfolding k interval_bounds_real[OF *]
```
```  5576         using xk(1)
```
```  5577         unfolding k
```
```  5578         by (auto simp add: dist_real_def field_simps)
```
```  5579       finally show "norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le>
```
```  5580         e * (Sup k - Inf k)"
```
```  5581         unfolding box_real k interval_bounds_real[OF *] content_real[OF *]
```
```  5582           interval_upperbound_real interval_lowerbound_real
```
```  5583           .
```
```  5584     qed
```
```  5585   qed
```
```  5586 qed
```
```  5587
```
```  5588 lemma ident_has_integral:
```
```  5589   fixes a::real
```
```  5590   assumes "a \<le> b"
```
```  5591   shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2) / 2) {a..b}"
```
```  5592 proof -
```
```  5593   have "((\<lambda>x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}"
```
```  5594     apply (rule fundamental_theorem_of_calculus [OF assms], clarify)
```
```  5595     unfolding power2_eq_square
```
```  5596     by (rule derivative_eq_intros | simp)+
```
```  5597   then show ?thesis
```
```  5598     by (simp add: field_simps)
```
```  5599 qed
```
```  5600
```
```  5601 lemma integral_ident [simp]:
```
```  5602   fixes a::real
```
```  5603   assumes "a \<le> b"
```
```  5604   shows "integral {a..b} (\<lambda>x. x) = (if a \<le> b then (b\<^sup>2 - a\<^sup>2) / 2 else 0)"
```
```  5605 using ident_has_integral integral_unique by fastforce
```
```  5606
```
```  5607 lemma ident_integrable_on:
```
```  5608   fixes a::real
```
```  5609   shows "(\<lambda>x. x) integrable_on {a..b}"
```
```  5610 by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
```
```  5611
```
```  5612
```
```  5613 subsection \<open>Taylor series expansion\<close>
```
```  5614
```
```  5615 lemma (in bounded_bilinear) setsum_prod_derivatives_has_vector_derivative:
```
```  5616   assumes "p>0"
```
```  5617   and f0: "Df 0 = f"
```
```  5618   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5619     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  5620   and g0: "Dg 0 = g"
```
```  5621   and Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5622     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  5623   and ivl: "a \<le> t" "t \<le> b"
```
```  5624   shows "((\<lambda>t. \<Sum>i<p. (-1)^i *\<^sub>R prod (Df i t) (Dg (p - Suc i) t))
```
```  5625     has_vector_derivative
```
```  5626       prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t))
```
```  5627     (at t within {a .. b})"
```
```  5628   using assms
```
```  5629 proof cases
```
```  5630   assume p: "p \<noteq> 1"
```
```  5631   define p' where "p' = p - 2"
```
```  5632   from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
```
```  5633     by (auto simp: p'_def)
```
```  5634   have *: "\<And>i. i \<le> p' \<Longrightarrow> Suc (Suc p' - i) = (Suc (Suc p') - i)"
```
```  5635     by auto
```
```  5636   let ?f = "\<lambda>i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))"
```
```  5637   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  5638     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
```
```  5639     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
```
```  5640     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
```
```  5641   also note setsum_telescope
```
```  5642   finally
```
```  5643   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  5644     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
```
```  5645     = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)"
```
```  5646     unfolding p'[symmetric]
```
```  5647     by (simp add: assms)
```
```  5648   thus ?thesis
```
```  5649     using assms
```
```  5650     by (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  5651 qed (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  5652
```
```  5653 lemma
```
```  5654   fixes f::"real\<Rightarrow>'a::banach"
```
```  5655   assumes "p>0"
```
```  5656   and f0: "Df 0 = f"
```
```  5657   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5658     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  5659   and ivl: "a \<le> b"
```
```  5660   defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"
```
```  5661   shows taylor_has_integral:
```
```  5662     "(i has_integral f b - (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
```
```  5663   and taylor_integral:
```
```  5664     "f b = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i"
```
```  5665   and taylor_integrable:
```
```  5666     "i integrable_on {a .. b}"
```
```  5667 proof goal_cases
```
```  5668   case 1
```
```  5669   interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a"
```
```  5670     by (rule bounded_bilinear_scaleR)
```
```  5671   define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
```
```  5672   define Dg where [abs_def]:
```
```  5673     "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
```
```  5674   have g0: "Dg 0 = g"
```
```  5675     using \<open>p > 0\<close>
```
```  5676     by (auto simp add: Dg_def divide_simps g_def split: if_split_asm)
```
```  5677   {
```
```  5678     fix m
```
```  5679     assume "p > Suc m"
```
```  5680     hence "p - Suc m = Suc (p - Suc (Suc m))"
```
```  5681       by auto
```
```  5682     hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
```
