src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
 author nipkow Mon Oct 17 17:33:07 2016 +0200 (2016-10-17) changeset 64272 f76b6dda2e56 parent 64267 b9a1486e79be child 64287 d85d88722745 permissions -rw-r--r--
setprod -> prod
```     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 Tagged_Division
```
```    10 begin
```
```    11
```
```    12 (* BEGIN MOVE *)
```
```    13 lemma swap_continuous:
```
```    14   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
```
```    15     shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
```
```    16 proof -
```
```    17   have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
```
```    18     by auto
```
```    19   then show ?thesis
```
```    20     apply (rule ssubst)
```
```    21     apply (rule continuous_on_compose)
```
```    22     apply (simp add: split_def)
```
```    23     apply (rule continuous_intros | simp add: assms)+
```
```    24     done
```
```    25 qed
```
```    26
```
```    27
```
```    28 lemma norm_minus2: "norm (x1-x2, y1-y2) = norm (x2-x1, y2-y1)"
```
```    29   by (simp add: norm_minus_eqI)
```
```    30
```
```    31 lemma norm_diff2: "\<lbrakk>y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \<le> e1; norm(y2 - x2) \<le> e2\<rbrakk>
```
```    32   \<Longrightarrow> norm(y - x) \<le> e"
```
```    33   using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
```
```    34   by (simp add: add_diff_add)
```
```    35
```
```    36 lemma setcomp_dot1: "{z. P (z \<bullet> (i,0))} = {(x,y). P(x \<bullet> i)}"
```
```    37   by auto
```
```    38
```
```    39 lemma setcomp_dot2: "{z. P (z \<bullet> (0,i))} = {(x,y). P(y \<bullet> i)}"
```
```    40   by auto
```
```    41
```
```    42 lemma Sigma_Int_Paircomp1: "(Sigma A B) \<inter> {(x, y). P x} = Sigma (A \<inter> {x. P x}) B"
```
```    43   by blast
```
```    44
```
```    45 lemma Sigma_Int_Paircomp2: "(Sigma A B) \<inter> {(x, y). P y} = Sigma A (\<lambda>z. B z \<inter> {y. P y})"
```
```    46   by blast
```
```    47
```
```    48 lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
```
```    49   using nonempty_Basis
```
```    50   by (fastforce simp add: set_eq_iff mem_box)
```
```    51 (* END MOVE *)
```
```    52
```
```    53 subsection \<open>Content (length, area, volume...) of an interval.\<close>
```
```    54
```
```    55 abbreviation content :: "'a::euclidean_space set \<Rightarrow> real"
```
```    56   where "content s \<equiv> measure lborel s"
```
```    57
```
```    58 lemma content_cbox_cases:
```
```    59   "content (cbox a b) = (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then prod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
```
```    60   by (simp add: measure_lborel_cbox_eq inner_diff)
```
```    61
```
```    62 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)"
```
```    63   unfolding content_cbox_cases by simp
```
```    64
```
```    65 lemma content_cbox': "cbox a b \<noteq> {} \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
```
```    66   by (simp add: box_ne_empty inner_diff)
```
```    67
```
```    68 lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
```
```    69   by simp
```
```    70
```
```    71 lemma abs_eq_content: "\<bar>y - x\<bar> = (if x\<le>y then content {x .. y} else content {y..x})"
```
```    72   by (auto simp: content_real)
```
```    73
```
```    74 lemma content_singleton: "content {a} = 0"
```
```    75   by simp
```
```    76
```
```    77 lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1"
```
```    78   by simp
```
```    79
```
```    80 lemma content_pos_le[intro]: "0 \<le> content (cbox a b)"
```
```    81   by simp
```
```    82
```
```    83 corollary content_nonneg [simp]: "~ content (cbox a b) < 0"
```
```    84   using not_le by blast
```
```    85
```
```    86 lemma content_pos_lt: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> 0 < content (cbox a b)"
```
```    87   by (auto simp: less_imp_le inner_diff box_eq_empty intro!: prod_pos)
```
```    88
```
```    89 lemma content_eq_0: "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
```
```    90   by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl)
```
```    91
```
```    92 lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
```
```    93   unfolding content_eq_0 interior_cbox box_eq_empty by auto
```
```    94
```
```    95 lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
```
```    96   by (auto simp add: content_cbox_cases less_le prod_nonneg)
```
```    97
```
```    98 lemma content_empty [simp]: "content {} = 0"
```
```    99   by simp
```
```   100
```
```   101 lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
```
```   102   by (simp add: content_real)
```
```   103
```
```   104 lemma content_subset: "cbox a b \<subseteq> cbox c d \<Longrightarrow> content (cbox a b) \<le> content (cbox c d)"
```
```   105   unfolding measure_def
```
```   106   by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq)
```
```   107
```
```   108 lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
```
```   109   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
```
```   110
```
```   111 lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
```
```   112   unfolding measure_lborel_cbox_eq Basis_prod_def
```
```   113   apply (subst prod.union_disjoint)
```
```   114   apply (auto simp: bex_Un ball_Un)
```
```   115   apply (subst (1 2) prod.reindex_nontrivial)
```
```   116   apply auto
```
```   117   done
```
```   118
```
```   119 lemma content_cbox_pair_eq0_D:
```
```   120    "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
```
```   121   by (simp add: content_Pair)
```
```   122
```
```   123 lemma content_0_subset: "content(cbox a b) = 0 \<Longrightarrow> s \<subseteq> cbox a b \<Longrightarrow> content s = 0"
```
```   124   using emeasure_mono[of s "cbox a b" lborel]
```
```   125   by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq)
```
```   126
```
```   127 lemma content_split:
```
```   128   fixes a :: "'a::euclidean_space"
```
```   129   assumes "k \<in> Basis"
```
```   130   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})"
```
```   131   -- \<open>Prove using measure theory\<close>
```
```   132 proof cases
```
```   133   note simps = interval_split[OF assms] content_cbox_cases
```
```   134   have *: "Basis = insert k (Basis - {k})" "\<And>x. finite (Basis-{x})" "\<And>x. x\<notin>Basis-{x}"
```
```   135     using assms by auto
```
```   136   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))"
```
```   137     "(\<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)"
```
```   138     apply (subst *(1))
```
```   139     defer
```
```   140     apply (subst *(1))
```
```   141     unfolding prod.insert[OF *(2-)]
```
```   142     apply auto
```
```   143     done
```
```   144   assume as: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
```
```   145   moreover
```
```   146   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
```
```   147     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)"
```
```   148     by  (auto simp add: field_simps)
```
```   149   moreover
```
```   150   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)) =
```
```   151       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
```
```   152     "(\<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)) =
```
```   153       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
```
```   154     by (auto intro!: prod.cong)
```
```   155   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
```
```   156     unfolding not_le
```
```   157     using as[unfolded ,rule_format,of k] assms
```
```   158     by auto
```
```   159   ultimately show ?thesis
```
```   160     using assms
```
```   161     unfolding simps **
```
```   162     unfolding *(1)[of "\<lambda>i x. b\<bullet>i - x"] *(1)[of "\<lambda>i x. x - a\<bullet>i"]
```
```   163     unfolding *(2)
```
```   164     by auto
```
```   165 next
```
```   166   assume "\<not> (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
```
```   167   then have "cbox a b = {}"
```
```   168     unfolding box_eq_empty by (auto simp: not_le)
```
```   169   then show ?thesis
```
```   170     by (auto simp: not_le)
```
```   171 qed
```
```   172
```
```   173 lemma division_of_content_0:
```
```   174   assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
```
```   175   shows "\<forall>k\<in>d. content k = 0"
```
```   176   unfolding forall_in_division[OF assms(2)]
```
```   177   by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
```
```   178
```
```   179 lemma sum_content_null:
```
```   180   assumes "content (cbox a b) = 0"
```
```   181     and "p tagged_division_of (cbox a b)"
```
```   182   shows "sum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
```
```   183 proof (rule sum.neutral, rule)
```
```   184   fix y
```
```   185   assume y: "y \<in> p"
```
```   186   obtain x k where xk: "y = (x, k)"
```
```   187     using surj_pair[of y] by blast
```
```   188   note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
```
```   189   from this(2) obtain c d where k: "k = cbox c d" by blast
```
```   190   have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
```
```   191     unfolding xk by auto
```
```   192   also have "\<dots> = 0"
```
```   193     using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
```
```   194     unfolding assms(1) k
```
```   195     by auto
```
```   196   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
```
```   197 qed
```
```   198
```
```   199 lemma operative_content[intro]: "add.operative content"
```
```   200   by (force simp add: add.operative_def content_split[symmetric] content_eq_0_interior)
```
```   201
```
```   202 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> sum content d = content (cbox a b)"
```
```   203   by (metis operative_content sum.operative_division)
```
```   204
```
```   205 lemma additive_content_tagged_division:
```
```   206   "d tagged_division_of (cbox a b) \<Longrightarrow> sum (\<lambda>(x,l). content l) d = content (cbox a b)"
```
```   207   unfolding sum.operative_tagged_division[OF operative_content, symmetric] by blast
```
```   208
```
```   209 lemma content_real_eq_0: "content {a .. b::real} = 0 \<longleftrightarrow> a \<ge> b"
```
```   210   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
```
```   211
```
```   212 lemma property_empty_interval: "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b) \<Longrightarrow> P {}"
```
```   213   using content_empty unfolding empty_as_interval by auto
```
```   214
```
```   215 subsection \<open>Gauge integral\<close>
```
```   216
```
```   217 text \<open>Case distinction to define it first on compact intervals first, then use a limit. This is only
```
```   218 much later unified. In Fremlin: Measure Theory, Volume 4I this is generalized using residual sets.\<close>
```
```   219
```
```   220 definition has_integral :: "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
```
```   221   (infixr "has'_integral" 46)
```
```   222   where "(f has_integral I) s \<longleftrightarrow>
```
```   223     (if \<exists>a b. s = cbox a b
```
```   224       then ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s)
```
```   225       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```   226         (\<exists>z. ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R (if x \<in> s then f x else 0)) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and>
```
```   227           norm (z - I) < e)))"
```
```   228
```
```   229 lemma has_integral_cbox:
```
```   230   "(f has_integral I) (cbox a b) \<longleftrightarrow> ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
```
```   231   by (auto simp add: has_integral_def)
```
```   232
```
```   233 lemma has_integral:
```
```   234   "(f has_integral y) (cbox a b) \<longleftrightarrow>
```
```   235     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```   236       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```   237         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```   238   by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)
```
```   239
```
```   240 lemma has_integral_real:
```
```   241   "(f has_integral y) {a .. b::real} \<longleftrightarrow>
```
```   242     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```   243       (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
```
```   244         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
```
```   245   unfolding box_real[symmetric]
```
```   246   by (rule has_integral)
```
```   247
```
```   248 lemma has_integralD[dest]:
```
```   249   assumes "(f has_integral y) (cbox a b)"
```
```   250     and "e > 0"
```
```   251   obtains d
```
```   252     where "gauge d"
```
```   253       and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
```
```   254         norm ((\<Sum>(x,k)\<in>p. content k *\<^sub>R f x) - y) < e"
```
```   255   using assms unfolding has_integral by auto
```
```   256
```
```   257 lemma has_integral_alt:
```
```   258   "(f has_integral y) i \<longleftrightarrow>
```
```   259     (if \<exists>a b. i = cbox a b
```
```   260      then (f has_integral y) i
```
```   261      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```   262       (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
```
```   263   by (subst has_integral_def) (auto simp add: has_integral_cbox)
```
```   264
```
```   265 lemma has_integral_altD:
```
```   266   assumes "(f has_integral y) i"
```
```   267     and "\<not> (\<exists>a b. i = cbox a b)"
```
```   268     and "e>0"
```
```   269   obtains B where "B > 0"
```
```   270     and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```   271       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
```
```   272   using assms has_integral_alt[of f y i] by auto
```
```   273
```
```   274 definition integrable_on (infixr "integrable'_on" 46)
```
```   275   where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
```
```   276
```
```   277 definition "integral i f = (SOME y. (f has_integral y) i \<or> ~ f integrable_on i \<and> y=0)"
```
```   278
```
```   279 lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
```
```   280   unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
```
```   281
```
```   282 lemma not_integrable_integral: "~ f integrable_on i \<Longrightarrow> integral i f = 0"
```
```   283   unfolding integrable_on_def integral_def by blast
```
```   284
```
```   285 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
```
```   286   unfolding integrable_on_def by auto
```
```   287
```
```   288 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
```
```   289   by auto
```
```   290
```
```   291 subsection \<open>Basic theorems about integrals.\<close>
```
```   292
```
```   293 lemma has_integral_unique:
```
```   294   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```   295   assumes "(f has_integral k1) i"
```
```   296     and "(f has_integral k2) i"
```
```   297   shows "k1 = k2"
```
```   298 proof (rule ccontr)
```
```   299   let ?e = "norm (k1 - k2) / 2"
```
```   300   assume as: "k1 \<noteq> k2"
```
```   301   then have e: "?e > 0"
```
```   302     by auto
```
```   303   have lem: "(f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 = k2"
```
```   304     for f :: "'n \<Rightarrow> 'a" and a b k1 k2
```
```   305     by (auto simp: has_integral_cbox intro: tendsto_unique[OF division_filter_not_empty])
```
```   306   {
```
```   307     presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
```
```   308     then show False
```
```   309       using as assms lem by blast
```
```   310   }
```
```   311   assume as: "\<not> (\<exists>a b. i = cbox a b)"
```
```   312   obtain B1 where B1:
```
```   313       "0 < B1"
```
```   314       "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```   315         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```   316           norm (z - k1) < norm (k1 - k2) / 2"
```
```   317     by (rule has_integral_altD[OF assms(1) as,OF e]) blast
```
```   318   obtain B2 where B2:
```
```   319       "0 < B2"
```
```   320       "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
```
```   321         \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
```
```   322           norm (z - k2) < norm (k1 - k2) / 2"
```
```   323     by (rule has_integral_altD[OF assms(2) as,OF e]) blast
```
```   324   have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
```
```   325     apply (rule bounded_subset_cbox)
```
```   326     using bounded_Un bounded_ball
```
```   327     apply auto
```
```   328     done
```
```   329   then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
```
```   330     by blast
```
```   331   obtain w where w:
```
```   332     "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
```
```   333     "norm (w - k1) < norm (k1 - k2) / 2"
```
```   334     using B1(2)[OF ab(1)] by blast
```
```   335   obtain z where z:
```
```   336     "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
```
```   337     "norm (z - k2) < norm (k1 - k2) / 2"
```
```   338     using B2(2)[OF ab(2)] by blast
```
```   339   have "z = w"
```
```   340     using lem[OF w(1) z(1)] by auto
```
```   341   then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
```
```   342     using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
```
```   343     by (auto simp add: norm_minus_commute)
```
```   344   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
```
```   345     apply (rule add_strict_mono)
```
```   346     apply (rule_tac[!] z(2) w(2))
```
```   347     done
```
```   348   finally show False by auto
```
```   349 qed
```
```   350
```
```   351 lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
```
```   352   unfolding integral_def
```
```   353   by (rule some_equality) (auto intro: has_integral_unique)
```
```   354
```
```   355 lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> ~ f integrable_on k \<and> y=0"
```
```   356   unfolding integral_def integrable_on_def
```
```   357   apply (erule subst)
```
```   358   apply (rule someI_ex)
```
```   359   by blast
```
```   360
```
```   361
```
```   362 lemma has_integral_const [intro]:
```
```   363   fixes a b :: "'a::euclidean_space"
```
```   364   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
```
```   365   using eventually_division_filter_tagged_division[of "cbox a b"]
```
```   366      additive_content_tagged_division[of _ a b]
```
```   367   by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric]
```
```   368            elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const])
```
```   369
```
```   370 lemma has_integral_const_real [intro]:
```
```   371   fixes a b :: real
```
```   372   shows "((\<lambda>x. c) has_integral (content {a .. b} *\<^sub>R c)) {a .. b}"
```
```   373   by (metis box_real(2) has_integral_const)
```
```   374
```
```   375 lemma integral_const [simp]:
```
```   376   fixes a b :: "'a::euclidean_space"
```
```   377   shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
```
```   378   by (rule integral_unique) (rule has_integral_const)
```
```   379
```
```   380 lemma integral_const_real [simp]:
```
```   381   fixes a b :: real
```
```   382   shows "integral {a .. b} (\<lambda>x. c) = content {a .. b} *\<^sub>R c"
```
```   383   by (metis box_real(2) integral_const)
```
```   384
```
```   385 lemma has_integral_is_0:
```
```   386   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```   387   assumes "\<forall>x\<in>s. f x = 0"
```
```   388   shows "(f has_integral 0) s"
```
```   389 proof -
```
```   390   have lem: "(\<forall>x\<in>cbox a b. f x = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)" for a  b and f :: "'n \<Rightarrow> 'a"
```
```   391     unfolding has_integral_cbox
```
```   392     using eventually_division_filter_tagged_division[of "cbox a b"]
```
```   393     by (subst tendsto_cong[where g="\<lambda>_. 0"])
```
```   394        (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval)
```
```   395   {
```
```   396     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```   397     with assms lem show ?thesis
```
```   398       by blast
```
```   399   }
```
```   400   have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
```
```   401     apply (rule ext)
```
```   402     using assms
```
```   403     apply auto
```
```   404     done
```
```   405   assume "\<not> (\<exists>a b. s = cbox a b)"
```
```   406   then show ?thesis
```
```   407     using lem
```
```   408     by (subst has_integral_alt) (force simp add: *)
```
```   409 qed
```
```   410
```
```   411 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
```
```   412   by (rule has_integral_is_0) auto
```
```   413
```
```   414 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
```
```   415   using has_integral_unique[OF has_integral_0] by auto
```
```   416
```
```   417 lemma has_integral_linear:
```
```   418   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```   419   assumes "(f has_integral y) s"
```
```   420     and "bounded_linear h"
```
```   421   shows "((h \<circ> f) has_integral ((h y))) s"
```
```   422 proof -
```
```   423   interpret bounded_linear h
```
```   424     using assms(2) .
```
```   425   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
```
```   426     by blast
```
```   427   have lem: "\<And>a b y f::'n\<Rightarrow>'a. (f has_integral y) (cbox a b) \<Longrightarrow> ((h \<circ> f) has_integral h y) (cbox a b)"
```
```   428     unfolding has_integral_cbox by (drule tendsto) (simp add: sum scaleR split_beta')
```
```   429   {
```
```   430     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```   431     then show ?thesis
```
```   432       using assms(1) lem by blast
```
```   433   }
```
```   434   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```   435   then show ?thesis
```
```   436   proof (subst has_integral_alt, clarsimp)
```
```   437     fix e :: real
```
```   438     assume e: "e > 0"
```
```   439     have *: "0 < e/B" using e B(1) by simp
```
```   440     obtain M where M:
```
```   441       "M > 0"
```
```   442       "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
```
```   443         \<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"
```
```   444       using has_integral_altD[OF assms(1) as *] by blast
```
```   445     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```   446       (\<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)"
```
```   447     proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
```
```   448       case prems: (1 a b)
```
```   449       obtain z where z:
```
```   450         "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
```
```   451         "norm (z - y) < e / B"
```
```   452         using M(2)[OF prems(1)] by blast
```
```   453       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)"
```
```   454         using zero by auto
```
```   455       show ?case
```
```   456         apply (rule_tac x="h z" in exI)
```
```   457         apply (simp add: * lem[OF z(1)])
```
```   458         apply (metis B diff le_less_trans pos_less_divide_eq z(2))
```
```   459         done
```
```   460     qed
```
```   461   qed
```
```   462 qed
```
```   463
```
```   464 lemma has_integral_scaleR_left:
```
```   465   "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
```
```   466   using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
```
```   467
```
```   468 lemma has_integral_mult_left:
```
```   469   fixes c :: "_ :: real_normed_algebra"
```
```   470   shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
```
```   471   using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
```
```   472
```
```   473 text\<open>The case analysis eliminates the condition @{term "f integrable_on s"} at the cost
```
```   474      of the type class constraint \<open>division_ring\<close>\<close>
```
```   475 corollary integral_mult_left [simp]:
```
```   476   fixes c:: "'a::{real_normed_algebra,division_ring}"
```
```   477   shows "integral s (\<lambda>x. f x * c) = integral s f * c"
```
```   478 proof (cases "f integrable_on s \<or> c = 0")
```
```   479   case True then show ?thesis
```
```   480     by (force intro: has_integral_mult_left)
```
```   481 next
```
```   482   case False then have "~ (\<lambda>x. f x * c) integrable_on s"
```
```   483     using has_integral_mult_left [of "(\<lambda>x. f x * c)" _ s "inverse c"]
```
```   484     by (force simp add: mult.assoc)
```
```   485   with False show ?thesis by (simp add: not_integrable_integral)
```
```   486 qed
```
```   487
```
```   488 corollary integral_mult_right [simp]:
```
```   489   fixes c:: "'a::{real_normed_field}"
```
```   490   shows "integral s (\<lambda>x. c * f x) = c * integral s f"
```
```   491 by (simp add: mult.commute [of c])
```
```   492
```
```   493 corollary integral_divide [simp]:
```
```   494   fixes z :: "'a::real_normed_field"
```
```   495   shows "integral S (\<lambda>x. f x / z) = integral S (\<lambda>x. f x) / z"
```
```   496 using integral_mult_left [of S f "inverse z"]
```
```   497   by (simp add: divide_inverse_commute)
```
```   498
```
```   499 lemma has_integral_mult_right:
```
```   500   fixes c :: "'a :: real_normed_algebra"
```
```   501   shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
```
```   502   using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
```
```   503
```
```   504 lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
```
```   505   unfolding o_def[symmetric]
```
```   506   by (metis has_integral_linear bounded_linear_scaleR_right)
```
```   507
```
```   508 lemma has_integral_cmult_real:
```
```   509   fixes c :: real
```
```   510   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
```
```   511   shows "((\<lambda>x. c * f x) has_integral c * x) A"
```
```   512 proof (cases "c = 0")
```
```   513   case True
```
```   514   then show ?thesis by simp
```
```   515 next
```
```   516   case False
```
```   517   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
```
```   518     unfolding real_scaleR_def .
```
```   519 qed
```
```   520
```
```   521 lemma has_integral_neg: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral -k) s"
```
```   522   by (drule_tac c="-1" in has_integral_cmul) auto
```
```   523
```
```   524 lemma has_integral_add:
```
```   525   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```   526   assumes "(f has_integral k) s"
```
```   527     and "(g has_integral l) s"
```
```   528   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"
```
```   529 proof -
```
```   530   have lem: "(f has_integral k) (cbox a b) \<Longrightarrow> (g has_integral l) (cbox a b) \<Longrightarrow>
```
```   531     ((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
```
```   532     for f :: "'n \<Rightarrow> 'a" and g a b k l
```
```   533     unfolding has_integral_cbox
```
```   534     by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)
```
```   535   {
```
```   536     presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
```
```   537     then show ?thesis
```
```   538       using assms lem by force
```
```   539   }
```
```   540   assume as: "\<not> (\<exists>a b. s = cbox a b)"
```
```   541   then show ?thesis
```
```   542   proof (subst has_integral_alt, clarsimp, goal_cases)
```
```   543     case (1 e)
```
```   544     then have *: "e / 2 > 0"
```
```   545       by auto
```
```   546     from has_integral_altD[OF assms(1) as *]
```
```   547     obtain B1 where B1:
```
```   548         "0 < B1"
```
```   549         "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
```
```   550           \<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"
```
```   551       by blast
```
```   552     from has_integral_altD[OF assms(2) as *]
```
```   553     obtain B2 where B2:
```
```   554         "0 < B2"
```
```   555         "\<And>a b. ball 0 B2 \<subseteq> (cbox a b) \<Longrightarrow>
```
```   556           \<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"
```
```   557       by blast
```
```   558     show ?case
```
```   559     proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
```
```   560       fix a b
```
```   561       assume "ball 0 (max B1 B2) \<subseteq> cbox a (b::'n)"
```
```   562       then have *: "ball 0 B1 \<subseteq> cbox a (b::'n)" "ball 0 B2 \<subseteq> cbox a (b::'n)"
```
```   563         by auto
```
```   564       obtain w where w:
```
```   565         "((\<lambda>x. if x \<in> s then f x else 0) has_integral w) (cbox a b)"
```
```   566         "norm (w - k) < e / 2"
```
```   567         using B1(2)[OF *(1)] by blast
```
```   568       obtain z where z:
```
```   569         "((\<lambda>x. if x \<in> s then g x else 0) has_integral z) (cbox a b)"
```
```   570         "norm (z - l) < e / 2"
```
```   571         using B2(2)[OF *(2)] by blast
```
```   572       have *: "\<And>x. (if x \<in> s then f x + g x else 0) =
```
```   573         (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)"
```
```   574         by auto
```
```   575       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"
```
```   576         apply (rule_tac x="w + z" in exI)
```
```   577         apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
```
```   578         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
```
```   579         apply (auto simp add: field_simps)
```
```   580         done
```
```   581     qed
```
```   582   qed
```
```   583 qed
```
```   584
```
```   585 lemma has_integral_sub:
```
```   586   "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow>
```
```   587     ((\<lambda>x. f x - g x) has_integral (k - l)) s"
```
```   588   using has_integral_add[OF _ has_integral_neg, of f k s g l]
```
```   589   by (auto simp: algebra_simps)
```
```   590
```
```   591 lemma integral_0 [simp]:
```
```   592   "integral s (\<lambda>x::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
```
```   593   by (rule integral_unique has_integral_0)+
```
```   594
```
```   595 lemma integral_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```   596     integral s (\<lambda>x. f x + g x) = integral s f + integral s g"
```
```   597   by (rule integral_unique) (metis integrable_integral has_integral_add)
```
```   598
```
```   599 lemma integral_cmul [simp]: "integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"
```
```   600 proof (cases "f integrable_on s \<or> c = 0")
```
```   601   case True with has_integral_cmul show ?thesis by force
```
```   602 next
```
```   603   case False then have "~ (\<lambda>x. c *\<^sub>R f x) integrable_on s"
```
```   604     using has_integral_cmul [of "(\<lambda>x. c *\<^sub>R f x)" _ s "inverse c"]
```
```   605     by force
```
```   606   with False show ?thesis by (simp add: not_integrable_integral)
```
```   607 qed
```
```   608
```
```   609 lemma integral_neg [simp]: "integral s (\<lambda>x. - f x) = - integral s f"
```
```   610 proof (cases "f integrable_on s")
```
```   611   case True then show ?thesis
```
```   612     by (simp add: has_integral_neg integrable_integral integral_unique)
```
```   613 next
```
```   614   case False then have "~ (\<lambda>x. - f x) integrable_on s"
```
```   615     using has_integral_neg [of "(\<lambda>x. - f x)" _ s ]
```
```   616     by force
```
```   617   with False show ?thesis by (simp add: not_integrable_integral)
```
```   618 qed
```
```   619
```
```   620 lemma integral_diff: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>
```
```   621     integral s (\<lambda>x. f x - g x) = integral s f - integral s g"
```
```   622   by (rule integral_unique) (metis integrable_integral has_integral_sub)
```
```   623
```
```   624 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"
```
```   625   unfolding integrable_on_def using has_integral_0 by auto
```
```   626
```
```   627 lemma integrable_add: "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"
```
```   628   unfolding integrable_on_def by(auto intro: has_integral_add)
```
```   629
```
```   630 lemma integrable_cmul: "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"
```
```   631   unfolding integrable_on_def by(auto intro: has_integral_cmul)
```
```   632
```
```   633 lemma integrable_on_cmult_iff:
```
```   634   fixes c :: real
```
```   635   assumes "c \<noteq> 0"
```
```   636   shows "(\<lambda>x. c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```   637   using integrable_cmul[of "\<lambda>x. c * f x" s "1 / c"] integrable_cmul[of f s c] \<open>c \<noteq> 0\<close>
```
```   638   by auto
```
```   639
```
```   640 lemma integrable_on_cmult_left:
```
```   641   assumes "f integrable_on s"
```
```   642   shows "(\<lambda>x. of_real c * f x) integrable_on s"
```
```   643     using integrable_cmul[of f s "of_real c"] assms
```
```   644     by (simp add: scaleR_conv_of_real)
```
```   645
```
```   646 lemma integrable_neg: "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"
```
```   647   unfolding integrable_on_def by(auto intro: has_integral_neg)
```
```   648
```
```   649 lemma integrable_diff:
```
```   650   "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"
```
```   651   unfolding integrable_on_def by(auto intro: has_integral_sub)
```
```   652
```
```   653 lemma integrable_linear:
```
```   654   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on s"
```
```   655   unfolding integrable_on_def by(auto intro: has_integral_linear)
```
```   656
```
```   657 lemma integral_linear:
```
```   658   "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h \<circ> f) = h (integral s f)"
```
```   659   apply (rule has_integral_unique [where i=s and f = "h \<circ> f"])
```
```   660   apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
```
```   661   done
```
```   662
```
```   663 lemma integral_component_eq[simp]:
```
```   664   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```   665   assumes "f integrable_on s"
```
```   666   shows "integral s (\<lambda>x. f x \<bullet> k) = integral s f \<bullet> k"
```
```   667   unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..
```
```   668
```
```   669 lemma has_integral_sum:
```
```   670   assumes "finite t"
```
```   671     and "\<forall>a\<in>t. ((f a) has_integral (i a)) s"
```
```   672   shows "((\<lambda>x. sum (\<lambda>a. f a x) t) has_integral (sum i t)) s"
```
```   673   using assms(1) subset_refl[of t]
```
```   674 proof (induct rule: finite_subset_induct)
```
```   675   case empty
```
```   676   then show ?case by auto
```
```   677 next
```
```   678   case (insert x F)
```
```   679   with assms show ?case
```
```   680     by (simp add: has_integral_add)
```
```   681 qed
```
```   682
```
```   683 lemma integral_sum:
```
```   684   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow>
```
```   685    integral s (\<lambda>x. sum (\<lambda>a. f a x) t) = sum (\<lambda>a. integral s (f a)) t"
```
```   686   by (auto intro: has_integral_sum integrable_integral)
```
```   687
```
```   688 lemma integrable_sum:
```
```   689   "\<lbrakk>finite t;  \<forall>a\<in>t. (f a) integrable_on s\<rbrakk> \<Longrightarrow> (\<lambda>x. sum (\<lambda>a. f a x) t) integrable_on s"
```
```   690   unfolding integrable_on_def
```
```   691   apply (drule bchoice)
```
```   692   using has_integral_sum[of t]
```
```   693   apply auto
```
```   694   done
```
```   695
```
```   696 lemma has_integral_eq:
```
```   697   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```   698     and "(f has_integral k) s"
```
```   699   shows "(g has_integral k) s"
```
```   700   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]
```
```   701   using has_integral_is_0[of s "\<lambda>x. f x - g x"]
```
```   702   using assms(1)
```
```   703   by auto
```
```   704
```
```   705 lemma integrable_eq: "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"
```
```   706   unfolding integrable_on_def
```
```   707   using has_integral_eq[of s f g] has_integral_eq by blast
```
```   708
```
```   709 lemma has_integral_cong:
```
```   710   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```   711   shows "(f has_integral i) s = (g has_integral i) s"
```
```   712   using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
```
```   713   by auto
```
```   714
```
```   715 lemma integral_cong:
```
```   716   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```   717   shows "integral s f = integral s g"
```
```   718   unfolding integral_def
```
```   719 by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
```
```   720
```
```   721 lemma integrable_on_cmult_left_iff [simp]:
```
```   722   assumes "c \<noteq> 0"
```
```   723   shows "(\<lambda>x. of_real c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```   724         (is "?lhs = ?rhs")
```
```   725 proof
```
```   726   assume ?lhs
```
```   727   then have "(\<lambda>x. of_real (1 / c) * (of_real c * f x)) integrable_on s"
```
```   728     using integrable_cmul[of "\<lambda>x. of_real c * f x" s "1 / of_real c"]
```
```   729     by (simp add: scaleR_conv_of_real)
```
```   730   then have "(\<lambda>x. (of_real (1 / c) * of_real c * f x)) integrable_on s"
```
```   731     by (simp add: algebra_simps)
```
```   732   with \<open>c \<noteq> 0\<close> show ?rhs
```
```   733     by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
```
```   734 qed (blast intro: integrable_on_cmult_left)
```
```   735
```
```   736 lemma integrable_on_cmult_right:
```
```   737   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```   738   assumes "f integrable_on s"
```
```   739   shows "(\<lambda>x. f x * of_real c) integrable_on s"
```
```   740 using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
```
```   741
```
```   742 lemma integrable_on_cmult_right_iff [simp]:
```
```   743   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
```
```   744   assumes "c \<noteq> 0"
```
```   745   shows "(\<lambda>x. f x * of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```   746 using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
```
```   747
```
```   748 lemma integrable_on_cdivide:
```
```   749   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```   750   assumes "f integrable_on s"
```
```   751   shows "(\<lambda>x. f x / of_real c) integrable_on s"
```
```   752 by (simp add: integrable_on_cmult_right divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```   753
```
```   754 lemma integrable_on_cdivide_iff [simp]:
```
```   755   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
```
```   756   assumes "c \<noteq> 0"
```
```   757   shows "(\<lambda>x. f x / of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
```
```   758 by (simp add: divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
```
```   759
```
```   760 lemma has_integral_null [intro]: "content(cbox a b) = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```   761   unfolding has_integral_cbox
```
```   762   using eventually_division_filter_tagged_division[of "cbox a b"]
```
```   763   by (subst tendsto_cong[where g="\<lambda>_. 0"]) (auto elim: eventually_mono intro: sum_content_null)
```
```   764
```
```   765 lemma has_integral_null_real [intro]: "content {a .. b::real} = 0 \<Longrightarrow> (f has_integral 0) {a .. b}"
```
```   766   by (metis box_real(2) has_integral_null)
```
```   767
```
```   768 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
```
```   769   by (auto simp add: has_integral_null dest!: integral_unique)
```
```   770
```
```   771 lemma integral_null [simp]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
```
```   772   by (metis has_integral_null integral_unique)
```
```   773
```
```   774 lemma integrable_on_null [intro]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
```
```   775   by (simp add: has_integral_integrable)
```
```   776
```
```   777 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
```
```   778   by (simp add: has_integral_is_0)
```
```   779
```
```   780 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
```
```   781   by (auto simp add: has_integral_empty has_integral_unique)
```
```   782
```
```   783 lemma integrable_on_empty[intro]: "f integrable_on {}"
```
```   784   unfolding integrable_on_def by auto
```
```   785
```
```   786 lemma integral_empty[simp]: "integral {} f = 0"
```
```   787   by (rule integral_unique) (rule has_integral_empty)
```
```   788
```
```   789 lemma has_integral_refl[intro]:
```
```   790   fixes a :: "'a::euclidean_space"
```
```   791   shows "(f has_integral 0) (cbox a a)"
```
```   792     and "(f has_integral 0) {a}"
```
```   793 proof -
```
```   794   have *: "{a} = cbox a a"
```
```   795     apply (rule set_eqI)
```
```   796     unfolding mem_box singleton_iff euclidean_eq_iff[where 'a='a]
```
```   797     apply safe
```
```   798     prefer 3
```
```   799     apply (erule_tac x=b in ballE)
```
```   800     apply (auto simp add: field_simps)
```
```   801     done
```
```   802   show "(f has_integral 0) (cbox a a)" "(f has_integral 0) {a}"
```
```   803     unfolding *
```
```   804     apply (rule_tac[!] has_integral_null)
```
```   805     unfolding content_eq_0_interior
```
```   806     unfolding interior_cbox
```
```   807     using box_sing
```
```   808     apply auto
```
```   809     done
```
```   810 qed
```
```   811
```
```   812 lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
```
```   813   unfolding integrable_on_def by auto
```
```   814
```
```   815 lemma integral_refl [simp]: "integral (cbox a a) f = 0"
```
```   816   by (rule integral_unique) auto
```
```   817
```
```   818 lemma integral_singleton [simp]: "integral {a} f = 0"
```
```   819   by auto
```
```   820
```
```   821 lemma integral_blinfun_apply:
```
```   822   assumes "f integrable_on s"
```
```   823   shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
```
```   824   by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
```
```   825
```
```   826 lemma blinfun_apply_integral:
```
```   827   assumes "f integrable_on s"
```
```   828   shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
```
```   829   by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
```
```   830
```
```   831 lemma has_integral_componentwise_iff:
```
```   832   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```   833   shows "(f has_integral y) A \<longleftrightarrow> (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```   834 proof safe
```
```   835   fix b :: 'b assume "(f has_integral y) A"
```
```   836   from has_integral_linear[OF this(1) bounded_linear_inner_left, of b]
```
```   837     show "((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A" by (simp add: o_def)
```
```   838 next
```
```   839   assume "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```   840   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"
```
```   841     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```   842   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"
```
```   843     by (intro has_integral_sum) (simp_all add: o_def)
```
```   844   thus "(f has_integral y) A" by (simp add: euclidean_representation)
```
```   845 qed
```
```   846
```
```   847 lemma has_integral_componentwise:
```
```   848   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```   849   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"
```
```   850   by (subst has_integral_componentwise_iff) blast
```
```   851
```
```   852 lemma integrable_componentwise_iff:
```
```   853   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```   854   shows "f integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```   855 proof
```
```   856   assume "f integrable_on A"
```
```   857   then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def)
```
```   858   hence "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
```
```   859     by (subst (asm) has_integral_componentwise_iff)
```
```   860   thus "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)" by (auto simp: integrable_on_def)
```
```   861 next
```
```   862   assume "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
```
```   863   then obtain y where "\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y b) A"
```
```   864     unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
```
```   865   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"
```
```   866     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
```
```   867   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"
```
```   868     by (intro has_integral_sum) (simp_all add: o_def)
```
```   869   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
```
```   870 qed
```
```   871
```
```   872 lemma integrable_componentwise:
```
```   873   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```   874   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A) \<Longrightarrow> f integrable_on A"
```
```   875   by (subst integrable_componentwise_iff) blast
```
```   876
```
```   877 lemma integral_componentwise:
```
```   878   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
```
```   879   assumes "f integrable_on A"
```
```   880   shows "integral A f = (\<Sum>b\<in>Basis. integral A (\<lambda>x. (f x \<bullet> b) *\<^sub>R b))"
```
```   881 proof -
```
```   882   from assms have integrable: "\<forall>b\<in>Basis. (\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. (f x \<bullet> b)) integrable_on A"
```
```   883     by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI)
```
```   884        (simp_all add: bounded_linear_scaleR_left)
```
```   885   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
```
```   886     by (simp add: euclidean_representation)
```
```   887   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
```
```   888     by (subst integral_sum) (simp_all add: o_def)
```
```   889   finally show ?thesis .
