src/HOL/Analysis/Interval_Integral.thy
 author wenzelm Sun Sep 18 20:33:48 2016 +0200 (2016-09-18) changeset 63915 bab633745c7f parent 63886 685fb01256af child 63941 f353674c2528 permissions -rw-r--r--
tuned proofs;
```     1 (*  Title:      HOL/Analysis/Interval_Integral.thy
```
```     2     Author:     Jeremy Avigad (CMU), Johannes Hölzl (TUM), Luke Serafin (CMU)
```
```     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 \<open>s \<subseteq> t\<close>] 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 \<longlonglongrightarrow> b"
```
```    66 proof (cases b)
```
```    67   case PInf
```
```    68   with \<open>a < b\<close> have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
```
```    69     by (cases a) auto
```
```    70   moreover have "(\<lambda>x. ereal (real (Suc x))) \<longlonglongrightarrow> \<infinity>"
```
```    71       apply (subst LIMSEQ_Suc_iff)
```
```    72       apply (simp add: Lim_PInfty)
```
```    73       using nat_ceiling_le_eq by blast
```
```    74   moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) \<longlonglongrightarrow> \<infinity>"
```
```    75     apply (subst LIMSEQ_Suc_iff)
```
```    76     apply (subst Lim_PInfty)
```
```    77     apply (metis add.commute diff_le_eq nat_ceiling_le_eq ereal_less_eq(3))
```
```    78     done
```
```    79   ultimately show thesis
```
```    80     by (intro that[of "\<lambda>i. real_of_ereal a + Suc i"])
```
```    81        (auto simp: incseq_def PInf)
```
```    82 next
```
```    83   case (real b')
```
```    84   define d where "d = b' - (if a = -\<infinity> then b' - 1 else real_of_ereal a)"
```
```    85   with \<open>a < b\<close> have a': "0 < d"
```
```    86     by (cases a) (auto simp: real)
```
```    87   moreover
```
```    88   have "\<And>i r. r < b' \<Longrightarrow> (b' - r) * 1 < (b' - r) * real (Suc (Suc i))"
```
```    89     by (intro mult_strict_left_mono) auto
```
```    90   with \<open>a < b\<close> a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
```
```    91     by (cases a) (auto simp: real d_def field_simps)
```
```    92   moreover have "(\<lambda>i. b' - d / Suc (Suc i)) \<longlonglongrightarrow> b'"
```
```    93     apply (subst filterlim_sequentially_Suc)
```
```    94     apply (subst filterlim_sequentially_Suc)
```
```    95     apply (rule tendsto_eq_intros)
```
```    96     apply (auto intro!: tendsto_divide_0[OF tendsto_const] filterlim_sup1
```
```    97                 simp: at_infinity_eq_at_top_bot filterlim_real_sequentially)
```
```    98     done
```
```    99   ultimately show thesis
```
```   100     by (intro that[of "\<lambda>i. b' - d / Suc (Suc i)"])
```
```   101        (auto simp add: real incseq_def intro!: divide_left_mono)
```
```   102 qed (insert \<open>a < b\<close>, auto)
```
```   103
```
```   104 lemma ereal_decseq_approx:
```
```   105   fixes a b :: ereal
```
```   106   assumes "a < b"
```
```   107   obtains X :: "nat \<Rightarrow> real" where
```
```   108     "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X \<longlonglongrightarrow> a"
```
```   109 proof -
```
```   110   have "-b < -a" using \<open>a < b\<close> by simp
```
```   111   from ereal_incseq_approx[OF this] guess X .
```
```   112   then show thesis
```
```   113     apply (intro that[of "\<lambda>i. - X i"])
```
```   114     apply (auto simp add: uminus_ereal.simps[symmetric] decseq_def incseq_def
```
```   115                 simp del: uminus_ereal.simps)
```
```   116     apply (metis ereal_minus_less_minus ereal_uminus_uminus ereal_Lim_uminus)+
```
```   117     done
```
```   118 qed
```
```   119
```
```   120 lemma einterval_Icc_approximation:
```
```   121   fixes a b :: ereal
```
```   122   assumes "a < b"
```
```   123   obtains u l :: "nat \<Rightarrow> real" where
```
```   124     "einterval a b = (\<Union>i. {l i .. u i})"
```
```   125     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   126     "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
```
```   127 proof -
```
```   128   from dense[OF \<open>a < b\<close>] obtain c where "a < c" "c < b" by safe
```
```   129   from ereal_incseq_approx[OF \<open>c < b\<close>] guess u . note u = this
```
```   130   from ereal_decseq_approx[OF \<open>a < c\<close>] guess l . note l = this
```
```   131   { fix i from less_trans[OF \<open>l i < c\<close> \<open>c < u i\<close>] have "l i < u i" by simp }
```
```   132   have "einterval a b = (\<Union>i. {l i .. u i})"
```
```   133   proof (auto simp: einterval_iff)
```
```   134     fix x assume "a < ereal x" "ereal x < b"
```
```   135     have "eventually (\<lambda>i. ereal (l i) < ereal x) sequentially"
```
```   136       using l(4) \<open>a < ereal x\<close> by (rule order_tendstoD)
```
```   137     moreover
```
```   138     have "eventually (\<lambda>i. ereal x < ereal (u i)) sequentially"
```
```   139       using u(4) \<open>ereal x< b\<close> by (rule order_tendstoD)
```
```   140     ultimately have "eventually (\<lambda>i. l i < x \<and> x < u i) sequentially"
```
```   141       by eventually_elim auto
```
```   142     then show "\<exists>i. l i \<le> x \<and> x \<le> u i"
```
```   143       by (auto intro: less_imp_le simp: eventually_sequentially)
```
```   144   next
```
```   145     fix x i assume "l i \<le> x" "x \<le> u i"
```
```   146     with \<open>a < ereal (l i)\<close> \<open>ereal (u i) < b\<close>
```
```   147     show "a < ereal x" "ereal x < b"
```
```   148       by (auto simp del: ereal_less_eq(3) simp add: ereal_less_eq(3)[symmetric])
```
```   149   qed
```
```   150   show thesis
```
```   151     by (intro that) fact+
```
```   152 qed
```
```   153
```
```   154 (* TODO: in this definition, it would be more natural if einterval a b included a and b when
```
```   155    they are real. *)
```
```   156 definition interval_lebesgue_integral :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> 'a::{banach, second_countable_topology}" where
```
```   157   "interval_lebesgue_integral M a b f =
```
```   158     (if a \<le> b then (LINT x:einterval a b|M. f x) else - (LINT x:einterval b a|M. f x))"
```
```   159
```
```   160 syntax
```
```   161   "_ascii_interval_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real measure \<Rightarrow> real \<Rightarrow> real"
```
```   162   ("(5LINT _=_.._|_. _)" [0,60,60,61,100] 60)
```
```   163
```
```   164 translations
```
```   165   "LINT x=a..b|M. f" == "CONST interval_lebesgue_integral M a b (\<lambda>x. f)"
```
```   166
```
```   167 definition interval_lebesgue_integrable :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a::{banach, second_countable_topology}) \<Rightarrow> bool" where
```
```   168   "interval_lebesgue_integrable M a b f =
```
```   169     (if a \<le> b then set_integrable M (einterval a b) f else set_integrable M (einterval b a) f)"
```
```   170
```
```   171 syntax
```
```   172   "_ascii_interval_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"
```
```   173   ("(4LBINT _=_.._. _)" [0,60,60,61] 60)
```
```   174
```
```   175 translations
```
```   176   "LBINT x=a..b. f" == "CONST interval_lebesgue_integral CONST lborel a b (\<lambda>x. f)"
```
```   177
```
```   178 (*
```
```   179     Basic properties of integration over an interval.
