src/HOL/Analysis/Interval_Integral.thy

tuned names;

(*  Title:      HOL/Analysis/Interval_Integral.thy
    Author:     Jeremy Avigad (CMU), Johannes Hölzl (TUM), Luke Serafin (CMU)

Lebesgue integral over an interval (with endpoints possibly +-\<infinity>)
*)

theory Interval_Integral (*FIX ME rename? Lebesgue  *)
  imports Equivalence_Lebesgue_Henstock_Integration
begin

definition "einterval a b = {x. a < ereal x \<and> ereal x < b}"

lemma einterval_eq[simp]:
  shows einterval_eq_Icc: "einterval (ereal a) (ereal b) = {a <..< b}"
    and einterval_eq_Ici: "einterval (ereal a) \<infinity> = {a <..}"
    and einterval_eq_Iic: "einterval (- \<infinity>) (ereal b) = {..< b}"
    and einterval_eq_UNIV: "einterval (- \<infinity>) \<infinity> = UNIV"
  by (auto simp: einterval_def)

lemma einterval_same: "einterval a a = {}"
  by (auto simp: einterval_def)

lemma einterval_iff: "x \<in> einterval a b \<longleftrightarrow> a < ereal x \<and> ereal x < b"
  by (simp add: einterval_def)

lemma einterval_nonempty: "a < b \<Longrightarrow> \<exists>c. c \<in> einterval a b"
  by (cases a b rule: ereal2_cases, auto simp: einterval_def intro!: dense gt_ex lt_ex)

lemma open_einterval[simp]: "open (einterval a b)"
  by (cases a b rule: ereal2_cases)
     (auto simp: einterval_def intro!: open_Collect_conj open_Collect_less continuous_intros)

lemma borel_einterval[measurable]: "einterval a b \<in> sets borel"
  unfolding einterval_def by measurable

subsection \<open>Approximating a (possibly infinite) interval\<close>

lemma filterlim_sup1: "(LIM x F. f x :> G1) \<Longrightarrow> (LIM x F. f x :> (sup G1 G2))"
 unfolding filterlim_def by (auto intro: le_supI1)

lemma ereal_incseq_approx:
  fixes a b :: ereal
  assumes "a < b"
  obtains X :: "nat \<Rightarrow> real" where "incseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X \<longlonglongrightarrow> b"
proof (cases b)
  case PInf
  with \<open>a < b\<close> have "a = -\<infinity> \<or> (\<exists>r. a = ereal r)"
    by (cases a) auto
  moreover have "(\<lambda>x. ereal (real (Suc x))) \<longlonglongrightarrow> \<infinity>"
    by (simp add: Lim_PInfty filterlim_sequentially_Suc) (metis le_SucI of_nat_Suc of_nat_mono order_trans real_arch_simple)
  moreover have "\<And>r. (\<lambda>x. ereal (r + real (Suc x))) \<longlonglongrightarrow> \<infinity>"
    by (simp add: filterlim_sequentially_Suc Lim_PInfty) (metis add.commute diff_le_eq nat_ceiling_le_eq)
  ultimately show thesis
    by (intro that[of "\<lambda>i. real_of_ereal a + Suc i"])
       (auto simp: incseq_def PInf)
next
  case (real b')
  define d where "d = b' - (if a = -\<infinity> then b' - 1 else real_of_ereal a)"
  with \<open>a < b\<close> have a': "0 < d"
    by (cases a) (auto simp: real)
  moreover
  have "\<And>i r. r < b' \<Longrightarrow> (b' - r) * 1 < (b' - r) * real (Suc (Suc i))"
    by (intro mult_strict_left_mono) auto
  with \<open>a < b\<close> a' have "\<And>i. a < ereal (b' - d / real (Suc (Suc i)))"
    by (cases a) (auto simp: real d_def field_simps)
  moreover
  have "(\<lambda>i. b' - d / real i) \<longlonglongrightarrow> b'"
    by (force intro: tendsto_eq_intros tendsto_divide_0[OF tendsto_const] filterlim_sup1
              simp: at_infinity_eq_at_top_bot filterlim_real_sequentially)
  then have "(\<lambda>i. b' - d / Suc (Suc i)) \<longlonglongrightarrow> b'"
    by (blast intro: dest: filterlim_sequentially_Suc [THEN iffD2])
  ultimately show thesis
    by (intro that[of "\<lambda>i. b' - d / Suc (Suc i)"])
       (auto simp: real incseq_def intro!: divide_left_mono)
qed (use \<open>a < b\<close> in auto)

lemma ereal_decseq_approx:
  fixes a b :: ereal
  assumes "a < b"
  obtains X :: "nat \<Rightarrow> real" where
    "decseq X" "\<And>i. a < X i" "\<And>i. X i < b" "X \<longlonglongrightarrow> a"
proof -
  have "-b < -a" using \<open>a < b\<close> by simp
  from ereal_incseq_approx[OF this] obtain X where
    "incseq X"
    "\<And>i. - b < ereal (X i)"
    "\<And>i. ereal (X i) < - a"
    "(\<lambda>x. ereal (X x)) \<longlonglongrightarrow> - a"
    by auto
  then show thesis
    apply (intro that[of "\<lambda>i. - X i"])
    apply (auto simp: decseq_def incseq_def simp flip: uminus_ereal.simps)
    apply (metis ereal_minus_less_minus ereal_uminus_uminus ereal_Lim_uminus)+
    done
qed