```
```   890 qed
```
```   891
```
```   892 lemma integrable_component:
```
```   893   "f integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (y :: 'b :: euclidean_space)) integrable_on A"
```
```   894   by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def)
```
```   895
```
```   896
```
```   897
```
```   898 subsection \<open>Cauchy-type criterion for integrability.\<close>
```
```   899
```
```   900 (* XXXXXXX *)
```
```   901 lemma integrable_cauchy:
```
```   902   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
```
```   903   shows "f integrable_on cbox a b \<longleftrightarrow>
```
```   904     (\<forall>e>0. \<exists>d. gauge d \<and>
```
```   905       (\<forall>p1 p2. p1 tagged_division_of (cbox a b) \<and> d fine p1 \<and>
```
```   906         p2 tagged_division_of (cbox a b) \<and> d fine p2 \<longrightarrow>
```
```   907         norm ((\<Sum>(x,k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x,k)\<in>p2. content k *\<^sub>R f x)) < e))"
```
```   908   (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")
```
```   909 proof
```
```   910   assume ?l
```
```   911   then guess y unfolding integrable_on_def has_integral .. note y=this
```
```   912   show "\<forall>e>0. \<exists>d. ?P e d"
```
```   913   proof (clarify, goal_cases)
```
```   914     case (1 e)
```
```   915     then have "e/2 > 0" by auto
```
```   916     then guess d
```
```   917       apply -
```
```   918       apply (drule y[rule_format])
```
```   919       apply (elim exE conjE)
```
```   920       done
```
```   921     note d=this[rule_format]
```
```   922     show ?case
```
```   923     proof (rule_tac x=d in exI, clarsimp simp: d)
```
```   924       fix p1 p2
```
```   925       assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
```
```   926                  "p2 tagged_division_of (cbox a b)" "d fine p2"
```
```   927       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"
```
```   928         apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
```
```   929         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
```
```   930     qed
```
```   931   qed
```
```   932 next
```
```   933   assume "\<forall>e>0. \<exists>d. ?P e d"
```
```   934   then have "\<forall>n::nat. \<exists>d. ?P (inverse(of_nat (n + 1))) d"
```
```   935     by auto
```
```   936   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
```
```   937   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})"
```
```   938     apply (rule gauge_inters)
```
```   939     using d(1)
```
```   940     apply auto
```
```   941     done
```
```   942   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"
```
```   943     by (meson fine_division_exists)
```
```   944   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
```
```   945   have dp: "\<And>i n. i\<le>n \<Longrightarrow> d i fine p n"
```
```   946     using p(2) unfolding fine_inters by auto
```
```   947   have "Cauchy (\<lambda>n. sum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"
```
```   948   proof (rule CauchyI, goal_cases)
```
```   949     case (1 e)
```
```   950     then guess N unfolding real_arch_inverse[of e] .. note N=this
```
```   951     show ?case
```
```   952       apply (rule_tac x=N in exI)
```
```   953     proof clarify
```
```   954       fix m n
```
```   955       assume mn: "N \<le> m" "N \<le> n"
```
```   956       have *: "N = (N - 1) + 1" using N by auto
```
```   957       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"
```
```   958         apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
```
```   959         apply(subst *)
```
```   960         using dp p(1) mn d(2) by auto
```
```   961     qed
```
```   962   qed
```
```   963   then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
```
```   964   show ?l
```
```   965     unfolding integrable_on_def has_integral
```
```   966   proof (rule_tac x=y in exI, clarify)
```
```   967     fix e :: real
```
```   968     assume "e>0"
```
```   969     then have *:"e/2 > 0" by auto
```
```   970     then guess N1 unfolding real_arch_inverse[of "e/2"] .. note N1=this
```
```   971     then have N1': "N1 = N1 - 1 + 1"
```
```   972       by auto
```
```   973     guess N2 using y[OF *] .. note N2=this
```
```   974     have "gauge (d (N1 + N2))"
```
```   975       using d by auto
```
```   976     moreover
```
```   977     {
```
```   978       fix q
```
```   979       assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
```
```   980       have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
```
```   981         apply (rule less_trans)
```
```   982         using N1
```
```   983         apply auto
```
```   984         done
```
```   985       have "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e"
```
```   986         apply (rule norm_triangle_half_r)
```
```   987         apply (rule less_trans[OF _ *])
```
```   988         apply (subst N1', rule d(2)[of "p (N1+N2)"])
```
```   989         using N1' as(1) as(2) dp
```
```   990         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>)
```
```   991         using N2 le_add2 by blast
```
```   992     }
```
```   993     ultimately show "\<exists>d. gauge d \<and>
```
```   994       (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```   995         norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"
```
```   996       by (rule_tac x="d (N1 + N2)" in exI) auto
```
```   997   qed
```
```   998 qed
```
```   999
```
```  1000
```
```  1001 subsection \<open>Additivity of integral on abutting intervals.\<close>
```
```  1002
```
```  1003 lemma tagged_division_split_left_inj_content:
```
```  1004   assumes d: "d tagged_division_of i"
```
```  1005     and "(x1, k1) \<in> d" "(x2, k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}" "k \<in> Basis"
```
```  1006   shows "content (k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
```
```  1007 proof -
```
```  1008   from tagged_division_ofD(4)[OF d \<open>(x1, k1) \<in> d\<close>] obtain a b where k1: "k1 = cbox a b"
```
```  1009     by auto
```
```  1010   show ?thesis
```
```  1011     unfolding k1 interval_split[OF \<open>k \<in> Basis\<close>]
```
```  1012     unfolding content_eq_0_interior
```
```  1013     unfolding interval_split[OF \<open>k \<in> Basis\<close>, symmetric] k1[symmetric]
```
```  1014     by (rule tagged_division_split_left_inj[OF assms])
```
```  1015 qed
```
```  1016
```
```  1017 lemma tagged_division_split_right_inj_content:
```
```  1018   assumes d: "d tagged_division_of i"
```
```  1019     and "(x1, k1) \<in> d" "(x2, k2) \<in> d" "k1 \<noteq> k2" "k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}" "k \<in> Basis"
```
```  1020   shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
```
```  1021 proof -
```
```  1022   from tagged_division_ofD(4)[OF d \<open>(x1, k1) \<in> d\<close>] obtain a b where k1: "k1 = cbox a b"
```
```  1023     by auto
```
```  1024   show ?thesis
```
```  1025     unfolding k1 interval_split[OF \<open>k \<in> Basis\<close>]
```
```  1026     unfolding content_eq_0_interior
```
```  1027     unfolding interval_split[OF \<open>k \<in> Basis\<close>, symmetric] k1[symmetric]
```
```  1028     by (rule tagged_division_split_right_inj[OF assms])
```
```  1029 qed
```
```  1030
```
```  1031 lemma has_integral_split:
```
```  1032   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1033   assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
```
```  1034       and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
```
```  1035       and k: "k \<in> Basis"
```
```  1036   shows "(f has_integral (i + j)) (cbox a b)"
```
```  1037 proof (unfold has_integral, rule, rule, goal_cases)
```
```  1038   case (1 e)
```
```  1039   then have e: "e/2 > 0"
```
```  1040     by auto
```
```  1041     obtain d1
```
```  1042     where d1: "gauge d1"
```
```  1043       and d1norm:
```
```  1044         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c};
```
```  1045                d1 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - i) < e / 2"
```
```  1046        apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
```
```  1047        apply (simp add: interval_split[symmetric] k)
```
```  1048        done
```
```  1049     obtain d2
```
```  1050     where d2: "gauge d2"
```
```  1051       and d2norm:
```
```  1052         "\<And>p. \<lbrakk>p tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k};
```
```  1053                d2 fine p\<rbrakk> \<Longrightarrow> norm ((\<Sum>(x, k) \<in> p. content k *\<^sub>R f x) - j) < e / 2"
```
```  1054        apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
```
```  1055        apply (simp add: interval_split[symmetric] k)
```
```  1056        done
```
```  1057   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"
```
```  1058   have "gauge ?d"
```
```  1059     using d1 d2 unfolding gauge_def by auto
```
```  1060   then show ?case
```
```  1061   proof (rule_tac x="?d" in exI, safe)
```
```  1062     fix p
```
```  1063     assume "p tagged_division_of (cbox a b)" "?d fine p"
```
```  1064     note p = this tagged_division_ofD[OF this(1)]
```
```  1065     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"
```
```  1066     proof -
```
```  1067       fix x kk
```
```  1068       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}"
```
```  1069       show "x\<bullet>k \<le> c"
```
```  1070       proof (rule ccontr)
```
```  1071         assume **: "\<not> ?thesis"
```
```  1072         from this[unfolded not_le]
```
```  1073         have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  1074           using p(2)[unfolded fine_def, rule_format,OF as] by auto
```
```  1075         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
```
```  1076           by blast
```
```  1077         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  1078           using Basis_le_norm[OF k, of "x - y"]
```
```  1079           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  1080         with y show False
```
```  1081           using ** by (auto simp add: field_simps)
```
```  1082       qed
```
```  1083     qed
```
```  1084     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"
```
```  1085     proof -
```
```  1086       fix x kk
```
```  1087       assume as: "(x, kk) \<in> p" and kk: "kk \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}"
```
```  1088       show "x\<bullet>k \<ge> c"
```
```  1089       proof (rule ccontr)
```
```  1090         assume **: "\<not> ?thesis"
```
```  1091         from this[unfolded not_le] have "kk \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
```
```  1092           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
```
```  1093         with kk obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
```
```  1094           by blast
```
```  1095         then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
```
```  1096           using Basis_le_norm[OF k, of "x - y"]
```
```  1097           by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
```
```  1098         with y show False
```
```  1099           using ** by (auto simp add: field_simps)
```
```  1100       qed
```
```  1101     qed
```
```  1102
```
```  1103     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>
```
```  1104                          (\<forall>x k. P x k \<longrightarrow> Q x (f k))"
```
```  1105       by auto
```
```  1106     have fin_finite: "finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"
```
```  1107       if "finite s" for s and f :: "'a set \<Rightarrow> 'a set" and P :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"
```
```  1108     proof -
```
```  1109       from that have "finite ((\<lambda>(x, k). (x, f k)) ` s)"
```
```  1110         by auto
```
```  1111       then show ?thesis
```
```  1112         by (rule rev_finite_subset) auto
```
```  1113     qed
```
```  1114     { fix g :: "'a set \<Rightarrow> 'a set"
```
```  1115       fix i :: "'a \<times> 'a set"
```
```  1116       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  1117       then obtain x k where xk:
```
```  1118               "i = (x, g k)"  "(x, k) \<in> p"
```
```  1119               "(x, g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"
```
```  1120           by auto
```
```  1121       have "content (g k) = 0"
```
```  1122         using xk using content_empty by auto
```
```  1123       then have "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0"
```
```  1124         unfolding xk split_conv by auto
```
```  1125     } note [simp] = this
```
```  1126     have lem3: "\<And>g :: 'a set \<Rightarrow> 'a set. finite p \<Longrightarrow>
```
```  1127                   sum (\<lambda>(x, k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> g k \<noteq> {}} =
```
```  1128                   sum (\<lambda>(x, k). content k *\<^sub>R f x) ((\<lambda>(x, k). (x, g k)) ` p)"
```
```  1129       by (rule sum.mono_neutral_left) auto
```
```  1130     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> {}}"
```
```  1131     have d1_fine: "d1 fine ?M1"
```
```  1132       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  1133     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
```
```  1134     proof (rule d1norm [OF tagged_division_ofI d1_fine])
```
```  1135       show "finite ?M1"
```
```  1136         by (rule fin_finite p(3))+
```
```  1137       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
```
```  1138         unfolding p(8)[symmetric] by auto
```
```  1139       fix x l
```
```  1140       assume xl: "(x, l) \<in> ?M1"
```
```  1141       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  1142       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  1143         unfolding xl'
```
```  1144         using p(4-6)[OF xl'(3)] using xl'(4)
```
```  1145         using xk_le_c[OF xl'(3-4)] by auto
```
```  1146       show "\<exists>a b. l = cbox a b"
```
```  1147         unfolding xl'
```
```  1148         using p(6)[OF xl'(3)]
```
```  1149         by (fastforce simp add: interval_split[OF k,where c=c])
```
```  1150       fix y r
```
```  1151       let ?goal = "interior l \<inter> interior r = {}"
```
```  1152       assume yr: "(y, r) \<in> ?M1"
```
```  1153       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  1154       assume as: "(x, l) \<noteq> (y, r)"
```
```  1155       show "interior l \<inter> interior r = {}"
```
```  1156       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  1157         case False
```
```  1158         then show ?thesis
```
```  1159           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1160       next
```
```  1161         case True
```
```  1162         then have "l' \<noteq> r'"
```
```  1163           using as unfolding xl' yr' by auto
```
```  1164         then show ?thesis
```
```  1165           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1166       qed
```
```  1167     qed
```
```  1168     moreover
```
```  1169     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> {}}"
```
```  1170     have d2_fine: "d2 fine ?M2"
```
```  1171       by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
```
```  1172     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
```
```  1173     proof (rule d2norm [OF tagged_division_ofI d2_fine])
```
```  1174       show "finite ?M2"
```
```  1175         by (rule fin_finite p(3))+
```
```  1176       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
```
```  1177         unfolding p(8)[symmetric] by auto
```
```  1178       fix x l
```
```  1179       assume xl: "(x, l) \<in> ?M2"
```
```  1180       then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
```
```  1181       show "x \<in> l" "l \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  1182         unfolding xl'
```
```  1183         using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
```
```  1184         by auto
```
```  1185       show "\<exists>a b. l = cbox a b"
```
```  1186         unfolding xl'
```
```  1187         using p(6)[OF xl'(3)]
```
```  1188         by (fastforce simp add: interval_split[OF k, where c=c])
```
```  1189       fix y r
```
```  1190       let ?goal = "interior l \<inter> interior r = {}"
```
```  1191       assume yr: "(y, r) \<in> ?M2"
```
```  1192       then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
```
```  1193       assume as: "(x, l) \<noteq> (y, r)"
```
```  1194       show "interior l \<inter> interior r = {}"
```
```  1195       proof (cases "l' = r' \<longrightarrow> x' = y'")
```
```  1196         case False
```
```  1197         then show ?thesis
```
```  1198           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1199       next
```
```  1200         case True
```
```  1201         then have "l' \<noteq> r'"
```
```  1202           using as unfolding xl' yr' by auto
```
```  1203         then show ?thesis
```
```  1204           using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
```
```  1205       qed
```
```  1206     qed
```
```  1207     ultimately
```
```  1208     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"
```
```  1209       using norm_add_less by blast
```
```  1210     also {
```
```  1211       have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
```
```  1212         using scaleR_zero_left by auto
```
```  1213       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)"
```
```  1214         by auto
```
```  1215       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) =
```
```  1216         (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
```
```  1217         by auto
```
```  1218       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
```
```  1219         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
```
```  1220         unfolding lem3[OF p(3)]
```
```  1221         by (subst (1 2) sum.reindex_nontrivial[OF p(3)])
```
```  1222            (auto intro!: k eq0 tagged_division_split_left_inj_content[OF p(1)] tagged_division_split_right_inj_content[OF p(1)]
```
```  1223                  simp: cont_eq)+
```
```  1224       also note sum.distrib[symmetric]
```
```  1225       also have "\<And>x. x \<in> p \<Longrightarrow>
```
```  1226                     (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
```
```  1227                     (\<lambda>(x,ka). content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) x =
```
```  1228                     (\<lambda>(x,ka). content ka *\<^sub>R f x) x"
```
```  1229       proof clarify
```
```  1230         fix a b
```
```  1231         assume "(a, b) \<in> p"
```
```  1232         from p(6)[OF this] guess u v by (elim exE) note uv=this
```
```  1233         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 =
```
```  1234           content b *\<^sub>R f a"
```
```  1235           unfolding scaleR_left_distrib[symmetric]
```
```  1236           unfolding uv content_split[OF k,of u v c]
```
```  1237           by auto
```
```  1238       qed
```
```  1239       note sum.cong [OF _ this]
```
```  1240       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 +
```
```  1241         ((\<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) =
```
```  1242         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
```
```  1243         by auto
```
```  1244     }
```
```  1245     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
```
```  1246       by auto
```
```  1247   qed
```
```  1248 qed
```
```  1249
```
```  1250
```
```  1251 subsection \<open>A sort of converse, integrability on subintervals.\<close>
```
```  1252
```
```  1253 lemma has_integral_separate_sides:
```
```  1254   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1255   assumes "(f has_integral i) (cbox a b)"
```
```  1256     and "e > 0"
```
```  1257     and k: "k \<in> Basis"
```
```  1258   obtains d where "gauge d"
```
```  1259     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
```
```  1260         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
```
```  1261         norm ((sum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + sum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
```
```  1262 proof -
```
```  1263   guess d using has_integralD[OF assms(1-2)] . note d=this
```
```  1264   { fix p1 p2
```
```  1265     assume "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  1266     note p1=tagged_division_ofD[OF this(1)] this
```
```  1267     assume "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" "d fine p2"
```
```  1268     note p2=tagged_division_ofD[OF this(1)] this
```
```  1269     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
```
```  1270     { fix a b
```
```  1271       assume ab: "(a, b) \<in> p1 \<inter> p2"
```
```  1272       have "(a, b) \<in> p1"
```
```  1273         using ab by auto
```
```  1274       with p1 obtain u v where uv: "b = cbox u v" by auto
```
```  1275       have "b \<subseteq> {x. x\<bullet>k = c}"
```
```  1276         using ab p1(3)[of a b] p2(3)[of a b] by fastforce
```
```  1277       moreover
```
```  1278       have "interior {x::'a. x \<bullet> k = c} = {}"
```
```  1279       proof (rule ccontr)
```
```  1280         assume "\<not> ?thesis"
```
```  1281         then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
```
```  1282           by auto
```
```  1283         then guess e unfolding mem_interior .. note e=this
```
```  1284         have x: "x\<bullet>k = c"
```
```  1285           using x interior_subset by fastforce
```
```  1286         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)"
```
```  1287           using e k by (auto simp: inner_simps inner_not_same_Basis)
```
```  1288         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (e / 2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
```
```  1289               (\<Sum>i\<in>Basis. (if i = k then e / 2 else 0))"
```
```  1290           using "*" by (blast intro: sum.cong)
```
```  1291         also have "\<dots> < e"
```
```  1292           apply (subst sum.delta)
```
```  1293           using e
```
```  1294           apply auto
```
```  1295           done
```
```  1296         finally have "x + (e/2) *\<^sub>R k \<in> ball x e"
```
```  1297           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
```
```  1298         then have "x + (e/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
```
```  1299           using e by auto
```
```  1300         then show False
```
```  1301           unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
```
```  1302       qed
```
```  1303       ultimately have "content b = 0"
```
```  1304         unfolding uv content_eq_0_interior
```
```  1305         using interior_mono by blast
```
```  1306       then have "content b *\<^sub>R f a = 0"
```
```  1307         by auto
```
```  1308     }
```
```  1309     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) =
```
```  1310                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
```
```  1311       by (subst sum.union_inter_neutral) (auto simp: p1 p2)
```
```  1312     also have "\<dots> < e"
```
```  1313       by (rule k d(2) p12 fine_union p1 p2)+
```
```  1314     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" .
```
```  1315    }
```
```  1316   then show ?thesis
```
```  1317     by (auto intro: that[of d] d elim: )
```
```  1318 qed
```
```  1319
```
```  1320 lemma integrable_split[intro]:
```
```  1321   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
```
```  1322   assumes "f integrable_on cbox a b"
```
```  1323     and k: "k \<in> Basis"
```
```  1324   shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})" (is ?t1)
```
```  1325     and "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})" (is ?t2)
```
```  1326 proof -
```
```  1327   guess y using assms(1) unfolding integrable_on_def .. note y=this
```
```  1328   define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
```
```  1329   define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
```
```  1330   show ?t1 ?t2
```
```  1331     unfolding interval_split[OF k] integrable_cauchy
```
```  1332     unfolding interval_split[symmetric,OF k]
```
```  1333   proof (rule_tac[!] allI impI)+
```
```  1334     fix e :: real
```
```  1335     assume "e > 0"
```
```  1336     then have "e/2>0"
```
```  1337       by auto
```
```  1338     from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
```
```  1339     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>
```
```  1340       p2 tagged_division_of (cbox a b) \<inter> A \<and> d fine p2 \<longrightarrow>
```
```  1341       norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"
```
```  1342     show "?P {x. x \<bullet> k \<le> c}"
```
```  1343     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  1344       fix p1 p2
```
```  1345       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
```
```  1346                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
```
```  1347       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"
```
```  1348       proof (rule fine_division_exists[OF d(1), of a' b] )
```
```  1349         fix p
```
```  1350         assume "p tagged_division_of cbox a' b" "d fine p"
```
```  1351         then show ?thesis
```
```  1352           using as norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
```
```  1353           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  1354           by (auto simp add: algebra_simps)
```
```  1355       qed
```
```  1356     qed
```
```  1357     show "?P {x. x \<bullet> k \<ge> c}"
```
```  1358     proof (rule_tac x=d in exI, clarsimp simp add: d)
```
```  1359       fix p1 p2
```
```  1360       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
```
```  1361                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
```
```  1362       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"
```
```  1363       proof (rule fine_division_exists[OF d(1), of a b'] )
```
```  1364         fix p
```
```  1365         assume "p tagged_division_of cbox a b'" "d fine p"
```
```  1366         then show ?thesis
```
```  1367           using as norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
```
```  1368           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
```
```  1369           by (auto simp add: algebra_simps)
```
```  1370       qed
```
```  1371     qed
```
```  1372   qed
```
```  1373 qed
```
```  1374
```
```  1375 lemma operative_integral:
```
```  1376   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  1377   shows "comm_monoid.operative (lift_option op +) (Some 0)
```
```  1378     (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
```
```  1379 proof -
```
```  1380   interpret comm_monoid "lift_option plus" "Some (0::'b)"
```
```  1381     by (rule comm_monoid_lift_option)
```
```  1382       (rule add.comm_monoid_axioms)
```
```  1383   show ?thesis
```
```  1384   proof (unfold operative_def, safe)
```
```  1385     fix a b c
```
```  1386     fix k :: 'a
```
```  1387     assume k: "k \<in> Basis"
```
```  1388     show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
```
```  1389           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)
```
```  1390           (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)"
```
```  1391     proof (cases "f integrable_on cbox a b")
```
```  1392       case True
```
```  1393       with k show ?thesis
```
```  1394         apply (simp add: integrable_split)
```
```  1395         apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
```
```  1396         apply (auto intro: integrable_integral)
```
```  1397         done
```
```  1398     next
```
```  1399     case False
```
```  1400       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})"
```
```  1401       proof (rule ccontr)
```
```  1402         assume "\<not> ?thesis"
```
```  1403         then have "f integrable_on cbox a b"
```
```  1404           unfolding integrable_on_def
```
```  1405           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)
```
```  1406           apply (rule has_integral_split[OF _ _ k])
```
```  1407           apply (auto intro: integrable_integral)
```
```  1408           done
```
```  1409         then show False
```
```  1410           using False by auto
```
```  1411       qed
```
```  1412       then show ?thesis
```
```  1413         using False by auto
```
```  1414     qed
```
```  1415   next
```
```  1416     fix a b :: 'a
```
```  1417     assume "box a b = {}"
```
```  1418     then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
```
```  1419       using has_integral_null_eq
```
```  1420       by (auto simp: integrable_on_null content_eq_0_interior)
```
```  1421   qed
```
```  1422 qed
```
```  1423
```
```  1424 subsection \<open>Bounds on the norm of Riemann sums and the integral itself.\<close>
```
```  1425
```
```  1426 lemma dsum_bound:
```
```  1427   assumes "p division_of (cbox a b)"
```
```  1428     and "norm c \<le> e"
```
```  1429   shows "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
```
```  1430 proof -
```
```  1431   have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = sum content p"
```
```  1432     apply (rule sum.cong)
```
```  1433     using assms
```
```  1434     apply simp
```
```  1435     apply (metis abs_of_nonneg assms(1) content_pos_le division_ofD(4))
```
```  1436     done
```
```  1437   have e: "0 \<le> e"
```
```  1438     using assms(2) norm_ge_zero order_trans by blast
```
```  1439   have "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
```
```  1440     using norm_sum by blast
```
```  1441   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
```
```  1442     by (simp add: sum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono sum_nonneg)
```
```  1443   also have "... \<le> e * content (cbox a b)"
```
```  1444     apply (rule mult_left_mono [OF _ e])
```
```  1445     apply (simp add: sumeq)
```
```  1446     using additive_content_division assms(1) eq_iff apply blast
```
```  1447     done
```
```  1448   finally show ?thesis .
```
```  1449 qed
```
```  1450
```
```  1451 lemma rsum_bound:
```
```  1452   assumes p: "p tagged_division_of (cbox a b)"
```
```  1453       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
```
```  1454     shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
```
```  1455 proof (cases "cbox a b = {}")
```
```  1456   case True show ?thesis
```
```  1457     using p unfolding True tagged_division_of_trivial by auto
```
```  1458 next
```
```  1459   case False
```
```  1460   then have e: "e \<ge> 0"
```
```  1461     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
```
```  1462   have sum_le: "sum (content \<circ> snd) p \<le> content (cbox a b)"
```
```  1463     unfolding additive_content_tagged_division[OF p, symmetric] split_def
```
```  1464     by (auto intro: eq_refl)
```
```  1465   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
```
```  1466     using tagged_division_ofD(4) [OF p] content_pos_le
```
```  1467     by force
```
```  1468   have norm: "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
```
```  1469     unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
```
```  1470     by (metis prod.collapse subset_eq)
```
```  1471   have "norm (sum (\<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))"
```
```  1472     by (rule norm_sum)
```
```  1473   also have "...  \<le> e * content (cbox a b)"
```
```  1474     unfolding split_def norm_scaleR
```
```  1475     apply (rule order_trans[OF sum_mono])
```
```  1476     apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
```
```  1477     apply (metis norm)
```
```  1478     unfolding sum_distrib_right[symmetric]
```
```  1479     using con sum_le
```
```  1480     apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
```
```  1481     done
```
```  1482   finally show ?thesis .