```
```   180 *)
```
```   181
```
```   182 lemma interval_lebesgue_integral_cong:
```
```   183   "a \<le> b \<Longrightarrow> (\<And>x. x \<in> einterval a b \<Longrightarrow> f x = g x) \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
```
```   184     interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   185   by (auto intro: set_lebesgue_integral_cong simp: interval_lebesgue_integral_def)
```
```   186
```
```   187 lemma interval_lebesgue_integral_cong_AE:
```
```   188   "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
```
```   189     a \<le> b \<Longrightarrow> AE x \<in> einterval a b in M. f x = g x \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
```
```   190     interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   191   by (auto intro: set_lebesgue_integral_cong_AE simp: interval_lebesgue_integral_def)
```
```   192
```
```   193 lemma interval_integrable_mirror:
```
```   194   shows "interval_lebesgue_integrable lborel a b (\<lambda>x. f (-x)) \<longleftrightarrow>
```
```   195     interval_lebesgue_integrable lborel (-b) (-a) f"
```
```   196 proof -
```
```   197   have *: "indicator (einterval a b) (- x) = (indicator (einterval (-b) (-a)) x :: real)"
```
```   198     for a b :: ereal and x :: real
```
```   199     by (cases a b rule: ereal2_cases) (auto simp: einterval_def split: split_indicator)
```
```   200   show ?thesis
```
```   201     unfolding interval_lebesgue_integrable_def
```
```   202     using lborel_integrable_real_affine_iff[symmetric, of "-1" "\<lambda>x. indicator (einterval _ _) x *\<^sub>R f x" 0]
```
```   203     by (simp add: *)
```
```   204 qed
```
```   205
```
```   206 lemma interval_lebesgue_integral_add [intro, simp]:
```
```   207   fixes M a b f
```
```   208   assumes "interval_lebesgue_integrable M a b f" "interval_lebesgue_integrable M a b g"
```
```   209   shows "interval_lebesgue_integrable M a b (\<lambda>x. f x + g x)" and
```
```   210     "interval_lebesgue_integral M a b (\<lambda>x. f x + g x) =
```
```   211    interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g"
```
```   212 using assms by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```   213     field_simps)
```
```   214
```
```   215 lemma interval_lebesgue_integral_diff [intro, simp]:
```
```   216   fixes M a b f
```
```   217   assumes "interval_lebesgue_integrable M a b f"
```
```   218     "interval_lebesgue_integrable M a b g"
```
```   219   shows "interval_lebesgue_integrable M a b (\<lambda>x. f x - g x)" and
```
```   220     "interval_lebesgue_integral M a b (\<lambda>x. f x - g x) =
```
```   221    interval_lebesgue_integral M a b f - interval_lebesgue_integral M a b g"
```
```   222 using assms by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```   223     field_simps)
```
```   224
```
```   225 lemma interval_lebesgue_integrable_mult_right [intro, simp]:
```
```   226   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   227   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   228     interval_lebesgue_integrable M a b (\<lambda>x. c * f x)"
```
```   229   by (simp add: interval_lebesgue_integrable_def)
```
```   230
```
```   231 lemma interval_lebesgue_integrable_mult_left [intro, simp]:
```
```   232   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   233   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   234     interval_lebesgue_integrable M a b (\<lambda>x. f x * c)"
```
```   235   by (simp add: interval_lebesgue_integrable_def)
```
```   236
```
```   237 lemma interval_lebesgue_integrable_divide [intro, simp]:
```
```   238   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
```
```   239   shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
```
```   240     interval_lebesgue_integrable M a b (\<lambda>x. f x / c)"
```
```   241   by (simp add: interval_lebesgue_integrable_def)
```
```   242
```
```   243 lemma interval_lebesgue_integral_mult_right [simp]:
```
```   244   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   245   shows "interval_lebesgue_integral M a b (\<lambda>x. c * f x) =
```
```   246     c * interval_lebesgue_integral M a b f"
```
```   247   by (simp add: interval_lebesgue_integral_def)
```
```   248
```
```   249 lemma interval_lebesgue_integral_mult_left [simp]:
```
```   250   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
```
```   251   shows "interval_lebesgue_integral M a b (\<lambda>x. f x * c) =
```
```   252     interval_lebesgue_integral M a b f * c"
```
```   253   by (simp add: interval_lebesgue_integral_def)
```
```   254
```
```   255 lemma interval_lebesgue_integral_divide [simp]:
```
```   256   fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
```
```   257   shows "interval_lebesgue_integral M a b (\<lambda>x. f x / c) =
```
```   258     interval_lebesgue_integral M a b f / c"
```
```   259   by (simp add: interval_lebesgue_integral_def)
```
```   260
```
```   261 lemma interval_lebesgue_integral_uminus:
```
```   262   "interval_lebesgue_integral M a b (\<lambda>x. - f x) = - interval_lebesgue_integral M a b f"
```
```   263   by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def)
```
```   264
```
```   265 lemma interval_lebesgue_integral_of_real:
```
```   266   "interval_lebesgue_integral M a b (\<lambda>x. complex_of_real (f x)) =
```
```   267     of_real (interval_lebesgue_integral M a b f)"
```
```   268   unfolding interval_lebesgue_integral_def
```
```   269   by (auto simp add: interval_lebesgue_integral_def set_integral_complex_of_real)
```
```   270
```
```   271 lemma interval_lebesgue_integral_le_eq:
```
```   272   fixes a b f
```
```   273   assumes "a \<le> b"
```
```   274   shows "interval_lebesgue_integral M a b f = (LINT x : einterval a b | M. f x)"
```
```   275 using assms by (auto simp add: interval_lebesgue_integral_def)
```
```   276
```
```   277 lemma interval_lebesgue_integral_gt_eq:
```
```   278   fixes a b f
```
```   279   assumes "a > b"
```
```   280   shows "interval_lebesgue_integral M a b f = -(LINT x : einterval b a | M. f x)"
```
```   281 using assms by (auto simp add: interval_lebesgue_integral_def less_imp_le einterval_def)
```
```   282
```
```   283 lemma interval_lebesgue_integral_gt_eq':
```
```   284   fixes a b f
```
```   285   assumes "a > b"
```
```   286   shows "interval_lebesgue_integral M a b f = - interval_lebesgue_integral M b a f"
```
```   287 using assms by (auto simp add: interval_lebesgue_integral_def less_imp_le einterval_def)
```
```   288
```
```   289 lemma interval_integral_endpoints_same [simp]: "(LBINT x=a..a. f x) = 0"
```
```   290   by (simp add: interval_lebesgue_integral_def einterval_same)
```
```   291
```
```   292 lemma interval_integral_endpoints_reverse: "(LBINT x=a..b. f x) = -(LBINT x=b..a. f x)"
```
```   293   by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integral_def einterval_same)
```
```   294
```
```   295 lemma interval_integrable_endpoints_reverse:
```
```   296   "interval_lebesgue_integrable lborel a b f \<longleftrightarrow>
```
```   297     interval_lebesgue_integrable lborel b a f"
```
```   298   by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integrable_def einterval_same)
```
```   299
```
```   300 lemma interval_integral_reflect:
```
```   301   "(LBINT x=a..b. f x) = (LBINT x=-b..-a. f (-x))"
```
```   302 proof (induct a b rule: linorder_wlog)
```
```   303   case (sym a b) then show ?case
```
```   304     by (auto simp add: interval_lebesgue_integral_def interval_integrable_endpoints_reverse
```
```   305              split: if_split_asm)
```
```   306 next
```
```   307   case (le a b) then show ?case
```
```   308     unfolding interval_lebesgue_integral_def
```
```   309     by (subst set_integral_reflect)
```
```   310        (auto simp: interval_lebesgue_integrable_def einterval_iff
```
```   311                    ereal_uminus_le_reorder ereal_uminus_less_reorder not_less
```
```   312                    uminus_ereal.simps[symmetric]
```
```   313              simp del: uminus_ereal.simps
```
```   314              intro!: Bochner_Integration.integral_cong
```
```   315              split: split_indicator)
```
```   316 qed
```
```   317
```
```   318 lemma interval_lebesgue_integral_0_infty:
```
```   319   "interval_lebesgue_integrable M 0 \<infinity> f \<longleftrightarrow> set_integrable M {0<..} f"
```
```   320   "interval_lebesgue_integral M 0 \<infinity> f = (LINT x:{0<..}|M. f x)"
```
```   321   unfolding zero_ereal_def
```
```   322   by (auto simp: interval_lebesgue_integral_le_eq interval_lebesgue_integrable_def)
```
```   323
```
```   324 lemma interval_integral_to_infinity_eq: "(LINT x=ereal a..\<infinity> | M. f x) = (LINT x : {a<..} | M. f x)"
```
```   325   unfolding interval_lebesgue_integral_def by auto
```
```   326
```
```   327 lemma interval_integrable_to_infinity_eq: "(interval_lebesgue_integrable M a \<infinity> f) =
```
```   328   (set_integrable M {a<..} f)"
```
```   329   unfolding interval_lebesgue_integrable_def by auto
```
```   330
```
```   331 (*
```
```   332     Basic properties of integration over an interval wrt lebesgue measure.
