src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
 author hoelzl Fri Sep 23 10:26:04 2016 +0200 (2016-09-23) changeset 63940 0d82c4c94014 parent 63938 f6ce08859d4c child 63941 f353674c2528 permissions -rw-r--r--
prove HK-integrable implies Lebesgue measurable; prove HK-integral equals Lebesgue integral for nonneg functions
```     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   Lebesgue_Measure
```
```    10 begin
```
```    11
```
```    12 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
```
```    13   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
```
```    14   scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
```
```    15
```
```    16
```
```    17 subsection \<open>Sundries\<close>
```
```    18
```
```    19 lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
```
```    20 lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
```
```    21 lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
```
```    22
```
```    23 lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
```
```    24   by auto
```
```    25
```
```    26 declare norm_triangle_ineq4[intro]
```
```    27
```
```    28 lemma transitive_stepwise_le:
```
```    29   assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)"
```
```    30   shows "\<forall>n\<ge>m. R m n"
```
```    31 proof (intro allI impI)
```
```    32   show "m \<le> n \<Longrightarrow> R m n" for n
```
```    33     by (induction rule: dec_induct)
```
```    34        (use assms in blast)+
```
```    35 qed
```
```    36
```
```    37 subsection \<open>Some useful lemmas about intervals.\<close>
```
```    38
```
```    39 lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
```
```    40   using nonempty_Basis
```
```    41   by (fastforce simp add: set_eq_iff mem_box)
```
```    42
```
```    43 lemma interior_subset_union_intervals:
```
```    44   assumes "i = cbox a b"
```
```    45     and "j = cbox c d"
```
```    46     and "interior j \<noteq> {}"
```
```    47     and "i \<subseteq> j \<union> s"
```
```    48     and "interior i \<inter> interior j = {}"
```
```    49   shows "interior i \<subseteq> interior s"
```
```    50 proof -
```
```    51   have "box a b \<inter> cbox c d = {}"
```
```    52      using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
```
```    53      unfolding assms(1,2) interior_cbox by auto
```
```    54   moreover
```
```    55   have "box a b \<subseteq> cbox c d \<union> s"
```
```    56     apply (rule order_trans,rule box_subset_cbox)
```
```    57     using assms(4) unfolding assms(1,2)
```
```    58     apply auto
```
```    59     done
```
```    60   ultimately
```
```    61   show ?thesis
```
```    62     unfolding assms interior_cbox
```
```    63       by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
```
```    64 qed
```
```    65
```
```    66 lemma interior_Union_subset_cbox:
```
```    67   assumes "finite f"
```
```    68   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"
```
```    69     and t: "closed t"
```
```    70   shows "interior (\<Union>f) \<subseteq> t"
```
```    71 proof -
```
```    72   have [simp]: "s \<in> f \<Longrightarrow> closed s" for s
```
```    73     using f by auto
```
```    74   define E where "E = {s\<in>f. interior s = {}}"
```
```    75   then have "finite E" "E \<subseteq> {s\<in>f. interior s = {}}"
```
```    76     using \<open>finite f\<close> by auto
```
```    77   then have "interior (\<Union>f) = interior (\<Union>(f - E))"
```
```    78   proof (induction E rule: finite_subset_induct')
```
```    79     case (insert s f')
```
```    80     have "interior (\<Union>(f - insert s f') \<union> s) = interior (\<Union>(f - insert s f'))"
```
```    81       using insert.hyps \<open>finite f\<close> by (intro interior_closed_Un_empty_interior) auto
```
```    82     also have "\<Union>(f - insert s f') \<union> s = \<Union>(f - f')"
```
```    83       using insert.hyps by auto
```
```    84     finally show ?case
```
```    85       by (simp add: insert.IH)
```
```    86   qed simp
```
```    87   also have "\<dots> \<subseteq> \<Union>(f - E)"
```
```    88     by (rule interior_subset)
```
```    89   also have "\<dots> \<subseteq> t"
```
```    90   proof (rule Union_least)
```
```    91     fix s assume "s \<in> f - E"
```
```    92     with f[of s] obtain a b where s: "s \<in> f" "s = cbox a b" "box a b \<noteq> {}"
```
```    93       by (fastforce simp: E_def)
```
```    94     have "closure (interior s) \<subseteq> closure t"
```
```    95       by (intro closure_mono f \<open>s \<in> f\<close>)
```
```    96     with s \<open>closed t\<close> show "s \<subseteq> t"
```
```    97       by (simp add: closure_box)
```
```    98   qed
```
```    99   finally show ?thesis .
```
```   100 qed
```
```   101
```
```   102 lemma inter_interior_unions_intervals:
```
```   103     "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) = {}"
```
```   104   using interior_Union_subset_cbox[of f "UNIV - s"] by auto
```
```   105
```
```   106 lemma interval_split:
```
```   107   fixes a :: "'a::euclidean_space"
```
```   108   assumes "k \<in> Basis"
```
```   109   shows
```
```   110     "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)"
```
```   111     "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"
```
```   112   apply (rule_tac[!] set_eqI)
```
```   113   unfolding Int_iff mem_box mem_Collect_eq
```
```   114   using assms
```
```   115   apply auto
```
```   116   done
```
```   117
```
```   118 lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
```
```   119   by (simp add: box_ne_empty)
```
```   120
```
```   121 subsection \<open>Bounds on intervals where they exist.\<close>
```
```   122
```
```   123 definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
```
```   124   where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
```
```   125
```
```   126 definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
```
```   127   where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
```
```   128
```
```   129 lemma interval_upperbound[simp]:
```
```   130   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
```
```   131     interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
```
```   132   unfolding interval_upperbound_def euclidean_representation_setsum cbox_def
```
```   133   by (safe intro!: cSup_eq) auto
```
```   134
```
```   135 lemma interval_lowerbound[simp]:
```
```   136   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
```
```   137     interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
```
```   138   unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def
```
```   139   by (safe intro!: cInf_eq) auto
```
```   140
```
```   141 lemmas interval_bounds = interval_upperbound interval_lowerbound
```
```   142
```
```   143 lemma
```
```   144   fixes X::"real set"
```
```   145   shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
```
```   146     and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
```
```   147   by (auto simp: interval_upperbound_def interval_lowerbound_def)
```
```   148
```
```   149 lemma interval_bounds'[simp]:
```
```   150   assumes "cbox a b \<noteq> {}"
```
```   151   shows "interval_upperbound (cbox a b) = b"
```
```   152     and "interval_lowerbound (cbox a b) = a"
```
```   153   using assms unfolding box_ne_empty by auto
```
```   154
```
```   155 lemma interval_upperbound_Times:
```
```   156   assumes "A \<noteq> {}" and "B \<noteq> {}"
```
```   157   shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
```
```   158 proof-
```
```   159   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
```
```   160   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)"
```
```   161       by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
```
```   162   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
```
```   163   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)"
```
```   164       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
```
```   165   ultimately show ?thesis unfolding interval_upperbound_def
```
```   166       by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
```
```   167 qed
```
```   168
```
```   169 lemma interval_lowerbound_Times:
```
```   170   assumes "A \<noteq> {}" and "B \<noteq> {}"
```
```   171   shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
```
```   172 proof-
```
```   173   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
```
```   174   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)"
```
```   175       by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
```
```   176   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
```
```   177   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)"
```
```   178       by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
```
```   179   ultimately show ?thesis unfolding interval_lowerbound_def
```
```   180       by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
```
```   181 qed
```
```   182
```
```   183 subsection \<open>Content (length, area, volume...) of an interval.\<close>
```
```   184
```
```   185 abbreviation content :: "'a::euclidean_space set \<Rightarrow> real"
```
```   186   where "content s \<equiv> measure lborel s"
```
```   187
```
```   188 lemma content_cbox_cases:
```
```   189   "content (cbox a b) = (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)"
```
```   190   by (simp add: measure_lborel_cbox_eq inner_diff)
```
```   191
```
```   192 lemma content_cbox: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
```
```   193   unfolding content_cbox_cases by simp
```
```   194
```
```   195 lemma content_cbox': "cbox a b \<noteq> {} \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
```
```   196   by (simp add: box_ne_empty inner_diff)
```
```   197
```
```   198 lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
```
```   199   by simp
```
```   200
```
```   201 lemma abs_eq_content: "\<bar>y - x\<bar> = (if x\<le>y then content {x .. y} else content {y..x})"
```
```   202   by (auto simp: content_real)
```
```   203
```
```   204 lemma content_singleton: "content {a} = 0"
```
```   205   by simp
```
```   206
```
```   207 lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1"
```
```   208   by simp
```
```   209
```
```   210 lemma content_pos_le[intro]: "0 \<le> content (cbox a b)"
```
```   211   by simp
```
```   212
```
```   213 corollary content_nonneg [simp]: "~ content (cbox a b) < 0"
```
```   214   using not_le by blast
```
```   215
```
```   216 lemma content_pos_lt: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> 0 < content (cbox a b)"
```
```   217   by (auto simp: less_imp_le inner_diff box_eq_empty intro!: setprod_pos)
```
```   218
```
```   219 lemma content_eq_0: "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
```
```   220   by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl)
```
```   221
```
```   222 lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
```
```   223   unfolding content_eq_0 interior_cbox box_eq_empty by auto
```
```   224
```
```   225 lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
```
```   226   by (auto simp add: content_cbox_cases less_le setprod_nonneg)
```
```   227
```
```   228 lemma content_empty [simp]: "content {} = 0"
```
```   229   by simp
```
```   230
```
```   231 lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
```
```   232   by (simp add: content_real)
```
```   233
```
```   234 lemma content_subset: "cbox a b \<subseteq> cbox c d \<Longrightarrow> content (cbox a b) \<le> content (cbox c d)"
```
```   235   unfolding measure_def
```
```   236   by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq)
```
```   237
```
```   238 lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
```
```   239   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
```
```   240
```
```   241 lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
```
```   242   unfolding measure_lborel_cbox_eq Basis_prod_def
```
```   243   apply (subst setprod.union_disjoint)
```
```   244   apply (auto simp: bex_Un ball_Un)
```
```   245   apply (subst (1 2) setprod.reindex_nontrivial)
```
```   246   apply auto
```
```   247   done
```
```   248
```
```   249 lemma content_cbox_pair_eq0_D:
```
```   250    "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
```
```   251   by (simp add: content_Pair)
```
```   252
```
```   253 lemma content_0_subset: "content(cbox a b) = 0 \<Longrightarrow> s \<subseteq> cbox a b \<Longrightarrow> content s = 0"
```
```   254   using emeasure_mono[of s "cbox a b" lborel]
```
```   255   by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq)
```
```   256
```
```   257 lemma content_split:
```
```   258   fixes a :: "'a::euclidean_space"
```
```   259   assumes "k \<in> Basis"
```
```   260   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})"
```
```   261   -- \<open>Prove using measure theory\<close>
```
```   262 proof cases
```
```   263   note simps = interval_split[OF assms] content_cbox_cases
```
```   264   have *: "Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
```
```   265     using assms by auto
```
```   266   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))"
```
```   267     "(\<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)"
```
```   268     apply (subst *(1))
```
```   269     defer
```
```   270     apply (subst *(1))
```
```   271     unfolding setprod.insert[OF *(2-)]
```
```   272     apply auto
```
```   273     done
```
```   274   assume as: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```   275   moreover
```
```   276   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
```
```   277     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)"
```
```   278     by  (auto simp add: field_simps)
```
```   279   moreover
```
```   280   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)) =
```
```   281       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
```
```   282     "(\<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)) =
```
```   283       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
```
```   284     by (auto intro!: setprod.cong)
```
```   285   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
```
```   286     unfolding not_le
```
```   287     using as[unfolded ,rule_format,of k] assms
```
```   288     by auto
```
```   289   ultimately show ?thesis
```
```   290     using assms
```
```   291     unfolding simps **
```
```   292     unfolding *(1)[of "\<lambda>i x. b\<bullet>i - x"] *(1)[of "\<lambda>i x. x - a\<bullet>i"]
```
```   293     unfolding *(2)
```
```   294     by auto
```
```   295 next
```
```   296   assume "\<not> (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
```
```   297   then have "cbox a b = {}"
```
```   298     unfolding box_eq_empty by (auto simp: not_le)
```
```   299   then show ?thesis
```
```   300     by (auto simp: not_le)
```
```   301 qed
```
```   302
```
```   303 subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
```
```   304
```
```   305 definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
```
```   306
```
```   307 lemma gaugeI:
```
```   308   assumes "\<And>x. x \<in> g x"
```
```   309     and "\<And>x. open (g x)"
```
```   310   shows "gauge g"
```
```   311   using assms unfolding gauge_def by auto
```
```   312
```
```   313 lemma gaugeD[dest]:
```
```   314   assumes "gauge d"
```
```   315   shows "x \<in> d x"
```
```   316     and "open (d x)"
```
```   317   using assms unfolding gauge_def by auto
```
```   318
```
```   319 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
```
```   320   unfolding gauge_def by auto
```
```   321
```
```   322 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
```
```   323   unfolding gauge_def by auto
```
```   324
```
```   325 lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
```
```   326   by (rule gauge_ball) auto
```
```   327
```
```   328 lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
```
```   329   unfolding gauge_def by auto
```
```   330
```
```   331 lemma gauge_inters:
```
```   332   assumes "finite s"
```
```   333     and "\<forall>d\<in>s. gauge (f d)"
```
```   334   shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
```
```   335 proof -
```
```   336   have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
```
```   337     by auto
```
```   338   show ?thesis
```
```   339     unfolding gauge_def unfolding *
```
```   340     using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
```
```   341 qed
```
```   342
```
```   343 lemma gauge_existence_lemma:
```
```   344   "(\<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)"
```
```   345   by (metis zero_less_one)
```
```   346
```
```   347
```
```   348 subsection \<open>Divisions.\<close>
```
```   349
```
```   350 definition division_of (infixl "division'_of" 40)
```
```   351 where
```
```   352   "s division_of i \<longleftrightarrow>
```
```   353     finite s \<and>
```
```   354     (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```   355     (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
```
```   356     (\<Union>s = i)"
```
```   357
```
```   358 lemma division_ofD[dest]:
```
```   359   assumes "s division_of i"
```
```   360   shows "finite s"
```
```   361     and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
```
```   362     and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
```
```   363     and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```   364     and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
```
```   365     and "\<Union>s = i"
```
```   366   using assms unfolding division_of_def by auto
```
```   367
```
```   368 lemma division_ofI:
```
```   369   assumes "finite s"
```
```   370     and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
```
```   371     and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
```
```   372     and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```   373     and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```   374     and "\<Union>s = i"
```
```   375   shows "s division_of i"
```
```   376   using assms unfolding division_of_def by auto
```
```   377
```
```   378 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
```
```   379   unfolding division_of_def by auto
```
```   380
```
```   381 lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
```
```   382   unfolding division_of_def by auto
```
```   383
```
```   384 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
```
```   385   unfolding division_of_def by auto
```
```   386
```
```   387 lemma division_of_sing[simp]:
```
```   388   "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
```
```   389   (is "?l = ?r")
```
```   390 proof
```
```   391   assume ?r
```
```   392   moreover
```
```   393   { fix k
```
```   394     assume "s = {{a}}" "k\<in>s"
```
```   395     then have "\<exists>x y. k = cbox x y"
```
```   396       apply (rule_tac x=a in exI)+
```
```   397       apply (force simp: cbox_sing)
```
```   398       done
```
```   399   }
```
```   400   ultimately show ?l
```
```   401     unfolding division_of_def cbox_sing by auto
```
```   402 next
```
```   403   assume ?l
```
```   404   note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
```
```   405   {
```
```   406     fix x
```
```   407     assume x: "x \<in> s" have "x = {a}"
```
```   408       using *(2)[rule_format,OF x] by auto
```
```   409   }
```
```   410   moreover have "s \<noteq> {}"
```
```   411     using *(4) by auto
```
```   412   ultimately show ?r
```
```   413     unfolding cbox_sing by auto
```
```   414 qed
```
```   415
```
```   416 lemma elementary_empty: obtains p where "p division_of {}"
```
```   417   unfolding division_of_trivial by auto
```
```   418
```
```   419 lemma elementary_interval: obtains p where "p division_of (cbox a b)"
```
```   420   by (metis division_of_trivial division_of_self)
```
```   421
```
```   422 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
```
```   423   unfolding division_of_def by auto
```
```   424
```
```   425 lemma forall_in_division:
```
```   426   "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))"
```
```   427   unfolding division_of_def by fastforce
```
```   428
```
```   429 lemma division_of_subset:
```
```   430   assumes "p division_of (\<Union>p)"
```
```   431     and "q \<subseteq> p"
```
```   432   shows "q division_of (\<Union>q)"
```
```   433 proof (rule division_ofI)
```
```   434   note * = division_ofD[OF assms(1)]
```
```   435   show "finite q"
```
```   436     using "*"(1) assms(2) infinite_super by auto
```
```   437   {
```
```   438     fix k
```
```   439     assume "k \<in> q"
```
```   440     then have kp: "k \<in> p"
```
```   441       using assms(2) by auto
```
```   442     show "k \<subseteq> \<Union>q"
```
```   443       using \<open>k \<in> q\<close> by auto
```
```   444     show "\<exists>a b. k = cbox a b"
```
```   445       using *(4)[OF kp] by auto
```
```   446     show "k \<noteq> {}"
```
```   447       using *(3)[OF kp] by auto
```
```   448   }
```
```   449   fix k1 k2
```
```   450   assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
```
```   451   then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
```
```   452     using assms(2) by auto
```
```   453   show "interior k1 \<inter> interior k2 = {}"
```
```   454     using *(5)[OF **] by auto
```
```   455 qed auto
```
```   456
```
```   457 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
```
```   458   unfolding division_of_def by auto
```
```   459
```
```   460 lemma division_of_content_0:
```
```   461   assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
```
```   462   shows "\<forall>k\<in>d. content k = 0"
```
```   463   unfolding forall_in_division[OF assms(2)]
```
```   464   by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
```
```   465
```
```   466 lemma division_inter:
```
```   467   fixes s1 s2 :: "'a::euclidean_space set"
```
```   468   assumes "p1 division_of s1"
```
```   469     and "p2 division_of s2"
```
```   470   shows "{k1 \<inter> k2 | k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
```
```   471   (is "?A' division_of _")
```
```   472 proof -
```
```   473   let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
```
```   474   have *: "?A' = ?A" by auto
```
```   475   show ?thesis
```
```   476     unfolding *
```
```   477   proof (rule division_ofI)
```
```   478     have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
```
```   479       by auto
```
```   480     moreover have "finite (p1 \<times> p2)"
```
```   481       using assms unfolding division_of_def by auto
```
```   482     ultimately show "finite ?A" by auto
```
```   483     have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
```
```   484       by auto
```
```   485     show "\<Union>?A = s1 \<inter> s2"
```
```   486       apply (rule set_eqI)
```
```   487       unfolding * and UN_iff
```
```   488       using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
```
```   489       apply auto
```
```   490       done
```
```   491     {
```
```   492       fix k
```
```   493       assume "k \<in> ?A"
```
```   494       then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
```
```   495         by auto
```
```   496       then show "k \<noteq> {}"
```
```   497         by auto
```
```   498       show "k \<subseteq> s1 \<inter> s2"
```
```   499         using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
```
```   500         unfolding k by auto
```
```   501       obtain a1 b1 where k1: "k1 = cbox a1 b1"
```
```   502         using division_ofD(4)[OF assms(1) k(2)] by blast
```
```   503       obtain a2 b2 where k2: "k2 = cbox a2 b2"
```
```   504         using division_ofD(4)[OF assms(2) k(3)] by blast
```
```   505       show "\<exists>a b. k = cbox a b"
```
```   506         unfolding k k1 k2 unfolding inter_interval by auto
```
```   507     }
```
```   508     fix k1 k2
```
```   509     assume "k1 \<in> ?A"
```
```   510     then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
```
```   511       by auto
```
```   512     assume "k2 \<in> ?A"
```
```   513     then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
```
```   514       by auto
```
```   515     assume "k1 \<noteq> k2"
```
```   516     then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
```
```   517       unfolding k1 k2 by auto
```
```   518     have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
```
```   519       interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
```
```   520       interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
```
```   521       interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
```
```   522     show "interior k1 \<inter> interior k2 = {}"
```
```   523       unfolding k1 k2
```
```   524       apply (rule *)
```
```   525       using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
```
```   526       done
```
```   527   qed
```
```   528 qed
```
```   529
```
```   530 lemma division_inter_1:
```
```   531   assumes "d division_of i"
```
```   532     and "cbox a (b::'a::euclidean_space) \<subseteq> i"
```
```   533   shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
```
```   534 proof (cases "cbox a b = {}")
```
```   535   case True
```
```   536   show ?thesis
```
```   537     unfolding True and division_of_trivial by auto
```
```   538 next
```
```   539   case False
```
```   540   have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
```
```   541   show ?thesis
```
```   542     using division_inter[OF division_of_self[OF False] assms(1)]
```
```   543     unfolding * by auto
```
```   544 qed
```
```   545
```
```   546 lemma elementary_inter:
```
```   547   fixes s t :: "'a::euclidean_space set"
```
```   548   assumes "p1 division_of s"
```
```   549     and "p2 division_of t"
```
```   550   shows "\<exists>p. p division_of (s \<inter> t)"
```
```   551 using assms division_inter by blast
```
```   552
```
```   553 lemma elementary_inters:
```
```   554   assumes "finite f"
```
```   555     and "f \<noteq> {}"
```
```   556     and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
```
```   557   shows "\<exists>p. p division_of (\<Inter>f)"
```
```   558   using assms
```
```   559 proof (induct f rule: finite_induct)
```
```   560   case (insert x f)
```
```   561   show ?case
```
```   562   proof (cases "f = {}")
```
```   563     case True
```
```   564     then show ?thesis
```
```   565       unfolding True using insert by auto
```
```   566   next
```
```   567     case False
```
```   568     obtain p where "p division_of \<Inter>f"
```
```   569       using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
```
```   570     moreover obtain px where "px division_of x"
```
```   571       using insert(5)[rule_format,OF insertI1] ..
```
```   572     ultimately show ?thesis
```
```   573       by (simp add: elementary_inter Inter_insert)
```
```   574   qed
```
```   575 qed auto
```
```   576
```
```   577 lemma division_disjoint_union:
```
```   578   assumes "p1 division_of s1"
```
```   579     and "p2 division_of s2"
```
```   580     and "interior s1 \<inter> interior s2 = {}"
```
```   581   shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
```
```   582 proof (rule division_ofI)
```
```   583   note d1 = division_ofD[OF assms(1)]
```
```   584   note d2 = division_ofD[OF assms(2)]
```
```   585   show "finite (p1 \<union> p2)"
```
```   586     using d1(1) d2(1) by auto
```
```   587   show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
```
```   588     using d1(6) d2(6) by auto
```
```   589   {
```
```   590     fix k1 k2
```
```   591     assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
```
```   592     moreover
```
```   593     let ?g="interior k1 \<inter> interior k2 = {}"
```
```   594     {
```
```   595       assume as: "k1\<in>p1" "k2\<in>p2"
```
```   596       have ?g
```
```   597         using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
```
```   598         using assms(3) by blast
```
```   599     }
```
```   600     moreover
```
```   601     {
```
```   602       assume as: "k1\<in>p2" "k2\<in>p1"
```
```   603       have ?g
```
```   604         using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
```
```   605         using assms(3) by blast
```
```   606     }
```
```   607     ultimately show ?g
```
```   608       using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
```
```   609   }
```
```   610   fix k
```
```   611   assume k: "k \<in> p1 \<union> p2"
```
```   612   show "k \<subseteq> s1 \<union> s2"
```
```   613     using k d1(2) d2(2) by auto
```
```   614   show "k \<noteq> {}"
```
```   615     using k d1(3) d2(3) by auto
```
```   616   show "\<exists>a b. k = cbox a b"
```
```   617     using k d1(4) d2(4) by auto
```
```   618 qed
```
```   619
```
```   620 lemma partial_division_extend_1:
```
```   621   fixes a b c d :: "'a::euclidean_space"
```
```   622   assumes incl: "cbox c d \<subseteq> cbox a b"
```
```   623     and nonempty: "cbox c d \<noteq> {}"
```
```   624   obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
```
```   625 proof
```
```   626   let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
```
```   627     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)"
```
```   628   define p where "p = ?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
```
```   629
```
```   630   show "cbox c d \<in> p"
```
```   631     unfolding p_def
```
```   632     by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
```
```   633   {
```
```   634     fix i :: 'a
```
```   635     assume "i \<in> Basis"
```
```   636     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"
```
```   637       unfolding box_eq_empty subset_box by (auto simp: not_le)
```
```   638   }
```
```   639   note ord = this
```
```   640
```
```   641   show "p division_of (cbox a b)"
```
```   642   proof (rule division_ofI)
```
```   643     show "finite p"
```
```   644       unfolding p_def by (auto intro!: finite_PiE)
```
```   645     {
```
```   646       fix k
```
```   647       assume "k \<in> p"
```
```   648       then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
```
```   649         by (auto simp: p_def)
```
```   650       then show "\<exists>a b. k = cbox a b"
```
```   651         by auto
```
```   652       have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
```
```   653       proof (simp add: k box_eq_empty subset_box not_less, safe)
```
```   654         fix i :: 'a
```
```   655         assume i: "i \<in> Basis"
```
```   656         with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
```
```   657           by (auto simp: PiE_iff)
```
```   658         with i ord[of i]
```
```   659         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"
```
```   660           by auto
```
```   661       qed
```
```   662       then show "k \<noteq> {}" "k \<subseteq> cbox a b"
```
```   663         by auto
```
```   664       {
```
```   665         fix l
```
```   666         assume "l \<in> p"
```
```   667         then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
```
```   668           by (auto simp: p_def)
```
```   669         assume "l \<noteq> k"
```
```   670         have "\<exists>i\<in>Basis. f i \<noteq> g i"
```
```   671         proof (rule ccontr)
```
```   672           assume "\<not> ?thesis"
```
```   673           with f g have "f = g"
```
```   674             by (auto simp: PiE_iff extensional_def intro!: ext)
```
```   675           with \<open>l \<noteq> k\<close> show False
```
```   676             by (simp add: l k)
```
```   677         qed
```
```   678         then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
```
```   679         then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
```
```   680                   "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
```
```   681           using f g by (auto simp: PiE_iff)
```
```   682         with * ord[of i] show "interior l \<inter> interior k = {}"
```
```   683           by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
```
```   684       }
```
```   685       note \<open>k \<subseteq> cbox a b\<close>
```
```   686     }
```
```   687     moreover
```
```   688     {
```
```   689       fix x assume x: "x \<in> cbox a b"
```
```   690       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)}"
```
```   691       proof
```
```   692         fix i :: 'a
```
```   693         assume "i \<in> Basis"
```
```   694         with x ord[of i]
```
```   695         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>
```
```   696             (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
```
```   697           by (auto simp: cbox_def)
```
```   698         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)}"
```
```   699           by auto
```
```   700       qed
```
```   701       then obtain f where
```
```   702         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)}"
```
```   703         unfolding bchoice_iff ..
```
```   704       moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
```
```   705         by auto
```
```   706       moreover from f have "x \<in> ?B (restrict f Basis)"
```
```   707         by (auto simp: mem_box)
```
```   708       ultimately have "\<exists>k\<in>p. x \<in> k"
```
```   709         unfolding p_def by blast
```
```   710     }
```
```   711     ultimately show "\<Union>p = cbox a b"
```
```   712       by auto
```
```   713   qed
```
```   714 qed
```
```   715
```
```   716 lemma partial_division_extend_interval:
```
```   717   assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
```
```   718   obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
```
```   719 proof (cases "p = {}")
```
```   720   case True
```
```   721   obtain q where "q division_of (cbox a b)"
```
```   722     by (rule elementary_interval)
```
```   723   then show ?thesis
```
```   724     using True that by blast
```
```   725 next
```
```   726   case False
```
```   727   note p = division_ofD[OF assms(1)]
```
```   728   have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
```
```   729   proof
```
```   730     fix k
```
```   731     assume kp: "k \<in> p"
```
```   732     obtain c d where k: "k = cbox c d"
```
```   733       using p(4)[OF kp] by blast
```
```   734     have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
```
```   735       using p(2,3)[OF kp, unfolded k] using assms(2)
```
```   736       by (blast intro: order.trans)+
```
```   737     obtain q where "q division_of cbox a b" "cbox c d \<in> q"
```
```   738       by (rule partial_division_extend_1[OF *])
```
```   739     then show "\<exists>q. q division_of cbox a b \<and> k \<in> q"
```
```   740       unfolding k by auto
```
```   741   qed
```
```   742   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"
```
```   743     using bchoice[OF div_cbox] by blast
```
```   744   { fix x
```
```   745     assume x: "x \<in> p"
```
```   746     have "q x division_of \<Union>q x"
```
```   747       apply (rule division_ofI)
```
```   748       using division_ofD[OF q(1)[OF x]]
```
```   749       apply auto
```
```   750       done }
```
```   751   then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
```
```   752     by (meson Diff_subset division_of_subset)
```
```   753   then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
```
```   754     apply -
```
```   755     apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
```
```   756     apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
```
```   757     done
```
```   758   then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
```
```   759   have "d \<union> p division_of cbox a b"
```
```   760   proof -
```
```   761     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
```
```   762     have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
```
```   763     proof (rule te[OF False], clarify)
```
```   764       fix i
```
```   765       assume i: "i \<in> p"
```
```   766       show "\<Union>(q i - {i}) \<union> i = cbox a b"
```
```   767         using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
```
```   768     qed
```
```   769     { fix k
```
```   770       assume k: "k \<in> p"
```
```   771       have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}"
```
```   772         by auto
```
```   773       have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}"
```
```   774       proof (rule *[OF inter_interior_unions_intervals])
```
```   775         note qk=division_ofD[OF q(1)[OF k]]
```
```   776         show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
```
```   777           using qk by auto
```
```   778         show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
```
```   779           using qk(5) using q(2)[OF k] by auto
```
```   780         show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))"
```
```   781           apply (rule interior_mono)+
```
```   782           using k
```
```   783           apply auto
```
```   784           done
```
```   785       qed } note [simp] = this
```
```   786     show "d \<union> p division_of (cbox a b)"
```
```   787       unfolding cbox_eq
```
```   788       apply (rule division_disjoint_union[OF d assms(1)])
```
```   789       apply (rule inter_interior_unions_intervals)
```
```   790       apply (rule p open_interior ballI)+
```
```   791       apply simp_all
```
```   792       done
```
```   793   qed
```
```   794   then show ?thesis
```
```   795     by (meson Un_upper2 that)
```
```   796 qed
```
```   797
```
```   798 lemma elementary_bounded[dest]:
```
```   799   fixes s :: "'a::euclidean_space set"
```
```   800   shows "p division_of s \<Longrightarrow> bounded s"
```
```   801   unfolding division_of_def by (metis bounded_Union bounded_cbox)
```
```   802
```
```   803 lemma elementary_subset_cbox:
```
```   804   "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
```
```   805   by (meson elementary_bounded bounded_subset_cbox)
```
```   806
```
```   807 lemma division_union_intervals_exists:
```
```   808   fixes a b :: "'a::euclidean_space"
```
```   809   assumes "cbox a b \<noteq> {}"
```
```   810   obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
```
```   811 proof (cases "cbox c d = {}")
```
```   812   case True
```
```   813   show ?thesis
```
```   814     apply (rule that[of "{}"])
```
```   815     unfolding True
```
```   816     using assms
```
```   817     apply auto
```
```   818     done
```
```   819 next
```
```   820   case False
```
```   821   show ?thesis
```
```   822   proof (cases "cbox a b \<inter> cbox c d = {}")
```
```   823     case True
```
```   824     then show ?thesis
```
```   825       by (metis that False assms division_disjoint_union division_of_self insert_is_Un interior_Int interior_empty)
```
```   826   next
```
```   827     case False
```
```   828     obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
```
```   829       unfolding inter_interval by auto
```
```   830     have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
```
```   831     obtain p where "p division_of cbox c d" "cbox u v \<in> p"
```
```   832       by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
```
```   833     note p = this division_ofD[OF this(1)]
```
```   834     have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
```
```   835       apply (rule arg_cong[of _ _ interior])
```
```   836       using p(8) uv by auto
```
```   837     also have "\<dots> = {}"
```
```   838       unfolding interior_Int
```
```   839       apply (rule inter_interior_unions_intervals)
```
```   840       using p(6) p(7)[OF p(2)] p(3)
```
```   841       apply auto
```
```   842       done
```
```   843     finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
```
```   844     have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
```
```   845       using p(8) unfolding uv[symmetric] by auto
```
```   846     have "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
```
```   847     proof -
```
```   848       have "{cbox a b} division_of cbox a b"
```
```   849         by (simp add: assms division_of_self)
```
```   850       then show "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
```
```   851         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))
```
```   852     qed
```
```   853     with that[of "p - {cbox u v}"] show ?thesis by (simp add: cbe)
```
```   854   qed
```
```   855 qed
```
```   856
```
```   857 lemma division_of_unions:
```
```   858   assumes "finite f"
```
```   859     and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
```
```   860     and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```   861   shows "\<Union>f division_of \<Union>\<Union>f"
```
```   862   using assms
```
```   863   by (auto intro!: division_ofI)
```
```   864
```
```   865 lemma elementary_union_interval:
```
```   866   fixes a b :: "'a::euclidean_space"
```
```   867   assumes "p division_of \<Union>p"
```
```   868   obtains q where "q division_of (cbox a b \<union> \<Union>p)"
```
```   869 proof -
```
```   870   note assm = division_ofD[OF assms]
```
```   871   have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
```
```   872     by auto
```
```   873   have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
```
```   874     by auto
```
```   875   {
```
```   876     presume "p = {} \<Longrightarrow> thesis"
```
```   877       "cbox a b = {} \<Longrightarrow> thesis"
```
```   878       "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
```
```   879       "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
```
```   880     then show thesis by auto
```
```   881   next
```
```   882     assume as: "p = {}"
```
```   883     obtain p where "p division_of (cbox a b)"
```
```   884       by (rule elementary_interval)
```
```   885     then show thesis
```
```   886       using as that by auto
```
```   887   next
```
```   888     assume as: "cbox a b = {}"
```
```   889     show thesis
```
```   890       using as assms that by auto
```
```   891   next
```
```   892     assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
```
```   893     show thesis
```
```   894       apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
```
```   895       unfolding finite_insert
```
```   896       apply (rule assm(1)) unfolding Union_insert
```
```   897       using assm(2-4) as
```
```   898       apply -
```
```   899       apply (fast dest: assm(5))+
```
```   900       done
```
```   901   next
```
```   902     assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
```
```   903     have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
```
```   904     proof
```
```   905       fix k
```
```   906       assume kp: "k \<in> p"
```
```   907       from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast
```
```   908       then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
```
```   909         by (meson as(3) division_union_intervals_exists)
```
```   910     qed
```
```   911     from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" ..