proposition einterval_Icc_approximation:
  fixes a b :: ereal
  assumes "a < b"
  obtains u l :: "nat \<Rightarrow> real" where
    "einterval a b = (\<Union>i. {l i .. u i})"
    "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
proof -
  from dense[OF \<open>a < b\<close>] obtain c where "a < c" "c < b" by safe
  from ereal_incseq_approx[OF \<open>c < b\<close>] obtain u where u:
    "incseq u"
    "\<And>i. c < ereal (u i)"
    "\<And>i. ereal (u i) < b"
    "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> b"
    by auto
  from ereal_decseq_approx[OF \<open>a < c\<close>] obtain l where l:
    "decseq l"
    "\<And>i. a < ereal (l i)"
    "\<And>i. ereal (l i) < c"
    "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"
    by auto
  have "einterval a b = (\<Union>i. {l i .. u i})"
  proof (auto simp: einterval_iff)
    fix x assume "a < ereal x" "ereal x < b"
    have "eventually (\<lambda>i. ereal (l i) < ereal x) sequentially"
      using l(4) \<open>a < ereal x\<close> by (rule order_tendstoD)
    moreover
    have "eventually (\<lambda>i. ereal x < ereal (u i)) sequentially"
      using u(4) \<open>ereal x< b\<close> by (rule order_tendstoD)
    ultimately have "eventually (\<lambda>i. l i < x \<and> x < u i) sequentially"
      by eventually_elim auto
    then show "\<exists>i. l i \<le> x \<and> x \<le> u i"
      by (auto intro: less_imp_le simp: eventually_sequentially)
  next
    fix x i assume "l i \<le> x" "x \<le> u i"
    with \<open>a < ereal (l i)\<close> \<open>ereal (u i) < b\<close>
    show "a < ereal x" "ereal x < b"
      by (auto simp flip: ereal_less_eq(3))
  qed
  moreover { fix i from less_trans[OF \<open>l i < c\<close> \<open>c < u i\<close>] have "l i < u i" by simp }
  ultimately show thesis
    by (simp add: l that u)
qed

(* TODO: in this definition, it would be more natural if einterval a b included a and b when
   they are real. *)
definition\<^marker>\<open>tag important\<close> interval_lebesgue_integral :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a) \<Rightarrow> 'a::{banach, second_countable_topology}" where
  "interval_lebesgue_integral M a b f =
    (if a \<le> b then (LINT x:einterval a b|M. f x) else - (LINT x:einterval b a|M. f x))"

syntax
  "_ascii_interval_lebesgue_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real measure \<Rightarrow> real \<Rightarrow> real"
  ("(5LINT _=_.._|_. _)" [0,60,60,61,100] 60)

translations
  "LINT x=a..b|M. f" == "CONST interval_lebesgue_integral M a b (\<lambda>x. f)"

definition\<^marker>\<open>tag important\<close> interval_lebesgue_integrable :: "real measure \<Rightarrow> ereal \<Rightarrow> ereal \<Rightarrow> (real \<Rightarrow> 'a::{banach, second_countable_topology}) \<Rightarrow> bool" where
  "interval_lebesgue_integrable M a b f =
    (if a \<le> b then set_integrable M (einterval a b) f else set_integrable M (einterval b a) f)"

syntax
  "_ascii_interval_lebesgue_borel_integral" :: "pttrn \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"
  ("(4LBINT _=_.._. _)" [0,60,60,61] 60)

translations
  "LBINT x=a..b. f" == "CONST interval_lebesgue_integral CONST lborel a b (\<lambda>x. f)"

subsection\<open>Basic properties of integration over an interval\<close>

lemma interval_lebesgue_integral_cong:
  "a \<le> b \<Longrightarrow> (\<And>x. x \<in> einterval a b \<Longrightarrow> f x = g x) \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
    interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
  by (auto intro: set_lebesgue_integral_cong simp: interval_lebesgue_integral_def)

lemma interval_lebesgue_integral_cong_AE:
  "f \<in> borel_measurable M \<Longrightarrow> g \<in> borel_measurable M \<Longrightarrow>
    a \<le> b \<Longrightarrow> AE x \<in> einterval a b in M. f x = g x \<Longrightarrow> einterval a b \<in> sets M \<Longrightarrow>
    interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
  by (auto intro: set_lebesgue_integral_cong_AE simp: interval_lebesgue_integral_def)

lemma interval_integrable_mirror:
  shows "interval_lebesgue_integrable lborel a b (\<lambda>x. f (-x)) \<longleftrightarrow>
    interval_lebesgue_integrable lborel (-b) (-a) f"
proof -
  have *: "indicator (einterval a b) (- x) = (indicator (einterval (-b) (-a)) x :: real)"
    for a b :: ereal and x :: real
    by (cases a b rule: ereal2_cases) (auto simp: einterval_def split: split_indicator)
  show ?thesis
    unfolding interval_lebesgue_integrable_def
    using lborel_integrable_real_affine_iff[symmetric, of "-1" "\<lambda>x. indicator (einterval _ _) x *\<^sub>R f x" 0]
    by (simp add: * set_integrable_def)
qed