```
```   333 *)
```
```   334
```
```   335 lemma interval_integral_zero [simp]:
```
```   336   fixes a b :: ereal
```
```   337   shows"LBINT x=a..b. 0 = 0"
```
```   338 unfolding interval_lebesgue_integral_def einterval_eq
```
```   339 by simp
```
```   340
```
```   341 lemma interval_integral_const [intro, simp]:
```
```   342   fixes a b c :: real
```
```   343   shows "interval_lebesgue_integrable lborel a b (\<lambda>x. c)" and "LBINT x=a..b. c = c * (b - a)"
```
```   344 unfolding interval_lebesgue_integral_def interval_lebesgue_integrable_def einterval_eq
```
```   345 by (auto simp add:  less_imp_le field_simps measure_def)
```
```   346
```
```   347 lemma interval_integral_cong_AE:
```
```   348   assumes [measurable]: "f \<in> borel_measurable borel" "g \<in> borel_measurable borel"
```
```   349   assumes "AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x"
```
```   350   shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
```
```   351   using assms
```
```   352 proof (induct a b rule: linorder_wlog)
```
```   353   case (sym a b) then show ?case
```
```   354     by (simp add: min.commute max.commute interval_integral_endpoints_reverse[of a b])
```
```   355 next
```
```   356   case (le a b) then show ?case
```
```   357     by (auto simp: interval_lebesgue_integral_def max_def min_def
```
```   358              intro!: set_lebesgue_integral_cong_AE)
```
```   359 qed
```
```   360
```
```   361 lemma interval_integral_cong:
```
```   362   assumes "\<And>x. x \<in> einterval (min a b) (max a b) \<Longrightarrow> f x = g x"
```
```   363   shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
```
```   364   using assms
```
```   365 proof (induct a b rule: linorder_wlog)
```
```   366   case (sym a b) then show ?case
```
```   367     by (simp add: min.commute max.commute interval_integral_endpoints_reverse[of a b])
```
```   368 next
```
```   369   case (le a b) then show ?case
```
```   370     by (auto simp: interval_lebesgue_integral_def max_def min_def
```
```   371              intro!: set_lebesgue_integral_cong)
```
```   372 qed
```
```   373
```
```   374 lemma interval_lebesgue_integrable_cong_AE:
```
```   375     "f \<in> borel_measurable lborel \<Longrightarrow> g \<in> borel_measurable lborel \<Longrightarrow>
```
```   376     AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x \<Longrightarrow>
```
```   377     interval_lebesgue_integrable lborel a b f = interval_lebesgue_integrable lborel a b g"
```
```   378   apply (simp add: interval_lebesgue_integrable_def )
```
```   379   apply (intro conjI impI set_integrable_cong_AE)
```
```   380   apply (auto simp: min_def max_def)
```
```   381   done
```
```   382
```
```   383 lemma interval_integrable_abs_iff:
```
```   384   fixes f :: "real \<Rightarrow> real"
```
```   385   shows  "f \<in> borel_measurable lborel \<Longrightarrow>
```
```   386     interval_lebesgue_integrable lborel a b (\<lambda>x. \<bar>f x\<bar>) = interval_lebesgue_integrable lborel a b f"
```
```   387   unfolding interval_lebesgue_integrable_def
```
```   388   by (subst (1 2) set_integrable_abs_iff') simp_all
```
```   389
```
```   390 lemma interval_integral_Icc:
```
```   391   fixes a b :: real
```
```   392   shows "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..b}. f x)"
```
```   393   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   394            simp add: interval_lebesgue_integral_def)
```
```   395
```
```   396 lemma interval_integral_Icc':
```
```   397   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a \<le> ereal x \<and> ereal x \<le> b}. f x)"
```
```   398   by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
```
```   399            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   400
```
```   401 lemma interval_integral_Ioc:
```
```   402   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a<..b}. f x)"
```
```   403   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   404            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   405
```
```   406 (* TODO: other versions as well? *) (* Yes: I need the Icc' version. *)
```
```   407 lemma interval_integral_Ioc':
```
```   408   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a < ereal x \<and> ereal x \<le> b}. f x)"
```
```   409   by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
```
```   410            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   411
```
```   412 lemma interval_integral_Ico:
```
```   413   "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..<b}. f x)"
```
```   414   by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
```
```   415            simp add: interval_lebesgue_integral_def einterval_iff)
```
```   416
```
```   417 lemma interval_integral_Ioi:
```
```   418   "\<bar>a\<bar> < \<infinity> \<Longrightarrow> (LBINT x=a..\<infinity>. f x) = (LBINT x : {real_of_ereal a <..}. f x)"
```
```   419   by (auto simp add: interval_lebesgue_integral_def einterval_iff)
```
```   420
```
```   421 lemma interval_integral_Ioo:
```
```   422   "a \<le> b \<Longrightarrow> \<bar>a\<bar> < \<infinity> ==> \<bar>b\<bar> < \<infinity> \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {real_of_ereal a <..< real_of_ereal b}. f x)"
```
```   423   by (auto simp add: interval_lebesgue_integral_def einterval_iff)
```
```   424
```
```   425 lemma interval_integral_discrete_difference:
```
```   426   fixes f :: "real \<Rightarrow> 'b::{banach, second_countable_topology}" and a b :: ereal
```
```   427   assumes "countable X"
```
```   428   and eq: "\<And>x. a \<le> b \<Longrightarrow> a < x \<Longrightarrow> x < b \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
```
```   429   and anti_eq: "\<And>x. b \<le> a \<Longrightarrow> b < x \<Longrightarrow> x < a \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
```
```   430   assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
```
```   431   shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
```
```   432   unfolding interval_lebesgue_integral_def
```
```   433   apply (intro if_cong refl arg_cong[where f="\<lambda>x. - x"] integral_discrete_difference[of X] assms)
```
```   434   apply (auto simp: eq anti_eq einterval_iff split: split_indicator)
```
```   435   done
```
```   436
```
```   437 lemma interval_integral_sum:
```
```   438   fixes a b c :: ereal
```
```   439   assumes integrable: "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f"
```
```   440   shows "(LBINT x=a..b. f x) + (LBINT x=b..c. f x) = (LBINT x=a..c. f x)"
```
```   441 proof -
```
```   442   let ?I = "\<lambda>a b. LBINT x=a..b. f x"
```
```   443   { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
```
```   444     then have ord: "a \<le> b" "b \<le> c" "a \<le> c" and f': "set_integrable lborel (einterval a c) f"
```
```   445       by (auto simp: interval_lebesgue_integrable_def)
```
```   446     then have f: "set_borel_measurable borel (einterval a c) f"
```
```   447       by (drule_tac borel_measurable_integrable) simp
```
```   448     have "(LBINT x:einterval a c. f x) = (LBINT x:einterval a b \<union> einterval b c. f x)"
```
```   449     proof (rule set_integral_cong_set)
```
```   450       show "AE x in lborel. (x \<in> einterval a b \<union> einterval b c) = (x \<in> einterval a c)"
```
```   451         using AE_lborel_singleton[of "real_of_ereal b"] ord
```
```   452         by (cases a b c rule: ereal3_cases) (auto simp: einterval_iff)
```
```   453     qed (insert ord, auto intro!: set_borel_measurable_subset[OF f] simp: einterval_iff)
```
```   454     also have "\<dots> = (LBINT x:einterval a b. f x) + (LBINT x:einterval b c. f x)"
```
```   455       using ord
```
```   456       by (intro set_integral_Un_AE) (auto intro!: set_integrable_subset[OF f'] simp: einterval_iff not_less)
```
```   457     finally have "?I a b + ?I b c = ?I a c"
```
```   458       using ord by (simp add: interval_lebesgue_integral_def)
```
```   459   } note 1 = this
```
```   460   { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
```
```   461     from 1[OF this] have "?I b c + ?I a b = ?I a c"
```
```   462       by (metis add.commute)
```
```   463   } note 2 = this
```
```   464   have 3: "\<And>a b. b \<le> a \<Longrightarrow> (LBINT x=a..b. f x) = - (LBINT x=b..a. f x)"
```
```   465     by (rule interval_integral_endpoints_reverse)
```
```   466   show ?thesis
```
```   467     using integrable
```
```   468     by (cases a b b c a c rule: linorder_le_cases[case_product linorder_le_cases linorder_cases])