```
```   912     note q = division_ofD[OF this[rule_format]]
```
```   913     let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
```
```   914     show thesis
```
```   915     proof (rule that[OF division_ofI])
```
```   916       have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
```
```   917         by auto
```
```   918       show "finite ?D"
```
```   919         using "*" assm(1) q(1) by auto
```
```   920       show "\<Union>?D = cbox a b \<union> \<Union>p"
```
```   921         unfolding * lem1
```
```   922         unfolding lem2[OF as(1), of "cbox a b", symmetric]
```
```   923         using q(6)
```
```   924         by auto
```
```   925       fix k
```
```   926       assume k: "k \<in> ?D"
```
```   927       then show "k \<subseteq> cbox a b \<union> \<Union>p"
```
```   928         using q(2) by auto
```
```   929       show "k \<noteq> {}"
```
```   930         using q(3) k by auto
```
```   931       show "\<exists>a b. k = cbox a b"
```
```   932         using q(4) k by auto
```
```   933       fix k'
```
```   934       assume k': "k' \<in> ?D" "k \<noteq> k'"
```
```   935       obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
```
```   936         using k by auto
```
```   937       obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
```
```   938         using k' by auto
```
```   939       show "interior k \<inter> interior k' = {}"
```
```   940       proof (cases "x = x'")
```
```   941         case True
```
```   942         show ?thesis
```
```   943           using True k' q(5) x' x by auto
```
```   944       next
```
```   945         case False
```
```   946         {
```
```   947           presume "k = cbox a b \<Longrightarrow> ?thesis"
```
```   948             and "k' = cbox a b \<Longrightarrow> ?thesis"
```
```   949             and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
```
```   950           then show ?thesis by linarith
```
```   951         next
```
```   952           assume as': "k  = cbox a b"
```
```   953           show ?thesis
```
```   954             using as' k' q(5) x' by blast
```
```   955         next
```
```   956           assume as': "k' = cbox a b"
```
```   957           show ?thesis
```
```   958             using as' k'(2) q(5) x by blast
```
```   959         }
```
```   960         assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
```
```   961         obtain c d where k: "k = cbox c d"
```
```   962           using q(4)[OF x(2,1)] by blast
```
```   963         have "interior k \<inter> interior (cbox a b) = {}"
```
```   964           using as' k'(2) q(5) x by blast
```
```   965         then have "interior k \<subseteq> interior x"
```
```   966         using interior_subset_union_intervals
```
```   967           by (metis as(2) k q(2) x interior_subset_union_intervals)
```
```   968         moreover
```
```   969         obtain c d where c_d: "k' = cbox c d"
```
```   970           using q(4)[OF x'(2,1)] by blast
```
```   971         have "interior k' \<inter> interior (cbox a b) = {}"
```
```   972           using as'(2) q(5) x' by blast
```
```   973         then have "interior k' \<subseteq> interior x'"
```
```   974           by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
```
```   975         ultimately show ?thesis
```
```   976           using assm(5)[OF x(2) x'(2) False] by auto
```
```   977       qed
```
```   978     qed
```
```   979   }
```
```   980 qed
```
```   981
```
```   982 lemma elementary_unions_intervals:
```
```   983   assumes fin: "finite f"
```
```   984     and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
```
```   985   obtains p where "p division_of (\<Union>f)"
```
```   986 proof -
```
```   987   have "\<exists>p. p division_of (\<Union>f)"
```
```   988   proof (induct_tac f rule:finite_subset_induct)
```
```   989     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
```
```   990   next
```
```   991     fix x F
```
```   992     assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
```
```   993     from this(3) obtain p where p: "p division_of \<Union>F" ..
```
```   994     from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
```
```   995     have *: "\<Union>F = \<Union>p"
```
```   996       using division_ofD[OF p] by auto
```
```   997     show "\<exists>p. p division_of \<Union>insert x F"
```
```   998       using elementary_union_interval[OF p[unfolded *], of a b]
```
```   999       unfolding Union_insert x * by metis
```
```  1000   qed (insert assms, auto)
```
```  1001   then show ?thesis
```
```  1002     using that by auto
```
```  1003 qed
```
```  1004
```
```  1005 lemma elementary_union:
```
```  1006   fixes s t :: "'a::euclidean_space set"
```
```  1007   assumes "ps division_of s" "pt division_of t"
```
```  1008   obtains p where "p division_of (s \<union> t)"
```
```  1009 proof -
```
```  1010   have *: "s \<union> t = \<Union>ps \<union> \<Union>pt"
```
```  1011     using assms unfolding division_of_def by auto
```
```  1012   show ?thesis
```
```  1013     apply (rule elementary_unions_intervals[of "ps \<union> pt"])
```
```  1014     using assms apply auto
```
```  1015     by (simp add: * that)
```
```  1016 qed
```
```  1017
```
```  1018 lemma partial_division_extend:
```
```  1019   fixes t :: "'a::euclidean_space set"
```
```  1020   assumes "p division_of s"
```
```  1021     and "q division_of t"
```
```  1022     and "s \<subseteq> t"
```
```  1023   obtains r where "p \<subseteq> r" and "r division_of t"
```
```  1024 proof -
```
```  1025   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
```
```  1026   obtain a b where ab: "t \<subseteq> cbox a b"
```
```  1027     using elementary_subset_cbox[OF assms(2)] by auto
```
```  1028   obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
```
```  1029     using assms
```
```  1030     by (metis ab dual_order.trans partial_division_extend_interval divp(6))
```
```  1031   note r1 = this division_ofD[OF this(2)]
```
```  1032   obtain p' where "p' division_of \<Union>(r1 - p)"
```
```  1033     apply (rule elementary_unions_intervals[of "r1 - p"])
```
```  1034     using r1(3,6)
```
```  1035     apply auto
```
```  1036     done
```
```  1037   then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
```
```  1038     by (metis assms(2) divq(6) elementary_inter)
```
```  1039   {
```
```  1040     fix x
```
```  1041     assume x: "x \<in> t" "x \<notin> s"
```
```  1042     then have "x\<in>\<Union>r1"
```
```  1043       unfolding r1 using ab by auto
```
```  1044     then obtain r where r: "r \<in> r1" "x \<in> r"
```
```  1045       unfolding Union_iff ..
```
```  1046     moreover
```
```  1047     have "r \<notin> p"
```
```  1048     proof
```
```  1049       assume "r \<in> p"
```
```  1050       then have "x \<in> s" using divp(2) r by auto
```
```  1051       then show False using x by auto
```
```  1052     qed
```
```  1053     ultimately have "x\<in>\<Union>(r1 - p)" by auto
```
```  1054   }
```
```  1055   then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
```
```  1056     unfolding divp divq using assms(3) by auto
```
```  1057   show ?thesis
```
```  1058     apply (rule that[of "p \<union> r2"])
```
```  1059     unfolding *
```
```  1060     defer
```
```  1061     apply (rule division_disjoint_union)
```
```  1062     unfolding divp(6)
```
```  1063     apply(rule assms r2)+
```
```  1064   proof -
```
```  1065     have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
```
```  1066     proof (rule inter_interior_unions_intervals)
```
```  1067       show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
```
```  1068         using r1 by auto
```
```  1069       have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
```
```  1070         by auto
```
```  1071       show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
```
```  1072       proof
```
```  1073         fix m x
```
```  1074         assume as: "m \<in> r1 - p"
```
```  1075         have "interior m \<inter> interior (\<Union>p) = {}"
```
```  1076         proof (rule inter_interior_unions_intervals)
```
```  1077           show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
```
```  1078             using divp by auto
```
```  1079           show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
```
```  1080             by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
```
```  1081         qed
```
```  1082         then show "interior s \<inter> interior m = {}"
```
```  1083           unfolding divp by auto
```
```  1084       qed
```
```  1085     qed
```
```  1086     then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
```
```  1087       using interior_subset by auto
```
```  1088   qed auto
```
```  1089 qed
```
```  1090
```
```  1091 lemma division_split_left_inj:
```
```  1092   fixes type :: "'a::euclidean_space"
```
```  1093   assumes "d division_of i"
```
```  1094     and "k1 \<in> d"
```
```  1095     and "k2 \<in> d"
```
```  1096     and "k1 \<noteq> k2"
```
```  1097     and "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
```
```  1098     and k: "k\<in>Basis"
```
```  1099   shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
```
```  1100 proof -
```
```  1101   note d=division_ofD[OF assms(1)]
```
```  1102   have *: "\<And>(a::'a) b c. content (cbox a b \<inter> {x. x\<bullet>k \<le> c}) = 0 \<longleftrightarrow>
```
```  1103     interior(cbox a b \<inter> {x. x\<bullet>k \<le> c}) = {}"
```
```  1104     unfolding  interval_split[OF k] content_eq_0_interior by auto
```
```  1105   guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
```
```  1106   guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
```
```  1107   have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
```
```  1108     by auto
```
```  1109   show ?thesis
```
```  1110     unfolding uv1 uv2 *
```
```  1111     apply (rule **[OF d(5)[OF assms(2-4)]])
```
```  1112     apply (simp add: uv1)
```
```  1113     using assms(5) uv1 by auto
```
```  1114 qed
```
```  1115
```
```  1116 lemma division_split_right_inj:
```
```  1117   fixes type :: "'a::euclidean_space"
```
```  1118   assumes "d division_of i"
```
```  1119     and "k1 \<in> d"
```
```  1120     and "k2 \<in> d"
```
```  1121     and "k1 \<noteq> k2"
```
```  1122     and "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
```
```  1123     and k: "k \<in> Basis"
```
```  1124   shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
```
```  1125 proof -
```
```  1126   note d=division_ofD[OF assms(1)]
```
```  1127   have *: "\<And>a b::'a. \<And>c. content(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = 0 \<longleftrightarrow>
```
```  1128     interior(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = {}"
```
```  1129     unfolding interval_split[OF k] content_eq_0_interior by auto
```
```  1130   guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
```
```  1131   guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
```
```  1132   have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
```
```  1133     by auto
```
```  1134   show ?thesis
```
```  1135     unfolding uv1 uv2 *
```
```  1136     apply (rule **[OF d(5)[OF assms(2-4)]])
```
```  1137     apply (simp add: uv1)
```
```  1138     using assms(5) uv1 by auto
```
```  1139 qed
```
```  1140
```
```  1141
```
```  1142 lemma division_split:
```
```  1143   fixes a :: "'a::euclidean_space"
```
```  1144   assumes "p division_of (cbox a b)"
```
```  1145     and k: "k\<in>Basis"
```
```  1146   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})"
```
```  1147       (is "?p1 division_of ?I1")
```
```  1148     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})"
```
```  1149       (is "?p2 division_of ?I2")
```
```  1150 proof (rule_tac[!] division_ofI)
```
```  1151   note p = division_ofD[OF assms(1)]
```
```  1152   show "finite ?p1" "finite ?p2"
```
```  1153     using p(1) by auto
```
```  1154   show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2"
```
```  1155     unfolding p(6)[symmetric] by auto
```
```  1156   {
```
```  1157     fix k
```
```  1158     assume "k \<in> ?p1"
```
```  1159     then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
```
```  1160     guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
```
```  1161     show "k \<subseteq> ?I1"
```
```  1162       using l p(2) uv by force
```
```  1163     show  "k \<noteq> {}"
```
```  1164       by (simp add: l)
```
```  1165     show  "\<exists>a b. k = cbox a b"
```
```  1166       apply (simp add: l uv p(2-3)[OF l(2)])
```
```  1167       apply (subst interval_split[OF k])
```
```  1168       apply (auto intro: order.trans)
```
```  1169       done
```
```  1170     fix k'
```
```  1171     assume "k' \<in> ?p1"
```
```  1172     then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
```
```  1173     assume "k \<noteq> k'"
```
```  1174     then show "interior k \<inter> interior k' = {}"
```
```  1175       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
```
```  1176   }
```
```  1177   {
```
```  1178     fix k
```
```  1179     assume "k \<in> ?p2"
```
```  1180     then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
```
```  1181     guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
```
```  1182     show "k \<subseteq> ?I2"
```
```  1183       using l p(2) uv by force
```
```  1184     show  "k \<noteq> {}"
```
```  1185       by (simp add: l)
```
```  1186     show  "\<exists>a b. k = cbox a b"
```
```  1187       apply (simp add: l uv p(2-3)[OF l(2)])
```
```  1188       apply (subst interval_split[OF k])
```
```  1189       apply (auto intro: order.trans)
```
```  1190       done
```
```  1191     fix k'
```
```  1192     assume "k' \<in> ?p2"
```
```  1193     then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
```
```  1194     assume "k \<noteq> k'"
```
```  1195     then show "interior k \<inter> interior k' = {}"
```
```  1196       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
```
```  1197   }
```
```  1198 qed
```
```  1199
```
```  1200 subsection \<open>Tagged (partial) divisions.\<close>
```
```  1201
```
```  1202 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
```
```  1203   where "s tagged_partial_division_of i \<longleftrightarrow>
```
```  1204     finite s \<and>
```
```  1205     (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```  1206     (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
```
```  1207       interior k1 \<inter> interior k2 = {})"
```
```  1208
```
```  1209 lemma tagged_partial_division_ofD[dest]:
```
```  1210   assumes "s tagged_partial_division_of i"
```
```  1211   shows "finite s"
```
```  1212     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1213     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1214     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1215     and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
```
```  1216       (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
```
```  1217   using assms unfolding tagged_partial_division_of_def by blast+
```
```  1218
```
```  1219 definition tagged_division_of (infixr "tagged'_division'_of" 40)
```
```  1220   where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1221
```
```  1222 lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
```
```  1223   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```  1224
```
```  1225 lemma tagged_division_of:
```
```  1226   "s tagged_division_of i \<longleftrightarrow>
```
```  1227     finite s \<and>
```
```  1228     (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
```
```  1229     (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
```
```  1230       interior k1 \<inter> interior k2 = {}) \<and>
```
```  1231     (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1232   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
```
```  1233
```
```  1234 lemma tagged_division_ofI:
```
```  1235   assumes "finite s"
```
```  1236     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1237     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1238     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1239     and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
```
```  1240       interior k1 \<inter> interior k2 = {}"
```
```  1241     and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1242   shows "s tagged_division_of i"
```
```  1243   unfolding tagged_division_of
```
```  1244   using assms
```
```  1245   apply auto
```
```  1246   apply fastforce+
```
```  1247   done
```
```  1248
```
```  1249 lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
```
```  1250   assumes "s tagged_division_of i"
```
```  1251   shows "finite s"
```
```  1252     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
```
```  1253     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
```
```  1254     and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
```
```  1255     and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
```
```  1256       interior k1 \<inter> interior k2 = {}"
```
```  1257     and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
```
```  1258   using assms unfolding tagged_division_of by blast+
```
```  1259
```
```  1260 lemma division_of_tagged_division:
```
```  1261   assumes "s tagged_division_of i"
```
```  1262   shows "(snd ` s) division_of i"
```
```  1263 proof (rule division_ofI)
```
```  1264   note assm = tagged_division_ofD[OF assms]
```
```  1265   show "\<Union>(snd ` s) = i" "finite (snd ` s)"
```
```  1266     using assm by auto
```
```  1267   fix k
```
```  1268   assume k: "k \<in> snd ` s"
```
```  1269   then obtain xk where xk: "(xk, k) \<in> s"
```
```  1270     by auto
```
```  1271   then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
```
```  1272     using assm by fastforce+
```
```  1273   fix k'
```
```  1274   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
```
```  1275   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
```
```  1276     by auto
```
```  1277   then show "interior k \<inter> interior k' = {}"
```
```  1278     using assm(5) k'(2) xk by blast
```
```  1279 qed
```
```  1280
```
```  1281 lemma partial_division_of_tagged_division:
```
```  1282   assumes "s tagged_partial_division_of i"
```
```  1283   shows "(snd ` s) division_of \<Union>(snd ` s)"
```
```  1284 proof (rule division_ofI)
```
```  1285   note assm = tagged_partial_division_ofD[OF assms]
```
```  1286   show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
```
```  1287     using assm by auto
```
```  1288   fix k
```
```  1289   assume k: "k \<in> snd ` s"
```
```  1290   then obtain xk where xk: "(xk, k) \<in> s"
```
```  1291     by auto
```
```  1292   then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
```
```  1293     using assm by auto
```
```  1294   fix k'
```
```  1295   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
```
```  1296   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
```
```  1297     by auto
```
```  1298   then show "interior k \<inter> interior k' = {}"
```
```  1299     using assm(5) k'(2) xk by auto
```
```  1300 qed
```
```  1301
```
```  1302 lemma tagged_partial_division_subset:
```
```  1303   assumes "s tagged_partial_division_of i"
```
```  1304     and "t \<subseteq> s"
```
```  1305   shows "t tagged_partial_division_of i"
```
```  1306   using assms
```
```  1307   unfolding tagged_partial_division_of_def
```
```  1308   using finite_subset[OF assms(2)]
```
```  1309   by blast
```
```  1310
```
```  1311 lemma (in comm_monoid_set) over_tagged_division_lemma:
```
```  1312   assumes "p tagged_division_of i"
```
```  1313     and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = \<^bold>1"
```
```  1314   shows "F (\<lambda>(x,k). d k) p = F d (snd ` p)"
```
```  1315 proof -
```
```  1316   have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
```
```  1317     unfolding o_def by (rule ext) auto
```
```  1318   note assm = tagged_division_ofD[OF assms(1)]
```
```  1319   show ?thesis
```
```  1320     unfolding *
```
```  1321   proof (rule reindex_nontrivial[symmetric])
```
```  1322     show "finite p"
```
```  1323       using assm by auto
```
```  1324     fix x y
```
```  1325     assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
```
```  1326     obtain a b where ab: "snd x = cbox a b"
```
```  1327       using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
```
```  1328     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
```
```  1329       by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
```
```  1330     with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
```
```  1331       by (intro assm(5)[of "fst x" _ "fst y"]) auto
```
```  1332     then have "content (cbox a b) = 0"
```
```  1333       unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior by auto
```
```  1334     then have "d (cbox a b) = \<^bold>1"
```
```  1335       using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
```
```  1336     then show "d (snd x) = \<^bold>1"
```
```  1337       unfolding ab by auto
```
```  1338   qed
```
```  1339 qed
```
```  1340
```
```  1341 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
```
```  1342   by auto
```
```  1343
```
```  1344 lemma tagged_division_of_empty: "{} tagged_division_of {}"
```
```  1345   unfolding tagged_division_of by auto
```
```  1346
```
```  1347 lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
```
```  1348   unfolding tagged_partial_division_of_def by auto
```
```  1349
```
```  1350 lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
```
```  1351   unfolding tagged_division_of by auto
```
```  1352
```
```  1353 lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
```
```  1354   by (rule tagged_division_ofI) auto
```
```  1355
```
```  1356 lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
```
```  1357   unfolding box_real[symmetric]
```
```  1358   by (rule tagged_division_of_self)
```
```  1359
```
```  1360 lemma tagged_division_union:
```
```  1361   assumes "p1 tagged_division_of s1"
```
```  1362     and "p2 tagged_division_of s2"
```
```  1363     and "interior s1 \<inter> interior s2 = {}"
```
```  1364   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
```
```  1365 proof (rule tagged_division_ofI)
```
```  1366   note p1 = tagged_division_ofD[OF assms(1)]
```
```  1367   note p2 = tagged_division_ofD[OF assms(2)]
```
```  1368   show "finite (p1 \<union> p2)"
```
```  1369     using p1(1) p2(1) by auto
```
```  1370   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
```
```  1371     using p1(6) p2(6) by blast
```
```  1372   fix x k
```
```  1373   assume xk: "(x, k) \<in> p1 \<union> p2"
```
```  1374   show "x \<in> k" "\<exists>a b. k = cbox a b"
```
```  1375     using xk p1(2,4) p2(2,4) by auto
```
```  1376   show "k \<subseteq> s1 \<union> s2"
```
```  1377     using xk p1(3) p2(3) by blast
```
```  1378   fix x' k'
```
```  1379   assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
```
```  1380   have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
```
```  1381     using assms(3) interior_mono by blast
```
```  1382   show "interior k \<inter> interior k' = {}"
```
```  1383     apply (cases "(x, k) \<in> p1")
```
```  1384     apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
```
```  1385     by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
```
```  1386 qed
```
```  1387
```
```  1388 lemma tagged_division_unions:
```
```  1389   assumes "finite iset"
```
```  1390     and "\<forall>i\<in>iset. pfn i tagged_division_of i"
```
```  1391     and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
```
```  1392   shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
```
```  1393 proof (rule tagged_division_ofI)
```
```  1394   note assm = tagged_division_ofD[OF assms(2)[rule_format]]
```
```  1395   show "finite (\<Union>(pfn ` iset))"
```
```  1396     using assms by auto
```
```  1397   have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
```
```  1398     by blast
```
```  1399   also have "\<dots> = \<Union>iset"
```
```  1400     using assm(6) by auto
```
```  1401   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
```
```  1402   fix x k
```
```  1403   assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
```
```  1404   then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
```
```  1405     by auto
```
```  1406   show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
```
```  1407     using assm(2-4)[OF i] using i(1) by auto
```
```  1408   fix x' k'
```
```  1409   assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
```
```  1410   then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
```
```  1411     by auto
```
```  1412   have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
```
```  1413     using i(1) i'(1)
```
```  1414     using assms(3)[rule_format] interior_mono
```
```  1415     by blast
```
```  1416   show "interior k \<inter> interior k' = {}"
```
```  1417     apply (cases "i = i'")
```
```  1418     using assm(5) i' i(2) xk'(2) apply blast
```
```  1419     using "*" assm(3) i' i by auto
```
```  1420 qed
```
```  1421
```
```  1422 lemma tagged_partial_division_of_union_self:
```
```  1423   assumes "p tagged_partial_division_of s"
```
```  1424   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```  1425   apply (rule tagged_division_ofI)
```
```  1426   using tagged_partial_division_ofD[OF assms]
```
```  1427   apply auto
```
```  1428   done
```
```  1429
```
```  1430 lemma tagged_division_of_union_self:
```
```  1431   assumes "p tagged_division_of s"
```
```  1432   shows "p tagged_division_of (\<Union>(snd ` p))"
```
```  1433   apply (rule tagged_division_ofI)
```
```  1434   using tagged_division_ofD[OF assms]
```
```  1435   apply auto
```
```  1436   done
```
```  1437
```
```  1438 subsection \<open>Functions closed on boxes: morphisms from boxes to monoids\<close>
```
```  1439
```
```  1440 text \<open>This auxiliary structure is used to sum up over the elements of a division. Main theorem is
```
```  1441   @{text operative_division}. Instances for the monoid are @{typ "'a option"}, @{typ real}, and
```
```  1442   @{typ bool}.\<close>
```
```  1443
```
```  1444 lemma property_empty_interval: "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b) \<Longrightarrow> P {}"
```
```  1445   using content_empty unfolding empty_as_interval by auto
```
```  1446
```
```  1447 paragraph \<open>Using additivity of lifted function to encode definedness.\<close>
```
```  1448
```
```  1449 definition lift_option :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> 'c option"
```
```  1450 where
```
```  1451   "lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))"
```
```  1452
```
```  1453 lemma lift_option_simps[simp]:
```
```  1454   "lift_option f (Some a) (Some b) = Some (f a b)"
```
```  1455   "lift_option f None b' = None"
```
```  1456   "lift_option f a' None = None"
```
```  1457   by (auto simp: lift_option_def)
```
```  1458
```
```  1459 lemma comm_monoid_lift_option:
```
```  1460   assumes "comm_monoid f z"
```
```  1461   shows "comm_monoid (lift_option f) (Some z)"
```
```  1462 proof -
```
```  1463   from assms interpret comm_monoid f z .
```
```  1464   show ?thesis
```
```  1465     by standard (auto simp: lift_option_def ac_simps split: bind_split)
```
```  1466 qed
```
```  1467
```
```  1468 lemma comm_monoid_and: "comm_monoid HOL.conj True"
```
```  1469   by standard auto
```
```  1470
```
```  1471 lemma comm_monoid_set_and: "comm_monoid_set HOL.conj True"
```
```  1472   by (rule comm_monoid_set.intro) (fact comm_monoid_and)
```
```  1473
```
```  1474 paragraph \<open>Operative\<close>
```
```  1475
```
```  1476 definition (in comm_monoid) operative :: "('b::euclidean_space set \<Rightarrow> 'a) \<Rightarrow> bool"
```
```  1477   where "operative g \<longleftrightarrow>
```
```  1478     (\<forall>a b. content (cbox a b) = 0 \<longrightarrow> g (cbox a b) = \<^bold>1) \<and>
```
```  1479     (\<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}))"
```
```  1480
```
```  1481 lemma (in comm_monoid) operativeD[dest]:
```
```  1482   assumes "operative g"
```
```  1483   shows "\<And>a b. content (cbox a b) = 0 \<Longrightarrow> g (cbox a b) = \<^bold>1"
```
```  1484     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})"
```
```  1485   using assms unfolding operative_def by auto
```
```  1486
```
```  1487 lemma (in comm_monoid) operative_empty: "operative g \<Longrightarrow> g {} = \<^bold>1"
```
```  1488   unfolding operative_def by (rule property_empty_interval) auto
```
```  1489
```
```  1490 lemma operative_content[intro]: "add.operative content"
```
```  1491   by (force simp add: add.operative_def content_split[symmetric])
```
```  1492
```
```  1493 definition "division_points (k::('a::euclidean_space) set) d =
```
```  1494    {(j,x). j \<in> Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
```
```  1495      (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
```
```  1496
```
```  1497 lemma division_points_finite:
```
```  1498   fixes i :: "'a::euclidean_space set"
```
```  1499   assumes "d division_of i"
```
```  1500   shows "finite (division_points i d)"
```
```  1501 proof -
```
```  1502   note assm = division_ofD[OF assms]
```
```  1503   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and>
```
```  1504     (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
```
```  1505   have *: "division_points i d = \<Union>(?M ` Basis)"
```
```  1506     unfolding division_points_def by auto
```
```  1507   show ?thesis
```
```  1508     unfolding * using assm by auto
```
```  1509 qed
```
```  1510
```
```  1511 lemma division_points_subset:
```
```  1512   fixes a :: "'a::euclidean_space"
```
```  1513   assumes "d division_of (cbox a b)"
```
```  1514     and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
```
```  1515     and k: "k \<in> Basis"
```
```  1516   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>
```
```  1517       division_points (cbox a b) d" (is ?t1)
```
```  1518     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>
```
```  1519       division_points (cbox a b) d" (is ?t2)
```
```  1520 proof -
```
```  1521   note assm = division_ofD[OF assms(1)]
```
```  1522   have *: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1523     "\<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"
```
```  1524     "\<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"
```
```  1525     "min (b \<bullet> k) c = c" "max (a \<bullet> k) c = c"
```
```  1526     using assms using less_imp_le by auto
```
```  1527   show ?t1 (*FIXME a horrible mess*)
```
```  1528     unfolding division_points_def interval_split[OF k, of a b]
```
```  1529     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
```
```  1530     unfolding *
```
```  1531     apply (rule subsetI)
```
```  1532     unfolding mem_Collect_eq split_beta
```
```  1533     apply (erule bexE conjE)+
```
```  1534     apply (simp add: )
```
```  1535     apply (erule exE conjE)+
```
```  1536   proof
```
```  1537     fix i l x
```
```  1538     assume as:
```
```  1539       "a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)"
```
```  1540       "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1541       "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
```
```  1542       and fstx: "fst x \<in> Basis"
```
```  1543     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
```
```  1544     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"
```
```  1545       using as(6) unfolding l interval_split[OF k] box_ne_empty as .
