src/HOL/Probability/Interval_Integral.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 59867 58043346ca64 child 61609 77b453bd616f permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     1 (*  Title:      HOL/Probability/Interval_Integral.thy
```
```     2     Author:     Jeremy Avigad, Johannes Hölzl, Luke Serafin
```
```     3
```
```     4 Lebesgue integral over an interval (with endpoints possibly +-\<infinity>)
```
```     5 *)
```
```     6
```
```     7 theory Interval_Integral
```
```     8   imports Set_Integral
```
```     9 begin
```
```    10
```
```    11 lemma continuous_on_vector_derivative:
```
```    12   "(\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)) \<Longrightarrow> continuous_on S f"
```
```    13   by (auto simp: continuous_on_eq_continuous_within intro!: has_vector_derivative_continuous)
```
```    14
```
```    15 lemma has_vector_derivative_weaken:
```
```    16   fixes x D and f g s t
```
```    17   assumes f: "(f has_vector_derivative D) (at x within t)"
```
```    18     and "x \<in> s" "s \<subseteq> t"
```
```    19     and "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
```
```    20   shows "(g has_vector_derivative D) (at x within s)"
```
```    21 proof -
```
```    22   have "(f has_vector_derivative D) (at x within s) \<longleftrightarrow> (g has_vector_derivative D) (at x within s)"
```
```    23     unfolding has_vector_derivative_def has_derivative_iff_norm
```
```    24     using assms by (intro conj_cong Lim_cong_within refl) auto
```
```    25   then show ?thesis
```
```    26     using has_vector_derivative_within_subset[OF f `s \<subseteq> t`] by simp
```
```    27 qed
```
```    28
```
```    29 definition "einterval a b = {x. a < ereal x \<and> ereal x < b}"
```
```    30
```
```    31 lemma einterval_eq[simp]:
```
```    32   shows einterval_eq_Icc: "einterval (ereal a) (ereal b) = {a <..< b}"
```
```    33     and einterval_eq_Ici: "einterval (ereal a) \<infinity> = {a <..}"
```
```    34     and einterval_eq_Iic: "einterval (- \<infinity>) (ereal b) = {..< b}"
```
```    35     and einterval_eq_UNIV: "einterval (- \<infinity>) \<infinity> = UNIV"
```
```    36   by (auto simp: einterval_def)
```
```    37
```
```    38 lemma einterval_same: "einterval a a = {}"
```
```    39   by (auto simp add: einterval_def)
```
```    40
```
```    41 lemma einterval_iff: "x \<in> einterval a b \<longleftrightarrow> a < ereal x \<and> ereal x < b"
```
```    42   by (simp add: einterval_def)
```
```    43
```
```    44 lemma einterval_nonempty: "a < b \<Longrightarrow> \<exists>c. c \<in> einterval a b"
```
```    45   by (cases a b rule: ereal2_cases, auto simp: einterval_def intro!: dense gt_ex lt_ex)
```
```    46
```
```    47 lemma open_einterval[simp]: "open (einterval a b)"
```
```    48   by (cases a b rule: ereal2_cases)
```
```    49      (auto simp: einterval_def intro!: open_Collect_conj open_Collect_less continuous_intros)
```
```    50
```
```    51 lemma borel_einterval[measurable]: "einterval a b \<in> sets borel"
```
```    52   unfolding einterval_def by measurable
```
```    53
```
```    54 (*
```
```    55     Approximating a (possibly infinite) interval
```
```    56 *)
```
```    57
```
```    58 lemma filterlim_sup1: "(LIM x F. f x :> G1) \<Longrightarrow> (LIM x F. f x :> (sup G1 G2))"
```
```    59  unfolding filterlim_def by (auto intro: le_supI1)
```
```    60
```
```    61 lemma ereal_incseq_approx:
```
```    62   fixes a b :: ereal
```
```    63   assumes "a < b"
```
```    64   obtains X :: "nat \<Rightarrow> real" where
```
```    65     "incseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> b"
```
```    66 proof (cases b)
```
```    67   case PInf
```
```    68   with `a < b` have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
```
```    69     by (cases a) auto
```
```    70   moreover have " (\<lambda>x. ereal (real (Suc x))) ----> \<infinity>"
```
```    71     using nat_ceiling_le_eq by (subst LIMSEQ_Suc_iff) (auto simp: Lim_PInfty)
```
```    72   moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) ----> \<infinity>"
```
```    73     apply (subst LIMSEQ_Suc_iff)
```
```    74     apply (subst Lim_PInfty)
```
```    75     apply (metis add.commute diff_le_eq nat_ceiling_le_eq ereal_less_eq(3))
```
```    76     done
```
```    77   ultimately show thesis
```
```    78     by (intro that[of "\<lambda>i. real a + Suc i"])
```
```    79        (auto simp: incseq_def PInf)
```
```    80 next
```
```    81   case (real b')
```
```    82   def d \<equiv> "b' - (if a = -\<infinity> then b' - 1 else real a)"
```
```    83   with `a < b` have a': "0 < d"
```
```    84     by (cases a) (auto simp: real)
```
```    85   moreover
```
```    86   have "\<And>i r. r < b' \<Longrightarrow> (b' - r) * 1 < (b' - r) * real (Suc (Suc i))"
```
```    87     by (intro mult_strict_left_mono) auto
```
```    88   with `a < b` a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
```
```    89     by (cases a) (auto simp: real d_def field_simps)
```
```    90   moreover have "(\<lambda>i. b' - d / Suc (Suc i)) ----> b'"
```
```    91     apply (subst filterlim_sequentially_Suc)
```
```    92     apply (subst filterlim_sequentially_Suc)
```
```    93     apply (rule tendsto_eq_intros)
```
```    94     apply (auto intro!: tendsto_divide_0[OF tendsto_const] filterlim_sup1
```
```    95                 simp: at_infinity_eq_at_top_bot filterlim_real_sequentially)
```
```    96     done
```
```    97   ultimately show thesis
```
```    98     by (intro that[of "\<lambda>i. b' - d / Suc (Suc i)"])
```
```    99        (auto simp add: real incseq_def intro!: divide_left_mono)
```
```   100 qed (insert `a < b`, auto)
```
```   101
```
```   102 lemma ereal_decseq_approx:
```
```   103   fixes a b :: ereal
```
```   104   assumes "a < b"
```
```   105   obtains X :: "nat \<Rightarrow> real" where
```
```   106     "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X ----> a"
```
```   107 proof -
```
```   108   have "-b < -a" using `a < b` by simp
```
```   109   from ereal_incseq_approx[OF this] guess X .