```
```   469        (simp_all add: min_absorb1 min_absorb2 max_absorb1 max_absorb2 field_simps 1 2 3)
```
```   470 qed
```
```   471
```
```   472 lemma interval_integrable_isCont:
```
```   473   fixes a b and f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
```
```   474   shows "(\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> isCont f x) \<Longrightarrow>
```
```   475     interval_lebesgue_integrable lborel a b f"
```
```   476 proof (induct a b rule: linorder_wlog)
```
```   477   case (le a b) then show ?case
```
```   478     by (auto simp: interval_lebesgue_integrable_def
```
```   479              intro!: set_integrable_subset[OF borel_integrable_compact[of "{a .. b}"]]
```
```   480                      continuous_at_imp_continuous_on)
```
```   481 qed (auto intro: interval_integrable_endpoints_reverse[THEN iffD1])
```
```   482
```
```   483 lemma interval_integrable_continuous_on:
```
```   484   fixes a b :: real and f
```
```   485   assumes "a \<le> b" and "continuous_on {a..b} f"
```
```   486   shows "interval_lebesgue_integrable lborel a b f"
```
```   487 using assms unfolding interval_lebesgue_integrable_def apply simp
```
```   488   by (rule set_integrable_subset, rule borel_integrable_atLeastAtMost' [of a b], auto)
```
```   489
```
```   490 lemma interval_integral_eq_integral:
```
```   491   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   492   shows "a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f \<Longrightarrow> LBINT x=a..b. f x = integral {a..b} f"
```
```   493   by (subst interval_integral_Icc, simp) (rule set_borel_integral_eq_integral)
```
```   494
```
```   495 lemma interval_integral_eq_integral':
```
```   496   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   497   shows "a \<le> b \<Longrightarrow> set_integrable lborel (einterval a b) f \<Longrightarrow> LBINT x=a..b. f x = integral (einterval a b) f"
```
```   498   by (subst interval_lebesgue_integral_le_eq, simp) (rule set_borel_integral_eq_integral)
```
```   499
```
```   500 (*
```
```   501     General limit approximation arguments
```
```   502 *)
```
```   503
```
```   504 lemma interval_integral_Icc_approx_nonneg:
```
```   505   fixes a b :: ereal
```
```   506   assumes "a < b"
```
```   507   fixes u l :: "nat \<Rightarrow> real"
```
```   508   assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
```
```   509     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   510     "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
```
```   511   fixes f :: "real \<Rightarrow> real"
```
```   512   assumes f_integrable: "\<And>i. set_integrable lborel {l i..u i} f"
```
```   513   assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
```
```   514   assumes f_measurable: "set_borel_measurable lborel (einterval a b) f"
```
```   515   assumes lbint_lim: "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> C"
```
```   516   shows
```
```   517     "set_integrable lborel (einterval a b) f"
```
```   518     "(LBINT x=a..b. f x) = C"
```
```   519 proof -
```
```   520   have 1: "\<And>i. set_integrable lborel {l i..u i} f" by (rule f_integrable)
```
```   521   have 2: "AE x in lborel. mono (\<lambda>n. indicator {l n..u n} x *\<^sub>R f x)"
```
```   522   proof -
```
```   523      from f_nonneg have "AE x in lborel. \<forall>i. l i \<le> x \<longrightarrow> x \<le> u i \<longrightarrow> 0 \<le> f x"
```
```   524       by eventually_elim
```
```   525          (metis approx(5) approx(6) dual_order.strict_trans1 ereal_less_eq(3) le_less_trans)
```
```   526     then show ?thesis
```
```   527       apply eventually_elim
```
```   528       apply (auto simp: mono_def split: split_indicator)
```
```   529       apply (metis approx(3) decseqD order_trans)
```
```   530       apply (metis approx(2) incseqD order_trans)
```
```   531       done
```
```   532   qed
```
```   533   have 3: "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> indicator (einterval a b) x *\<^sub>R f x"
```
```   534   proof -
```
```   535     { fix x i assume "l i \<le> x" "x \<le> u i"
```
```   536       then have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
```
```   537         apply (auto simp: eventually_sequentially intro!: exI[of _ i])
```
```   538         apply (metis approx(3) decseqD order_trans)
```
```   539         apply (metis approx(2) incseqD order_trans)
```
```   540         done
```
```   541       then have "eventually (\<lambda>i. f x * indicator {l i..u i} x = f x) sequentially"
```
```   542         by eventually_elim auto }
```
```   543     then show ?thesis
```
```   544       unfolding approx(1) by (auto intro!: AE_I2 Lim_eventually split: split_indicator)
```
```   545   qed
```
```   546   have 4: "(\<lambda>i. \<integral> x. indicator {l i..u i} x *\<^sub>R f x \<partial>lborel) \<longlonglongrightarrow> C"
```
```   547     using lbint_lim by (simp add: interval_integral_Icc approx less_imp_le)
```
```   548   have 5: "set_borel_measurable lborel (einterval a b) f" by (rule assms)
```
```   549   have "(LBINT x=a..b. f x) = lebesgue_integral lborel (\<lambda>x. indicator (einterval a b) x *\<^sub>R f x)"
```
```   550     using assms by (simp add: interval_lebesgue_integral_def less_imp_le)
```
```   551   also have "... = C" by (rule integral_monotone_convergence [OF 1 2 3 4 5])
```
```   552   finally show "(LBINT x=a..b. f x) = C" .
```
```   553
```
```   554   show "set_integrable lborel (einterval a b) f"
```
```   555     by (rule integrable_monotone_convergence[OF 1 2 3 4 5])
```
```   556 qed
```
```   557
```
```   558 lemma interval_integral_Icc_approx_integrable:
```
```   559   fixes u l :: "nat \<Rightarrow> real" and a b :: ereal
```
```   560   fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
```
```   561   assumes "a < b"
```
```   562   assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
```
```   563     "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
```
```   564     "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
```
```   565   assumes f_integrable: "set_integrable lborel (einterval a b) f"
```
```   566   shows "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
```
```   567 proof -
```
```   568   have "(\<lambda>i. LBINT x:{l i.. u i}. f x) \<longlonglongrightarrow> (LBINT x:einterval a b. f x)"
```
```   569   proof (rule integral_dominated_convergence)
```
```   570     show "integrable lborel (\<lambda>x. norm (indicator (einterval a b) x *\<^sub>R f x))"
```
```   571       by (rule integrable_norm) fact
```
```   572     show "set_borel_measurable lborel (einterval a b) f"
```
```   573       using f_integrable by (rule borel_measurable_integrable)
```
```   574     then show "\<And>i. set_borel_measurable lborel {l i..u i} f"
```
```   575       by (rule set_borel_measurable_subset) (auto simp: approx)
```
```   576     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)"
```
```   577       by (intro AE_I2) (auto simp: approx split: split_indicator)
```
```   578     show "AE x in lborel. (\<lambda>i. indicator {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> indicator (einterval a b) x *\<^sub>R f x"
```
```   579     proof (intro AE_I2 tendsto_intros Lim_eventually)
```
```   580       fix x
```
```   581       { fix i assume "l i \<le> x" "x \<le> u i"
```
```   582         with \<open>incseq u\<close>[THEN incseqD, of i] \<open>decseq l\<close>[THEN decseqD, of i]
```
```   583         have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
```
```   584           by (auto simp: eventually_sequentially decseq_def incseq_def intro: order_trans) }
```
```   585       then show "eventually (\<lambda>xa. indicator {l xa..u xa} x = (indicator (einterval a b) x::real)) sequentially"
```
```   586         using approx order_tendstoD(2)[OF \<open>l \<longlonglongrightarrow> a\<close>, of x] order_tendstoD(1)[OF \<open>u \<longlonglongrightarrow> b\<close>, of x]
```
```   587         by (auto split: split_indicator)
```
```   588     qed
```
```   589   qed
```
```   590   with \<open>a < b\<close> \<open>\<And>i. l i < u i\<close> show ?thesis
```
```   591     by (simp add: interval_lebesgue_integral_le_eq[symmetric] interval_integral_Icc less_imp_le)
```
```   592 qed
```
```   593
```
```   594 (*
```
```   595   A slightly stronger version of integral_FTC_atLeastAtMost and related facts,
```
```   596   with continuous_on instead of isCont
```
```   597
```
```   598   TODO: make the older versions corollaries of these (using continuous_at_imp_continuous_on, etc.)