```
```  1546     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
```
```  1547       using l using as(6) unfolding box_ne_empty[symmetric] by auto
```
```  1548     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1549       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
```
```  1550       using as(1-3,5) fstx
```
```  1551       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
```
```  1552       apply (auto split: if_split_asm)
```
```  1553       done
```
```  1554     show "snd x < b \<bullet> fst x"
```
```  1555       using as(2) \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm)
```
```  1556   qed
```
```  1557   show ?t2
```
```  1558     unfolding division_points_def interval_split[OF k, of a b]
```
```  1559     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
```
```  1560     unfolding *
```
```  1561     unfolding subset_eq
```
```  1562     apply rule
```
```  1563     unfolding mem_Collect_eq split_beta
```
```  1564     apply (erule bexE conjE)+
```
```  1565     apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
```
```  1566     apply (erule exE conjE)+
```
```  1567   proof
```
```  1568     fix i l x
```
```  1569     assume as:
```
```  1570       "(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
```
```  1571       "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1572       "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
```
```  1573       and fstx: "fst x \<in> Basis"
```
```  1574     from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
```
```  1575     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"
```
```  1576       using as(6) unfolding l interval_split[OF k] box_ne_empty as .
```
```  1577     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
```
```  1578       using l using as(6) unfolding box_ne_empty[symmetric] by auto
```
```  1579     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
```
```  1580       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
```
```  1581       using as(1-3,5) fstx
```
```  1582       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
```
```  1583       apply (auto split: if_split_asm)
```
```  1584       done
```
```  1585     show "a \<bullet> fst x < snd x"
```
```  1586       using as(1) \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm)
```
```  1587    qed
```
```  1588 qed
```
```  1589
```
```  1590 lemma division_points_psubset:
```
```  1591   fixes a :: "'a::euclidean_space"
```
```  1592   assumes "d division_of (cbox a b)"
```
```  1593       and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
```
```  1594       and "l \<in> d"
```
```  1595       and "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c"
```
```  1596       and k: "k \<in> Basis"
```
```  1597   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>
```
```  1598          division_points (cbox a b) d" (is "?D1 \<subset> ?D")
```
```  1599     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>
```
```  1600          division_points (cbox a b) d" (is "?D2 \<subset> ?D")
```
```  1601 proof -
```
```  1602   have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1603     using assms(2) by (auto intro!:less_imp_le)
```
```  1604   guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
```
```  1605   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"
```
```  1606     using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
```
```  1607     using subset_box(1)
```
```  1608     apply auto
```
```  1609     apply blast+
```
```  1610     done
```
```  1611   have *: "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
```
```  1612           "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
```
```  1613     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
```
```  1614     using uv[rule_format, of k] ab k
```
```  1615     by auto
```
```  1616   have "\<exists>x. x \<in> ?D - ?D1"
```
```  1617     using assms(3-)
```
```  1618     unfolding division_points_def interval_bounds[OF ab]
```
```  1619     apply -
```
```  1620     apply (erule disjE)
```
```  1621     apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1622     apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1623     done
```
```  1624   moreover have "?D1 \<subseteq> ?D"
```
```  1625     by (auto simp add: assms division_points_subset)
```
```  1626   ultimately show "?D1 \<subset> ?D"
```
```  1627     by blast
```
```  1628   have *: "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
```
```  1629     "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
```
```  1630     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
```
```  1631     using uv[rule_format, of k] ab k
```
```  1632     by auto
```
```  1633   have "\<exists>x. x \<in> ?D - ?D2"
```
```  1634     using assms(3-)
```
```  1635     unfolding division_points_def interval_bounds[OF ab]
```
```  1636     apply -
```
```  1637     apply (erule disjE)
```
```  1638     apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1639     apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
```
```  1640     done
```
```  1641   moreover have "?D2 \<subseteq> ?D"
```
```  1642     by (auto simp add: assms division_points_subset)
```
```  1643   ultimately show "?D2 \<subset> ?D"
```
```  1644     by blast
```
```  1645 qed
```
```  1646
```
```  1647 lemma (in comm_monoid_set) operative_division:
```
```  1648   fixes g :: "'b::euclidean_space set \<Rightarrow> 'a"
```
```  1649   assumes g: "operative g" and d: "d division_of (cbox a b)" shows "F g d = g (cbox a b)"
```
```  1650 proof -
```
```  1651   define C where [abs_def]: "C = card (division_points (cbox a b) d)"
```
```  1652   then show ?thesis
```
```  1653     using d
```
```  1654   proof (induction C arbitrary: a b d rule: less_induct)
```
```  1655     case (less a b d)
```
```  1656     show ?case
```
```  1657     proof cases
```
```  1658       show "content (cbox a b) = 0 \<Longrightarrow> F g d = g (cbox a b)"
```
```  1659         using division_of_content_0[OF _ less.prems] operativeD(1)[OF  g] division_ofD(4)[OF less.prems]
```
```  1660         by (fastforce intro!: neutral)
```
```  1661     next
```
```  1662       assume "content (cbox a b) \<noteq> 0"
```
```  1663       note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
```
```  1664       then have ab': "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
```
```  1665         by (auto intro!: less_imp_le)
```
```  1666       show "F g d = g (cbox a b)"
```
```  1667       proof (cases "division_points (cbox a b) d = {}")
```
```  1668         case True
```
```  1669         { fix u v and j :: 'b
```
```  1670           assume j: "j \<in> Basis" and as: "cbox u v \<in> d"
```
```  1671           then have "cbox u v \<noteq> {}"
```
```  1672             using less.prems by blast
```
```  1673           then have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "u\<bullet>j \<le> v\<bullet>j"
```
```  1674             using j unfolding box_ne_empty by auto
```
```  1675           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)"
```
```  1676             using as j by auto
```
```  1677           have "(j, u\<bullet>j) \<notin> division_points (cbox a b) d"
```
```  1678                "(j, v\<bullet>j) \<notin> division_points (cbox a b) d" using True by auto
```
```  1679           note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
```
```  1680           note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
```
```  1681           moreover
```
```  1682           have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j"
```
```  1683             using division_ofD(2,2,3)[OF \<open>d division_of cbox a b\<close> as]
```
```  1684             apply (metis j subset_box(1) uv(1))
```
```  1685             by (metis \<open>cbox u v \<subseteq> cbox a b\<close> j subset_box(1) uv(1))
```
```  1686           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"
```
```  1687             unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force }
```
```  1688         then have d': "\<forall>i\<in>d. \<exists>u v. i = cbox u v \<and>
```
```  1689           (\<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)"
```
```  1690           unfolding forall_in_division[OF less.prems] by blast
```
```  1691         have "(1/2) *\<^sub>R (a+b) \<in> cbox a b"
```
```  1692           unfolding mem_box using ab by(auto intro!: less_imp_le simp: inner_simps)
```
```  1693         note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff]
```
```  1694         then guess i .. note i=this
```
```  1695         guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
```
```  1696         have "cbox a b \<in> d"
```
```  1697         proof -
```
```  1698           have "u = a" "v = b"
```
```  1699             unfolding euclidean_eq_iff[where 'a='b]
```
```  1700           proof safe
```
```  1701             fix j :: 'b
```
```  1702             assume j: "j \<in> Basis"
```
```  1703             note i(2)[unfolded uv mem_box,rule_format,of j]
```
```  1704             then show "u \<bullet> j = a \<bullet> j" and "v \<bullet> j = b \<bullet> j"
```
```  1705               using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
```
```  1706           qed
```
```  1707           then have "i = cbox a b" using uv by auto
```
```  1708           then show ?thesis using i by auto
```
```  1709         qed
```
```  1710         then have deq: "d = insert (cbox a b) (d - {cbox a b})"
```
```  1711           by auto
```
```  1712         have "F g (d - {cbox a b}) = \<^bold>1"
```
```  1713         proof (intro neutral ballI)
```
```  1714           fix x
```
```  1715           assume x: "x \<in> d - {cbox a b}"
```
```  1716           then have "x\<in>d"
```
```  1717             by auto note d'[rule_format,OF this]
```
```  1718           then guess u v by (elim exE conjE) note uv=this
```
```  1719           have "u \<noteq> a \<or> v \<noteq> b"
```
```  1720             using x[unfolded uv] by auto
```
```  1721           then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j: "j \<in> Basis"
```
```  1722             unfolding euclidean_eq_iff[where 'a='b] by auto
```
```  1723           then have "u\<bullet>j = v\<bullet>j"
```
```  1724             using uv(2)[rule_format,OF j] by auto
```
```  1725           then have "content (cbox u v) = 0"
```
```  1726             unfolding content_eq_0 using j
```
```  1727             by force
```
```  1728           then show "g x = \<^bold>1"
```
```  1729             unfolding uv(1) by (rule operativeD(1)[OF g])
```
```  1730         qed
```
```  1731         then show "F g d = g (cbox a b)"
```
```  1732           using division_ofD[OF less.prems]
```
```  1733           apply (subst deq)
```
```  1734           apply (subst insert)
```
```  1735           apply auto
```
```  1736           done
```
```  1737       next
```
```  1738         case False
```
```  1739         then have "\<exists>x. x \<in> division_points (cbox a b) d"
```
```  1740           by auto
```
```  1741         then guess k c
```
```  1742           unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
```
```  1743           apply (elim exE conjE)
```
```  1744           done
```
```  1745         note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
```
```  1746         from this(3) guess j .. note j=this
```
```  1747         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> {}}"
```
```  1748         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> {}}"
```
```  1749         define cb where "cb = (\<Sum>i\<in>Basis. (if i = k then c else b\<bullet>i) *\<^sub>R i)"
```
```  1750         define ca where "ca = (\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)"
```
```  1751         note division_points_psubset[OF \<open>d division_of cbox a b\<close> ab kc(1-2) j]
```
```  1752         note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
```
```  1753         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})"
```
```  1754           unfolding interval_split[OF kc(4)]
```
```  1755           apply (rule_tac[!] "less.hyps"[rule_format])
```
```  1756           using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c]
```
```  1757           apply (simp_all add: interval_split kc d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>])
```
```  1758           done
```
```  1759         { fix l y
```
```  1760           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"
```
```  1761           from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
```
```  1762           have "g (l \<inter> {x. x \<bullet> k \<le> c}) = \<^bold>1"
```
```  1763             unfolding leq interval_split[OF kc(4)]
```
```  1764             apply (rule operativeD[OF g])
```
```  1765             unfolding interval_split[symmetric, OF kc(4)]
```
```  1766             using division_split_left_inj less as kc leq by blast
```
```  1767         } note fxk_le = this
```
```  1768         { fix l y
```
```  1769           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"
```
```  1770           from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
```
```  1771           have "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1"
```
```  1772             unfolding leq interval_split[OF kc(4)]
```
```  1773             apply (rule operativeD(1)[OF g])
```
```  1774             unfolding interval_split[symmetric,OF kc(4)]
```
```  1775             using division_split_right_inj less leq as kc by blast
```
```  1776         } note fxk_ge = this
```
```  1777         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> {}}"
```
```  1778           using d1_def by auto
```
```  1779         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> {}}"
```
```  1780           using d2_def by auto
```
```  1781         have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev")
```
```  1782           unfolding * using g kc(4) by blast
```
```  1783         also have "F g d1 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d"
```
```  1784           unfolding d1_alt using division_of_finite[OF less.prems] fxk_le
```
```  1785           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
```
```  1786         also have "F g d2 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d"
```
```  1787           unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge
```
```  1788           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
```
```  1789         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})"
```
```  1790           unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>]
```
```  1791           using g kc(4) by blast
```
```  1792         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"
```
```  1793           using * by (simp add: distrib)
```
```  1794         finally show ?thesis by auto
```
```  1795       qed
```
```  1796     qed
```
```  1797   qed
```
```  1798 qed
```
```  1799
```
```  1800 lemma (in comm_monoid_set) operative_tagged_division:
```
```  1801   assumes f: "operative g" and d: "d tagged_division_of (cbox a b)"
```
```  1802   shows "F (\<lambda>(x, l). g l) d = g (cbox a b)"
```
```  1803   unfolding d[THEN division_of_tagged_division, THEN operative_division[OF f], symmetric]
```
```  1804   by (simp add: f[THEN operativeD(1)] over_tagged_division_lemma[OF d])
```
```  1805
```
```  1806 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> setsum content d = content (cbox a b)"
```
```  1807   by (metis operative_content setsum.operative_division)
```
```  1808
```
```  1809 lemma additive_content_tagged_division:
```
```  1810   "d tagged_division_of (cbox a b) \<Longrightarrow> setsum (\<lambda>(x,l). content l) d = content (cbox a b)"
```
```  1811   unfolding setsum.operative_tagged_division[OF operative_content, symmetric] by blast
```
```  1812
```
```  1813 lemma content_real_eq_0: "content {a .. b::real} = 0 \<longleftrightarrow> a \<ge> b"
```
```  1814   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
```
```  1815
```
```  1816 lemma interval_real_split:
```
```  1817   "{a .. b::real} \<inter> {x. x \<le> c} = {a .. min b c}"
```
```  1818   "{a .. b} \<inter> {x. c \<le> x} = {max a c .. b}"
```
```  1819   apply (metis Int_atLeastAtMostL1 atMost_def)
```
```  1820   apply (metis Int_atLeastAtMostL2 atLeast_def)
```
```  1821   done
```
```  1822
```
```  1823 lemma (in comm_monoid) operative_1_lt:
```
```  1824   "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
```
```  1825     ((\<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}))"
```
```  1826   apply (simp add: operative_def content_real_eq_0 atMost_def[symmetric] atLeast_def[symmetric]
```
```  1827               del: content_real_if)
```
```  1828 proof safe
```
```  1829   fix a b c :: real
```
```  1830   assume *: "\<forall>a b c. g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
```
```  1831   assume "a < c" "c < b"
```
```  1832   with *[rule_format, of a b c] show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1833     by (simp add: less_imp_le min.absorb2 max.absorb2)
```
```  1834 next
```
```  1835   fix a b c :: real
```
```  1836   assume as: "\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1"
```
```  1837     "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1838   from as(1)[rule_format, of 0 1] as(1)[rule_format, of a a for a] as(2)
```
```  1839   have [simp]: "g {} = \<^bold>1" "\<And>a. g {a} = \<^bold>1"
```
```  1840     "\<And>a b c. a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1841     by auto
```
```  1842   show "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
```
```  1843     by (auto simp: min_def max_def le_less)
```
```  1844 qed
```
```  1845
```
```  1846 lemma (in comm_monoid) operative_1_le:
```
```  1847   "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
```
```  1848     ((\<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}))"
```
```  1849   unfolding operative_1_lt
```
```  1850 proof safe
```
```  1851   fix a b c :: real
```
```  1852   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"
```
```  1853   show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1854     apply (rule as(1)[rule_format])
```
```  1855     using as(2-)
```
```  1856     apply auto
```
```  1857     done
```
```  1858 next
```
```  1859   fix a b c :: real
```
```  1860   assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
```
```  1861     and "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1862     and "a \<le> c"
```
```  1863     and "c \<le> b"
```
```  1864   note as = this[rule_format]
```
```  1865   show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
```
```  1866   proof (cases "c = a \<or> c = b")
```
```  1867     case False
```
```  1868     then show ?thesis
```
```  1869       apply -
```
```  1870       apply (subst as(2))
```
```  1871       using as(3-)
```
```  1872       apply auto
```
```  1873       done
```
```  1874   next
```
```  1875     case True
```
```  1876     then show ?thesis
```
```  1877     proof
```
```  1878       assume *: "c = a"
```
```  1879       then have "g {a .. c} = \<^bold>1"
```
```  1880         apply -
```
```  1881         apply (rule as(1)[rule_format])
```
```  1882         apply auto
```
```  1883         done
```
```  1884       then show ?thesis
```
```  1885         unfolding * by auto
```
```  1886     next
```
```  1887       assume *: "c = b"
```
```  1888       then have "g {c .. b} = \<^bold>1"
```
```  1889         apply -
```
```  1890         apply (rule as(1)[rule_format])
```
```  1891         apply auto
```
```  1892         done
```
```  1893       then show ?thesis
```
```  1894         unfolding * by auto
```
```  1895     qed
```
```  1896   qed
```
```  1897 qed
```
```  1898
```
```  1899 subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
```
```  1900
```
```  1901 definition fine  (infixr "fine" 46)
```
```  1902   where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
```
```  1903
```
```  1904 lemma fineI:
```
```  1905   assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```  1906   shows "d fine s"
```
```  1907   using assms unfolding fine_def by auto
```
```  1908
```
```  1909 lemma fineD[dest]:
```
```  1910   assumes "d fine s"
```
```  1911   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
```
```  1912   using assms unfolding fine_def by auto
```
```  1913
```
```  1914 lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
```
```  1915   unfolding fine_def by auto
```
```  1916
```
```  1917 lemma fine_inters:
```
```  1918  "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
```
```  1919   unfolding fine_def by blast
```
```  1920
```
```  1921 lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
```
```  1922   unfolding fine_def by blast
```
```  1923
```
```  1924 lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
```
```  1925   unfolding fine_def by auto
```
```  1926
```
```  1927 lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
```
```  1928   unfolding fine_def by blast
```
```  1929
```
```  1930
```
```  1931 subsection \<open>Gauge integral. Define on compact intervals first, then use a limit.\<close>
```
```  1932
```
```  1933 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
```
```  1934   where "(f has_integral_compact_interval y) i \<longleftrightarrow>
```
```  1935     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  1936       (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow> norm ((\<Sum>(x,k)\<in>p. content k *\<^sub>R f x) - y) < e))"
```
```  1937
```
```  1938 definition has_integral ::
```
```  1939     "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
```
```  1940   (infixr "has'_integral" 46)
```
```  1941   where "(f has_integral y) i \<longleftrightarrow>
```
```  1942     (if \<exists>a b. i = cbox a b
```
```  1943      then (f has_integral_compact_interval y) i
```
```  1944      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  1945       (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
```
```  1946         norm (z - y) < e)))"
```
```  1947
```
```  1948 lemma has_integral:
```
```  1949   "(f has_integral y) (cbox a b) \<longleftrightarrow>
```
```  1950     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  1951       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  1952         norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```  1953   unfolding has_integral_def has_integral_compact_interval_def
```
```  1954   by auto
```
```  1955
```
```  1956 lemma has_integral_real:
```
```  1957   "(f has_integral y) {a .. b::real} \<longleftrightarrow>
```
```  1958     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```  1959       (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
```
```  1960         norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```  1961   unfolding box_real[symmetric]
```
```  1962   by (rule has_integral)
```
```  1963
```
```  1964 lemma has_integralD[dest]:
```
```  1965   assumes "(f has_integral y) (cbox a b)"
```
```  1966     and "e > 0"
```
```  1967   obtains d where "gauge d"
```
```  1968     and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
```
```  1969       norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
```
```  1970   using assms unfolding has_integral by auto
```
```  1971
```
```  1972 lemma has_integral_alt:
```
```  1973   "(f has_integral y) i \<longleftrightarrow>
```
```  1974     (if \<exists>a b. i = cbox a b
```
```  1975      then (f has_integral y) i
```
```  1976      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  1977       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
```
```  1978   unfolding has_integral
```
```  1979   unfolding has_integral_compact_interval_def has_integral_def
```
```  1980   by auto
```
```  1981
```
```  1982 lemma has_integral_altD:
```
```  1983   assumes "(f has_integral y) i"
```
```  1984     and "\<not> (\<exists>a b. i = cbox a b)"
```
```  1985     and "e>0"
```
```  1986   obtains B where "B > 0"
```
```  1987     and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  1988       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
```
```  1989   using assms
```
```  1990   unfolding has_integral
```
```  1991   unfolding has_integral_compact_interval_def has_integral_def
```
```  1992   by auto
```
```  1993
```
```  1994 definition integrable_on (infixr "integrable'_on" 46)
```
```  1995   where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
```
```  1996
```
```  1997 definition "integral i f = (SOME y. (f has_integral y) i \<or> ~ f integrable_on i \<and> y=0)"
```
```  1998
```
```  1999 lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
```
```  2000   unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
```
```  2001
```
```  2002 lemma not_integrable_integral: "~ f integrable_on i \<Longrightarrow> integral i f = 0"
```
```  2003   unfolding integrable_on_def integral_def by blast
```
```  2004
```
```  2005 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
```
```  2006   unfolding integrable_on_def by auto
```
```  2007
```
```  2008 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
```
```  2009   by auto
```
```  2010
```
```  2011 lemma setsum_content_null:
```
```  2012   assumes "content (cbox a b) = 0"
```
```  2013     and "p tagged_division_of (cbox a b)"
```
```  2014   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
```
```  2015 proof (rule setsum.neutral, rule)
```
```  2016   fix y
```
```  2017   assume y: "y \<in> p"
```
```  2018   obtain x k where xk: "y = (x, k)"
```
```  2019     using surj_pair[of y] by blast
```
```  2020   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
```
```  2021   from this(2) obtain c d where k: "k = cbox c d" by blast
```
```  2022   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
```
```  2023     unfolding xk by auto
```
```  2024   also have "\<dots> = 0"
```
```  2025     using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
```
```  2026     unfolding assms(1) k
```
```  2027     by auto
```
```  2028   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
```
```  2029 qed
```
```  2030
```
```  2031
```
```  2032 subsection \<open>Some basic combining lemmas.\<close>
```
```  2033
```
```  2034 lemma tagged_division_unions_exists:
```
```  2035   assumes "finite iset"
```
```  2036     and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
```
```  2037     and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
```
```  2038     and "\<Union>iset = i"
```
```  2039    obtains p where "p tagged_division_of i" and "d fine p"
```
```  2040 proof -
```
```  2041   obtain pfn where pfn:
```
```  2042     "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
```
```  2043     "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
```
```  2044     using bchoice[OF assms(2)] by auto
```
```  2045   show thesis
```
```  2046     apply (rule_tac p="\<Union>(pfn ` iset)" in that)
```
```  2047     using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
```
```  2048     by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
```
```  2049 qed
```
```  2050
```
```  2051
```
```  2052 subsection \<open>The set we're concerned with must be closed.\<close>
```
```  2053
```
```  2054 lemma division_of_closed:
```
```  2055   fixes i :: "'n::euclidean_space set"
```
```  2056   shows "s division_of i \<Longrightarrow> closed i"
```
```  2057   unfolding division_of_def by fastforce
```
```  2058
```
```  2059 subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
```
```  2060
```
```  2061 lemma interval_bisection_step:
```
```  2062   fixes type :: "'a::euclidean_space"
```
```  2063   assumes "P {}"
```
```  2064     and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
```
```  2065     and "\<not> P (cbox a (b::'a))"
```
```  2066   obtains c d where "\<not> P (cbox c d)"
```
```  2067     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"
```
```  2068 proof -
```
```  2069   have "cbox a b \<noteq> {}"
```
```  2070     using assms(1,3) by metis
```
```  2071   then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
```
```  2072     by (force simp: mem_box)
```
```  2073   { fix f
```
```  2074     have "\<lbrakk>finite f;
```
```  2075            \<And>s. s\<in>f \<Longrightarrow> P s;
```
```  2076            \<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
```
```  2077            \<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)"
```
```  2078     proof (induct f rule: finite_induct)
```
```  2079       case empty
```
```  2080       show ?case
```
```  2081         using assms(1) by auto
```
```  2082     next
```
```  2083       case (insert x f)
```
```  2084       show ?case
```
```  2085         unfolding Union_insert
```
```  2086         apply (rule assms(2)[rule_format])
```
```  2087         using inter_interior_unions_intervals [of f "interior x"]
```
```  2088         apply (auto simp: insert)
```
```  2089         by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
```
```  2090     qed
```
```  2091   } note UN_cases = this
```
```  2092   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>
```
```  2093     (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
```
```  2094   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"
```
```  2095   {
```
```  2096     presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
```
```  2097     then show thesis
```
```  2098       unfolding atomize_not not_all
```
```  2099       by (blast intro: that)
```
```  2100   }
```
```  2101   assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
```
```  2102   have "P (\<Union>?A)"
```
```  2103   proof (rule UN_cases)
```
```  2104     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)
```
```  2105       (\<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}"
```
```  2106     have "?A \<subseteq> ?B"
```
```  2107     proof
```
```  2108       fix x
```
```  2109       assume "x \<in> ?A"
```
```  2110       then obtain c d
```
```  2111         where x:  "x = cbox c d"
```
```  2112                   "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2113                         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2114                         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
```
```  2115       show "x \<in> ?B"
```
```  2116         unfolding image_iff x
```
```  2117         apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
```
```  2118         apply (rule arg_cong2 [where f = cbox])
```
```  2119         using x(2) ab
```
```  2120         apply (auto simp add: euclidean_eq_iff[where 'a='a])
```
```  2121         by fastforce
```
```  2122     qed
```
```  2123     then show "finite ?A"
```
```  2124       by (rule finite_subset) auto
```
```  2125   next
```
```  2126     fix s
```
```  2127     assume "s \<in> ?A"
```
```  2128     then obtain c d
```
```  2129       where s: "s = cbox c d"
```
```  2130                "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2131                      c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2132                      c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
```
```  2133       by blast
```
```  2134     show "P s"
```
```  2135       unfolding s
```
```  2136       apply (rule as[rule_format])
```
```  2137       using ab s(2) by force
```
```  2138     show "\<exists>a b. s = cbox a b"
```
```  2139       unfolding s by auto
```
```  2140     fix t
```
```  2141     assume "t \<in> ?A"
```
```  2142     then obtain e f where t:
```
```  2143       "t = cbox e f"
```
```  2144       "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2145         e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2146         e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
```
```  2147       by blast
```
```  2148     assume "s \<noteq> t"
```
```  2149     then have "\<not> (c = e \<and> d = f)"
```
```  2150       unfolding s t by auto
```
```  2151     then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
```
```  2152       unfolding euclidean_eq_iff[where 'a='a] by auto
```
```  2153     then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
```
```  2154       using s(2) t(2) apply fastforce
```
```  2155       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
```
```  2156     have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
```
```  2157       by auto
```
```  2158     show "interior s \<inter> interior t = {}"
```
```  2159       unfolding s t interior_cbox
```
```  2160     proof (rule *)
```
```  2161       fix x
```
```  2162       assume "x \<in> box c d" "x \<in> box e f"
```
```  2163       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"
```
```  2164         unfolding mem_box using i'
```
```  2165         by force+
```
```  2166       show False  using s(2)[OF i']
```
```  2167       proof safe
```
```  2168         assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
```
```  2169         show False
```
```  2170           using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
```
```  2171       next
```
```  2172         assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
```
```  2173         show False
```
```  2174           using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
```
```  2175       qed
```
```  2176     qed
```
```  2177   qed
```
```  2178   also have "\<Union>?A = cbox a b"
```
```  2179   proof (rule set_eqI,rule)
```
```  2180     fix x
```
```  2181     assume "x \<in> \<Union>?A"
```
```  2182     then obtain c d where x:
```
```  2183       "x \<in> cbox c d"
```
```  2184       "\<And>i. i \<in> Basis \<Longrightarrow>
```
```  2185         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
```
```  2186         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
```
```  2187       by blast
```
```  2188     show "x\<in>cbox a b"
```
```  2189       unfolding mem_box
```
```  2190     proof safe
```
```  2191       fix i :: 'a
```
```  2192       assume i: "i \<in> Basis"
```
```  2193       then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
```
```  2194         using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
```
```  2195     qed
```
```  2196   next
```
```  2197     fix x
```
```  2198     assume x: "x \<in> cbox a b"
```
```  2199     have "\<forall>i\<in>Basis.
```
```  2200       \<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"
```
```  2201       (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
```
```  2202       unfolding mem_box
```
```  2203     proof
```
```  2204       fix i :: 'a
```
```  2205       assume i: "i \<in> Basis"
```
```  2206       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)"
```
```  2207         using x[unfolded mem_box,THEN bspec, OF i] by auto
```
```  2208       then show "\<exists>c d. ?P i c d"
```
```  2209         by blast
```
```  2210     qed
```
```  2211     then show "x\<in>\<Union>?A"
```
```  2212       unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
```
```  2213       apply auto
```
```  2214       apply (rule_tac x="cbox xa xaa" in exI)
```
```  2215       unfolding mem_box
```
```  2216       apply auto
```
```  2217       done
```
```  2218   qed
```
```  2219   finally show False
```
```  2220     using assms by auto
```
```  2221 qed
```
```  2222
```
```  2223 lemma interval_bisection:
```
```  2224   fixes type :: "'a::euclidean_space"
```
```  2225   assumes "P {}"
```
```  2226     and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
```
```  2227     and "\<not> P (cbox a (b::'a))"
```
```  2228   obtains x where "x \<in> cbox a b"
```
```  2229     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)"
```
```  2230 proof -
```
```  2231   have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
```
```  2232     (\<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>
```
```  2233        2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x")
```
```  2234   proof
```
```  2235     show "?P x" for x
```
```  2236     proof (cases "P (cbox (fst x) (snd x))")
```
```  2237       case True
```
```  2238       then show ?thesis by auto
```
```  2239     next
```
```  2240       case as: False
```
```  2241       obtain c d where "\<not> P (cbox c d)"
```
```  2242         "\<forall>i\<in>Basis.
```
```  2243            fst x \<bullet> i \<le> c \<bullet> i \<and>
```
```  2244            c \<bullet> i \<le> d \<bullet> i \<and>
```
```  2245            d \<bullet> i \<le> snd x \<bullet> i \<and>
```
```  2246            2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
```
```  2247         by (rule interval_bisection_step[of P, OF assms(1-2) as])
```
```  2248       then show ?thesis
```
```  2249         apply -
```
```  2250         apply (rule_tac x="(c,d)" in exI)
```
```  2251         apply auto
```
```  2252         done
```
```  2253     qed
```
```  2254   qed
```
```  2255   then obtain f where f:
```
```  2256     "\<forall>x.
```
```  2257       \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
```
```  2258       \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
```
```  2259         (\<forall>i\<in>Basis.
```
```  2260             fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
```
```  2261             fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
```
```  2262             snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
```
```  2263             2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
```
```  2264     apply -
```
```  2265     apply (drule choice)
```
```  2266     apply blast
```
```  2267     done
```
```  2268   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
```
```  2269   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
```
```  2270     (\<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>
```
```  2271     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")
```
```  2272   proof -
```
```  2273     show "A 0 = a" "B 0 = b"
```
```  2274       unfolding ab_def by auto
```
```  2275     note S = ab_def funpow.simps o_def id_apply
```
```  2276     show "?P n" for n
```
```  2277     proof (induct n)
```
```  2278       case 0
```
```  2279       then show ?case
```
```  2280         unfolding S
```
```  2281         apply (rule f[rule_format]) using assms(3)
```
```  2282         apply auto
```
```  2283         done
```
```  2284     next
```
```  2285       case (Suc n)
```
```  2286       show ?case
```
```  2287         unfolding S
```
```  2288         apply (rule f[rule_format])
```
```  2289         using Suc
```
```  2290         unfolding S
```
```  2291         apply auto
```
```  2292         done
```
```  2293     qed
```
```  2294   qed
```
```  2295   note AB = this(1-2) conjunctD2[OF this(3),rule_format]
```
```  2296
```
```  2297   have interv: "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
```
```  2298     if e: "0 < e" for e
```
```  2299   proof -
```
```  2300     obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
```
```  2301       using real_arch_pow[of 2 "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto
```
```  2302     show ?thesis
```
```  2303     proof (rule exI [where x=n], clarify)
```
```  2304       fix x y
```
```  2305       assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
```
```  2306       have "dist x y \<le> setsum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
```
```  2307         unfolding dist_norm by(rule norm_le_l1)
```
```  2308       also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
```
```  2309       proof (rule setsum_mono)
```
```  2310         fix i :: 'a
```
```  2311         assume i: "i \<in> Basis"
```
```  2312         show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
```
```  2313           using xy[unfolded mem_box,THEN bspec, OF i]
```
```  2314           by (auto simp: inner_diff_left)
```
```  2315       qed
```
```  2316       also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
```
```  2317         unfolding setsum_divide_distrib
```
```  2318       proof (rule setsum_mono)
```
```  2319         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
```
```  2320         proof (induct n)
```
```  2321           case 0
```
```  2322           then show ?case
```
```  2323             unfolding AB by auto
```
```  2324         next
```
```  2325           case (Suc n)
```
```  2326           have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
```
```  2327             using AB(4)[of i n] using i by auto
```
```  2328           also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
```
```  2329             using Suc by (auto simp add: field_simps)
```
```  2330           finally show ?case .