```
```  1483 qed
```
```  1484
```
```  1485 lemma rsum_diff_bound:
```
```  1486   assumes "p tagged_division_of (cbox a b)"
```
```  1487     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
```
```  1488   shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - sum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le>
```
```  1489          e * content (cbox a b)"
```
```  1490   apply (rule order_trans[OF _ rsum_bound[OF assms]])
```
```  1491   apply (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)
```
```  1492   done
```
```  1493
```
```  1494 lemma has_integral_bound:
```
```  1495   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1496   assumes "0 \<le> B"
```
```  1497       and *: "(f has_integral i) (cbox a b)"
```
```  1498       and "\<forall>x\<in>cbox a b. norm (f x) \<le> B"
```
```  1499     shows "norm i \<le> B * content (cbox a b)"
```
```  1500 proof (rule ccontr)
```
```  1501   assume "\<not> ?thesis"
```
```  1502   then have *: "norm i - B * content (cbox a b) > 0"
```
```  1503     by auto
```
```  1504   from assms(2)[unfolded has_integral,rule_format,OF *]
```
```  1505   guess d by (elim exE conjE) note d=this[rule_format]
```
```  1506   from fine_division_exists[OF this(1), of a b] guess p . note p=this
```
```  1507   have *: "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
```
```  1508     unfolding not_less
```
```  1509     by (metis norm_triangle_sub[of i] add.commute le_less_trans less_diff_eq linorder_not_le norm_minus_commute)
```
```  1510   show False
```
```  1511     using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
```
```  1512 qed
```
```  1513
```
```  1514 corollary has_integral_bound_real:
```
```  1515   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
```
```  1516   assumes "0 \<le> B"
```
```  1517       and "(f has_integral i) {a .. b}"
```
```  1518       and "\<forall>x\<in>{a .. b}. norm (f x) \<le> B"
```
```  1519     shows "norm i \<le> B * content {a .. b}"
```
```  1520   by (metis assms box_real(2) has_integral_bound)
```
```  1521
```
```  1522 corollary integrable_bound:
```
```  1523   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  1524   assumes "0 \<le> B"
```
```  1525       and "f integrable_on (cbox a b)"
```
```  1526       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
```
```  1527     shows "norm (integral (cbox a b) f) \<le> B * content (cbox a b)"
```
```  1528 by (metis integrable_integral has_integral_bound assms)
```
```  1529
```
```  1530
```
```  1531 subsection \<open>Similar theorems about relationship among components.\<close>
```
```  1532
```
```  1533 lemma rsum_component_le:
```
```  1534   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1535   assumes "p tagged_division_of (cbox a b)"
```
```  1536       and "\<forall>x\<in>cbox a b. (f x)\<bullet>i \<le> (g x)\<bullet>i"
```
```  1537     shows "(sum (\<lambda>(x,k). content k *\<^sub>R f x) p)\<bullet>i \<le> (sum (\<lambda>(x,k). content k *\<^sub>R g x) p)\<bullet>i"
```
```  1538 unfolding inner_sum_left
```
```  1539 proof (rule sum_mono, clarify)
```
```  1540   fix a b
```
```  1541   assume ab: "(a, b) \<in> p"
```
```  1542   note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
```
```  1543   from this(3) guess u v by (elim exE) note b=this
```
```  1544   show "(content b *\<^sub>R f a) \<bullet> i \<le> (content b *\<^sub>R g a) \<bullet> i"
```
```  1545     unfolding b inner_simps real_scaleR_def
```
```  1546     apply (rule mult_left_mono)
```
```  1547     using assms(2) tagged
```
```  1548     by (auto simp add: content_pos_le)
```
```  1549 qed
```
```  1550
```
```  1551 lemma has_integral_component_le:
```
```  1552   fixes f g :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1553   assumes k: "k \<in> Basis"
```
```  1554   assumes "(f has_integral i) s" "(g has_integral j) s"
```
```  1555     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  1556   shows "i\<bullet>k \<le> j\<bullet>k"
```
```  1557 proof -
```
```  1558   have lem: "i\<bullet>k \<le> j\<bullet>k"
```
```  1559     if f_i: "(f has_integral i) (cbox a b)"
```
```  1560     and g_j: "(g has_integral j) (cbox a b)"
```
```  1561     and le: "\<forall>x\<in>cbox a b. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  1562     for a b i and j :: 'b and f g :: "'a \<Rightarrow> 'b"
```
```  1563   proof (rule ccontr)
```
```  1564     assume "\<not> ?thesis"
```
```  1565     then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3"
```
```  1566       by auto
```
```  1567     guess d1 using f_i[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
```
```  1568     guess d2 using g_j[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
```
```  1569     obtain p where p: "p tagged_division_of cbox a b" "d1 fine p" "d2 fine p"
```
```  1570        using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter
```
```  1571        by metis
```
```  1572     note le_less_trans[OF Basis_le_norm[OF k]]
```
```  1573     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"
```
```  1574               "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
```
```  1575       using  k norm_bound_Basis_lt d1 d2 p
```
```  1576       by blast+
```
```  1577     then show False
```
```  1578       unfolding inner_simps
```
```  1579       using rsum_component_le[OF p(1) le]
```
```  1580       by (simp add: abs_real_def split: if_split_asm)
```
```  1581   qed
```
```  1582   show ?thesis
```
```  1583   proof (cases "\<exists>a b. s = cbox a b")
```
```  1584     case True
```
```  1585     with lem assms show ?thesis
```
```  1586       by auto
```
```  1587   next
```
```  1588     case False
```
```  1589     show ?thesis
```
```  1590     proof (rule ccontr)
```
```  1591       assume "\<not> i\<bullet>k \<le> j\<bullet>k"
```
```  1592       then have ij: "(i\<bullet>k - j\<bullet>k) / 3 > 0"
```
```  1593         by auto
```
```  1594       note has_integral_altD[OF _ False this]
```
```  1595       from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
```
```  1596       have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
```
```  1597         unfolding bounded_Un by(rule conjI bounded_ball)+
```
```  1598       from bounded_subset_cbox[OF this] guess a b by (elim exE)
```
```  1599       note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
```
```  1600       guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
```
```  1601       guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
```
```  1602       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"
```
```  1603         by (simp add: abs_real_def split: if_split_asm)
```
```  1604       note le_less_trans[OF Basis_le_norm[OF k]]
```
```  1605       note this[OF w1(2)] this[OF w2(2)]
```
```  1606       moreover
```
```  1607       have "w1\<bullet>k \<le> w2\<bullet>k"
```
```  1608         by (rule lem[OF w1(1) w2(1)]) (simp add: assms(4))
```
```  1609       ultimately show False
```
```  1610         unfolding inner_simps by(rule *)
```
```  1611     qed
```
```  1612   qed
```
```  1613 qed
```
```  1614
```
```  1615 lemma integral_component_le:
```
```  1616   fixes g f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1617   assumes "k \<in> Basis"
```
```  1618     and "f integrable_on s" "g integrable_on s"
```
```  1619     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> (g x)\<bullet>k"
```
```  1620   shows "(integral s f)\<bullet>k \<le> (integral s g)\<bullet>k"
```
```  1621   apply (rule has_integral_component_le)
```
```  1622   using integrable_integral assms
```
```  1623   apply auto
```
```  1624   done
```
```  1625
```
```  1626 lemma has_integral_component_nonneg:
```
```  1627   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1628   assumes "k \<in> Basis"
```
```  1629     and "(f has_integral i) s"
```
```  1630     and "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  1631   shows "0 \<le> i\<bullet>k"
```
```  1632   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
```
```  1633   using assms(3-)
```
```  1634   by auto
```
```  1635
```
```  1636 lemma integral_component_nonneg:
```
```  1637   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1638   assumes "k \<in> Basis"
```
```  1639     and  "\<forall>x\<in>s. 0 \<le> (f x)\<bullet>k"
```
```  1640   shows "0 \<le> (integral s f)\<bullet>k"
```
```  1641 proof (cases "f integrable_on s")
```
```  1642   case True show ?thesis
```
```  1643     apply (rule has_integral_component_nonneg)
```
```  1644     using assms True
```
```  1645     apply auto
```
```  1646     done
```
```  1647 next
```
```  1648   case False then show ?thesis by (simp add: not_integrable_integral)
```
```  1649 qed
```
```  1650
```
```  1651 lemma has_integral_component_neg:
```
```  1652   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1653   assumes "k \<in> Basis"
```
```  1654     and "(f has_integral i) s"
```
```  1655     and "\<forall>x\<in>s. (f x)\<bullet>k \<le> 0"
```
```  1656   shows "i\<bullet>k \<le> 0"
```
```  1657   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
```
```  1658   by auto
```
```  1659
```
```  1660 lemma has_integral_component_lbound:
```
```  1661   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1662   assumes "(f has_integral i) (cbox a b)"
```
```  1663     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  1664     and "k \<in> Basis"
```
```  1665   shows "B * content (cbox a b) \<le> i\<bullet>k"
```
```  1666   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-)
```
```  1667   by (auto simp add: field_simps)
```
```  1668
```
```  1669 lemma has_integral_component_ubound:
```
```  1670   fixes f::"'a::euclidean_space => 'b::euclidean_space"
```
```  1671   assumes "(f has_integral i) (cbox a b)"
```
```  1672     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  1673     and "k \<in> Basis"
```
```  1674   shows "i\<bullet>k \<le> B * content (cbox a b)"
```
```  1675   using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"] assms(2-)
```
```  1676   by (auto simp add: field_simps)
```
```  1677
```
```  1678 lemma integral_component_lbound:
```
```  1679   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1680   assumes "f integrable_on cbox a b"
```
```  1681     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
```
```  1682     and "k \<in> Basis"
```
```  1683   shows "B * content (cbox a b) \<le> (integral(cbox a b) f)\<bullet>k"
```
```  1684   apply (rule has_integral_component_lbound)
```
```  1685   using assms
```
```  1686   unfolding has_integral_integral
```
```  1687   apply auto
```
```  1688   done
```
```  1689
```
```  1690 lemma integral_component_lbound_real:
```
```  1691   assumes "f integrable_on {a ::real .. b}"
```
```  1692     and "\<forall>x\<in>{a .. b}. B \<le> f(x)\<bullet>k"
```
```  1693     and "k \<in> Basis"
```
```  1694   shows "B * content {a .. b} \<le> (integral {a .. b} f)\<bullet>k"
```
```  1695   using assms
```
```  1696   by (metis box_real(2) integral_component_lbound)
```
```  1697
```
```  1698 lemma integral_component_ubound:
```
```  1699   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
```
```  1700   assumes "f integrable_on cbox a b"
```
```  1701     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
```
```  1702     and "k \<in> Basis"
```
```  1703   shows "(integral (cbox a b) f)\<bullet>k \<le> B * content (cbox a b)"
```
```  1704   apply (rule has_integral_component_ubound)
```
```  1705   using assms
```
```  1706   unfolding has_integral_integral
```
```  1707   apply auto
```
```  1708   done
```
```  1709
```
```  1710 lemma integral_component_ubound_real:
```
```  1711   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```  1712   assumes "f integrable_on {a .. b}"
```
```  1713     and "\<forall>x\<in>{a .. b}. f x\<bullet>k \<le> B"
```
```  1714     and "k \<in> Basis"
```
```  1715   shows "(integral {a .. b} f)\<bullet>k \<le> B * content {a .. b}"
```
```  1716   using assms
```
```  1717   by (metis box_real(2) integral_component_ubound)
```
```  1718
```
```  1719 subsection \<open>Uniform limit of integrable functions is integrable.\<close>
```
```  1720
```
```  1721 lemma real_arch_invD:
```
```  1722   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
```
```  1723   by (subst(asm) real_arch_inverse)
```
```  1724
```
```  1725 lemma integrable_uniform_limit:
```
```  1726   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  1727   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"
```
```  1728   shows "f integrable_on cbox a b"
```
```  1729 proof (cases "content (cbox a b) > 0")
```
```  1730   case False then show ?thesis
```
```  1731       using has_integral_null
```
```  1732       by (simp add: content_lt_nz integrable_on_def)
```
```  1733 next
```
```  1734   case True
```
```  1735   have *: "\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n + 1))"
```
```  1736     by auto
```
```  1737   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
```
```  1738   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]]
```
```  1739   obtain i where i: "\<And>x. (g x has_integral i x) (cbox a b)"
```
```  1740       by auto
```
```  1741   have "Cauchy i"
```
```  1742     unfolding Cauchy_def
```
```  1743   proof clarify
```
```  1744     fix e :: real
```
```  1745     assume "e>0"
```
```  1746     then have "e / 4 / content (cbox a b) > 0"
```
```  1747       using True by (auto simp add: field_simps)
```
```  1748     then obtain M :: nat
```
```  1749          where M: "M \<noteq> 0" "0 < inverse (real_of_nat M)" "inverse (of_nat M) < e / 4 / content (cbox a b)"
```
```  1750       by (subst (asm) real_arch_inverse) auto
```
```  1751     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e"
```
```  1752     proof (rule exI [where x=M], clarify)
```
```  1753       fix m n
```
```  1754       assume m: "M \<le> m" and n: "M \<le> n"
```
```  1755       have "e/4>0" using \<open>e>0\<close> by auto
```
```  1756       note * = i[unfolded has_integral,rule_format,OF this]
```
```  1757       from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
```
```  1758       from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
```
```  1759       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b]
```
```  1760       obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine p"
```
```  1761         by auto
```
```  1762       { fix s1 s2 i1 and i2::'b
```
```  1763         assume no: "norm(s2 - s1) \<le> e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4"
```
```  1764         have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
```
```  1765           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
```
```  1766           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
```
```  1767           by (auto simp add: algebra_simps)
```
```  1768         also have "\<dots> < e"
```
```  1769           using no
```
```  1770           unfolding norm_minus_commute
```
```  1771           by (auto simp add: algebra_simps)
```
```  1772         finally have "norm (i1 - i2) < e" .
```
```  1773       } note triangle3 = this
```
```  1774       have finep: "gm fine p" "gn fine p"
```
```  1775         using fine_inter p  by auto
```
```  1776       { fix x
```
```  1777         assume x: "x \<in> cbox a b"
```
```  1778         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)"
```
```  1779           using g(1)[OF x, of n] g(1)[OF x, of m] by auto
```
```  1780         also have "\<dots> \<le> inverse (real M) + inverse (real M)"
```
```  1781           apply (rule add_mono)
```
```  1782           using M(2) m n by auto
```
```  1783         also have "\<dots> = 2 / real M"
```
```  1784           unfolding divide_inverse by auto
```
```  1785         finally have "norm (g n x - g m x) \<le> 2 / real M"
```
```  1786           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
```
```  1787           by (auto simp add: algebra_simps simp add: norm_minus_commute)
```
```  1788       } note norm_le = this
```
```  1789       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"
```
```  1790         apply (rule order_trans [OF rsum_diff_bound[OF p(1), where e="2 / real M"]])
```
```  1791         apply (blast intro: norm_le)
```
```  1792         using M True
```
```  1793         by (auto simp add: field_simps)
```
```  1794       then show "dist (i m) (i n) < e"
```
```  1795         unfolding dist_norm
```
```  1796         using gm gn p finep
```
```  1797         by (auto intro!: triangle3)
```
```  1798     qed
```
```  1799   qed
```
```  1800   then obtain s where s: "i \<longlonglongrightarrow> s"
```
```  1801     using convergent_eq_cauchy[symmetric] by blast
```
```  1802   show ?thesis
```
```  1803     unfolding integrable_on_def has_integral
```
```  1804   proof (rule_tac x=s in exI, clarify)
```
```  1805     fix e::real
```
```  1806     assume e: "0 < e"
```
```  1807     then have *: "e/3 > 0" by auto
```
```  1808     then obtain N1 where N1: "\<forall>n\<ge>N1. norm (i n - s) < e / 3"
```
```  1809       using LIMSEQ_D [OF s] by metis
```
```  1810     from e True have "e / 3 / content (cbox a b) > 0"
```
```  1811       by (auto simp add: field_simps)
```
```  1812     from real_arch_invD[OF this] guess N2 by (elim exE conjE) note N2=this
```
```  1813     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
```
```  1814     { fix sf sg i
```
```  1815       assume no: "norm (sf - sg) \<le> e / 3"
```
```  1816                  "norm(i - s) < e / 3"
```
```  1817                  "norm (sg - i) < e / 3"
```
```  1818       have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"
```
```  1819         using norm_triangle_ineq[of "sf - sg" "sg - s"]
```
```  1820         using norm_triangle_ineq[of "sg -  i" " i - s"]
```
```  1821         by (auto simp add: algebra_simps)
```
```  1822       also have "\<dots> < e"
```
```  1823         using no
```
```  1824         unfolding norm_minus_commute
```
```  1825         by (auto simp add: algebra_simps)
```
```  1826       finally have "norm (sf - s) < e" .
```
```  1827     } note lem = this
```
```  1828     { fix p
```
```  1829       assume p: "p tagged_division_of (cbox a b) \<and> g' fine p"
```
```  1830       then have norm_less: "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g (N1 + N2) x) - i (N1 + N2)) < e / 3"
```
```  1831         using g' by blast
```
```  1832       have "content (cbox a b) < e / 3 * (of_nat N2)"
```
```  1833         using N2 unfolding inverse_eq_divide using True by (auto simp add: field_simps)
```
```  1834       moreover have "e / 3 * of_nat N2 \<le> e / 3 * (of_nat (N1 + N2) + 1)"
```
```  1835         using \<open>e>0\<close> by auto
```
```  1836       ultimately have "content (cbox a b) < e / 3 * (of_nat (N1 + N2) + 1)"
```
```  1837         by linarith
```
```  1838       then have le_e3: "inverse (real (N1 + N2) + 1) * content (cbox a b) \<le> e / 3"
```
```  1839         unfolding inverse_eq_divide
```
```  1840         by (auto simp add: field_simps)
```
```  1841       have ne3: "norm (i (N1 + N2) - s) < e / 3"
```
```  1842         using N1 by auto
```
```  1843       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e"
```
```  1844         apply (rule lem[OF order_trans [OF _ le_e3] ne3 norm_less])
```
```  1845         apply (rule rsum_diff_bound[OF p[THEN conjunct1]])
```
```  1846         apply (blast intro: g)
```
```  1847         done }
```
```  1848     then show "\<exists>d. gauge d \<and>
```
```  1849              (\<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)"
```
```  1850       by (blast intro: g')
```
```  1851   qed
```
```  1852 qed
```
```  1853
```
```  1854 lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
```
```  1855
```
```  1856
```
```  1857 subsection \<open>Negligible sets.\<close>
```
```  1858
```
```  1859 definition "negligible (s:: 'a::euclidean_space set) \<longleftrightarrow>
```
```  1860   (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) (cbox a b))"
```
```  1861
```
```  1862
```
```  1863 subsection \<open>Negligibility of hyperplane.\<close>
```
```  1864
```
```  1865 lemma content_doublesplit:
```
```  1866   fixes a :: "'a::euclidean_space"
```
```  1867   assumes "0 < e"
```
```  1868     and k: "k \<in> Basis"
```
```  1869   obtains d where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) < e"
```
```  1870 proof cases
```
```  1871   assume *: "a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j)"
```
```  1872   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
```
```  1873   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
```
```  1874
```
```  1875   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)"
```
```  1876     by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros)
```
```  1877   also have "(\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j) = 0"
```
```  1878     using k *
```
```  1879     by (intro prod_zero bexI[OF _ k])
```
```  1880        (auto simp: b'_def a'_def inner_diff inner_sum_left inner_not_same_Basis intro!: sum.cong)
```
```  1881   also have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> 0) (at_right 0) =
```
```  1882     ((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)"
```
```  1883   proof (intro tendsto_cong eventually_at_rightI)
```
```  1884     fix d :: real assume d: "d \<in> {0<..<1}"
```
```  1885     have "cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d} = cbox (a' d) (b' d)" for d
```
```  1886       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)
```
```  1887     moreover have "j \<in> Basis \<Longrightarrow> a' d \<bullet> j \<le> b' d \<bullet> j" for j
```
```  1888       using * d k by (auto simp: a'_def b'_def)
```
```  1889     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})"
```
```  1890       by simp
```
```  1891   qed simp
```
```  1892   finally have "((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)" .
```
```  1893   from order_tendstoD(2)[OF this \<open>0<e\<close>]
```
```  1894   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"
```
```  1895     by (subst (asm) eventually_at_right[of _ 1]) auto
```
```  1896   show ?thesis
```
```  1897     by (rule that[of "d'/2"], insert \<open>0<d'\<close> d'[of "d'/2"], auto)
```
```  1898 next
```
```  1899   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))"
```
```  1900   then have "(\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j) \<or> (c < a \<bullet> k \<or> b \<bullet> k < c)"
```
```  1901     by (auto simp: not_le)
```
```  1902   show thesis
```
```  1903   proof cases
```
```  1904     assume "\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j"
```
```  1905     then have [simp]: "cbox a b = {}"
```
```  1906       using box_ne_empty(1)[of a b] by auto
```
```  1907     show ?thesis
```
```  1908       by (rule that[of 1]) (simp_all add: \<open>0<e\<close>)
```
```  1909   next
```
```  1910     assume "\<not> (\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j)"
```
```  1911     with * have "c < a \<bullet> k \<or> b \<bullet> k < c"
```
```  1912       by auto
```
```  1913     then show thesis
```
```  1914     proof
```
```  1915       assume c: "c < a \<bullet> k"
```
```  1916       moreover have "x \<in> cbox a b \<Longrightarrow> c \<le> x \<bullet> k" for x
```
```  1917         using k c by (auto simp: cbox_def)
```
```  1918       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (a \<bullet> k - c) / 2} = {}"
```
```  1919         using k by (auto simp: cbox_def)
```
```  1920       with \<open>0<e\<close> c that[of "(a \<bullet> k - c) / 2"] show ?thesis
```
```  1921         by auto
```
```  1922     next
```
```  1923       assume c: "b \<bullet> k < c"
```
```  1924       moreover have "x \<in> cbox a b \<Longrightarrow> x \<bullet> k \<le> c" for x
```
```  1925         using k c by (auto simp: cbox_def)
```
```  1926       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (c - b \<bullet> k) / 2} = {}"
```
```  1927         using k by (auto simp: cbox_def)
```
```  1928       with \<open>0<e\<close> c that[of "(c - b \<bullet> k) / 2"] show ?thesis
```
```  1929         by auto
```
```  1930     qed
```
```  1931   qed
```
```  1932 qed
```
```  1933
```
```  1934
```
```  1935 lemma negligible_standard_hyperplane[intro]:
```
```  1936   fixes k :: "'a::euclidean_space"
```
```  1937   assumes k: "k \<in> Basis"
```
```  1938   shows "negligible {x. x\<bullet>k = c}"
```
```  1939   unfolding negligible_def has_integral
```
```  1940 proof (clarify, goal_cases)
```
```  1941   case (1 a b e)
```
```  1942   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"
```
```  1943     by (rule content_doublesplit)
```
```  1944   let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
```
```  1945   show ?case
```
```  1946     apply (rule_tac x="\<lambda>x. ball x d" in exI)
```
```  1947     apply rule
```
```  1948     apply (rule gauge_ball)
```
```  1949     apply (rule d)
```
```  1950   proof (rule, rule)
```
```  1951     fix p
```
```  1952     assume p: "p tagged_division_of (cbox a b) \<and> (\<lambda>x. ball x d) fine p"
```
```  1953     have *: "(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) =
```
```  1954       (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
```
```  1955       apply (rule sum.cong)
```
```  1956       apply (rule refl)
```
```  1957       unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
```
```  1958       apply cases
```
```  1959       apply (rule disjI1)
```
```  1960       apply assumption
```
```  1961       apply (rule disjI2)
```
```  1962     proof -
```
```  1963       fix x l
```
```  1964       assume as: "(x, l) \<in> p" "?i x \<noteq> 0"
```
```  1965       then have xk: "x\<bullet>k = c"
```
```  1966         unfolding indicator_def
```
```  1967         apply -
```
```  1968         apply (rule ccontr)
```
```  1969         apply auto
```
```  1970         done
```
```  1971       show "content l = content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  1972         apply (rule arg_cong[where f=content])
```
```  1973         apply (rule set_eqI)
```
```  1974         apply rule
```
```  1975         apply rule
```
```  1976         unfolding mem_Collect_eq
```
```  1977       proof -
```
```  1978         fix y
```
```  1979         assume y: "y \<in> l"
```
```  1980         note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
```
```  1981         note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y]
```
```  1982         note le_less_trans[OF Basis_le_norm[OF k] this]
```
```  1983         then show "\<bar>y \<bullet> k - c\<bar> \<le> d"
```
```  1984           unfolding inner_simps xk by auto
```
```  1985       qed auto
```
```  1986     qed
```
```  1987     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
```
```  1988     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e"
```
```  1989       unfolding diff_0_right *
```
```  1990       unfolding real_scaleR_def real_norm_def
```
```  1991       apply (subst abs_of_nonneg)
```
```  1992       apply (rule sum_nonneg)
```
```  1993       apply rule
```
```  1994       unfolding split_paired_all split_conv
```
```  1995       apply (rule mult_nonneg_nonneg)
```
```  1996       apply (drule p'(4))
```
```  1997       apply (erule exE)+
```
```  1998       apply(rule_tac b=b in back_subst)
```
```  1999       prefer 2
```
```  2000       apply (subst(asm) eq_commute)
```
```  2001       apply assumption
```
```  2002       apply (subst interval_doublesplit[OF k])
```
```  2003       apply (rule content_pos_le)
```
```  2004       apply (rule indicator_pos_le)
```
```  2005     proof -
```
```  2006       have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le>
```
```  2007         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
```
```  2008         apply (rule sum_mono)
```
```  2009         unfolding split_paired_all split_conv
```
```  2010         apply (rule mult_right_le_one_le)
```
```  2011         apply (drule p'(4))
```
```  2012         apply (auto simp add:interval_doublesplit[OF k])
```
```  2013         done
```
```  2014       also have "\<dots> < e"
```
```  2015       proof (subst sum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
```
```  2016         case prems: (1 u v)
```
```  2017         then have *: "content (cbox u v) = 0"
```
```  2018           unfolding content_eq_0_interior by simp
```
```  2019         have "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content (cbox u v)"
```
```  2020           unfolding interval_doublesplit[OF k]
```
```  2021           apply (rule content_subset)
```
```  2022           unfolding interval_doublesplit[symmetric,OF k]
```
```  2023           apply auto
```
```  2024           done
```
```  2025         then show ?case
```
```  2026           unfolding * interval_doublesplit[OF k]
```
```  2027           by (blast intro: antisym)
```
```  2028       next
```
```  2029         have "(\<Sum>l\<in>snd ` p. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
```
```  2030           sum 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> {}})"
```
```  2031         proof (subst (2) sum.reindex_nontrivial)
```
```  2032           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> {}}"
```
```  2033             "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}"
```
```  2034           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> {}"
```
```  2035             by (auto)
```
```  2036           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) = {}"
```
```  2037             by auto
```
```  2038           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)"
```
```  2039             by (auto intro: interior_mono)
```
```  2040           ultimately have "interior (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}"
```
```  2041             by (auto simp: eq)
```
```  2042           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
```
```  2043             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)
```
```  2044         qed (insert p'(1), auto intro!: sum.mono_neutral_right)
```
```  2045         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)"
```
```  2046           by simp
```
```  2047         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
```
```  2048           using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
```
```  2049           unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
```
```  2050         also have "\<dots> < e"
```
```  2051           using d(2) by simp
```
```  2052         finally show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e" .
```
```  2053       qed
```
```  2054       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
```
```  2055     qed
```
```  2056   qed
```
```  2057 qed
```
```  2058
```
```  2059
```
```  2060
```
```  2061 subsection \<open>Hence the main theorem about negligible sets.\<close>
```
```  2062
```
```  2063 lemma has_integral_negligible:
```
```  2064   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2065   assumes "negligible s"
```
```  2066     and "\<forall>x\<in>(t - s). f x = 0"
```
```  2067   shows "(f has_integral 0) t"
```
```  2068 proof -
```
```  2069   presume P: "\<And>f::'b::euclidean_space \<Rightarrow> 'a.
```
```  2070     \<And>a b. \<forall>x. x \<notin> s \<longrightarrow> f x = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
```
```  2071   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"
```
```  2072   show ?thesis
```
```  2073     apply (rule_tac f="?f" in has_integral_eq)
```
```  2074     unfolding if_P
```
```  2075     apply (rule refl)
```
```  2076     apply (subst has_integral_alt)
```
```  2077     apply cases
```
```  2078     apply (subst if_P, assumption)
```
```  2079     unfolding if_not_P
```
```  2080   proof -
```
```  2081     assume "\<exists>a b. t = cbox a b"
```
```  2082     then guess a b apply - by (erule exE)+ note t = this
```
```  2083     show "(?f has_integral 0) t"
```
```  2084       unfolding t
```
```  2085       apply (rule P)
```
```  2086       using assms(2)
```
```  2087       unfolding t
```
```  2088       apply auto
```
```  2089       done
```
```  2090   next
```
```  2091     show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  2092       (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) (cbox a b) \<and> norm (z - 0) < e)"
```
```  2093       apply safe
```
```  2094       apply (rule_tac x=1 in exI)
```
```  2095       apply rule
```
```  2096       apply (rule zero_less_one)
```
```  2097       apply safe
```
```  2098       apply (rule_tac x=0 in exI)
```
```  2099       apply rule
```
```  2100       apply (rule P)
```
```  2101       using assms(2)
```
```  2102       apply auto
```
```  2103       done
```
```  2104   qed
```
```  2105 next
```
```  2106   fix f :: "'b \<Rightarrow> 'a"
```
```  2107   fix a b :: 'b
```
```  2108   assume assm: "\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  2109   show "(f has_integral 0) (cbox a b)"
```
```  2110     unfolding has_integral
```
```  2111   proof (safe, goal_cases)
```
```  2112     case prems: (1 e)
```
```  2113     then have "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
```
```  2114       apply -
```
```  2115       apply (rule divide_pos_pos)
```
```  2116       defer
```
```  2117       apply (rule mult_pos_pos)
```
```  2118       apply (auto simp add:field_simps)
```
```  2119       done
```
```  2120     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b]
```
```  2121     note allI[OF this,of "\<lambda>x. x"]
```
```  2122     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
```
```  2123     show ?case
```
```  2124       apply (rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI)
```
```  2125     proof safe
```
```  2126       show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)"
```
```  2127         using d(1) unfolding gauge_def by auto
```
```  2128       fix p
```
```  2129       assume as: "p tagged_division_of (cbox a b)" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p"
```
```  2130       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
```
```  2131       {
```
```  2132         presume "p \<noteq> {} \<Longrightarrow> ?goal"
```
```  2133         then show ?goal
```
```  2134           apply (cases "p = {}")
```
```  2135           using prems
```
```  2136           apply auto
```
```  2137           done
```
```  2138       }
```
```  2139       assume as': "p \<noteq> {}"
```
```  2140       from real_arch_simple[of "Max((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..