```
```   599 *)
```
```   600
```
```   601 (*
```
```   602 TODO: many proofs below require inferences like
```
```   603
```
```   604   a < ereal x \<Longrightarrow> x < y \<Longrightarrow> a < ereal y
```
```   605
```
```   606 where x and y are real. These should be better automated.
```
```   607 *)
```
```   608
```
```   609 (*
```
```   610     The first Fundamental Theorem of Calculus
```
```   611
```
```   612     First, for finite intervals, and then two versions for arbitrary intervals.
```
```   613 *)
```
```   614
```
```   615 lemma interval_integral_FTC_finite:
```
```   616   fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: real
```
```   617   assumes f: "continuous_on {min a b..max a b} f"
```
```   618   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
```
```   619     {min a b..max a b})"
```
```   620   shows "(LBINT x=a..b. f x) = F b - F a"
```
```   621   apply (case_tac "a \<le> b")
```
```   622   apply (subst interval_integral_Icc, simp)
```
```   623   apply (rule integral_FTC_atLeastAtMost, assumption)
```
```   624   apply (metis F max_def min_def)
```
```   625   using f apply (simp add: min_absorb1 max_absorb2)
```
```   626   apply (subst interval_integral_endpoints_reverse)
```
```   627   apply (subst interval_integral_Icc, simp)
```
```   628   apply (subst integral_FTC_atLeastAtMost, auto)
```
```   629   apply (metis F max_def min_def)
```
```   630 using f by (simp add: min_absorb2 max_absorb1)
```
```   631
```
```   632 lemma interval_integral_FTC_nonneg:
```
```   633   fixes f F :: "real \<Rightarrow> real" and a b :: ereal
```
```   634   assumes "a < b"
```
```   635   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV F x :> f x"
```
```   636   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
```
```   637   assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
```
```   638   assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
```
```   639   assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
```
```   640   shows
```
```   641     "set_integrable lborel (einterval a b) f"
```
```   642     "(LBINT x=a..b. f x) = B - A"
```
```   643 proof -
```
```   644   from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx = this
```
```   645   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   646     by (rule order_less_le_trans, rule approx, force)
```
```   647   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   648     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   649   have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
```
```   650     using assms approx apply (intro interval_integral_FTC_finite)
```
```   651     apply (auto simp add: less_imp_le min_def max_def
```
```   652       has_field_derivative_iff_has_vector_derivative[symmetric])
```
```   653     apply (rule continuous_at_imp_continuous_on, auto intro!: f)
```
```   654     by (rule DERIV_subset [OF F], auto)
```
```   655   have 1: "\<And>i. set_integrable lborel {l i..u i} f"
```
```   656   proof -
```
```   657     fix i show "set_integrable lborel {l i .. u i} f"
```
```   658       using \<open>a < l i\<close> \<open>u i < b\<close>
```
```   659       by (intro borel_integrable_compact f continuous_at_imp_continuous_on compact_Icc ballI)
```
```   660          (auto simp del: ereal_less_eq simp add: ereal_less_eq(3)[symmetric])
```
```   661   qed
```
```   662   have 2: "set_borel_measurable lborel (einterval a b) f"
```
```   663     by (auto simp del: real_scaleR_def intro!: set_borel_measurable_continuous
```
```   664              simp: continuous_on_eq_continuous_at einterval_iff f)
```
```   665   have 3: "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
```
```   666     apply (subst FTCi)
```
```   667     apply (intro tendsto_intros)
```
```   668     using B approx unfolding tendsto_at_iff_sequentially comp_def
```
```   669     using tendsto_at_iff_sequentially[where 'a=real]
```
```   670     apply (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
```
```   671     using A approx unfolding tendsto_at_iff_sequentially comp_def
```
```   672     by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
```
```   673   show "(LBINT x=a..b. f x) = B - A"
```
```   674     by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
```
```   675   show "set_integrable lborel (einterval a b) f"
```
```   676     by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
```
```   677 qed
```
```   678
```
```   679 lemma interval_integral_FTC_integrable:
```
```   680   fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: ereal
```
```   681   assumes "a < b"
```
```   682   assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> (F has_vector_derivative f x) (at x)"
```
```   683   assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
```
```   684   assumes f_integrable: "set_integrable lborel (einterval a b) f"
```
```   685   assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
```
```   686   assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
```
```   687   shows "(LBINT x=a..b. f x) = B - A"
```
```   688 proof -
```
```   689   from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx = this
```
```   690   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   691     by (rule order_less_le_trans, rule approx, force)
```
```   692   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   693     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   694   have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
```
```   695     using assms approx
```
```   696     by (auto simp add: less_imp_le min_def max_def
```
```   697              intro!: f continuous_at_imp_continuous_on interval_integral_FTC_finite
```
```   698              intro: has_vector_derivative_at_within)
```
```   699   have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
```
```   700     apply (subst FTCi)
```
```   701     apply (intro tendsto_intros)
```
```   702     using B approx unfolding tendsto_at_iff_sequentially comp_def
```
```   703     apply (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
```
```   704     using A approx unfolding tendsto_at_iff_sequentially comp_def
```
```   705     by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
```
```   706   moreover have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
```
```   707     by (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx f_integrable])
```
```   708   ultimately show ?thesis
```
```   709     by (elim LIMSEQ_unique)
```
```   710 qed
```
```   711
```
```   712 (*
```
```   713   The second Fundamental Theorem of Calculus and existence of antiderivatives on an
```
```   714   einterval.