lemma interval_lebesgue_integral_add [intro, simp]:
  fixes M a b f
  assumes "interval_lebesgue_integrable M a b f" "interval_lebesgue_integrable M a b g"
  shows "interval_lebesgue_integrable M a b (\<lambda>x. f x + g x)"
    and "interval_lebesgue_integral M a b (\<lambda>x. f x + g x) =
         interval_lebesgue_integral M a b f + interval_lebesgue_integral M a b g"
using assms by (auto simp: interval_lebesgue_integral_def interval_lebesgue_integrable_def
    field_simps)

lemma interval_lebesgue_integral_diff [intro, simp]:
  fixes M a b f
  assumes "interval_lebesgue_integrable M a b f"
    "interval_lebesgue_integrable M a b g"
  shows "interval_lebesgue_integrable M a b (\<lambda>x. f x - g x)" and
    "interval_lebesgue_integral M a b (\<lambda>x. f x - g x) =
   interval_lebesgue_integral M a b f - interval_lebesgue_integral M a b g"
using assms by (auto simp: interval_lebesgue_integral_def interval_lebesgue_integrable_def
    field_simps)

lemma interval_lebesgue_integrable_mult_right [intro, simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
  shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
    interval_lebesgue_integrable M a b (\<lambda>x. c * f x)"
  by (simp add: interval_lebesgue_integrable_def)

lemma interval_lebesgue_integrable_mult_left [intro, simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
  shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
    interval_lebesgue_integrable M a b (\<lambda>x. f x * c)"
  by (simp add: interval_lebesgue_integrable_def)

lemma interval_lebesgue_integrable_divide [intro, simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
  shows "(c \<noteq> 0 \<Longrightarrow> interval_lebesgue_integrable M a b f) \<Longrightarrow>
    interval_lebesgue_integrable M a b (\<lambda>x. f x / c)"
  by (simp add: interval_lebesgue_integrable_def)

lemma interval_lebesgue_integral_mult_right [simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
  shows "interval_lebesgue_integral M a b (\<lambda>x. c * f x) =
    c * interval_lebesgue_integral M a b f"
  by (simp add: interval_lebesgue_integral_def)

lemma interval_lebesgue_integral_mult_left [simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, second_countable_topology}"
  shows "interval_lebesgue_integral M a b (\<lambda>x. f x * c) =
    interval_lebesgue_integral M a b f * c"
  by (simp add: interval_lebesgue_integral_def)

lemma interval_lebesgue_integral_divide [simp]:
  fixes M a b c and f :: "real \<Rightarrow> 'a::{banach, real_normed_field, field, second_countable_topology}"
  shows "interval_lebesgue_integral M a b (\<lambda>x. f x / c) =
    interval_lebesgue_integral M a b f / c"
  by (simp add: interval_lebesgue_integral_def)

lemma interval_lebesgue_integral_uminus:
  "interval_lebesgue_integral M a b (\<lambda>x. - f x) = - interval_lebesgue_integral M a b f"
  by (auto simp: interval_lebesgue_integral_def interval_lebesgue_integrable_def set_lebesgue_integral_def)

lemma interval_lebesgue_integral_of_real:
  "interval_lebesgue_integral M a b (\<lambda>x. complex_of_real (f x)) =
    of_real (interval_lebesgue_integral M a b f)"
  unfolding interval_lebesgue_integral_def
  by (auto simp: interval_lebesgue_integral_def set_integral_complex_of_real)

lemma interval_lebesgue_integral_le_eq:
  fixes a b f
  assumes "a \<le> b"
  shows "interval_lebesgue_integral M a b f = (LINT x : einterval a b | M. f x)"
  using assms by (auto simp: interval_lebesgue_integral_def)

lemma interval_lebesgue_integral_gt_eq:
  fixes a b f
  assumes "a > b"
  shows "interval_lebesgue_integral M a b f = -(LINT x : einterval b a | M. f x)"
using assms by (auto simp: interval_lebesgue_integral_def less_imp_le einterval_def)

lemma interval_lebesgue_integral_gt_eq':
  fixes a b f
  assumes "a > b"
  shows "interval_lebesgue_integral M a b f = - interval_lebesgue_integral M b a f"
using assms by (auto simp: interval_lebesgue_integral_def less_imp_le einterval_def)

lemma interval_integral_endpoints_same [simp]: "(LBINT x=a..a. f x) = 0"
  by (simp add: interval_lebesgue_integral_def set_lebesgue_integral_def einterval_same)

lemma interval_integral_endpoints_reverse: "(LBINT x=a..b. f x) = -(LBINT x=b..a. f x)"
  by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integral_def set_lebesgue_integral_def einterval_same)