```
```  2141       then have N: "\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N"
```
```  2142         by (meson Max_ge as(1) dual_order.trans finite_imageI tagged_division_of_finite)
```
```  2143       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)"
```
```  2144         by (auto intro: tagged_division_finer[OF as(1) d(1)])
```
```  2145       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
```
```  2146       have *: "\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> (0::real)"
```
```  2147         apply (rule sum_nonneg)
```
```  2148         apply safe
```
```  2149         unfolding real_scaleR_def
```
```  2150         apply (drule tagged_division_ofD(4)[OF q(1)])
```
```  2151         apply (auto intro: mult_nonneg_nonneg)
```
```  2152         done
```
```  2153       have **: "finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow>
```
```  2154         (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> sum f s \<le> sum g t" for f g s t
```
```  2155         apply (rule sum_le_included[of s t g snd f])
```
```  2156         prefer 4
```
```  2157         apply safe
```
```  2158         apply (erule_tac x=x in ballE)
```
```  2159         apply (erule exE)
```
```  2160         apply (rule_tac x="(xa,x)" in bexI)
```
```  2161         apply auto
```
```  2162         done
```
```  2163       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> sum (\<lambda>i. (real i + 1) *
```
```  2164         norm (sum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
```
```  2165         unfolding real_norm_def sum_distrib_left abs_of_nonneg[OF *] diff_0_right
```
```  2166         apply (rule order_trans)
```
```  2167         apply (rule norm_sum)
```
```  2168         apply (subst sum_sum_product)
```
```  2169         prefer 3
```
```  2170       proof (rule **, safe)
```
```  2171         show "finite {(i, j) |i j. i \<in> {..N + 1} \<and> j \<in> q i}"
```
```  2172           apply (rule finite_product_dependent)
```
```  2173           using q
```
```  2174           apply auto
```
```  2175           done
```
```  2176         fix i a b
```
```  2177         assume as'': "(a, b) \<in> q i"
```
```  2178         show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"
```
```  2179           unfolding real_scaleR_def
```
```  2180           using tagged_division_ofD(4)[OF q(1) as'']
```
```  2181           by (auto intro!: mult_nonneg_nonneg)
```
```  2182       next
```
```  2183         fix i :: nat
```
```  2184         show "finite (q i)"
```
```  2185           using q by auto
```
```  2186       next
```
```  2187         fix x k
```
```  2188         assume xk: "(x, k) \<in> p"
```
```  2189         define n where "n = nat \<lfloor>norm (f x)\<rfloor>"
```
```  2190         have *: "norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p"
```
```  2191           using xk by auto
```
```  2192         have nfx: "real n \<le> norm (f x)" "norm (f x) \<le> real n + 1"
```
```  2193           unfolding n_def by auto
```
```  2194         then have "n \<in> {0..N + 1}"
```
```  2195           using N[rule_format,OF *] by auto
```
```  2196         moreover
```
```  2197         note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
```
```  2198         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this]
```
```  2199         note this[unfolded n_def[symmetric]]
```
```  2200         moreover
```
```  2201         have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"
```
```  2202         proof (cases "x \<in> s")
```
```  2203           case False
```
```  2204           then show ?thesis
```
```  2205             using assm by auto
```
```  2206         next
```
```  2207           case True
```
```  2208           have *: "content k \<ge> 0"
```
```  2209             using tagged_division_ofD(4)[OF as(1) xk] by auto
```
```  2210           moreover
```
```  2211           have "content k * norm (f x) \<le> content k * (real n + 1)"
```
```  2212             apply (rule mult_mono)
```
```  2213             using nfx *
```
```  2214             apply auto
```
```  2215             done
```
```  2216           ultimately
```
```  2217           show ?thesis
```
```  2218             unfolding abs_mult
```
```  2219             using nfx True
```
```  2220             by (auto simp add: field_simps)
```
```  2221         qed
```
```  2222         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>
```
```  2223           (real y + 1) * (content k *\<^sub>R indicator s x)"
```
```  2224           apply (rule_tac x=n in exI)
```
```  2225           apply safe
```
```  2226           apply (rule_tac x=n in exI)
```
```  2227           apply (rule_tac x="(x,k)" in exI)
```
```  2228           apply safe
```
```  2229           apply auto
```
```  2230           done
```
```  2231       qed (insert as, auto)
```
```  2232       also have "\<dots> \<le> sum (\<lambda>i. e / 2 / 2 ^ i) {..N+1}"
```
```  2233       proof (rule sum_mono, goal_cases)
```
```  2234         case (1 i)
```
```  2235         then show ?case
```
```  2236           apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
```
```  2237           using d(2)[rule_format, of "q i" i]
```
```  2238           using q[rule_format]
```
```  2239           apply (auto simp add: field_simps)
```
```  2240           done
```
```  2241       qed
```
```  2242       also have "\<dots> < e * inverse 2 * 2"
```
```  2243         unfolding divide_inverse sum_distrib_left[symmetric]
```
```  2244         apply (rule mult_strict_left_mono)
```
```  2245         unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
```
```  2246         apply (subst geometric_sum)
```
```  2247         using prems
```
```  2248         apply auto
```
```  2249         done
```
```  2250       finally show "?goal" by auto
```
```  2251     qed
```
```  2252   qed
```
```  2253 qed
```
```  2254
```
```  2255 lemma has_integral_spike:
```
```  2256   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
```
```  2257   assumes "negligible s"
```
```  2258     and "(\<forall>x\<in>(t - s). g x = f x)"
```
```  2259     and "(f has_integral y) t"
```
```  2260   shows "(g has_integral y) t"
```
```  2261 proof -
```
```  2262   {
```
```  2263     fix a b :: 'b
```
```  2264     fix f g :: "'b \<Rightarrow> 'a"
```
```  2265     fix y :: 'a
```
```  2266     assume as: "\<forall>x \<in> cbox a b - s. g x = f x" "(f has_integral y) (cbox a b)"
```
```  2267     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
```
```  2268       apply (rule has_integral_add[OF as(2)])
```
```  2269       apply (rule has_integral_negligible[OF assms(1)])
```
```  2270       using as
```
```  2271       apply auto
```
```  2272       done
```
```  2273     then have "(g has_integral y) (cbox a b)"
```
```  2274       by auto
```
```  2275   } note * = this
```
```  2276   show ?thesis
```
```  2277     apply (subst has_integral_alt)
```
```  2278     using assms(2-)
```
```  2279     apply -
```
```  2280     apply (rule cond_cases)
```
```  2281     apply safe
```
```  2282     apply (rule *)
```
```  2283     apply assumption+
```
```  2284     apply (subst(asm) has_integral_alt)
```
```  2285     unfolding if_not_P
```
```  2286     apply (erule_tac x=e in allE)
```
```  2287     apply safe
```
```  2288     apply (rule_tac x=B in exI)
```
```  2289     apply safe
```
```  2290     apply (erule_tac x=a in allE)
```
```  2291     apply (erule_tac x=b in allE)
```
```  2292     apply safe
```
```  2293     apply (rule_tac x=z in exI)
```
```  2294     apply safe
```
```  2295     apply (rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"])
```
```  2296     apply auto
```
```  2297     done
```
```  2298 qed
```
```  2299
```
```  2300 lemma has_integral_spike_eq:
```
```  2301   assumes "negligible s"
```
```  2302     and "\<forall>x\<in>(t - s). g x = f x"
```
```  2303   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2304   apply rule
```
```  2305   apply (rule_tac[!] has_integral_spike[OF assms(1)])
```
```  2306   using assms(2)
```
```  2307   apply auto
```
```  2308   done
```
```  2309
```
```  2310 lemma integrable_spike:
```
```  2311   assumes "negligible s"
```
```  2312     and "\<forall>x\<in>(t - s). g x = f x"
```
```  2313     and "f integrable_on t"
```
```  2314   shows "g integrable_on  t"
```
```  2315   using assms
```
```  2316   unfolding integrable_on_def
```
```  2317   apply -
```
```  2318   apply (erule exE)
```
```  2319   apply rule
```
```  2320   apply (rule has_integral_spike)
```
```  2321   apply fastforce+
```
```  2322   done
```
```  2323
```
```  2324 lemma integral_spike:
```
```  2325   assumes "negligible s"
```
```  2326     and "\<forall>x\<in>(t - s). g x = f x"
```
```  2327   shows "integral t f = integral t g"
```
```  2328   using has_integral_spike_eq[OF assms] by (simp add: integral_def integrable_on_def)
```
```  2329
```
```  2330
```
```  2331 subsection \<open>Some other trivialities about negligible sets.\<close>
```
```  2332
```
```  2333 lemma negligible_subset:
```
```  2334   assumes "negligible s" "t \<subseteq> s"
```
```  2335   shows "negligible t"
```
```  2336   unfolding negligible_def
```
```  2337     by (metis (no_types) Diff_iff assms contra_subsetD has_integral_negligible indicator_simps(2))
```
```  2338
```
```  2339 lemma negligible_diff[intro?]:
```
```  2340   assumes "negligible s"
```
```  2341   shows "negligible (s - t)"
```
```  2342   using assms by (meson Diff_subset negligible_subset)
```
```  2343
```
```  2344 lemma negligible_Int:
```
```  2345   assumes "negligible s \<or> negligible t"
```
```  2346   shows "negligible (s \<inter> t)"
```
```  2347   using assms negligible_subset by force
```
```  2348
```
```  2349 lemma negligible_Un:
```
```  2350   assumes "negligible s"
```
```  2351     and "negligible t"
```
```  2352   shows "negligible (s \<union> t)"
```
```  2353   unfolding negligible_def
```
```  2354 proof (safe, goal_cases)
```
```  2355   case (1 a b)
```
```  2356   note assm = assms[unfolded negligible_def,rule_format,of a b]
```
```  2357   then show ?case
```
```  2358     apply (subst has_integral_spike_eq[OF assms(2)])
```
```  2359     defer
```
```  2360     apply assumption
```
```  2361     unfolding indicator_def
```
```  2362     apply auto
```
```  2363     done
```
```  2364 qed
```
```  2365
```
```  2366 lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
```
```  2367   using negligible_Un negligible_subset by blast
```
```  2368
```
```  2369 lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
```
```  2370   using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] negligible_subset by blast
```
```  2371
```
```  2372 lemma negligible_insert[simp]: "negligible (insert a s) \<longleftrightarrow> negligible s"
```
```  2373   apply (subst insert_is_Un)
```
```  2374   unfolding negligible_Un_eq
```
```  2375   apply auto
```
```  2376   done
```
```  2377
```
```  2378 lemma negligible_empty[iff]: "negligible {}"
```
```  2379   using negligible_insert by blast
```
```  2380
```
```  2381 lemma negligible_finite[intro]:
```
```  2382   assumes "finite s"
```
```  2383   shows "negligible s"
```
```  2384   using assms by (induct s) auto
```
```  2385
```
```  2386 lemma negligible_Union[intro]:
```
```  2387   assumes "finite s"
```
```  2388     and "\<forall>t\<in>s. negligible t"
```
```  2389   shows "negligible(\<Union>s)"
```
```  2390   using assms by induct auto
```
```  2391
```
```  2392 lemma negligible:
```
```  2393   "negligible s \<longleftrightarrow> (\<forall>t::('a::euclidean_space) set. ((indicator s::'a\<Rightarrow>real) has_integral 0) t)"
```
```  2394   apply safe
```
```  2395   defer
```
```  2396   apply (subst negligible_def)
```
```  2397 proof -
```
```  2398   fix t :: "'a set"
```
```  2399   assume as: "negligible s"
```
```  2400   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)"
```
```  2401     by auto
```
```  2402   show "((indicator s::'a\<Rightarrow>real) has_integral 0) t"
```
```  2403     apply (subst has_integral_alt)
```
```  2404     apply cases
```
```  2405     apply (subst if_P,assumption)
```
```  2406     unfolding if_not_P
```
```  2407     apply safe
```
```  2408     apply (rule as[unfolded negligible_def,rule_format])
```
```  2409     apply (rule_tac x=1 in exI)
```
```  2410     apply safe
```
```  2411     apply (rule zero_less_one)
```
```  2412     apply (rule_tac x=0 in exI)
```
```  2413     using negligible_subset[OF as,of "s \<inter> t"]
```
```  2414     unfolding negligible_def indicator_def [abs_def]
```
```  2415     unfolding *
```
```  2416     apply auto
```
```  2417     done
```
```  2418 qed auto
```
```  2419
```
```  2420
```
```  2421 subsection \<open>Finite case of the spike theorem is quite commonly needed.\<close>
```
```  2422
```
```  2423 lemma has_integral_spike_finite:
```
```  2424   assumes "finite s"
```
```  2425     and "\<forall>x\<in>t-s. g x = f x"
```
```  2426     and "(f has_integral y) t"
```
```  2427   shows "(g has_integral y) t"
```
```  2428   apply (rule has_integral_spike)
```
```  2429   using assms
```
```  2430   apply auto
```
```  2431   done
```
```  2432
```
```  2433 lemma has_integral_spike_finite_eq:
```
```  2434   assumes "finite s"
```
```  2435     and "\<forall>x\<in>t-s. g x = f x"
```
```  2436   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"
```
```  2437   apply rule
```
```  2438   apply (rule_tac[!] has_integral_spike_finite)
```
```  2439   using assms
```
```  2440   apply auto
```
```  2441   done
```
```  2442
```
```  2443 lemma integrable_spike_finite:
```
```  2444   assumes "finite s"
```
```  2445     and "\<forall>x\<in>t-s. g x = f x"
```
```  2446     and "f integrable_on t"
```
```  2447   shows "g integrable_on  t"
```
```  2448   using assms
```
```  2449   unfolding integrable_on_def
```
```  2450   apply safe
```
```  2451   apply (rule_tac x=y in exI)
```
```  2452   apply (rule has_integral_spike_finite)
```
```  2453   apply auto
```
```  2454   done
```
```  2455
```
```  2456
```
```  2457 subsection \<open>In particular, the boundary of an interval is negligible.\<close>
```
```  2458
```
```  2459 lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
```
```  2460 proof -
```
```  2461   let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
```
```  2462   have "cbox a b - box a b \<subseteq> ?A"
```
```  2463     apply rule unfolding Diff_iff mem_box
```
```  2464     apply simp
```
```  2465     apply(erule conjE bexE)+
```
```  2466     apply(rule_tac x=i in bexI)
```
```  2467     apply auto
```
```  2468     done
```
```  2469   then show ?thesis
```
```  2470     apply -
```
```  2471     apply (rule negligible_subset[of ?A])
```
```  2472     apply (rule negligible_Union[OF finite_imageI])
```
```  2473     apply auto
```
```  2474     done
```
```  2475 qed
```
```  2476
```
```  2477 lemma has_integral_spike_interior:
```
```  2478   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  2479     and "(f has_integral y) (cbox a b)"
```
```  2480   shows "(g has_integral y) (cbox a b)"
```
```  2481   apply (rule has_integral_spike[OF negligible_frontier_interval _ assms(2)])
```
```  2482   using assms(1)
```
```  2483   apply auto
```
```  2484   done
```
```  2485
```
```  2486 lemma has_integral_spike_interior_eq:
```
```  2487   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  2488   shows "(f has_integral y) (cbox a b) \<longleftrightarrow> (g has_integral y) (cbox a b)"
```
```  2489   apply rule
```
```  2490   apply (rule_tac[!] has_integral_spike_interior)
```
```  2491   using assms
```
```  2492   apply auto
```
```  2493   done
```
```  2494
```
```  2495 lemma integrable_spike_interior:
```
```  2496   assumes "\<forall>x\<in>box a b. g x = f x"
```
```  2497     and "f integrable_on cbox a b"
```
```  2498   shows "g integrable_on cbox a b"
```
```  2499   using assms
```
```  2500   unfolding integrable_on_def
```
```  2501   using has_integral_spike_interior[OF assms(1)]
```
```  2502   by auto
```
```  2503
```
```  2504
```
```  2505 subsection \<open>Integrability of continuous functions.\<close>
```
```  2506
```
```  2507 lemma operative_approximable:
```
```  2508   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  2509   assumes "0 \<le> e"
```
```  2510   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)"
```
```  2511   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  2512 proof safe
```
```  2513   fix a b :: 'b
```
```  2514   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  2515     if "box a b = {}"
```
```  2516     apply (rule_tac x=f in exI)
```
```  2517     using assms that
```
```  2518     apply (auto simp: content_eq_0_interior)
```
```  2519     done
```
```  2520   {
```
```  2521     fix c g
```
```  2522     fix k :: 'b
```
```  2523     assume as: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  2524     assume k: "k \<in> Basis"
```
```  2525     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}"
```
```  2526       "\<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}"
```
```  2527       apply (rule_tac[!] x=g in exI)
```
```  2528       using as(1) integrable_split[OF as(2) k]
```
```  2529       apply auto
```
```  2530       done
```
```  2531   }
```
```  2532   fix c k g1 g2
```
```  2533   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}"
```
```  2534     "\<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}"
```
```  2535   assume k: "k \<in> Basis"
```
```  2536   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"
```
```  2537   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  2538     apply (rule_tac x="?g" in exI)
```
```  2539     apply safe
```
```  2540   proof goal_cases
```
```  2541     case (1 x)
```
```  2542     then show ?case
```
```  2543       apply -
```
```  2544       apply (cases "x\<bullet>k=c")
```
```  2545       apply (case_tac "x\<bullet>k < c")
```
```  2546       using as assms
```
```  2547       apply auto
```
```  2548       done
```
```  2549   next
```
```  2550     case 2
```
```  2551     presume "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
```
```  2552       and "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
```
```  2553     then guess h1 h2 unfolding integrable_on_def by auto
```
```  2554     from has_integral_split[OF this k] show ?case
```
```  2555       unfolding integrable_on_def by auto
```
```  2556   next
```
```  2557     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}"
```
```  2558       apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
```
```  2559       using k as(2,4)
```
```  2560       apply auto
```
```  2561       done
```
```  2562   qed
```
```  2563 qed
```
```  2564
```
```  2565 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))"
```
```  2566 proof -
```
```  2567   interpret bool: comm_monoid_set "op \<and>" True
```
```  2568     proof qed auto
```
```  2569   show ?thesis
```
```  2570     by (induction s rule: infinite_finite_induct) auto
```
```  2571 qed
```
```  2572
```
```  2573 lemma approximable_on_division:
```
```  2574   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  2575   assumes "0 \<le> e"
```
```  2576     and "d division_of (cbox a b)"
```
```  2577     and "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  2578   obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
```
```  2579 proof -
```
```  2580   note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_approximable[OF assms(1)] assms(2)]
```
```  2581   from assms(3) this[unfolded comm_monoid_set_F_and, of f] division_of_finite[OF assms(2)]
```
```  2582   guess g by auto
```
```  2583   then show thesis
```
```  2584     apply -
```
```  2585     apply (rule that[of g])
```
```  2586     apply auto
```
```  2587     done
```
```  2588 qed
```
```  2589
```
```  2590 lemma integrable_continuous:
```
```  2591   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  2592   assumes "continuous_on (cbox a b) f"
```
```  2593   shows "f integrable_on cbox a b"
```
```  2594 proof (rule integrable_uniform_limit, safe)
```
```  2595   fix e :: real
```
```  2596   assume e: "e > 0"
```
```  2597   from compact_uniformly_continuous[OF assms compact_cbox,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
```
```  2598   note d=conjunctD2[OF this,rule_format]
```
```  2599   from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
```
```  2600   note p' = tagged_division_ofD[OF p(1)]
```
```  2601   have *: "\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
```
```  2602   proof (safe, unfold snd_conv)
```
```  2603     fix x l
```
```  2604     assume as: "(x, l) \<in> p"
```
```  2605     from p'(4)[OF this] guess a b by (elim exE) note l=this
```
```  2606     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l"
```
```  2607       apply (rule_tac x="\<lambda>y. f x" in exI)
```
```  2608     proof safe
```
```  2609       show "(\<lambda>y. f x) integrable_on l"
```
```  2610         unfolding integrable_on_def l
```
```  2611         apply rule
```
```  2612         apply (rule has_integral_const)
```
```  2613         done
```
```  2614       fix y
```
```  2615       assume y: "y \<in> l"
```
```  2616       note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
```
```  2617       note d(2)[OF _ _ this[unfolded mem_ball]]
```
```  2618       then show "norm (f y - f x) \<le> e"
```
```  2619         using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce
```
```  2620     qed
```
```  2621   qed
```
```  2622   from e have "e \<ge> 0"
```
```  2623     by auto
```
```  2624   from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
```
```  2625   then show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
```
```  2626     by auto
```
```  2627 qed
```
```  2628
```
```  2629 lemma integrable_continuous_real:
```
```  2630   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  2631   assumes "continuous_on {a .. b} f"
```
```  2632   shows "f integrable_on {a .. b}"
```
```  2633   by (metis assms box_real(2) integrable_continuous)
```
```  2634
```
```  2635 subsection \<open>Specialization of additivity to one dimension.\<close>
```
```  2636
```
```  2637
```
```  2638 subsection \<open>A useful lemma allowing us to factor out the content size.\<close>
```
```  2639
```
```  2640 lemma has_integral_factor_content:
```
```  2641   "(f has_integral i) (cbox a b) \<longleftrightarrow>
```
```  2642     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  2643       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
```
```  2644 proof (cases "content (cbox a b) = 0")
```
```  2645   case True
```
```  2646   show ?thesis
```
```  2647     unfolding has_integral_null_eq[OF True]
```
```  2648     apply safe
```
```  2649     apply (rule, rule, rule gauge_trivial, safe)
```
```  2650     unfolding sum_content_null[OF True] True
```
```  2651     defer
```
```  2652     apply (erule_tac x=1 in allE)
```
```  2653     apply safe
```
```  2654     defer
```
```  2655     apply (rule fine_division_exists[of _ a b])
```
```  2656     apply assumption
```
```  2657     apply (erule_tac x=p in allE)
```
```  2658     unfolding sum_content_null[OF True]
```
```  2659     apply auto
```
```  2660     done
```
```  2661 next
```
```  2662   case False
```
```  2663   note F = this[unfolded content_lt_nz[symmetric]]
```
```  2664   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
```
```  2665     (\<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)"
```
```  2666   show ?thesis
```
```  2667     apply (subst has_integral)
```
```  2668   proof safe
```
```  2669     fix e :: real
```
```  2670     assume e: "e > 0"
```
```  2671     {
```
```  2672       assume "\<forall>e>0. ?P e op <"
```
```  2673       then show "?P (e * content (cbox a b)) op \<le>"
```
```  2674         apply (erule_tac x="e * content (cbox a b)" in allE)
```
```  2675         apply (erule impE)
```
```  2676         defer
```
```  2677         apply (erule exE,rule_tac x=d in exI)
```
```  2678         using F e
```
```  2679         apply (auto simp add:field_simps)
```
```  2680         done
```
```  2681     }
```
```  2682     {
```
```  2683       assume "\<forall>e>0. ?P (e * content (cbox a b)) op \<le>"
```
```  2684       then show "?P e op <"
```
```  2685         apply (erule_tac x="e / 2 / content (cbox a b)" in allE)
```
```  2686         apply (erule impE)
```
```  2687         defer
```
```  2688         apply (erule exE,rule_tac x=d in exI)
```
```  2689         using F e
```
```  2690         apply (auto simp add: field_simps)
```
```  2691         done
```
```  2692     }
```
```  2693   qed
```
```  2694 qed
```
```  2695
```
```  2696 lemma has_integral_factor_content_real:
```
```  2697   "(f has_integral i) {a .. b::real} \<longleftrightarrow>
```
```  2698     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a .. b}  \<and> d fine p \<longrightarrow>
```
```  2699       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a .. b} ))"
```
```  2700   unfolding box_real[symmetric]
```
```  2701   by (rule has_integral_factor_content)
```
```  2702
```
```  2703
```
```  2704 subsection \<open>Fundamental theorem of calculus.\<close>
```
```  2705
```
```  2706 lemma interval_bounds_real:
```
```  2707   fixes q b :: real
```
```  2708   assumes "a \<le> b"
```
```  2709   shows "Sup {a..b} = b"
```
```  2710     and "Inf {a..b} = a"
```
```  2711   using assms by auto
```
```  2712
```
```  2713 lemma fundamental_theorem_of_calculus:
```
```  2714   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  2715   assumes "a \<le> b"
```
```  2716     and "\<forall>x\<in>{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
```
```  2717   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  2718   unfolding has_integral_factor_content box_real[symmetric]
```
```  2719 proof safe
```
```  2720   fix e :: real
```
```  2721   assume e: "e > 0"
```
```  2722   note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
```
```  2723   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"
```
```  2724     using e by blast
```
```  2725   note this[OF assm,unfolded gauge_existence_lemma]
```
```  2726   from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
```
```  2727   note d=conjunctD2[OF this[rule_format],rule_format]
```
```  2728   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  2729     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
```
```  2730     apply (rule_tac x="\<lambda>x. ball x (d x)" in exI)
```
```  2731     apply safe
```
```  2732     apply (rule gauge_ball_dependent)
```
```  2733     apply rule
```
```  2734     apply (rule d(1))
```
```  2735   proof -
```
```  2736     fix p
```
```  2737     assume as: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
```
```  2738     show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
```
```  2739       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]
```
```  2740       unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "\<lambda>x. x",symmetric]
```
```  2741       unfolding sum_distrib_left
```
```  2742       defer
```
```  2743       unfolding sum_subtractf[symmetric]
```
```  2744     proof (rule sum_norm_le,safe)
```
```  2745       fix x k
```
```  2746       assume "(x, k) \<in> p"
```
```  2747       note xk = tagged_division_ofD(2-4)[OF as(1) this]
```
```  2748       from this(3) guess u v by (elim exE) note k=this
```
```  2749       have *: "u \<le> v"
```
```  2750         using xk unfolding k by auto
```
```  2751       have ball: "\<forall>xa\<in>k. xa \<in> ball x (d x)"
```
```  2752         using as(2)[unfolded fine_def,rule_format,OF \<open>(x,k)\<in>p\<close>,unfolded split_conv subset_eq] .
```
```  2753       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
```
```  2754         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
```
```  2755         apply (rule order_trans[OF _ norm_triangle_ineq4])
```
```  2756         apply (rule eq_refl)
```
```  2757         apply (rule arg_cong[where f=norm])
```
```  2758         unfolding scaleR_diff_left
```
```  2759         apply (auto simp add:algebra_simps)
```
```  2760         done
```
```  2761       also have "\<dots> \<le> e * norm (u - x) + e * norm (v - x)"
```
```  2762         apply (rule add_mono)
```
```  2763         apply (rule d(2)[of "x" "u",unfolded o_def])
```
```  2764         prefer 4
```
```  2765         apply (rule d(2)[of "x" "v",unfolded o_def])
```
```  2766         using ball[rule_format,of u] ball[rule_format,of v]
```
```  2767         using xk(1-2)
```
```  2768         unfolding k subset_eq
```
```  2769         apply (auto simp add:dist_real_def)
```
```  2770         done
```
```  2771       also have "\<dots> \<le> e * (Sup k - Inf k)"
```
```  2772         unfolding k interval_bounds_real[OF *]
```
```  2773         using xk(1)
```
```  2774         unfolding k
```
```  2775         by (auto simp add: dist_real_def field_simps)
```
```  2776       finally show "norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le>
```
```  2777         e * (Sup k - Inf k)"
```
```  2778         unfolding box_real k interval_bounds_real[OF *] content_real[OF *]
```
```  2779           interval_upperbound_real interval_lowerbound_real
```
```  2780           .
```
```  2781     qed
```
```  2782   qed
```
```  2783 qed
```
```  2784
```
```  2785 lemma ident_has_integral:
```
```  2786   fixes a::real
```
```  2787   assumes "a \<le> b"
```
```  2788   shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2) / 2) {a..b}"
```
```  2789 proof -
```
```  2790   have "((\<lambda>x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}"
```
```  2791     apply (rule fundamental_theorem_of_calculus [OF assms], clarify)
```
```  2792     unfolding power2_eq_square
```
```  2793     by (rule derivative_eq_intros | simp)+
```
```  2794   then show ?thesis
```
```  2795     by (simp add: field_simps)
```
```  2796 qed
```
```  2797
```
```  2798 lemma integral_ident [simp]:
```
```  2799   fixes a::real
```
```  2800   assumes "a \<le> b"
```
```  2801   shows "integral {a..b} (\<lambda>x. x) = (if a \<le> b then (b\<^sup>2 - a\<^sup>2) / 2 else 0)"
```
```  2802 using ident_has_integral integral_unique by fastforce
```
```  2803
```
```  2804 lemma ident_integrable_on:
```
```  2805   fixes a::real
```
```  2806   shows "(\<lambda>x. x) integrable_on {a..b}"
```
```  2807 by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
```
```  2808
```
```  2809
```
```  2810 subsection \<open>Taylor series expansion\<close>
```
```  2811
```
```  2812 lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative:
```
```  2813   assumes "p>0"
```
```  2814   and f0: "Df 0 = f"
```
```  2815   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  2816     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  2817   and g0: "Dg 0 = g"
```
```  2818   and Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  2819     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  2820   and ivl: "a \<le> t" "t \<le> b"
```
```  2821   shows "((\<lambda>t. \<Sum>i<p. (-1)^i *\<^sub>R prod (Df i t) (Dg (p - Suc i) t))
```
```  2822     has_vector_derivative
```
```  2823       prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t))
```
```  2824     (at t within {a .. b})"
```
```  2825   using assms
```
```  2826 proof cases
```
```  2827   assume p: "p \<noteq> 1"
```
```  2828   define p' where "p' = p - 2"
```
```  2829   from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
```
```  2830     by (auto simp: p'_def)
```
```  2831   have *: "\<And>i. i \<le> p' \<Longrightarrow> Suc (Suc p' - i) = (Suc (Suc p') - i)"
```
```  2832     by auto
```
```  2833   let ?f = "\<lambda>i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))"
```
```  2834   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  2835     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
```
```  2836     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
```
```  2837     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
```
```  2838   also note sum_telescope
```
```  2839   finally
```
```  2840   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
```
```  2841     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
```
```  2842     = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)"
```
```  2843     unfolding p'[symmetric]
```
```  2844     by (simp add: assms)
```
```  2845   thus ?thesis
```
```  2846     using assms
```
```  2847     by (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  2848 qed (auto intro!: derivative_eq_intros has_vector_derivative)
```
```  2849
```
```  2850 lemma
```
```  2851   fixes f::"real\<Rightarrow>'a::banach"
```
```  2852   assumes "p>0"
```
```  2853   and f0: "Df 0 = f"
```
```  2854   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  2855     (Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
```
```  2856   and ivl: "a \<le> b"
```
```  2857   defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"
```
```  2858   shows taylor_has_integral:
```
```  2859     "(i has_integral f b - (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
```
```  2860   and taylor_integral:
```
```  2861     "f b = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i"
```
```  2862   and taylor_integrable:
```
```  2863     "i integrable_on {a .. b}"
```
```  2864 proof goal_cases
```
```  2865   case 1
```
```  2866   interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a"
```
```  2867     by (rule bounded_bilinear_scaleR)
```
```  2868   define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
```
```  2869   define Dg where [abs_def]:
```
```  2870     "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
```
```  2871   have g0: "Dg 0 = g"
```
```  2872     using \<open>p > 0\<close>
```
```  2873     by (auto simp add: Dg_def divide_simps g_def split: if_split_asm)
```
```  2874   {
```
```  2875     fix m
```
```  2876     assume "p > Suc m"
```
```  2877     hence "p - Suc m = Suc (p - Suc (Suc m))"
```
```  2878       by auto
```
```  2879     hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
```
```  2880       by auto
```
```  2881   } note fact_eq = this
```
```  2882   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  2883     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
```
```  2884     unfolding Dg_def
```
```  2885     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
```
```  2886   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
```
```  2887   from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
```
```  2888       OF \<open>p > 0\<close> g0 Dg f0 Df]
```
```  2889   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
```
```  2890     (?sum has_vector_derivative
```
```  2891       g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})"
```
```  2892     by auto
```
```  2893   from fundamental_theorem_of_calculus[rule_format, OF \<open>a \<le> b\<close> deriv]
```
```  2894   have "(i has_integral ?sum b - ?sum a) {a .. b}"
```
```  2895     using atLeastatMost_empty'[simp del]
```
```  2896     by (simp add: i_def g_def Dg_def)
```
```  2897   also
```
```  2898   have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
```
```  2899     and "{..<p} \<inter> {i. p = Suc i} = {p - 1}"
```
```  2900     for p'
```
```  2901     using \<open>p > 0\<close>
```
```  2902     by (auto simp: power_mult_distrib[symmetric])
```
```  2903   then have "?sum b = f b"
```
```  2904     using Suc_pred'[OF \<open>p > 0\<close>]
```
```  2905     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
```
```  2906         cond_application_beta sum.If_cases f0)
```
```  2907   also
```
```  2908   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
```
```  2909   proof safe
```
```  2910     fix x
```
```  2911     assume "x < p"
```
```  2912     thus "x \<in> (\<lambda>x. p - x - 1) ` {..<p}"
```
```  2913       by (auto intro!: image_eqI[where x = "p - x - 1"])
```
```  2914   qed simp
```
```  2915   from _ this
```
```  2916   have "?sum a = (\<Sum>i<p. ((b - a) ^ i / fact i) *\<^sub>R Df i a)"
```
```  2917     by (rule sum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
```
```  2918   finally show c: ?case .
```
```  2919   case 2 show ?case using c integral_unique by force
```
```  2920   case 3 show ?case using c by force
```
```  2921 qed
```
```  2922
```
```  2923
```
```  2924
```
```  2925 subsection \<open>Only need trivial subintervals if the interval itself is trivial.\<close>
```
```  2926
```
```  2927 lemma division_of_nontrivial:
```
```  2928   fixes s :: "'a::euclidean_space set set"
```
```  2929   assumes "s division_of (cbox a b)"
```
```  2930     and "content (cbox a b) \<noteq> 0"
```
```  2931   shows "{k. k \<in> s \<and> content k \<noteq> 0} division_of (cbox a b)"
```
```  2932   using assms(1)
```
```  2933   apply -
```
```  2934 proof (induct "card s" arbitrary: s rule: nat_less_induct)
```
```  2935   fix s::"'a set set"
```
```  2936   assume assm: "s division_of (cbox a b)"
```
```  2937     "\<forall>m<card s. \<forall>x. m = card x \<longrightarrow>
```
```  2938       x division_of (cbox a b) \<longrightarrow> {k \<in> x. content k \<noteq> 0} division_of (cbox a b)"
```
```  2939   note s = division_ofD[OF assm(1)]
```
```  2940   let ?thesis = "{k \<in> s. content k \<noteq> 0} division_of (cbox a b)"
```
```  2941   {
```
```  2942     presume *: "{k \<in> s. content k \<noteq> 0} \<noteq> s \<Longrightarrow> ?thesis"
```
```  2943     show ?thesis
```
```  2944       apply cases
```
```  2945       defer
```
```  2946       apply (rule *)
```
```  2947       apply assumption
```
```  2948       using assm(1)
```
```  2949       apply auto
```
```  2950       done
```
```  2951   }
```
```  2952   assume noteq: "{k \<in> s. content k \<noteq> 0} \<noteq> s"
```
```  2953   then obtain k where k: "k \<in> s" "content k = 0"
```
```  2954     by auto
```
```  2955   from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
```
```  2956   from k have "card s > 0"
```
```  2957     unfolding card_gt_0_iff using assm(1) by auto
```
```  2958   then have card: "card (s - {k}) < card s"
```
```  2959     using assm(1) k(1)
```
```  2960     apply (subst card_Diff_singleton_if)
```
```  2961     apply auto
```
```  2962     done
```
```  2963   have *: "closed (\<Union>(s - {k}))"
```
```  2964     apply (rule closed_Union)
```
```  2965     defer
```
```  2966     apply rule
```
```  2967     apply (drule DiffD1,drule s(4))
```
```  2968     using assm(1)
```
```  2969     apply auto
```
```  2970     done
```
```  2971   have "k \<subseteq> \<Union>(s - {k})"
```
```  2972     apply safe
```
```  2973     apply (rule *[unfolded closed_limpt,rule_format])
```
```  2974     unfolding islimpt_approachable
```
```  2975   proof safe
```
```  2976     fix x
```
```  2977     fix e :: real
```
```  2978     assume as: "x \<in> k" "e > 0"
```
```  2979     from k(2)[unfolded k content_eq_0] guess i ..
```
```  2980     then have i:"c\<bullet>i = d\<bullet>i" "i\<in>Basis"
```
```  2981       using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
```
```  2982     then have xi: "x\<bullet>i = d\<bullet>i"
```
```  2983       using as unfolding k mem_box by (metis antisym)
```
```  2984     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 +
```
```  2985       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)"
```
```  2986     show "\<exists>x'\<in>\<Union>(s - {k}). x' \<noteq> x \<and> dist x' x < e"
```
```  2987       apply (rule_tac x=y in bexI)
```
```  2988     proof
```
```  2989       have "d \<in> cbox c d"
```
```  2990         using s(3)[OF k(1)]
```
```  2991         unfolding k box_eq_empty mem_box
```
```  2992         by (fastforce simp add: not_less)
```
```  2993       then have "d \<in> cbox a b"
```
```  2994         using s(2)[OF k(1)]
```
```  2995         unfolding k
```
```  2996         by auto
```
```  2997       note di = this[unfolded mem_box,THEN bspec[where x=i]]
```
```  2998       then have xyi: "y\<bullet>i \<noteq> x\<bullet>i"
```
```  2999         unfolding y_def i xi
```
```  3000         using as(2) assms(2)[unfolded content_eq_0] i(2)
```
```  3001         by (auto elim!: ballE[of _ _ i])
```
```  3002       then show "y \<noteq> x"
```
```  3003         unfolding euclidean_eq_iff[where 'a='a] using i by auto
```
```  3004       have *: "Basis = insert i (Basis - {i})"
```
```  3005         using i by auto
```
```  3006       have "norm (y - x) < e + sum (\<lambda>i. 0) Basis"
```
```  3007         apply (rule le_less_trans[OF norm_le_l1])
```
```  3008         apply (subst *)
```
```  3009         apply (subst sum.insert)
```
```  3010         prefer 3
```
```  3011         apply (rule add_less_le_mono)
```
```  3012       proof -
```
```  3013         show "\<bar>(y - x) \<bullet> i\<bar> < e"
```
```  3014           using di as(2) y_def i xi by (auto simp: inner_simps)
```
```  3015         show "(\<Sum>i\<in>Basis - {i}. \<bar>(y - x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
```
```  3016           unfolding y_def by (auto simp: inner_simps)
```
```  3017       qed auto
```
```  3018       then show "dist y x < e"
```
```  3019         unfolding dist_norm by auto
```
```  3020       have "y \<notin> k"
```
```  3021         unfolding k mem_box
```
```  3022         apply rule
```
```  3023         apply (erule_tac x=i in ballE)
```
```  3024         using xyi k i xi
```
```  3025         apply auto
```
```  3026         done
```
```  3027       moreover
```
```  3028       have "y \<in> \<Union>s"
```
```  3029         using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i
```
```  3030         unfolding s mem_box y_def
```
```  3031         by (auto simp: field_simps elim!: ballE[of _ _ i])
```
```  3032       ultimately
```
```  3033       show "y \<in> \<Union>(s - {k})" by auto
```
```  3034     qed
```
```  3035   qed
```
```  3036   then have "\<Union>(s - {k}) = cbox a b"
```
```  3037     unfolding s(6)[symmetric] by auto
```
```  3038   then have  "{ka \<in> s - {k}. content ka \<noteq> 0} division_of (cbox a b)"
```
```  3039     apply -
```
```  3040     apply (rule assm(2)[rule_format,OF card refl])
```
```  3041     apply (rule division_ofI)
```
```  3042     defer
```
```  3043     apply (rule_tac[1-4] s)
```
```  3044     using assm(1)
```
```  3045     apply auto
```
```  3046     done
```
```  3047   moreover
```
```  3048   have "{ka \<in> s - {k}. content ka \<noteq> 0} = {k \<in> s. content k \<noteq> 0}"
```
```  3049     using k by auto
```
```  3050   ultimately show ?thesis by auto
```
```  3051 qed
```
```  3052
```
```  3053
```
```  3054 subsection \<open>Integrability on subintervals.\<close>
```
```  3055
```
```  3056 lemma operative_integrable:
```
```  3057   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  3058   shows "comm_monoid.operative op \<and> True (\<lambda>i. f integrable_on i)"
```
```  3059   unfolding comm_monoid.operative_def[OF comm_monoid_and]
```
```  3060   apply safe
```
```  3061      apply (subst integrable_on_def)
```
```  3062      apply rule
```
```  3063      apply (rule has_integral_null_eq[where i=0, THEN iffD2])
```
```  3064       apply (simp add: content_eq_0_interior)
```
```  3065      apply rule
```
```  3066     apply (rule, assumption, assumption)+
```
```  3067   unfolding integrable_on_def
```
```  3068   by (auto intro!: has_integral_split)
```
```  3069
```
```  3070 lemma integrable_subinterval:
```
```  3071   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  3072   assumes "f integrable_on cbox a b"
```
```  3073     and "cbox c d \<subseteq> cbox a b"
```
```  3074   shows "f integrable_on cbox c d"
```
```  3075   apply (cases "cbox c d = {}")
```
```  3076   defer
```
```  3077   apply (rule partial_division_extend_1[OF assms(2)],assumption)
```
```  3078   using comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable,symmetric,of _ _ _ f] assms(1)
```
```  3079   apply (auto simp: comm_monoid_set_F_and)
```
```  3080   done
```
```  3081
```
```  3082 lemma integrable_subinterval_real:
```
```  3083   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3084   assumes "f integrable_on {a .. b}"
```
```  3085     and "{c .. d} \<subseteq> {a .. b}"
```
```  3086   shows "f integrable_on {c .. d}"
```
```  3087   by (metis assms(1) assms(2) box_real(2) integrable_subinterval)
```
```  3088
```
```  3089
```
```  3090 subsection \<open>Combining adjacent intervals in 1 dimension.\<close>
```
```  3091
```
```  3092 lemma has_integral_combine:
```
```  3093   fixes a b c :: real
```
```  3094   assumes "a \<le> c"
```
```  3095     and "c \<le> b"
```
```  3096     and "(f has_integral i) {a .. c}"
```
```  3097     and "(f has_integral (j::'a::banach)) {c .. b}"
```
```  3098   shows "(f has_integral (i + j)) {a .. b}"
```
```  3099 proof -
```
```  3100   interpret comm_monoid "lift_option plus" "Some (0::'a)"
```
```  3101     by (rule comm_monoid_lift_option)
```
```  3102       (rule add.comm_monoid_axioms)
```
```  3103   note operative_integral [of f, unfolded operative_1_le]
```
```  3104   note conjunctD2 [OF this, rule_format]
```
```  3105   note * = this(2) [OF conjI [OF assms(1-2)],
```
```  3106     unfolded if_P [OF assms(3)]]
```
```  3107   then have "f integrable_on cbox a b"
```
```  3108     apply -
```
```  3109     apply (rule ccontr)
```
```  3110     apply (subst(asm) if_P)
```
```  3111     defer
```
```  3112     apply (subst(asm) if_P)
```
```  3113     using assms(3-)
```
```  3114     apply auto
```
```  3115     done
```
```  3116   with *
```
```  3117   show ?thesis
```
```  3118     apply -
```
```  3119     apply (subst(asm) if_P)
```
```  3120     defer
```
```  3121     apply (subst(asm) if_P)
```
```  3122     defer
```
```  3123     apply (subst(asm) if_P)
```
```  3124     using assms(3-)
```
```  3125     apply (auto simp add: integrable_on_def integral_unique)
```
```  3126     done
```
```  3127 qed
```
```  3128
```
```  3129 lemma integral_combine:
```
```  3130   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3131   assumes "a \<le> c"
```
```  3132     and "c \<le> b"
```
```  3133     and "f integrable_on {a .. b}"
```
```  3134   shows "integral {a .. c} f + integral {c .. b} f = integral {a .. b} f"
```
```  3135   apply (rule integral_unique[symmetric])
```
```  3136   apply (rule has_integral_combine[OF assms(1-2)])
```
```  3137   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)
```
```  3138   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)
```
```  3139
```
```  3140 lemma integrable_combine:
```
```  3141   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3142   assumes "a \<le> c"
```
```  3143     and "c \<le> b"
```
```  3144     and "f integrable_on {a .. c}"
```
```  3145     and "f integrable_on {c .. b}"
```
```  3146   shows "f integrable_on {a .. b}"
```
```  3147   using assms
```
```  3148   unfolding integrable_on_def
```
```  3149   by (fastforce intro!:has_integral_combine)
```
```  3150
```
```  3151
```
```  3152 subsection \<open>Reduce integrability to "local" integrability.\<close>
```
```  3153
```
```  3154 lemma integrable_on_little_subintervals:
```
```  3155   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
```
```  3156   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>
```
```  3157     f integrable_on cbox u v"
```
```  3158   shows "f integrable_on cbox a b"
```
```  3159 proof -
```
```  3160   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>
```
```  3161     f integrable_on cbox u v)"
```
```  3162     using assms by auto
```
```  3163   note this[unfolded gauge_existence_lemma]
```
```  3164   from choice[OF this] guess d .. note d=this[rule_format]
```
```  3165   guess p
```
```  3166     apply (rule fine_division_exists[OF gauge_ball_dependent,of d a b])
```
```  3167     using d
```
```  3168     by auto
```
```  3169   note p=this(1-2)
```
```  3170   note division_of_tagged_division[OF this(1)]
```
```  3171   note * = comm_monoid_set.operative_division[OF comm_monoid_set_and operative_integrable, OF this, symmetric, of f]
```
```  3172   show ?thesis
```
```  3173     unfolding * comm_monoid_set_F_and
```
```  3174     apply safe
```
```  3175     unfolding snd_conv
```
```  3176   proof -
```
```  3177     fix x k
```
```  3178     assume "(x, k) \<in> p"
```
```  3179     note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
```
```  3180     then show "f integrable_on k"
```
```  3181       apply safe
```
```  3182       apply (rule d[THEN conjunct2,rule_format,of x])
```
```  3183       apply (auto intro: order.trans)
```
```  3184       done
```
```  3185   qed
```
```  3186 qed
```
```  3187
```
```  3188
```
```  3189 subsection \<open>Second FTC or existence of antiderivative.\<close>
```
```  3190
```
```  3191 lemma integrable_const[intro]: "(\<lambda>x. c) integrable_on cbox a b"
```
```  3192   unfolding integrable_on_def
```
```  3193   apply rule
```
```  3194   apply (rule has_integral_const)
```
```  3195   done
```
```  3196
```
```  3197 lemma integral_has_vector_derivative_continuous_at:
```
```  3198   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3199   assumes f: "f integrable_on {a..b}"
```
```  3200       and x: "x \<in> {a..b}"
```
```  3201       and fx: "continuous (at x within {a..b}) f"
```
```  3202   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
```
```  3203 proof -
```
```  3204   let ?I = "\<lambda>a b. integral {a..b} f"
```
```  3205   { fix e::real
```
```  3206     assume "e > 0"
```
```  3207     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"
```
```  3208       using \<open>e>0\<close> fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
```
```  3209     have "norm (integral {a..y} f - integral {a..x} f - (y - x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
```
```  3210            if y: "y \<in> {a..b}" and yx: "\<bar>y - x\<bar> < d" for y
```
```  3211     proof (cases "y < x")
```
```  3212       case False
```
```  3213       have "f integrable_on {a..y}"
```
```  3214         using f y by (simp add: integrable_subinterval_real)
```
```  3215       then have Idiff: "?I a y - ?I a x = ?I x y"
```
```  3216         using False x by (simp add: algebra_simps integral_combine)
```
```  3217       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {x..y} f - (y - x) *\<^sub>R f x) {x..y}"
```
```  3218         apply (rule has_integral_sub)
```
```  3219         using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
```
```  3220         using has_integral_const_real [of "f x" x y] False
```
```  3221         apply (simp add: )
```
```  3222         done
```
```  3223       show ?thesis
```
```  3224         using False
```
```  3225         apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
```
```  3226         apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
```
```  3227         using yx False d x y \<open>e>0\<close> apply (auto simp add: Idiff fux_int)
```
```  3228         done
```
```  3229     next
```
```  3230       case True
```
```  3231       have "f integrable_on {a..x}"
```
```  3232         using f x by (simp add: integrable_subinterval_real)
```
```  3233       then have Idiff: "?I a x - ?I a y = ?I y x"
```
```  3234         using True x y by (simp add: algebra_simps integral_combine)
```
```  3235       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}"
```
```  3236         apply (rule has_integral_sub)
```
```  3237         using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
```
```  3238         using has_integral_const_real [of "f x" y x] True
```
```  3239         apply (simp add: )
```
```  3240         done
```
```  3241       have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
```
```  3242         using True
```
```  3243         apply (simp add: abs_eq_content del: content_real_if measure_lborel_Icc)
```
```  3244         apply (rule has_integral_bound_real[where f="(\<lambda>u. f u - f x)"])
```
```  3245         using yx True d x y \<open>e>0\<close> apply (auto simp add: Idiff fux_int)
```
```  3246         done
```
```  3247       then show ?thesis
```
```  3248         by (simp add: algebra_simps norm_minus_commute)
```
```  3249     qed
```
```  3250     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>"
```
```  3251       using \<open>d>0\<close> by blast
```
```  3252   }
```
```  3253   then show ?thesis
```
```  3254     by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
```
```  3255 qed
```
```  3256
```
```  3257 lemma integral_has_vector_derivative:
```
```  3258   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3259   assumes "continuous_on {a .. b} f"
```
```  3260     and "x \<in> {a .. b}"
```
```  3261   shows "((\<lambda>u. integral {a .. u} f) has_vector_derivative f(x)) (at x within {a .. b})"
```
```  3262 apply (rule integral_has_vector_derivative_continuous_at [OF integrable_continuous_real])
```
```  3263 using assms
```
```  3264 apply (auto simp: continuous_on_eq_continuous_within)
```
```  3265 done
```
```  3266
```
```  3267 lemma antiderivative_continuous:
```
```  3268   fixes q b :: real
```
```  3269   assumes "continuous_on {a .. b} f"
```
```  3270   obtains g where "\<forall>x\<in>{a .. b}. (g has_vector_derivative (f x::_::banach)) (at x within {a .. b})"
```
```  3271   apply (rule that)
```
```  3272   apply rule
```
```  3273   using integral_has_vector_derivative[OF assms]
```
```  3274   apply auto
```
```  3275   done
```
```  3276
```
```  3277
```
```  3278 subsection \<open>Combined fundamental theorem of calculus.\<close>
```
```  3279
```
```  3280 lemma antiderivative_integral_continuous:
```
```  3281   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  3282   assumes "continuous_on {a .. b} f"
```
```  3283   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}"
```
```  3284 proof -
```
```  3285   from antiderivative_continuous[OF assms] guess g . note g=this
```
```  3286   show ?thesis
```
```  3287     apply (rule that[of g])
```
```  3288     apply safe
```
```  3289   proof goal_cases
```
```  3290     case prems: (1 u v)
```
```  3291     have "\<forall>x\<in>cbox u v. (g has_vector_derivative f x) (at x within cbox u v)"
```
```  3292       apply rule
```
```  3293       apply (rule has_vector_derivative_within_subset)
```
```  3294       apply (rule g[rule_format])
```
```  3295       using prems(1,2)
```
```  3296       apply auto
```
```  3297       done
```
```  3298     then show ?case
```
```  3299       using fundamental_theorem_of_calculus[OF prems(3), of g f] by auto
```
```  3300   qed
```
```  3301 qed
```
```  3302
```
```  3303
```
```  3304 subsection \<open>General "twiddling" for interval-to-interval function image.\<close>
```
```  3305
```
```  3306 lemma has_integral_twiddle:
```
```  3307   assumes "0 < r"
```
```  3308     and "\<forall>x. h(g x) = x"
```
```  3309     and "\<forall>x. g(h x) = x"
```
```  3310     and contg: "\<And>x. continuous (at x) g"
```
```  3311     and "\<forall>u v. \<exists>w z. g ` cbox u v = cbox w z"
```
```  3312     and h: "\<forall>u v. \<exists>w z. h ` cbox u v = cbox w z"
```
```  3313     and "\<forall>u v. content(g ` cbox u v) = r * content (cbox u v)"
```
```  3314     and "(f has_integral i) (cbox a b)"
```
```  3315   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
```
```  3316 proof -
```
```  3317   show ?thesis when *: "cbox a b \<noteq> {} \<Longrightarrow> ?thesis"
```
```  3318     apply cases
```
```  3319     defer
```
```  3320     apply (rule *)
```
```  3321     apply assumption
```
```  3322   proof goal_cases
```
```  3323     case prems: 1
```
```  3324     then show ?thesis
```
```  3325       unfolding prems assms(8)[unfolded prems has_integral_empty_eq] by auto
```
```  3326   qed
```
```  3327   assume "cbox a b \<noteq> {}"
```
```  3328   from assms(6)[rule_format,of a b] guess w z by (elim exE) note wz=this
```
```  3329   have inj: "inj g" "inj h"
```
```  3330     unfolding inj_on_def
```
```  3331     apply safe
```
```  3332     apply(rule_tac[!] ccontr)
```
```  3333     using assms(2)
```
```  3334     apply(erule_tac x=x in allE)
```
```  3335     using assms(2)
```
```  3336     apply(erule_tac x=y in allE)
```
```  3337     defer
```
```  3338     using assms(3)
```
```  3339     apply (erule_tac x=x in allE)
```
```  3340     using assms(3)
```
```  3341     apply(erule_tac x=y in allE)
```
```  3342     apply auto
```
```  3343     done
```
```  3344   from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast
```
```  3345   show ?thesis
```
```  3346     unfolding h_eq has_integral
```
```  3347     unfolding h_eq[symmetric]
```
```  3348   proof safe
```
```  3349     fix e :: real
```
```  3350     assume e: "e > 0"
```
```  3351     with assms(1) have "e * r > 0" by simp
```
```  3352     from assms(8)[unfolded has_integral,rule_format,OF this] guess d by (elim exE conjE) note d=this[rule_format]
```
```  3353     define d' where "d' x = {y. g y \<in> d (g x)}" for x
```
```  3354     have d': "\<And>x. d' x = {y. g y \<in> (d (g x))}"
```
```  3355       unfolding d'_def ..