```
```   110   then show thesis
```
```   111     apply (intro that[of "\<lambda>i. - X i"])
```
```   112     apply (auto simp add: uminus_ereal.simps[symmetric] decseq_def incseq_def
```
```   113                 simp del: uminus_ereal.simps)
```
```   114     apply (metis ereal_minus_less_minus ereal_uminus_uminus ereal_Lim_uminus)+
```
```   115     done
```
```   116 qed
```
```   117
```
```   118 lemma einterval_Icc_approximation:
```
```   119   fixes a b :: ereal
```
```   120   assumes "a < b"
```
```   121   obtains u l :: "nat \<Rightarrow> real" where
```
```   122     "einterval a b = (\<Union>i. {l i .. u i})"
```
```   123     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   124     "l ----> a" "u ----> b"
```
```   125 proof -
```
```   126   from dense[OF `a < b`] obtain c where "a < c" "c < b" by safe
```
```   127   from ereal_incseq_approx[OF `c < b`] guess u . note u = this
```
```   128   from ereal_decseq_approx[OF `a < c`] guess l . note l = this
```
```   129   { fix i from less_trans[OF `l i < c` `c < u i`] have "l i < u i" by simp }
```
```   130   have "einterval a b = (\<Union>i. {l i .. u i})"
```
```   131   proof (auto simp: einterval_iff)
```
```   132     fix x assume "a < ereal x" "ereal x < b"
```
```   133     have "eventually (\<lambda>i. ereal (l i) < ereal x) sequentially"
```
```   134       using l(4) `a < ereal x` by (rule order_tendstoD)
```
```   135     moreover
```
```   136     have "eventually (\<lambda>i. ereal x < ereal (u i)) sequentially"
```
```   137       using u(4) `ereal x< b` by (rule order_tendstoD)
```
```   138     ultimately have "eventually (\<lambda>i. l i < x \<and> x < u i) sequentially"
```
```   139       by eventually_elim auto
```
```   140     then show "\<exists>i. l i \<le> x \<and> x \<le> u i"
```
```   141       by (auto intro: less_imp_le simp: eventually_sequentially)
```
```   142   next
```
```   143     fix x i assume "l i \<le> x" "x \<le> u i"
```
```   144     with `a < ereal (l i)` `ereal (u i) < b`
```
```   145     show "a < ereal x" "ereal x < b"
```
```   146       by (auto simp del: ereal_less_eq(3) simp add: ereal_less_eq(3)[symmetric])
```
```   147   qed
```
```   148   show thesis
```
```   149     by (intro that) fact+
```
```   150 qed
```
```   151
```
```   152 (* TODO: in this definition, it would be more natural if einterval a b included a and b when
```
```   153    they are real. *)
```
```   154 definition interval_lebesgue_integral :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> 'a::{banach, second_countable_topology}" where
```
```   155   "interval_lebesgue_integral M a b f =
```
```   156     (if a \<le> b then (LINT x:einterval a b|M. f x) else - (LINT x:einterval b a|M. f x))"
```
```   157
```
```   158 syntax
```
```   159   "_ascii_interval_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real measure \<Rightarrow> real \<Rightarrow> real"
```
```   160   ("(5LINT _=_.._|_. _)" [0,60,60,61,100] 60)
```
```   161
```
```   162 translations
```
```   163   "LINT x=a..b|M. f" == "CONST interval_lebesgue_integral M a b (\<lambda>x. f)"
```
```   164
```
```   165 definition interval_lebesgue_integrable :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a::{banach, second_countable_topology}) \<Rightarrow> bool" where
```
```   166   "interval_lebesgue_integrable M a b f =
```
```   167     (if a \<le> b then set_integrable M (einterval a b) f else set_integrable M (einterval b a) f)"
```
```   168
```
```   169 syntax
```
```   170   "_ascii_interval_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"
```
```   171   ("(4LBINT _=_.._. _)" [0,60,60,61] 60)
```
```   172
```
```   173 translations
```
```   174   "LBINT x=a..b. f" == "CONST interval_lebesgue_integral CONST lborel a b (\<lambda>x. f)"
```
```   175
```
```   176 (*
```
```   177     Basic properties of integration over an interval.
```
```   178 *)
```
```   179
```
```   180 lemma interval_lebesgue_integral_cong:
```
```   181   "a \<le> b \<Longrightarrow> (\<And>x. x \<in> einterval a b \<Longrightarrow> f x = g x) \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
```
```   182     interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   183   by (auto intro: set_lebesgue_integral_cong simp: interval_lebesgue_integral_def)
```
```   184
```
```   185 lemma interval_lebesgue_integral_cong_AE:
```
```   186   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```   187     a \<le> b \<Longrightarrow> AE x \<in> einterval a b in M. f x = g x \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
```
```   188     interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   189   by (auto intro: set_lebesgue_integral_cong_AE simp: interval_lebesgue_integral_def)
```
```   190
```
```   191 lemma interval_lebesgue_integral_add [intro, simp]:
```
```   192   fixes M a b f
```
```   193   assumes "interval_lebesgue_integrable M a b f" "interval_lebesgue_integrable M a b g"
```
```   194   shows "interval_lebesgue_integrable M a b (\<lambda>x. f x + g x)" and
```
```   195     "interval_lebesgue_integral M a b (\<lambda>x. f x + g x) =
```
```   196    interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g"
```
```   197 using assms by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```   198     field_simps)
```
```   199
```
```   200 lemma interval_lebesgue_integral_diff [intro, simp]:
```
```   201   fixes M a b f
```
```   202   assumes "interval_lebesgue_integrable M a b f"
```
```   203     "interval_lebesgue_integrable M a b g"
```
```   204   shows "interval_lebesgue_integrable M a b (\<lambda>x. f x - g x)" and
```
```   205     "interval_lebesgue_integral M a b (\<lambda>x. f x - g x) =
```
```   206    interval_lebesgue_integral M a b f - interval_lebesgue_integral M a b g"
```
```   207 using assms by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```   208     field_simps)
```
```   209
```
```   210 lemma interval_lebesgue_integrable_mult_right [intro, simp]:
```
```   211   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   212   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   213     interval_lebesgue_integrable M a b (\<lambda>x. c * f x)"
```
```   214   by (simp add: interval_lebesgue_integrable_def)
```
```   215
```
```   216 lemma interval_lebesgue_integrable_mult_left [intro, simp]:
```
```   217   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   218   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   219     interval_lebesgue_integrable M a b (\<lambda>x. f x * c)"
```
```   220   by (simp add: interval_lebesgue_integrable_def)
```
```   221
```
```   222 lemma interval_lebesgue_integrable_divide [intro, simp]:
```
```   223   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
```
```   224   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   225     interval_lebesgue_integrable M a b (\<lambda>x. f x / c)"
```
```   226   by (simp add: interval_lebesgue_integrable_def)
```
```   227
```
```   228 lemma interval_lebesgue_integral_mult_right [simp]:
```
```   229   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   230   shows "interval_lebesgue_integral M a b (\<lambda>x. c * f x) =
```
```   231     c * interval_lebesgue_integral M a b f"
```
```   232   by (simp add: interval_lebesgue_integral_def)
```
```   233
```
```   234 lemma interval_lebesgue_integral_mult_left [simp]:
```
```   235   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   236   shows "interval_lebesgue_integral M a b (\<lambda>x. f x * c) =
```
```   237     interval_lebesgue_integral M a b f * c"
```
```   238   by (simp add: interval_lebesgue_integral_def)
```
```   239
```
```   240 lemma interval_lebesgue_integral_divide [simp]:
```
```   241   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
```
```   242   shows "interval_lebesgue_integral M a b (\<lambda>x. f x / c) =
```
```   243     interval_lebesgue_integral M a b f / c"
```
```   244   by (simp add: interval_lebesgue_integral_def)
```
```   245
```
```   246 lemma interval_lebesgue_integral_uminus:
```
```   247   "interval_lebesgue_integral M a b (\<lambda>x. - f x) = - interval_lebesgue_integral M a b f"
```
```   248   by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def)
```
```   249
```
```   250 lemma interval_lebesgue_integral_of_real:
```
```   251   "interval_lebesgue_integral M a b (\<lambda>x. complex_of_real (f x)) =
```
```   252     of_real (interval_lebesgue_integral M a b f)"
```
```   253   unfolding interval_lebesgue_integral_def
```
```   254   by (auto simp add: interval_lebesgue_integral_def set_integral_complex_of_real)
```
```   255
```
```   256 lemma interval_lebesgue_integral_le_eq:
```
```   257   fixes a b f
```
```   258   assumes "a \<le> b"
```
```   259   shows "interval_lebesgue_integral M a b f = (LINT x : einterval a b | M. f x)"
```
```   260 using assms by (auto simp add: interval_lebesgue_integral_def)
```
```   261
```
```   262 lemma interval_lebesgue_integral_gt_eq:
```
```   263   fixes a b f
```
```   264   assumes "a > b"
```
```   265   shows "interval_lebesgue_integral M a b f = -(LINT x : einterval b a | M. f x)"
```
```   266 using assms by (auto simp add: interval_lebesgue_integral_def less_imp_le einterval_def)
```
```   267
```
```   268 lemma interval_lebesgue_integral_gt_eq':
```
```   269   fixes a b f
```
```   270   assumes "a > b"
```
```   271   shows "interval_lebesgue_integral M a b f = - interval_lebesgue_integral M b a f"
```
```   272 using assms by (auto simp add: interval_lebesgue_integral_def less_imp_le einterval_def)
```
```   273
```
```   274 lemma interval_integral_endpoints_same [simp]: "(LBINT x=a..a. f x) = 0"
```
```   275   by (simp add: interval_lebesgue_integral_def einterval_same)
```
```   276
```
```   277 lemma interval_integral_endpoints_reverse: "(LBINT x=a..b. f x) = -(LBINT x=b..a. f x)"
```
```   278   by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integral_def einterval_same)
```
```   279
```
```   280 lemma interval_integrable_endpoints_reverse:
```
```   281   "interval_lebesgue_integrable lborel a b f \<longleftrightarrow>
```
```   282     interval_lebesgue_integrable lborel b a f"
```
```   283   by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integrable_def einterval_same)
```
```   284
```
```   285 lemma interval_integral_reflect:
```
```   286   "(LBINT x=a..b. f x) = (LBINT x=-b..-a. f (-x))"
```
```   287 proof (induct a b rule: linorder_wlog)
```
```   288   case (sym a b) then show ?case
```
```   289     by (auto simp add: interval_lebesgue_integral_def interval_integrable_endpoints_reverse
```
```   290              split: split_if_asm)
```
```   291 next
```
```   292   case (le a b) then show ?case
```
```   293     unfolding interval_lebesgue_integral_def
```
```   294     by (subst set_integral_reflect)
```
```   295        (auto simp: interval_lebesgue_integrable_def einterval_iff
```
```   296                    ereal_uminus_le_reorder ereal_uminus_less_reorder not_less
```
```   297                    uminus_ereal.simps[symmetric]
```
```   298              simp del: uminus_ereal.simps
```
```   299              intro!: integral_cong
```
```   300              split: split_indicator)
```
```   301 qed
```
```   302
```
```   303 (*
```
```   304     Basic properties of integration over an interval wrt lebesgue measure.