lemma interval_integrable_endpoints_reverse:
  "interval_lebesgue_integrable lborel a b f \<longleftrightarrow>
    interval_lebesgue_integrable lborel b a f"
  by (cases a b rule: linorder_cases) (auto simp: interval_lebesgue_integrable_def einterval_same)

lemma interval_integral_reflect:
  "(LBINT x=a..b. f x) = (LBINT x=-b..-a. f (-x))"
proof (induct a b rule: linorder_wlog)
  case (sym a b) then show ?case
    by (auto simp: interval_lebesgue_integral_def interval_integrable_endpoints_reverse
             split: if_split_asm)
next
  case (le a b) 
  have "(LBINT x:{x. - x \<in> einterval a b}. f (- x)) = (LBINT x:einterval (- b) (- a). f (- x))"
    unfolding interval_lebesgue_integrable_def set_lebesgue_integral_def einterval_def
    by (metis (lifting) ereal_less_uminus_reorder ereal_uminus_less_reorder indicator_simps mem_Collect_eq uminus_ereal.simps(1))
  then show ?case
    unfolding interval_lebesgue_integral_def 
    by (subst set_integral_reflect) (simp add: le)
qed

lemma interval_lebesgue_integral_0_infty:
  "interval_lebesgue_integrable M 0 \<infinity> f \<longleftrightarrow> set_integrable M {0<..} f"
  "interval_lebesgue_integral M 0 \<infinity> f = (LINT x:{0<..}|M. f x)"
  unfolding zero_ereal_def
  by (auto simp: interval_lebesgue_integral_le_eq interval_lebesgue_integrable_def)

lemma interval_integral_to_infinity_eq: "(LINT x=ereal a..\<infinity> | M. f x) = (LINT x : {a<..} | M. f x)"
  unfolding interval_lebesgue_integral_def by auto

proposition interval_integrable_to_infinity_eq: "(interval_lebesgue_integrable M a \<infinity> f) =
  (set_integrable M {a<..} f)"
  unfolding interval_lebesgue_integrable_def by auto

subsection\<open>Basic properties of integration over an interval wrt lebesgue measure\<close>

lemma interval_integral_zero [simp]:
  fixes a b :: ereal
  shows "(LBINT x=a..b. 0) = 0"
unfolding interval_lebesgue_integral_def set_lebesgue_integral_def einterval_eq
by simp

lemma interval_integral_const [intro, simp]:
  fixes a b c :: real
  shows "interval_lebesgue_integrable lborel a b (\<lambda>x. c)" and "(LBINT x=a..b. c) = c * (b - a)"
  unfolding interval_lebesgue_integral_def interval_lebesgue_integrable_def einterval_eq
  by (auto simp: less_imp_le field_simps measure_def set_integrable_def set_lebesgue_integral_def)

lemma interval_integral_cong_AE:
  assumes [measurable]: "f \<in> borel_measurable borel" "g \<in> borel_measurable borel"
  assumes "AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x"
  shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
  using assms
  by (auto simp: interval_lebesgue_integral_def max_def min_def intro!: set_lebesgue_integral_cong_AE)

lemma interval_integral_cong:
  assumes "\<And>x. x \<in> einterval (min a b) (max a b) \<Longrightarrow> f x = g x"
  shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
  using assms by (simp add: interval_lebesgue_integral_def set_lebesgue_integral_cong)

lemma interval_lebesgue_integrable_cong_AE:
    "f \<in> borel_measurable lborel \<Longrightarrow> g \<in> borel_measurable lborel \<Longrightarrow>
    AE x \<in> einterval (min a b) (max a b) in lborel. f x = g x \<Longrightarrow>
    interval_lebesgue_integrable lborel a b f = interval_lebesgue_integrable lborel a b g"
  apply (simp add: interval_lebesgue_integrable_def)
  apply (intro conjI impI set_integrable_cong_AE)
  apply (auto simp: min_def max_def)
  done

lemma interval_integrable_abs_iff:
  fixes f :: "real \<Rightarrow> real"
  shows  "f \<in> borel_measurable lborel \<Longrightarrow>
    interval_lebesgue_integrable lborel a b (\<lambda>x. \<bar>f x\<bar>) = interval_lebesgue_integrable lborel a b f"
  unfolding interval_lebesgue_integrable_def
  by (simp add: set_integrable_abs_iff')

lemma interval_integral_Icc:
  fixes a b :: real
  shows "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..b}. f x)"
  by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
           simp add: interval_lebesgue_integral_def)

lemma interval_integral_Icc':
  "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a \<le> ereal x \<and> ereal x \<le> b}. f x)"
  by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
           simp add: interval_lebesgue_integral_def einterval_iff)

lemma interval_integral_Ioc:
  "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a<..b}. f x)"
  by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
           simp add: interval_lebesgue_integral_def einterval_iff)