```
```  3356     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` cbox a b \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
```
```  3357     proof (rule_tac x=d' in exI, safe)
```
```  3358       show "gauge d'"
```
```  3359         using d(1)
```
```  3360         unfolding gauge_def d'
```
```  3361         using continuous_open_preimage_univ[OF _ contg]
```
```  3362         by auto
```
```  3363       fix p
```
```  3364       assume as: "p tagged_division_of h ` cbox a b" "d' fine p"
```
```  3365       note p = tagged_division_ofD[OF as(1)]
```
```  3366       have "(\<lambda>(x, k). (g x, g ` k)) ` p tagged_division_of (cbox a b) \<and> d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
```
```  3367         unfolding tagged_division_of
```
```  3368       proof safe
```
```  3369         show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)"
```
```  3370           using as by auto
```
```  3371         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
```
```  3372           using as(2) unfolding fine_def d' by auto
```
```  3373         fix x k
```
```  3374         assume xk[intro]: "(x, k) \<in> p"
```
```  3375         show "g x \<in> g ` k"
```
```  3376           using p(2)[OF xk] by auto
```
```  3377         show "\<exists>u v. g ` k = cbox u v"
```
```  3378           using p(4)[OF xk] using assms(5-6) by auto
```
```  3379         {
```
```  3380           fix y
```
```  3381           assume "y \<in> k"
```
```  3382           then show "g y \<in> cbox a b" "g y \<in> cbox a b"
```
```  3383             using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
```
```  3384             using assms(2)[rule_format,of y]
```
```  3385             unfolding inj_image_mem_iff[OF inj(2)]
```
```  3386             by auto
```
```  3387         }
```
```  3388         fix x' k'
```
```  3389         assume xk': "(x', k') \<in> p"
```
```  3390         fix z
```
```  3391         assume z: "z \<in> interior (g ` k)" "z \<in> interior (g ` k')"
```
```  3392         have same: "(x, k) = (x', k')"
```
```  3393           apply -
```
```  3394           apply (rule ccontr)
```
```  3395           apply (drule p(5)[OF xk xk'])
```
```  3396         proof -
```
```  3397           assume as: "interior k \<inter> interior k' = {}"
```
```  3398           have "z \<in> g ` (interior k \<inter> interior k')"
```
```  3399             using interior_image_subset[OF \<open>inj g\<close> contg] z
```
```  3400             unfolding image_Int[OF inj(1)] by blast
```
```  3401           then show False
```
```  3402             using as by blast
```
```  3403         qed
```
```  3404         then show "g x = g x'"
```
```  3405           by auto
```
```  3406         {
```
```  3407           fix z
```
```  3408           assume "z \<in> k"
```
```  3409           then show "g z \<in> g ` k'"
```
```  3410             using same by auto
```
```  3411         }
```
```  3412         {
```
```  3413           fix z
```
```  3414           assume "z \<in> k'"
```
```  3415           then show "g z \<in> g ` k"
```
```  3416             using same by auto
```
```  3417         }
```
```  3418       next
```
```  3419         fix x
```
```  3420         assume "x \<in> cbox a b"
```
```  3421         then have "h x \<in>  \<Union>{k. \<exists>x. (x, k) \<in> p}"
```
```  3422           using p(6) by auto
```
```  3423         then guess X unfolding Union_iff .. note X=this
```
```  3424         from this(1) guess y unfolding mem_Collect_eq ..
```
```  3425         then show "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}"
```
```  3426           apply -
```
```  3427           apply (rule_tac X="g ` X" in UnionI)
```
```  3428           defer
```
```  3429           apply (rule_tac x="h x" in image_eqI)
```
```  3430           using X(2) assms(3)[rule_format,of x]
```
```  3431           apply auto
```
```  3432           done
```
```  3433       qed
```
```  3434         note ** = d(2)[OF this]
```
```  3435         have *: "inj_on (\<lambda>(x, k). (g x, g ` k)) p"
```
```  3436           using inj(1) unfolding inj_on_def by fastforce
```
```  3437         have "(\<Sum>(x, k)\<in>(\<lambda>(x, k). (g x, g ` k)) ` p. content k *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - i" (is "?l = _")
```
```  3438           using assms(7)
```
```  3439           apply (simp only: algebra_simps add_left_cancel scaleR_right.sum)
```
```  3440           apply (subst sum.reindex_bij_betw[symmetric, where h="\<lambda>(x, k). (g x, g ` k)" and S=p])
```
```  3441           apply (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
```
```  3442           done
```
```  3443       also have "\<dots> = r *\<^sub>R ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" (is "_ = ?r")
```
```  3444         unfolding scaleR_diff_right scaleR_scaleR
```
```  3445         using assms(1)
```
```  3446         by auto
```
```  3447       finally have *: "?l = ?r" .
```
```  3448       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e"
```
```  3449         using **
```
```  3450         unfolding *
```
```  3451         unfolding norm_scaleR
```
```  3452         using assms(1)
```
```  3453         by (auto simp add:field_simps)
```
```  3454     qed
```
```  3455   qed
```
```  3456 qed
```
```  3457
```
```  3458
```
```  3459 subsection \<open>Special case of a basic affine transformation.\<close>
```
```  3460
```
```  3461 lemma AE_lborel_inner_neq:
```
```  3462   assumes k: "k \<in> Basis"
```
```  3463   shows "AE x in lborel. x \<bullet> k \<noteq> c"
```
```  3464 proof -
```
```  3465   interpret finite_product_sigma_finite "\<lambda>_. lborel" Basis
```
```  3466     proof qed simp
```
```  3467
```
```  3468   have "emeasure lborel {x\<in>space lborel. x \<bullet> k = c} = emeasure (\<Pi>\<^sub>M j::'a\<in>Basis. lborel) (\<Pi>\<^sub>E j\<in>Basis. if j = k then {c} else UNIV)"
```
```  3469     using k
```
```  3470     by (auto simp add: lborel_eq[where 'a='a] emeasure_distr intro!: arg_cong2[where f=emeasure])
```
```  3471        (auto simp: space_PiM PiE_iff extensional_def split: if_split_asm)
```
```  3472   also have "\<dots> = (\<Prod>j\<in>Basis. emeasure lborel (if j = k then {c} else UNIV))"
```
```  3473     by (intro measure_times) auto
```
```  3474   also have "\<dots> = 0"
```
```  3475     by (intro prod_zero bexI[OF _ k]) auto
```
```  3476   finally show ?thesis
```
```  3477     by (subst AE_iff_measurable[OF _ refl]) auto
```
```  3478 qed
```
```  3479
```
```  3480 lemma content_image_stretch_interval:
```
```  3481   fixes m :: "'a::euclidean_space \<Rightarrow> real"
```
```  3482   defines "s f x \<equiv> (\<Sum>k::'a\<in>Basis. (f k * (x\<bullet>k)) *\<^sub>R k)"
```
```  3483   shows "content (s m ` cbox a b) = \<bar>\<Prod>k\<in>Basis. m k\<bar> * content (cbox a b)"
```
```  3484 proof cases
```
```  3485   have s[measurable]: "s f \<in> borel \<rightarrow>\<^sub>M borel" for f
```
```  3486     by (auto simp: s_def[abs_def])
```
```  3487   assume m: "\<forall>k\<in>Basis. m k \<noteq> 0"
```
```  3488   then have s_comp_s: "s (\<lambda>k. 1 / m k) \<circ> s m = id" "s m \<circ> s (\<lambda>k. 1 / m k) = id"
```
```  3489     by (auto simp: s_def[abs_def] fun_eq_iff euclidean_representation)
```
```  3490   then have "inv (s (\<lambda>k. 1 / m k)) = s m" "bij (s (\<lambda>k. 1 / m k))"
```
```  3491     by (auto intro: inv_unique_comp o_bij)
```
```  3492   then have eq: "s m ` cbox a b = s (\<lambda>k. 1 / m k) -` cbox a b"
```
```  3493     using bij_vimage_eq_inv_image[OF \<open>bij (s (\<lambda>k. 1 / m k))\<close>, of "cbox a b"] by auto
```
```  3494   show ?thesis
```
```  3495     using m unfolding eq measure_def
```
```  3496     by (subst lborel_affine_euclidean[where c=m and t=0])
```
```  3497        (simp_all add: emeasure_density measurable_sets_borel[OF s] abs_prod nn_integral_cmult
```
```  3498                       s_def[symmetric] emeasure_distr vimage_comp s_comp_s enn2real_mult prod_nonneg)
```
```  3499 next
```
```  3500   assume "\<not> (\<forall>k\<in>Basis. m k \<noteq> 0)"
```
```  3501   then obtain k where k: "k \<in> Basis" "m k = 0" by auto
```
```  3502   then have [simp]: "(\<Prod>k\<in>Basis. m k) = 0"
```
```  3503     by (intro prod_zero) auto
```
```  3504   have "emeasure lborel {x\<in>space lborel. x \<in> s m ` cbox a b} = 0"
```
```  3505   proof (rule emeasure_eq_0_AE)
```
```  3506     show "AE x in lborel. x \<notin> s m ` cbox a b"
```
```  3507       using AE_lborel_inner_neq[OF \<open>k\<in>Basis\<close>]
```
```  3508     proof eventually_elim
```
```  3509       show "x \<bullet> k \<noteq> 0 \<Longrightarrow> x \<notin> s m ` cbox a b " for x
```
```  3510         using k by (auto simp: s_def[abs_def] cbox_def)
```
```  3511     qed
```
```  3512   qed
```
```  3513   then show ?thesis
```
```  3514     by (simp add: measure_def)
```
```  3515 qed
```
```  3516
```
```  3517 lemma interval_image_affinity_interval:
```
```  3518   "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v"
```
```  3519   unfolding image_affinity_cbox
```
```  3520   by auto
```
```  3521
```
```  3522 lemma content_image_affinity_cbox:
```
```  3523   "content((\<lambda>x::'a::euclidean_space. m *\<^sub>R x + c) ` cbox a b) =
```
```  3524     \<bar>m\<bar> ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
```
```  3525 proof (cases "cbox a b = {}")
```
```  3526   case True then show ?thesis by simp
```
```  3527 next
```
```  3528   case False
```
```  3529   show ?thesis
```
```  3530   proof (cases "m \<ge> 0")
```
```  3531     case True
```
```  3532     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) \<noteq> {}"
```
```  3533       unfolding box_ne_empty
```
```  3534       apply (intro ballI)
```
```  3535       apply (erule_tac x=i in ballE)
```
```  3536       apply (auto simp: inner_simps mult_left_mono)
```
```  3537       done
```
```  3538     moreover from True have *: "\<And>i. (m *\<^sub>R b + c) \<bullet> i - (m *\<^sub>R a + c) \<bullet> i = m *\<^sub>R (b - a) \<bullet> i"
```
```  3539       by (simp add: inner_simps field_simps)
```
```  3540     ultimately show ?thesis
```
```  3541       by (simp add: image_affinity_cbox True content_cbox'
```
```  3542         prod.distrib prod_constant inner_diff_left)
```
```  3543   next
```
```  3544     case False
```
```  3545     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c) \<noteq> {}"
```
```  3546       unfolding box_ne_empty
```
```  3547       apply (intro ballI)
```
```  3548       apply (erule_tac x=i in ballE)
```
```  3549       apply (auto simp: inner_simps mult_left_mono)
```
```  3550       done
```
```  3551     moreover from False have *: "\<And>i. (m *\<^sub>R a + c) \<bullet> i - (m *\<^sub>R b + c) \<bullet> i = (-m) *\<^sub>R (b - a) \<bullet> i"
```
```  3552       by (simp add: inner_simps field_simps)
```
```  3553     ultimately show ?thesis using False
```
```  3554       by (simp add: image_affinity_cbox content_cbox'
```
```  3555         prod.distrib[symmetric] prod_constant[symmetric] inner_diff_left)
```
```  3556   qed
```
```  3557 qed
```
```  3558
```
```  3559 lemma has_integral_affinity:
```
```  3560   fixes a :: "'a::euclidean_space"
```
```  3561   assumes "(f has_integral i) (cbox a b)"
```
```  3562       and "m \<noteq> 0"
```
```  3563   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral ((1 / (\<bar>m\<bar> ^ DIM('a))) *\<^sub>R i)) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` cbox a b)"
```
```  3564   apply (rule has_integral_twiddle)
```
```  3565   using assms
```
```  3566   apply (safe intro!: interval_image_affinity_interval content_image_affinity_cbox)
```
```  3567   apply (rule zero_less_power)
```
```  3568   unfolding scaleR_right_distrib
```
```  3569   apply auto
```
```  3570   done
```
```  3571
```
```  3572 lemma integrable_affinity:
```
```  3573   assumes "f integrable_on cbox a b"
```
```  3574     and "m \<noteq> 0"
```
```  3575   shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` cbox a b)"
```
```  3576   using assms
```
```  3577   unfolding integrable_on_def
```
```  3578   apply safe
```
```  3579   apply (drule has_integral_affinity)
```
```  3580   apply auto
```
```  3581   done
```
```  3582
```
```  3583 lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified]
```
```  3584
```
```  3585 subsection \<open>Special case of stretching coordinate axes separately.\<close>
```
```  3586
```
```  3587 lemma has_integral_stretch:
```
```  3588   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3589   assumes "(f has_integral i) (cbox a b)"
```
```  3590     and "\<forall>k\<in>Basis. m k \<noteq> 0"
```
```  3591   shows "((\<lambda>x. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) has_integral
```
```  3592          ((1/ \<bar>prod m Basis\<bar>) *\<^sub>R i)) ((\<lambda>x. (\<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k)) ` cbox a b)"
```
```  3593 apply (rule has_integral_twiddle[where f=f])
```
```  3594 unfolding zero_less_abs_iff content_image_stretch_interval
```
```  3595 unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
```
```  3596 using assms
```
```  3597 by auto
```
```  3598
```
```  3599
```
```  3600 lemma integrable_stretch:
```
```  3601   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
```
```  3602   assumes "f integrable_on cbox a b"
```
```  3603     and "\<forall>k\<in>Basis. m k \<noteq> 0"
```
```  3604   shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on
```
```  3605     ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` cbox a b)"
```
```  3606   using assms unfolding integrable_on_def
```
```  3607   by (force dest: has_integral_stretch)
```
```  3608
```
```  3609
```
```  3610 subsection \<open>even more special cases.\<close>
```
```  3611
```
```  3612 lemma uminus_interval_vector[simp]:
```
```  3613   fixes a b :: "'a::euclidean_space"
```
```  3614   shows "uminus ` cbox a b = cbox (-b) (-a)"
```
```  3615   apply (rule set_eqI)
```
```  3616   apply rule
```
```  3617   defer
```
```  3618   unfolding image_iff
```
```  3619   apply (rule_tac x="-x" in bexI)
```
```  3620   apply (auto simp add:minus_le_iff le_minus_iff mem_box)
```
```  3621   done
```
```  3622
```
```  3623 lemma has_integral_reflect_lemma[intro]:
```
```  3624   assumes "(f has_integral i) (cbox a b)"
```
```  3625   shows "((\<lambda>x. f(-x)) has_integral i) (cbox (-b) (-a))"
```
```  3626   using has_integral_affinity[OF assms, of "-1" 0]
```
```  3627   by auto
```
```  3628
```
```  3629 lemma has_integral_reflect_lemma_real[intro]:
```
```  3630   assumes "(f has_integral i) {a .. b::real}"
```
```  3631   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
```
```  3632   using assms
```
```  3633   unfolding box_real[symmetric]
```
```  3634   by (rule has_integral_reflect_lemma)
```
```  3635
```
```  3636 lemma has_integral_reflect[simp]:
```
```  3637   "((\<lambda>x. f (-x)) has_integral i) (cbox (-b) (-a)) \<longleftrightarrow> (f has_integral i) (cbox a b)"
```
```  3638   apply rule
```
```  3639   apply (drule_tac[!] has_integral_reflect_lemma)
```
```  3640   apply auto
```
```  3641   done
```
```  3642
```
```  3643 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on cbox (-b) (-a) \<longleftrightarrow> f integrable_on cbox a b"
```
```  3644   unfolding integrable_on_def by auto
```
```  3645
```
```  3646 lemma integrable_reflect_real[simp]: "(\<lambda>x. f(-x)) integrable_on {-b .. -a} \<longleftrightarrow> f integrable_on {a .. b::real}"
```
```  3647   unfolding box_real[symmetric]
```
```  3648   by (rule integrable_reflect)
```
```  3649
```
```  3650 lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (\<lambda>x. f (-x)) = integral (cbox a b) f"
```
```  3651   unfolding integral_def by auto
```
```  3652
```
```  3653 lemma integral_reflect_real[simp]: "integral {-b .. -a} (\<lambda>x. f (-x)) = integral {a .. b::real} f"
```
```  3654   unfolding box_real[symmetric]
```
```  3655   by (rule integral_reflect)
```
```  3656
```
```  3657
```
```  3658 subsection \<open>Stronger form of FCT; quite a tedious proof.\<close>
```
```  3659
```
```  3660 lemma split_minus[simp]: "(\<lambda>(x, k). f x k) x - (\<lambda>(x, k). g x k) x = (\<lambda>(x, k). f x k - g x k) x"
```
```  3661   by (simp add: split_def)
```
```  3662
```
```  3663 lemma norm_triangle_le_sub: "norm x + norm y \<le> e \<Longrightarrow> norm (x - y) \<le> e"
```
```  3664   apply (subst(asm)(2) norm_minus_cancel[symmetric])
```
```  3665   apply (drule norm_triangle_le)
```
```  3666   apply (auto simp add: algebra_simps)
```
```  3667   done
```
```  3668
```
```  3669 lemma fundamental_theorem_of_calculus_interior:
```
```  3670   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
```
```  3671   assumes "a \<le> b"
```
```  3672     and "continuous_on {a .. b} f"
```
```  3673     and "\<forall>x\<in>{a <..< b}. (f has_vector_derivative f'(x)) (at x)"
```
```  3674   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  3675 proof -
```
```  3676   {
```
```  3677     presume *: "a < b \<Longrightarrow> ?thesis"
```
```  3678     show ?thesis
```
```  3679     proof (cases "a < b")
```
```  3680       case True
```
```  3681       then show ?thesis by (rule *)
```
```  3682     next
```
```  3683       case False
```
```  3684       then have "a = b"
```
```  3685         using assms(1) by auto
```
```  3686       then have *: "cbox a b = {b}" "f b - f a = 0"
```
```  3687         by (auto simp add:  order_antisym)
```
```  3688       show ?thesis
```
```  3689         unfolding *(2)
```
```  3690         unfolding content_eq_0
```
```  3691         using * \<open>a = b\<close>
```
```  3692         by (auto simp: ex_in_conv)
```
```  3693     qed
```
```  3694   }
```
```  3695   assume ab: "a < b"
```
```  3696   let ?P = "\<lambda>e. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
```
```  3697     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a .. b})"
```
```  3698   { presume "\<And>e. e > 0 \<Longrightarrow> ?P e" then show ?thesis unfolding has_integral_factor_content_real by auto }
```
```  3699   fix e :: real
```
```  3700   assume e: "e > 0"
```
```  3701   note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
```
```  3702   note conjunctD2[OF this]
```
```  3703   note bounded=this(1) and this(2)
```
```  3704   from this(2) have "\<forall>x\<in>box a b. \<exists>d>0. \<forall>y. norm (y - x) < d \<longrightarrow>
```
```  3705     norm (f y - f x - (y - x) *\<^sub>R f' x) \<le> e/2 * norm (y - x)"
```
```  3706     apply -
```
```  3707     apply safe
```
```  3708     apply (erule_tac x=x in ballE)
```
```  3709     apply (erule_tac x="e/2" in allE)
```
```  3710     using e
```
```  3711     apply auto
```
```  3712     done
```
```  3713   note this[unfolded bgauge_existence_lemma]
```
```  3714   from choice[OF this] guess d ..