```
```   305 *)
```
```   306
```
```   307 lemma interval_integral_zero [simp]:
```
```   308   fixes a b :: ereal
```
```   309   shows"LBINT x=a..b. 0 = 0"
```
```   310 using assms unfolding interval_lebesgue_integral_def einterval_eq
```
```   311 by simp
```
```   312
```
```   313 lemma interval_integral_const [intro, simp]:
```
```   314   fixes a b c :: real
```
```   315   shows "interval_lebesgue_integrable lborel a b (\<lambda>x. c)" and "LBINT x=a..b. c = c * (b - a)"
```
```   316 using assms unfolding interval_lebesgue_integral_def interval_lebesgue_integrable_def einterval_eq
```
```   317 by (auto simp add:  less_imp_le field_simps measure_def)
```
```   318
```
```   319 lemma interval_integral_cong_AE:
```
```   320   assumes [measurable]: "f \<in> borel_measurable borel" "g \<in> borel_measurable borel"
```
```   321   assumes "AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x"
```
```   322   shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
```
```   323   using assms
```
```   324 proof (induct a b rule: linorder_wlog)
```
```   325   case (sym a b) then show ?case
```
```   326     by (simp add: min.commute max.commute interval_integral_endpoints_reverse[of a b])
```
```   327 next
```
```   328   case (le a b) then show ?case
```
```   329     by (auto simp: interval_lebesgue_integral_def max_def min_def
```
```   330              intro!: set_lebesgue_integral_cong_AE)
```
```   331 qed
```
```   332
```
```   333 lemma interval_integral_cong:
```
```   334   assumes "\<And>x. x \<in> einterval (min a b) (max a b) \<Longrightarrow> f x = g x"
```
```   335   shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
```
```   336   using assms
```
```   337 proof (induct a b rule: linorder_wlog)
```
```   338   case (sym a b) then show ?case
```
```   339     by (simp add: min.commute max.commute interval_integral_endpoints_reverse[of a b])
```
```   340 next
```
```   341   case (le a b) then show ?case
```
```   342     by (auto simp: interval_lebesgue_integral_def max_def min_def
```
```   343              intro!: set_lebesgue_integral_cong)
```
```   344 qed
```
```   345
```
```   346 lemma interval_lebesgue_integrable_cong_AE:
```
```   347     "f \<in> borel_measurable lborel \<Longrightarrow> g \<in> borel_measurable lborel \<Longrightarrow>
```
```   348     AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x \<Longrightarrow>
```
```   349     interval_lebesgue_integrable lborel a b f = interval_lebesgue_integrable lborel a b g"
```
```   350   apply (simp add: interval_lebesgue_integrable_def )
```
```   351   apply (intro conjI impI set_integrable_cong_AE)
```
```   352   apply (auto simp: min_def max_def)
```
```   353   done
```
```   354
```
```   355 lemma interval_integrable_abs_iff:
```
```   356   fixes f :: "real \<Rightarrow> real"
```
```   357   shows  "f \<in> borel_measurable lborel \<Longrightarrow>
```
```   358     interval_lebesgue_integrable lborel a b (\<lambda>x. \<bar>f x\<bar>) = interval_lebesgue_integrable lborel a b f"
```
```   359   unfolding interval_lebesgue_integrable_def
```
```   360   by (subst (1 2) set_integrable_abs_iff') simp_all
```
```   361
```
```   362 lemma interval_integral_Icc:
```
```   363   fixes a b :: real
```
```   364   shows "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..b}. f x)"
```
```   365   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   366            simp add: interval_lebesgue_integral_def)
```
```   367
```
```   368 lemma interval_integral_Icc':
```
```   369   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a \<le> ereal x \<and> ereal x \<le> b}. f x)"
```
```   370   by (auto intro!: set_integral_discrete_difference[where X="{real a, real b}"]
```
```   371            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   372
```
```   373 lemma interval_integral_Ioc:
```
```   374   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a<..b}. f x)"
```
```   375   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   376            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   377
```
```   378 (* TODO: other versions as well? *) (* Yes: I need the Icc' version. *)
```
```   379 lemma interval_integral_Ioc':
```
```   380   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a < ereal x \<and> ereal x \<le> b}. f x)"
```
```   381   by (auto intro!: set_integral_discrete_difference[where X="{real a, real b}"]
```
```   382            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   383
```
```   384 lemma interval_integral_Ico:
```
```   385   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..<b}. f x)"
```
```   386   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   387            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   388
```
```   389 lemma interval_integral_Ioi:
```
```   390   "\<bar>a\<bar> < \<infinity> \<Longrightarrow> (LBINT x=a..\<infinity>. f x) = (LBINT x : {real a <..}. f x)"
```
```   391   by (auto simp add: interval_lebesgue_integral_def einterval_iff)
```
```   392
```
```   393 lemma interval_integral_Ioo:
```
```   394   "a \<le> b \<Longrightarrow> \<bar>a\<bar> < \<infinity> ==> \<bar>b\<bar> < \<infinity> \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {real a <..< real b}. f x)"
```
```   395   by (auto simp add: interval_lebesgue_integral_def einterval_iff)
```
```   396
```
```   397 lemma interval_integral_discrete_difference:
```
```   398   fixes f :: "real \<Rightarrow> 'b::{banach, second_countable_topology}" and a b :: ereal
```
```   399   assumes "countable X"
```
```   400   and eq: "\<And>x. a \<le> b \<Longrightarrow> a < x \<Longrightarrow> x < b \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
```
```   401   and anti_eq: "\<And>x. b \<le> a \<Longrightarrow> b < x \<Longrightarrow> x < a \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
```
```   402   assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
```
```   403   shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   404   unfolding interval_lebesgue_integral_def
```
```   405   apply (intro if_cong refl arg_cong[where f="\<lambda>x. - x"] integral_discrete_difference[of X] assms)
```
```   406   apply (auto simp: eq anti_eq einterval_iff split: split_indicator)
```
```   407   done
```
```   408
```
```   409 lemma interval_integral_sum:
```
```   410   fixes a b c :: ereal
```
```   411   assumes integrable: "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f"
```
```   412   shows "(LBINT x=a..b. f x) + (LBINT x=b..c. f x) = (LBINT x=a..c. f x)"
```
```   413 proof -
```
```   414   let ?I = "\<lambda>a b. LBINT x=a..b. f x"
```
```   415   { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
```
```   416     then have ord: "a \<le> b" "b \<le> c" "a \<le> c" and f': "set_integrable lborel (einterval a c) f"
```
```   417       by (auto simp: interval_lebesgue_integrable_def)
```
```   418     then have f: "set_borel_measurable borel (einterval a c) f"