(* TODO: other versions as well? *) (* Yes: I need the Icc' version. *)
lemma interval_integral_Ioc':
  "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {x. a < ereal x \<and> ereal x \<le> b}. f x)"
  by (auto intro!: set_integral_discrete_difference[where X="{real_of_ereal a, real_of_ereal b}"]
           simp add: interval_lebesgue_integral_def einterval_iff)

lemma interval_integral_Ico:
  "a \<le> b \<Longrightarrow> (LBINT x=a..b. f x) = (LBINT x : {a..<b}. f x)"
  by (auto intro!: set_integral_discrete_difference[where X="{a, b}"]
           simp add: interval_lebesgue_integral_def einterval_iff)

lemma interval_integral_Ioi:
  "\<bar>a\<bar> < \<infinity> \<Longrightarrow> (LBINT x=a..\<infinity>. f x) = (LBINT x : {real_of_ereal a <..}. f x)"
  by (auto simp: interval_lebesgue_integral_def einterval_iff)

lemma interval_integral_Ioo:
  "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)"
  by (auto simp: interval_lebesgue_integral_def einterval_iff)

lemma interval_integral_discrete_difference:
  fixes f :: "real \<Rightarrow> 'b::{banach, second_countable_topology}" and a b :: ereal
  assumes "countable X"
  and eq: "\<And>x. a \<le> b \<Longrightarrow> a < x \<Longrightarrow> x < b \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
  and anti_eq: "\<And>x. b \<le> a \<Longrightarrow> b < x \<Longrightarrow> x < a \<Longrightarrow> x \<notin> X \<Longrightarrow> f x = g x"
  assumes "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0" "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"
  shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
  unfolding interval_lebesgue_integral_def set_lebesgue_integral_def
  apply (intro if_cong refl arg_cong[where f="\<lambda>x. - x"] integral_discrete_difference[of X] assms)
  apply (auto simp: eq anti_eq einterval_iff split: split_indicator)
  done

lemma interval_integral_sum:
  fixes a b c :: ereal
  assumes integrable: "interval_lebesgue_integrable lborel (min a (min b c)) (max a (max b c)) f"
  shows "(LBINT x=a..b. f x) + (LBINT x=b..c. f x) = (LBINT x=a..c. f x)"
proof -
  let ?I = "\<lambda>a b. LBINT x=a..b. f x"
  { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
    then have ord: "a \<le> b" "b \<le> c" "a \<le> c" and f': "set_integrable lborel (einterval a c) f"
      by (auto simp: interval_lebesgue_integrable_def)
    then have f: "set_borel_measurable borel (einterval a c) f"
      unfolding set_integrable_def set_borel_measurable_def
      by (drule_tac borel_measurable_integrable) simp
    have "(LBINT x:einterval a c. f x) = (LBINT x:einterval a b \<union> einterval b c. f x)"
    proof (rule set_integral_cong_set)
      show "AE x in lborel. (x \<in> einterval a b \<union> einterval b c) = (x \<in> einterval a c)"
        using AE_lborel_singleton[of "real_of_ereal b"] ord
        by (cases a b c rule: ereal3_cases) (auto simp: einterval_iff)
      show "set_borel_measurable lborel (einterval a c) f" "set_borel_measurable lborel (einterval a b \<union> einterval b c) f"
        unfolding set_borel_measurable_def
        using ord by (auto simp: einterval_iff intro!: set_borel_measurable_subset[OF f, unfolded set_borel_measurable_def])
    qed
    also have "\<dots> = (LBINT x:einterval a b. f x) + (LBINT x:einterval b c. f x)"
      using ord
      by (intro set_integral_Un_AE) (auto intro!: set_integrable_subset[OF f'] simp: einterval_iff not_less)
    finally have "?I a b + ?I b c = ?I a c"
      using ord by (simp add: interval_lebesgue_integral_def)
  } note 1 = this
  { fix a b c :: ereal assume "interval_lebesgue_integrable lborel a c f" "a \<le> b" "b \<le> c"
    from 1[OF this] have "?I b c + ?I a b = ?I a c"
      by (metis add.commute)
  } note 2 = this
  have 3: "\<And>a b. b \<le> a \<Longrightarrow> (LBINT x=a..b. f x) = - (LBINT x=b..a. f x)"
    by (rule interval_integral_endpoints_reverse)
  show ?thesis
    using integrable
    apply (cases a b b c a c rule: linorder_le_cases[case_product linorder_le_cases linorder_cases])
    apply simp_all (* remove some looping cases *)
    by (simp_all add: min_absorb1 min_absorb2 max_absorb1 max_absorb2 field_simps 1 2 3)
qed

lemma interval_integrable_isCont:
  fixes a b and f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
  shows "(\<And>x. min a b \<le> x \<Longrightarrow> x \<le> max a b \<Longrightarrow> isCont f x) \<Longrightarrow>
    interval_lebesgue_integrable lborel a b f"
proof (induct a b rule: linorder_wlog)
  case (le a b) then show ?case
    unfolding interval_lebesgue_integrable_def set_integrable_def
    by (auto simp: interval_lebesgue_integrable_def
        intro!: set_integrable_subset[unfolded set_integrable_def, OF borel_integrable_compact[of "{a .. b}"]]
        continuous_at_imp_continuous_on)
qed (auto intro: interval_integrable_endpoints_reverse[THEN iffD1])