```
```  2331         qed
```
```  2332       qed
```
```  2333       also have "\<dots> < e"
```
```  2334         using n using e by (auto simp add: field_simps)
```
```  2335       finally show "dist x y < e" .
```
```  2336     qed
```
```  2337   qed
```
```  2338   {
```
```  2339     fix n m :: nat
```
```  2340     assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
```
```  2341     proof (induction rule: inc_induct)
```
```  2342       case (step i)
```
```  2343       show ?case
```
```  2344         using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
```
```  2345     qed simp
```
```  2346   } note ABsubset = this
```
```  2347   have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
```
```  2348     by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
```
```  2349       (metis nat.exhaust AB(1-3) assms(1,3))
```
```  2350   then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
```
```  2351     by blast
```
```  2352   show thesis
```
```  2353   proof (rule that[rule_format, of x0])
```
```  2354     show "x0\<in>cbox a b"
```
```  2355       using x0[of 0] unfolding AB .
```
```  2356     fix e :: real
```
```  2357     assume "e > 0"
```
```  2358     from interv[OF this] obtain n
```
```  2359       where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
```
```  2360     have "\<not> P (cbox (A n) (B n))"
```
```  2361       apply (cases "0 < n")
```
```  2362       using AB(3)[of "n - 1"] assms(3) AB(1-2)
```
```  2363       apply auto
```
```  2364       done
```
```  2365     moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
```
```  2366       using n using x0[of n] by auto
```
```  2367     moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
```
```  2368       unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
```
```  2369     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)"
```
```  2370       apply (rule_tac x="A n" in exI)
```
```  2371       apply (rule_tac x="B n" in exI)
```
```  2372       apply (auto simp: x0)
```
```  2373       done
```
```  2374   qed
```
```  2375 qed
```
```  2376
```
```  2377
```
```  2378 subsection \<open>Cousin's lemma.\<close>
```
```  2379
```
```  2380 lemma fine_division_exists:
```
```  2381   fixes a b :: "'a::euclidean_space"
```
```  2382   assumes "gauge g"
```
```  2383   obtains p where "p tagged_division_of (cbox a b)" "g fine p"
```
```  2384 proof -
```
```  2385   presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
```
```  2386   then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
```
```  2387     by blast
```
```  2388   then show thesis ..
```
```  2389 next
```
```  2390   assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
```
```  2391   obtain x where x:
```
```  2392       "x \<in> (cbox a b)"
```
```  2393       "\<And>e. 0 < e \<Longrightarrow>
```
```  2394         \<exists>c d.
```
```  2395           x \<in> cbox c d \<and>
```
```  2396           cbox c d \<subseteq> ball x e \<and>
```
```  2397           cbox c d \<subseteq> (cbox a b) \<and>
```
```  2398           \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
```
```  2399     apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
```
```  2400     apply (simp add: fine_def)
```
```  2401     apply (metis tagged_division_union fine_union)
```
```  2402     apply (auto simp: )
```
```  2403     done
```
```  2404   obtain e where e: "e > 0" "ball x e \<subseteq> g x"
```
```  2405     using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
```
```  2406   from x(2)[OF e(1)]
```
```  2407   obtain c d where c_d: "x \<in> cbox c d"
```
```  2408                         "cbox c d \<subseteq> ball x e"
```
```  2409                         "cbox c d \<subseteq> cbox a b"
```
```  2410                         "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
```
```  2411     by blast
```
```  2412   have "g fine {(x, cbox c d)}"
```
```  2413     unfolding fine_def using e using c_d(2) by auto
```
```  2414   then show False
```
```  2415     using tagged_division_of_self[OF c_d(1)] using c_d by auto
```
```  2416 qed
```
```  2417
```
```  2418 lemma fine_division_exists_real:
```
```  2419   fixes a b :: real
```
```  2420   assumes "gauge g"
```
```  2421   obtains p where "p tagged_division_of {a .. b}" "g fine p"
```
```  2422   by (metis assms box_real(2) fine_division_exists)
```
```  2423
```
```  2424 subsection \<open>Basic theorems about integrals.\<close>
```
```  2425
```
```  2426 lemma has_integral_unique:
```
```  2427   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2428   assumes "(f has_integral k1) i"
```
```  2429     and "(f has_integral k2) i"
```
```  2430   shows "k1 = k2"
```
```  2431 proof (rule ccontr)
```
```  2432   let ?e = "norm (k1 - k2) / 2"
```
```  2433   assume as: "k1 \<noteq> k2"
```
```  2434   then have e: "?e > 0"
```
```  2435     by auto
```
```  2436   have lem: False
```
```  2437     if f_k1: "(f has_integral k1) (cbox a b)"
```
```  2438     and f_k2: "(f has_integral k2) (cbox a b)"
```
```  2439     and "k1 \<noteq> k2"
```
```  2440     for f :: "'n \<Rightarrow> 'a" and a b k1 k2
```
```  2441   proof -
```
```  2442     let ?e = "norm (k1 - k2) / 2"
```
```  2443     from \<open>k1 \<noteq> k2\<close> have e: "?e > 0" by auto
```
```  2444     obtain d1 where d1:
```
```  2445         "gauge d1"
```
```  2446         "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
```
```  2447           d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
```
```  2448       by (rule has_integralD[OF f_k1 e]) blast
```
```  2449     obtain d2 where d2:
```
```  2450         "gauge d2"
```
```  2451         "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
```
```  2452           d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
```
```  2453       by (rule has_integralD[OF f_k2 e]) blast
```
```  2454     obtain p where p:
```
```  2455         "p tagged_division_of cbox a b"
```
```  2456         "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  2457       by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
```
```  2458     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  2459     have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
```
```  2460       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
```
```  2461       by (auto simp add:algebra_simps norm_minus_commute)
```
```  2462     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  2463       apply (rule add_strict_mono)
```
```  2464       apply (rule_tac[!] d2(2) d1(2))
```
```  2465       using p unfolding fine_def
```
```  2466       apply auto
```
```  2467       done
```
```  2468     finally show False by auto
```
```  2469   qed
```
```  2470   {
```
```  2471     presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
```
```  2472     then show False
```
```  2473       using as assms lem by blast
```
```  2474   }
```
```  2475   assume as: "\<not> (\<exists>a b. i = cbox a b)"
```
```  2476   obtain B1 where B1:
```
```  2477       "0 < B1"
```
```  2478       "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```  2479         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```  2480           norm (z - k1) < norm (k1 - k2) / 2"
```
```  2481     by (rule has_integral_altD[OF assms(1) as,OF e]) blast
```
```  2482   obtain B2 where B2:
```
```  2483       "0 < B2"
```
```  2484       "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
```
```  2485         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```  2486           norm (z - k2) < norm (k1 - k2) / 2"
```
```  2487     by (rule has_integral_altD[OF assms(2) as,OF e]) blast
```
```  2488   have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
```
```  2489     apply (rule bounded_subset_cbox)
```
```  2490     using bounded_Un bounded_ball
```
```  2491     apply auto
```
```  2492     done
```
```  2493   then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
```
```  2494     by blast
```
```  2495   obtain w where w:
```
```  2496     "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
```
```  2497     "norm (w - k1) < norm (k1 - k2) / 2"
```
```  2498     using B1(2)[OF ab(1)] by blast
```
```  2499   obtain z where z:
```
```  2500     "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
```
```  2501     "norm (z - k2) < norm (k1 - k2) / 2"
```
```  2502     using B2(2)[OF ab(2)] by blast
```
```  2503   have "z = w"
```
```  2504     using lem[OF w(1) z(1)] by auto
```
```  2505   then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
```
```  2506     using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
```
```  2507     by (auto simp add: norm_minus_commute)
```
```  2508   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```  2509     apply (rule add_strict_mono)
```
```  2510     apply (rule_tac[!] z(2) w(2))
```
```  2511     done
```
```  2512   finally show False by auto
```
```  2513 qed
```
```  2514
```
```  2515 lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
```
```  2516   unfolding integral_def
```
```  2517   by (rule some_equality) (auto intro: has_integral_unique)
```
```  2518
```
```  2519 lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> ~ f integrable_on k \<and> y=0"
```
```  2520   unfolding integral_def integrable_on_def
```
```  2521   apply (erule subst)
```
```  2522   apply (rule someI_ex)
```
```  2523   by blast
```
```  2524
```
```  2525 lemma has_integral_is_0:
```
```  2526   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2527   assumes "\<forall>x\<in>s. f x = 0"
```
```  2528   shows "(f has_integral 0) s"
```
```  2529 proof -
```
```  2530   have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
```
```  2531     (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```  2532     unfolding has_integral
```
```  2533   proof clarify
```
```  2534     fix a b e
```
```  2535     fix f :: "'n \<Rightarrow> 'a"
```
```  2536     assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
```
```  2537     have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  2538       if p: "p tagged_division_of cbox a b" for p
```
```  2539     proof -
```
```  2540       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
```
```  2541       proof (rule setsum.neutral, rule)
```
```  2542         fix x
```
```  2543         assume x: "x \<in> p"
```
```  2544         have "f (fst x) = 0"
```
```  2545           using tagged_division_ofD(2-3)[OF p, of "fst x" "snd x"] using as x by auto
```
```  2546         then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
```
```  2547           apply (subst surjective_pairing[of x])
```
```  2548           unfolding split_conv
```
```  2549           apply auto
```
```  2550           done
```
```  2551       qed
```
```  2552       then show ?thesis
```
```  2553         using as by auto
```
```  2554     qed
```
```  2555     then show "\<exists>d. gauge d \<and>
```
```  2556         (\<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)"
```
```  2557       by auto
```
```  2558   qed
```
```  2559   {
```
```  2560     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  2561     with assms lem show ?thesis
```
```  2562       by blast
```
```  2563   }
```
```  2564   have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
```
```  2565     apply (rule ext)
```
```  2566     using assms
```
```  2567     apply auto
```
```  2568     done
```
```  2569   assume "\<not> (\<exists>a b. s = cbox a b)"
```
```  2570   then show ?thesis
```
```  2571     using lem
```
```  2572     by (subst has_integral_alt) (force simp add: *)
```
```  2573 qed
```
```  2574
```
```  2575 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
```
```  2576   by (rule has_integral_is_0) auto
```
```  2577
```
```  2578 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
```
```  2579   using has_integral_unique[OF has_integral_0] by auto
```
```  2580
```
```  2581 lemma has_integral_linear:
```
```  2582   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2583   assumes "(f has_integral y) s"
```
```  2584     and "bounded_linear h"
```
```  2585   shows "((h \<circ> f) has_integral ((h y))) s"
```
```  2586 proof -
```
```  2587   interpret bounded_linear h
```
```  2588     using assms(2) .
```
```  2589   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
```
```  2590     by blast
```
```  2591   have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
```
```  2592     (f has_integral y) (cbox a b) \<Longrightarrow> ((h \<circ> f) has_integral h y) (cbox a b)"
```
```  2593     unfolding has_integral
```
```  2594   proof (clarify, goal_cases)
```
```  2595     case prems: (1 f y a b e)
```
```  2596     from pos_bounded
```
```  2597     obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
```
```  2598       by blast
```
```  2599     have "e / B > 0" using prems(2) B by simp
```
```  2600     then obtain g
```
```  2601       where g: "gauge g"
```
```  2602                "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
```
```  2603                     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
```
```  2604         using prems(1) by auto
```
```  2605     {
```
```  2606       fix p
```
```  2607       assume as: "p tagged_division_of (cbox a b)" "g fine p"
```
```  2608       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"
```
```  2609         by auto
```
```  2610       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"
```
```  2611         unfolding o_def unfolding scaleR[symmetric] hc by simp
```
```  2612       also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  2613         using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
```
```  2614       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)" .
```
```  2615       then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
```
```  2616         apply (simp add: diff[symmetric])
```
```  2617         apply (rule le_less_trans[OF B(2)])
```
```  2618         using g(2)[OF as] B(1)
```
```  2619         apply (auto simp add: field_simps)
```
```  2620         done
```
```  2621     }
```
```  2622     with g show ?case
```
```  2623       by (rule_tac x=g in exI) auto
```
```  2624   qed
```
```  2625   {
```
```  2626     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  2627     then show ?thesis
```
```  2628       using assms(1) lem by blast
```
```  2629   }
```
```  2630   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```  2631   then show ?thesis
```
```  2632   proof (subst has_integral_alt, clarsimp)
```
```  2633     fix e :: real
```
```  2634     assume e: "e > 0"
```
```  2635     have *: "0 < e/B" using e B(1) by simp
```
```  2636     obtain M where M:
```
```  2637       "M > 0"
```
```  2638       "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
```
```  2639         \<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"
```
```  2640       using has_integral_altD[OF assms(1) as *] by blast
```
```  2641     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2642       (\<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)"
```
```  2643     proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
```
```  2644       case prems: (1 a b)
```
```  2645       obtain z where z:
```
```  2646         "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
```
```  2647         "norm (z - y) < e / B"
```
```  2648         using M(2)[OF prems(1)] by blast
```
```  2649       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)"
```
```  2650         using zero by auto
```
```  2651       show ?case
```
```  2652         apply (rule_tac x="h z" in exI)
```
```  2653         apply (simp add: * lem z(1))
```
```  2654         apply (metis B diff le_less_trans pos_less_divide_eq z(2))
```
```  2655         done
```
```  2656     qed
```
```  2657   qed
```
```  2658 qed
```
```  2659
```
```  2660 lemma has_integral_scaleR_left:
```
```  2661   "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
```
```  2662   using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
```
```  2663
```
```  2664 lemma has_integral_mult_left:
```
```  2665   fixes c :: "_ :: real_normed_algebra"
```
```  2666   shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
```
```  2667   using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
```
```  2668
```
```  2669 text\<open>The case analysis eliminates the condition @{term "f integrable_on s"} at the cost
```
```  2670      of the type class constraint \<open>division_ring\<close>\<close>
```
```  2671 corollary integral_mult_left [simp]:
```
```  2672   fixes c:: "'a::{real_normed_algebra,division_ring}"
```
```  2673   shows "integral s (\<lambda>x. f x * c) = integral s f * c"
```
```  2674 proof (cases "f integrable_on s \<or> c = 0")
```
```  2675   case True then show ?thesis
```
```  2676     by (force intro: has_integral_mult_left)
```
```  2677 next
```
```  2678   case False then have "~ (\<lambda>x. f x * c) integrable_on s"
```
```  2679     using has_integral_mult_left [of "(\<lambda>x. f x * c)" _ s "inverse c"]
```
```  2680     by (force simp add: mult.assoc)
```
```  2681   with False show ?thesis by (simp add: not_integrable_integral)
```
```  2682 qed
```
```  2683
```
```  2684 corollary integral_mult_right [simp]:
```
```  2685   fixes c:: "'a::{real_normed_field}"
```
```  2686   shows "integral s (\<lambda>x. c * f x) = c * integral s f"
```
```  2687 by (simp add: mult.commute [of c])
```
```  2688
```
```  2689 corollary integral_divide [simp]:
```
```  2690   fixes z :: "'a::real_normed_field"
```
```  2691   shows "integral S (\<lambda>x. f x / z) = integral S (\<lambda>x. f x) / z"
```
```  2692 using integral_mult_left [of S f "inverse z"]
```
```  2693   by (simp add: divide_inverse_commute)
```
```  2694
```
```  2695 lemma has_integral_mult_right:
```
```  2696   fixes c :: "'a :: real_normed_algebra"
```
```  2697   shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
```
```  2698   using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
```
```  2699
```
```  2700 lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
```
```  2701   unfolding o_def[symmetric]
```
```  2702   by (metis has_integral_linear bounded_linear_scaleR_right)
```
```  2703
```
```  2704 lemma has_integral_cmult_real:
```
```  2705   fixes c :: real
```
```  2706   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
```
```  2707   shows "((\<lambda>x. c * f x) has_integral c * x) A"
```
```  2708 proof (cases "c = 0")
```
```  2709   case True
```
```  2710   then show ?thesis by simp
```
```  2711 next
```
```  2712   case False
```
```  2713   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
```
```  2714     unfolding real_scaleR_def .
```
```  2715 qed
```
```  2716
```
```  2717 lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral -k) s"
```
```  2718   by (drule_tac c="-1" in has_integral_cmul) auto
```
```  2719
```
```  2720 lemma has_integral_add:
```
```  2721   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2722   assumes "(f has_integral k) s"
```
```  2723     and "(g has_integral l) s"
```
```  2724   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
```
```  2725 proof -
```
```  2726   have lem: "((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
```
```  2727     if f_k: "(f has_integral k) (cbox a b)"
```
```  2728     and g_l: "(g has_integral l) (cbox a b)"
```
```  2729     for f :: "'n \<Rightarrow> 'a" and g a b k l
```
```  2730     unfolding has_integral
```
```  2731   proof clarify
```
```  2732     fix e :: real
```
```  2733     assume e: "e > 0"
```
```  2734     then have *: "e / 2 > 0"
```
```  2735       by auto
```
```  2736     obtain d1 where d1:
```
```  2737       "gauge d1"
```
```  2738       "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d1 fine p \<Longrightarrow>
```
```  2739         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) < e / 2"
```
```  2740       using has_integralD[OF f_k *] by blast
```
```  2741     obtain d2 where d2:
```
```  2742       "gauge d2"
```
```  2743       "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d2 fine p \<Longrightarrow>
```
```  2744         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l) < e / 2"
```
```  2745       using has_integralD[OF g_l *] by blast
```
```  2746     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  2747               norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"
```
```  2748     proof (rule exI [where x="\<lambda>x. (d1 x) \<inter> (d2 x)"], clarsimp simp add: gauge_inter[OF d1(1) d2(1)])
```
```  2749       fix p
```
```  2750       assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d1 x \<inter> d2 x) fine p"
```
```  2751       have *: "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) =
```
```  2752         (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"
```
```  2753         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]
```
```  2754         by (rule setsum.cong) auto
```
```  2755       from as have fine: "d1 fine p" "d2 fine p"
```
```  2756         unfolding fine_inter by auto
```
```  2757       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) =
```
```  2758             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))"
```
```  2759         unfolding * by (auto simp add: algebra_simps)
```
```  2760       also have "\<dots> < e/2 + e/2"
```
```  2761         apply (rule le_less_trans[OF norm_triangle_ineq])
```
```  2762         using as d1 d2 fine
```
```  2763         apply (blast intro: add_strict_mono)
```
```  2764         done
```
```  2765       finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e"
```
```  2766         by auto
```
```  2767     qed
```
```  2768   qed
```
```  2769   {
```
```  2770     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```  2771     then show ?thesis
```
```  2772       using assms lem by force
```
```  2773   }
```
```  2774   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```  2775   then show ?thesis
```
```  2776   proof (subst has_integral_alt, clarsimp, goal_cases)
```
```  2777     case (1 e)
```
```  2778     then have *: "e / 2 > 0"
```
```  2779       by auto
```
```  2780     from has_integral_altD[OF assms(1) as *]
```
```  2781     obtain B1 where B1:
```
```  2782         "0 < B1"
```
```  2783         "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```  2784           \<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"
```
```  2785       by blast
```
```  2786     from has_integral_altD[OF assms(2) as *]
```
```  2787     obtain B2 where B2:
```
```  2788         "0 < B2"
```
```  2789         "\<And>a b. ball 0 B2 \<subseteq> (cbox a b) \<Longrightarrow>
```
```  2790           \<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"
```
```  2791       by blast
```
```  2792     show ?case
```
```  2793     proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
```
```  2794       fix a b
```
```  2795       assume "ball 0 (max B1 B2) \<subseteq> cbox a (b::'n)"
```
```  2796       then have *: "ball 0 B1 \<subseteq> cbox a (b::'n)" "ball 0 B2 \<subseteq> cbox a (b::'n)"
```
```  2797         by auto
```
```  2798       obtain w where w:
```
```  2799         "((\<lambda>x. if x \<in> s then f x else 0) has_integral w) (cbox a b)"
```
```  2800         "norm (w - k) < e / 2"
```
```  2801         using B1(2)[OF *(1)] by blast
```
```  2802       obtain z where z:
```
```  2803         "((\<lambda>x. if x \<in> s then g x else 0) has_integral z) (cbox a b)"
```
```  2804         "norm (z - l) < e / 2"
```
```  2805         using B2(2)[OF *(2)] by blast
```
```  2806       have *: "\<And>x. (if x \<in> s then f x + g x else 0) =
```
```  2807         (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)"
```
```  2808         by auto
```
```  2809       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"
```
```  2810         apply (rule_tac x="w + z" in exI)
```
```  2811         apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
```
```  2812         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
```
```  2813         apply (auto simp add: field_simps)
```
```  2814         done
```
```  2815     qed
```
```  2816   qed
```
```  2817 qed
```
```  2818
```
```  2819 lemma has_integral_sub:
```
```  2820   "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow>
```
```  2821     ((\<lambda>x. f x - g x) has_integral (k - l)) s"
```
```  2822   using has_integral_add[OF _ has_integral_neg, of f k s g l]
```
```  2823   by (auto simp: algebra_simps)
```
```  2824
```
```  2825 lemma integral_0 [simp]:
```
```  2826   "integral s (\<lambda>x::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
```
```  2827   by (rule integral_unique has_integral_0)+
```
```  2828
```
```  2829 lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  2830     integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
```
```  2831   by (rule integral_unique) (metis integrable_integral has_integral_add)
```
```  2832
```
```  2833 lemma integral_cmul [simp]: "integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
```
```  2834 proof (cases "f integrable_on s \<or> c = 0")
```
```  2835   case True with has_integral_cmul show ?thesis by force
```
```  2836 next
```
```  2837   case False then have "~ (\<lambda>x. c *\<^sub>R f x) integrable_on s"
```
```  2838     using has_integral_cmul [of "(\<lambda>x. c *\<^sub>R f x)" _ s "inverse c"]
```
```  2839     by force
```
```  2840   with False show ?thesis by (simp add: not_integrable_integral)
```
```  2841 qed
```
```  2842
```
```  2843 lemma integral_neg [simp]: "integral s (\<lambda>x. - f x) = - integral s f"
```
```  2844 proof (cases "f integrable_on s")
```
```  2845   case True then show ?thesis
```
```  2846     by (simp add: has_integral_neg integrable_integral integral_unique)
```
```  2847 next
```
```  2848   case False then have "~ (\<lambda>x. - f x) integrable_on s"
```
```  2849     using has_integral_neg [of "(\<lambda>x. - f x)" _ s ]
```
```  2850     by force
```
```  2851   with False show ?thesis by (simp add: not_integrable_integral)
```
```  2852 qed
```
```  2853
```
```  2854 lemma integral_diff: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```  2855     integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
```
```  2856   by (rule integral_unique) (metis integrable_integral has_integral_sub)
```
```  2857
```
```  2858 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
```
```  2859   unfolding integrable_on_def using has_integral_0 by auto
```
```  2860
```
```  2861 lemma integrable_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
```
```  2862   unfolding integrable_on_def by(auto intro: has_integral_add)
```
```  2863
```
```  2864 lemma integrable_cmul: "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
```
```  2865   unfolding integrable_on_def by(auto intro: has_integral_cmul)
```
```  2866
```
```  2867 lemma integrable_on_cmult_iff:
```
```  2868   fixes c :: real
```
```  2869   assumes "c \<noteq> 0"
```
```  2870   shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  2871   using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] \<open>c \<noteq> 0\<close>
```
```  2872   by auto
```
```  2873
```
```  2874 lemma integrable_on_cmult_left:
```
```  2875   assumes "f integrable_on s"
```
```  2876   shows "(\<lambda>x. of_real c * f x) integrable_on s"
```
```  2877     using integrable_cmul[of f s "of_real c"] assms
```
```  2878     by (simp add: scaleR_conv_of_real)
```
```  2879
```
```  2880 lemma integrable_neg: "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
```
```  2881   unfolding integrable_on_def by(auto intro: has_integral_neg)
```
```  2882
```
```  2883 lemma integrable_diff:
```
```  2884   "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
```
```  2885   unfolding integrable_on_def by(auto intro: has_integral_sub)
```
```  2886
```
```  2887 lemma integrable_linear:
```
```  2888   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on s"
```
```  2889   unfolding integrable_on_def by(auto intro: has_integral_linear)
```
```  2890
```
```  2891 lemma integral_linear:
```
```  2892   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
```
```  2893   apply (rule has_integral_unique [where i=s and f = "h \<circ> f"])
```
```  2894   apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
```
```  2895   done
```
```  2896
```
```  2897 lemma integral_component_eq[simp]:
```
```  2898   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  2899   assumes "f integrable_on s"
```
```  2900   shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
```
```  2901   unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..
```
```  2902
```
```  2903 lemma has_integral_setsum:
```
```  2904   assumes "finite t"
```
```  2905     and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
```
```  2906   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"
```
```  2907   using assms(1) subset_refl[of t]
```
```  2908 proof (induct rule: finite_subset_induct)
```
```  2909   case empty
```
```  2910   then show ?case by auto
```
```  2911 next
```
```  2912   case (insert x F)
```
```  2913   with assms show ?case
```
```  2914     by (simp add: has_integral_add)
```
```  2915 qed
```
```  2916
```
```  2917 lemma integral_setsum:
```
```  2918   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow>
```
```  2919    integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"
```
```  2920   by (auto intro: has_integral_setsum integrable_integral)
```
```  2921
```
```  2922 lemma integrable_setsum:
```
```  2923   "\<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"
```
```  2924   unfolding integrable_on_def
```
```  2925   apply (drule bchoice)
```
```  2926   using has_integral_setsum[of t]
```
```  2927   apply auto
```
```  2928   done
```
```  2929
```
```  2930 lemma has_integral_eq:
```
```  2931   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  2932     and "(f has_integral k) s"
```
```  2933   shows "(g has_integral k) s"
```
```  2934   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
```
```  2935   using has_integral_is_0[of s "\<lambda>x. f x - g x"]
```
```  2936   using assms(1)
```
```  2937   by auto
```
```  2938
```
```  2939 lemma integrable_eq: "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
```
```  2940   unfolding integrable_on_def
```
```  2941   using has_integral_eq[of s f g] has_integral_eq by blast
```
```  2942
```
```  2943 lemma has_integral_cong:
```
```  2944   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  2945   shows "(f has_integral i) s = (g has_integral i) s"
```
```  2946   using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
```
```  2947   by auto
```
```  2948
```
```  2949 lemma integral_cong:
```
```  2950   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```  2951   shows "integral s f = integral s g"
```
```  2952   unfolding integral_def
```
```  2953 by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
```
```  2954
```
```  2955 lemma integrable_on_cmult_left_iff [simp]:
```
```  2956   assumes "c \<noteq> 0"
```
```  2957   shows "(\<lambda>x. of_real c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  2958         (is "?lhs = ?rhs")
```
```  2959 proof
```
```  2960   assume ?lhs
```
```  2961   then have "(\<lambda>x. of_real (1 / c) * (of_real c * f x)) integrable_on s"
```
```  2962     using integrable_cmul[of "\<lambda>x. of_real c * f x" s "1 / of_real c"]
```
```  2963     by (simp add: scaleR_conv_of_real)
```
```  2964   then have "(\<lambda>x. (of_real (1 / c) * of_real c * f x)) integrable_on s"
```
```  2965     by (simp add: algebra_simps)
```
```  2966   with \<open>c \<noteq> 0\<close> show ?rhs
```
```  2967     by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
```
```  2968 qed (blast intro: integrable_on_cmult_left)
```
```  2969
```
```  2970 lemma integrable_on_cmult_right:
```
```  2971   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```  2972   assumes "f integrable_on s"
```
```  2973   shows "(\<lambda>x. f x * of_real c) integrable_on s"
```
```  2974 using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
```
```  2975
```
```  2976 lemma integrable_on_cmult_right_iff [simp]:
```
```  2977   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```  2978   assumes "c \<noteq> 0"
```
```  2979   shows "(\<lambda>x. f x * of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  2980 using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
```
```  2981
```
```  2982 lemma integrable_on_cdivide:
```
```  2983   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```  2984   assumes "f integrable_on s"
```
```  2985   shows "(\<lambda>x. f x / of_real c) integrable_on s"
```
```  2986 by (simp add: integrable_on_cmult_right divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```  2987
```
```  2988 lemma integrable_on_cdivide_iff [simp]:
```
```  2989   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```  2990   assumes "c \<noteq> 0"
```
```  2991   shows "(\<lambda>x. f x / of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```  2992 by (simp add: divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```  2993
```
```  2994 lemma has_integral_null [intro]:
```
```  2995   assumes "content(cbox a b) = 0"
```
```  2996   shows "(f has_integral 0) (cbox a b)"
```
```  2997 proof -
```
```  2998   have "gauge (\<lambda>x. ball x 1)"
```
```  2999     by auto
```
```  3000   moreover
```
```  3001   {
```
```  3002     fix e :: real
```
```  3003     fix p
```
```  3004     assume e: "e > 0"
```
```  3005     assume p: "p tagged_division_of (cbox a b)"
```
```  3006     have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0"
```
```  3007       unfolding norm_eq_zero diff_0_right
```
```  3008       using setsum_content_null[OF assms(1) p, of f] .