```
```   419       by (drule_tac borel_measurable_integrable) simp
```
```   420     have "(LBINT x:einterval a c. f x) = (LBINT x:einterval a b \<union> einterval b c. f x)"
```
```   421     proof (rule set_integral_cong_set)
```
```   422       show "AE x in lborel. (x \<in> einterval a b \<union> einterval b c) = (x \<in> einterval a c)"
```
```   423         using AE_lborel_singleton[of "real b"] ord
```
```   424         by (cases a b c rule: ereal3_cases) (auto simp: einterval_iff)
```
```   425     qed (insert ord, auto intro!: set_borel_measurable_subset[OF f] simp: einterval_iff)
```
```   426     also have "\<dots> = (LBINT x:einterval a b. f x) + (LBINT x:einterval b c. f x)"
```
```   427       using ord
```
```   428       by (intro set_integral_Un_AE) (auto intro!: set_integrable_subset[OF f'] simp: einterval_iff not_less)
```
```   429     finally have "?I a b + ?I b c = ?I a c"
```
```   430       using ord by (simp add: interval_lebesgue_integral_def)
```
```   431   } note 1 = this
```
```   432   { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
```
```   433     from 1[OF this] have "?I b c + ?I a b = ?I a c"
```
```   434       by (metis add.commute)
```
```   435   } note 2 = this
```
```   436   have 3: "\<And>a b. b \<le> a \<Longrightarrow> (LBINT x=a..b. f x) = - (LBINT x=b..a. f x)"
```
```   437     by (rule interval_integral_endpoints_reverse)
```
```   438   show ?thesis
```
```   439     using integrable
```
```   440     by (cases a b b c a c rule: linorder_le_cases[case_product linorder_le_cases linorder_cases])
```
```   441        (simp_all add: min_absorb1 min_absorb2 max_absorb1 max_absorb2 field_simps 1 2 3)
```
```   442 qed
```
```   443
```
```   444 lemma interval_integrable_isCont:
```
```   445   fixes a b and f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
```
```   446   shows "(\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> isCont f x) \<Longrightarrow>
```
```   447     interval_lebesgue_integrable lborel a b f"
```
```   448 proof (induct a b rule: linorder_wlog)
```
```   449   case (le a b) then show ?case
```
```   450     by (auto simp: interval_lebesgue_integrable_def
```
```   451              intro!: set_integrable_subset[OF borel_integrable_compact[of "{a .. b}"]]
```
```   452                      continuous_at_imp_continuous_on)
```
```   453 qed (auto intro: interval_integrable_endpoints_reverse[THEN iffD1])
```
```   454
```
```   455 lemma interval_integrable_continuous_on:
```
```   456   fixes a b :: real and f
```
```   457   assumes "a \<le> b" and "continuous_on {a..b} f"
```
```   458   shows "interval_lebesgue_integrable lborel a b f"
```
```   459 using assms unfolding interval_lebesgue_integrable_def apply simp
```
```   460   by (rule set_integrable_subset, rule borel_integrable_atLeastAtMost' [of a b], auto)
```
```   461
```
```   462 lemma interval_integral_eq_integral:
```
```   463   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   464   shows "a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f \<Longrightarrow> LBINT x=a..b. f x = integral {a..b} f"
```
```   465   by (subst interval_integral_Icc, simp) (rule set_borel_integral_eq_integral)
```
```   466
```
```   467 lemma interval_integral_eq_integral':
```
```   468   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   469   shows "a \<le> b \<Longrightarrow> set_integrable lborel (einterval a b) f \<Longrightarrow> LBINT x=a..b. f x = integral (einterval a b) f"
```
```   470   by (subst interval_lebesgue_integral_le_eq, simp) (rule set_borel_integral_eq_integral)
```
```   471
```
```   472 (*
```
```   473     General limit approximation arguments
```
```   474 *)
```
```   475
```
```   476 lemma interval_integral_Icc_approx_nonneg:
```
```   477   fixes a b :: ereal
```
```   478   assumes "a < b"
```
```   479   fixes u l :: "nat \<Rightarrow> real"
```
```   480   assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
```
```   481     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   482     "l ----> a" "u ----> b"
```
```   483   fixes f :: "real \<Rightarrow> real"
```
```   484   assumes f_integrable: "\<And>i. set_integrable lborel {l i..u i} f"
```
```   485   assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
```
```   486   assumes f_measurable: "set_borel_measurable lborel (einterval a b) f"
```
```   487   assumes lbint_lim: "(\<lambda>i. LBINT x=l i.. u i. f x) ----> C"
```
```   488   shows
```
```   489     "set_integrable lborel (einterval a b) f"
```
```   490     "(LBINT x=a..b. f x) = C"
```
```   491 proof -
```
```   492   have 1: "\<And>i. set_integrable lborel {l i..u i} f" by (rule f_integrable)
```
```   493   have 2: "AE x in lborel. mono (\<lambda>n. indicator {l n..u n} x *\<^sub>R f x)"
```
```   494   proof -
```
```   495      from f_nonneg have "AE x in lborel. \<forall>i. l i \<le> x \<longrightarrow> x \<le> u i \<longrightarrow> 0 \<le> f x"
```
```   496       by eventually_elim
```
```   497          (metis approx(5) approx(6) dual_order.strict_trans1 ereal_less_eq(3) le_less_trans)
```
```   498     then show ?thesis
```
```   499       apply eventually_elim
```
```   500       apply (auto simp: mono_def split: split_indicator)
```
```   501       apply (metis approx(3) decseqD order_trans)
```
```   502       apply (metis approx(2) incseqD order_trans)
```
```   503       done
```
```   504   qed
```
```   505   have 3: "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) ----> indicator (einterval a b) x *\<^sub>R f x"
```
```   506   proof -
```
```   507     { fix x i assume "l i \<le> x" "x \<le> u i"
```
```   508       then have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
```
```   509         apply (auto simp: eventually_sequentially intro!: exI[of _ i])
```
```   510         apply (metis approx(3) decseqD order_trans)
```
```   511         apply (metis approx(2) incseqD order_trans)
```
```   512         done
```
```   513       then have "eventually (\<lambda>i. f x * indicator {l i..u i} x = f x) sequentially"
```
```   514         by eventually_elim auto }
```
```   515     then show ?thesis
```
```   516       unfolding approx(1) by (auto intro!: AE_I2 Lim_eventually split: split_indicator)
```
```   517   qed
```
```   518   have 4: "(\<lambda>i. \<integral> x. indicator {l i..u i} x *\<^sub>R f x \<partial>lborel) ----> C"
```
```   519     using lbint_lim by (simp add: interval_integral_Icc approx less_imp_le)
```
```   520   have 5: "set_borel_measurable lborel (einterval a b) f" by (rule assms)
```
```   521   have "(LBINT x=a..b. f x) = lebesgue_integral lborel (\<lambda>x. indicator (einterval a b) x *\<^sub>R f x)"
```
```   522     using assms by (simp add: interval_lebesgue_integral_def less_imp_le)
```
```   523   also have "... = C" by (rule integral_monotone_convergence [OF 1 2 3 4 5])
```
```   524   finally show "(LBINT x=a..b. f x) = C" .