lemma interval_integrable_continuous_on:
  fixes a b :: real and f
  assumes "a \<le> b" and "continuous_on {a..b} f"
  shows "interval_lebesgue_integrable lborel a b f"
using assms unfolding interval_lebesgue_integrable_def apply simp
  by (rule set_integrable_subset, rule borel_integrable_atLeastAtMost' [of a b], auto)

lemma interval_integral_eq_integral:
  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  shows "a \<le> b \<Longrightarrow> set_integrable lborel {a..b} f \<Longrightarrow> LBINT x=a..b. f x = integral {a..b} f"
  by (subst interval_integral_Icc, simp) (rule set_borel_integral_eq_integral)

lemma interval_integral_eq_integral':
  fixes f :: "real \<Rightarrow> 'a::euclidean_space"
  shows "a \<le> b \<Longrightarrow> set_integrable lborel (einterval a b) f \<Longrightarrow> LBINT x=a..b. f x = integral (einterval a b) f"
  by (subst interval_lebesgue_integral_le_eq, simp) (rule set_borel_integral_eq_integral)


subsection\<open>General limit approximation arguments\<close>

proposition interval_integral_Icc_approx_nonneg:
  fixes a b :: ereal
  assumes "a < b"
  fixes u l :: "nat \<Rightarrow> real"
  assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
    "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
  fixes f :: "real \<Rightarrow> real"
  assumes f_integrable: "\<And>i. set_integrable lborel {l i..u i} f"
  assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
  assumes f_measurable: "set_borel_measurable lborel (einterval a b) f"
  assumes lbint_lim: "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> C"
  shows
    "set_integrable lborel (einterval a b) f"
    "(LBINT x=a..b. f x) = C"
proof -
  have 1 [unfolded set_integrable_def]: "\<And>i. set_integrable lborel {l i..u i} f" by (rule f_integrable)
  have 2: "AE x in lborel. mono (\<lambda>n. indicator {l n..u n} x *\<^sub>R f x)"
  proof -
     from f_nonneg have "AE x in lborel. \<forall>i. l i \<le> x \<longrightarrow> x \<le> u i \<longrightarrow> 0 \<le> f x"
      by eventually_elim
         (metis approx(5) approx(6) dual_order.strict_trans1 ereal_less_eq(3) le_less_trans)
    then show ?thesis
      apply eventually_elim
      apply (auto simp: mono_def split: split_indicator)
      apply (metis approx(3) decseqD order_trans)
      apply (metis approx(2) incseqD order_trans)
      done
  qed
  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"
  proof -
    { fix x i assume "l i \<le> x" "x \<le> u i"
      then have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
        apply (auto simp: eventually_sequentially intro!: exI[of _ i])
        apply (metis approx(3) decseqD order_trans)
        apply (metis approx(2) incseqD order_trans)
        done
      then have "eventually (\<lambda>i. f x * indicator {l i..u i} x = f x) sequentially"
        by eventually_elim auto }
    then show ?thesis
      unfolding approx(1) by (auto intro!: AE_I2 tendsto_eventually split: split_indicator)
  qed
  have 4: "(\<lambda>i. \<integral> x. indicator {l i..u i} x *\<^sub>R f x \<partial>lborel) \<longlonglongrightarrow> C"
    using lbint_lim by (simp add: interval_integral_Icc [unfolded set_lebesgue_integral_def] approx less_imp_le)
  have 5: "(\<lambda>x. indicat_real (einterval a b) x *\<^sub>R f x) \<in> borel_measurable lborel"
    using f_measurable set_borel_measurable_def by blast
  have "(LBINT x=a..b. f x) = lebesgue_integral lborel (\<lambda>x. indicator (einterval a b) x *\<^sub>R f x)"
    using assms by (simp add: interval_lebesgue_integral_def set_lebesgue_integral_def less_imp_le)
  also have "\<dots> = C"
    by (rule integral_monotone_convergence [OF 1 2 3 4 5])
  finally show "(LBINT x=a..b. f x) = C" .
  show "set_integrable lborel (einterval a b) f"
    unfolding set_integrable_def
    by (rule integrable_monotone_convergence[OF 1 2 3 4 5])
qed