```  5683       by auto
```
```  5684   } note fact_eq = this
```
```  5685   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5686     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  5687     unfolding Dg_def
```
```  5688     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
```
```  5689   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
```
```  5690   from setsum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
```
```  5691       OF \<open>p > 0\<close> g0 Dg f0 Df]
```
```  5692   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5693     (?sum has_vector_derivative
```
```  5694       g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})"
```
```  5695     by auto
```
```  5696   from fundamental_theorem_of_calculus[rule_format, OF \<open>a \<le> b\<close> deriv]
```
```  5697   have "(i has_integral ?sum b - ?sum a) {a .. b}"
```
```  5698     by (simp add: i_def g_def Dg_def)
```
```  5699   also
```
```  5700   have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
```
```  5701     and "{..<p} \<inter> {i. p = Suc i} = {p - 1}"
```
```  5702     for p'
```
```  5703     using \<open>p > 0\<close>
```
```  5704     by (auto simp: power_mult_distrib[symmetric])
```
```  5705   then have "?sum b = f b"
```
```  5706     using Suc_pred'[OF \<open>p > 0\<close>]
```
```  5707     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
```
```  5708         cond_application_beta setsum.If_cases f0)
```
```  5709   also
```
```  5710   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
```
```  5711   proof safe
```
```  5712     fix x
```
```  5713     assume "x < p"
```
```  5714     thus "x \<in> (\<lambda>x. p - x - 1) ` {..<p}"
```
```  5715       by (auto intro!: image_eqI[where x = "p - x - 1"])
```
```  5716   qed simp
```
```  5717   from _ this
```
```  5718   have "?sum a = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)"
```
```  5719     by (rule setsum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
```
```  5720   finally show c: ?case .
```
```  5721   case 2 show ?case using c integral_unique by force
```
```  5722   case 3 show ?case using c by force
```
```  5723 qed
```
```  5724
```
```  5725
```
```  5726 subsection \<open>Attempt a systematic general set of "offset" results for components.\<close>
```
```  5727
```
```  5728 lemma gauge_modify:
```
```  5729   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
```
```  5730   shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})"
```
```  5731   using assms
```
```  5732   unfolding gauge_def
```
```  5733   apply safe
```
```  5734   defer
```
```  5735   apply (erule_tac x="f x" in allE)
```
```  5736   apply (erule_tac x="d (f x)" in allE)
```
```  5737   apply auto
```
```  5738   done
```
```  5739
```
```  5740
```
```  5741 subsection \<open>Only need trivial subintervals if the interval itself is trivial.\<close>
```
```  5742
```
```  5743 lemma division_of_nontrivial:
```
```  5744   fixes s :: "'a::euclidean_space set set"
```
```  5745   assumes "s division_of (cbox a b)"
```
```  5746     and "content (cbox a b) \<noteq> 0"
```
```  5747   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of (cbox a b)"
```
```  5748   using assms(1)
```
```  5749   apply -
```
```  5750 proof (induct "card s" arbitrary: s rule: nat_less_induct)
```
```  5751   fix s::"'a set set"
```
```  5752   assume assm: "s division_of (cbox a b)"
```
```  5753     "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow>
```
```  5754       x division_of (cbox a b) \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of (cbox a b)"
```
```  5755   note s = division_ofD[OF assm(1)]
```
```  5756   let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of (cbox a b)"
```
```  5757   {
```
```  5758     presume *: "{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
```
```  5759     show ?thesis
```
```  5760       apply cases
```
```  5761       defer
```
```  5762       apply (rule *)
```
```  5763       apply assumption
```
```  5764       using assm(1)
```
```  5765       apply auto
```
```  5766       done
```
```  5767   }
```
```  5768   assume noteq: "{k \<in> s. content k \<noteq> 0} \<noteq> s"
```
```  5769   then obtain k where k: "k \<in> s" "content k = 0"
```
```  5770     by auto
```
```  5771   from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
```
```  5772   from k have "card s > 0"
```
```  5773     unfolding card_gt_0_iff using assm(1) by auto
```
```  5774   then have card: "card (s - {k}) < card s"
```
```  5775     using assm(1) k(1)
```
```  5776     apply (subst card_Diff_singleton_if)
```
```  5777     apply auto
```
```  5778     done
```
```  5779   have *: "closed (\<Union>(s - {k}))"
```
```  5780     apply (rule closed_Union)
```
```  5781     defer
```
```  5782     apply rule
```
```  5783     apply (drule DiffD1,drule s(4))
```
```  5784     using assm(1)
```
```  5785     apply auto
```
```  5786     done
```
```  5787   have "k \<subseteq> \<Union>(s - {k})"
```
```  5788     apply safe
```
```  5789     apply (rule *[unfolded closed_limpt,rule_format])
```
```  5790     unfolding islimpt_approachable
```
```  5791   proof safe
```
```  5792     fix x
```
```  5793     fix e :: real
```
```  5794     assume as: "x \<in> k" "e > 0"
```
```  5795     from k(2)[unfolded k content_eq_0] guess i ..