```
```  3009     then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  3010       using e by auto
```
```  3011   }
```
```  3012   ultimately show ?thesis
```
```  3013     by (auto simp: has_integral)
```
```  3014 qed
```
```  3015
```
```  3016 lemma has_integral_null_real [intro]:
```
```  3017   assumes "content {a .. b::real} = 0"
```
```  3018   shows "(f has_integral 0) {a .. b}"
```
```  3019   by (metis assms box_real(2) has_integral_null)
```
```  3020
```
```  3021 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
```
```  3022   by (auto simp add: has_integral_null dest!: integral_unique)
```
```  3023
```
```  3024 lemma integral_null [simp]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
```
```  3025   by (metis has_integral_null integral_unique)
```
```  3026
```
```  3027 lemma integrable_on_null [intro]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
```
```  3028   by (simp add: has_integral_integrable)
```
```  3029
```
```  3030 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
```
```  3031   by (simp add: has_integral_is_0)
```
```  3032
```
```  3033 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
```
```  3034   by (auto simp add: has_integral_empty has_integral_unique)
```
```  3035
```
```  3036 lemma integrable_on_empty[intro]: "f integrable_on {}"
```
```  3037   unfolding integrable_on_def by auto
```
```  3038
```
```  3039 lemma integral_empty[simp]: "integral {} f = 0"
```
```  3040   by (rule integral_unique) (rule has_integral_empty)
```
```  3041
```
```  3042 lemma has_integral_refl[intro]:
```
```  3043   fixes a :: "'a::euclidean_space"
```
```  3044   shows "(f has_integral 0) (cbox a a)"
```
```  3045     and "(f has_integral 0) {a}"
```
```  3046 proof -
```
```  3047   have *: "{a} = cbox a a"
```
```  3048     apply (rule set_eqI)
```
```  3049     unfolding mem_box singleton_iff euclidean_eq_iff[where 'a='a]
```
```  3050     apply safe
```
```  3051     prefer 3
```
```  3052     apply (erule_tac x=b in ballE)
```
```  3053     apply (auto simp add: field_simps)
```
```  3054     done
```
```  3055   show "(f has_integral 0) (cbox a a)" "(f has_integral 0) {a}"
```
```  3056     unfolding *
```
```  3057     apply (rule_tac[!] has_integral_null)
```
```  3058     unfolding content_eq_0_interior
```
```  3059     unfolding interior_cbox
```
```  3060     using box_sing
```
```  3061     apply auto
```
```  3062     done
```
```  3063 qed
```
```  3064
```
```  3065 lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
```
```  3066   unfolding integrable_on_def by auto
```
```  3067
```
```  3068 lemma integral_refl [simp]: "integral (cbox a a) f = 0"
```
```  3069   by (rule integral_unique) auto
```
```  3070
```
```  3071 lemma integral_singleton [simp]: "integral {a} f = 0"
```
```  3072   by auto
```
```  3073
```
```  3074 lemma integral_blinfun_apply:
```
```  3075   assumes "f integrable_on s"
```
```  3076   shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
```
```  3077   by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
```
```  3078
```
```  3079 lemma blinfun_apply_integral:
```
```  3080   assumes "f integrable_on s"
```
```  3081   shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
```
```  3082   by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
```
```  3083
```
```  3084 lemma has_integral_componentwise_iff:
```
```  3085   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3086   shows "(f has_integral y) A \<longleftrightarrow> (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3087 proof safe
```
```  3088   fix b :: 'b assume "(f has_integral y) A"
```
```  3089   from has_integral_linear[OF this(1) bounded_linear_inner_left, of b]
```
```  3090     show "((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A" by (simp add: o_def)
```
```  3091 next
```
```  3092   assume "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3093   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"
```
```  3094     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```  3095   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"
```
```  3096     by (intro has_integral_setsum) (simp_all add: o_def)
```
```  3097   thus "(f has_integral y) A" by (simp add: euclidean_representation)
```
```  3098 qed
```
```  3099
```
```  3100 lemma has_integral_componentwise:
```
```  3101   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3102   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"
```
```  3103   by (subst has_integral_componentwise_iff) blast
```
```  3104
```
```  3105 lemma integrable_componentwise_iff:
```
```  3106   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3107   shows "f integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```  3108 proof
```
```  3109   assume "f integrable_on A"
```
```  3110   then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def)
```
```  3111   hence "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```  3112     by (subst (asm) has_integral_componentwise_iff)
```
```  3113   thus "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)" by (auto simp: integrable_on_def)
```
```  3114 next
```
```  3115   assume "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```  3116   then obtain y where "\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y b) A"
```
```  3117     unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
```
```  3118   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"
```
```  3119     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```  3120   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"
```
```  3121     by (intro has_integral_setsum) (simp_all add: o_def)
```
```  3122   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
```
```  3123 qed
```
```  3124
```
```  3125 lemma integrable_componentwise:
```
```  3126   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3127   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A) \<Longrightarrow> f integrable_on A"
```
```  3128   by (subst integrable_componentwise_iff) blast
```
```  3129
```
```  3130 lemma integral_componentwise:
```
```  3131   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```  3132   assumes "f integrable_on A"
```
```  3133   shows "integral A f = (\<Sum>b\<in>Basis. integral A (\<lambda>x. (f x \<bullet> b) *\<^sub>R b))"
```
```  3134 proof -
```
```  3135   from assms have integrable: "\<forall>b\<in>Basis. (\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. (f x \<bullet> b)) integrable_on A"
```
```  3136     by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI)
```
```  3137        (simp_all add: bounded_linear_scaleR_left)
```
```  3138   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
```
```  3139     by (simp add: euclidean_representation)
```
```  3140   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
```
```  3141     by (subst integral_setsum) (simp_all add: o_def)
```
```  3142   finally show ?thesis .
```
```  3143 qed
```
```  3144
```
```  3145 lemma integrable_component:
```
```  3146   "f integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (y :: 'b :: euclidean_space)) integrable_on A"
```
```  3147   by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def)
```
```  3148
```
```  3149
```
```  3150
```
```  3151 subsection \<open>Cauchy-type criterion for integrability.\<close>
```
```  3152
```
```  3153 (* XXXXXXX *)
```
```  3154 lemma integrable_cauchy:
```
```  3155   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
```
```  3156   shows "f integrable_on cbox a b \<longleftrightarrow>
```
```  3157     (\<forall>e>0.\<exists>d. gauge d \<and>
```
```  3158       (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<and> d fine p1 \<and>
```
```  3159         p2 tagged_division_of (cbox a b) \<and> d fine p2 \<longrightarrow>
```
```  3160         norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -
```
```  3161         setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))"
```
```  3162   (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
```
```  3163 proof
```
```  3164   assume ?l
```
```  3165   then guess y unfolding integrable_on_def has_integral .. note y=this
```
```  3166   show "\<forall>e>0. \<exists>d. ?P e d"
```
```  3167   proof (clarify, goal_cases)
```
```  3168     case (1 e)
```
```  3169     then have "e/2 > 0" by auto
```
```  3170     then guess d
```
```  3171       apply -
```
```  3172       apply (drule y[rule_format])
```
```  3173       apply (elim exE conjE)
```
```  3174       done
```
```  3175     note d=this[rule_format]
```
```  3176     show ?case
```
```  3177     proof (rule_tac x=d in exI, clarsimp simp: d)
```
```  3178       fix p1 p2
```
```  3179       assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
```
```  3180                  "p2 tagged_division_of (cbox a b)" "d fine p2"
```
```  3181       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"
```
```  3182         apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
```
```  3183         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
```
```  3184     qed
```
```  3185   qed
```
```  3186 next
```
```  3187   assume "\<forall>e>0. \<exists>d. ?P e d"
```
```  3188   then have "\<forall>n::nat. \<exists>d. ?P (inverse(of_nat (n + 1))) d"
```
```  3189     by auto
```
```  3190   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```  3191   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})"
```
```  3192     apply (rule gauge_inters)
```
```  3193     using d(1)
```
```  3194     apply auto
```
```  3195     done
```
```  3196   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"
```
```  3197     by (meson fine_division_exists)
```
```  3198   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
```
```  3199   have dp: "\<And>i n. i\<le>n \<Longrightarrow> d i fine p n"
```
```  3200     using p(2) unfolding fine_inters by auto
```
```  3201   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
```
```  3202   proof (rule CauchyI, goal_cases)
```
```  3203     case (1 e)
```
```  3204     then guess N unfolding real_arch_inverse[of e] .. note N=this
```
```  3205     show ?case
```
```  3206       apply (rule_tac x=N in exI)
```
```  3207     proof clarify
```
```  3208       fix m n
```
```  3209       assume mn: "N \<le> m" "N \<le> n"
```
```  3210       have *: "N = (N - 1) + 1" using N by auto
```
```  3211       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"
```
```  3212         apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
```
```  3213         apply(subst *)
```
```  3214         using dp p(1) mn d(2) by auto
```
```  3215     qed
```
```  3216   qed
```
```  3217   then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
```
```  3218   show ?l
```
```  3219     unfolding integrable_on_def has_integral
```
```  3220   proof (rule_tac x=y in exI, clarify)
```
```  3221     fix e :: real
```
```  3222     assume "e>0"
```
```  3223     then have *:"e/2 > 0" by auto
```
```  3224     then guess N1 unfolding real_arch_inverse[of "e/2"] .. note N1=this
```
```  3225     then have N1': "N1 = N1 - 1 + 1"
```
```  3226       by auto
```
```  3227     guess N2 using y[OF *] .. note N2=this
```
```  3228     have "gauge (d (N1 + N2))"
```
```  3229       using d by auto
```
```  3230     moreover
```
```  3231     {
```
```  3232       fix q
```
```  3233       assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
```
```  3234       have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
```
```  3235         apply (rule less_trans)
```
```  3236         using N1
```
```  3237         apply auto
```
```  3238         done
```
```  3239       have "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e"
```
```  3240         apply (rule norm_triangle_half_r)
```
```  3241         apply (rule less_trans[OF _ *])
```
```  3242         apply (subst N1', rule d(2)[of "p (N1+N2)"])
```
```  3243         using N1' as(1) as(2) dp
```
```  3244         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>)
```
```  3245         using N2 le_add2 by blast
```
```  3246     }
```
```  3247     ultimately show "\<exists>d. gauge d \<and>
```
```  3248       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  3249         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
```
```  3250       by (rule_tac x="d (N1 + N2)" in exI) auto
```
```  3251   qed
```
```  3252 qed
```
```  3253
```
```  3254
```
```  3255 subsection \<open>Additivity of integral on abutting intervals.\<close>
```
```  3256
```
```  3257 lemma tagged_division_split_left_inj:
```
```  3258   fixes x1 :: "'a::euclidean_space"
```
```  3259   assumes d: "d tagged_division_of i"
```
```  3260     and k12: "(x1, k1) \<in> d"
```
```  3261              "(x2, k2) \<in> d"
```
```  3262              "k1 \<noteq> k2"
```
```  3263              "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
```
```  3264              "k \<in> Basis"
```
```  3265   shows "content (k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
```
```  3266 proof -
```
```  3267   have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
```
```  3268     by force
```
```  3269   show ?thesis
```
```  3270     using k12
```
```  3271     by (fastforce intro!:  division_split_left_inj[OF division_of_tagged_division[OF d]] *)
```
```  3272 qed
```
```  3273
```
```  3274 lemma tagged_division_split_right_inj:
```
```  3275   fixes x1 :: "'a::euclidean_space"
```
```  3276   assumes d: "d tagged_division_of i"
```
```  3277     and k12: "(x1, k1) \<in> d"
```
```  3278              "(x2, k2) \<in> d"
```
```  3279              "k1 \<noteq> k2"
```
```  3280              "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
```
```  3281              "k \<in> Basis"
```
```  3282   shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
```
```  3283 proof -
```
```  3284   have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
```
```  3285     by force
```
```  3286   show ?thesis
```
```  3287     using k12
```
```  3288     by (fastforce intro!:  division_split_right_inj[OF division_of_tagged_division[OF d]] *)
```
```  3289 qed
```
```  3290
```
```  3291 lemma has_integral_split:
```
```  3292   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3293   assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
```
```  3294       and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3295       and k: "k \<in> Basis"
```
```  3296   shows "(f has_integral (i + j)) (cbox a b)"
```
```  3297 proof (unfold has_integral, rule, rule, goal_cases)
```
```  3298   case (1 e)
```
```  3299   then have e: "e/2 > 0"
```
```  3300     by auto
```
```  3301     obtain d1
```
```  3302     where d1: "gauge d1"
```
```  3303       and d1norm:
```
```  3304         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c};
```
```  3305                d1 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - i) < e / 2"
```
```  3306        apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
```
```  3307        apply (simp add: interval_split[symmetric] k)
```
```  3308        done
```
```  3309     obtain d2
```
```  3310     where d2: "gauge d2"
```
```  3311       and d2norm:
```
```  3312         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k};
```
```  3313                d2 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - j) < e / 2"
```
```  3314        apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
```
```  3315        apply (simp add: interval_split[symmetric] k)
```
```  3316        done
```
```  3317   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"
```
```  3318   have "gauge ?d"
```
```  3319     using d1 d2 unfolding gauge_def by auto
```
```  3320   then show ?case
```
```  3321   proof (rule_tac x="?d" in exI, safe)
```
```  3322     fix p
```
```  3323     assume "p tagged_division_of (cbox a b)" "?d fine p"
```
```  3324     note p = this tagged_division_ofD[OF this(1)]
```
```  3325     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"
```
```  3326     proof -
```
```  3327       fix x kk
```
```  3328       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}"
```
```  3329       show "x\<bullet>k \<le> c"
```
```  3330       proof (rule ccontr)
```
```  3331         assume **: "\<not> ?thesis"
```
```  3332         from this[unfolded not_le]
```
```  3333         have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  3334           using p(2)[unfolded fine_def, rule_format,OF as] by auto
```
```  3335         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
```
```  3336           by blast
```
```  3337         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  3338           using Basis_le_norm[OF k, of "x - y"]
```
```  3339           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  3340         with y show False
```
```  3341           using ** by (auto simp add: field_simps)
```
```  3342       qed
```
```  3343     qed
```
```  3344     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"
```
```  3345     proof -
```
```  3346       fix x kk
```
```  3347       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}"
```
```  3348       show "x\<bullet>k \<ge> c"
```
```  3349       proof (rule ccontr)
```
```  3350         assume **: "\<not> ?thesis"
```
```  3351         from this[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  3352           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  3353         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
```
```  3354           by blast
```
```  3355         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  3356           using Basis_le_norm[OF k, of "x - y"]
```
```  3357           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  3358         with y show False
```
```  3359           using ** by (auto simp add: field_simps)
```
```  3360       qed
```
```  3361     qed
```
```  3362
```
```  3363     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>
```
```  3364                          (\<forall>x k. P x k \<longrightarrow> Q x (f k))"
```
```  3365       by auto
```
```  3366     have fin_finite: "finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}" if "finite s" for f s P
```
```  3367     proof -
```
```  3368       from that have "finite ((\<lambda>(x, k). (x, f k)) ` s)"
```
```  3369         by auto
```
```  3370       then show ?thesis
```
```  3371         by (rule rev_finite_subset) auto
```
```  3372     qed
```
```  3373     { fix g :: "'a set \<Rightarrow> 'a set"
```
```  3374       fix i :: "'a \<times> 'a set"
```
```  3375       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  3376       then obtain x k where xk:
```
```  3377               "i = (x, g k)"  "(x, k) \<in> p"
```
```  3378               "(x, g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  3379           by auto
```
```  3380       have "content (g k) = 0"
```
```  3381         using xk using content_empty by auto
```
```  3382       then have "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0"
```
```  3383         unfolding xk split_conv by auto
```
```  3384     } note [simp] = this
```
```  3385     have lem3: "\<And>g :: 'a set \<Rightarrow> 'a set. finite p \<Longrightarrow>
```
```  3386                   setsum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
```
```  3387                   setsum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
```
```  3388       by (rule setsum.mono_neutral_left) auto
```
```  3389     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> {}}"
```
```  3390     have d1_fine: "d1 fine ?M1"
```
```  3391       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  3392     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
```
```  3393     proof (rule d1norm [OF tagged_division_ofI d1_fine])
```
```  3394       show "finite ?M1"
```
```  3395         by (rule fin_finite p(3))+
```
```  3396       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
```
```  3397         unfolding p(8)[symmetric] by auto
```
```  3398       fix x l
```
```  3399       assume xl: "(x, l) \<in> ?M1"
```
```  3400       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  3401       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  3402         unfolding xl'
```
```  3403         using p(4-6)[OF xl'(3)] using xl'(4)
```
```  3404         using xk_le_c[OF xl'(3-4)] by auto
```
```  3405       show "\<exists>a b. l = cbox a b"
```
```  3406         unfolding xl'
```
```  3407         using p(6)[OF xl'(3)]
```
```  3408         by (fastforce simp add: interval_split[OF k,where c=c])
```
```  3409       fix y r
```
```  3410       let ?goal = "interior l \<inter> interior r = {}"
```
```  3411       assume yr: "(y, r) \<in> ?M1"
```
```  3412       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  3413       assume as: "(x, l) \<noteq> (y, r)"
```
```  3414       show "interior l \<inter> interior r = {}"
```
```  3415       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  3416         case False
```
```  3417         then show ?thesis
```
```  3418           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3419       next
```
```  3420         case True
```
```  3421         then have "l' \<noteq> r'"
```
```  3422           using as unfolding xl' yr' by auto
```
```  3423         then show ?thesis
```
```  3424           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3425       qed
```
```  3426     qed
```
```  3427     moreover
```
```  3428     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> {}}"
```
```  3429     have d2_fine: "d2 fine ?M2"
```
```  3430       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  3431     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
```
```  3432     proof (rule d2norm [OF tagged_division_ofI d2_fine])
```
```  3433       show "finite ?M2"
```
```  3434         by (rule fin_finite p(3))+
```
```  3435       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
```
```  3436         unfolding p(8)[symmetric] by auto
```
```  3437       fix x l
```
```  3438       assume xl: "(x, l) \<in> ?M2"
```
```  3439       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  3440       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  3441         unfolding xl'
```
```  3442         using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
```
```  3443         by auto
```
```  3444       show "\<exists>a b. l = cbox a b"
```
```  3445         unfolding xl'
```
```  3446         using p(6)[OF xl'(3)]
```
```  3447         by (fastforce simp add: interval_split[OF k, where c=c])
```
```  3448       fix y r
```
```  3449       let ?goal = "interior l \<inter> interior r = {}"
```
```  3450       assume yr: "(y, r) \<in> ?M2"
```
```  3451       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  3452       assume as: "(x, l) \<noteq> (y, r)"
```
```  3453       show "interior l \<inter> interior r = {}"
```
```  3454       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  3455         case False
```
```  3456         then show ?thesis
```
```  3457           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3458       next
```
```  3459         case True
```
```  3460         then have "l' \<noteq> r'"
```
```  3461           using as unfolding xl' yr' by auto
```
```  3462         then show ?thesis
```
```  3463           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  3464       qed
```
```  3465     qed
```
```  3466     ultimately
```
```  3467     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"
```
```  3468       using norm_add_less by blast
```
```  3469     also {
```
```  3470       have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
```
```  3471         using scaleR_zero_left by auto
```
```  3472       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)"
```
```  3473         by auto
```
```  3474       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) =
```
```  3475         (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
```
```  3476         by auto
```
```  3477       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
```
```  3478         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
```
```  3479         unfolding lem3[OF p(3)]
```
```  3480         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)]
```
```  3481               simp: cont_eq)+
```
```  3482       also note setsum.distrib[symmetric]
```
```  3483       also have "\<And>x. x \<in> p \<Longrightarrow>
```
```  3484                     (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
```
```  3485                     (\<lambda>(x,ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x =
```
```  3486                     (\<lambda>(x,ka). content ka *\<^sub>R f x) x"
```
```  3487       proof clarify
```
```  3488         fix a b
```
```  3489         assume "(a, b) \<in> p"
```
```  3490         from p(6)[OF this] guess u v by (elim exE) note uv=this
```
```  3491         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 =
```
```  3492           content b *\<^sub>R f a"
```
```  3493           unfolding scaleR_left_distrib[symmetric]
```
```  3494           unfolding uv content_split[OF k,of u v c]
```
```  3495           by auto
```
```  3496       qed
```
```  3497       note setsum.cong [OF _ this]
```
```  3498       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 +
```
```  3499         ((\<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) =
```
```  3500         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
```
```  3501         by auto
```
```  3502     }
```
```  3503     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
```
```  3504       by auto
```
```  3505   qed
```
```  3506 qed
```
```  3507
```
```  3508
```
```  3509 subsection \<open>A sort of converse, integrability on subintervals.\<close>
```
```  3510
```
```  3511 lemma tagged_division_union_interval:
```
```  3512   fixes a :: "'a::euclidean_space"
```
```  3513   assumes "p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> (c::real)})"
```
```  3514     and "p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3515     and k: "k \<in> Basis"
```
```  3516   shows "(p1 \<union> p2) tagged_division_of (cbox a b)"
```
```  3517 proof -
```
```  3518   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})"
```
```  3519     by auto
```
```  3520   show ?thesis
```
```  3521     apply (subst *)
```
```  3522     apply (rule tagged_division_union[OF assms(1-2)])
```
```  3523     unfolding interval_split[OF k] interior_cbox
```
```  3524     using k
```
```  3525     apply (auto simp add: box_def elim!: ballE[where x=k])
```
```  3526     done
```
```  3527 qed
```
```  3528
```
```  3529 lemma tagged_division_union_interval_real:
```
```  3530   fixes a :: real
```
```  3531   assumes "p1 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<le> (c::real)})"
```
```  3532     and "p2 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<ge> c})"
```
```  3533     and k: "k \<in> Basis"
```
```  3534   shows "(p1 \<union> p2) tagged_division_of {a .. b}"
```
```  3535   using assms
```
```  3536   unfolding box_real[symmetric]
```
```  3537   by (rule tagged_division_union_interval)
```
```  3538
```
```  3539 lemma has_integral_separate_sides:
```
```  3540   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3541   assumes "(f has_integral i) (cbox a b)"
```
```  3542     and "e > 0"
```
```  3543     and k: "k \<in> Basis"
```
```  3544   obtains d where "gauge d"
```
```  3545     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
```
```  3546         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
```
```  3547         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"
```
```  3548 proof -
```
```  3549   guess d using has_integralD[OF assms(1-2)] . note d=this
```
```  3550   { fix p1 p2
```
```  3551     assume "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  3552     note p1=tagged_division_ofD[OF this(1)] this
```
```  3553     assume "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" "d fine p2"
```
```  3554     note p2=tagged_division_ofD[OF this(1)] this
```
```  3555     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
```
```  3556     { fix a b
```
```  3557       assume ab: "(a, b) \<in> p1 \<inter> p2"
```
```  3558       have "(a, b) \<in> p1"
```
```  3559         using ab by auto
```
```  3560       with p1 obtain u v where uv: "b = cbox u v" by auto
```
```  3561       have "b \<subseteq> {x. x\<bullet>k = c}"
```
```  3562         using ab p1(3)[of a b] p2(3)[of a b] by fastforce
```
```  3563       moreover
```
```  3564       have "interior {x::'a. x \<bullet> k = c} = {}"
```
```  3565       proof (rule ccontr)
```
```  3566         assume "\<not> ?thesis"
```
```  3567         then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
```
```  3568           by auto
```
```  3569         then guess e unfolding mem_interior .. note e=this
```
```  3570         have x: "x\<bullet>k = c"
```
```  3571           using x interior_subset by fastforce
```
```  3572         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)"
```
```  3573           using e k by (auto simp: inner_simps inner_not_same_Basis)
```
```  3574         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
```
```  3575               (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))"
```
```  3576           using "*" by (blast intro: setsum.cong)
```
```  3577         also have "\<dots> < e"
```
```  3578           apply (subst setsum.delta)
```
```  3579           using e
```
```  3580           apply auto
```
```  3581           done
```
```  3582         finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
```
```  3583           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
```
```  3584         then have "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
```
```  3585           using e by auto
```
```  3586         then show False
```
```  3587           unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
```
```  3588       qed
```
```  3589       ultimately have "content b = 0"
```
```  3590         unfolding uv content_eq_0_interior
```
```  3591         using interior_mono by blast
```
```  3592       then have "content b *\<^sub>R f a = 0"
```
```  3593         by auto
```
```  3594     }
```
```  3595     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) =
```
```  3596                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
```
```  3597       by (subst setsum.union_inter_neutral) (auto simp: p1 p2)
```
```  3598     also have "\<dots> < e"
```
```  3599       by (rule k d(2) p12 fine_union p1 p2)+
```
```  3600     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" .
```
```  3601    }
```
```  3602   then show ?thesis
```
```  3603     by (auto intro: that[of d] d elim: )
```
```  3604 qed
```
```  3605
```
```  3606 lemma integrable_split[intro]:
```
```  3607   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
```
```  3608   assumes "f integrable_on cbox a b"
```
```  3609     and k: "k \<in> Basis"
```
```  3610   shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})" (is ?t1)
```
```  3611     and "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
```
```  3612 proof -
```
```  3613   guess y using assms(1) unfolding integrable_on_def .. note y=this
```
```  3614   define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
```
```  3615   define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
```
```  3616   show ?t1 ?t2
```
```  3617     unfolding interval_split[OF k] integrable_cauchy
```
```  3618     unfolding interval_split[symmetric,OF k]
```
```  3619   proof (rule_tac[!] allI impI)+
```
```  3620     fix e :: real
```
```  3621     assume "e > 0"
```
```  3622     then have "e/2>0"
```
```  3623       by auto
```
```  3624     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
```
```  3625     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>
```
```  3626       p2 tagged_division_of (cbox a b) \<inter> A \<and> d fine p2 \<longrightarrow>
```
```  3627       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
```
```  3628     show "?P {x. x \<bullet> k \<le> c}"
```
```  3629     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  3630       fix p1 p2
```
```  3631       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  3632                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
```
```  3633       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"
```
```  3634       proof (rule fine_division_exists[OF d(1), of a' b] )
```
```  3635         fix p
```
```  3636         assume "p tagged_division_of cbox a' b" "d fine p"
```
```  3637         then show ?thesis
```
```  3638           using as norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
```
```  3639           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  3640           by (auto simp add: algebra_simps)
```
```  3641       qed
```
```  3642     qed
```
```  3643     show "?P {x. x \<bullet> k \<ge> c}"
```
```  3644     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  3645       fix p1 p2
```
```  3646       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
```
```  3647                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
```
```  3648       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"
```
```  3649       proof (rule fine_division_exists[OF d(1), of a b'] )
```
```  3650         fix p
```
```  3651         assume "p tagged_division_of cbox a b'" "d fine p"
```
```  3652         then show ?thesis
```
```  3653           using as norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
```
```  3654           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  3655           by (auto simp add: algebra_simps)
```
```  3656       qed
```
```  3657     qed
```
```  3658   qed
```
```  3659 qed
```
```  3660
```
```  3661 lemma operative_integral:
```
```  3662   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  3663   shows "comm_monoid.operative (lift_option op +) (Some 0)
```
```  3664     (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
```
```  3665 proof -
```
```  3666   interpret comm_monoid "lift_option plus" "Some (0::'b)"
```
```  3667     by (rule comm_monoid_lift_option)
```
```  3668       (rule add.comm_monoid_axioms)
```
```  3669   show ?thesis
```
```  3670   proof (unfold operative_def, safe)
```
```  3671     fix a b c
```
```  3672     fix k :: 'a
```
```  3673     assume k: "k \<in> Basis"
```
```  3674     show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
```
```  3675           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)
```
```  3676           (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)"
```
```  3677     proof (cases "f integrable_on cbox a b")
```
```  3678       case True
```
```  3679       with k show ?thesis
```
```  3680         apply (simp add: integrable_split)
```
```  3681         apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
```
```  3682         apply (auto intro: integrable_integral)
```
```  3683         done
```
```  3684     next
```
```  3685     case False
```
```  3686       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})"
```
```  3687       proof (rule ccontr)
```
```  3688         assume "\<not> ?thesis"
```
```  3689         then have "f integrable_on cbox a b"
```
```  3690           unfolding integrable_on_def
```
```  3691           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)
```
```  3692           apply (rule has_integral_split[OF _ _ k])
```
```  3693           apply (auto intro: integrable_integral)
```
```  3694           done
```
```  3695         then show False
```
```  3696           using False by auto
```
```  3697       qed
```
```  3698       then show ?thesis
```
```  3699         using False by auto
```
```  3700     qed
```
```  3701   next
```
```  3702     fix a b :: 'a
```
```  3703     assume "content (cbox a b) = 0"
```
```  3704     then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
```
```  3705       using has_integral_null_eq
```
```  3706       by (auto simp: integrable_on_null)
```
```  3707   qed
```
```  3708 qed
```
```  3709
```
```  3710 subsection \<open>Finally, the integral of a constant\<close>
```
```  3711
```
```  3712 lemma has_integral_const [intro]:
```
```  3713   fixes a b :: "'a::euclidean_space"
```
```  3714   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
```
```  3715   apply (auto intro!: exI [where x="\<lambda>x. ball x 1"] simp: split_def has_integral)
```
```  3716   apply (subst scaleR_left.setsum[symmetric, unfolded o_def])
```
```  3717   apply (subst additive_content_tagged_division[unfolded split_def])
```
```  3718   apply auto
```
```  3719   done
```
```  3720
```
```  3721 lemma has_integral_const_real [intro]:
```
```  3722   fixes a b :: real
```
```  3723   shows "((\<lambda>x. c) has_integral (content {a .. b} *\<^sub>R c)) {a .. b}"
```
```  3724   by (metis box_real(2) has_integral_const)
```
```  3725
```
```  3726 lemma integral_const [simp]:
```
```  3727   fixes a b :: "'a::euclidean_space"
```
```  3728   shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
```
```  3729   by (rule integral_unique) (rule has_integral_const)
```
```  3730
```
```  3731 lemma integral_const_real [simp]:
```
```  3732   fixes a b :: real
```
```  3733   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
```
```  3734   by (metis box_real(2) integral_const)
```
```  3735
```
```  3736
```
```  3737 subsection \<open>Bounds on the norm of Riemann sums and the integral itself.\<close>
```
```  3738
```
```  3739 lemma dsum_bound:
```
```  3740   assumes "p division_of (cbox a b)"
```
```  3741     and "norm c \<le> e"
```
```  3742   shows "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
```
```  3743 proof -
```
```  3744   have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = setsum content p"
```
```  3745     apply (rule setsum.cong)
```
```  3746     using assms
```
```  3747     apply simp
```
```  3748     apply (metis abs_of_nonneg assms(1) content_pos_le division_ofD(4))
```
```  3749     done
```
```  3750   have e: "0 \<le> e"
```
```  3751     using assms(2) norm_ge_zero order_trans by blast
```
```  3752   have "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
```
```  3753     using norm_setsum by blast
```
```  3754   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
```
```  3755     by (simp add: setsum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono setsum_nonneg)
```
```  3756   also have "... \<le> e * content (cbox a b)"
```
```  3757     apply (rule mult_left_mono [OF _ e])
```
```  3758     apply (simp add: sumeq)
```
```  3759     using additive_content_division assms(1) eq_iff apply blast
```
```  3760     done
```
```  3761   finally show ?thesis .
```
```  3762 qed
```
```  3763
```
```  3764 lemma rsum_bound:
```
```  3765   assumes p: "p tagged_division_of (cbox a b)"
```
```  3766       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
```
```  3767     shows "norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
```
```  3768 proof (cases "cbox a b = {}")
```
```  3769   case True show ?thesis
```
```  3770     using p unfolding True tagged_division_of_trivial by auto
```
```  3771 next
```
```  3772   case False
```
```  3773   then have e: "e \<ge> 0"
```
```  3774     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
```
```  3775   have setsum_le: "setsum (content \<circ> snd) p \<le> content (cbox a b)"
```
```  3776     unfolding additive_content_tagged_division[OF p, symmetric] split_def
```
```  3777     by (auto intro: eq_refl)
```
```  3778   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
```
```  3779     using tagged_division_ofD(4) [OF p] content_pos_le
```
```  3780     by force
```
```  3781   have norm: "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
```
```  3782     unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
```
```  3783     by (metis prod.collapse subset_eq)
```
```  3784   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))"
```
```  3785     by (rule norm_setsum)
```
```  3786   also have "...  \<le> e * content (cbox a b)"
```
```  3787     unfolding split_def norm_scaleR
```
```  3788     apply (rule order_trans[OF setsum_mono])
```
```  3789     apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
```
```  3790     apply (metis norm)
```
```  3791     unfolding setsum_distrib_right[symmetric]
```
```  3792     using con setsum_le
```
```  3793     apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
```
```  3794     done
```
```  3795   finally show ?thesis .