```
```  3715   note conjunctD2[OF this[rule_format]]
```
```  3716   note d = this[rule_format]
```
```  3717   have "bounded (f ` cbox a b)"
```
```  3718     apply (rule compact_imp_bounded compact_continuous_image)+
```
```  3719     using compact_cbox assms
```
```  3720     apply auto
```
```  3721     done
```
```  3722   from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
```
```  3723
```
```  3724   have "\<exists>da. 0 < da \<and> (\<forall>c. a \<le> c \<and> {a .. c} \<subseteq> {a .. b} \<and> {a .. c} \<subseteq> ball a da \<longrightarrow>
```
```  3725     norm (content {a .. c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b - a)) / 4)"
```
```  3726   proof -
```
```  3727     have "a \<in> {a .. b}"
```
```  3728       using ab by auto
```
```  3729     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3730     note * = this[unfolded continuous_within Lim_within,rule_format]
```
```  3731     have "(e * (b - a)) / 8 > 0"
```
```  3732       using e ab by (auto simp add: field_simps)
```
```  3733     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3734     have "\<exists>l. 0 < l \<and> norm(l *\<^sub>R f' a) \<le> (e * (b - a)) / 8"
```
```  3735     proof (cases "f' a = 0")
```
```  3736       case True
```
```  3737       thus ?thesis using ab e by auto
```
```  3738     next
```
```  3739       case False
```
```  3740       then show ?thesis
```
```  3741         apply (rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
```
```  3742         using ab e
```
```  3743         apply (auto simp add: field_simps)
```
```  3744         done
```
```  3745     qed
```
```  3746     then guess l .. note l = conjunctD2[OF this]
```
```  3747     show ?thesis
```
```  3748       apply (rule_tac x="min k l" in exI)
```
```  3749       apply safe
```
```  3750       unfolding min_less_iff_conj
```
```  3751       apply rule
```
```  3752       apply (rule l k)+
```
```  3753     proof -
```
```  3754       fix c
```
```  3755       assume as: "a \<le> c" "{a .. c} \<subseteq> {a .. b}" "{a .. c} \<subseteq> ball a (min k l)"
```
```  3756       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
```
```  3757       have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)"
```
```  3758         by (rule norm_triangle_ineq4)
```
```  3759       also have "\<dots> \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3760       proof (rule add_mono)
```
```  3761         have "\<bar>c - a\<bar> \<le> \<bar>l\<bar>"
```
```  3762           using as' by auto
```
```  3763         then show "norm ((c - a) *\<^sub>R f' a) \<le> e * (b - a) / 8"
```
```  3764           apply -
```
```  3765           apply (rule order_trans[OF _ l(2)])
```
```  3766           unfolding norm_scaleR
```
```  3767           apply (rule mult_right_mono)
```
```  3768           apply auto
```
```  3769           done
```
```  3770       next
```
```  3771         show "norm (f c - f a) \<le> e * (b - a) / 8"
```
```  3772           apply (rule less_imp_le)
```
```  3773           apply (cases "a = c")
```
```  3774           defer
```
```  3775           apply (rule k(2)[unfolded dist_norm])
```
```  3776           using as' e ab
```
```  3777           apply (auto simp add: field_simps)
```
```  3778           done
```
```  3779       qed
```
```  3780       finally show "norm (content {a .. c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b - a) / 4"
```
```  3781         unfolding content_real[OF as(1)] by auto
```
```  3782     qed
```
```  3783   qed
```
```  3784   then guess da .. note da=conjunctD2[OF this,rule_format]
```
```  3785
```
```  3786   have "\<exists>db>0. \<forall>c\<le>b. {c .. b} \<subseteq> {a .. b} \<and> {c .. b} \<subseteq> ball b db \<longrightarrow>
```
```  3787     norm (content {c .. b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b - a)) / 4"
```
```  3788   proof -
```
```  3789     have "b \<in> {a .. b}"
```
```  3790       using ab by auto
```
```  3791     note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
```
```  3792     note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
```
```  3793       using e ab by (auto simp add: field_simps)
```
```  3794     from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
```
```  3795     have "\<exists>l. 0 < l \<and> norm (l *\<^sub>R f' b) \<le> (e * (b - a)) / 8"
```
```  3796     proof (cases "f' b = 0")
```
```  3797       case True
```
```  3798       thus ?thesis using ab e by auto
```
```  3799     next
```
```  3800       case False
```
```  3801       then show ?thesis
```
```  3802         apply (rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
```
```  3803         using ab e
```
```  3804         apply (auto simp add: field_simps)
```
```  3805         done
```
```  3806     qed
```
```  3807     then guess l .. note l = conjunctD2[OF this]
```
```  3808     show ?thesis
```
```  3809       apply (rule_tac x="min k l" in exI)
```
```  3810       apply safe
```
```  3811       unfolding min_less_iff_conj
```
```  3812       apply rule
```
```  3813       apply (rule l k)+
```
```  3814     proof -
```
```  3815       fix c
```
```  3816       assume as: "c \<le> b" "{c..b} \<subseteq> {a..b}" "{c..b} \<subseteq> ball b (min k l)"
```
```  3817       note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
```
```  3818       have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)"
```
```  3819         by (rule norm_triangle_ineq4)
```
```  3820       also have "\<dots> \<le> e * (b - a) / 8 + e * (b - a) / 8"
```
```  3821       proof (rule add_mono)
```
```  3822         have "\<bar>c - b\<bar> \<le> \<bar>l\<bar>"
```
```  3823           using as' by auto
```
```  3824         then show "norm ((b - c) *\<^sub>R f' b) \<le> e * (b - a) / 8"
```
```  3825           apply -
```
```  3826           apply (rule order_trans[OF _ l(2)])
```
```  3827           unfolding norm_scaleR
```
```  3828           apply (rule mult_right_mono)
```
```  3829           apply auto
```
```  3830           done
```
```  3831       next
```
```  3832         show "norm (f b - f c) \<le> e * (b - a) / 8"
```
```  3833           apply (rule less_imp_le)
```
```  3834           apply (cases "b = c")
```
```  3835           defer
```
```  3836           apply (subst norm_minus_commute)
```
```  3837           apply (rule k(2)[unfolded dist_norm])
```
```  3838           using as' e ab
```
```  3839           apply (auto simp add: field_simps)
```
```  3840           done
```
```  3841       qed
```
```  3842       finally show "norm (content {c .. b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b - a) / 4"
```
```  3843         unfolding content_real[OF as(1)] by auto
```
```  3844     qed
```
```  3845   qed
```
```  3846   then guess db .. note db=conjunctD2[OF this,rule_format]
```
```  3847
```
```  3848   let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
```
```  3849   show "?P e"
```
```  3850     apply (rule_tac x="?d" in exI)
```
```  3851   proof (safe, goal_cases)
```
```  3852     case 1
```
```  3853     show ?case
```
```  3854       apply (rule gauge_ball_dependent)
```
```  3855       using ab db(1) da(1) d(1)
```
```  3856       apply auto
```
```  3857       done
```
```  3858   next
```
```  3859     case as: (2 p)
```
```  3860     let ?A = "{t. fst t \<in> {a, b}}"
```
```  3861     note p = tagged_division_ofD[OF as(1)]
```
```  3862     have pA: "p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"
```
```  3863       using as by auto
```
```  3864     note * = additive_tagged_division_1'[OF assms(1) as(1), symmetric]
```
```  3865     have **: "\<And>n1 s1 n2 s2::real. n2 \<le> s2 / 2 \<Longrightarrow> n1 - s1 \<le> s2 / 2 \<Longrightarrow> n1 + n2 \<le> s1 + s2"
```
```  3866       by arith
```
```  3867     show ?case
```
```  3868       unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] sum_subtractf[symmetric] split_minus
```
```  3869       unfolding sum_distrib_left
```
```  3870       apply (subst(2) pA)
```
```  3871       apply (subst pA)
```
```  3872       unfolding sum.union_disjoint[OF pA(2-)]
```
```  3873     proof (rule norm_triangle_le, rule **, goal_cases)
```
```  3874       case 1
```
```  3875       show ?case
```
```  3876         apply (rule order_trans)
```
```  3877         apply (rule sum_norm_le)
```
```  3878         defer
```
```  3879         apply (subst sum_divide_distrib)
```
```  3880         apply (rule order_refl)
```
```  3881         apply safe
```
```  3882         apply (unfold not_le o_def split_conv fst_conv)
```
```  3883       proof (rule ccontr)
```
```  3884         fix x k
```
```  3885         assume xk: "(x, k) \<in> p"
```
```  3886           "e * (Sup k -  Inf k) / 2 <
```
```  3887             norm (content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))"
```
```  3888         from p(4)[OF this(1)] guess u v by (elim exE) note k=this
```
```  3889         then have "u \<le> v" and uv: "{u, v} \<subseteq> cbox u v"
```
```  3890           using p(2)[OF xk(1)] by auto
```
```  3891         note result = xk(2)[unfolded k box_real interval_bounds_real[OF this(1)] content_real[OF this(1)]]
```
```  3892
```
```  3893         assume as': "x \<noteq> a" "x \<noteq> b"
```
```  3894         then have "x \<in> box a b"
```
```  3895           using p(2-3)[OF xk(1)] by (auto simp: mem_box)
```
```  3896         note  * = d(2)[OF this]
```
```  3897         have "norm ((v - u) *\<^sub>R f' (x) - (f (v) - f (u))) =
```
```  3898           norm ((f (u) - f (x) - (u - x) *\<^sub>R f' (x)) - (f (v) - f (x) - (v - x) *\<^sub>R f' (x)))"
```
```  3899           apply (rule arg_cong[of _ _ norm])
```
```  3900           unfolding scaleR_left.diff
```
```  3901           apply auto
```
```  3902           done
```
```  3903         also have "\<dots> \<le> e / 2 * norm (u - x) + e / 2 * norm (v - x)"
```
```  3904           apply (rule norm_triangle_le_sub)
```
```  3905           apply (rule add_mono)
```
```  3906           apply (rule_tac[!] *)
```
```  3907           using fineD[OF as(2) xk(1)] as'
```
```  3908           unfolding k subset_eq
```
```  3909           apply -
```
```  3910           apply (erule_tac x=u in ballE)
```
```  3911           apply (erule_tac[3] x=v in ballE)
```
```  3912           using uv
```
```  3913           apply (auto simp:dist_real_def)
```
```  3914           done
```
```  3915         also have "\<dots> \<le> e / 2 * norm (v - u)"
```
```  3916           using p(2)[OF xk(1)]
```
```  3917           unfolding k
```
```  3918           by (auto simp add: field_simps)
```
```  3919         finally have "e * (v - u) / 2 < e * (v - u) / 2"
```
```  3920           apply -
```
```  3921           apply (rule less_le_trans[OF result])
```
```  3922           using uv
```
```  3923           apply auto
```
```  3924           done
```
```  3925         then show False by auto
```
```  3926       qed
```
```  3927     next
```
```  3928       have *: "\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2) / 2 \<Longrightarrow> x - s1 \<le> s2 / 2"
```
```  3929         by auto
```
```  3930       case 2
```
```  3931       show ?case
```
```  3932         apply (rule *)
```
```  3933         apply (rule sum_nonneg)
```
```  3934         apply rule
```
```  3935         apply (unfold split_paired_all split_conv)
```
```  3936         defer
```
```  3937         unfolding sum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
```
```  3938         unfolding sum_distrib_left[symmetric]
```
```  3939         apply (subst additive_tagged_division_1[OF _ as(1)])
```
```  3940         apply (rule assms)
```
```  3941       proof -
```
```  3942         fix x k
```
```  3943         assume "(x, k) \<in> p \<inter> {t. fst t \<in> {a, b}}"
```
```  3944         note xk=IntD1[OF this]
```
```  3945         from p(4)[OF this] guess u v by (elim exE) note uv=this
```
```  3946         with p(2)[OF xk] have "cbox u v \<noteq> {}"
```
```  3947           by auto
```
```  3948         then show "0 \<le> e * ((Sup k) - (Inf k))"
```
```  3949           unfolding uv using e by (auto simp add: field_simps)
```
```  3950       next
```
```  3951         have *: "\<And>s f t e. sum f s = sum f t \<Longrightarrow> norm (sum f t) \<le> e \<Longrightarrow> norm (sum f s) \<le> e"
```
```  3952           by auto
```
```  3953         show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x -
```
```  3954           (f ((Sup k)) - f ((Inf k)))) \<le> e * (b - a) / 2"
```
```  3955           apply (rule *[where t1="p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0}"])
```
```  3956           apply (rule sum.mono_neutral_right[OF pA(2)])
```
```  3957           defer
```
```  3958           apply rule
```
```  3959           unfolding split_paired_all split_conv o_def
```
```  3960         proof goal_cases
```
```  3961           fix x k
```
```  3962           assume "(x, k) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> {t. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
```
```  3963           then have xk: "(x, k) \<in> p" "content k = 0"
```
```  3964             by auto
```
```  3965           from p(4)[OF xk(1)] guess u v by (elim exE) note uv=this
```
```  3966           have "k \<noteq> {}"
```
```  3967             using p(2)[OF xk(1)] by auto
```
```  3968           then have *: "u = v"
```
```  3969             using xk
```
```  3970             unfolding uv content_eq_0 box_eq_empty
```
```  3971             by auto
```
```  3972           then show "content k *\<^sub>R (f' (x)) - (f ((Sup k)) - f ((Inf k))) = 0"
```
```  3973             using xk unfolding uv by auto
```
```  3974         next
```
```  3975           have *: "p \<inter> {t. fst t \<in> {a, b} \<and> content(snd t) \<noteq> 0} =
```
```  3976             {t. t\<in>p \<and> fst t = a \<and> content(snd t) \<noteq> 0} \<union> {t. t\<in>p \<and> fst t = b \<and> content(snd t) \<noteq> 0}"
```
```  3977             by blast
```
```  3978           have **: "norm (sum f s) \<le> e"
```
```  3979             if "\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> x = y"
```
```  3980             and "\<forall>x. x \<in> s \<longrightarrow> norm (f x) \<le> e"
```
```  3981             and "e > 0"
```
```  3982             for s f and e :: real
```
```  3983           proof (cases "s = {}")
```
```  3984             case True
```
```  3985             with that show ?thesis by auto
```
```  3986           next
```
```  3987             case False
```
```  3988             then obtain x where "x \<in> s"
```
```  3989               by auto
```
```  3990             then have *: "s = {x}"
```
```  3991               using that(1) by auto
```
```  3992             then show ?thesis
```
```  3993               using \<open>x \<in> s\<close> that(2) by auto
```
```  3994           qed
```
```  3995           case 2
```
```  3996           show ?case
```
```  3997             apply (subst *)
```
```  3998             apply (subst sum.union_disjoint)
```
```  3999             prefer 4
```
```  4000             apply (rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
```
```  4001             apply (rule norm_triangle_le,rule add_mono)
```
```  4002             apply (rule_tac[1-2] **)
```
```  4003           proof -
```
```  4004             let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
```
```  4005             have pa: "\<exists>v. k = cbox a v \<and> a \<le> v" if "(a, k) \<in> p" for k
```
```  4006             proof -
```
```  4007               guess u v using p(4)[OF that] by (elim exE) note uv=this
```
```  4008               have *: "u \<le> v"
```
```  4009                 using p(2)[OF that] unfolding uv by auto
```
```  4010               have u: "u = a"
```
```  4011               proof (rule ccontr)
```
```  4012                 have "u \<in> cbox u v"
```
```  4013                   using p(2-3)[OF that(1)] unfolding uv by auto
```
```  4014                 have "u \<ge> a"
```
```  4015                   using p(2-3)[OF that(1)] unfolding uv subset_eq by auto
```
```  4016                 moreover assume "\<not> ?thesis"
```
```  4017                 ultimately have "u > a" by auto
```
```  4018                 then show False
```
```  4019                   using p(2)[OF that(1)] unfolding uv by (auto simp add:)
```
```  4020               qed
```
```  4021               then show ?thesis
```
```  4022                 apply (rule_tac x=v in exI)
```
```  4023                 unfolding uv
```
```  4024                 using *
```
```  4025                 apply auto
```
```  4026                 done
```
```  4027             qed
```
```  4028             have pb: "\<exists>v. k = cbox v b \<and> b \<ge> v" if "(b, k) \<in> p" for k
```
```  4029             proof -
```
```  4030               guess u v using p(4)[OF that] by (elim exE) note uv=this
```
```  4031               have *: "u \<le> v"
```
```  4032                 using p(2)[OF that] unfolding uv by auto
```
```  4033               have u: "v = b"
```
```  4034               proof (rule ccontr)
```
```  4035                 have "u \<in> cbox u v"
```
```  4036                   using p(2-3)[OF that(1)] unfolding uv by auto
```
```  4037                 have "v \<le> b"
```
```  4038                   using p(2-3)[OF that(1)] unfolding uv subset_eq by auto
```
```  4039                 moreover assume "\<not> ?thesis"
```
```  4040                 ultimately have "v < b" by auto
```
```  4041                 then show False
```
```  4042                   using p(2)[OF that(1)] unfolding uv by (auto simp add:)
```
```  4043               qed
```
```  4044               then show ?thesis
```
```  4045                 apply (rule_tac x=u in exI)
```
```  4046                 unfolding uv
```
```  4047                 using *
```
```  4048                 apply auto
```
```  4049                 done
```
```  4050             qed
```
```  4051             show "\<forall>x y. x \<in> ?B a \<and> y \<in> ?B a \<longrightarrow> x = y"
```
```  4052               apply (rule,rule,rule,unfold split_paired_all)
```
```  4053               unfolding mem_Collect_eq fst_conv snd_conv
```
```  4054               apply safe
```
```  4055             proof -
```
```  4056               fix x k k'
```
```  4057               assume k: "(a, k) \<in> p" "(a, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
```
```  4058               guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
```
```  4059               guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "min v v'"
```
```  4060               have "box a ?v \<subseteq> k \<inter> k'"
```
```  4061                 unfolding v v' by (auto simp add: mem_box)
```
```  4062               note interior_mono[OF this,unfolded interior_Int]
```
```  4063               moreover have "(a + ?v)/2 \<in> box a ?v"
```
```  4064                 using k(3-)
```
```  4065                 unfolding v v' content_eq_0 not_le
```
```  4066                 by (auto simp add: mem_box)
```
```  4067               ultimately have "(a + ?v)/2 \<in> interior k \<inter> interior k'"
```
```  4068                 unfolding interior_open[OF open_box] by auto
```
```  4069               then have *: "k = k'"
```
```  4070                 apply -
```
```  4071                 apply (rule ccontr)
```
```  4072                 using p(5)[OF k(1-2)]
```
```  4073                 apply auto
```
```  4074                 done
```
```  4075               { assume "x \<in> k" then show "x \<in> k'" unfolding * . }
```
```  4076               { assume "x \<in> k'" then show "x \<in> k" unfolding * . }
```
```  4077             qed
```
```  4078             show "\<forall>x y. x \<in> ?B b \<and> y \<in> ?B b \<longrightarrow> x = y"
```
```  4079               apply rule
```
```  4080               apply rule
```
```  4081               apply rule
```
```  4082               apply (unfold split_paired_all)
```
```  4083               unfolding mem_Collect_eq fst_conv snd_conv
```
```  4084               apply safe
```
```  4085             proof -
```
```  4086               fix x k k'
```
```  4087               assume k: "(b, k) \<in> p" "(b, k') \<in> p" "content k \<noteq> 0" "content k' \<noteq> 0"
```
```  4088               guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
```
```  4089               guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this]
```
```  4090               let ?v = "max v v'"
```
```  4091               have "box ?v b \<subseteq> k \<inter> k'"
```
```  4092                 unfolding v v' by (auto simp: mem_box)
```
```  4093                 note interior_mono[OF this,unfolded interior_Int]
```
```  4094               moreover have " ((b + ?v)/2) \<in> box ?v b"
```
```  4095                 using k(3-) unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
```
```  4096               ultimately have " ((b + ?v)/2) \<in> interior k \<inter> interior k'"
```
```  4097                 unfolding interior_open[OF open_box] by auto
```
```  4098               then have *: "k = k'"
```
```  4099                 apply -
```
```  4100                 apply (rule ccontr)
```
```  4101                 using p(5)[OF k(1-2)]
```
```  4102                 apply auto
```
```  4103                 done
```
```  4104               { assume "x \<in> k" then show "x \<in> k'" unfolding * . }
```
```  4105               { assume "x \<in> k'" then show "x\<in>k" unfolding * . }
```
```  4106             qed
```
```  4107
```
```  4108             let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
```
```  4109             show "\<forall>x. x \<in> ?B a \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' x - (f (Sup k) -
```
```  4110               f (Inf k))) x) \<le> e * (b - a) / 4"
```
```  4111               apply rule
```
```  4112               apply rule
```
```  4113               unfolding mem_Collect_eq
```
```  4114               unfolding split_paired_all fst_conv snd_conv
```
```  4115             proof (safe, goal_cases)
```
```  4116               case prems: 1
```
```  4117               guess v using pa[OF prems(1)] .. note v = conjunctD2[OF this]
```
```  4118               have "?a \<in> {?a..v}"
```
```  4119                 using v(2) by auto
```
```  4120               then have "v \<le> ?b"
```
```  4121                 using p(3)[OF prems(1)] unfolding subset_eq v by auto
```
```  4122               moreover have "{?a..v} \<subseteq> ball ?a da"
```
```  4123                 using fineD[OF as(2) prems(1)]
```
```  4124                 apply -
```
```  4125                 apply (subst(asm) if_P)
```
```  4126                 apply (rule refl)
```
```  4127                 unfolding subset_eq
```
```  4128                 apply safe
```
```  4129                 apply (erule_tac x=" x" in ballE)
```
```  4130                 apply (auto simp add:subset_eq dist_real_def v)
```
```  4131                 done
```
```  4132               ultimately show ?case
```
```  4133                 unfolding v interval_bounds_real[OF v(2)] box_real
```
```  4134                 apply -
```
```  4135                 apply(rule da(2)[of "v"])
```
```  4136                 using prems fineD[OF as(2) prems(1)]
```
```  4137                 unfolding v content_eq_0
```
```  4138                 apply auto
```
```  4139                 done
```
```  4140             qed
```
```  4141             show "\<forall>x. x \<in> ?B b \<longrightarrow> norm ((\<lambda>(x, k). content k *\<^sub>R f' x -
```
```  4142               (f (Sup k) - f (Inf k))) x) \<le> e * (b - a) / 4"
```
```  4143               apply rule
```
```  4144               apply rule
```
```  4145               unfolding mem_Collect_eq
```
```  4146               unfolding split_paired_all fst_conv snd_conv
```
```  4147             proof (safe, goal_cases)
```
```  4148               case prems: 1
```
```  4149               guess v using pb[OF prems(1)] .. note v = conjunctD2[OF this]
```
```  4150               have "?b \<in> {v.. ?b}"
```
```  4151                 using v(2) by auto
```
```  4152               then have "v \<ge> ?a" using p(3)[OF prems(1)]
```
```  4153                 unfolding subset_eq v by auto
```
```  4154               moreover have "{v..?b} \<subseteq> ball ?b db"
```
```  4155                 using fineD[OF as(2) prems(1)]
```
```  4156                 apply -
```
```  4157                 apply (subst(asm) if_P, rule refl)
```
```  4158                 unfolding subset_eq
```
```  4159                 apply safe
```
```  4160                 apply (erule_tac x=" x" in ballE)
```
```  4161                 using ab
```
```  4162                 apply (auto simp add:subset_eq v dist_real_def)
```
```  4163                 done
```
```  4164               ultimately show ?case
```
```  4165                 unfolding v
```
```  4166                 unfolding interval_bounds_real[OF v(2)] box_real
```
```  4167                 apply -
```
```  4168                 apply(rule db(2)[of "v"])
```
```  4169                 using prems fineD[OF as(2) prems(1)]
```
```  4170                 unfolding v content_eq_0
```
```  4171                 apply auto
```
```  4172                 done
```
```  4173             qed
```
```  4174           qed (insert p(1) ab e, auto simp add: field_simps)
```
```  4175         qed auto
```
```  4176       qed
```
```  4177     qed
```
```  4178   qed
```
```  4179 qed
```
```  4180
```
```  4181
```
```  4182 subsection \<open>Stronger form with finite number of exceptional points.\<close>
```
```  4183
```
```  4184 lemma fundamental_theorem_of_calculus_interior_strong:
```
```  4185   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  4186   assumes "finite s"
```
```  4187     and "a \<le> b"
```
```  4188     and "continuous_on {a .. b} f"
```
```  4189     and "\<forall>x\<in>{a <..< b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  4190   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  4191   using assms
```
```  4192 proof (induct "card s" arbitrary: s a b)
```
```  4193   case 0
```
```  4194   show ?case
```
```  4195     apply (rule fundamental_theorem_of_calculus_interior)
```
```  4196     using 0
```
```  4197     apply auto
```
```  4198     done
```
```  4199 next
```
```  4200   case (Suc n)
```
```  4201   from this(2) guess c s'
```
```  4202     apply -
```
```  4203     apply (subst(asm) eq_commute)
```
```  4204     unfolding card_Suc_eq
```
```  4205     apply (subst(asm)(2) eq_commute)
```
```  4206     apply (elim exE conjE)
```
```  4207     done
```
```  4208   note cs = this[rule_format]
```
```  4209   show ?case
```
```  4210   proof (cases "c \<in> box a b")
```
```  4211     case False
```
```  4212     then show ?thesis
```
```  4213       apply -
```
```  4214       apply (rule Suc(1)[OF cs(3) _ Suc(4,5)])
```
```  4215       apply safe
```
```  4216       defer
```
```  4217       apply (rule Suc(6)[rule_format])
```
```  4218       using Suc(3)
```
```  4219       unfolding cs
```
```  4220       apply auto
```
```  4221       done
```
```  4222   next
```
```  4223     have *: "f b - f a = (f c - f a) + (f b - f c)"
```
```  4224       by auto
```
```  4225     case True
```
```  4226     then have "a \<le> c" "c \<le> b"
```
```  4227       by (auto simp: mem_box)
```
```  4228     then show ?thesis
```
```  4229       apply (subst *)
```
```  4230       apply (rule has_integral_combine)
```
```  4231       apply assumption+
```
```  4232       apply (rule_tac[!] Suc(1)[OF cs(3)])
```
```  4233       using Suc(3)
```
```  4234       unfolding cs
```
```  4235     proof -
```
```  4236       show "continuous_on {a .. c} f" "continuous_on {c .. b} f"
```
```  4237         apply (rule_tac[!] continuous_on_subset[OF Suc(5)])
```
```  4238         using True
```
```  4239         apply (auto simp: mem_box)
```
```  4240         done
```
```  4241       let ?P = "\<lambda>i j. \<forall>x\<in>{i <..< j} - s'. (f has_vector_derivative f' x) (at x)"
```
```  4242       show "?P a c" "?P c b"
```
```  4243         apply safe
```
```  4244         apply (rule_tac[!] Suc(6)[rule_format])
```
```  4245         using True
```
```  4246         unfolding cs
```
```  4247         apply (auto simp: mem_box)
```
```  4248         done
```
```  4249     qed auto
```
```  4250   qed
```
```  4251 qed
```
```  4252
```
```  4253 lemma fundamental_theorem_of_calculus_strong:
```
```  4254   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  4255   assumes "finite s"
```
```  4256     and "a \<le> b"
```
```  4257     and "continuous_on {a .. b} f"
```
```  4258     and "\<forall>x\<in>{a .. b} - s. (f has_vector_derivative f'(x)) (at x)"
```
```  4259   shows "(f' has_integral (f b - f a)) {a .. b}"
```
```  4260   apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
```
```  4261   using assms(4)
```
```  4262   apply (auto simp: mem_box)
```
```  4263   done
```
```  4264
```
```  4265 lemma indefinite_integral_continuous_left:
```
```  4266   fixes f:: "real \<Rightarrow> 'a::banach"
```
```  4267   assumes "f integrable_on {a .. b}"
```
```  4268     and "a < c"
```
```  4269     and "c \<le> b"
```
```  4270     and "e > 0"
```
```  4271   obtains d where "d > 0"
```
```  4272     and "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm (integral {a .. c} f - integral {a .. t} f) < e"
```
```  4273 proof -
```
```  4274   have "\<exists>w>0. \<forall>t. c - w < t \<and> t < c \<longrightarrow> norm (f c) * norm(c - t) < e / 3"
```
```  4275   proof (cases "f c = 0")
```
```  4276     case False
```
```  4277     hence "0 < e / 3 / norm (f c)" using \<open>e>0\<close> by simp
```
```  4278     then show ?thesis
```
```  4279       apply -
```
```  4280       apply rule
```
```  4281       apply rule
```
```  4282       apply assumption
```
```  4283       apply safe
```
```  4284     proof -
```
```  4285       fix t
```
```  4286       assume as: "t < c" and "c - e / 3 / norm (f c) < t"
```
```  4287       then have "c - t < e / 3 / norm (f c)"
```
```  4288         by auto
```
```  4289       then have "norm (c - t) < e / 3 / norm (f c)"
```
```  4290         using as by auto
```
```  4291       then show "norm (f c) * norm (c - t) < e / 3"
```
```  4292         using False
```
```  4293         apply -
```
```  4294         apply (subst mult.commute)
```
```  4295         apply (subst pos_less_divide_eq[symmetric])
```
```  4296         apply auto
```
```  4297         done
```
```  4298     qed
```
```  4299   next
```
```  4300     case True
```
```  4301     show ?thesis
```
```  4302       apply (rule_tac x=1 in exI)
```
```  4303       unfolding True
```
```  4304       using \<open>e > 0\<close>
```
```  4305       apply auto
```
```  4306       done
```
```  4307   qed
```
```  4308   then guess w .. note w = conjunctD2[OF this,rule_format]
```
```  4309
```
```  4310   have *: "e / 3 > 0"
```
```  4311     using assms by auto
```
```  4312   have "f integrable_on {a .. c}"
```
```  4313     apply (rule integrable_subinterval_real[OF assms(1)])
```
```  4314     using assms(2-3)
```
```  4315     apply auto
```
```  4316     done
```
```  4317   from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d1 ..
```
```  4318   note d1 = conjunctD2[OF this,rule_format]
```
```  4319   define d where [abs_def]: "d x = ball x w \<inter> d1 x" for x
```
```  4320   have "gauge d"
```
```  4321     unfolding d_def using w(1) d1 by auto
```
```  4322   note this[unfolded gauge_def,rule_format,of c]
```
```  4323   note conjunctD2[OF this]
```
```  4324   from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k ..
```
```  4325   note k=conjunctD2[OF this]
```
```  4326
```
```  4327   let ?d = "min k (c - a) / 2"
```
```  4328   show ?thesis
```
```  4329     apply (rule that[of ?d])
```
```  4330     apply safe
```
```  4331   proof -
```
```  4332     show "?d > 0"
```
```  4333       using k(1) using assms(2) by auto
```
```  4334     fix t
```
```  4335     assume as: "c - ?d < t" "t \<le> c"
```
```  4336     let ?thesis = "norm (integral ({a .. c}) f - integral ({a .. t}) f) < e"
```
```  4337     {
```
```  4338       presume *: "t < c \<Longrightarrow> ?thesis"
```
```  4339       show ?thesis
```
```  4340         apply (cases "t = c")
```
```  4341         defer
```
```  4342         apply (rule *)
```
```  4343         apply (subst less_le)
```
```  4344         using \<open>e > 0\<close> as(2)
```
```  4345         apply auto
```
```  4346         done
```
```  4347     }
```
```  4348     assume "t < c"
```
```  4349
```
```  4350     have "f integrable_on {a .. t}"
```
```  4351       apply (rule integrable_subinterval_real[OF assms(1)])
```
```  4352       using assms(2-3) as(2)
```
```  4353       apply auto
```
```  4354       done
```
```  4355     from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d2 ..
```
```  4356     note d2 = conjunctD2[OF this,rule_format]
```
```  4357     define d3 where "d3 x = (if x \<le> t then d1 x \<inter> d2 x else d1 x)" for x
```
```  4358     have "gauge d3"
```
```  4359       using d2(1) d1(1) unfolding d3_def gauge_def by auto
```
```  4360     from fine_division_exists_real[OF this, of a t] guess p . note p=this
```
```  4361     note p'=tagged_division_ofD[OF this(1)]
```
```  4362     have pt: "\<forall>(x,k)\<in>p. x \<le> t"
```
```  4363     proof (safe, goal_cases)
```
```  4364       case prems: 1
```
```  4365       from p'(2,3)[OF prems] show ?case
```
```  4366         by auto
```
```  4367     qed
```
```  4368     with p(2) have "d2 fine p"
```
```  4369       unfolding fine_def d3_def
```
```  4370       apply safe
```
```  4371       apply (erule_tac x="(a,b)" in ballE)+
```
```  4372       apply auto
```
```  4373       done
```
```  4374     note d2_fin = d2(2)[OF conjI[OF p(1) this]]
```
```  4375
```
```  4376     have *: "{a .. c} \<inter> {x. x \<bullet> 1 \<le> t} = {a .. t}" "{a .. c} \<inter> {x. x \<bullet> 1 \<ge> t} = {t .. c}"
```
```  4377       using assms(2-3) as by (auto simp add: field_simps)
```
```  4378     have "p \<union> {(c, {t .. c})} tagged_division_of {a .. c} \<and> d1 fine p \<union> {(c, {t .. c})}"
```
```  4379       apply rule
```
```  4380       apply (rule tagged_division_union_interval_real[of _ _ _ 1 "t"])
```
```  4381       unfolding *
```
```  4382       apply (rule p)
```
```  4383       apply (rule tagged_division_of_self_real)
```
```  4384       unfolding fine_def
```
```  4385       apply safe
```
```  4386     proof -
```
```  4387       fix x k y
```
```  4388       assume "(x,k) \<in> p" and "y \<in> k"
```
```  4389       then show "y \<in> d1 x"
```
```  4390         using p(2) pt
```
```  4391         unfolding fine_def d3_def
```
```  4392         apply -
```
```  4393         apply (erule_tac x="(x,k)" in ballE)+
```
```  4394         apply auto
```
```  4395         done
```
```  4396     next
```
```  4397       fix x assume "x \<in> {t..c}"
```
```  4398       then have "dist c x < k"
```
```  4399         unfolding dist_real_def
```
```  4400         using as(1)
```
```  4401         by (auto simp add: field_simps)
```
```  4402       then show "x \<in> d1 c"
```
```  4403         using k(2)
```
```  4404         unfolding d_def
```
```  4405         by auto
```
```  4406     qed (insert as(2), auto) note d1_fin = d1(2)[OF this]
```
```  4407
```
```  4408     have *: "integral {a .. c} f - integral {a .. t} f = -(((c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) -
```
```  4409       integral {a .. c} f) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - integral {a .. t} f) + (c - t) *\<^sub>R f c"
```
```  4410       "e = (e/3 + e/3) + e/3"
```
```  4411       by auto
```
```  4412     have **: "(\<Sum>(x, k)\<in>p \<union> {(c, {t .. c})}. content k *\<^sub>R f x) =
```
```  4413       (c - t) *\<^sub>R f c + (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
```
```  4414     proof -
```
```  4415       have **: "\<And>x F. F \<union> {x} = insert x F"
```
```  4416         by auto
```
```  4417       have "(c, cbox t c) \<notin> p"
```
```  4418       proof (safe, goal_cases)
```
```  4419         case prems: 1
```
```  4420         from p'(2-3)[OF prems] have "c \<in> cbox a t"
```
```  4421           by auto
```
```  4422         then show False using \<open>t < c\<close>
```
```  4423           by auto
```
```  4424       qed
```
```  4425       then show ?thesis
```
```  4426         unfolding ** box_real
```
```  4427         apply -
```
```  4428         apply (subst sum.insert)
```
```  4429         apply (rule p')
```
```  4430         unfolding split_conv
```
```  4431         defer
```
```  4432         apply (subst content_real)
```
```  4433         using as(2)
```
```  4434         apply auto
```
```  4435         done
```
```  4436     qed
```
```  4437     have ***: "c - w < t \<and> t < c"
```
```  4438     proof -
```
```  4439       have "c - k < t"
```
```  4440         using \<open>k>0\<close> as(1) by (auto simp add: field_simps)
```
```  4441       moreover have "k \<le> w"
```
```  4442         apply (rule ccontr)
```
```  4443         using k(2)
```
```  4444         unfolding subset_eq
```
```  4445         apply (erule_tac x="c + ((k + w)/2)" in ballE)
```
```  4446         unfolding d_def
```
```  4447         using \<open>k > 0\<close> \<open>w > 0\<close>
```
```  4448         apply (auto simp add: field_simps not_le not_less dist_real_def)
```
```  4449         done
```
```  4450       ultimately show ?thesis using \<open>t < c\<close>
```
```  4451         by (auto simp add: field_simps)
```
```  4452     qed
```
```  4453     show ?thesis
```
```  4454       unfolding *(1)
```
```  4455       apply (subst *(2))
```
```  4456       apply (rule norm_triangle_lt add_strict_mono)+
```
```  4457       unfolding norm_minus_cancel
```
```  4458       apply (rule d1_fin[unfolded **])
```
```  4459       apply (rule d2_fin)
```
```  4460       using w(2)[OF ***]
```
```  4461       unfolding norm_scaleR
```
```  4462       apply (auto simp add: field_simps)
```
```  4463       done
```
```  4464   qed
```
```  4465 qed
```
```  4466
```
```  4467 lemma indefinite_integral_continuous_right:
```
```  4468   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  4469   assumes "f integrable_on {a .. b}"
```
```  4470     and "a \<le> c"
```
```  4471     and "c < b"
```
```  4472     and "e > 0"
```
```  4473   obtains d where "0 < d"
```
```  4474     and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a .. c} f - integral {a .. t} f) < e"
```
```  4475 proof -
```
```  4476   have *: "(\<lambda>x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \<le> - a"
```
```  4477     using assms by auto
```
```  4478   from indefinite_integral_continuous_left[OF * \<open>e>0\<close>] guess d . note d = this
```
```  4479   let ?d = "min d (b - c)"
```
```  4480   show ?thesis
```
```  4481     apply (rule that[of "?d"])
```
```  4482     apply safe
```
```  4483   proof -
```
```  4484     show "0 < ?d"
```
```  4485       using d(1) assms(3) by auto
```
```  4486     fix t :: real
```
```  4487     assume as: "c \<le> t" "t < c + ?d"
```
```  4488     have *: "integral {a .. c} f = integral {a .. b} f - integral {c .. b} f"
```
```  4489       "integral {a .. t} f = integral {a .. b} f - integral {t .. b} f"
```
```  4490       apply (simp_all only: algebra_simps)
```
```  4491       apply (rule_tac[!] integral_combine)
```
```  4492       using assms as
```
```  4493       apply auto
```
```  4494       done
```
```  4495     have "(- c) - d < (- t) \<and> - t \<le> - c"
```
```  4496       using as by auto note d(2)[rule_format,OF this]
```
```  4497     then show "norm (integral {a .. c} f - integral {a .. t} f) < e"
```
```  4498       unfolding *
```
```  4499       unfolding integral_reflect
```
```  4500       apply (subst norm_minus_commute)
```
```  4501       apply (auto simp add: algebra_simps)
```
```  4502       done
```
```  4503   qed
```
```  4504 qed
```
```  4505
```
```  4506 lemma indefinite_integral_continuous:
```
```  4507   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  4508   assumes "f integrable_on {a .. b}"
```
```  4509   shows "continuous_on {a .. b} (\<lambda>x. integral {a .. x} f)"
```
```  4510 proof (unfold continuous_on_iff, safe)
```
```  4511   fix x e :: real
```
```  4512   assume as: "x \<in> {a .. b}" "e > 0"
```
```  4513   let ?thesis = "\<exists>d>0. \<forall>x'\<in>{a .. b}. dist x' x < d \<longrightarrow> dist (integral {a .. x'} f) (integral {a .. x} f) < e"
```
```  4514   {
```
```  4515     presume *: "a < b \<Longrightarrow> ?thesis"
```
```  4516     show ?thesis
```
```  4517       apply cases
```
```  4518       apply (rule *)
```
```  4519       apply assumption
```
```  4520     proof goal_cases
```
```  4521       case 1
```
```  4522       then have "cbox a b = {x}"
```
```  4523         using as(1)
```
```  4524         apply -
```
```  4525         apply (rule set_eqI)
```
```  4526         apply auto
```
```  4527         done
```
```  4528       then show ?case using \<open>e > 0\<close> by auto
```
```  4529     qed
```
```  4530   }
```
```  4531   assume "a < b"
```
```  4532   have "(x = a \<or> x = b) \<or> (a < x \<and> x < b)"
```
```  4533     using as(1) by auto
```
```  4534   then show ?thesis
```
```  4535     apply (elim disjE)
```
```  4536   proof -
```
```  4537     assume "x = a"
```
```  4538     have "a \<le> a" ..
```
```  4539     from indefinite_integral_continuous_right[OF assms(1) this \<open>a<b\<close> \<open>e>0\<close>] guess d . note d=this
```
```  4540     show ?thesis
```
```  4541       apply rule
```
```  4542       apply rule
```
```  4543       apply (rule d)
```
```  4544       apply safe
```
```  4545       apply (subst dist_commute)
```
```  4546       unfolding \<open>x = a\<close> dist_norm
```
```  4547       apply (rule d(2)[rule_format])
```
```  4548       apply auto
```
```  4549       done
```
```  4550   next
```
```  4551     assume "x = b"
```
```  4552     have "b \<le> b" ..