```
```  5796     then have i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis"
```
```  5797       using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
```
```  5798     then have xi: "x\<bullet>i = d\<bullet>i"
```
```  5799       using as unfolding k mem_box by (metis antisym)
```
```  5800     define y where "y = (\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i) / 2 then c\<bullet>i +
```
```  5801       min e (b\<bullet>i - c\<bullet>i) / 2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i) / 2 else x\<bullet>j) *\<^sub>R j)"
```
```  5802     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e"
```
```  5803       apply (rule_tac x=y in bexI)
```
```  5804     proof
```
```  5805       have "d \<in> cbox c d"
```
```  5806         using s(3)[OF k(1)]
```
```  5807         unfolding k box_eq_empty mem_box
```
```  5808         by (fastforce simp add: not_less)
```
```  5809       then have "d \<in> cbox a b"
```
```  5810         using s(2)[OF k(1)]
```
```  5811         unfolding k
```
```  5812         by auto
```
```  5813       note di = this[unfolded mem_box,THEN bspec[where x=i]]
```
```  5814       then have xyi: "y\<bullet>i \<noteq> x\<bullet>i"
```
```  5815         unfolding y_def i xi
```
```  5816         using as(2) assms(2)[unfolded content_eq_0] i(2)
```
```  5817         by (auto elim!: ballE[of _ _ i])
```
```  5818       then show "y \<noteq> x"
```
```  5819         unfolding euclidean_eq_iff[where 'a='a] using i by auto
```
```  5820       have *: "Basis = insert i (Basis - {i})"
```
```  5821         using i by auto
```
```  5822       have "norm (y - x) < e + setsum (\<lambda>i. 0) Basis"
```
```  5823         apply (rule le_less_trans[OF norm_le_l1])
```
```  5824         apply (subst *)
```
```  5825         apply (subst setsum.insert)
```
```  5826         prefer 3
```
```  5827         apply (rule add_less_le_mono)
```
```  5828       proof -
```
```  5829         show "\<bar>(y - x) \<bullet> i\<bar> < e"
```
```  5830           using di as(2) y_def i xi by (auto simp: inner_simps)
```
```  5831         show "(\<Sum>i\<in>Basis - {i}. \<bar>(y - x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
```
```  5832           unfolding y_def by (auto simp: inner_simps)
```
```  5833       qed auto
```
```  5834       then show "dist y x < e"
```
```  5835         unfolding dist_norm by auto
```
```  5836       have "y \<notin> k"
```
```  5837         unfolding k mem_box
```
```  5838         apply rule
```
```  5839         apply (erule_tac x=i in ballE)
```
```  5840         using xyi k i xi
```
```  5841         apply auto
```
```  5842         done
```
```  5843       moreover
```
```  5844       have "y \<in> \<Union>s"
```
```  5845         using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i
```
```  5846         unfolding s mem_box y_def
```
```  5847         by (auto simp: field_simps elim!: ballE[of _ _ i])
```
```  5848       ultimately
```
```  5849       show "y \<in> \<Union>(s - {k})" by auto
```
```  5850     qed
```
```  5851   qed
```
```  5852   then have "\<Union>(s - {k}) = cbox a b"
```
```  5853     unfolding s(6)[symmetric] by auto
```
```  5854   then have  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of (cbox a b)"
```
```  5855     apply -
```
```  5856     apply (rule assm(2)[rule_format,OF card refl])
```
```  5857     apply (rule division_ofI)
```
```  5858     defer
```
```  5859     apply (rule_tac[1-4] s)
```
```  5860     using assm(1)
```
```  5861     apply auto
```
```  5862     done
```
```  5863   moreover
```
```  5864   have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}"
```
```  5865     using k by auto
```
```  5866   ultimately show ?thesis by auto
```
```  5867 qed
```
```  5868
```
```  5869
```
```  5870 subsection \<open>Integrability on subintervals.\<close>
```
```  5871
```
```  5872 lemma operative_integrable:
```