```
```  3796 qed
```
```  3797
```
```  3798 lemma rsum_diff_bound:
```
```  3799   assumes "p tagged_division_of (cbox a b)"
```
```  3800     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
```
```  3801   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>
```
```  3802          e * content (cbox a b)"
```
```  3803   apply (rule order_trans[OF _ rsum_bound[OF assms]])
```
```  3804   apply (simp add: split_def scaleR_diff_right setsum_subtractf eq_refl)
```
```  3805   done
```
```  3806
```
```  3807 lemma has_integral_bound:
```
```  3808   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3809   assumes "0 \<le> B"
```
```  3810       and "(f has_integral i) (cbox a b)"
```
```  3811       and "\<forall>x\<in>cbox a b. norm (f x) \<le> B"
```
```  3812     shows "norm i \<le> B * content (cbox a b)"
```
```  3813 proof (rule ccontr)
```
```  3814   assume "\<not> ?thesis"
```
```  3815   then have *: "norm i - B * content (cbox a b) > 0"
```
```  3816     by auto
```
```  3817   from assms(2)[unfolded has_integral,rule_format,OF *]
```
```  3818   guess d by (elim exE conjE) note d=this[rule_format]
```
```  3819   from fine_division_exists[OF this(1), of a b] guess p . note p=this
```
```  3820   have *: "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
```
```  3821     unfolding not_less
```
```  3822     by (metis norm_triangle_sub[of i] add.commute le_less_trans less_diff_eq linorder_not_le norm_minus_commute)
```
```  3823   show False
```
```  3824     using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
```
```  3825 qed
```
```  3826
```
```  3827 corollary has_integral_bound_real:
```
```  3828   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
```
```  3829   assumes "0 \<le> B"
```
```  3830       and "(f has_integral i) {a .. b}"
```
```  3831       and "\<forall>x\<in>{a .. b}. norm (f x) \<le> B"
```
```  3832     shows "norm i \<le> B * content {a .. b}"
```
```  3833   by (metis assms box_real(2) has_integral_bound)
```
```  3834
```
```  3835 corollary integrable_bound:
```
```  3836   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3837   assumes "0 \<le> B"
```
```  3838       and "f integrable_on (cbox a b)"
```
```  3839       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
```
```  3840     shows "norm (integral (cbox a b) f) \<le> B * content (cbox a b)"
```
```  3841 by (metis integrable_integral has_integral_bound assms)
```
```  3842
```
```  3843
```
```  3844 subsection \<open>Similar theorems about relationship among components.\<close>
```
```  3845
```
```  3846 lemma rsum_component_le:
```
```  3847   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3848   assumes "p tagged_division_of (cbox a b)"
```
```  3849       and "\<forall>x\<in>cbox a b. (f x)\<bullet>i \<le> (g x)\<bullet>i"
```
```  3850     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"
```
```  3851 unfolding inner_setsum_left
```
```  3852 proof (rule setsum_mono, clarify)
```
```  3853   fix a b
```
```  3854   assume ab: "(a, b) \<in> p"
```
```  3855   note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
```
```  3856   from this(3) guess u v by (elim exE) note b=this
```
```  3857   show "(content b *\<^sub>R f a) \<bullet> i \<le> (content b *\<^sub>R g a) \<bullet> i"
```
```  3858     unfolding b inner_simps real_scaleR_def
```
```  3859     apply (rule mult_left_mono)
```
```  3860     using assms(2) tagged
```
```  3861     by (auto simp add: content_pos_le)
```
```  3862 qed
```
```  3863
```
```  3864 lemma has_integral_component_le:
```
```  3865   fixes f g :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3866   assumes k: "k \<in> Basis"
```
```  3867   assumes "(f has_integral i) s" "(g has_integral j) s"
```
```  3868     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  3869   shows "i\<bullet>k \<le> j\<bullet>k"
```
```  3870 proof -
```
```  3871   have lem: "i\<bullet>k \<le> j\<bullet>k"
```
```  3872     if f_i: "(f has_integral i) (cbox a b)"
```
```  3873     and g_j: "(g has_integral j) (cbox a b)"
```
```  3874     and le: "\<forall>x\<in>cbox a b. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  3875     for a b i and j :: 'b and f g :: "'a \<Rightarrow> 'b"
```
```  3876   proof (rule ccontr)
```
```  3877     assume "\<not> ?thesis"
```
```  3878     then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3"
```
```  3879       by auto
```
```  3880     guess d1 using f_i[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
```
```  3881     guess d2 using g_j[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
```
```  3882     obtain p where p: "p tagged_division_of cbox a b" "d1 fine p" "d2 fine p"
```
```  3883        using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter
```
```  3884        by metis
```
```  3885     note le_less_trans[OF Basis_le_norm[OF k]]
```
```  3886     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"
```
```  3887               "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
```
```  3888       using  k norm_bound_Basis_lt d1 d2 p
```
```  3889       by blast+
```
```  3890     then show False
```
```  3891       unfolding inner_simps
```
```  3892       using rsum_component_le[OF p(1) le]
```
```  3893       by (simp add: abs_real_def split: if_split_asm)
```
```  3894   qed
```
```  3895   show ?thesis
```
```  3896   proof (cases "\<exists>a b. s = cbox a b")
```
```  3897     case True
```
```  3898     with lem assms show ?thesis
```
```  3899       by auto
```
```  3900   next
```
```  3901     case False
```
```  3902     show ?thesis
```
```  3903     proof (rule ccontr)
```
```  3904       assume "\<not> i\<bullet>k \<le> j\<bullet>k"
```
```  3905       then have ij: "(i\<bullet>k - j\<bullet>k) / 3 > 0"
```
```  3906         by auto
```
```  3907       note has_integral_altD[OF _ False this]
```
```  3908       from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
```
```  3909       have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
```
```  3910         unfolding bounded_Un by(rule conjI bounded_ball)+
```
```  3911       from bounded_subset_cbox[OF this] guess a b by (elim exE)
```
```  3912       note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  3913       guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
```
```  3914       guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
```
```  3915       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"
```
```  3916         by (simp add: abs_real_def split: if_split_asm)
```
```  3917       note le_less_trans[OF Basis_le_norm[OF k]]
```
```  3918       note this[OF w1(2)] this[OF w2(2)]
```
```  3919       moreover
```
```  3920       have "w1\<bullet>k \<le> w2\<bullet>k"
```
```  3921         by (rule lem[OF w1(1) w2(1)]) (simp add: assms(4))
```
```  3922       ultimately show False
```
```  3923         unfolding inner_simps by(rule *)
```
```  3924     qed
```
```  3925   qed
```
```  3926 qed
```
```  3927
```
```  3928 lemma integral_component_le:
```
```  3929   fixes g f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3930   assumes "k \<in> Basis"
```
```  3931     and "f integrable_on s" "g integrable_on s"
```
```  3932     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  3933   shows "(integral s f)\<bullet>k \<le> (integral s g)\<bullet>k"
```
```  3934   apply (rule has_integral_component_le)
```
```  3935   using integrable_integral assms
```
```  3936   apply auto
```
```  3937   done
```
```  3938
```
```  3939 lemma has_integral_component_nonneg:
```
```  3940   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3941   assumes "k \<in> Basis"
```
```  3942     and "(f has_integral i) s"
```
```  3943     and "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  3944   shows "0 \<le> i\<bullet>k"
```
```  3945   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
```
```  3946   using assms(3-)
```
```  3947   by auto
```
```  3948
```
```  3949 lemma integral_component_nonneg:
```
```  3950   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3951   assumes "k \<in> Basis"
```
```  3952     and  "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  3953   shows "0 \<le> (integral s f)\<bullet>k"
```
```  3954 proof (cases "f integrable_on s")
```
```  3955   case True show ?thesis
```
```  3956     apply (rule has_integral_component_nonneg)
```
```  3957     using assms True
```
```  3958     apply auto
```
```  3959     done
```
```  3960 next
```
```  3961   case False then show ?thesis by (simp add: not_integrable_integral)
```
```  3962 qed
```
```  3963
```
```  3964 lemma has_integral_component_neg:
```
```  3965   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3966   assumes "k \<in> Basis"
```
```  3967     and "(f has_integral i) s"
```
```  3968     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"
```
```  3969   shows "i\<bullet>k \<le> 0"
```
```  3970   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
```
```  3971   by auto
```
```  3972
```
```  3973 lemma has_integral_component_lbound:
```
```  3974   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3975   assumes "(f has_integral i) (cbox a b)"
```
```  3976     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  3977     and "k \<in> Basis"
```
```  3978   shows "B * content (cbox a b) \<le> i\<bullet>k"
```
```  3979   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-)
```
```  3980   by (auto simp add: field_simps)
```
```  3981
```
```  3982 lemma has_integral_component_ubound:
```
```  3983   fixes f::"'a::euclidean_space => 'b::euclidean_space"
```
```  3984   assumes "(f has_integral i) (cbox a b)"
```
```  3985     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  3986     and "k \<in> Basis"
```
```  3987   shows "i\<bullet>k \<le> B * content (cbox a b)"
```
```  3988   using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"] assms(2-)
```
```  3989   by (auto simp add: field_simps)
```
```  3990
```
```  3991 lemma integral_component_lbound:
```
```  3992   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  3993   assumes "f integrable_on cbox a b"
```
```  3994     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  3995     and "k \<in> Basis"
```
```  3996   shows "B * content (cbox a b) \<le> (integral(cbox a b) f)\<bullet>k"
```
```  3997   apply (rule has_integral_component_lbound)
```
```  3998   using assms
```
```  3999   unfolding has_integral_integral
```
```  4000   apply auto
```
```  4001   done
```
```  4002
```
```  4003 lemma integral_component_lbound_real:
```
```  4004   assumes "f integrable_on {a ::real .. b}"
```
```  4005     and "\<forall>x\<in>{a .. b}. B \<le> f(x)\<bullet>k"
```
```  4006     and "k \<in> Basis"
```
```  4007   shows "B * content {a .. b} \<le> (integral {a .. b} f)\<bullet>k"
```
```  4008   using assms
```
```  4009   by (metis box_real(2) integral_component_lbound)
```
```  4010
```
```  4011 lemma integral_component_ubound:
```
```  4012   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  4013   assumes "f integrable_on cbox a b"
```
```  4014     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  4015     and "k \<in> Basis"
```
```  4016   shows "(integral (cbox a b) f)\<bullet>k \<le> B * content (cbox a b)"
```
```  4017   apply (rule has_integral_component_ubound)
```
```  4018   using assms
```
```  4019   unfolding has_integral_integral
```
```  4020   apply auto
```
```  4021   done
```
```  4022
```
```  4023 lemma integral_component_ubound_real:
```
```  4024   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```  4025   assumes "f integrable_on {a .. b}"
```
```  4026     and "\<forall>x\<in>{a .. b}. f x\<bullet>k \<le> B"
```
```  4027     and "k \<in> Basis"
```
```  4028   shows "(integral {a .. b} f)\<bullet>k \<le> B * content {a .. b}"
```
```  4029   using assms
```
```  4030   by (metis box_real(2) integral_component_ubound)
```
```  4031
```
```  4032 subsection \<open>Uniform limit of integrable functions is integrable.\<close>
```
```  4033
```
```  4034 lemma real_arch_invD:
```
```  4035   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```  4036   by (subst(asm) real_arch_inverse)
```
```  4037
```
```  4038 lemma integrable_uniform_limit:
```
```  4039   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4040   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"
```
```  4041   shows "f integrable_on cbox a b"
```
```  4042 proof (cases "content (cbox a b) > 0")
```
```  4043   case False then show ?thesis
```
```  4044       using has_integral_null
```
```  4045       by (simp add: content_lt_nz integrable_on_def)
```
```  4046 next
```
```  4047   case True
```
```  4048   have *: "\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n + 1))"
```
```  4049     by auto
```
```  4050   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
```
```  4051   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]]
```
```  4052   obtain i where i: "\<And>x. (g x has_integral i x) (cbox a b)"
```
```  4053       by auto
```
```  4054   have "Cauchy i"
```
```  4055     unfolding Cauchy_def
```
```  4056   proof clarify
```
```  4057     fix e :: real
```
```  4058     assume "e>0"
```
```  4059     then have "e / 4 / content (cbox a b) > 0"
```
```  4060       using True by (auto simp add: field_simps)
```
```  4061     then obtain M :: nat
```
```  4062          where M: "M \<noteq> 0" "0 < inverse (real_of_nat M)" "inverse (of_nat M) < e / 4 / content (cbox a b)"
```
```  4063       by (subst (asm) real_arch_inverse) auto
```
```  4064     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e"
```
```  4065     proof (rule exI [where x=M], clarify)
```
```  4066       fix m n
```
```  4067       assume m: "M \<le> m" and n: "M \<le> n"
```
```  4068       have "e/4>0" using \<open>e>0\<close> by auto
```
```  4069       note * = i[unfolded has_integral,rule_format,OF this]
```
```  4070       from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
```
```  4071       from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
```
```  4072       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b]
```
```  4073       obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine p"
```
```  4074         by auto
```
```  4075       { fix s1 s2 i1 and i2::'b
```
```  4076         assume no: "norm(s2 - s1) \<le> e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4"
```
```  4077         have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
```
```  4078           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
```
```  4079           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
```
```  4080           by (auto simp add: algebra_simps)
```
```  4081         also have "\<dots> < e"
```
```  4082           using no
```
```  4083           unfolding norm_minus_commute
```
```  4084           by (auto simp add: algebra_simps)
```
```  4085         finally have "norm (i1 - i2) < e" .
```
```  4086       } note triangle3 = this
```
```  4087       have finep: "gm fine p" "gn fine p"
```
```  4088         using fine_inter p  by auto
```
```  4089       { fix x
```
```  4090         assume x: "x \<in> cbox a b"
```
```  4091         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
```
```  4092           using g(1)[OF x, of n] g(1)[OF x, of m] by auto
```
```  4093         also have "\<dots> \<le> inverse (real M) + inverse (real M)"
```
```  4094           apply (rule add_mono)
```
```  4095           using M(2) m n by auto
```
```  4096         also have "\<dots> = 2 / real M"
```
```  4097           unfolding divide_inverse by auto
```
```  4098         finally have "norm (g n x - g m x) \<le> 2 / real M"
```
```  4099           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
```
```  4100           by (auto simp add: algebra_simps simp add: norm_minus_commute)
```
```  4101       } note norm_le = this
```
```  4102       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"
```
```  4103         apply (rule order_trans [OF rsum_diff_bound[OF p(1), where e="2 / real M"]])
```
```  4104         apply (blast intro: norm_le)
```
```  4105         using M True
```
```  4106         by (auto simp add: field_simps)
```
```  4107       then show "dist (i m) (i n) < e"
```
```  4108         unfolding dist_norm
```
```  4109         using gm gn p finep
```
```  4110         by (auto intro!: triangle3)
```
```  4111     qed
```
```  4112   qed
```
```  4113   then obtain s where s: "i \<longlonglongrightarrow> s"
```
```  4114     using convergent_eq_cauchy[symmetric] by blast
```
```  4115   show ?thesis
```
```  4116     unfolding integrable_on_def has_integral
```
```  4117   proof (rule_tac x=s in exI, clarify)
```
```  4118     fix e::real
```
```  4119     assume e: "0 < e"
```
```  4120     then have *: "e/3 > 0" by auto
```
```  4121     then obtain N1 where N1: "\<forall>n\<ge>N1. norm (i n - s) < e / 3"
```
```  4122       using LIMSEQ_D [OF s] by metis
```
```  4123     from e True have "e / 3 / content (cbox a b) > 0"
```
```  4124       by (auto simp add: field_simps)
```
```  4125     from real_arch_invD[OF this] guess N2 by (elim exE conjE) note N2=this
```
```  4126     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
```
```  4127     { fix sf sg i
```
```  4128       assume no: "norm (sf - sg) \<le> e / 3"
```
```  4129                  "norm(i - s) < e / 3"
```
```  4130                  "norm (sg - i) < e / 3"
```
```  4131       have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
```
```  4132         using norm_triangle_ineq[of "sf - sg" "sg - s"]
```
```  4133         using norm_triangle_ineq[of "sg -  i" " i - s"]
```
```  4134         by (auto simp add: algebra_simps)
```
```  4135       also have "\<dots> < e"
```
```  4136         using no
```
```  4137         unfolding norm_minus_commute
```
```  4138         by (auto simp add: algebra_simps)
```
```  4139       finally have "norm (sf - s) < e" .
```
```  4140     } note lem = this
```
```  4141     { fix p
```
```  4142       assume p: "p tagged_division_of (cbox a b) \<and> g' fine p"
```
```  4143       then have norm_less: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g (N1 + N2) x) - i (N1 + N2)) < e / 3"
```
```  4144         using g' by blast
```
```  4145       have "content (cbox a b) < e / 3 * (of_nat N2)"
```
```  4146         using N2 unfolding inverse_eq_divide using True by (auto simp add: field_simps)
```
```  4147       moreover have "e / 3 * of_nat N2 \<le> e / 3 * (of_nat (N1 + N2) + 1)"
```
```  4148         using \<open>e>0\<close> by auto
```
```  4149       ultimately have "content (cbox a b) < e / 3 * (of_nat (N1 + N2) + 1)"
```
```  4150         by linarith
```
```  4151       then have le_e3: "inverse (real (N1 + N2) + 1) * content (cbox a b) \<le> e / 3"
```
```  4152         unfolding inverse_eq_divide
```
```  4153         by (auto simp add: field_simps)
```
```  4154       have ne3: "norm (i (N1 + N2) - s) < e / 3"
```
```  4155         using N1 by auto
```
```  4156       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e"
```
```  4157         apply (rule lem[OF order_trans [OF _ le_e3] ne3 norm_less])
```
```  4158         apply (rule rsum_diff_bound[OF p[THEN conjunct1]])
```
```  4159         apply (blast intro: g)
```
```  4160         done }
```
```  4161     then show "\<exists>d. gauge d \<and>
```
```  4162              (\<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)"
```
```  4163       by (blast intro: g')
```
```  4164   qed
```
```  4165 qed
```
```  4166
```
```  4167 lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
```
```  4168
```
```  4169
```
```  4170 subsection \<open>Negligible sets.\<close>
```
```  4171
```
```  4172 definition "negligible (s:: 'a::euclidean_space set) \<longleftrightarrow>
```
```  4173   (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) (cbox a b))"
```
```  4174
```
```  4175
```
```  4176 subsection \<open>Negligibility of hyperplane.\<close>
```
```  4177
```
```  4178 lemma interval_doublesplit:
```
```  4179   fixes a :: "'a::euclidean_space"
```
```  4180   assumes "k \<in> Basis"
```
```  4181   shows "cbox a b \<inter> {x . \<bar>x\<bullet>k - c\<bar> \<le> (e::real)} =
```
```  4182     cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i)
```
```  4183      (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)"
```
```  4184 proof -
```
```  4185   have *: "\<And>x c e::real. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
```
```  4186     by auto
```
```  4187   have **: "\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}"
```
```  4188     by blast
```
```  4189   show ?thesis
```
```  4190     unfolding * ** interval_split[OF assms] by (rule refl)
```
```  4191 qed
```
```  4192
```
```  4193 lemma division_doublesplit:
```
```  4194   fixes a :: "'a::euclidean_space"
```
```  4195   assumes "p division_of (cbox a b)"
```
```  4196     and k: "k \<in> Basis"
```
```  4197   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> {}}
```
```  4198          division_of  (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e})"
```
```  4199 proof -
```
```  4200   have *: "\<And>x c. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
```
```  4201     by auto
```
```  4202   have **: "\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'"
```
```  4203     by auto
```
```  4204   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
```
```  4205   note division_split(2)[OF this, where c="c-e" and k=k,OF k]
```
```  4206   then show ?thesis
```
```  4207     apply (rule **)
```
```  4208     subgoal
```
```  4209       apply (simp add: abs_diff_le_iff field_simps Collect_conj_eq setcompr_eq_image[symmetric])
```
```  4210       apply (rule equalityI)
```
```  4211       apply blast
```
```  4212       apply clarsimp
```
```  4213       apply (rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI)
```
```  4214       apply auto
```
```  4215       done
```
```  4216     by (simp add: interval_split k interval_doublesplit)
```
```  4217 qed
```
```  4218
```
```  4219 lemma content_doublesplit:
```
```  4220   fixes a :: "'a::euclidean_space"
```
```  4221   assumes "0 < e"
```
```  4222     and k: "k \<in> Basis"
```
```  4223   obtains d where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) < e"
```
```  4224 proof cases
```
```  4225   assume *: "a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j)"
```
```  4226   define a' where "a' d = (\<Sum>j\<in>Basis. (if j = k then max (a\<bullet>j) (c - d) else a\<bullet>j) *\<^sub>R j)" for d
```
```  4227   define b' where "b' d = (\<Sum>j\<in>Basis. (if j = k then min (b\<bullet>j) (c + d) else b\<bullet>j) *\<^sub>R j)" for d
```
```  4228
```
```  4229   have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> (\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j)) (at_right 0)"
```
```  4230     by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros)
```
```  4231   also have "(\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j) = 0"
```
```  4232     using k *
```
```  4233     by (intro setprod_zero bexI[OF _ k])
```
```  4234        (auto simp: b'_def a'_def inner_diff inner_setsum_left inner_not_same_Basis intro!: setsum.cong)
```
```  4235   also have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> 0) (at_right 0) =
```
```  4236     ((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)"
```
```  4237   proof (intro tendsto_cong eventually_at_rightI)
```
```  4238     fix d :: real assume d: "d \<in> {0<..<1}"
```
```  4239     have "cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d} = cbox (a' d) (b' d)" for d
```
```  4240       using * d k by (auto simp add: cbox_def set_eq_iff Int_def ball_conj_distrib abs_diff_le_iff a'_def b'_def)
```
```  4241     moreover have "j \<in> Basis \<Longrightarrow> a' d \<bullet> j \<le> b' d \<bullet> j" for j
```
```  4242       using * d k by (auto simp: a'_def b'_def)
```
```  4243     ultimately show "(\<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) = content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})"
```
```  4244       by simp
```
```  4245   qed simp
```
```  4246   finally have "((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)" .
```
```  4247   from order_tendstoD(2)[OF this \<open>0<e\<close>]
```
```  4248   obtain d' where "0 < d'" and d': "\<And>y. y > 0 \<Longrightarrow> y < d' \<Longrightarrow> content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> y}) < e"
```
```  4249     by (subst (asm) eventually_at_right[of _ 1]) auto
```
```  4250   show ?thesis
```
```  4251     by (rule that[of "d'/2"], insert \<open>0<d'\<close> d'[of "d'/2"], auto)
```
```  4252 next
```
```  4253   assume *: "\<not> (a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j))"
```
```  4254   then have "(\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j) \<or> (c < a \<bullet> k \<or> b \<bullet> k < c)"
```
```  4255     by (auto simp: not_le)
```
```  4256   show thesis
```
```  4257   proof cases
```
```  4258     assume "\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j"
```
```  4259     then have [simp]: "cbox a b = {}"
```
```  4260       using box_ne_empty(1)[of a b] by auto
```
```  4261     show ?thesis
```
```  4262       by (rule that[of 1]) (simp_all add: \<open>0<e\<close>)
```
```  4263   next
```
```  4264     assume "\<not> (\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j)"
```
```  4265     with * have "c < a \<bullet> k \<or> b \<bullet> k < c"
```
```  4266       by auto
```
```  4267     then show thesis
```
```  4268     proof
```
```  4269       assume c: "c < a \<bullet> k"
```
```  4270       moreover have "x \<in> cbox a b \<Longrightarrow> c \<le> x \<bullet> k" for x
```
```  4271         using k c by (auto simp: cbox_def)
```
```  4272       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (a \<bullet> k - c) / 2} = {}"
```
```  4273         using k by (auto simp: cbox_def)
```
```  4274       with \<open>0<e\<close> c that[of "(a \<bullet> k - c) / 2"] show ?thesis
```
```  4275         by auto
```
```  4276     next
```
```  4277       assume c: "b \<bullet> k < c"
```
```  4278       moreover have "x \<in> cbox a b \<Longrightarrow> x \<bullet> k \<le> c" for x
```
```  4279         using k c by (auto simp: cbox_def)
```
```  4280       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (c - b \<bullet> k) / 2} = {}"
```
```  4281         using k by (auto simp: cbox_def)
```
```  4282       with \<open>0<e\<close> c that[of "(c - b \<bullet> k) / 2"] show ?thesis
```
```  4283         by auto
```
```  4284     qed
```
```  4285   qed
```
```  4286 qed
```
```  4287
```
```  4288
```
```  4289 lemma negligible_standard_hyperplane[intro]:
```
```  4290   fixes k :: "'a::euclidean_space"
```
```  4291   assumes k: "k \<in> Basis"
```
```  4292   shows "negligible {x. x\<bullet>k = c}"
```
```  4293   unfolding negligible_def has_integral
```
```  4294 proof (clarify, goal_cases)
```
```  4295   case (1 a b e)
```
```  4296   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"
```
```  4297     by (rule content_doublesplit)
```
```  4298   let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
```
```  4299   show ?case
```
```  4300     apply (rule_tac x="\<lambda>x. ball x d" in exI)
```
```  4301     apply rule
```
```  4302     apply (rule gauge_ball)
```
```  4303     apply (rule d)
```
```  4304   proof (rule, rule)
```
```  4305     fix p
```
```  4306     assume p: "p tagged_division_of (cbox a b) \<and> (\<lambda>x. ball x d) fine p"
```
```  4307     have *: "(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) =
```
```  4308       (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
```
```  4309       apply (rule setsum.cong)
```
```  4310       apply (rule refl)
```
```  4311       unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
```
```  4312       apply cases
```
```  4313       apply (rule disjI1)
```
```  4314       apply assumption
```
```  4315       apply (rule disjI2)
```
```  4316     proof -
```
```  4317       fix x l
```
```  4318       assume as: "(x, l) \<in> p" "?i x \<noteq> 0"
```
```  4319       then have xk: "x\<bullet>k = c"
```
```  4320         unfolding indicator_def
```
```  4321         apply -
```
```  4322         apply (rule ccontr)
```
```  4323         apply auto
```
```  4324         done
```
```  4325       show "content l = content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  4326         apply (rule arg_cong[where f=content])
```
```  4327         apply (rule set_eqI)
```
```  4328         apply rule
```
```  4329         apply rule
```
```  4330         unfolding mem_Collect_eq
```
```  4331       proof -
```
```  4332         fix y
```
```  4333         assume y: "y \<in> l"
```
```  4334         note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
```
```  4335         note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y]
```
```  4336         note le_less_trans[OF Basis_le_norm[OF k] this]
```
```  4337         then show "\<bar>y \<bullet> k - c\<bar> \<le> d"
```
```  4338           unfolding inner_simps xk by auto
```
```  4339       qed auto
```
```  4340     qed
```
```  4341     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
```
```  4342     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e"
```
```  4343       unfolding diff_0_right *
```
```  4344       unfolding real_scaleR_def real_norm_def
```
```  4345       apply (subst abs_of_nonneg)
```
```  4346       apply (rule setsum_nonneg)
```
```  4347       apply rule
```
```  4348       unfolding split_paired_all split_conv
```
```  4349       apply (rule mult_nonneg_nonneg)
```
```  4350       apply (drule p'(4))
```
```  4351       apply (erule exE)+
```
```  4352       apply(rule_tac b=b in back_subst)
```
```  4353       prefer 2
```
```  4354       apply (subst(asm) eq_commute)
```
```  4355       apply assumption
```
```  4356       apply (subst interval_doublesplit[OF k])
```
```  4357       apply (rule content_pos_le)
```
```  4358       apply (rule indicator_pos_le)
```
```  4359     proof -
```
```  4360       have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le>
```
```  4361         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
```
```  4362         apply (rule setsum_mono)
```
```  4363         unfolding split_paired_all split_conv
```
```  4364         apply (rule mult_right_le_one_le)
```
```  4365         apply (drule p'(4))
```
```  4366         apply (auto simp add:interval_doublesplit[OF k])
```
```  4367         done
```
```  4368       also have "\<dots> < e"
```
```  4369       proof (subst setsum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
```
```  4370         case prems: (1 u v)
```
```  4371         have "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content (cbox u v)"
```
```  4372           unfolding interval_doublesplit[OF k]
```
```  4373           apply (rule content_subset)
```
```  4374           unfolding interval_doublesplit[symmetric,OF k]
```
```  4375           apply auto
```
```  4376           done
```
```  4377         then show ?case
```
```  4378           unfolding prems interval_doublesplit[OF k]
```
```  4379           by (blast intro: antisym)
```
```  4380       next
```
```  4381         have "(\<Sum>l\<in>snd ` p. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
```
```  4382           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> {}})"
```
```  4383         proof (subst (2) setsum.reindex_nontrivial)
```
```  4384           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> {}}"
```
```  4385             "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}"
```
```  4386           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> {}"
```
```  4387             by (auto)
```
```  4388           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) = {}"
```
```  4389             by auto
```
```  4390           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)"
```
```  4391             by (auto intro: interior_mono)
```
```  4392           ultimately have "interior (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}"
```
```  4393             by (auto simp: eq)
```
```  4394           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
```
```  4395             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)
```
```  4396         qed (insert p'(1), auto intro!: setsum.mono_neutral_right)
```
```  4397         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)"
```
```  4398           by simp
```
```  4399         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  4400           using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
```
```  4401           unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
```
```  4402         also have "\<dots> < e"
```
```  4403           using d(2) by simp
```
```  4404         finally show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e" .