```
```   525
```
```   526   show "set_integrable lborel (einterval a b) f"
```
```   527     by (rule integrable_monotone_convergence[OF 1 2 3 4 5])
```
```   528 qed
```
```   529
```
```   530 lemma interval_integral_Icc_approx_integrable:
```
```   531   fixes u l :: "nat \<Rightarrow> real" and a b :: ereal
```
```   532   fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
```
```   533   assumes "a < b"
```
```   534   assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
```
```   535     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   536     "l ----> a" "u ----> b"
```
```   537   assumes f_integrable: "set_integrable lborel (einterval a b) f"
```
```   538   shows "(\<lambda>i. LBINT x=l i.. u i. f x) ----> (LBINT x=a..b. f x)"
```
```   539 proof -
```
```   540   have "(\<lambda>i. LBINT x:{l i.. u i}. f x) ----> (LBINT x:einterval a b. f x)"
```
```   541   proof (rule integral_dominated_convergence)
```
```   542     show "integrable lborel (\<lambda>x. norm (indicator (einterval a b) x *\<^sub>R f x))"
```
```   543       by (rule integrable_norm) fact
```
```   544     show "set_borel_measurable lborel (einterval a b) f"
```
```   545       using f_integrable by (rule borel_measurable_integrable)
```
```   546     then show "\<And>i. set_borel_measurable lborel {l i..u i} f"
```
```   547       by (rule set_borel_measurable_subset) (auto simp: approx)
```
```   548     show "\<And>i. AE x in lborel. norm (indicator {l i..u i} x *\<^sub>R f x) \<le> norm (indicator (einterval a b) x *\<^sub>R f x)"
```
```   549       by (intro AE_I2) (auto simp: approx split: split_indicator)
```
```   550     show "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) ----> indicator (einterval a b) x *\<^sub>R f x"
```
```   551     proof (intro AE_I2 tendsto_intros Lim_eventually)
```
```   552       fix x
```
```   553       { fix i assume "l i \<le> x" "x \<le> u i"
```
```   554         with `incseq u`[THEN incseqD, of i] `decseq l`[THEN decseqD, of i]
```
```   555         have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
```
```   556           by (auto simp: eventually_sequentially decseq_def incseq_def intro: order_trans) }
```
```   557       then show "eventually (\<lambda>xa. indicator {l xa..u xa} x = (indicator (einterval a b) x::real)) sequentially"
```
```   558         using approx order_tendstoD(2)[OF `l ----> a`, of x] order_tendstoD(1)[OF `u ----> b`, of x]
```
```   559         by (auto split: split_indicator)
```
```   560     qed
```
```   561   qed
```
```   562   with `a < b` `\<And>i. l i < u i` show ?thesis
```
```   563     by (simp add: interval_lebesgue_integral_le_eq[symmetric] interval_integral_Icc less_imp_le)
```
```   564 qed
```
```   565
```
```   566 (*
```
```   567   A slightly stronger version of integral_FTC_atLeastAtMost and related facts,
```
```   568   with continuous_on instead of isCont
```
```   569
```
```   570   TODO: make the older versions corollaries of these (using continuous_at_imp_continuous_on, etc.)
```
```   571 *)
```
```   572
```
```   573 (*
```
```   574 TODO: many proofs below require inferences like
```
```   575
```
```   576   a < ereal x \<Longrightarrow> x < y \<Longrightarrow> a < ereal y
```
```   577
```
```   578 where x and y are real. These should be better automated.
```
```   579 *)
```
```   580
```
```   581 (*
```
```   582     The first Fundamental Theorem of Calculus
```
```   583
```
```   584     First, for finite intervals, and then two versions for arbitrary intervals.
```
```   585 *)
```
```   586
```
```   587 lemma interval_integral_FTC_finite:
```
```   588   fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: real
```
```   589   assumes f: "continuous_on {min a b..max a b} f"
```
```   590   assumes F: "\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> (F has_vector_derivative (f x)) (at x within
```
```   591     {min a b..max a b})"
```
```   592   shows "(LBINT x=a..b. f x) = F b - F a"
```
```   593   apply (case_tac "a \<le> b")
```
```   594   apply (subst interval_integral_Icc, simp)
```
```   595   apply (rule integral_FTC_atLeastAtMost, assumption)
```
```   596   apply (metis F max_def min_def)
```
```   597   using f apply (simp add: min_absorb1 max_absorb2)
```
```   598   apply (subst interval_integral_endpoints_reverse)
```
```   599   apply (subst interval_integral_Icc, simp)
```
```   600   apply (subst integral_FTC_atLeastAtMost, auto)
```
```   601   apply (metis F max_def min_def)
```
```   602 using f by (simp add: min_absorb2 max_absorb1)
```
```   603
```
```   604 lemma interval_integral_FTC_nonneg:
```
```   605   fixes f F :: "real \<Rightarrow> real" and a b :: ereal
```
```   606   assumes "a < b"
```
```   607   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV F x :> f x"
```
```   608   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
```
```   609   assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
```
```   610   assumes A: "((F \<circ> real) ---> A) (at_right a)"
```
```   611   assumes B: "((F \<circ> real) ---> B) (at_left b)"
```
```   612   shows
```
```   613     "set_integrable lborel (einterval a b) f"
```
```   614     "(LBINT x=a..b. f x) = B - A"
```
```   615 proof -
```
```   616   from einterval_Icc_approximation[OF `a < b`] guess u l . note approx = this
```
```   617   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   618     by (rule order_less_le_trans, rule approx, force)
```
```   619   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   620     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   621   have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
```
```   622     using assms approx apply (intro interval_integral_FTC_finite)
```
```   623     apply (auto simp add: less_imp_le min_def max_def
```
```   624       has_field_derivative_iff_has_vector_derivative[symmetric])
```
```   625     apply (rule continuous_at_imp_continuous_on, auto intro!: f)
```
```   626     by (rule DERIV_subset [OF F], auto)
```
```   627   have 1: "\<And>i. set_integrable lborel {l i..u i} f"
```
```   628   proof -
```
```   629     fix i show "set_integrable lborel {l i .. u i} f"
```
```   630       using `a < l i` `u i < b`
```
```   631       by (intro borel_integrable_compact f continuous_at_imp_continuous_on compact_Icc ballI)
```
```   632          (auto simp del: ereal_less_eq simp add: ereal_less_eq(3)[symmetric])
```
```   633   qed
```
```   634   have 2: "set_borel_measurable lborel (einterval a b) f"
```
```   635     by (auto simp del: real_scaleR_def intro!: set_borel_measurable_continuous
```
```   636              simp: continuous_on_eq_continuous_at einterval_iff f)
```
```   637   have 3: "(\<lambda>i. LBINT x=l i..u i. f x) ----> B - A"
```
```   638     apply (subst FTCi)
```
```   639     apply (intro tendsto_intros)
```
```   640     using B approx unfolding tendsto_at_iff_sequentially comp_def
```
```   641     using tendsto_at_iff_sequentially[where 'a=real]
```
```   642     apply (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
```
```   643     using A approx unfolding tendsto_at_iff_sequentially comp_def
```
```   644     by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
```
```   645   show "(LBINT x=a..b. f x) = B - A"
```
```   646     by (rule interval_integral_Icc_approx_nonneg [OF `a < b` approx 1 f_nonneg 2 3])
```
```   647   show "set_integrable lborel (einterval a b) f"
```
```   648     by (rule interval_integral_Icc_approx_nonneg [OF `a < b` approx 1 f_nonneg 2 3])
```
```   649 qed
```
```   650
```
```   651 lemma interval_integral_FTC_integrable:
```
```   652   fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: ereal
```
```   653   assumes "a < b"
```
```   654   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> (F has_vector_derivative f x) (at x)"
```
```   655   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
```
```   656   assumes f_integrable: "set_integrable lborel (einterval a b) f"
```
```   657   assumes A: "((F \<circ> real) ---> A) (at_right a)"
```
```   658   assumes B: "((F \<circ> real) ---> B) (at_left b)"
```
```   659   shows "(LBINT x=a..b. f x) = B - A"
```
```   660 proof -
```
```   661   from einterval_Icc_approximation[OF `a < b`] guess u l . note approx = this
```
```   662   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   663     by (rule order_less_le_trans, rule approx, force)
```
```   664   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   665     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   666   have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
```
```   667     using assms approx
```
```   668     by (auto simp add: less_imp_le min_def max_def
```
```   669              intro!: f continuous_at_imp_continuous_on interval_integral_FTC_finite
```
```   670              intro: has_vector_derivative_at_within)
```
```   671   have "(\<lambda>i. LBINT x=l i..u i. f x) ----> B - A"
```
```   672     apply (subst FTCi)
```
```   673     apply (intro tendsto_intros)
```
```   674     using B approx unfolding tendsto_at_iff_sequentially comp_def
```
```   675     apply (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
```
```   676     using A approx unfolding tendsto_at_iff_sequentially comp_def
```
```   677     by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
```
```   678   moreover have "(\<lambda>i. LBINT x=l i..u i. f x) ----> (LBINT x=a..b. f x)"
```
```   679     by (rule interval_integral_Icc_approx_integrable [OF `a < b` approx f_integrable])
```
```   680   ultimately show ?thesis
```
```   681     by (elim LIMSEQ_unique)
```
```   682 qed
```
```   683
```
```   684 (*
```
```   685   The second Fundamental Theorem of Calculus and existence of antiderivatives on an
```
```   686   einterval.