```  5873   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5874   shows "comm_monoid.operative op \<and> True (\<lambda>i. f integrable_on i)"
```
```  5875   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  5876   apply safe
```
```  5877   apply (subst integrable_on_def)
```
```  5878   unfolding has_integral_null_eq
```
```  5879   apply (rule, rule refl)
```
```  5880   apply (rule, assumption, assumption)+
```
```  5881   unfolding integrable_on_def
```
```  5882   by (auto intro!: has_integral_split)
```
```  5883
```
```  5884 lemma integrable_subinterval:
```
```  5885   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5886   assumes "f integrable_on cbox a b"
```
```  5887     and "cbox c d \<subseteq> cbox a b"
```
```  5888   shows "f integrable_on cbox c d"
```
```  5889   apply (cases "cbox c d = {}")
```
```  5890   defer
```
```  5891   apply (rule partial_division_extend_1[OF assms(2)],assumption)
```
```  5892   using comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable,symmetric,of _ _ _ f] assms(1)
```
```  5893   apply (auto simp: comm_monoid_set_F_and)
```
```  5894   done
```
```  5895
```
```  5896 lemma integrable_subinterval_real:
```
```  5897   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5898   assumes "f integrable_on {a .. b}"
```
```  5899     and "{c .. d} \<subseteq> {a .. b}"
```
```  5900   shows "f integrable_on {c .. d}"
```
```  5901   by (metis assms(1) assms(2) box_real(2) integrable_subinterval)
```
```  5902
```
```  5903
```
```  5904 subsection \<open>Combining adjacent intervals in 1 dimension.\<close>
```
```  5905
```
```  5906 lemma has_integral_combine:
```
```  5907   fixes a b c :: real
```
```  5908   assumes "a \<le> c"
```
```  5909     and "c \<le> b"
```
```  5910     and "(f has_integral i) {a .. c}"
```
```  5911     and "(f has_integral (j::'a::banach)) {c .. b}"
```
```  5912   shows "(f has_integral (i + j)) {a .. b}"
```
```  5913 proof -
```
```  5914   interpret comm_monoid "lift_option plus" "Some (0::'a)"
```
```  5915     by (rule comm_monoid_lift_option)
```
```  5916       (rule add.comm_monoid_axioms)
```
```  5917   note operative_integral [of f, unfolded operative_1_le]
```
```  5918   note conjunctD2 [OF this, rule_format]
```
```  5919   note * = this(2) [OF conjI [OF assms(1-2)],
```
```  5920     unfolded if_P [OF assms(3)]]
```
```  5921   then have "f integrable_on cbox a b"
```
```  5922     apply -
```
```  5923     apply (rule ccontr)
```
```  5924     apply (subst(asm) if_P)
```
```  5925     defer
```
```  5926     apply (subst(asm) if_P)
```
```  5927     using assms(3-)
```
```  5928     apply auto
```
```  5929     done
```
```  5930   with *
```
```  5931   show ?thesis
```
```  5932     apply -
```
```  5933     apply (subst(asm) if_P)
```
```  5934     defer
```
```  5935     apply (subst(asm) if_P)
```
```  5936     defer
```
```  5937     apply (subst(asm) if_P)
```
```  5938     using assms(3-)
```
```  5939     apply (auto simp add: integrable_on_def integral_unique)
```
```  5940     done
```
```  5941 qed
```
```  5942
```
```  5943 lemma integral_combine:
```
```  5944   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5945   assumes "a \<le> c"
```
```  5946     and "c \<le> b"
```
```  5947     and "f integrable_on {a .. b}"
```
```  5948   shows "integral {a .. c} f + integral {c .. b} f = integral {a .. b} f"
```
```  5949   apply (rule integral_unique[symmetric])
```
```  5950   apply (rule has_integral_combine[OF assms(1-2)])
```
```  5951   apply (metis assms(2) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel2 monoid_add_class.add.left_neutral)
```
```  5952   by (metis assms(1) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel1 monoid_add_class.add.right_neutral)
```
```  5953
```
```  5954 lemma integrable_combine:
```
```  5955   fixes f :: "real \<Rightarrow> 'a::banach"
```
`  5956   assumes "a \<le> c"`