```
```  4553     from indefinite_integral_continuous_left[OF assms(1) \<open>a<b\<close> this \<open>e>0\<close>] guess d . note d=this
```
```  4554     show ?thesis
```
```  4555       apply rule
```
```  4556       apply rule
```
```  4557       apply (rule d)
```
```  4558       apply safe
```
```  4559       apply (subst dist_commute)
```
```  4560       unfolding \<open>x = b\<close> dist_norm
```
```  4561       apply (rule d(2)[rule_format])
```
```  4562       apply auto
```
```  4563       done
```
```  4564   next
```
```  4565     assume "a < x \<and> x < b"
```
```  4566     then have xl: "a < x" "x \<le> b" and xr: "a \<le> x" "x < b"
```
```  4567       by auto
```
```  4568     from indefinite_integral_continuous_left [OF assms(1) xl \<open>e>0\<close>] guess d1 . note d1=this
```
```  4569     from indefinite_integral_continuous_right[OF assms(1) xr \<open>e>0\<close>] guess d2 . note d2=this
```
```  4570     show ?thesis
```
```  4571       apply (rule_tac x="min d1 d2" in exI)
```
```  4572     proof safe
```
```  4573       show "0 < min d1 d2"
```
```  4574         using d1 d2 by auto
```
```  4575       fix y
```
```  4576       assume "y \<in> {a .. b}" and "dist y x < min d1 d2"
```
```  4577       then show "dist (integral {a .. y} f) (integral {a .. x} f) < e"
```
```  4578         apply (subst dist_commute)
```
```  4579         apply (cases "y < x")
```
```  4580         unfolding dist_norm
```
```  4581         apply (rule d1(2)[rule_format])
```
```  4582         defer
```
```  4583         apply (rule d2(2)[rule_format])
```
```  4584         unfolding not_less
```
```  4585         apply (auto simp add: field_simps)
```
```  4586         done
```
```  4587     qed
```
```  4588   qed
```
```  4589 qed
```
```  4590
```
```  4591
```
```  4592 subsection \<open>This doesn't directly involve integration, but that gives an easy proof.\<close>
```
```  4593
```
```  4594 lemma has_derivative_zero_unique_strong_interval:
```
```  4595   fixes f :: "real \<Rightarrow> 'a::banach"
```
```  4596   assumes "finite k"
```
```  4597     and "continuous_on {a .. b} f"
```
```  4598     and "f a = y"
```
```  4599     and "\<forall>x\<in>({a .. b} - k). (f has_derivative (\<lambda>h. 0)) (at x within {a .. b})" "x \<in> {a .. b}"
```
```  4600   shows "f x = y"
```
```  4601 proof -
```
```  4602   have ab: "a \<le> b"
```
```  4603     using assms by auto
```
```  4604   have *: "a \<le> x"
```
```  4605     using assms(5) by auto
```
```  4606   have "((\<lambda>x. 0::'a) has_integral f x - f a) {a .. x}"
```
```  4607     apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
```
```  4608     apply (rule continuous_on_subset[OF assms(2)])
```
```  4609     defer
```
```  4610     apply safe
```
```  4611     unfolding has_vector_derivative_def
```
```  4612     apply (subst has_derivative_within_open[symmetric])
```
```  4613     apply assumption
```
```  4614     apply (rule open_greaterThanLessThan)
```
```  4615     apply (rule has_derivative_within_subset[where s="{a .. b}"])
```
```  4616     using assms(4) assms(5)
```
```  4617     apply (auto simp: mem_box)
```
```  4618     done
```
```  4619   note this[unfolded *]
```
```  4620   note has_integral_unique[OF has_integral_0 this]
```
```  4621   then show ?thesis
```
```  4622     unfolding assms by auto
```
```  4623 qed
```
```  4624
```
```  4625
```
```  4626 subsection \<open>Generalize a bit to any convex set.\<close>
```
```  4627
```
```  4628 lemma has_derivative_zero_unique_strong_convex:
```
```  4629   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4630   assumes "convex s"
```
```  4631     and "finite k"
```
```  4632     and "continuous_on s f"
```
```  4633     and "c \<in> s"
```
```  4634     and "f c = y"
```
```  4635     and "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)"
```
```  4636     and "x \<in> s"
```
```  4637   shows "f x = y"
```
```  4638 proof -
```
```  4639   {
```
```  4640     presume *: "x \<noteq> c \<Longrightarrow> ?thesis"
```
```  4641     show ?thesis
```
```  4642       apply cases
```
```  4643       apply (rule *)
```
```  4644       apply assumption
```
```  4645       unfolding assms(5)[symmetric]
```
```  4646       apply auto
```
```  4647       done
```
```  4648   }
```
```  4649   assume "x \<noteq> c"
```
```  4650   note conv = assms(1)[unfolded convex_alt,rule_format]
```
```  4651   have as1: "continuous_on {0 ..1} (f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x))"
```
```  4652     apply (rule continuous_intros)+
```
```  4653     apply (rule continuous_on_subset[OF assms(3)])
```
```  4654     apply safe
```
```  4655     apply (rule conv)
```
```  4656     using assms(4,7)
```
```  4657     apply auto
```
```  4658     done
```
```  4659   have *: "t = xa" if "(1 - t) *\<^sub>R c + t *\<^sub>R x = (1 - xa) *\<^sub>R c + xa *\<^sub>R x" for t xa
```
```  4660   proof -
```
```  4661     from that have "(t - xa) *\<^sub>R x = (t - xa) *\<^sub>R c"
```
```  4662       unfolding scaleR_simps by (auto simp add: algebra_simps)
```
```  4663     then show ?thesis
```
```  4664       using \<open>x \<noteq> c\<close> by auto
```
```  4665   qed
```
```  4666   have as2: "finite {t. ((1 - t) *\<^sub>R c + t *\<^sub>R x) \<in> k}"
```
```  4667     using assms(2)
```
```  4668     apply (rule finite_surj[where f="\<lambda>z. SOME t. (1-t) *\<^sub>R c + t *\<^sub>R x = z"])
```
```  4669     apply safe
```
```  4670     unfolding image_iff
```
```  4671     apply rule
```
```  4672     defer
```
```  4673     apply assumption
```
```  4674     apply (rule sym)
```
```  4675     apply (rule some_equality)
```
```  4676     defer
```
```  4677     apply (drule *)
```
```  4678     apply auto
```
```  4679     done
```
```  4680   have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x)) 1 = y"
```
```  4681     apply (rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
```
```  4682     unfolding o_def
```
```  4683     using assms(5)
```
```  4684     defer
```
```  4685     apply -
```
```  4686     apply rule
```
```  4687   proof -
```
```  4688     fix t
```
```  4689     assume as: "t \<in> {0 .. 1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> k}"
```
```  4690     have *: "c - t *\<^sub>R c + t *\<^sub>R x \<in> s - k"
```
```  4691       apply safe
```
```  4692       apply (rule conv[unfolded scaleR_simps])
```
```  4693       using \<open>x \<in> s\<close> \<open>c \<in> s\<close> as
```
```  4694       by (auto simp add: algebra_simps)
```
```  4695     have "(f \<circ> (\<lambda>t. (1 - t) *\<^sub>R c + t *\<^sub>R x) has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x))
```
```  4696       (at t within {0 .. 1})"
```
```  4697       apply (intro derivative_eq_intros)
```
```  4698       apply simp_all
```
```  4699       apply (simp add: field_simps)
```
```  4700       unfolding scaleR_simps
```
```  4701       apply (rule has_derivative_within_subset,rule assms(6)[rule_format])
```
```  4702       apply (rule *)
```
```  4703       apply safe
```
```  4704       apply (rule conv[unfolded scaleR_simps])
```
```  4705       using \<open>x \<in> s\<close> \<open>c \<in> s\<close>
```
```  4706       apply auto
```
```  4707       done
```
```  4708     then show "((\<lambda>xa. f ((1 - xa) *\<^sub>R c + xa *\<^sub>R x)) has_derivative (\<lambda>h. 0)) (at t within {0 .. 1})"
```
```  4709       unfolding o_def .
```
```  4710   qed auto
```
```  4711   then show ?thesis
```
```  4712     by auto
```
```  4713 qed
```
```  4714
```
```  4715
```
```  4716 text \<open>Also to any open connected set with finite set of exceptions. Could
```
```  4717  generalize to locally convex set with limpt-free set of exceptions.\<close>
```
```  4718
```
```  4719 lemma has_derivative_zero_unique_strong_connected:
```
```  4720   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4721   assumes "connected s"
```
```  4722     and "open s"
```
```  4723     and "finite k"
```
```  4724     and "continuous_on s f"
```
```  4725     and "c \<in> s"
```
```  4726     and "f c = y"
```
```  4727     and "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)"
```
```  4728     and "x\<in>s"
```
```  4729   shows "f x = y"
```
```  4730 proof -
```
```  4731   have "{x \<in> s. f x \<in> {y}} = {} \<or> {x \<in> s. f x \<in> {y}} = s"
```
```  4732     apply (rule assms(1)[unfolded connected_clopen,rule_format])
```
```  4733     apply rule
```
```  4734     defer
```
```  4735     apply (rule continuous_closedin_preimage[OF assms(4) closed_singleton])
```
```  4736     apply (rule open_openin_trans[OF assms(2)])
```
```  4737     unfolding open_contains_ball
```
```  4738   proof safe
```
```  4739     fix x
```
```  4740     assume "x \<in> s"
```
```  4741     from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
```
```  4742     show "\<exists>e>0. ball x e \<subseteq> {xa \<in> s. f xa \<in> {f x}}"
```
```  4743       apply rule
```
```  4744       apply rule
```
```  4745       apply (rule e)
```
```  4746     proof safe
```
```  4747       fix y
```
```  4748       assume y: "y \<in> ball x e"
```
```  4749       then show "y \<in> s"
```
```  4750         using e by auto
```
```  4751       show "f y = f x"
```
```  4752         apply (rule has_derivative_zero_unique_strong_convex[OF convex_ball])
```
```  4753         apply (rule assms)
```
```  4754         apply (rule continuous_on_subset)
```
```  4755         apply (rule assms)
```
```  4756         apply (rule e)+
```
```  4757         apply (subst centre_in_ball)
```
```  4758         apply (rule e)
```
```  4759         apply rule
```
```  4760         apply safe
```
```  4761         apply (rule has_derivative_within_subset)
```
```  4762         apply (rule assms(7)[rule_format])
```
```  4763         using y e
```
```  4764         apply auto
```
```  4765         done
```
```  4766     qed
```
```  4767   qed
```
```  4768   then show ?thesis
```
```  4769     using \<open>x \<in> s\<close> \<open>f c = y\<close> \<open>c \<in> s\<close> by auto
```
```  4770 qed
```
```  4771
```
```  4772 lemma has_derivative_zero_connected_constant:
```
```  4773   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4774   assumes "connected s"
```
```  4775       and "open s"
```
```  4776       and "finite k"
```
```  4777       and "continuous_on s f"
```
```  4778       and "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)"
```
```  4779     obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
```
```  4780 proof (cases "s = {}")
```
```  4781   case True
```
```  4782   then show ?thesis
```
```  4783 by (metis empty_iff that)
```
```  4784 next
```
```  4785   case False
```
```  4786   then obtain c where "c \<in> s"
```
```  4787     by (metis equals0I)
```
```  4788   then show ?thesis
```
```  4789     by (metis has_derivative_zero_unique_strong_connected assms that)
```
```  4790 qed
```
```  4791
```
```  4792
```
```  4793 subsection \<open>Integrating characteristic function of an interval\<close>
```
```  4794
```
```  4795 lemma has_integral_restrict_open_subinterval:
```
```  4796   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4797   assumes "(f has_integral i) (cbox c d)"
```
```  4798     and "cbox c d \<subseteq> cbox a b"
```
```  4799   shows "((\<lambda>x. if x \<in> box c d then f x else 0) has_integral i) (cbox a b)"
```
```  4800 proof -
```
```  4801   define g where [abs_def]: "g x = (if x \<in>box c d then f x else 0)" for x
```
```  4802   {
```
```  4803     presume *: "cbox c d \<noteq> {} \<Longrightarrow> ?thesis"
```
```  4804     show ?thesis
```
```  4805       apply cases
```
```  4806       apply (rule *)
```
```  4807       apply assumption
```
```  4808     proof goal_cases
```
```  4809       case prems: 1
```
```  4810       then have *: "box c d = {}"
```
```  4811         by (metis bot.extremum_uniqueI box_subset_cbox)
```
```  4812       show ?thesis
```
```  4813         using assms(1)
```
```  4814         unfolding *
```
```  4815         using prems
```
```  4816         by auto
```
```  4817     qed
```
```  4818   }
```
```  4819   assume "cbox c d \<noteq> {}"
```
```  4820   from partial_division_extend_1 [OF assms(2) this] guess p . note p=this
```
```  4821   interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)"
```
```  4822     apply (rule comm_monoid_set.intro)
```
```  4823     apply (rule comm_monoid_lift_option)
```
```  4824     apply (rule add.comm_monoid_axioms)
```
```  4825     done
```
```  4826   note operat = operative_division
```
```  4827     [OF operative_integral p(1), symmetric]
```
```  4828   let ?P = "(if g integrable_on cbox a b then Some (integral (cbox a b) g) else None) = Some i"
```
```  4829   {
```
```  4830     presume "?P"
```
```  4831     then have "g integrable_on cbox a b \<and> integral (cbox a b) g = i"
```
```  4832       apply -
```
```  4833       apply cases
```
```  4834       apply (subst(asm) if_P)
```
```  4835       apply assumption
```
```  4836       apply auto
```
```  4837       done
```
```  4838     then show ?thesis
```
```  4839       using integrable_integral
```
```  4840       unfolding g_def
```
```  4841       by auto
```
```  4842   }
```
```  4843   let ?F = F
```
```  4844   have iterate:"?F (\<lambda>i. if g integrable_on i then Some (integral i g) else None) (p - {cbox c d}) = Some 0"
```
```  4845   proof (intro neutral ballI)
```
```  4846     fix x
```
```  4847     assume x: "x \<in> p - {cbox c d}"
```
```  4848     then have "x \<in> p"
```
```  4849       by auto
```
```  4850     note div = division_ofD(2-5)[OF p(1) this]
```
```  4851     from div(3) guess u v by (elim exE) note uv=this
```
```  4852     have "interior x \<inter> interior (cbox c d) = {}"
```
```  4853       using div(4)[OF p(2)] x by auto
```
```  4854     then have "(g has_integral 0) x"
```
```  4855       unfolding uv
```
```  4856       apply -
```
```  4857       apply (rule has_integral_spike_interior[where f="\<lambda>x. 0"])
```
```  4858       unfolding g_def interior_cbox
```
```  4859       apply auto
```
```  4860       done
```
```  4861     then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
```
```  4862       by auto
```
```  4863   qed
```
```  4864
```
```  4865   have *: "p = insert (cbox c d) (p - {cbox c d})"
```
```  4866     using p by auto
```
```  4867   interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)"
```
```  4868     apply (rule comm_monoid_set.intro)
```
```  4869     apply (rule comm_monoid_lift_option)
```
```  4870     apply (rule add.comm_monoid_axioms)
```
```  4871     done
```
```  4872   have **: "g integrable_on cbox c d"
```
```  4873     apply (rule integrable_spike_interior[where f=f])
```
```  4874     unfolding g_def  using assms(1)
```
```  4875     apply auto
```
```  4876     done
```
```  4877   moreover
```
```  4878   have "integral (cbox c d) g = i"
```
```  4879     apply (rule has_integral_unique[OF _ assms(1)])
```
```  4880     apply (rule has_integral_spike_interior[where f=g])
```
```  4881     defer
```
```  4882     apply (rule integrable_integral[OF **])
```
```  4883     unfolding g_def
```
```  4884     apply auto
```
```  4885     done
```
```  4886   ultimately show ?P
```
```  4887     unfolding operat
```
```  4888     using p
```
```  4889     apply (subst *)
```
```  4890     apply (subst insert)
```
```  4891     apply (simp_all add: division_of_finite iterate)
```
```  4892     done
```
```  4893 qed
```
```  4894
```
```  4895 lemma has_integral_restrict_closed_subinterval:
```
```  4896   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4897   assumes "(f has_integral i) (cbox c d)"
```
```  4898     and "cbox c d \<subseteq> cbox a b"
```
```  4899   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b)"
```
```  4900 proof -
```
```  4901   note has_integral_restrict_open_subinterval[OF assms]
```
```  4902   note * = has_integral_spike[OF negligible_frontier_interval _ this]
```
```  4903   show ?thesis
```
```  4904     apply (rule *[of c d])
```
```  4905     using box_subset_cbox[of c d]
```
```  4906     apply auto
```
```  4907     done
```
```  4908 qed
```
```  4909
```
```  4910 lemma has_integral_restrict_closed_subintervals_eq:
```
```  4911   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
```
```  4912   assumes "cbox c d \<subseteq> cbox a b"
```
```  4913   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b) \<longleftrightarrow> (f has_integral i) (cbox c d)"
```
```  4914   (is "?l = ?r")
```
```  4915 proof (cases "cbox c d = {}")
```
```  4916   case False
```
```  4917   let ?g = "\<lambda>x. if x \<in> cbox c d then f x else 0"
```
```  4918   show ?thesis
```
```  4919     apply rule
```
```  4920     defer
```
```  4921     apply (rule has_integral_restrict_closed_subinterval[OF _ assms])
```
```  4922     apply assumption
```
```  4923   proof -
```
```  4924     assume ?l
```
```  4925     then have "?g integrable_on cbox c d"
```
```  4926       using assms has_integral_integrable integrable_subinterval by blast
```
```  4927     then have *: "f integrable_on cbox c d"
```
```  4928       apply -
```
```  4929       apply (rule integrable_eq)
```
```  4930       apply auto
```
```  4931       done
```
```  4932     then have "i = integral (cbox c d) f"
```
```  4933       apply -
```
```  4934       apply (rule has_integral_unique)
```
```  4935       apply (rule \<open>?l\<close>)
```
```  4936       apply (rule has_integral_restrict_closed_subinterval[OF _ assms])
```
```  4937       apply auto
```
```  4938       done
```
```  4939     then show ?r
```
```  4940       using * by auto
```
```  4941   qed
```
```  4942 qed auto
```
```  4943
```
```  4944
```
```  4945 text \<open>Hence we can apply the limit process uniformly to all integrals.\<close>
```
```  4946
```
```  4947 lemma has_integral':
```
```  4948   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  4949   shows "(f has_integral i) s \<longleftrightarrow>
```
```  4950     (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  4951       (\<exists>z. ((\<lambda>x. if x \<in> s then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - i) < e))"
```
```  4952   (is "?l \<longleftrightarrow> (\<forall>e>0. ?r e)")
```
```  4953 proof -
```
```  4954   {
```
```  4955     presume *: "\<exists>a b. s = cbox a b \<Longrightarrow> ?thesis"
```
```  4956     show ?thesis
```
```  4957       apply cases
```
```  4958       apply (rule *)
```
```  4959       apply assumption
```
```  4960       apply (subst has_integral_alt)
```
```  4961       apply auto
```
```  4962       done
```
```  4963   }
```
```  4964   assume "\<exists>a b. s = cbox a b"
```
```  4965   then guess a b by (elim exE) note s=this
```
```  4966   from bounded_cbox[of a b, unfolded bounded_pos] guess B ..
```
```  4967   note B = conjunctD2[OF this,rule_format] show ?thesis
```
```  4968     apply safe
```
```  4969   proof -
```
```  4970     fix e :: real
```
```  4971     assume ?l and "e > 0"
```
```  4972     show "?r e"
```
```  4973       apply (rule_tac x="B+1" in exI)
```
```  4974       apply safe
```
```  4975       defer
```
```  4976       apply (rule_tac x=i in exI)
```
```  4977     proof
```
```  4978       fix c d :: 'n
```
```  4979       assume as: "ball 0 (B+1) \<subseteq> cbox c d"
```
```  4980       then show "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) (cbox c d)"
```
```  4981         unfolding s
```
```  4982         apply -
```
```  4983         apply (rule has_integral_restrict_closed_subinterval)
```
```  4984         apply (rule \<open>?l\<close>[unfolded s])
```
```  4985         apply safe
```
```  4986         apply (drule B(2)[rule_format])
```
```  4987         unfolding subset_eq
```
```  4988         apply (erule_tac x=x in ballE)
```
```  4989         apply (auto simp add: dist_norm)
```
```  4990         done
```
```  4991     qed (insert B \<open>e>0\<close>, auto)
```
```  4992   next
```
```  4993     assume as: "\<forall>e>0. ?r e"
```
```  4994     from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
```
```  4995     define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
```
```  4996     define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
```
```  4997     have c_d: "cbox a b \<subseteq> cbox c d"
```
```  4998       apply safe
```
```  4999       apply (drule B(2))
```
```  5000       unfolding mem_box
```
```  5001     proof
```
```  5002       fix x i
```
```  5003       show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm x \<le> B" and "i \<in> Basis"
```
```  5004         using that and Basis_le_norm[OF \<open>i\<in>Basis\<close>, of x]
```
```  5005         unfolding c_def d_def
```
```  5006         by (auto simp add: field_simps sum_negf)
```
```  5007     qed
```
```  5008     have "ball 0 C \<subseteq> cbox c d"
```
```  5009       apply (rule subsetI)
```
```  5010       unfolding mem_box mem_ball dist_norm
```
```  5011     proof
```
```  5012       fix x i :: 'n
```
```  5013       assume x: "norm (0 - x) < C" and i: "i \<in> Basis"
```
```  5014       show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
```
```  5015         using Basis_le_norm[OF i, of x] and x i
```
```  5016         unfolding c_def d_def
```
```  5017         by (auto simp: sum_negf)
```
```  5018     qed
```
```  5019     from C(2)[OF this] have "\<exists>y. (f has_integral y) (cbox a b)"
```
```  5020       unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,symmetric]
```
```  5021       unfolding s
```
```  5022       by auto
```
```  5023     then guess y .. note y=this
```
```  5024
```
```  5025     have "y = i"
```
```  5026     proof (rule ccontr)
```
```  5027       assume "\<not> ?thesis"
```
```  5028       then have "0 < norm (y - i)"
```
```  5029         by auto
```
```  5030       from as[rule_format,OF this] guess C ..  note C=conjunctD2[OF this,rule_format]
```
```  5031       define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
```
```  5032       define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
```
```  5033       have c_d: "cbox a b \<subseteq> cbox c d"
```
```  5034         apply safe
```
```  5035         apply (drule B(2))
```
```  5036         unfolding mem_box
```
```  5037       proof
```
```  5038         fix x i :: 'n
```
```  5039         assume "norm x \<le> B" and "i \<in> Basis"
```
```  5040         then show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
```
```  5041           using Basis_le_norm[of i x]
```
```  5042           unfolding c_def d_def
```
```  5043           by (auto simp add: field_simps sum_negf)
```
```  5044       qed
```
```  5045       have "ball 0 C \<subseteq> cbox c d"
```
```  5046         apply (rule subsetI)
```
```  5047         unfolding mem_box mem_ball dist_norm
```
```  5048       proof
```
```  5049         fix x i :: 'n
```
```  5050         assume "norm (0 - x) < C" and "i \<in> Basis"
```
```  5051         then show "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
```
```  5052           using Basis_le_norm[of i x]
```
```  5053           unfolding c_def d_def
```
```  5054           by (auto simp: sum_negf)
```
```  5055       qed
```
```  5056       note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
```
```  5057       note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
```
```  5058       then have "z = y" and "norm (z - i) < norm (y - i)"
```
```  5059         apply -
```
```  5060         apply (rule has_integral_unique[OF _ y(1)])
```
```  5061         apply assumption
```
```  5062         apply assumption
```
```  5063         done
```
```  5064       then show False
```
```  5065         by auto
```
```  5066     qed
```
```  5067     then show ?l
```
```  5068       using y
```
```  5069       unfolding s
```
```  5070       by auto
```
```  5071   qed
```
```  5072 qed
```
```  5073
```
```  5074 lemma has_integral_le:
```
```  5075   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5076   assumes "(f has_integral i) s"
```
```  5077     and "(g has_integral j) s"
```
```  5078     and "\<forall>x\<in>s. f x \<le> g x"
```
```  5079   shows "i \<le> j"
```
```  5080   using has_integral_component_le[OF _ assms(1-2), of 1]
```
```  5081   using assms(3)
```
```  5082   by auto
```
```  5083
```
```  5084 lemma integral_le:
```
```  5085   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5086   assumes "f integrable_on s"
```
```  5087     and "g integrable_on s"
```
```  5088     and "\<forall>x\<in>s. f x \<le> g x"
```
```  5089   shows "integral s f \<le> integral s g"
```
```  5090   by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])
```
```  5091
```
```  5092 lemma has_integral_nonneg:
```
```  5093   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5094   assumes "(f has_integral i) s"
```
```  5095     and "\<forall>x\<in>s. 0 \<le> f x"
```
```  5096   shows "0 \<le> i"
```
```  5097   using has_integral_component_nonneg[of 1 f i s]
```
```  5098   unfolding o_def
```
```  5099   using assms
```
```  5100   by auto
```
```  5101
```
```  5102 lemma integral_nonneg:
```
```  5103   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5104   assumes "f integrable_on s"
```
```  5105     and "\<forall>x\<in>s. 0 \<le> f x"
```
```  5106   shows "0 \<le> integral s f"
```
```  5107   by (rule has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)])
```
```  5108
```
```  5109
```
```  5110 text \<open>Hence a general restriction property.\<close>
```
```  5111
```
```  5112 lemma has_integral_restrict[simp]:
```
```  5113   assumes "s \<subseteq> t"
```
```  5114   shows "((\<lambda>x. if x \<in> s then f x else (0::'a::banach)) has_integral i) t \<longleftrightarrow> (f has_integral i) s"
```
```  5115 proof -
```
```  5116   have *: "\<And>x. (if x \<in> t then if x \<in> s then f x else 0 else 0) =  (if x\<in>s then f x else 0)"
```
```  5117     using assms by auto
```
```  5118   show ?thesis
```
```  5119     apply (subst(2) has_integral')
```
```  5120     apply (subst has_integral')
```
```  5121     unfolding *
```
```  5122     apply rule
```
```  5123     done
```
```  5124 qed
```
```  5125
```
```  5126 lemma has_integral_restrict_univ:
```
```  5127   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5128   shows "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s"
```
```  5129   by auto
```
```  5130
```
```  5131 lemma has_integral_on_superset:
```
```  5132   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5133   assumes "\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  5134     and "s \<subseteq> t"
```
```  5135     and "(f has_integral i) s"
```
```  5136   shows "(f has_integral i) t"
```
```  5137 proof -
```
```  5138   have "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. if x \<in> t then f x else 0)"
```
```  5139     apply rule
```
```  5140     using assms(1-2)
```
```  5141     apply auto
```
```  5142     done
```
```  5143   then show ?thesis
```
```  5144     using assms(3)
```
```  5145     apply (subst has_integral_restrict_univ[symmetric])
```
```  5146     apply (subst(asm) has_integral_restrict_univ[symmetric])
```
```  5147     apply auto
```
```  5148     done
```
```  5149 qed
```
```  5150
```
```  5151 lemma integrable_on_superset:
```
```  5152   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5153   assumes "\<forall>x. x \<notin> s \<longrightarrow> f x = 0"
```
```  5154     and "s \<subseteq> t"
```
```  5155     and "f integrable_on s"
```
```  5156   shows "f integrable_on t"
```
```  5157   using assms
```
```  5158   unfolding integrable_on_def
```
```  5159   by (auto intro:has_integral_on_superset)
```
```  5160
```
```  5161 lemma integral_restrict_univ[intro]:
```
```  5162   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5163   shows "f integrable_on s \<Longrightarrow> integral UNIV (\<lambda>x. if x \<in> s then f x else 0) = integral s f"
```
```  5164   apply (rule integral_unique)
```
```  5165   unfolding has_integral_restrict_univ
```
```  5166   apply auto
```
```  5167   done
```
```  5168
```
```  5169 lemma integrable_restrict_univ:
```
```  5170   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5171   shows "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
```
```  5172   unfolding integrable_on_def
```
```  5173   by auto
```
```  5174
```
```  5175 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> cbox a b))" (is "?l \<longleftrightarrow> ?r")
```
```  5176 proof
```
```  5177   assume ?r
```
```  5178   show ?l
```
```  5179     unfolding negligible_def
```
```  5180   proof safe
```
```  5181     fix a b
```
```  5182     show "(indicator s has_integral 0) (cbox a b)"
```
```  5183       apply (rule has_integral_negligible[OF \<open>?r\<close>[rule_format,of a b]])
```
```  5184       unfolding indicator_def
```
```  5185       apply auto
```
```  5186       done
```
```  5187   qed
```
```  5188 qed (simp add: negligible_Int)
```
```  5189
```
```  5190 lemma negligible_translation:
```
```  5191   assumes "negligible S"
```
```  5192     shows "negligible (op + c ` S)"
```
```  5193 proof -
```
```  5194   have inj: "inj (op + c)"
```
```  5195     by simp
```
```  5196   show ?thesis
```
```  5197   using assms
```
```  5198   proof (clarsimp simp: negligible_def)
```
```  5199     fix a b
```
```  5200     assume "\<forall>x y. (indicator S has_integral 0) (cbox x y)"
```
```  5201     then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))"
```
```  5202       by (meson Diff_iff assms has_integral_negligible indicator_simps(2))
```
```  5203     have eq: "indicator (op + c ` S) = (\<lambda>x. indicator S (x - c))"
```
```  5204       by (force simp add: indicator_def)
```
```  5205     show "(indicator (op + c ` S) has_integral 0) (cbox a b)"
```
```  5206       using has_integral_affinity [OF *, of 1 "-c"]
```
```  5207             cbox_translation [of "c" "-c+a" "-c+b"]
```
```  5208       by (simp add: eq add.commute)
```
```  5209   qed
```
```  5210 qed
```
```  5211
```
```  5212 lemma negligible_translation_rev:
```
```  5213   assumes "negligible (op + c ` S)"
```
```  5214     shows "negligible S"
```
```  5215 by (metis negligible_translation [OF assms, of "-c"] translation_galois)
```
```  5216
```
```  5217 lemma has_integral_spike_set_eq:
```
```  5218   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5219   assumes "negligible ((s - t) \<union> (t - s))"
```
```  5220   shows "(f has_integral y) s \<longleftrightarrow> (f has_integral y) t"
```
```  5221   unfolding has_integral_restrict_univ[symmetric,of f]
```
```  5222   apply (rule has_integral_spike_eq[OF assms])
```
```  5223   by (auto split: if_split_asm)
```
```  5224
```
```  5225 lemma has_integral_spike_set:
```
```  5226   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5227   assumes "(f has_integral y) s" "negligible ((s - t) \<union> (t - s))"
```
```  5228   shows "(f has_integral y) t"
```
```  5229   using assms has_integral_spike_set_eq
```
```  5230   by auto
```
```  5231
```
```  5232 lemma integrable_spike_set:
```
```  5233   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5234   assumes "f integrable_on s" and "negligible ((s - t) \<union> (t - s))"
```
```  5235     shows "f integrable_on t"
```
```  5236   using assms by (simp add: integrable_on_def has_integral_spike_set_eq)
```
```  5237
```
```  5238 lemma integrable_spike_set_eq:
```
```  5239   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5240   assumes "negligible ((s - t) \<union> (t - s))"
```
```  5241   shows "f integrable_on s \<longleftrightarrow> f integrable_on t"
```
```  5242 by (blast intro: integrable_spike_set assms negligible_subset)
```
```  5243
```
```  5244 subsection \<open>More lemmas that are useful later\<close>
```
```  5245
```
```  5246 lemma has_integral_subset_component_le:
```
```  5247   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  5248   assumes k: "k \<in> Basis"
```
```  5249     and as: "s \<subseteq> t" "(f has_integral i) s" "(f has_integral j) t" "\<forall>x\<in>t. 0 \<le> f(x)\<bullet>k"
```
```  5250   shows "i\<bullet>k \<le> j\<bullet>k"
```
```  5251 proof -
```
```  5252   note has_integral_restrict_univ[symmetric, of f]
```
```  5253   note as(2-3)[unfolded this] note * = has_integral_component_le[OF k this]
```
```  5254   show ?thesis
```
```  5255     apply (rule *)
```
```  5256     using as(1,4)
```
```  5257     apply auto
```
```  5258     done
```
```  5259 qed
```
```  5260
```
```  5261 lemma has_integral_subset_le:
```
```  5262   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5263   assumes "s \<subseteq> t"
```
```  5264     and "(f has_integral i) s"
```
```  5265     and "(f has_integral j) t"
```
```  5266     and "\<forall>x\<in>t. 0 \<le> f x"
```
```  5267   shows "i \<le> j"
```
```  5268   using has_integral_subset_component_le[OF _ assms(1), of 1 f i j]
```
```  5269   using assms
```
```  5270   by auto
```
```  5271
```
```  5272 lemma integral_subset_component_le:
```
```  5273   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
```
```  5274   assumes "k \<in> Basis"
```
```  5275     and "s \<subseteq> t"
```
```  5276     and "f integrable_on s"
```
```  5277     and "f integrable_on t"
```
```  5278     and "\<forall>x \<in> t. 0 \<le> f x \<bullet> k"
```
```  5279   shows "(integral s f)\<bullet>k \<le> (integral t f)\<bullet>k"
```
```  5280   apply (rule has_integral_subset_component_le)
```
```  5281   using assms
```
```  5282   apply auto
```
```  5283   done
```
```  5284
```
```  5285 lemma integral_subset_le:
```
```  5286   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5287   assumes "s \<subseteq> t"
```
```  5288     and "f integrable_on s"
```
```  5289     and "f integrable_on t"
```
```  5290     and "\<forall>x \<in> t. 0 \<le> f x"
```
```  5291   shows "integral s f \<le> integral t f"
```
```  5292   apply (rule has_integral_subset_le)
```
```  5293   using assms
```
```  5294   apply auto
```
```  5295   done
```
```  5296
```
```  5297 lemma has_integral_alt':
```
```  5298   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5299   shows "(f has_integral i) s \<longleftrightarrow> (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
```
```  5300     (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  5301       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e)"
```
```  5302   (is "?l = ?r")
```
```  5303 proof
```
```  5304   assume ?r
```
```  5305   show ?l
```
```  5306     apply (subst has_integral')
```
```  5307     apply safe
```
```  5308   proof goal_cases
```
```  5309     case (1 e)
```
```  5310     from \<open>?r\<close>[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
```
```  5311     show ?case
```
```  5312       apply rule
```
```  5313       apply rule
```
```  5314       apply (rule B)
```
```  5315       apply safe
```
```  5316       apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0)" in exI)
```
```  5317       apply (drule B(2)[rule_format])
```
```  5318       using integrable_integral[OF \<open>?r\<close>[THEN conjunct1,rule_format]]
```
```  5319       apply auto
```
```  5320       done
```
```  5321   qed
```
```  5322 next
```
```  5323   assume ?l note as = this[unfolded has_integral'[of f],rule_format]
```
```  5324   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
```
```  5325   show ?r
```
```  5326   proof safe
```
```  5327     fix a b :: 'n
```
```  5328     from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  5329     let ?a = "\<Sum>i\<in>Basis. min (a\<bullet>i) (-B) *\<^sub>R i::'n"
```
```  5330     let ?b = "\<Sum>i\<in>Basis. max (b\<bullet>i) B *\<^sub>R i::'n"
```
```  5331     show "?f integrable_on cbox a b"
```
```  5332     proof (rule integrable_subinterval[of _ ?a ?b])
```
```  5333       have "ball 0 B \<subseteq> cbox ?a ?b"
```
```  5334         apply (rule subsetI)
```
```  5335         unfolding mem_ball mem_box dist_norm
```
```  5336       proof (rule, goal_cases)
```
```  5337         case (1 x i)
```
```  5338         then show ?case using Basis_le_norm[of i x]
```
```  5339           by (auto simp add:field_simps)
```
```  5340       qed
```
```  5341       from B(2)[OF this] guess z .. note conjunct1[OF this]
```
```  5342       then show "?f integrable_on cbox ?a ?b"
```
```  5343         unfolding integrable_on_def by auto
```
```  5344       show "cbox a b \<subseteq> cbox ?a ?b"
```
```  5345         apply safe
```
```  5346         unfolding mem_box
```
```  5347         apply rule
```
```  5348         apply (erule_tac x=i in ballE)
```
```  5349         apply auto
```
```  5350         done
```
```  5351     qed
```
```  5352
```
```  5353     fix e :: real
```
```  5354     assume "e > 0"
```
```  5355     from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  5356     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
```
```  5357       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
```
```  5358       apply rule
```
```  5359       apply rule
```
```  5360       apply (rule B)
```
```  5361       apply safe
```
```  5362     proof goal_cases
```
```  5363       case 1
```
```  5364       from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
```
```  5365       from integral_unique[OF this(1)] show ?case
```
```  5366         using z(2) by auto
```
```  5367     qed
```
```  5368   qed
```
```  5369 qed
```
```  5370
```
```  5371
```
```  5372 subsection \<open>Continuity of the integral (for a 1-dimensional interval).\<close>
```
```  5373
```
```  5374 lemma integrable_alt:
```
```  5375   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5376   shows "f integrable_on s \<longleftrightarrow>
```
```  5377     (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
```
```  5378     (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
```
```  5379     norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) -
```
```  5380       integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e)"
```
```  5381   (is "?l = ?r")
```
```  5382 proof
```
```  5383   assume ?l
```
```  5384   then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
```
```  5385   note y=conjunctD2[OF this,rule_format]
```
```  5386   show ?r
```
```  5387     apply safe
```
```  5388     apply (rule y)
```
```  5389   proof goal_cases
```
```  5390     case (1 e)
```
```  5391     then have "e/2 > 0"
```
```  5392       by auto
```
```  5393     from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  5394     show ?case
```
```  5395       apply rule
```
```  5396       apply rule
```
```  5397       apply (rule B)
```
```  5398       apply safe
```
```  5399     proof goal_cases
```
```  5400       case prems: (1 a b c d)
```
```  5401       show ?case
```
```  5402         apply (rule norm_triangle_half_l)
```
```  5403         using B(2)[OF prems(1)] B(2)[OF prems(2)]
```
```  5404         apply auto
```
```  5405         done
```
```  5406     qed
```
```  5407   qed
```
```  5408 next
```
```  5409   assume ?r
```
```  5410   note as = conjunctD2[OF this,rule_format]
```
```  5411   let ?cube = "\<lambda>n. cbox (\<Sum>i\<in>Basis. - real n *\<^sub>R i::'n) (\<Sum>i\<in>Basis. real n *\<^sub>R i)"
```
```  5412   have "Cauchy (\<lambda>n. integral (?cube n) (\<lambda>x. if x \<in> s then f x else 0))"
```
```  5413   proof (unfold Cauchy_def, safe, goal_cases)
```
```  5414     case (1 e)
```
```  5415     from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
```
```  5416     from real_arch_simple[of B] guess N .. note N = this
```
```  5417     {
```
```  5418       fix n
```
```  5419       assume n: "n \<ge> N"
```
```  5420       have "ball 0 B \<subseteq> ?cube n"
```
```  5421         apply (rule subsetI)
```
```  5422         unfolding mem_ball mem_box dist_norm
```
```  5423       proof (rule, goal_cases)
```
```  5424         case (1 x i)
```
```  5425         then show ?case
```
```  5426           using Basis_le_norm[of i x] \<open>i\<in>Basis\<close>
```
```  5427           using n N
```
```  5428           by (auto simp add: field_simps sum_negf)
```
```  5429       qed
```
```  5430     }
```
```  5431     then show ?case
```
```  5432       apply -
```
```  5433       apply (rule_tac x=N in exI)
```
```  5434       apply safe
```
```  5435       unfolding dist_norm
```
```  5436       apply (rule B(2))
```
```  5437       apply auto
```
```  5438       done
```
```  5439   qed
```
```  5440   from this[unfolded convergent_eq_cauchy[symmetric]] guess i ..