proposition interval_integral_Icc_approx_integrable:
  fixes u l :: "nat \<Rightarrow> real" and a b :: ereal
  fixes f :: "real \<Rightarrow> 'a::{banach, second_countable_topology}"
  assumes "a < b"
  assumes  approx: "einterval a b = (\<Union>i. {l i .. u i})"
    "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b"
  assumes f_integrable: "set_integrable lborel (einterval a b) f"
  shows "(\<lambda>i. LBINT x=l i.. u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
proof -
  have "(\<lambda>i. LBINT x:{l i.. u i}. f x) \<longlonglongrightarrow> (LBINT x:einterval a b. f x)"
    unfolding set_lebesgue_integral_def
  proof (rule integral_dominated_convergence)
    show "integrable lborel (\<lambda>x. norm (indicator (einterval a b) x *\<^sub>R f x))"
      using f_integrable integrable_norm set_integrable_def by blast
    show "(\<lambda>x. indicat_real (einterval a b) x *\<^sub>R f x) \<in> borel_measurable lborel"
      using f_integrable by (simp add: set_integrable_def)
    then show "\<And>i. (\<lambda>x. indicat_real {l i..u i} x *\<^sub>R f x) \<in> borel_measurable lborel"
      by (rule set_borel_measurable_subset [unfolded set_borel_measurable_def]) (auto simp: approx)
    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)"
      by (intro AE_I2) (auto simp: approx split: split_indicator)
    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"
    proof (intro AE_I2 tendsto_intros tendsto_eventually)
      fix x
      { fix i assume "l i \<le> x" "x \<le> u i"
        with \<open>incseq u\<close>[THEN incseqD, of i] \<open>decseq l\<close>[THEN decseqD, of i]
        have "eventually (\<lambda>i. l i \<le> x \<and> x \<le> u i) sequentially"
          by (auto simp: eventually_sequentially decseq_def incseq_def intro: order_trans) }
      then show "eventually (\<lambda>xa. indicator {l xa..u xa} x = (indicator (einterval a b) x::real)) sequentially"
        using approx order_tendstoD(2)[OF \<open>l \<longlonglongrightarrow> a\<close>, of x] order_tendstoD(1)[OF \<open>u \<longlonglongrightarrow> b\<close>, of x]
        by (auto split: split_indicator)
    qed
  qed
  with \<open>a < b\<close> \<open>\<And>i. l i < u i\<close> show ?thesis
    by (simp add: interval_lebesgue_integral_le_eq[symmetric] interval_integral_Icc less_imp_le)
qed

subsection\<open>A slightly stronger Fundamental Theorem of Calculus\<close>

text\<open>Three versions: first, for finite intervals, and then two versions for
    arbitrary intervals.\<close>

(*
  TODO: make the older versions corollaries of these (using continuous_at_imp_continuous_on, etc.)
*)

lemma interval_integral_FTC_finite:
  fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: real
  assumes f: "continuous_on {min a b..max a b} f"
  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
    {min a b..max a b})"
  shows "(LBINT x=a..b. f x) = F b - F a"
proof (cases "a \<le> b")
  case True
  have "(LBINT x=a..b. f x) = (LBINT x. indicat_real {a..b} x *\<^sub>R f x)"
    by (simp add: True interval_integral_Icc set_lebesgue_integral_def)
  also have "\<dots> = F b - F a"
  proof (rule integral_FTC_atLeastAtMost [OF True])
    show "continuous_on {a..b} f"
      using True f by linarith
    show "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"
      by (metis F True max.commute max_absorb1 min_def)
  qed
  finally show ?thesis .
next
  case False
  then have "b \<le> a"
    by simp
  have "- interval_lebesgue_integral lborel (ereal b) (ereal a) f = - (LBINT x. indicat_real {b..a} x *\<^sub>R f x)"
    by (simp add: \<open>b \<le> a\<close> interval_integral_Icc set_lebesgue_integral_def)
  also have "\<dots> = F b - F a"
  proof (subst integral_FTC_atLeastAtMost [OF \<open>b \<le> a\<close>])
    show "continuous_on {b..a} f"
      using False f by linarith
    show "\<And>x. \<lbrakk>b \<le> x; x \<le> a\<rbrakk>
         \<Longrightarrow> (F has_vector_derivative f x) (at x within {b..a})"
      by (metis F False max_def min_def)
  qed auto
  finally show ?thesis
    by (metis interval_integral_endpoints_reverse)
qed


lemma interval_integral_FTC_nonneg:
  fixes f F :: "real \<Rightarrow> real" and a b :: ereal
  assumes "a < b"
  assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> DERIV F x :> f x"
  assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
  assumes f_nonneg: "AE x in lborel. a < ereal x \<longrightarrow> ereal x < b \<longrightarrow> 0 \<le> f x"
  assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
  assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
  shows
    "set_integrable lborel (einterval a b) f"
    "(LBINT x=a..b. f x) = B - A"
proof -
  obtain u l where approx:
    "einterval a b = (\<Union>i. {l i .. u i})"
    "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b" 
    by (blast intro: einterval_Icc_approximation[OF \<open>a < b\<close>])
  have aless[simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
    by (rule order_less_le_trans, rule approx, force)
  have lessb[simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
    by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
  have cf: "\<And>i. continuous_on {min (l i) (u i)..max (l i) (u i)} f"
    using approx f by (intro continuous_at_imp_continuous_on strip) auto
  have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
    apply (intro interval_integral_FTC_finite cf DERIV_subset [OF F])
    by (smt (verit) F aless approx(4) has_real_derivative_iff_has_vector_derivative has_vector_derivative_at_within lessb)
  have 1: "\<And>i. set_integrable lborel {l i..u i} f"
    by (meson aless lessb assms(3) atLeastAtMost_iff borel_integrable_atLeastAtMost' continuous_at_imp_continuous_on)
  have 2: "set_borel_measurable lborel (einterval a b) f"
    unfolding set_borel_measurable_def
    by (auto simp del: real_scaleR_def intro!: borel_measurable_continuous_on_indicator
             simp: continuous_on_eq_continuous_at einterval_iff f)
  have "(\<lambda>x. F (l x)) \<longlonglongrightarrow> A"
    using A approx unfolding tendsto_at_iff_sequentially comp_def
    by (force elim!: allE[of _ "\<lambda>i. ereal (l i)"])
  moreover have "(\<lambda>x. F (u x)) \<longlonglongrightarrow> B"
    using B approx unfolding tendsto_at_iff_sequentially comp_def
    by (force elim!: allE[of _ "\<lambda>i. ereal (u i)"])
  ultimately have 3: "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
    by (simp add: FTCi tendsto_diff)
  show "(LBINT x=a..b. f x) = B - A"
    by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
  show "set_integrable lborel (einterval a b) f"
    by (rule interval_integral_Icc_approx_nonneg [OF \<open>a < b\<close> approx 1 f_nonneg 2 3])
qed