```
```  4405       qed
```
```  4406       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
```
```  4407     qed
```
```  4408   qed
```
```  4409 qed
```
```  4410
```
```  4411
```
```  4412 subsection \<open>A technical lemma about "refinement" of division.\<close>
```
```  4413
```
```  4414 lemma tagged_division_finer:
```
```  4415   fixes p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
```
```  4416   assumes "p tagged_division_of (cbox a b)"
```
```  4417     and "gauge d"
```
```  4418   obtains q where "q tagged_division_of (cbox a b)"
```
```  4419     and "d fine q"
```
```  4420     and "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
```
```  4421 proof -
```
```  4422   let ?P = "\<lambda>p. p tagged_partial_division_of (cbox a b) \<longrightarrow> gauge d \<longrightarrow>
```
```  4423     (\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and>
```
```  4424       (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
```
```  4425   {
```
```  4426     have *: "finite p" "p tagged_partial_division_of (cbox a b)"
```
```  4427       using assms(1)
```
```  4428       unfolding tagged_division_of_def
```
```  4429       by auto
```
```  4430     presume "\<And>p. finite p \<Longrightarrow> ?P p"
```
```  4431     from this[rule_format,OF * assms(2)] guess q .. note q=this
```
```  4432     then show ?thesis
```
```  4433       apply -
```
```  4434       apply (rule that[of q])
```
```  4435       unfolding tagged_division_ofD[OF assms(1)]
```
```  4436       apply auto
```
```  4437       done
```
```  4438   }
```
```  4439   fix p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
```
```  4440   assume as: "finite p"
```
```  4441   show "?P p"
```
```  4442     apply rule
```
```  4443     apply rule
```
```  4444     using as
```
```  4445   proof (induct p)
```
```  4446     case empty
```
```  4447     show ?case
```
```  4448       apply (rule_tac x="{}" in exI)
```
```  4449       unfolding fine_def
```
```  4450       apply auto
```
```  4451       done
```
```  4452   next
```
```  4453     case (insert xk p)
```
```  4454     guess x k using surj_pair[of xk] by (elim exE) note xk=this
```
```  4455     note tagged_partial_division_subset[OF insert(4) subset_insertI]
```
```  4456     from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
```
```  4457     have *: "\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}"
```
```  4458       unfolding xk by auto
```
```  4459     note p = tagged_partial_division_ofD[OF insert(4)]
```
```  4460     from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
```
```  4461
```
```  4462     have "finite {k. \<exists>x. (x, k) \<in> p}"
```
```  4463       apply (rule finite_subset[of _ "snd ` p"])
```
```  4464       using p
```
```  4465       apply safe
```
```  4466       apply (metis image_iff snd_conv)
```
```  4467       apply auto
```
```  4468       done
```
```  4469     then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
```
```  4470       apply (rule inter_interior_unions_intervals)
```
```  4471       apply (rule open_interior)
```
```  4472       apply (rule_tac[!] ballI)
```
```  4473       unfolding mem_Collect_eq
```
```  4474       apply (erule_tac[!] exE)
```
```  4475       apply (drule p(4)[OF insertI2])
```
```  4476       apply assumption
```
```  4477       apply (rule p(5))
```
```  4478       unfolding uv xk
```
```  4479       apply (rule insertI1)
```
```  4480       apply (rule insertI2)
```
```  4481       apply assumption
```
```  4482       using insert(2)
```
```  4483       unfolding uv xk
```
```  4484       apply auto
```
```  4485       done
```
```  4486     show ?case
```
```  4487     proof (cases "cbox u v \<subseteq> d x")
```
```  4488       case True
```
```  4489       then show ?thesis
```
```  4490         apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
```
```  4491         apply rule
```
```  4492         unfolding * uv
```
```  4493         apply (rule tagged_division_union)
```
```  4494         apply (rule tagged_division_of_self)
```
```  4495         apply (rule p[unfolded xk uv] insertI1)+
```
```  4496         apply (rule q1)
```
```  4497         apply (rule int)
```
```  4498         apply rule
```
```  4499         apply (rule fine_union)
```
```  4500         apply (subst fine_def)
```
```  4501         defer
```
```  4502         apply (rule q1)
```
```  4503         unfolding Ball_def split_paired_All split_conv
```
```  4504         apply rule
```
```  4505         apply rule
```
```  4506         apply rule
```
```  4507         apply rule
```
```  4508         apply (erule insertE)
```
```  4509         apply (simp add: uv xk)
```
```  4510         apply (rule UnI2)
```
```  4511         apply (drule q1(3)[rule_format])
```
```  4512         unfolding xk uv
```
```  4513         apply auto
```
```  4514         done
```
```  4515     next
```
```  4516       case False
```
```  4517       from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
```
```  4518       show ?thesis
```
```  4519         apply (rule_tac x="q2 \<union> q1" in exI)
```
```  4520         apply rule
```
```  4521         unfolding * uv
```
```  4522         apply (rule tagged_division_union q2 q1 int fine_union)+
```
```  4523         unfolding Ball_def split_paired_All split_conv
```
```  4524         apply rule
```
```  4525         apply (rule fine_union)
```
```  4526         apply (rule q1 q2)+
```
```  4527         apply rule
```
```  4528         apply rule
```
```  4529         apply rule
```
```  4530         apply rule
```
```  4531         apply (erule insertE)
```
```  4532         apply (rule UnI2)
```
```  4533         apply (simp add: False uv xk)
```
```  4534         apply (drule q1(3)[rule_format])
```
```  4535         using False
```
```  4536         unfolding xk uv
```
```  4537         apply auto
```
```  4538         done
```
```  4539     qed
```
```  4540   qed
```
```  4541 qed
```
```  4542
```
```  4543
```
```  4544 subsection \<open>Hence the main theorem about negligible sets.\<close>
```
```  4545
```
```  4546 lemma finite_product_dependent:
```
```  4547   assumes "finite s"
```
```  4548     and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
```
```  4549   shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
```
```  4550   using assms
```
```  4551 proof induct
```
```  4552   case (insert x s)
```
```  4553   have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
```
```  4554     (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
```
```  4555   show ?case
```
```  4556     unfolding *
```
```  4557     apply (rule finite_UnI)
```
```  4558     using insert
```
```  4559     apply auto
```
```  4560     done
```
```  4561 qed auto
```
```  4562
```
```  4563 lemma sum_sum_product:
```
```  4564   assumes "finite s"
```
```  4565     and "\<forall>i\<in>s. finite (t i)"
```
```  4566   shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s =
```
```  4567     setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
```
```  4568   using assms
```
```  4569 proof induct
```
```  4570   case (insert a s)
```
```  4571   have *: "{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} =
```
```  4572     (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
```
```  4573   show ?case
```
```  4574     unfolding *
```
```  4575     apply (subst setsum.union_disjoint)
```
```  4576     unfolding setsum.insert[OF insert(1-2)]
```
```  4577     prefer 4
```
```  4578     apply (subst insert(3))
```
```  4579     unfolding add_right_cancel
```
```  4580   proof -
```
```  4581     show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
```
```  4582       apply (subst setsum.reindex)
```
```  4583       unfolding inj_on_def
```
```  4584       apply auto
```
```  4585       done
```
```  4586     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
```
```  4587       apply (rule finite_product_dependent)
```
```  4588       using insert
```
```  4589       apply auto
```
```  4590       done
```
```  4591   qed (insert insert, auto)
```
```  4592 qed auto
```
```  4593
```
```  4594 lemma has_integral_negligible:
```
```  4595   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  4596   assumes "negligible s"
```
```  4597     and "\<forall>x\<in>(t - s). f x = 0"
```
```  4598   shows "(f has_integral 0) t"
```
```  4599 proof -
```
```  4600   presume P: "\<And>f::'b::euclidean_space \<Rightarrow> 'a.
```
```  4601     \<And>a b. \<forall>x. x \<notin> s \<longrightarrow> f x = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```  4602   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
```
```  4603   show ?thesis
```
```  4604     apply (rule_tac f="?f" in has_integral_eq)
```
```  4605     unfolding if_P
```
```  4606     apply (rule refl)
```
```  4607     apply (subst has_integral_alt)
```
```  4608     apply cases
```
```  4609     apply (subst if_P, assumption)
```
```  4610     unfolding if_not_P
```
```  4611   proof -
```
```  4612     assume "\<exists>a b. t = cbox a b"
```
```  4613     then guess a b apply - by (erule exE)+ note t = this
```
```  4614     show "(?f has_integral 0) t"
```
```  4615       unfolding t
```
```  4616       apply (rule P)
```
```  4617       using assms(2)
```
```  4618       unfolding t
```
```  4619       apply auto
```
```  4620       done
```
```  4621   next
```
```  4622     show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  4623       (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) (cbox a b) \<and> norm (z - 0) < e)"
```
```  4624       apply safe
```
```  4625       apply (rule_tac x=1 in exI)
```
```  4626       apply rule
```
```  4627       apply (rule zero_less_one)
```
```  4628       apply safe
```
```  4629       apply (rule_tac x=0 in exI)
```
```  4630       apply rule
```
```  4631       apply (rule P)
```
```  4632       using assms(2)
```
```  4633       apply auto
```
```  4634       done
```
```  4635   qed
```
```  4636 next
```
```  4637   fix f :: "'b \<Rightarrow> 'a"
```
```  4638   fix a b :: 'b
```
```  4639   assume assm: "\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  4640   show "(f has_integral 0) (cbox a b)"
```
```  4641     unfolding has_integral
```
```  4642   proof (safe, goal_cases)
```
```  4643     case prems: (1 e)
```
```  4644     then have "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
```
```  4645       apply -
```
```  4646       apply (rule divide_pos_pos)
```
```  4647       defer
```
```  4648       apply (rule mult_pos_pos)
```
```  4649       apply (auto simp add:field_simps)
```
```  4650       done
```
```  4651     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b]
```
```  4652     note allI[OF this,of "\<lambda>x. x"]
```
```  4653     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
```
```  4654     show ?case
```
```  4655       apply (rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
```
```  4656     proof safe
```
```  4657       show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)"
```
```  4658         using d(1) unfolding gauge_def by auto
```
```  4659       fix p
```
```  4660       assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
```
```  4661       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  4662       {
```
```  4663         presume "p \<noteq> {} \<Longrightarrow> ?goal"
```
```  4664         then show ?goal
```
```  4665           apply (cases "p = {}")
```
```  4666           using prems
```
```  4667           apply auto
```
```  4668           done
```
```  4669       }
```
```  4670       assume as': "p \<noteq> {}"
```
```  4671       from real_arch_simple[of "Max((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
```
```  4672       then have N: "\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N"
```
```  4673         by (meson Max_ge as(1) dual_order.trans finite_imageI tagged_division_of_finite)
```
```  4674       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)"
```
```  4675         by (auto intro: tagged_division_finer[OF as(1) d(1)])
```
```  4676       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
```
```  4677       have *: "\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)"
```
```  4678         apply (rule setsum_nonneg)
```
```  4679         apply safe
```
```  4680         unfolding real_scaleR_def
```
```  4681         apply (drule tagged_division_ofD(4)[OF q(1)])
```
```  4682         apply (auto intro: mult_nonneg_nonneg)
```
```  4683         done
```
```  4684       have **: "finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow>
```
```  4685         (\<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
```
```  4686         apply (rule setsum_le_included[of s t g snd f])
```
```  4687         prefer 4
```
```  4688         apply safe
```
```  4689         apply (erule_tac x=x in ballE)
```
```  4690         apply (erule exE)
```
```  4691         apply (rule_tac x="(xa,x)" in bexI)
```
```  4692         apply auto
```
```  4693         done
```
```  4694       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
```
```  4695         norm (setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
```
```  4696         unfolding real_norm_def setsum_distrib_left abs_of_nonneg[OF *] diff_0_right
```
```  4697         apply (rule order_trans)
```
```  4698         apply (rule norm_setsum)
```
```  4699         apply (subst sum_sum_product)
```
```  4700         prefer 3
```
```  4701       proof (rule **, safe)
```
```  4702         show "finite {(i, j) |i j. i \<in> {..N + 1} \<and> j \<in> q i}"
```
```  4703           apply (rule finite_product_dependent)
```
```  4704           using q
```
```  4705           apply auto
```
```  4706           done
```
```  4707         fix i a b
```
```  4708         assume as'': "(a, b) \<in> q i"
```
```  4709         show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
```
```  4710           unfolding real_scaleR_def
```
```  4711           using tagged_division_ofD(4)[OF q(1) as'']
```
```  4712           by (auto intro!: mult_nonneg_nonneg)
```
```  4713       next
```
```  4714         fix i :: nat
```
```  4715         show "finite (q i)"
```
```  4716           using q by auto
```
```  4717       next
```
```  4718         fix x k
```
```  4719         assume xk: "(x, k) \<in> p"
```
```  4720         define n where "n = nat \<lfloor>norm (f x)\<rfloor>"
```
```  4721         have *: "norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p"
```
```  4722           using xk by auto
```
```  4723         have nfx: "real n \<le> norm (f x)" "norm (f x) \<le> real n + 1"
```
```  4724           unfolding n_def by auto
```
```  4725         then have "n \<in> {0..N + 1}"
```
```  4726           using N[rule_format,OF *] by auto
```
```  4727         moreover
```
```  4728         note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
```
```  4729         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this]
```
```  4730         note this[unfolded n_def[symmetric]]
```
```  4731         moreover
```
```  4732         have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
```
```  4733         proof (cases "x \<in> s")
```
```  4734           case False
```
```  4735           then show ?thesis
```
```  4736             using assm by auto
```
```  4737         next
```
```  4738           case True
```
```  4739           have *: "content k \<ge> 0"
```
```  4740             using tagged_division_ofD(4)[OF as(1) xk] by auto
```
```  4741           moreover
```
```  4742           have "content k * norm (f x) \<le> content k * (real n + 1)"
```
```  4743             apply (rule mult_mono)
```
```  4744             using nfx *
```
```  4745             apply auto
```
```  4746             done
```
```  4747           ultimately
```
```  4748           show ?thesis
```
```  4749             unfolding abs_mult
```
```  4750             using nfx True
```
```  4751             by (auto simp add: field_simps)
```
```  4752         qed
```
```  4753         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>
```
```  4754           (real y + 1) * (content k *\<^sub>R indicator s x)"
```
```  4755           apply (rule_tac x=n in exI)
```
```  4756           apply safe
```
```  4757           apply (rule_tac x=n in exI)
```
```  4758           apply (rule_tac x="(x,k)" in exI)
```
```  4759           apply safe
```
```  4760           apply auto
```
```  4761           done
```
```  4762       qed (insert as, auto)
```
```  4763       also have "\<dots> \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {..N+1}"
```
```  4764       proof (rule setsum_mono, goal_cases)
```
```  4765         case (1 i)
```
```  4766         then show ?case
```
```  4767           apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
```
```  4768           using d(2)[rule_format, of "q i" i]
```
```  4769           using q[rule_format]
```
```  4770           apply (auto simp add: field_simps)
```
```  4771           done
```
```  4772       qed
```
```  4773       also have "\<dots> < e * inverse 2 * 2"
```
```  4774         unfolding divide_inverse setsum_distrib_left[symmetric]
```
```  4775         apply (rule mult_strict_left_mono)
```
```  4776         unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
```
```  4777         apply (subst geometric_sum)
```
```  4778         using prems
```
```  4779         apply auto
```
```  4780         done
```
```  4781       finally show "?goal" by auto
```
```  4782     qed
```
```  4783   qed
```
```  4784 qed
```
```  4785
```
```  4786 lemma has_integral_spike:
```
```  4787   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  4788   assumes "negligible s"
```
```  4789     and "(\<forall>x\<in>(t - s). g x = f x)"
```
```  4790     and "(f has_integral y) t"
```
```  4791   shows "(g has_integral y) t"
```
```  4792 proof -
```
```  4793   {
```
```  4794     fix a b :: 'b
```
```  4795     fix f g :: "'b \<Rightarrow> 'a"
```
```  4796     fix y :: 'a
```
```  4797     assume as: "\<forall>x \<in> cbox a b - s. g x = f x" "(f has_integral y) (cbox a b)"
```
```  4798     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
```
```  4799       apply (rule has_integral_add[OF as(2)])
```
```  4800       apply (rule has_integral_negligible[OF assms(1)])
```
```  4801       using as
```
```  4802       apply auto
```
```  4803       done
```
```  4804     then have "(g has_integral y) (cbox a b)"
```
```  4805       by auto
```
```  4806   } note * = this
```
```  4807   show ?thesis
```
```  4808     apply (subst has_integral_alt)
```
```  4809     using assms(2-)
```
```  4810     apply -
```
```  4811     apply (rule cond_cases)
```
```  4812     apply safe
```
```  4813     apply (rule *)
```
```  4814     apply assumption+
```
```  4815     apply (subst(asm) has_integral_alt)
```
```  4816     unfolding if_not_P
```
```  4817     apply (erule_tac x=e in allE)
```
```  4818     apply safe
```
```  4819     apply (rule_tac x=B in exI)
```
```  4820     apply safe
```
```  4821     apply (erule_tac x=a in allE)
```
```  4822     apply (erule_tac x=b in allE)
```
```  4823     apply safe
```
```  4824     apply (rule_tac x=z in exI)
```
```  4825     apply safe
```
```  4826     apply (rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"])
```
```  4827     apply auto
```
```  4828     done
```
```  4829 qed
```
```  4830
```
```  4831 lemma has_integral_spike_eq:
```
```  4832   assumes "negligible s"
```
```  4833     and "\<forall>x\<in>(t - s). g x = f x"
```
```  4834   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  4835   apply rule
```
```  4836   apply (rule_tac[!] has_integral_spike[OF assms(1)])
```
```  4837   using assms(2)
```
```  4838   apply auto
```
```  4839   done
```
```  4840
```
```  4841 lemma integrable_spike:
```
```  4842   assumes "negligible s"
```
```  4843     and "\<forall>x\<in>(t - s). g x = f x"
```
```  4844     and "f integrable_on t"
```
```  4845   shows "g integrable_on  t"
```
```  4846   using assms
```
```  4847   unfolding integrable_on_def
```
```  4848   apply -
```
```  4849   apply (erule exE)
```
```  4850   apply rule
```
```  4851   apply (rule has_integral_spike)
```
```  4852   apply fastforce+
```
```  4853   done
```
```  4854
```
```  4855 lemma integral_spike:
```
```  4856   assumes "negligible s"
```
```  4857     and "\<forall>x\<in>(t - s). g x = f x"
```
```  4858   shows "integral t f = integral t g"
```
```  4859   using has_integral_spike_eq[OF assms] by (simp add: integral_def integrable_on_def)
```
```  4860
```
```  4861
```
```  4862 subsection \<open>Some other trivialities about negligible sets.\<close>
```
```  4863
```
```  4864 lemma negligible_subset[intro]:
```
```  4865   assumes "negligible s"
```
```  4866     and "t \<subseteq> s"
```
```  4867   shows "negligible t"
```
```  4868   unfolding negligible_def
```
```  4869 proof (safe, goal_cases)
```
```  4870   case (1 a b)
```
```  4871   show ?case
```
```  4872     using assms(1)[unfolded negligible_def,rule_format,of a b]
```
```  4873     apply -
```
```  4874     apply (rule has_integral_spike[OF assms(1)])
```
```  4875     defer
```
```  4876     apply assumption
```
```  4877     using assms(2)
```
```  4878     unfolding indicator_def
```
```  4879     apply auto
```
```  4880     done
```
```  4881 qed
```
```  4882
```
```  4883 lemma negligible_diff[intro?]:
```
```  4884   assumes "negligible s"
```
```  4885   shows "negligible (s - t)"
```
```  4886   using assms by auto
```
```  4887
```
```  4888 lemma negligible_Int:
```
```  4889   assumes "negligible s \<or> negligible t"
```
```  4890   shows "negligible (s \<inter> t)"
```
```  4891   using assms by auto
```
```  4892
```
```  4893 lemma negligible_Un:
```
```  4894   assumes "negligible s"
```
```  4895     and "negligible t"
```
```  4896   shows "negligible (s \<union> t)"
```
```  4897   unfolding negligible_def
```
```  4898 proof (safe, goal_cases)
```
```  4899   case (1 a b)
```
```  4900   note assm = assms[unfolded negligible_def,rule_format,of a b]
```
```  4901   then show ?case
```
```  4902     apply (subst has_integral_spike_eq[OF assms(2)])
```
```  4903     defer
```
```  4904     apply assumption
```
```  4905     unfolding indicator_def
```
```  4906     apply auto
```
```  4907     done
```
```  4908 qed
```
```  4909
```
```  4910 lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
```
```  4911   using negligible_Un by auto
```
```  4912
```
```  4913 lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
```
```  4914   using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] by auto
```
```  4915
```
```  4916 lemma negligible_insert[simp]: "negligible (insert a s) \<longleftrightarrow> negligible s"
```
```  4917   apply (subst insert_is_Un)
```
```  4918   unfolding negligible_Un_eq
```
```  4919   apply auto
```
```  4920   done
```
```  4921
```
```  4922 lemma negligible_empty[iff]: "negligible {}"
```
```  4923   by auto
```
```  4924
```
```  4925 lemma negligible_finite[intro]:
```
```  4926   assumes "finite s"
```
```  4927   shows "negligible s"
```
```  4928   using assms by (induct s) auto
```
```  4929
```
```  4930 lemma negligible_Union[intro]:
```
```  4931   assumes "finite s"
```
```  4932     and "\<forall>t\<in>s. negligible t"
```
```  4933   shows "negligible(\<Union>s)"
```
```  4934   using assms by induct auto
```
```  4935
```
```  4936 lemma negligible:
```
```  4937   "negligible s \<longleftrightarrow> (\<forall>t::('a::euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
```
```  4938   apply safe
```
```  4939   defer
```
```  4940   apply (subst negligible_def)
```
```  4941 proof -
```
```  4942   fix t :: "'a set"
```
```  4943   assume as: "negligible s"
```
```  4944   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)"
```
```  4945     by auto
```
```  4946   show "((indicator s::'a\<Rightarrow>real) has_integral 0) t"
```
```  4947     apply (subst has_integral_alt)
```
```  4948     apply cases
```
```  4949     apply (subst if_P,assumption)
```
```  4950     unfolding if_not_P
```
```  4951     apply safe
```
```  4952     apply (rule as[unfolded negligible_def,rule_format])
```
```  4953     apply (rule_tac x=1 in exI)
```
```  4954     apply safe
```
```  4955     apply (rule zero_less_one)
```
```  4956     apply (rule_tac x=0 in exI)
```
```  4957     using negligible_subset[OF as,of "s \<inter> t"]
```
```  4958     unfolding negligible_def indicator_def [abs_def]
```
```  4959     unfolding *
```
```  4960     apply auto
```
```  4961     done
```
```  4962 qed auto
```
```  4963
```
```  4964
```
```  4965 subsection \<open>Finite case of the spike theorem is quite commonly needed.\<close>
```
```  4966
```
```  4967 lemma has_integral_spike_finite:
```
```  4968   assumes "finite s"
```
```  4969     and "\<forall>x\<in>t-s. g x = f x"
```
```  4970     and "(f has_integral y) t"
```
```  4971   shows "(g has_integral y) t"
```
```  4972   apply (rule has_integral_spike)
```
```  4973   using assms
```
```  4974   apply auto
```
```  4975   done
```
```  4976
```
```  4977 lemma has_integral_spike_finite_eq:
```
```  4978   assumes "finite s"
```
```  4979     and "\<forall>x\<in>t-s. g x = f x"
```
```  4980   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  4981   apply rule
```
```  4982   apply (rule_tac[!] has_integral_spike_finite)
```
```  4983   using assms
```
```  4984   apply auto
```
```  4985   done
```
```  4986
```
```  4987 lemma integrable_spike_finite:
```
```  4988   assumes "finite s"
```
```  4989     and "\<forall>x\<in>t-s. g x = f x"
```
```  4990     and "f integrable_on t"
```
```  4991   shows "g integrable_on  t"
```
```  4992   using assms
```
```  4993   unfolding integrable_on_def
```
```  4994   apply safe
```
```  4995   apply (rule_tac x=y in exI)
```
```  4996   apply (rule has_integral_spike_finite)
```
```  4997   apply auto
```
```  4998   done
```
```  4999
```
```  5000
```
```  5001 subsection \<open>In particular, the boundary of an interval is negligible.\<close>
```
```  5002
```
```  5003 lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
```
```  5004 proof -
```
```  5005   let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
```
```  5006   have "cbox a b - box a b \<subseteq> ?A"
```
```  5007     apply rule unfolding Diff_iff mem_box
```
```  5008     apply simp
```
```  5009     apply(erule conjE bexE)+
```
```  5010     apply(rule_tac x=i in bexI)
```
```  5011     apply auto
```
```  5012     done
```
```  5013   then show ?thesis
```
```  5014     apply -
```
```  5015     apply (rule negligible_subset[of ?A])
```
```  5016     apply (rule negligible_Union[OF finite_imageI])
```
```  5017     apply auto
```
```  5018     done
```
```  5019 qed
```
```  5020
```
```  5021 lemma has_integral_spike_interior:
```
```  5022   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5023     and "(f has_integral y) (cbox a b)"
```
```  5024   shows "(g has_integral y) (cbox a b)"
```
```  5025   apply (rule has_integral_spike[OF negligible_frontier_interval _ assms(2)])
```
```  5026   using assms(1)
```
```  5027   apply auto
```
```  5028   done
```
```  5029
```
```  5030 lemma has_integral_spike_interior_eq:
```
```  5031   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5032   shows "(f has_integral y) (cbox a b) \<longleftrightarrow> (g has_integral y) (cbox a b)"
```
```  5033   apply rule
```
```  5034   apply (rule_tac[!] has_integral_spike_interior)
```
```  5035   using assms
```
```  5036   apply auto
```
```  5037   done
```
```  5038
```
```  5039 lemma integrable_spike_interior:
```
```  5040   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  5041     and "f integrable_on cbox a b"
```
```  5042   shows "g integrable_on cbox a b"
```
```  5043   using assms
```
```  5044   unfolding integrable_on_def
```
```  5045   using has_integral_spike_interior[OF assms(1)]
```
```  5046   by auto
```
```  5047
```
```  5048
```
```  5049 subsection \<open>Integrability of continuous functions.\<close>
```
```  5050
```
```  5051 lemma operative_approximable:
```
```  5052   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5053   assumes "0 \<le> e"
```
```  5054   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)"
```
```  5055   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  5056 proof safe
```
```  5057   fix a b :: 'b
```
```  5058   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5059     if "content (cbox a b) = 0"
```
```  5060     apply (rule_tac x=f in exI)
```
```  5061     using assms that
```
```  5062     apply (auto intro!: integrable_on_null)
```
```  5063     done
```
```  5064   {
```
```  5065     fix c g
```
```  5066     fix k :: 'b
```
```  5067     assume as: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  5068     assume k: "k \<in> Basis"
```
```  5069     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}"
```
```  5070       "\<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}"
```
```  5071       apply (rule_tac[!] x=g in exI)
```
```  5072       using as(1) integrable_split[OF as(2) k]
```
```  5073       apply auto
```
```  5074       done
```
```  5075   }
```
```  5076   fix c k g1 g2
```
```  5077   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}"
```
```  5078     "\<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}"
```
```  5079   assume k: "k \<in> Basis"
```
```  5080   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"
```
```  5081   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5082     apply (rule_tac x="?g" in exI)
```
```  5083     apply safe
```
```  5084   proof goal_cases
```
```  5085     case (1 x)
```
```  5086     then show ?case
```
```  5087       apply -
```
```  5088       apply (cases "x\<bullet>k=c")
```
```  5089       apply (case_tac "x\<bullet>k < c")
```
```  5090       using as assms
```
```  5091       apply auto
```
```  5092       done
```
```  5093   next
```
```  5094     case 2
```
```  5095     presume "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  5096       and "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  5097     then guess h1 h2 unfolding integrable_on_def by auto
```
```  5098     from has_integral_split[OF this k] show ?case
```
```  5099       unfolding integrable_on_def by auto
```
```  5100   next
```
```  5101     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}"
```
```  5102       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
```
```  5103       using k as(2,4)
```
```  5104       apply auto
```
```  5105       done
```
```  5106   qed
```
```  5107 qed
```
```  5108
```
```  5109 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))"
```
```  5110 proof -
```
```  5111   interpret bool: comm_monoid_set "op \<and>" True
```
```  5112     proof qed auto
```
```  5113   show ?thesis
```
```  5114     by (induction s rule: infinite_finite_induct) auto
```
```  5115 qed
```
```  5116
```
```  5117 lemma approximable_on_division:
```
```  5118   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5119   assumes "0 \<le> e"
```
```  5120     and "d division_of (cbox a b)"
```
```  5121     and "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  5122   obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  5123 proof -
```
```  5124   note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_approximable[OF assms(1)] assms(2)]
```
```  5125   from assms(3) this[unfolded comm_monoid_set_F_and, of f] division_of_finite[OF assms(2)]
```
```  5126   guess g by auto
```
```  5127   then show thesis
```
```  5128     apply -
```
```  5129     apply (rule that[of g])
```
```  5130     apply auto
```
```  5131     done
```
```  5132 qed
```
```  5133
```
```  5134 lemma integrable_continuous:
```
```  5135   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5136   assumes "continuous_on (cbox a b) f"
```
```  5137   shows "f integrable_on cbox a b"
```
```  5138 proof (rule integrable_uniform_limit, safe)
```
```  5139   fix e :: real
```
```  5140   assume e: "e > 0"
```
```  5141   from compact_uniformly_continuous[OF assms compact_cbox,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
```
```  5142   note d=conjunctD2[OF this,rule_format]
```
```  5143   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
```
```  5144   note p' = tagged_division_ofD[OF p(1)]
```
```  5145   have *: "\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  5146   proof (safe, unfold snd_conv)
```
```  5147     fix x l
```
```  5148     assume as: "(x, l) \<in> p"
```
```  5149     from p'(4)[OF this] guess a b by (elim exE) note l=this
```
```  5150     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l"
```
```  5151       apply (rule_tac x="\<lambda>y. f x" in exI)
```
```  5152     proof safe
```
```  5153       show "(\<lambda>y. f x) integrable_on l"
```
```  5154         unfolding integrable_on_def l
```
```  5155         apply rule
```
```  5156         apply (rule has_integral_const)
```
```  5157         done
```
```  5158       fix y
```
```  5159       assume y: "y \<in> l"
```
```  5160       note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
```
```  5161       note d(2)[OF _ _ this[unfolded mem_ball]]
```
```  5162       then show "norm (f y - f x) \<le> e"
```
```  5163         using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce
```
```  5164     qed
```
```  5165   qed
```
```  5166   from e have "e \<ge> 0"
```
```  5167     by auto
```
```  5168   from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
```
```  5169   then show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  5170     by auto
```
```  5171 qed
```
```  5172
```
```  5173 lemma integrable_continuous_real:
```
```  5174   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5175   assumes "continuous_on {a .. b} f"
```
```  5176   shows "f integrable_on {a .. b}"
```
```  5177   by (metis assms box_real(2) integrable_continuous)
```
```  5178
```
```  5179 subsection \<open>Specialization of additivity to one dimension.\<close>
```
```  5180
```
```  5181 subsection \<open>Special case of additivity we need for the FTC.\<close>
```
```  5182
```
```  5183 lemma additive_tagged_division_1:
```
```  5184   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
```
```  5185   assumes "a \<le> b"
```
```  5186     and "p tagged_division_of {a..b}"
```
```  5187   shows "setsum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
```
```  5188 proof -
```
```  5189   let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
```
```  5190   have ***: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```  5191     using assms by auto
```
```  5192   have *: "add.operative ?f"
```
```  5193     unfolding add.operative_1_lt box_eq_empty
```
```  5194     by auto
```
```  5195   have **: "cbox a b \<noteq> {}"
```
```  5196     using assms(1) by auto
```
```  5197   note setsum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
```
```  5198   note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
```
```  5199   show ?thesis
```
```  5200     unfolding *
```
```  5201     apply (rule setsum.cong)
```
```  5202     unfolding split_paired_all split_conv
```
```  5203     using assms(2)
```
```  5204     apply auto
```
```  5205     done
```
```  5206 qed
```
```  5207
```
```  5208
```
```  5209 subsection \<open>A useful lemma allowing us to factor out the content size.\<close>
```
```  5210
```
```  5211 lemma has_integral_factor_content:
```
```  5212   "(f has_integral i) (cbox a b) \<longleftrightarrow>
```
```  5213     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  5214       norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
```
```  5215 proof (cases "content (cbox a b) = 0")
```
```  5216   case True
```
```  5217   show ?thesis
```
```  5218     unfolding has_integral_null_eq[OF True]
```
```  5219     apply safe
```
```  5220     apply (rule, rule, rule gauge_trivial, safe)
```
```  5221     unfolding setsum_content_null[OF True] True
```
```  5222     defer
```
```  5223     apply (erule_tac x=1 in allE)
```
```  5224     apply safe
```
```  5225     defer
```
```  5226     apply (rule fine_division_exists[of _ a b])
```
```  5227     apply assumption
```
```  5228     apply (erule_tac x=p in allE)
```
```  5229     unfolding setsum_content_null[OF True]
```
```  5230     apply auto
```
```  5231     done
```
```  5232 next
```
```  5233   case False
```
```  5234   note F = this[unfolded content_lt_nz[symmetric]]
```
```  5235   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
```
```  5236     (\<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)"
```
```  5237   show ?thesis
```
```  5238     apply (subst has_integral)
```
```  5239   proof safe
```
```  5240     fix e :: real
```
```  5241     assume e: "e > 0"
```
```  5242     {
```
```  5243       assume "\<forall>e>0. ?P e op <"
```
```  5244       then show "?P (e * content (cbox a b)) op \<le>"
```
```  5245         apply (erule_tac x="e * content (cbox a b)" in allE)
```
```  5246         apply (erule impE)
```
```  5247         defer
```
```  5248         apply (erule exE,rule_tac x=d in exI)
```
```  5249         using F e
```
```  5250         apply (auto simp add:field_simps)
```
```  5251         done
```
```  5252     }
```
```  5253     {
```
```  5254       assume "\<forall>e>0. ?P (e * content (cbox a b)) op \<le>"
```
```  5255       then show "?P e op <"
```
```  5256         apply (erule_tac x="e / 2 / content (cbox a b)" in allE)
```
```  5257         apply (erule impE)
```
```  5258         defer
```
```  5259         apply (erule exE,rule_tac x=d in exI)
```
```  5260         using F e
```
```  5261         apply (auto simp add: field_simps)
```
```  5262         done
```
```  5263     }
```
```  5264   qed
```
```  5265 qed
```
```  5266
```
```  5267 lemma has_integral_factor_content_real:
```
```  5268   "(f has_integral i) {a .. b::real} \<longleftrightarrow>
```
```  5269     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a .. b}  \<and> d fine p \<longrightarrow>
```
```  5270       norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a .. b} ))"
```
```  5271   unfolding box_real[symmetric]
```
```  5272   by (rule has_integral_factor_content)
```
```  5273
```
```  5274
```
```  5275 subsection \<open>Fundamental theorem of calculus.\<close>
```
```  5276
```
```  5277 lemma interval_bounds_real:
```
```  5278   fixes q b :: real
```
```  5279   assumes "a \<le> b"
```
```  5280   shows "Sup {a..b} = b"
```
```  5281     and "Inf {a..b} = a"
```
```  5282   using assms by auto
```
```  5283
```
```  5284 lemma fundamental_theorem_of_calculus:
```
```  5285   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5286   assumes "a \<le> b"
```
```  5287     and "\<forall>x\<in>{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
```
```  5288   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  5289   unfolding has_integral_factor_content box_real[symmetric]
```
```  5290 proof safe
```
```  5291   fix e :: real
```
```  5292   assume e: "e > 0"
```
```  5293   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
```
```  5294   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"
```
```  5295     using e by blast
```
```  5296   note this[OF assm,unfolded gauge_existence_lemma]
```
```  5297   from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
```
```  5298   note d=conjunctD2[OF this[rule_format],rule_format]
```
```  5299   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  5300     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
```
```  5301     apply (rule_tac x="\<lambda>x. ball x (d x)" in exI)
```
```  5302     apply safe
```
```  5303     apply (rule gauge_ball_dependent)
```
```  5304     apply rule
```
```  5305     apply (rule d(1))
```
```  5306   proof -
```
```  5307     fix p
```
```  5308     assume as: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
```
```  5309     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
```
```  5310       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]
```
```  5311       unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "\<lambda>x. x",symmetric]
```
```  5312       unfolding setsum_distrib_left
```
```  5313       defer
```
```  5314       unfolding setsum_subtractf[symmetric]
```
```  5315     proof (rule setsum_norm_le,safe)
```
```  5316       fix x k
```
```  5317       assume "(x, k) \<in> p"
```
```  5318       note xk = tagged_division_ofD(2-4)[OF as(1) this]
```
```  5319       from this(3) guess u v by (elim exE) note k=this
```
```  5320       have *: "u \<le> v"
```
```  5321         using xk unfolding k by auto
```
```  5322       have ball: "\<forall>xa\<in>k. xa \<in> ball x (d x)"
```
```  5323         using as(2)[unfolded fine_def,rule_format,OF \<open>(x,k)\<in>p\<close>,unfolded split_conv subset_eq] .