```
```   715 *)
```
```   716
```
```   717 lemma interval_integral_FTC2:
```
```   718   fixes a b c :: real and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   719   assumes "a \<le> c" "c \<le> b"
```
```   720   and contf: "continuous_on {a..b} f"
```
```   721   fixes x :: real
```
```   722   assumes "a \<le> x" and "x \<le> b"
```
```   723   shows "((\<lambda>u. LBINT y=c..u. f y) has_vector_derivative (f x)) (at x within {a..b})"
```
```   724 proof -
```
```   725   let ?F = "(\<lambda>u. LBINT y=a..u. f y)"
```
```   726   have intf: "set_integrable lborel {a..b} f"
```
```   727     by (rule borel_integrable_atLeastAtMost', rule contf)
```
```   728   have "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
```
```   729     apply (intro integral_has_vector_derivative)
```
```   730     using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by (intro continuous_on_subset [OF contf], auto)
```
```   731   then have "((\<lambda>u. integral {a..u} f) has_vector_derivative (f x)) (at x within {a..b})"
```
```   732     by simp
```
```   733   then have "(?F has_vector_derivative (f x)) (at x within {a..b})"
```
```   734     by (rule has_vector_derivative_weaken)
```
```   735        (auto intro!: assms interval_integral_eq_integral[symmetric] set_integrable_subset [OF intf])
```
```   736   then have "((\<lambda>x. (LBINT y=c..a. f y) + ?F x) has_vector_derivative (f x)) (at x within {a..b})"
```
```   737     by (auto intro!: derivative_eq_intros)
```
```   738   then show ?thesis
```
```   739   proof (rule has_vector_derivative_weaken)
```
```   740     fix u assume "u \<in> {a .. b}"
```
```   741     then show "(LBINT y=c..a. f y) + (LBINT y=a..u. f y) = (LBINT y=c..u. f y)"
```
```   742       using assms
```
```   743       apply (intro interval_integral_sum)
```
```   744       apply (auto simp add: interval_lebesgue_integrable_def simp del: real_scaleR_def)
```
```   745       by (rule set_integrable_subset [OF intf], auto simp add: min_def max_def)
```
```   746   qed (insert assms, auto)
```
```   747 qed
```
```   748
```
```   749 lemma einterval_antiderivative:
```
```   750   fixes a b :: ereal and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   751   assumes "a < b" and contf: "\<And>x :: real. a < x \<Longrightarrow> x < b \<Longrightarrow> isCont f x"
```
```   752   shows "\<exists>F. \<forall>x :: real. a < x \<longrightarrow> x < b \<longrightarrow> (F has_vector_derivative f x) (at x)"
```
```   753 proof -
```
```   754   from einterval_nonempty [OF \<open>a < b\<close>] obtain c :: real where [simp]: "a < c" "c < b"
```
```   755     by (auto simp add: einterval_def)
```
```   756   let ?F = "(\<lambda>u. LBINT y=c..u. f y)"
```
```   757   show ?thesis
```
```   758   proof (rule exI, clarsimp)
```
```   759     fix x :: real
```
```   760     assume [simp]: "a < x" "x < b"
```
```   761     have 1: "a < min c x" by simp
```
```   762     from einterval_nonempty [OF 1] obtain d :: real where [simp]: "a < d" "d < c" "d < x"
```
```   763       by (auto simp add: einterval_def)
```
```   764     have 2: "max c x < b" by simp
```
```   765     from einterval_nonempty [OF 2] obtain e :: real where [simp]: "c < e" "x < e" "e < b"
```
```   766       by (auto simp add: einterval_def)
```
```   767     show "(?F has_vector_derivative f x) (at x)"
```
```   768       (* TODO: factor out the next three lines to has_field_derivative_within_open *)
```
```   769       unfolding has_vector_derivative_def
```
```   770       apply (subst has_derivative_within_open [of _ "{d<..<e}", symmetric], auto)
```
```   771       apply (subst has_vector_derivative_def [symmetric])
```
```   772       apply (rule has_vector_derivative_within_subset [of _ _ _ "{d..e}"])
```
```   773       apply (rule interval_integral_FTC2, auto simp add: less_imp_le)
```
```   774       apply (rule continuous_at_imp_continuous_on)
```
```   775       apply (auto intro!: contf)
```
```   776       apply (rule order_less_le_trans, rule \<open>a < d\<close>, auto)
```
```   777       apply (rule order_le_less_trans) prefer 2
```
```   778       by (rule \<open>e < b\<close>, auto)
```
```   779   qed
```
```   780 qed
```
```   781
```
```   782 (*
```
```   783     The substitution theorem
```
```   784
```
```   785     Once again, three versions: first, for finite intervals, and then two versions for
```
```   786     arbitrary intervals.
```
```   787 *)
```
```   788
```
```   789 lemma interval_integral_substitution_finite:
```
```   790   fixes a b :: real and f :: "real \<Rightarrow> 'a::euclidean_space"
```
```   791   assumes "a \<le> b"
```
```   792   and derivg: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (g has_real_derivative (g' x)) (at x within {a..b})"
```
```   793   and contf : "continuous_on (g ` {a..b}) f"
```
```   794   and contg': "continuous_on {a..b} g'"
```
```   795   shows "LBINT x=a..b. g' x *\<^sub>R f (g x) = LBINT y=g a..g b. f y"
```
```   796 proof-
```
```   797   have v_derivg: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> (g has_vector_derivative (g' x)) (at x within {a..b})"
```
```   798     using derivg unfolding has_field_derivative_iff_has_vector_derivative .
```
```   799   then have contg [simp]: "continuous_on {a..b} g"
```
```   800     by (rule continuous_on_vector_derivative) auto
```
```   801   have 1: "\<And>u. min (g a) (g b) \<le> u \<Longrightarrow> u \<le> max (g a) (g b) \<Longrightarrow>
```
```   802       \<exists>x\<in>{a..b}. u = g x"
```
```   803     apply (case_tac "g a \<le> g b")
```
```   804     apply (auto simp add: min_def max_def less_imp_le)
```
```   805     apply (frule (1) IVT' [of g], auto simp add: assms)
```
```   806     by (frule (1) IVT2' [of g], auto simp add: assms)
```
```   807   from contg \<open>a \<le> b\<close> have "\<exists>c d. g ` {a..b} = {c..d} \<and> c \<le> d"
```
```   808     by (elim continuous_image_closed_interval)
```
```   809   then obtain c d where g_im: "g ` {a..b} = {c..d}" and "c \<le> d" by auto
```
```   810   have "\<exists>F. \<forall>x\<in>{a..b}. (F has_vector_derivative (f (g x))) (at (g x) within (g ` {a..b}))"
```
```   811     apply (rule exI, auto, subst g_im)
```
```   812     apply (rule interval_integral_FTC2 [of c c d])
```
```   813     using \<open>c \<le> d\<close> apply auto
```
```   814     apply (rule continuous_on_subset [OF contf])
```
```   815     using g_im by auto
```
```   816   then guess F ..
```
```   817   then have derivF: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow>
```
```   818     (F has_vector_derivative (f (g x))) (at (g x) within (g ` {a..b}))" by auto
```
```   819   have contf2: "continuous_on {min (g a) (g b)..max (g a) (g b)} f"
```
```   820     apply (rule continuous_on_subset [OF contf])
```
```   821     apply (auto simp add: image_def)
```
```   822     by (erule 1)
```
```   823   have contfg: "continuous_on {a..b} (\<lambda>x. f (g x))"
```
```   824     by (blast intro: continuous_on_compose2 contf contg)
```
```   825   have "LBINT x=a..b. g' x *\<^sub>R f (g x) = F (g b) - F (g a)"
```
```   826     apply (subst interval_integral_Icc, simp add: assms)
```
```   827     apply (rule integral_FTC_atLeastAtMost[of a b "\<lambda>x. F (g x)", OF \<open>a \<le> b\<close>])
```
```   828     apply (rule vector_diff_chain_within[OF v_derivg derivF, unfolded comp_def])
```
```   829     apply (auto intro!: continuous_on_scaleR contg' contfg)
```
```   830     done
```
```   831   moreover have "LBINT y=(g a)..(g b). f y = F (g b) - F (g a)"
```
```   832     apply (rule interval_integral_FTC_finite)
```
```   833     apply (rule contf2)
```
```   834     apply (frule (1) 1, auto)
```
```   835     apply (rule has_vector_derivative_within_subset [OF derivF])
```
```   836     apply (auto simp add: image_def)
```
```   837     by (rule 1, auto)
```
```   838   ultimately show ?thesis by simp
```
```   839 qed
```
```   840
```
```   841 (* TODO: is it possible to lift the assumption here that g' is nonnegative? *)
```
```   842
```
```   843 lemma interval_integral_substitution_integrable:
```
```   844   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and a b u v :: ereal
```
```   845   assumes "a < b"
```
```   846   and deriv_g: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV g x :> g' x"
```
```   847   and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
```
```   848   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
```
```   849   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
```
```   850   and A: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
```
```   851   and B: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