```
```   687 *)
```
```   688
```
```   689 lemma interval_integral_FTC2:
```
```   690   fixes a b c :: real and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   691   assumes "a \<le> c" "c \<le> b"
```
```   692   and contf: "continuous_on {a..b} f"
```
```   693   fixes x :: real
```
```   694   assumes "a \<le> x" and "x \<le> b"
```
```   695   shows "((\<lambda>u. LBINT y=c..u. f y) has_vector_derivative (f x)) (at x within {a..b})"
```
```   696 proof -
```
```   697   let ?F = "(\<lambda>u. LBINT y=a..u. f y)"
```
```   698   have intf: "set_integrable lborel {a..b} f"
```
```   699     by (rule borel_integrable_atLeastAtMost', rule contf)
```
```   700   have "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
```
```   701     apply (intro integral_has_vector_derivative)
```
```   702     using `a \<le> x` `x \<le> b` by (intro continuous_on_subset [OF contf], auto)
```
```   703   then have "((\<lambda>u. integral {a..u} f) has_vector_derivative (f x)) (at x within {a..b})"
```
```   704     by simp
```
```   705   then have "(?F has_vector_derivative (f x)) (at x within {a..b})"
```
```   706     by (rule has_vector_derivative_weaken)
```
```   707        (auto intro!: assms interval_integral_eq_integral[symmetric] set_integrable_subset [OF intf])
```
```   708   then have "((\<lambda>x. (LBINT y=c..a. f y) + ?F x) has_vector_derivative (f x)) (at x within {a..b})"
```
```   709     by (auto intro!: derivative_eq_intros)
```
```   710   then show ?thesis
```
```   711   proof (rule has_vector_derivative_weaken)
```
```   712     fix u assume "u \<in> {a .. b}"
```
```   713     then show "(LBINT y=c..a. f y) + (LBINT y=a..u. f y) = (LBINT y=c..u. f y)"
```
```   714       using assms
```
```   715       apply (intro interval_integral_sum)
```
```   716       apply (auto simp add: interval_lebesgue_integrable_def simp del: real_scaleR_def)
```
```   717       by (rule set_integrable_subset [OF intf], auto simp add: min_def max_def)
```
```   718   qed (insert assms, auto)
```
```   719 qed
```
```   720
```
```   721 lemma einterval_antiderivative:
```
```   722   fixes a b :: ereal and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   723   assumes "a < b" and contf: "\<And>x :: real. a < x \<Longrightarrow> x < b \<Longrightarrow> isCont f x"
```
```   724   shows "\<exists>F. \<forall>x :: real. a < x \<longrightarrow> x < b \<longrightarrow> (F has_vector_derivative f x) (at x)"
```
```   725 proof -
```
```   726   from einterval_nonempty [OF `a < b`] obtain c :: real where [simp]: "a < c" "c < b"
```
```   727     by (auto simp add: einterval_def)
```
```   728   let ?F = "(\<lambda>u. LBINT y=c..u. f y)"
```
```   729   show ?thesis
```
```   730   proof (rule exI, clarsimp)
```
```   731     fix x :: real
```
```   732     assume [simp]: "a < x" "x < b"
```
```   733     have 1: "a < min c x" by simp
```
```   734     from einterval_nonempty [OF 1] obtain d :: real where [simp]: "a < d" "d < c" "d < x"
```
```   735       by (auto simp add: einterval_def)
```
```   736     have 2: "max c x < b" by simp
```
```   737     from einterval_nonempty [OF 2] obtain e :: real where [simp]: "c < e" "x < e" "e < b"
```
```   738       by (auto simp add: einterval_def)
```
```   739     show "(?F has_vector_derivative f x) (at x)"
```
```   740       (* TODO: factor out the next three lines to has_field_derivative_within_open *)
```
```   741       unfolding has_vector_derivative_def
```
```   742       apply (subst has_derivative_within_open [of _ "{d<..<e}", symmetric], auto)
```
```   743       apply (subst has_vector_derivative_def [symmetric])
```
```   744       apply (rule has_vector_derivative_within_subset [of _ _ _ "{d..e}"])
```
```   745       apply (rule interval_integral_FTC2, auto simp add: less_imp_le)
```
```   746       apply (rule continuous_at_imp_continuous_on)
```
```   747       apply (auto intro!: contf)
```
```   748       apply (rule order_less_le_trans, rule `a < d`, auto)
```
```   749       apply (rule order_le_less_trans) prefer 2
```
```   750       by (rule `e < b`, auto)
```
```   751   qed
```
```   752 qed
```
```   753
```
```   754 (*
```
```   755     The substitution theorem
```
```   756
```
```   757     Once again, three versions: first, for finite intervals, and then two versions for
```
```   758     arbitrary intervals.
```
```   759 *)
```
```   760
```
```   761 lemma interval_integral_substitution_finite:
```
```   762   fixes a b :: real and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   763   assumes "a \<le> b"
```
```   764   and derivg: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (g has_real_derivative (g' x)) (at x within {a..b})"
```
```   765   and contf : "continuous_on (g ` {a..b}) f"
```
```   766   and contg': "continuous_on {a..b} g'"
```
```   767   shows "LBINT x=a..b. g' x *\<^sub>R f (g x) = LBINT y=g a..g b. f y"
```
```   768 proof-
```
```   769   have v_derivg: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (g has_vector_derivative (g' x)) (at x within {a..b})"
```
```   770     using derivg unfolding has_field_derivative_iff_has_vector_derivative .
```
```   771   then have contg [simp]: "continuous_on {a..b} g"
```
```   772     by (rule continuous_on_vector_derivative) auto
```
```   773   have 1: "\<And>u. min (g a) (g b) \<le> u \<Longrightarrow> u \<le> max (g a) (g b) \<Longrightarrow>
```
```   774       \<exists>x\<in>{a..b}. u = g x"
```
```   775     apply (case_tac "g a \<le> g b")
```
```   776     apply (auto simp add: min_def max_def less_imp_le)
```
```   777     apply (frule (1) IVT' [of g], auto simp add: assms)
```
```   778     by (frule (1) IVT2' [of g], auto simp add: assms)
```
```   779   from contg `a \<le> b` have "\<exists>c d. g ` {a..b} = {c..d} \<and> c \<le> d"
```
```   780     by (elim continuous_image_closed_interval)
```
```   781   then obtain c d where g_im: "g ` {a..b} = {c..d}" and "c \<le> d" by auto
```
```   782   have "\<exists>F. \<forall>x\<in>{a..b}. (F has_vector_derivative (f (g x))) (at (g x) within (g ` {a..b}))"
```
```   783     apply (rule exI, auto, subst g_im)
```
```   784     apply (rule interval_integral_FTC2 [of c c d])
```
```   785     using `c \<le> d` apply auto
```
```   786     apply (rule continuous_on_subset [OF contf])
```
```   787     using g_im by auto
```
```   788   then guess F ..