```
```  5324       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
```
```  5325         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
```
```  5326         apply (rule order_trans[OF _ norm_triangle_ineq4])
```
```  5327         apply (rule eq_refl)
```
```  5328         apply (rule arg_cong[where f=norm])
```
```  5329         unfolding scaleR_diff_left
```
```  5330         apply (auto simp add:algebra_simps)
```
```  5331         done
```
```  5332       also have "\<dots> \<le> e * norm (u - x) + e * norm (v - x)"
```
```  5333         apply (rule add_mono)
```
```  5334         apply (rule d(2)[of "x" "u",unfolded o_def])
```
```  5335         prefer 4
```
```  5336         apply (rule d(2)[of "x" "v",unfolded o_def])
```
```  5337         using ball[rule_format,of u] ball[rule_format,of v]
```
```  5338         using xk(1-2)
```
```  5339         unfolding k subset_eq
```
```  5340         apply (auto simp add:dist_real_def)
```
```  5341         done
```
```  5342       also have "\<dots> \<le> e * (Sup k - Inf k)"
```
```  5343         unfolding k interval_bounds_real[OF *]
```
```  5344         using xk(1)
```
```  5345         unfolding k
```
```  5346         by (auto simp add: dist_real_def field_simps)
```
```  5347       finally show "norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le>
```
```  5348         e * (Sup k - Inf k)"
```
```  5349         unfolding box_real k interval_bounds_real[OF *] content_real[OF *]
```
```  5350           interval_upperbound_real interval_lowerbound_real
```
```  5351           .
```
```  5352     qed
```
```  5353   qed
```
```  5354 qed
```
```  5355
```
```  5356 lemma ident_has_integral:
```
```  5357   fixes a::real
```
```  5358   assumes "a \<le> b"
```
```  5359   shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2) / 2) {a..b}"
```
```  5360 proof -
```
```  5361   have "((\<lambda>x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}"
```
```  5362     apply (rule fundamental_theorem_of_calculus [OF assms], clarify)
```
```  5363     unfolding power2_eq_square
```
```  5364     by (rule derivative_eq_intros | simp)+
```
```  5365   then show ?thesis
```
```  5366     by (simp add: field_simps)
```
```  5367 qed
```
```  5368
```
```  5369 lemma integral_ident [simp]:
```
```  5370   fixes a::real
```
```  5371   assumes "a \<le> b"
```
```  5372   shows "integral {a..b} (\<lambda>x. x) = (if a \<le> b then (b\<^sup>2 - a\<^sup>2) / 2 else 0)"
```
```  5373 using ident_has_integral integral_unique by fastforce
```
```  5374
```
```  5375 lemma ident_integrable_on:
```
```  5376   fixes a::real
```
```  5377   shows "(\<lambda>x. x) integrable_on {a..b}"
```
```  5378 by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
```
```  5379
```
```  5380
```
```  5381 subsection \<open>Taylor series expansion\<close>
```
```  5382
```
```  5383 lemma (in bounded_bilinear) setsum_prod_derivatives_has_vector_derivative:
```
```  5384   assumes "p>0"
```
```  5385   and f0: "Df 0 = f"
```
```  5386   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5387     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  5388   and g0: "Dg 0 = g"
```
```  5389   and Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5390     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  5391   and ivl: "a \<le> t" "t \<le> b"
```
```  5392   shows "((\<lambda>t. \<Sum>i<p. (-1)^i *\<^sub>R prod (Df i t) (Dg (p - Suc i) t))
```
```  5393     has_vector_derivative
```
```  5394       prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t))
```
```  5395     (at t within {a .. b})"
```
```  5396   using assms
```
```  5397 proof cases
```
```  5398   assume p: "p \<noteq> 1"
```
```  5399   define p' where "p' = p - 2"
```
```  5400   from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
```
```  5401     by (auto simp: p'_def)
```
```  5402   have *: "\<And>i. i \<le> p' \<Longrightarrow> Suc (Suc p' - i) = (Suc (Suc p') - i)"
```
```  5403     by auto
```
```  5404   let ?f = "\<lambda>i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))"
```
```  5405   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  5406     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
```
```  5407     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
```
```  5408     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
```
```  5409   also note setsum_telescope
```
```  5410   finally
```
```  5411   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  5412     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
```
```  5413     = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)"
```
```  5414     unfolding p'[symmetric]
```
```  5415     by (simp add: assms)
```
```  5416   thus ?thesis
```
```  5417     using assms
```
```  5418     by (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  5419 qed (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  5420
```
```  5421 lemma
```
```  5422   fixes f::"real\<Rightarrow>'a::banach"
```
```  5423   assumes "p>0"
```
```  5424   and f0: "Df 0 = f"
```
```  5425   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5426     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  5427   and ivl: "a \<le> b"
```
```  5428   defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"
```
```  5429   shows taylor_has_integral:
```
```  5430     "(i has_integral f b - (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
```
```  5431   and taylor_integral:
```
```  5432     "f b = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i"
```
```  5433   and taylor_integrable:
```
```  5434     "i integrable_on {a .. b}"
```
```  5435 proof goal_cases
```
```  5436   case 1
```
```  5437   interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a"
```
```  5438     by (rule bounded_bilinear_scaleR)
```
```  5439   define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
```
```  5440   define Dg where [abs_def]:
```
```  5441     "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
```
```  5442   have g0: "Dg 0 = g"
```
```  5443     using \<open>p > 0\<close>
```
```  5444     by (auto simp add: Dg_def divide_simps g_def split: if_split_asm)
```
```  5445   {
```
```  5446     fix m
```
```  5447     assume "p > Suc m"
```
```  5448     hence "p - Suc m = Suc (p - Suc (Suc m))"
```
```  5449       by auto
```
```  5450     hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
```
```  5451       by auto
```
```  5452   } note fact_eq = this
```
```  5453   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5454     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  5455     unfolding Dg_def
```
```  5456     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
```
```  5457   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
```
```  5458   from setsum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
```
```  5459       OF \<open>p > 0\<close> g0 Dg f0 Df]
```
```  5460   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  5461     (?sum has_vector_derivative
```
```  5462       g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})"
```
```  5463     by auto
```
```  5464   from fundamental_theorem_of_calculus[rule_format, OF \<open>a \<le> b\<close> deriv]
```
```  5465   have "(i has_integral ?sum b - ?sum a) {a .. b}"
```
```  5466     using atLeastatMost_empty'[simp del]
```
```  5467     by (simp add: i_def g_def Dg_def)
```
```  5468   also
```
```  5469   have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
```
```  5470     and "{..<p} \<inter> {i. p = Suc i} = {p - 1}"
```
```  5471     for p'
```
```  5472     using \<open>p > 0\<close>
```
```  5473     by (auto simp: power_mult_distrib[symmetric])
```
```  5474   then have "?sum b = f b"
```
```  5475     using Suc_pred'[OF \<open>p > 0\<close>]
```
```  5476     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
```
```  5477         cond_application_beta setsum.If_cases f0)
```
```  5478   also
```
```  5479   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
```
```  5480   proof safe
```
```  5481     fix x
```
```  5482     assume "x < p"
```
```  5483     thus "x \<in> (\<lambda>x. p - x - 1) ` {..<p}"
```
```  5484       by (auto intro!: image_eqI[where x = "p - x - 1"])
```
```  5485   qed simp
```
```  5486   from _ this
```
```  5487   have "?sum a = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)"
```
```  5488     by (rule setsum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
```
```  5489   finally show c: ?case .
```
```  5490   case 2 show ?case using c integral_unique by force
```
```  5491   case 3 show ?case using c by force
```
```  5492 qed
```
```  5493
```
```  5494
```
```  5495 subsection \<open>Attempt a systematic general set of "offset" results for components.\<close>
```
```  5496
```
```  5497 lemma gauge_modify:
```
```  5498   assumes "(\<forall>s. open s \<longrightarrow> open {x. f(x) \<in> s})" "gauge d"
```
```  5499   shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})"
```
```  5500   using assms
```
```  5501   unfolding gauge_def
```
```  5502   apply safe
```
```  5503   defer
```
```  5504   apply (erule_tac x="f x" in allE)
```
```  5505   apply (erule_tac x="d (f x)" in allE)
```
```  5506   apply auto
```
```  5507   done
```
```  5508
```
```  5509
```
```  5510 subsection \<open>Only need trivial subintervals if the interval itself is trivial.\<close>
```
```  5511
```
```  5512 lemma division_of_nontrivial:
```
```  5513   fixes s :: "'a::euclidean_space set set"
```
```  5514   assumes "s division_of (cbox a b)"
```
```  5515     and "content (cbox a b) \<noteq> 0"
```
```  5516   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of (cbox a b)"
```
```  5517   using assms(1)
```
```  5518   apply -
```
```  5519 proof (induct "card s" arbitrary: s rule: nat_less_induct)
```
```  5520   fix s::"'a set set"
```
```  5521   assume assm: "s division_of (cbox a b)"
```
```  5522     "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow>
```
```  5523       x division_of (cbox a b) \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of (cbox a b)"
```
```  5524   note s = division_ofD[OF assm(1)]
```
```  5525   let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of (cbox a b)"
```
```  5526   {
```
```  5527     presume *: "{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
```
```  5528     show ?thesis
```
```  5529       apply cases
```
```  5530       defer
```
```  5531       apply (rule *)
```
```  5532       apply assumption
```
```  5533       using assm(1)
```
```  5534       apply auto
```
```  5535       done
```
```  5536   }
```
```  5537   assume noteq: "{k \<in> s. content k \<noteq> 0} \<noteq> s"
```
```  5538   then obtain k where k: "k \<in> s" "content k = 0"
```
```  5539     by auto
```
```  5540   from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
```
```  5541   from k have "card s > 0"
```
```  5542     unfolding card_gt_0_iff using assm(1) by auto
```
```  5543   then have card: "card (s - {k}) < card s"
```
```  5544     using assm(1) k(1)
```
```  5545     apply (subst card_Diff_singleton_if)
```
```  5546     apply auto
```
```  5547     done
```
```  5548   have *: "closed (\<Union>(s - {k}))"
```
```  5549     apply (rule closed_Union)
```
```  5550     defer
```
```  5551     apply rule
```
```  5552     apply (drule DiffD1,drule s(4))
```
```  5553     using assm(1)
```
```  5554     apply auto
```
```  5555     done
```
```  5556   have "k \<subseteq> \<Union>(s - {k})"
```
```  5557     apply safe
```
```  5558     apply (rule *[unfolded closed_limpt,rule_format])
```
```  5559     unfolding islimpt_approachable
```
```  5560   proof safe
```
```  5561     fix x
```
```  5562     fix e :: real
```
```  5563     assume as: "x \<in> k" "e > 0"
```
```  5564     from k(2)[unfolded k content_eq_0] guess i ..
```
```  5565     then have i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis"
```
```  5566       using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
```
```  5567     then have xi: "x\<bullet>i = d\<bullet>i"
```
```  5568       using as unfolding k mem_box by (metis antisym)
```
```  5569     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 +
```
```  5570       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)"
```
```  5571     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e"
```
```  5572       apply (rule_tac x=y in bexI)
```
```  5573     proof
```
```  5574       have "d \<in> cbox c d"
```
```  5575         using s(3)[OF k(1)]
```
```  5576         unfolding k box_eq_empty mem_box
```
```  5577         by (fastforce simp add: not_less)
```
```  5578       then have "d \<in> cbox a b"
```
```  5579         using s(2)[OF k(1)]
```
```  5580         unfolding k
```
```  5581         by auto
```
```  5582       note di = this[unfolded mem_box,THEN bspec[where x=i]]
```
```  5583       then have xyi: "y\<bullet>i \<noteq> x\<bullet>i"
```
```  5584         unfolding y_def i xi
```
```  5585         using as(2) assms(2)[unfolded content_eq_0] i(2)
```
```  5586         by (auto elim!: ballE[of _ _ i])
```
```  5587       then show "y \<noteq> x"
```
```  5588         unfolding euclidean_eq_iff[where 'a='a] using i by auto
```
```  5589       have *: "Basis = insert i (Basis - {i})"
```
```  5590         using i by auto
```
```  5591       have "norm (y - x) < e + setsum (\<lambda>i. 0) Basis"
```
```  5592         apply (rule le_less_trans[OF norm_le_l1])
```
```  5593         apply (subst *)
```
```  5594         apply (subst setsum.insert)
```
```  5595         prefer 3
```
```  5596         apply (rule add_less_le_mono)
```
```  5597       proof -
```
```  5598         show "\<bar>(y - x) \<bullet> i\<bar> < e"
```
```  5599           using di as(2) y_def i xi by (auto simp: inner_simps)
```
```  5600         show "(\<Sum>i\<in>Basis - {i}. \<bar>(y - x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
```
```  5601           unfolding y_def by (auto simp: inner_simps)
```
```  5602       qed auto
```
```  5603       then show "dist y x < e"
```
```  5604         unfolding dist_norm by auto
```
```  5605       have "y \<notin> k"
```
```  5606         unfolding k mem_box
```
```  5607         apply rule
```
```  5608         apply (erule_tac x=i in ballE)
```
```  5609         using xyi k i xi
```
```  5610         apply auto
```
```  5611         done
```
```  5612       moreover
```
```  5613       have "y \<in> \<Union>s"
```
```  5614         using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i
```
```  5615         unfolding s mem_box y_def
```
```  5616         by (auto simp: field_simps elim!: ballE[of _ _ i])
```
```  5617       ultimately
```
```  5618       show "y \<in> \<Union>(s - {k})" by auto
```
```  5619     qed
```
```  5620   qed
```
```  5621   then have "\<Union>(s - {k}) = cbox a b"
```
```  5622     unfolding s(6)[symmetric] by auto
```
```  5623   then have  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of (cbox a b)"
```
```  5624     apply -
```
```  5625     apply (rule assm(2)[rule_format,OF card refl])
```
```  5626     apply (rule division_ofI)
```
```  5627     defer
```
```  5628     apply (rule_tac[1-4] s)
```
```  5629     using assm(1)
```
```  5630     apply auto
```
```  5631     done
```
```  5632   moreover
```
```  5633   have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}"
```
```  5634     using k by auto
```
```  5635   ultimately show ?thesis by auto
```
```  5636 qed
```
```  5637
```
```  5638
```
```  5639 subsection \<open>Integrability on subintervals.\<close>
```
```  5640
```
```  5641 lemma operative_integrable:
```
```  5642   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5643   shows "comm_monoid.operative op \<and> True (\<lambda>i. f integrable_on i)"
```
```  5644   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  5645   apply safe
```
```  5646   apply (subst integrable_on_def)
```
```  5647   unfolding has_integral_null_eq
```
```  5648   apply (rule, rule refl)
```
```  5649   apply (rule, assumption, assumption)+
```
```  5650   unfolding integrable_on_def
```
```  5651   by (auto intro!: has_integral_split)
```
```  5652
```
```  5653 lemma integrable_subinterval:
```
```  5654   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5655   assumes "f integrable_on cbox a b"
```
```  5656     and "cbox c d \<subseteq> cbox a b"
```
```  5657   shows "f integrable_on cbox c d"
```
```  5658   apply (cases "cbox c d = {}")
```
```  5659   defer
```
```  5660   apply (rule partial_division_extend_1[OF assms(2)],assumption)
```
```  5661   using comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable,symmetric,of _ _ _ f] assms(1)
```
```  5662   apply (auto simp: comm_monoid_set_F_and)
```
```  5663   done
```
```  5664
```
```  5665 lemma integrable_subinterval_real:
```
```  5666   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5667   assumes "f integrable_on {a .. b}"
```
```  5668     and "{c .. d} \<subseteq> {a .. b}"
```
```  5669   shows "f integrable_on {c .. d}"
```
```  5670   by (metis assms(1) assms(2) box_real(2) integrable_subinterval)
```
```  5671
```
```  5672
```
```  5673 subsection \<open>Combining adjacent intervals in 1 dimension.\<close>
```
```  5674
```
```  5675 lemma has_integral_combine:
```
```  5676   fixes a b c :: real
```
```  5677   assumes "a \<le> c"
```
```  5678     and "c \<le> b"
```
```  5679     and "(f has_integral i) {a .. c}"
```
```  5680     and "(f has_integral (j::'a::banach)) {c .. b}"
```
```  5681   shows "(f has_integral (i + j)) {a .. b}"
```
```  5682 proof -
```
```  5683   interpret comm_monoid "lift_option plus" "Some (0::'a)"
```
```  5684     by (rule comm_monoid_lift_option)
```
```  5685       (rule add.comm_monoid_axioms)
```
```  5686   note operative_integral [of f, unfolded operative_1_le]
```
```  5687   note conjunctD2 [OF this, rule_format]
```
```  5688   note * = this(2) [OF conjI [OF assms(1-2)],
```
```  5689     unfolded if_P [OF assms(3)]]
```
```  5690   then have "f integrable_on cbox a b"
```
```  5691     apply -
```
```  5692     apply (rule ccontr)
```
```  5693     apply (subst(asm) if_P)
```
```  5694     defer
```
```  5695     apply (subst(asm) if_P)
```
```  5696     using assms(3-)
```
```  5697     apply auto
```
```  5698     done
```
```  5699   with *
```
```  5700   show ?thesis
```
```  5701     apply -
```
```  5702     apply (subst(asm) if_P)
```
```  5703     defer
```
```  5704     apply (subst(asm) if_P)
```
```  5705     defer
```
```  5706     apply (subst(asm) if_P)
```
```  5707     using assms(3-)
```
```  5708     apply (auto simp add: integrable_on_def integral_unique)
```
```  5709     done
```
```  5710 qed
```
```  5711
```
```  5712 lemma integral_combine:
```
```  5713   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5714   assumes "a \<le> c"
```
```  5715     and "c \<le> b"
```
```  5716     and "f integrable_on {a .. b}"
```
```  5717   shows "integral {a .. c} f + integral {c .. b} f = integral {a .. b} f"
```
```  5718   apply (rule integral_unique[symmetric])
```
```  5719   apply (rule has_integral_combine[OF assms(1-2)])
```
```  5720   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)
```
```  5721   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)
```
```  5722
```
```  5723 lemma integrable_combine:
```
```  5724   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5725   assumes "a \<le> c"
```
```  5726     and "c \<le> b"
```
```  5727     and "f integrable_on {a .. c}"
```
```  5728     and "f integrable_on {c .. b}"
```
```  5729   shows "f integrable_on {a .. b}"
```
```  5730   using assms
```
```  5731   unfolding integrable_on_def
```
```  5732   by (fastforce intro!:has_integral_combine)
```
```  5733
```
```  5734
```
```  5735 subsection \<open>Reduce integrability to "local" integrability.\<close>
```
```  5736
```
```  5737 lemma integrable_on_little_subintervals:
```
```  5738   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  5739   assumes "\<forall>x\<in>cbox a b. \<exists>d>0. \<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
```
```  5740     f integrable_on cbox u v"
```
```  5741   shows "f integrable_on cbox a b"
```
```  5742 proof -
```
```  5743   have "\<forall>x. \<exists>d. x\<in>cbox a b \<longrightarrow> d>0 \<and> (\<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
```
```  5744     f integrable_on cbox u v)"
```
```  5745     using assms by auto
```
```  5746   note this[unfolded gauge_existence_lemma]
```
```  5747   from choice[OF this] guess d .. note d=this[rule_format]
```
```  5748   guess p
```
```  5749     apply (rule fine_division_exists[OF gauge_ball_dependent,of d a b])
```
```  5750     using d
```
```  5751     by auto
```
```  5752   note p=this(1-2)
```
```  5753   note division_of_tagged_division[OF this(1)]
```
```  5754   note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable, OF this, symmetric, of f]
```
```  5755   show ?thesis
```
```  5756     unfolding * comm_monoid_set_F_and
```
```  5757     apply safe
```
```  5758     unfolding snd_conv
```
```  5759   proof -
```
```  5760     fix x k
```
```  5761     assume "(x, k) \<in> p"
```
```  5762     note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
```
```  5763     then show "f integrable_on k"
```
```  5764       apply safe
```
```  5765       apply (rule d[THEN conjunct2,rule_format,of x])
```
```  5766       apply (auto intro: order.trans)
```
```  5767       done
```
```  5768   qed
```
```  5769 qed
```
```  5770
```
```  5771
```
```  5772 subsection \<open>Second FTC or existence of antiderivative.\<close>
```
```  5773
```
```  5774 lemma integrable_const[intro]: "(\<lambda>x. c) integrable_on cbox a b"
```
```  5775   unfolding integrable_on_def
```
```  5776   apply rule
```
```  5777   apply (rule has_integral_const)
```
```  5778   done
```
```  5779
```
```  5780 lemma integral_has_vector_derivative_continuous_at:
```
```  5781   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5782   assumes f: "f integrable_on {a..b}"
```
```  5783       and x: "x \<in> {a..b}"
```
```  5784       and fx: "continuous (at x within {a..b}) f"
```
```  5785   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
```
```  5786 proof -
```
```  5787   let ?I = "\<lambda>a b. integral {a..b} f"
```
```  5788   { fix e::real
```
```  5789     assume "e > 0"
```
```  5790     obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {a..b}; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> norm(f x' - f x) \<le> e"
```
```  5791       using \<open>e>0\<close> fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
```
```  5792     have "norm (integral {a..y} f - integral {a..x} f - (y - x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
```
```  5793            if y: "y \<in> {a..b}" and yx: "\<bar>y - x\<bar> < d" for y
```
```  5794     proof (cases "y < x")
```
```  5795       case False
```
```  5796       have "f integrable_on {a..y}"
```
```  5797         using f y by (simp add: integrable_subinterval_real)
```
```  5798       then have Idiff: "?I a y - ?I a x = ?I x y"
```
```  5799         using False x by (simp add: algebra_simps integral_combine)
```
```  5800       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {x..y} f - (y - x) *\<^sub>R f x) {x..y}"
```
```  5801         apply (rule has_integral_sub)
```
```  5802         using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
```
```  5803         using has_integral_const_real [of "f x" x y] False
```
```  5804         apply (simp add: )
```
```  5805         done
```
```  5806       show ?thesis
```
```  5807         using False
```
```  5808         apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
```
```  5809         apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
```
```  5810         using yx False d x y \<open>e>0\<close> apply (auto simp add: Idiff fux_int)
```
```  5811         done
```
```  5812     next
```
```  5813       case True
```
```  5814       have "f integrable_on {a..x}"
```
```  5815         using f x by (simp add: integrable_subinterval_real)
```
```  5816       then have Idiff: "?I a x - ?I a y = ?I y x"
```
```  5817         using True x y by (simp add: algebra_simps integral_combine)
```
```  5818       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}"
```
```  5819         apply (rule has_integral_sub)
```
```  5820         using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
```
```  5821         using has_integral_const_real [of "f x" y x] True
```
```  5822         apply (simp add: )
```
```  5823         done
```
```  5824       have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
```
```  5825         using True
```
```  5826         apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
```
```  5827         apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
```
```  5828         using yx True d x y \<open>e>0\<close> apply (auto simp add: Idiff fux_int)
```
```  5829         done
```
```  5830       then show ?thesis
```
```  5831         by (simp add: algebra_simps norm_minus_commute)
```
```  5832     qed
```
```  5833     then have "\<exists>d>0. \<forall>y\<in>{a..b}. \<bar>y - x\<bar> < d \<longrightarrow> norm (integral {a..y} f - integral {a..x} f - (y - x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
```
```  5834       using \<open>d>0\<close> by blast
```
```  5835   }
```
```  5836   then show ?thesis
```
```  5837     by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
```
```  5838 qed
```
```  5839
```
```  5840 lemma integral_has_vector_derivative:
```
```  5841   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5842   assumes "continuous_on {a .. b} f"
```
```  5843     and "x \<in> {a .. b}"
```
```  5844   shows "((\<lambda>u. integral {a .. u} f) has_vector_derivative f(x)) (at x within {a .. b})"
```
```  5845 apply (rule integral_has_vector_derivative_continuous_at [OF integrable_continuous_real])
```
```  5846 using assms
```
```  5847 apply (auto simp: continuous_on_eq_continuous_within)
```
```  5848 done
```
```  5849
```
```  5850 lemma antiderivative_continuous:
```
```  5851   fixes q b :: real
```
```  5852   assumes "continuous_on {a .. b} f"
```
```  5853   obtains g where "\<forall>x\<in>{a .. b}. (g has_vector_derivative (f x::_::banach)) (at x within {a .. b})"
```
```  5854   apply (rule that)
```
```  5855   apply rule
```
```  5856   using integral_has_vector_derivative[OF assms]
```
```  5857   apply auto
```
```  5858   done
```
```  5859
```
```  5860
```
```  5861 subsection \<open>Combined fundamental theorem of calculus.\<close>
```
```  5862
```
```  5863 lemma antiderivative_integral_continuous:
```
```  5864   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  5865   assumes "continuous_on {a .. b} f"
```
```  5866   obtains g where "\<forall>u\<in>{a .. b}. \<forall>v \<in> {a .. b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u .. v}"
```
```  5867 proof -
```
```  5868   from antiderivative_continuous[OF assms] guess g . note g=this
```
```  5869   show ?thesis
```
```  5870     apply (rule that[of g])
```
```  5871     apply safe
```
```  5872   proof goal_cases
```
```  5873     case prems: (1 u v)
```
```  5874     have "\<forall>x\<in>cbox u v. (g has_vector_derivative f x) (at x within cbox u v)"
```
```  5875       apply rule
```
```  5876       apply (rule has_vector_derivative_within_subset)
```
```  5877       apply (rule g[rule_format])
```
```  5878       using prems(1,2)
```
```  5879       apply auto
```
```  5880       done
```
```  5881     then show ?case
```
```  5882       using fundamental_theorem_of_calculus[OF prems(3), of g f] by auto
```
```  5883   qed
```
```  5884 qed
```
```  5885
```
```  5886
```
```  5887 subsection \<open>General "twiddling" for interval-to-interval function image.\<close>
```
```  5888
```
```  5889 lemma has_integral_twiddle:
```
```  5890   assumes "0 < r"
```
```  5891     and "\<forall>x. h(g x) = x"
```
```  5892     and "\<forall>x. g(h x) = x"
```
```  5893     and contg: "\<And>x. continuous (at x) g"
```
```  5894     and "\<forall>u v. \<exists>w z. g ` cbox u v = cbox w z"
```
```  5895     and "\<forall>u v. \<exists>w z. h ` cbox u v = cbox w z"
```
```  5896     and "\<forall>u v. content(g ` cbox u v) = r * content (cbox u v)"
```
```  5897     and "(f has_integral i) (cbox a b)"
```
```  5898   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
```
```  5899 proof -
```
```  5900   show ?thesis when *: "cbox a b \<noteq> {} \<Longrightarrow> ?thesis"
```
```  5901     apply cases
```
```  5902     defer
```
```  5903     apply (rule *)
```
```  5904     apply assumption
```
```  5905   proof goal_cases
```
```  5906     case prems: 1
```
```  5907     then show ?thesis
```
```  5908       unfolding prems assms(8)[unfolded prems has_integral_empty_eq] by auto
```
```  5909   qed
```
```  5910   assume "cbox a b \<noteq> {}"
```
```  5911   from assms(6)[rule_format,of a b] guess w z by (elim exE) note wz=this
```
```  5912   have inj: "inj g" "inj h"
```
```  5913     unfolding inj_on_def
```
```  5914     apply safe
```
```  5915     apply(rule_tac[!] ccontr)
```
```  5916     using assms(2)
```
```  5917     apply(erule_tac x=x in allE)
```
```  5918     using assms(2)
```
```  5919     apply(erule_tac x=y in allE)
```
```  5920     defer
```
```  5921     using assms(3)
```
```  5922     apply (erule_tac x=x in allE)
```
```  5923     using assms(3)
```
```  5924     apply(erule_tac x=y in allE)
```
```  5925     apply auto
```
```  5926     done
```
```  5927   show ?thesis
```
```  5928     unfolding has_integral_def has_integral_compact_interval_def
```
```  5929     apply (subst if_P)
```
```  5930     apply rule
```
```  5931     apply rule
```
```  5932     apply (rule wz)
```
```  5933   proof safe
```
```  5934     fix e :: real
```
```  5935     assume e: "e > 0"
```
```  5936     with assms(1) have "e * r > 0" by simp
```
`  5937     from assms(`