```
```   852   and integrable: "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
```
```   853   and integrable2: "set_integrable lborel (einterval A B) (\<lambda>x. f x)"
```
```   854   shows "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   855 proof -
```
```   856   from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx [simp] = this
```
```   857   note less_imp_le [simp]
```
```   858   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   859     by (rule order_less_le_trans, rule approx, force)
```
```   860   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   861     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   862   have [simp]: "\<And>i. l i < b"
```
```   863     apply (rule order_less_trans) prefer 2
```
```   864     by (rule approx, auto, rule approx)
```
```   865   have [simp]: "\<And>i. a < u i"
```
```   866     by (rule order_less_trans, rule approx, auto, rule approx)
```
```   867   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> l j \<le> l i" by (rule decseqD, rule approx)
```
```   868   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> u i \<le> u j" by (rule incseqD, rule approx)
```
```   869   have g_nondec [simp]: "\<And>x y. a < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < b \<Longrightarrow> g x \<le> g y"
```
```   870     apply (erule DERIV_nonneg_imp_nondecreasing, auto)
```
```   871     apply (rule exI, rule conjI, rule deriv_g)
```
```   872     apply (erule order_less_le_trans, auto)
```
```   873     apply (rule order_le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   874     apply (rule g'_nonneg)
```
```   875     apply (rule less_imp_le, erule order_less_le_trans, auto)
```
```   876     by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   877   have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   878   proof -
```
```   879     have A2: "(\<lambda>i. g (l i)) \<longlonglongrightarrow> A"
```
```   880       using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   881       by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
```
```   882     hence A3: "\<And>i. g (l i) \<ge> A"
```
```   883       by (intro decseq_le, auto simp add: decseq_def)
```
```   884     have B2: "(\<lambda>i. g (u i)) \<longlonglongrightarrow> B"
```
```   885       using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   886       by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
```
```   887     hence B3: "\<And>i. g (u i) \<le> B"
```
```   888       by (intro incseq_le, auto simp add: incseq_def)
```
```   889     show "A \<le> B"
```
```   890       apply (rule order_trans [OF A3 [of 0]])
```
```   891       apply (rule order_trans [OF _ B3 [of 0]])
```
```   892       by auto
```
```   893     { fix x :: real
```
```   894       assume "A < x" and "x < B"
```
```   895       then have "eventually (\<lambda>i. ereal (g (l i)) < x \<and> x < ereal (g (u i))) sequentially"
```
```   896         apply (intro eventually_conj order_tendstoD)
```
```   897         by (rule A2, assumption, rule B2, assumption)
```
```   898       hence "\<exists>i. g (l i) < x \<and> x < g (u i)"
```
```   899         by (simp add: eventually_sequentially, auto)
```
```   900     } note AB = this
```
```   901     show "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   902       apply (auto simp add: einterval_def)
```
```   903       apply (erule (1) AB)
```
```   904       apply (rule order_le_less_trans, rule A3, simp)
```
```   905       apply (rule order_less_le_trans) prefer 2
```
```   906       by (rule B3, simp)
```
```   907   qed
```
```   908   (* finally, the main argument *)
```
```   909   {
```
```   910      fix i
```
```   911      have "(LBINT x=l i.. u i. g' x *\<^sub>R f (g x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```   912         apply (rule interval_integral_substitution_finite, auto)
```
```   913         apply (rule DERIV_subset)
```
```   914         unfolding has_field_derivative_iff_has_vector_derivative[symmetric]
```
```   915         apply (rule deriv_g)
```
```   916         apply (auto intro!: continuous_at_imp_continuous_on contf contg')
```
```   917         done
```
```   918   } note eq1 = this
```
```   919   have "(\<lambda>i. LBINT x=l i..u i. g' x *\<^sub>R f (g x)) \<longlonglongrightarrow> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   920     apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
```
```   921     by (rule assms)
```
```   922   hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```   923     by (simp add: eq1)
```
```   924   have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
```
```   925     apply (auto simp add: incseq_def)
```
```   926     apply (rule order_le_less_trans)
```
```   927     prefer 2 apply (assumption, auto)
```
```   928     by (erule order_less_le_trans, rule g_nondec, auto)
```
```   929   have "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x = A..B. f x)"
```
```   930     apply (subst interval_lebesgue_integral_le_eq, auto simp del: real_scaleR_def)
```
```   931     apply (subst interval_lebesgue_integral_le_eq, rule \<open>A \<le> B\<close>)
```
```   932     apply (subst un, rule set_integral_cont_up, auto simp del: real_scaleR_def)
```
```   933     apply (rule incseq)
```
```   934     apply (subst un [symmetric])
```
```   935     by (rule integrable2)
```
```   936   thus ?thesis by (intro LIMSEQ_unique [OF _ 2])
```
```   937 qed
```
```   938
```
```   939 (* TODO: the last two proofs are only slightly different. Factor out common part?
```
```   940    An alternative: make the second one the main one, and then have another lemma
```
```   941    that says that if f is nonnegative and all the other hypotheses hold, then it is integrable. *)
```
```   942
```
```   943 lemma interval_integral_substitution_nonneg:
```
```   944   fixes f g g':: "real \<Rightarrow> real" and a b u v :: ereal
```
```   945   assumes "a < b"
```
```   946   and deriv_g: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV g x :> g' x"
```
```   947   and contf: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f (g x)"
```
```   948   and contg': "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont g' x"
```
```   949   and f_nonneg: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> 0 \<le> f (g x)" (* TODO: make this AE? *)
```
```   950   and g'_nonneg: "\<And>x. a \<le> ereal x \<Longrightarrow> ereal x \<le> b \<Longrightarrow> 0 \<le> g' x"
```
```   951   and A: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
```
```   952   and B: "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
```
```   953   and integrable_fg: "set_integrable lborel (einterval a b) (\<lambda>x. f (g x) * g' x)"
```
```   954   shows
```
```   955     "set_integrable lborel (einterval A B) f"
```
```   956     "(LBINT x=A..B. f x) = (LBINT x=a..b. (f (g x) * g' x))"
```
```   957 proof -
```
```   958   from einterval_Icc_approximation[OF \<open>a < b\<close>] guess u l . note approx [simp] = this
```
```   959   note less_imp_le [simp]
```
```   960   have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
```
```   961     by (rule order_less_le_trans, rule approx, force)
```
```   962   have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
```
```   963     by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
```
```   964   have [simp]: "\<And>i. l i < b"
```
```   965     apply (rule order_less_trans) prefer 2
```
```   966     by (rule approx, auto, rule approx)
```
```   967   have [simp]: "\<And>i. a < u i"
```
```   968     by (rule order_less_trans, rule approx, auto, rule approx)
```
```   969   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> l j \<le> l i" by (rule decseqD, rule approx)
```
```   970   have [simp]: "\<And>i j. i \<le> j \<Longrightarrow> u i \<le> u j" by (rule incseqD, rule approx)
```
```   971   have g_nondec [simp]: "\<And>x y. a < x \<Longrightarrow> x \<le> y \<Longrightarrow> y < b \<Longrightarrow> g x \<le> g y"
```
```   972     apply (erule DERIV_nonneg_imp_nondecreasing, auto)
```
```   973     apply (rule exI, rule conjI, rule deriv_g)
```
```   974     apply (erule order_less_le_trans, auto)
```
```   975     apply (rule order_le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   976     apply (rule g'_nonneg)
```
```   977     apply (rule less_imp_le, erule order_less_le_trans, auto)
```
```   978     by (rule less_imp_le, rule le_less_trans, subst ereal_less_eq(3), assumption, auto)
```
```   979   have "A \<le> B" and un: "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```   980   proof -
```
```   981     have A2: "(\<lambda>i. g (l i)) \<longlonglongrightarrow> A"