```
```  5441   note i = this[THEN LIMSEQ_D]
```
```  5442
```
```  5443   show ?l unfolding integrable_on_def has_integral_alt'[of f]
```
```  5444     apply (rule_tac x=i in exI)
```
```  5445     apply safe
```
```  5446     apply (rule as(1)[unfolded integrable_on_def])
```
```  5447   proof goal_cases
```
```  5448     case (1 e)
```
```  5449     then have *: "e/2 > 0" by auto
```
```  5450     from i[OF this] guess N .. note N =this[rule_format]
```
```  5451     from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format]
```
```  5452     let ?B = "max (real N) B"
```
```  5453     show ?case
```
```  5454       apply (rule_tac x="?B" in exI)
```
```  5455     proof safe
```
```  5456       show "0 < ?B"
```
```  5457         using B(1) by auto
```
```  5458       fix a b :: 'n
```
```  5459       assume ab: "ball 0 ?B \<subseteq> cbox a b"
```
```  5460       from real_arch_simple[of ?B] guess n .. note n=this
```
```  5461       show "norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
```
```  5462         apply (rule norm_triangle_half_l)
```
```  5463         apply (rule B(2))
```
```  5464         defer
```
```  5465         apply (subst norm_minus_commute)
```
```  5466         apply (rule N[of n])
```
```  5467       proof safe
```
```  5468         show "N \<le> n"
```
```  5469           using n by auto
```
```  5470         fix x :: 'n
```
```  5471         assume x: "x \<in> ball 0 B"
```
```  5472         then have "x \<in> ball 0 ?B"
```
```  5473           by auto
```
```  5474         then show "x \<in> cbox a b"
```
```  5475           using ab by blast
```
```  5476         show "x \<in> ?cube n"
```
```  5477           using x
```
```  5478           unfolding mem_box mem_ball dist_norm
```
```  5479           apply -
```
```  5480         proof (rule, goal_cases)
```
```  5481           case (1 i)
```
```  5482           then show ?case
```
```  5483             using Basis_le_norm[of i x] \<open>i \<in> Basis\<close>
```
```  5484             using n
```
```  5485             by (auto simp add: field_simps sum_negf)
```
```  5486         qed
```
```  5487       qed
```
```  5488     qed
```
```  5489   qed
```
```  5490 qed
```
```  5491
```
```  5492 lemma integrable_altD:
```
```  5493   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5494   assumes "f integrable_on s"
```
```  5495   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
```
```  5496     and "\<And>e. e > 0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
```
```  5497       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e"
```
```  5498   using assms[unfolded integrable_alt[of f]] by auto
```
```  5499
```
```  5500 lemma integrable_on_subcbox:
```
```  5501   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5502   assumes "f integrable_on s"
```
```  5503     and "cbox a b \<subseteq> s"
```
```  5504   shows "f integrable_on cbox a b"
```
```  5505   apply (rule integrable_eq)
```
```  5506   defer
```
```  5507   apply (rule integrable_altD(1)[OF assms(1)])
```
```  5508   using assms(2)
```
```  5509   apply auto
```
```  5510   done
```
```  5511
```
```  5512
```
```  5513 subsection \<open>A straddling criterion for integrability\<close>
```
```  5514
```
```  5515 lemma integrable_straddle_interval:
```
```  5516   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5517   assumes "\<forall>e>0. \<exists>g  h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and>
```
```  5518     norm (i - j) < e \<and> (\<forall>x\<in>cbox a b. (g x) \<le> f x \<and> f x \<le> h x)"
```
```  5519   shows "f integrable_on cbox a b"
```
```  5520 proof (subst integrable_cauchy, safe, goal_cases)
```
```  5521   case (1 e)
```
```  5522   then have e: "e/3 > 0"
```
```  5523     by auto
```
```  5524   note assms[rule_format,OF this]
```
```  5525   then guess g h i j by (elim exE conjE) note obt = this
```
```  5526   from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
```
```  5527   from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
```
```  5528   show ?case
```
```  5529     apply (rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI)
```
```  5530     apply (rule conjI gauge_inter d1 d2)+
```
```  5531     unfolding fine_inter
```
```  5532   proof (safe, goal_cases)
```
```  5533     have **: "\<And>i j g1 g2 h1 h2 f1 f2. g1 - h2 \<le> f1 - f2 \<Longrightarrow> f1 - f2 \<le> h1 - g2 \<Longrightarrow>
```
```  5534       \<bar>i - j\<bar> < e / 3 \<Longrightarrow> \<bar>g2 - i\<bar> < e / 3 \<Longrightarrow> \<bar>g1 - i\<bar> < e / 3 \<Longrightarrow>
```
```  5535       \<bar>h2 - j\<bar> < e / 3 \<Longrightarrow> \<bar>h1 - j\<bar> < e / 3 \<Longrightarrow> \<bar>f1 - f2\<bar> < e"
```
```  5536     using \<open>e > 0\<close> by arith
```
```  5537     case prems: (1 p1 p2)
```
```  5538     note tagged_division_ofD(2-4) note * = this[OF prems(1)] this[OF prems(4)]
```
```  5539
```
```  5540     have "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
```
```  5541       and "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
```
```  5542       and "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
```
```  5543       and "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
```
```  5544       unfolding sum_subtractf[symmetric]
```
```  5545       apply -
```
```  5546       apply (rule_tac[!] sum_nonneg)
```
```  5547       apply safe
```
```  5548       unfolding real_scaleR_def right_diff_distrib[symmetric]
```
```  5549       apply (rule_tac[!] mult_nonneg_nonneg)
```
```  5550     proof -
```
```  5551       fix a b
```
```  5552       assume ab: "(a, b) \<in> p1"
```
```  5553       show "0 \<le> content b"
```
```  5554         using *(3)[OF ab]
```
```  5555         apply safe
```
```  5556         apply (rule content_pos_le)
```
```  5557         done
```
```  5558       then show "0 \<le> content b" .
```
```  5559       show "0 \<le> f a - g a" "0 \<le> h a - f a"
```
```  5560         using *(1-2)[OF ab]
```
```  5561         using obt(4)[rule_format,of a]
```
```  5562         by auto
```
```  5563     next
```
```  5564       fix a b
```
```  5565       assume ab: "(a, b) \<in> p2"
```
```  5566       show "0 \<le> content b"
```
```  5567         using *(6)[OF ab]
```
```  5568         apply safe
```
```  5569         apply (rule content_pos_le)
```
```  5570         done
```
```  5571       then show "0 \<le> content b" .
```
```  5572       show "0 \<le> f a - g a" and "0 \<le> h a - f a"
```
```  5573         using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto
```
```  5574     qed
```
```  5575     then show ?case
```
```  5576       apply -
```
```  5577       unfolding real_norm_def
```
```  5578       apply (rule **)
```
```  5579       defer
```
```  5580       defer
```
```  5581       unfolding real_norm_def[symmetric]
```
```  5582       apply (rule obt(3))
```
```  5583       apply (rule d1(2)[OF conjI[OF prems(4,5)]])
```
```  5584       apply (rule d1(2)[OF conjI[OF prems(1,2)]])
```
```  5585       apply (rule d2(2)[OF conjI[OF prems(4,6)]])
```
```  5586       apply (rule d2(2)[OF conjI[OF prems(1,3)]])
```
```  5587       apply auto
```
```  5588       done
```
```  5589   qed
```
```  5590 qed
```
```  5591
```
```  5592 lemma integrable_straddle:
```
```  5593   fixes f :: "'n::euclidean_space \<Rightarrow> real"
```
```  5594   assumes "\<forall>e>0. \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
```
```  5595     norm (i - j) < e \<and> (\<forall>x\<in>s. g x \<le> f x \<and> f x \<le> h x)"
```
```  5596   shows "f integrable_on s"
```
```  5597 proof -
```
```  5598   have "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
```
```  5599   proof (rule integrable_straddle_interval, safe, goal_cases)
```
```  5600     case (1 a b e)
```
```  5601     then have *: "e/4 > 0"
```
```  5602       by auto
```
```  5603     from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
```
```  5604     note obt(1)[unfolded has_integral_alt'[of g]]
```
```  5605     note conjunctD2[OF this, rule_format]
```
```  5606     note g = this(1) and this(2)[OF *]
```
```  5607     from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
```
```  5608     note obt(2)[unfolded has_integral_alt'[of h]]
```
```  5609     note conjunctD2[OF this, rule_format]
```
```  5610     note h = this(1) and this(2)[OF *]
```
```  5611     from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
```
```  5612     define c :: 'n where "c = (\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max B1 B2)) *\<^sub>R i)"
```
```  5613     define d :: 'n where "d = (\<Sum>i\<in>Basis. max (b\<bullet>i) (max B1 B2) *\<^sub>R i)"
```
```  5614     have *: "ball 0 B1 \<subseteq> cbox c d" "ball 0 B2 \<subseteq> cbox c d"
```
```  5615       apply safe
```
```  5616       unfolding mem_ball mem_box dist_norm
```
```  5617       apply (rule_tac[!] ballI)
```
```  5618     proof goal_cases
```
```  5619       case (1 x i)
```
```  5620       then show ?case using Basis_le_norm[of i x]
```
```  5621         unfolding c_def d_def by auto
```
```  5622     next
```
```  5623       case (2 x i)
```
```  5624       then show ?case using Basis_le_norm[of i x]
```
```  5625         unfolding c_def d_def by auto
```
```  5626     qed
```
```  5627     have **: "\<And>ch cg ag ah::real. norm (ah - ag) \<le> norm (ch - cg) \<Longrightarrow> norm (cg - i) < e / 4 \<Longrightarrow>
```
```  5628       norm (ch - j) < e / 4 \<Longrightarrow> norm (ag - ah) < e"
```
```  5629       using obt(3)
```
```  5630       unfolding real_norm_def
```
```  5631       by arith
```
```  5632     show ?case
```
```  5633       apply (rule_tac x="\<lambda>x. if x \<in> s then g x else 0" in exI)
```
```  5634       apply (rule_tac x="\<lambda>x. if x \<in> s then h x else 0" in exI)
```
```  5635       apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then g x else 0)" in exI)
```
```  5636       apply (rule_tac x="integral (cbox a b) (\<lambda>x. if x \<in> s then h x else 0)" in exI)
```
```  5637       apply safe
```
```  5638       apply (rule_tac[1-2] integrable_integral,rule g)
```
```  5639       apply (rule h)
```
```  5640       apply (rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
```
```  5641     proof -
```
```  5642       have *: "\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
```
```  5643         (if x \<in> s then f x - g x else (0::real))"
```
```  5644         by auto
```
```  5645       note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF h g]]
```
```  5646       show "norm (integral (cbox a b) (\<lambda>x. if x \<in> s then h x else 0) -
```
```  5647           integral (cbox a b) (\<lambda>x. if x \<in> s then g x else 0)) \<le>
```
```  5648         norm (integral (cbox c d) (\<lambda>x. if x \<in> s then h x else 0) -
```
```  5649           integral (cbox c d) (\<lambda>x. if x \<in> s then g x else 0))"
```
```  5650         unfolding integral_diff[OF h g,symmetric] real_norm_def
```
```  5651         apply (subst **)
```
```  5652         defer
```
```  5653         apply (subst **)
```
```  5654         defer
```
```  5655         apply (rule has_integral_subset_le)
```
```  5656         defer
```
```  5657         apply (rule integrable_integral integrable_diff h g)+
```
```  5658       proof safe
```
```  5659         fix x
```
```  5660         assume "x \<in> cbox a b"
```
```  5661         then show "x \<in> cbox c d"
```
```  5662           unfolding mem_box c_def d_def
```
```  5663           apply -
```
```  5664           apply rule
```
```  5665           apply (erule_tac x=i in ballE)
```
```  5666           apply auto
```
```  5667           done
```
```  5668       qed (insert obt(4), auto)
```
```  5669     qed (insert obt(4), auto)
```
```  5670   qed
```
```  5671   note interv = this
```
```  5672
```
```  5673   show ?thesis
```
```  5674     unfolding integrable_alt[of f]
```
```  5675     apply safe
```
```  5676     apply (rule interv)
```
```  5677   proof goal_cases
```
```  5678     case (1 e)
```
```  5679     then have *: "e/3 > 0"
```
```  5680       by auto
```
```  5681     from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
```
```  5682     note obt(1)[unfolded has_integral_alt'[of g]]
```
```  5683     note conjunctD2[OF this, rule_format]
```
```  5684     note g = this(1) and this(2)[OF *]
```
```  5685     from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
```
```  5686     note obt(2)[unfolded has_integral_alt'[of h]]
```
```  5687     note conjunctD2[OF this, rule_format]
```
```  5688     note h = this(1) and this(2)[OF *]
```
```  5689     from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
```
```  5690     show ?case
```
```  5691       apply (rule_tac x="max B1 B2" in exI)
```
```  5692       apply safe
```
```  5693       apply (rule max.strict_coboundedI1)
```
```  5694       apply (rule B1)
```
```  5695     proof -
```
```  5696       fix a b c d :: 'n
```
```  5697       assume as: "ball 0 (max B1 B2) \<subseteq> cbox a b" "ball 0 (max B1 B2) \<subseteq> cbox c d"
```
```  5698       have **: "ball 0 B1 \<subseteq> ball (0::'n) (max B1 B2)" "ball 0 B2 \<subseteq> ball (0::'n) (max B1 B2)"
```
```  5699         by auto
```
```  5700       have *: "\<And>ga gc ha hc fa fc::real.
```
```  5701         \<bar>ga - i\<bar> < e / 3 \<and> \<bar>gc - i\<bar> < e / 3 \<and> \<bar>ha - j\<bar> < e / 3 \<and>
```
```  5702         \<bar>hc - j\<bar> < e / 3 \<and> \<bar>i - j\<bar> < e / 3 \<and> ga \<le> fa \<and> fa \<le> ha \<and> gc \<le> fc \<and> fc \<le> hc \<Longrightarrow>
```
```  5703         \<bar>fa - fc\<bar> < e"
```
```  5704         by (simp add: abs_real_def split: if_split_asm)
```
```  5705       show "norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)
```
```  5706         (\<lambda>x. if x \<in> s then f x else 0)) < e"
```
```  5707         unfolding real_norm_def
```
```  5708         apply (rule *)
```
```  5709         apply safe
```
```  5710         unfolding real_norm_def[symmetric]
```
```  5711         apply (rule B1(2))
```
```  5712         apply (rule order_trans)
```
```  5713         apply (rule **)
```
```  5714         apply (rule as(1))
```
```  5715         apply (rule B1(2))
```
```  5716         apply (rule order_trans)
```
```  5717         apply (rule **)
```
```  5718         apply (rule as(2))
```
```  5719         apply (rule B2(2))
```
```  5720         apply (rule order_trans)
```
```  5721         apply (rule **)
```
```  5722         apply (rule as(1))
```
```  5723         apply (rule B2(2))
```
```  5724         apply (rule order_trans)
```
```  5725         apply (rule **)
```
```  5726         apply (rule as(2))
```
```  5727         apply (rule obt)
```
```  5728         apply (rule_tac[!] integral_le)
```
```  5729         using obt
```
```  5730         apply (auto intro!: h g interv)
```
```  5731         done
```
```  5732     qed
```
```  5733   qed
```
```  5734 qed
```
```  5735
```
```  5736
```
```  5737 subsection \<open>Adding integrals over several sets\<close>
```
```  5738
```
```  5739 lemma has_integral_union:
```
```  5740   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5741   assumes "(f has_integral i) s"
```
```  5742     and "(f has_integral j) t"
```
```  5743     and "negligible (s \<inter> t)"
```
```  5744   shows "(f has_integral (i + j)) (s \<union> t)"
```
```  5745 proof -
```
```  5746   note * = has_integral_restrict_univ[symmetric, of f]
```
```  5747   show ?thesis
```
```  5748     unfolding *
```
```  5749     apply (rule has_integral_spike[OF assms(3)])
```
```  5750     defer
```
```  5751     apply (rule has_integral_add[OF assms(1-2)[unfolded *]])
```
```  5752     apply auto
```
```  5753     done
```
```  5754 qed
```
```  5755
```
```  5756 lemma integrable_union:
```
```  5757   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
```
```  5758   assumes "negligible (A \<inter> B)" "f integrable_on A" "f integrable_on B"
```
```  5759   shows   "f integrable_on (A \<union> B)"
```
```  5760 proof -
```
```  5761   from assms obtain y z where "(f has_integral y) A" "(f has_integral z) B"
```
```  5762      by (auto simp: integrable_on_def)
```
```  5763   from has_integral_union[OF this assms(1)] show ?thesis by (auto simp: integrable_on_def)
```
```  5764 qed
```
```  5765
```
```  5766 lemma integrable_union':
```
```  5767   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
```
```  5768   assumes "f integrable_on A" "f integrable_on B" "negligible (A \<inter> B)" "C = A \<union> B"
```
```  5769   shows   "f integrable_on C"
```
```  5770   using integrable_union[of A B f] assms by simp
```
```  5771
```
```  5772 lemma has_integral_unions:
```
```  5773   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5774   assumes "finite t"
```
```  5775     and "\<forall>s\<in>t. (f has_integral (i s)) s"
```
```  5776     and "\<forall>s\<in>t. \<forall>s'\<in>t. s \<noteq> s' \<longrightarrow> negligible (s \<inter> s')"
```
```  5777   shows "(f has_integral (sum i t)) (\<Union>t)"
```
```  5778 proof -
```
```  5779   note * = has_integral_restrict_univ[symmetric, of f]
```
```  5780   have **: "negligible (\<Union>((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> t \<and> b \<in> {y. y \<in> t \<and> a \<noteq> y}}))"
```
```  5781     apply (rule negligible_Union)
```
```  5782     apply (rule finite_imageI)
```
```  5783     apply (rule finite_subset[of _ "t \<times> t"])
```
```  5784     defer
```
```  5785     apply (rule finite_cartesian_product[OF assms(1,1)])
```
```  5786     using assms(3)
```
```  5787     apply auto
```
```  5788     done
```
```  5789   note assms(2)[unfolded *]
```
```  5790   note has_integral_sum[OF assms(1) this]
```
```  5791   then show ?thesis
```
```  5792     unfolding *
```
```  5793     apply -
```
```  5794     apply (rule has_integral_spike[OF **])
```
```  5795     defer
```
```  5796     apply assumption
```
```  5797     apply safe
```
```  5798   proof goal_cases
```
```  5799     case prems: (1 x)
```
```  5800     then show ?case
```
```  5801     proof (cases "x \<in> \<Union>t")
```
```  5802       case True
```
```  5803       then guess s unfolding Union_iff .. note s=this
```
```  5804       then have *: "\<forall>b\<in>t. x \<in> b \<longleftrightarrow> b = s"
```
```  5805         using prems(3) by blast
```
```  5806       show ?thesis
```
```  5807         unfolding if_P[OF True]
```
```  5808         apply (rule trans)
```
```  5809         defer
```
```  5810         apply (rule sum.cong)
```
```  5811         apply (rule refl)
```
```  5812         apply (subst *)
```
```  5813         apply assumption
```
```  5814         apply (rule refl)
```
```  5815         unfolding sum.delta[OF assms(1)]
```
```  5816         using s
```
```  5817         apply auto
```
```  5818         done
```
```  5819     qed auto
```
```  5820   qed
```
```  5821 qed
```
```  5822
```
```  5823
```
```  5824 text \<open>In particular adding integrals over a division, maybe not of an interval.\<close>
```
```  5825
```
```  5826 lemma has_integral_combine_division:
```
```  5827   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5828   assumes "d division_of s"
```
```  5829     and "\<forall>k\<in>d. (f has_integral (i k)) k"
```
```  5830   shows "(f has_integral (sum i d)) s"
```
```  5831 proof -
```
```  5832   note d = division_ofD[OF assms(1)]
```
```  5833   show ?thesis
```
```  5834     unfolding d(6)[symmetric]
```
```  5835     apply (rule has_integral_unions)
```
```  5836     apply (rule d assms)+
```
```  5837     apply rule
```
```  5838     apply rule
```
```  5839     apply rule
```
```  5840   proof goal_cases
```
```  5841     case prems: (1 s s')
```
```  5842     from d(4)[OF this(1)] d(4)[OF this(2)] guess a c b d by (elim exE) note obt=this
```
```  5843     from d(5)[OF prems] show ?case
```
```  5844       unfolding obt interior_cbox
```
```  5845       apply -
```
```  5846       apply (rule negligible_subset[of "(cbox a b-box a b) \<union> (cbox c d-box c d)"])
```
```  5847       apply (rule negligible_Un negligible_frontier_interval)+
```
```  5848       apply auto
```
```  5849       done
```
```  5850   qed
```
```  5851 qed
```
```  5852
```
```  5853 lemma integral_combine_division_bottomup:
```
```  5854   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5855   assumes "d division_of s"
```
```  5856     and "\<forall>k\<in>d. f integrable_on k"
```
```  5857   shows "integral s f = sum (\<lambda>i. integral i f) d"
```
```  5858   apply (rule integral_unique)
```
```  5859   apply (rule has_integral_combine_division[OF assms(1)])
```
```  5860   using assms(2)
```
```  5861   unfolding has_integral_integral
```
```  5862   apply assumption
```
```  5863   done
```
```  5864
```
```  5865 lemma has_integral_combine_division_topdown:
```
```  5866   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5867   assumes "f integrable_on s"
```
```  5868     and "d division_of k"
```
```  5869     and "k \<subseteq> s"
```
```  5870   shows "(f has_integral (sum (\<lambda>i. integral i f) d)) k"
```
```  5871   apply (rule has_integral_combine_division[OF assms(2)])
```
```  5872   apply safe
```
```  5873   unfolding has_integral_integral[symmetric]
```
```  5874 proof goal_cases
```
```  5875   case (1 k)
```
```  5876   from division_ofD(2,4)[OF assms(2) this]
```
```  5877   show ?case
```
```  5878     apply safe
```
```  5879     apply (rule integrable_on_subcbox)
```
```  5880     apply (rule assms)
```
```  5881     using assms(3)
```
```  5882     apply auto
```
```  5883     done
```
```  5884 qed
```
```  5885
```
```  5886 lemma integral_combine_division_topdown:
```
```  5887   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5888   assumes "f integrable_on s"
```
```  5889     and "d division_of s"
```
```  5890   shows "integral s f = sum (\<lambda>i. integral i f) d"
```
```  5891   apply (rule integral_unique)
```
```  5892   apply (rule has_integral_combine_division_topdown)
```
```  5893   using assms
```
```  5894   apply auto
```
```  5895   done
```
```  5896
```
```  5897 lemma integrable_combine_division:
```
```  5898   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5899   assumes "d division_of s"
```
```  5900     and "\<forall>i\<in>d. f integrable_on i"
```
```  5901   shows "f integrable_on s"
```
```  5902   using assms(2)
```
```  5903   unfolding integrable_on_def
```
```  5904   by (metis has_integral_combine_division[OF assms(1)])
```
```  5905
```
```  5906 lemma integrable_on_subdivision:
```
```  5907   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5908   assumes "d division_of i"
```
```  5909     and "f integrable_on s"
```
```  5910     and "i \<subseteq> s"
```
```  5911   shows "f integrable_on i"
```
```  5912   apply (rule integrable_combine_division assms)+
```
```  5913   apply safe
```
```  5914 proof goal_cases
```
```  5915   case 1
```
```  5916   note division_ofD(2,4)[OF assms(1) this]
```
```  5917   then show ?case
```
```  5918     apply safe
```
```  5919     apply (rule integrable_on_subcbox[OF assms(2)])
```
```  5920     using assms(3)
```
```  5921     apply auto
```
```  5922     done
```
```  5923 qed
```
```  5924
```
```  5925
```
```  5926 subsection \<open>Also tagged divisions\<close>
```
```  5927
```
```  5928 lemma has_integral_iff: "(f has_integral i) s \<longleftrightarrow> (f integrable_on s \<and> integral s f = i)"
```
```  5929   by blast
```
```  5930
```
```  5931 lemma has_integral_combine_tagged_division:
```
```  5932   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5933   assumes "p tagged_division_of s"
```
```  5934     and "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
```
```  5935   shows "(f has_integral (\<Sum>(x,k)\<in>p. i k)) s"
```
```  5936 proof -
```
```  5937   have *: "(f has_integral (\<Sum>k\<in>snd`p. integral k f)) s"
```
```  5938     using assms(2)
```
```  5939     apply (intro has_integral_combine_division)
```
```  5940     apply (auto simp: has_integral_integral[symmetric] intro: division_of_tagged_division[OF assms(1)])
```
```  5941     apply auto
```
```  5942     done
```
```  5943   also have "(\<Sum>k\<in>snd`p. integral k f) = (\<Sum>(x, k)\<in>p. integral k f)"
```
```  5944     by (intro sum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null)
```
```  5945        (simp add: content_eq_0_interior)
```
```  5946   finally show ?thesis
```
```  5947     using assms by (auto simp add: has_integral_iff intro!: sum.cong)
```
```  5948 qed
```
```  5949
```
```  5950 lemma integral_combine_tagged_division_bottomup:
```
```  5951   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5952   assumes "p tagged_division_of (cbox a b)"
```
```  5953     and "\<forall>(x,k)\<in>p. f integrable_on k"
```
```  5954   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
```
```  5955   apply (rule integral_unique)
```
```  5956   apply (rule has_integral_combine_tagged_division[OF assms(1)])
```
```  5957   using assms(2)
```
```  5958   apply auto
```
```  5959   done
```
```  5960
```
```  5961 lemma has_integral_combine_tagged_division_topdown:
```
```  5962   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5963   assumes "f integrable_on cbox a b"
```
```  5964     and "p tagged_division_of (cbox a b)"
```
```  5965   shows "(f has_integral (sum (\<lambda>(x,k). integral k f) p)) (cbox a b)"
```
```  5966   apply (rule has_integral_combine_tagged_division[OF assms(2)])
```
```  5967   apply safe
```
```  5968 proof goal_cases
```
```  5969   case 1
```
```  5970   note tagged_division_ofD(3-4)[OF assms(2) this]
```
```  5971   then show ?case
```
```  5972     using integrable_subinterval[OF assms(1)] by blast
```
```  5973 qed
```
```  5974
```
```  5975 lemma integral_combine_tagged_division_topdown:
```
```  5976   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5977   assumes "f integrable_on cbox a b"
```
```  5978     and "p tagged_division_of (cbox a b)"
```
```  5979   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
```
```  5980   apply (rule integral_unique)
```
```  5981   apply (rule has_integral_combine_tagged_division_topdown)
```
```  5982   using assms
```
```  5983   apply auto
```
```  5984   done
```
```  5985
```
```  5986
```
```  5987 subsection \<open>Henstock's lemma\<close>
```
```  5988
```
```  5989 lemma henstock_lemma_part1:
```
```  5990   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
```
```  5991   assumes "f integrable_on cbox a b"
```
```  5992     and "e > 0"
```
```  5993     and "gauge d"
```
```  5994     and "(\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
```
```  5995       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - integral(cbox a b) f) < e)"
```
```  5996     and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
```
```  5997   shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x - integral k f) p) \<le> e"
```
```  5998   (is "?x \<le> e")
```
```  5999 proof -
```
```  6000   { presume "\<And>k. 0<k \<Longrightarrow> ?x \<le> e + k" then show ?thesis by (blast intro: field_le_epsilon) }
```
```  6001   fix k :: real
```
```  6002   assume k: "k > 0"
```
```  6003   note p' = tagged_partial_division_ofD[OF p(1)]
```
```  6004   have "\<Union>(snd ` p) \<subseteq> cbox a b"
```
```  6005     using p'(3) by fastforce
```
```  6006   note partial_division_of_tagged_division[OF p(1)] this
```
```  6007   from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
```
```  6008   define r where "r = q - snd ` p"
```
```  6009   have "snd ` p \<inter> r = {}"
```
```  6010     unfolding r_def by auto
```
```  6011   have r: "finite r"
```
```  6012     using q' unfolding r_def by auto
```
```  6013
```
```  6014   have "\<forall>i\<in>r. \<exists>p. p tagged_division_of i \<and> d fine p \<and>
```
```  6015     norm (sum (\<lambda>(x,j). content j *\<^sub>R f x) p - integral i f) < k / (real (card r) + 1)"
```
```  6016     apply safe
```
```  6017   proof goal_cases
```
```  6018     case (1 i)
```
```  6019     then have i: "i \<in> q"
```
```  6020       unfolding r_def by auto
```
```  6021     from q'(4)[OF this] guess u v by (elim exE) note uv=this
```
```  6022     have *: "k / (real (card r) + 1) > 0" using k by simp
```
```  6023     have "f integrable_on cbox u v"
```
```  6024       apply (rule integrable_subinterval[OF assms(1)])
```
```  6025       using q'(2)[OF i]
```
```  6026       unfolding uv
```
```  6027       apply auto
```
```  6028       done
```
```  6029     note integrable_integral[OF this, unfolded has_integral[of f]]
```
```  6030     from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
```
```  6031     note gauge_inter[OF \<open>gauge d\<close> dd(1)]
```
```  6032     from fine_division_exists[OF this,of u v] guess qq .
```
```  6033     then show ?case
```
```  6034       apply (rule_tac x=qq in exI)
```
```  6035       using dd(2)[of qq]
```
```  6036       unfolding fine_inter uv
```
```  6037       apply auto
```
```  6038       done
```
```  6039   qed
```
```  6040   from bchoice[OF this] guess qq .. note qq=this[rule_format]
```
```  6041
```
```  6042   let ?p = "p \<union> \<Union>(qq ` r)"
```
```  6043   have "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - integral (cbox a b) f) < e"
```
```  6044     apply (rule assms(4)[rule_format])
```
```  6045   proof
```
```  6046     show "d fine ?p"
```
```  6047       apply (rule fine_union)
```
```  6048       apply (rule p)
```
```  6049       apply (rule fine_unions)
```
```  6050       using qq
```
```  6051       apply auto
```
```  6052       done
```
```  6053     note * = tagged_partial_division_of_union_self[OF p(1)]
```
```  6054     have "p \<union> \<Union>(qq ` r) tagged_division_of \<Union>(snd ` p) \<union> \<Union>r"
```
```  6055       using r
```
```  6056     proof (rule tagged_division_union[OF * tagged_division_unions], goal_cases)
```
```  6057       case 1
```
```  6058       then show ?case
```
```  6059         using qq by auto
```
```  6060     next
```
```  6061       case 2
```
```  6062       then show ?case
```
```  6063         apply rule
```
```  6064         apply rule
```
```  6065         apply rule
```
```  6066         apply(rule q'(5))
```
```  6067         unfolding r_def
```
```  6068         apply auto
```
```  6069         done
```
```  6070     next
```
```  6071       case 3
```
```  6072       then show ?case
```
```  6073         apply (rule inter_interior_unions_intervals)
```
```  6074         apply fact
```
```  6075         apply rule
```
```  6076         apply rule
```
```  6077         apply (rule q')
```
```  6078         defer
```
```  6079         apply rule
```
```  6080         apply (subst Int_commute)
```
```  6081         apply (rule inter_interior_unions_intervals)
```
```  6082         apply (rule finite_imageI)
```
```  6083         apply (rule p')
```
```  6084         apply rule
```
```  6085         defer
```
```  6086         apply rule
```
```  6087         apply (rule q')
```
```  6088         using q(1) p'
```
```  6089         unfolding r_def
```
```  6090         apply auto
```
```  6091         done
```
```  6092     qed
```
```  6093     moreover have "\<Union>(snd ` p) \<union> \<Union>r = cbox a b" and "{qq i |i. i \<in> r} = qq ` r"
```
`  6094       unfolding Union_Un_distrib[symmetri`