theorem interval_integral_FTC_integrable:
  fixes f F :: "real \<Rightarrow> 'a::euclidean_space" and a b :: ereal
  assumes "a < b"
  assumes F: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> (F has_vector_derivative f x) (at x)"
  assumes f: "\<And>x. a < ereal x \<Longrightarrow> ereal x < b \<Longrightarrow> isCont f x"
  assumes f_integrable: "set_integrable lborel (einterval a b) f"
  assumes A: "((F \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"
  assumes B: "((F \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"
  shows "(LBINT x=a..b. f x) = B - A"
proof -
  obtain u l where approx:
    "einterval a b = (\<Union>i. {l i .. u i})"
    "incseq u" "decseq l" "\<And>i. l i < u i" "\<And>i. a < l i" "\<And>i. u i < b"
    "l \<longlonglongrightarrow> a" "u \<longlonglongrightarrow> b" 
    by (blast intro: einterval_Icc_approximation[OF \<open>a < b\<close>])
  have [simp]: "\<And>x i. l i \<le> x \<Longrightarrow> a < ereal x"
    by (rule order_less_le_trans, rule approx, force)
  have [simp]: "\<And>x i. x \<le> u i \<Longrightarrow> ereal x < b"
    by (rule order_le_less_trans, subst ereal_less_eq(3), assumption, rule approx)
  have FTCi: "\<And>i. (LBINT x=l i..u i. f x) = F (u i) - F (l i)"
    using assms approx
    by (auto simp: less_imp_le min_def max_def
             intro!: f continuous_at_imp_continuous_on interval_integral_FTC_finite
             intro: has_vector_derivative_at_within)
  have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> B - A"
    unfolding FTCi
  proof (intro tendsto_intros)
    show "(\<lambda>x. F (l x)) \<longlonglongrightarrow> A"
      using A approx unfolding tendsto_at_iff_sequentially comp_def
      by (elim allE[of _ "\<lambda>i. ereal (l i)"], auto)
    show "(\<lambda>x. F (u x)) \<longlonglongrightarrow> B"
      using B approx unfolding tendsto_at_iff_sequentially comp_def
      by (elim allE[of _ "\<lambda>i. ereal (u i)"], auto)
  qed
  moreover have "(\<lambda>i. LBINT x=l i..u i. f x) \<longlonglongrightarrow> (LBINT x=a..b. f x)"
    by (rule interval_integral_Icc_approx_integrable [OF \<open>a < b\<close> approx f_integrable])
  ultimately show ?thesis
    by (elim LIMSEQ_unique)
qed

(*
  The second Fundamental Theorem of Calculus and existence of antiderivatives on an
  einterval.
*)

theorem interval_integral_FTC2:
  fixes a b c :: real and f :: "real \<Rightarrow> 'a::euclidean_space"
  assumes "a \<le> c" "c \<le> b"
  and contf: "continuous_on {a..b} f"
  fixes x :: real
  assumes "a \<le> x" and "x \<le> b"
  shows "((\<lambda>u. LBINT y=c..u. f y) has_vector_derivative (f x)) (at x within {a..b})"
proof -
  let ?F = "(\<lambda>u. LBINT y=a..u. f y)"
  have intf: "set_integrable lborel {a..b} f"
    by (rule borel_integrable_atLeastAtMost', rule contf)
  have "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
    using \<open>a \<le> x\<close> \<open>x \<le> b\<close> 
    by (auto intro: integral_has_vector_derivative continuous_on_subset [OF contf])
  then have "((\<lambda>u. integral {a..u} f) has_vector_derivative (f x)) (at x within {a..b})"
    by simp
  then have "(?F has_vector_derivative (f x)) (at x within {a..b})"
    by (rule has_vector_derivative_weaken)
       (auto intro!: assms interval_integral_eq_integral[symmetric] set_integrable_subset [OF intf])
  then have "((\<lambda>x. (LBINT y=c..a. f y) + ?F x) has_vector_derivative (f x)) (at x within {a..b})"
    by (auto intro!: derivative_eq_intros)
  then show ?thesis
  proof (rule has_vector_derivative_weaken)
    fix u assume "u \<in> {a .. b}"
    then show "(LBINT y=c..a. f y) + (LBINT y=a..u. f y) = (LBINT y=c..u. f y)"
      using assms
      apply (intro interval_integral_sum)
      apply (auto simp: interval_lebesgue_integrable_def simp del: real_scaleR_def)