```
```   789   then have derivF: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow>
```
```   790     (F has_vector_derivative (f (g x))) (at (g x) within (g ` {a..b}))" by auto
```
```   791   have contf2: "continuous_on {min (g a) (g b)..max (g a) (g b)} f"
```
```   792     apply (rule continuous_on_subset [OF contf])
```
```   793     apply (auto simp add: image_def)
```
```   794     by (erule 1)
```
```   795   have contfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
```
```   796     by (blast intro: continuous_on_compose2 contf contg)
```
```   797   have "LBINT x=a..b. g' x *\<^sub>R f (g x) = F (g b) - F (g a)"
```
```   798     apply (subst interval_integral_Icc, simp add: assms)
```
```   799     apply (rule integral_FTC_atLeastAtMost[of a b "\<lambda>x. F (g x)", OF `a \<le> b`])
```
```   800     apply (rule vector_diff_chain_within[OF v_derivg derivF, unfolded comp_def])
```
```   801     apply (auto intro!: continuous_on_scaleR contg' contfg)
```
```   802     done
```
```   803   moreover have "LBINT y=(g a)..(g b). f y = F (g b) - F (g a)"
```
```   804     apply (rule interval_integral_FTC_finite)
```
```   805     apply (rule contf2)
```
```   806     apply (frule (1) 1, auto)
```
```   807     apply (rule has_vector_derivative_within_subset [OF derivF])
```
```   808     apply (auto simp add: image_def)
```
```   809     by (rule 1, auto)
```
```   810   ultimately show ?thesis by simp
```
```   811 qed
```
```   812
```
```   813 (* TODO: is it possible to lift the assumption here that g' is nonnegative? *)
```
```   814
```
```   815 lemma interval_integral_substitution_integrable:
```
```   816   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and a b u v :: ereal
```
```   817   assumes "a < b"
```
```   818   and deriv_g: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV g x :> g' x"
```
```   819   and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
```
```   820   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
```
```   821   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
```
```   822   and A: "((ereal \<circ> g \<circ> real) ---> A) (at_right a)"
```
```   823   and B: "((ereal \<circ> g \<circ> real) ---> B) (at_left b)"
```
```   824   and integrable: "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
```
```   825   and integrable2: "set_integrable lborel (einterval A B) (\<lambda>x. f x)"
```
```   826   shows "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   827 proof -
```
```   828   from einterval_Icc_approximation[OF `a < b`] guess u l . note approx [simp] = this
```
```   829   note less_imp_le [simp]
```
```   830   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   831     by (rule order_less_le_trans, rule approx, force)
```
```   832   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   833     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   834   have [simp]: "\<And>i. l i < b"
```
```   835     apply (rule order_less_trans) prefer 2
```
```   836     by (rule approx, auto, rule approx)
```
```   837   have [simp]: "\<And>i. a < u i"
```
```   838     by (rule order_less_trans, rule approx, auto, rule approx)
```
```   839   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> l j \<le> l i" by (rule decseqD, rule approx)
```
```   840   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> u i \<le> u j" by (rule incseqD, rule approx)
```
```   841   have g_nondec [simp]: "\<And>x y. a < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < b \<Longrightarrow> g x \<le> g y"
```
```   842     apply (erule DERIV_nonneg_imp_nondecreasing, auto)
```
```   843     apply (rule exI, rule conjI, rule deriv_g)
```
```   844     apply (erule order_less_le_trans, auto)
```
```   845     apply (rule order_le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   846     apply (rule g'_nonneg)
```
```   847     apply (rule less_imp_le, erule order_less_le_trans, auto)
```
```   848     by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   849   have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   850   proof -
```
```   851     have A2: "(\<lambda>i. g (l i)) ----> A"
```
```   852       using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   853       by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
```
```   854     hence A3: "\<And>i. g (l i) \<ge> A"
```
```   855       by (intro decseq_le, auto simp add: decseq_def)
```
```   856     have B2: "(\<lambda>i. g (u i)) ----> B"
```
```   857       using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   858       by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
```
```   859     hence B3: "\<And>i. g (u i) \<le> B"
```
```   860       by (intro incseq_le, auto simp add: incseq_def)
```
```   861     show "A \<le> B"
```
```   862       apply (rule order_trans [OF A3 [of 0]])
```
```   863       apply (rule order_trans [OF _ B3 [of 0]])
```
```   864       by auto
```
```   865     { fix x :: real
```
```   866       assume "A < x" and "x < B"
```
```   867       then have "eventually (\<lambda>i. ereal (g (l i)) < x \<and> x < ereal (g (u i))) sequentially"
```
```   868         apply (intro eventually_conj order_tendstoD)
```
```   869         by (rule A2, assumption, rule B2, assumption)
```
```   870       hence "\<exists>i. g (l i) < x \<and> x < g (u i)"
```
```   871         by (simp add: eventually_sequentially, auto)
```
```   872     } note AB = this
```
```   873     show "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   874       apply (auto simp add: einterval_def)
```
```   875       apply (erule (1) AB)
```
```   876       apply (rule order_le_less_trans, rule A3, simp)
```
```   877       apply (rule order_less_le_trans) prefer 2
```
```   878       by (rule B3, simp)
```
```   879   qed
```
```   880   (* finally, the main argument *)
```
```   881   {
```
```   882      fix i
```
```   883      have "(LBINT x=l i.. u i. g' x *\<^sub>R f (g x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```   884         apply (rule interval_integral_substitution_finite, auto)
```
```   885         apply (rule DERIV_subset)
```
```   886         unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
```
```   887         apply (rule deriv_g)
```
```   888         apply (auto intro!: continuous_at_imp_continuous_on contf contg')
```
```   889         done
```
```   890   } note eq1 = this
```
```   891   have "(\<lambda>i. LBINT x=l i..u i. g' x *\<^sub>R f (g x)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   892     apply (rule interval_integral_Icc_approx_integrable [OF `a < b` approx])
```
```   893     by (rule assms)
```
```   894   hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   895     by (simp add: eq1)
```
```   896   have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
```
```   897     apply (auto simp add: incseq_def)
```
```   898     apply (rule order_le_less_trans)
```
```   899     prefer 2 apply (assumption, auto)
```
```   900     by (erule order_less_le_trans, rule g_nondec, auto)
```
```   901   have "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x = A..B. f x)"
```
```   902     apply (subst interval_lebesgue_integral_le_eq, auto simp del: real_scaleR_def)
```
```   903     apply (subst interval_lebesgue_integral_le_eq, rule `A \<le> B`)
```
```   904     apply (subst un, rule set_integral_cont_up, auto simp del: real_scaleR_def)
```
```   905     apply (rule incseq)
```
```   906     apply (subst un [symmetric])
```
```   907     by (rule integrable2)
```
```   908   thus ?thesis by (intro LIMSEQ_unique [OF _ 2])
```
```   909 qed
```
```   910
```
```   911 (* TODO: the last two proofs are only slightly different. Factor out common part?
```
```   912    An alternative: make the second one the main one, and then have another lemma
```
```   913    that says that if f is nonnegative and all the other hypotheses hold, then it is integrable. *)
```
```   914
```
```   915 lemma interval_integral_substitution_nonneg:
```
```   916   fixes f g g':: "real \<Rightarrow> real" and a b u v :: ereal
```
```   917   assumes "a < b"
```
```   918   and deriv_g: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV g x :> g' x"
```
```   919   and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
```
```   920   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
```
```   921   and f_nonneg: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> 0 \<le> f (g x)" (* TODO: make this AE? *)
```
```   922   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
```
```   923   and A: "((ereal \<circ> g \<circ> real) ---> A) (at_right a)"
```
```   924   and B: "((ereal \<circ> g \<circ> real) ---> B) (at_left b)"
```
```   925   and integrable_fg: "set_integrable lborel (einterval a b) (\<lambda>x. f (g x) * g' x)"
```
```   926   shows
```
```   927     "set_integrable lborel (einterval A B) f"
```
```   928     "(LBINT x=A..B. f x) = (LBINT x=a..b. (f (g x) * g' x))"
```
```   929 proof -
```
```   930   from einterval_Icc_approximation[OF `a < b`] guess u l . note approx [simp] = this
```
```   931   note less_imp_le [simp]
```
```   932   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   933     by (rule order_less_le_trans, rule approx, force)
```
```   934   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   935     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   936   have [simp]: "\<And>i. l i < b"
```
```   937     apply (rule order_less_trans) prefer 2
```
```   938     by (rule approx, auto, rule approx)
```
```   939   have [simp]: "\<And>i. a < u i"