```
```   982       using A apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   983       by (drule_tac x = "\<lambda>i. ereal (l i)" in spec, auto)
```
```   984     hence A3: "\<And>i. g (l i) \<ge> A"
```
```   985       by (intro decseq_le, auto simp add: decseq_def)
```
```   986     have B2: "(\<lambda>i. g (u i)) \<longlonglongrightarrow> B"
```
```   987       using B apply (auto simp add: einterval_def tendsto_at_iff_sequentially comp_def)
```
```   988       by (drule_tac x = "\<lambda>i. ereal (u i)" in spec, auto)
```
```   989     hence B3: "\<And>i. g (u i) \<le> B"
```
```   990       by (intro incseq_le, auto simp add: incseq_def)
```
```   991     show "A \<le> B"
```
```   992       apply (rule order_trans [OF A3 [of 0]])
```
```   993       apply (rule order_trans [OF _ B3 [of 0]])
```
```   994       by auto
```
```   995     { fix x :: real
```
```   996       assume "A < x" and "x < B"
```
```   997       then have "eventually (\<lambda>i. ereal (g (l i)) < x \<and> x < ereal (g (u i))) sequentially"
```
```   998         apply (intro eventually_conj order_tendstoD)
```
```   999         by (rule A2, assumption, rule B2, assumption)
```
```  1000       hence "\<exists>i. g (l i) < x \<and> x < g (u i)"
```
```  1001         by (simp add: eventually_sequentially, auto)
```
```  1002     } note AB = this
```
```  1003     show "einterval A B = (\<Union>i. {g(l i)<..<g(u i)})"
```
```  1004       apply (auto simp add: einterval_def)
```
```  1005       apply (erule (1) AB)
```
```  1006       apply (rule order_le_less_trans, rule A3, simp)
```
```  1007       apply (rule order_less_le_trans) prefer 2
```
```  1008       by (rule B3, simp)
```
```  1009   qed
```
```  1010   (* finally, the main argument *)
```
```  1011   {
```
```  1012      fix i
```
```  1013      have "(LBINT x=l i.. u i. g' x *\<^sub>R f (g x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```  1014         apply (rule interval_integral_substitution_finite, auto)
```
```  1015         apply (rule DERIV_subset, rule deriv_g, auto)
```
```  1016         apply (rule continuous_at_imp_continuous_on, auto, rule contf, auto)
```
```  1017         by (rule continuous_at_imp_continuous_on, auto, rule contg', auto)
```
```  1018      then have "(LBINT x=l i.. u i. (f (g x) * g' x)) = (LBINT y=g (l i)..g (u i). f y)"
```
```  1019        by (simp add: ac_simps)
```
```  1020   } note eq1 = this
```
```  1021   have "(\<lambda>i. LBINT x=l i..u i. f (g x) * g' x)
```
```  1022       \<longlonglongrightarrow> (LBINT x=a..b. f (g x) * g' x)"
```
```  1023     apply (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx])
```
```  1024     by (rule assms)
```
```  1025   hence 2: "(\<lambda>i. (LBINT y=g (l i)..g (u i). f y)) \<longlonglongrightarrow> (LBINT x=a..b. f (g x) * g' x)"
```
```  1026     by (simp add: eq1)
```
```  1027   have incseq: "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
```
```  1028     apply (auto simp add: incseq_def)
```
```  1029     apply (rule order_le_less_trans)
```
```  1030     prefer 2 apply assumption
```
```  1031     apply (rule g_nondec, auto)
```
```  1032     by (erule order_less_le_trans, rule g_nondec, auto)
```
```  1033   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"
```
```  1034     apply (frule (1) IVT' [of g], auto)
```
```  1035     apply (rule continuous_at_imp_continuous_on, auto)
```
```  1036     by (rule DERIV_isCont, rule deriv_g, auto)
```
```  1037   have nonneg_f2: "\<And>x i. g (l i) \<le> x \<Longrightarrow> x \<le> g (u i) \<Longrightarrow> 0 \<le> f x"
```
```  1038     by (frule (1) img, auto, rule f_nonneg, auto)
```
```  1039   have contf_2: "\<And>x i. g (l i) \<le> x \<Longrightarrow> x \<le> g (u i) \<Longrightarrow> isCont f x"
```
```  1040     by (frule (1) img, auto, rule contf, auto)
```
```  1041   have integrable: "set_integrable lborel (\<Union>i. {g (l i)<..<g (u i)}) f"
```
```  1042     apply (rule pos_integrable_to_top, auto simp del: real_scaleR_def)
```
```  1043     apply (rule incseq)
```
```  1044     apply (rule nonneg_f2, erule less_imp_le, erule less_imp_le)
```
```  1045     apply (rule set_integrable_subset)
```
```  1046     apply (rule borel_integrable_atLeastAtMost')
```
```  1047     apply (rule continuous_at_imp_continuous_on)
```
```  1048     apply (clarsimp, erule (1) contf_2, auto)
```
```  1049     apply (erule less_imp_le)+
```
```  1050     using 2 unfolding interval_lebesgue_integral_def
```
```  1051     by auto
```
```  1052   thus "set_integrable lborel (einterval A B) f"
```
```  1053     by (simp add: un)
```
```  1054
```
```  1055   have "(LBINT x=A..B. f x) = (LBINT x=a..b. g' x *\<^sub>R f (g x))"
```
```  1056   proof (rule interval_integral_substitution_integrable)
```
```  1057     show "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
```
```  1058       using integrable_fg by (simp add: ac_simps)
```
```  1059   qed fact+
```
```  1060   then show "(LBINT x=A..B. f x) = (LBINT x=a..b. (f (g x) * g' x))"
```
```  1061     by (simp add: ac_simps)
```
```  1062 qed
```
```  1063
```
```  1064
```
```  1065 syntax
```
```  1066 "_complex_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> complex"
```
```  1067 ("(2CLBINT _. _)" [0,60] 60)
```
```  1068
```
```  1069 translations
```
```  1070 "CLBINT x. f" == "CONST complex_lebesgue_integral CONST lborel (\<lambda>x. f)"
```
```  1071
```
```  1072 syntax
```
```  1073 "_complex_set_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real set \<Rightarrow> real \<Rightarrow> complex"
```
```  1074 ("(3CLBINT _:_. _)" [0,60,61] 60)
```
```  1075
```
```  1076 translations
```
```  1077 "CLBINT x:A. f" == "CONST complex_set_lebesgue_integral CONST lborel A (\<lambda>x. f)"
```
```  1078
```
```  1079 abbreviation complex_interval_lebesgue_integral ::
```
```  1080     "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> complex" where
```
```  1081   "complex_interval_lebesgue_integral M a b f \<equiv> interval_lebesgue_integral M a b f"
```
```  1082
```
```  1083 abbreviation complex_interval_lebesgue_integrable ::
```
```  1084   "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> complex) \<Rightarrow> bool" where
```
```  1085   "complex_interval_lebesgue_integrable M a b f \<equiv> interval_lebesgue_integrable M a b f"
```
```  1086
```
```  1087 syntax
```
```  1088   "_ascii_complex_interval_lebesgue_borel_integral" :: "pttrn \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> real \<Rightarrow> complex"
```
```  1089   ("(4CLBINT _=_.._. _)" [0,60,60,61] 60)
```
```  1090
```
```  1091 translations
```
```  1092   "CLBINT x=a..b. f" == "CONST complex_interval_lebesgue_integral CONST lborel a b (\<lambda>x. f)"
```
```  1093
```
```  1094 lemma interval_integral_norm:
```
```  1095   fixes f :: "real \<Rightarrow> 'a :: {banach, second_countable_topology}"
```
```  1096   shows "interval_lebesgue_integrable lborel a b f \<Longrightarrow> a \<le> b \<Longrightarrow>
```
```  1097     norm (LBINT t=a..b. f t) \<le> LBINT t=a..b. norm (f t)"
```
```  1098   using integral_norm_bound[of lborel "\<lambda>x. indicator (einterval a b) x *\<^sub>R f x"]
```
```  1099   by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def)
```
```  1100
```
```  1101 lemma interval_integral_norm2:
```
```  1102   "interval_lebesgue_integrable lborel a b f \<Longrightarrow>
```
```  1103     norm (LBINT t=a..b. f t) \<le> \<bar>LBINT t=a..b. norm (f t)\<bar>"
```
```  1104 proof (induct a b rule: linorder_wlog)
```
```  1105   case (sym a b) then show ?case
```
```  1106     by (simp add: interval_integral_endpoints_reverse[of a b] interval_integrable_endpoints_reverse[of a b])
```
```  1107 next
```
```  1108   case (le a b)
```
```  1109   then have "\<bar>LBINT t=a..b. norm (f t)\<bar> = LBINT t=a..b. norm (f t)"
```
```  1110     using integrable_norm[of lborel "\<lambda>x. indicator (einterval a b) x *\<^sub>R f x"]
```
```  1111     by (auto simp add: interval_lebesgue_integral_def interval_lebesgue_integrable_def
```
```  1112              intro!: integral_nonneg_AE abs_of_nonneg)
```
```  1113   then show ?case
```
```  1114     using le by (simp add: interval_integral_norm)
```
```  1115 qed
```
```  1116
```
```  1117 (* TODO: should we have a library of facts like these? *)
```
```  1118 lemma integral_cos: "t \<noteq> 0 \<Longrightarrow> LBINT x=a..b. cos (t * x) = sin (t * b) / t - sin (t * a) / t"
```
```  1119   apply (intro interval_integral_FTC_finite continuous_intros)
```
```  1120   by (auto intro!: derivative_eq_intros simp: has_field_derivative_iff_has_vector_derivative[symmetric])
```
```  1121
```
```  1122
```
```  1123 end
```