```
```   940     by (rule order_less_trans, rule approx, auto, rule approx)
```
```   941   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> l j \<le> l i" by (rule decseqD, rule approx)
```
```   942   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> u i \<le> u j" by (rule incseqD, rule approx)
```
```   943   have g_nondec [simp]: "\<And>x y. a < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < b \<Longrightarrow> g x \<le> g y"
```
```   944     apply (erule DERIV_nonneg_imp_nondecreasing, auto)
```
```   945     apply (rule exI, rule conjI, rule deriv_g)
```
```   946     apply (erule order_less_le_trans, auto)
```
```   947     apply (rule order_le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   948     apply (rule g'_nonneg)
```
```   949     apply (rule less_imp_le, erule order_less_le_trans, auto)
```
```   950     by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   951   have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   952   proof -
```
```   953     have A2: "(\<lambda>i. g (l i)) ----> A"
```
```   954       using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   955       by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
```
```   956     hence A3: "\<And>i. g (l i) \<ge> A"
```
```   957       by (intro decseq_le, auto simp add: decseq_def)
```
```   958     have B2: "(\<lambda>i. g (u i)) ----> B"
```
```   959       using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   960       by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
```
```   961     hence B3: "\<And>i. g (u i) \<le> B"
```
```   962       by (intro incseq_le, auto simp add: incseq_def)
```
```   963     show "A \<le> B"
```
```   964       apply (rule order_trans [OF A3 [of 0]])
```
```   965       apply (rule order_trans [OF _ B3 [of 0]])
```
```   966       by auto
```
```   967     { fix x :: real
```
```   968       assume "A < x" and "x < B"
```
```   969       then have "eventually (\<lambda>i. ereal (g (l i)) < x \<and> x < ereal (g (u i))) sequentially"
```
```   970         apply (intro eventually_conj order_tendstoD)
```
```   971         by (rule A2, assumption, rule B2, assumption)
```
```   972       hence "\<exists>i. g (l i) < x \<and> x < g (u i)"
```
```   973         by (simp add: eventually_sequentially, auto)
```
```   974     } note AB = this
```
```   975     show "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   976       apply (auto simp add: einterval_def)
```
```   977       apply (erule (1) AB)
```
```   978       apply (rule order_le_less_trans, rule A3, simp)
```
```   979       apply (rule order_less_le_trans) prefer 2
```
```   980       by (rule B3, simp)
```
```   981   qed
```
```   982   (* finally, the main argument *)
```
```   983   {
```
```   984      fix i
```
```   985      have "(LBINT x=l i.. u i. g' x *\<^sub>R f (g x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```   986         apply (rule interval_integral_substitution_finite, auto)
```
```   987         apply (rule DERIV_subset, rule deriv_g, auto)
```
```   988         apply (rule continuous_at_imp_continuous_on, auto, rule contf, auto)
```
```   989         by (rule continuous_at_imp_continuous_on, auto, rule contg', auto)
```
```   990      then have "(LBINT x=l i.. u i. (f (g x) * g' x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```   991        by (simp add: ac_simps)
```
```   992   } note eq1 = this
```
```   993   have "(\<lambda>i. LBINT x=l i..u i. f (g x) * g' x)
```
```   994       ----> (LBINT x=a..b. f (g x) * g' x)"
```
```   995     apply (rule interval_integral_Icc_approx_integrable [OF `a < b` approx])
```
```   996     by (rule assms)
```
```   997   hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) ----> (LBINT x=a..b. f (g x) * g' x)"
```
```   998     by (simp add: eq1)
```
```   999   have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
```
```  1000     apply (auto simp add: incseq_def)
```
```  1001     apply (rule order_le_less_trans)
```
```  1002     prefer 2 apply assumption
```
```  1003     apply (rule g_nondec, auto)
```
```  1004     by (erule order_less_le_trans, rule g_nondec, auto)
```
```  1005   have img: "\<And>x i. g (l i) \<le> x \<Longrightarrow> x \<le> g (u i) \<Longrightarrow> \<exists>c \<ge> l i. c \<le> u i \<and> x = g c"
```
```  1006     apply (frule (1) IVT' [of g], auto)
```
```  1007     apply (rule continuous_at_imp_continuous_on, auto)
```
```  1008     by (rule DERIV_isCont, rule deriv_g, auto)
```
```  1009   have nonneg_f2: "\<And>x i. g (l i) \<le> x \<Longrightarrow> x \<le> g (u i) \<Longrightarrow> 0 \<le> f x"
```
```  1010     by (frule (1) img, auto, rule f_nonneg, auto)
```
```  1011   have contf_2: "\<And>x i. g (l i) \<le> x \<Longrightarrow> x \<le> g (u i) \<Longrightarrow> isCont f x"
```
```  1012     by (frule (1) img, auto, rule contf, auto)
```
```  1013   have integrable: "set_integrable lborel (\<Union>i. {g (l i)<..<g (u i)}) f"
```
```  1014     apply (rule pos_integrable_to_top, auto simp del: real_scaleR_def)
```
```  1015     apply (rule incseq)
```
```  1016     apply (rule nonneg_f2, erule less_imp_le, erule less_imp_le)
```
```  1017     apply (rule set_integrable_subset)
```
```  1018     apply (rule borel_integrable_atLeastAtMost')
```
```  1019     apply (rule continuous_at_imp_continuous_on)
```
```  1020     apply (clarsimp, erule (1) contf_2, auto)
```
```  1021     apply (erule less_imp_le)+
```
```  1022     using 2 unfolding interval_lebesgue_integral_def
```
```  1023     by auto
```
```  1024   thus "set_integrable lborel (einterval A B) f"
```
```  1025     by (simp add: un)
```
```  1026
```
```  1027   have "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```  1028   proof (rule interval_integral_substitution_integrable)
```
```  1029     show "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
```
```  1030       using integrable_fg by (simp add: ac_simps)
```
```  1031   qed fact+
```
```  1032   then show "(LBINT x=A..B. f x) = (LBINT x=a..b. (f (g x) * g' x))"
```
```  1033     by (simp add: ac_simps)
```
```  1034 qed
```
```  1035
```
```  1036
```
```  1037 syntax
```
```  1038 "_complex_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> complex"
```
```  1039 ("(2CLBINT _. _)" [0,60] 60)
```
```  1040
```
```  1041 translations
```
```  1042 "CLBINT x. f" == "CONST complex_lebesgue_integral CONST lborel (\<lambda>x. f)"
```
```  1043
```
```  1044 syntax
```
```  1045 "_complex_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> complex"
```
```  1046 ("(3CLBINT _:_. _)" [0,60,61] 60)
```
```  1047
```
```  1048 translations
```
```  1049 "CLBINT x:A. f" == "CONST complex_set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
```
```  1050
```
```  1051 abbreviation complex_interval_lebesgue_integral ::
```
```  1052     "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> complex" where
```
```  1053   "complex_interval_lebesgue_integral M a b f \<equiv> interval_lebesgue_integral M a b f"
```
```  1054
```
```  1055 abbreviation complex_interval_lebesgue_integrable ::
```
```  1056   "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool" where
```
```  1057   "complex_interval_lebesgue_integrable M a b f \<equiv> interval_lebesgue_integrable M a b f"
```
```  1058
```
```  1059 syntax
```
```  1060   "_ascii_complex_interval_lebesgue_borel_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> real \<Rightarrow> complex"
```
```  1061   ("(4CLBINT _=_.._. _)" [0,60,60,61] 60)
```
```  1062
```
```  1063 translations
```
```  1064   "CLBINT x=a..b. f" == "CONST complex_interval_lebesgue_integral CONST lborel a b (\<lambda>x. f)"
```
```  1065
```
```  1066 lemma interval_integral_norm:
```
```  1067   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```  1068   shows "interval_lebesgue_integrable lborel a b f \<Longrightarrow> a \<le> b \<Longrightarrow>
```
```  1069     norm (LBINT t=a..b. f t) \<le> LBINT t=a..b. norm (f t)"
```
```  1070   using integral_norm_bound[of lborel "\<lambda>x. indicator (einterval a b) x *\<^sub>R f x"]
```
```  1071   by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def)
```
```  1072
```
```  1073 lemma interval_integral_norm2:
```
```  1074   "interval_lebesgue_integrable lborel a b f \<Longrightarrow>
```
```  1075     norm (LBINT t=a..b. f t) \<le> abs (LBINT t=a..b. norm (f t))"
```
```  1076 proof (induct a b rule: linorder_wlog)
```
```  1077   case (sym a b) then show ?case
```
```  1078     by (simp add: interval_integral_endpoints_reverse[of a b] interval_integrable_endpoints_reverse[of a b])
```
```  1079 next
```
```  1080   case (le a b)
```
```  1081   then have "\<bar>LBINT t=a..b. norm (f t)\<bar> = LBINT t=a..b. norm (f t)"
```
```  1082     using integrable_norm[of lborel "\<lambda>x. indicator (einterval a b) x *\<^sub>R f x"]
```
```  1083     by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```  1084              intro!: integral_nonneg_AE abs_of_nonneg)
```
```  1085   then show ?case
```
```  1086     using le by (simp add: interval_integral_norm)
```
```  1087 qed
```
```  1088
```
```  1089 (* TODO: should we have a library of facts like these? *)
```
```  1090 lemma integral_cos: "t \<noteq> 0 \<Longrightarrow> LBINT x=a..b. cos (t * x) = sin (t * b) / t - sin (t * a) / t"
```
```  1091   apply (intro interval_integral_FTC_finite continuous_intros)
```
```  1092   by (auto intro!: derivative_eq_intros simp: has_field_derivative_iff_has_vector_derivative[symmetric])
```
```  1093
```
```  1094
```
```  1095 end
```