Quellcode-Bibliothek Henstock_Kurzweil_Integration.thy

(*  Author:     John Harrison
  Author: Robert Himmelmann, TU Muenchen (Translation from HOL light)
  Huge cleanup by LCP
*)

section ‹Henstock-Kurzweil Gauge Integration in Many Dimensions›

theory Henstock_Kurzweil_Integration
imports
  Lebesgue_Measure Tagged_Division "HOL-Real_Asymp.Real_Asymp"

begin

lemma norm_diff2: "[y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) ≤ e1; norm(y2 - x2) ≤ e2]
  ==> norm(y-x) ≤ e"
  by (simp add: add_diff_add norm_add_rule_thm)

lemma setcomp_dot1: "{z. P (z ∙ (i,0))} = {(x,y). P(x ∙ i)}"
  by auto

lemma setcomp_dot2: "{z. P (z ∙ (0,i))} = {(x,y). P(y ∙ i)}"
  by auto

lemma Sigma_Int_Paircomp1: "(Sigma A B) ∩ {(x, y). P x} = Sigma (A ∩ {x. P x}) B"
  by blast

lemma Sigma_Int_Paircomp2: "(Sigma A B) ∩ {(x, y). P y} = Sigma A (λz. B z ∩ {y. P y})"
  by blast
(* END MOVE *)

subsection ‹Content (length, area, volume, etc.) of an interval›

abbreviation content :: "'a::euclidean_space set ==> real"
  where "content s ≡ measure lborel s"

lemma content_cbox_cases:
  "content (cbox a b) = (if ∀i∈Basis. a∙i ≤ b∙i then prod (λi. b∙i - a∙i) Basis else 0)"
  by (simp add: measure_lborel_cbox_eq inner_diff)

lemma content_cbox: "∀i∈Basis. a∙i ≤ b∙i ==> content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
  unfolding content_cbox_cases by simp

lemma content_cbox': "cbox a b ≠ {} ==> content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
  by (simp add: box_ne_empty inner_diff)

lemma content_cbox_if: "content (cbox a b) = (if cbox a b = {} then 0 else ∏i∈Basis. b∙i - a∙i)"
  by (simp add: content_cbox')

lemma content_cbox_cart:
   "cbox a b ≠ {} ==> content(cbox a b) = prod (λi. b$i - a$i) UNIV"
  by (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint)

lemma content_cbox_if_cart:
   "content(cbox a b) = (if cbox a b = {} then 0 else prod (λi. b$i - a$i) UNIV)"
  by (simp add: content_cbox_cart)

lemma content_division_of:
  assumes "K ∈ D" "D division_of S"
  shows "content K = (∏i ∈ Basis. interval_upperbound K ∙ i - interval_lowerbound K ∙ i)"
proof -
  obtain a b where "K = cbox a b"
    using cbox_division_memE assms by metis
  then show ?thesis
    using assms by (force simp: division_of_def content_cbox')
qed

lemma content_real: "a ≤ b ==> content {a..b} = b - a"
  by simp

lemma abs_eq_content: "∣y - x∣ = (if x≤y then content {x..y} else content {y..x})"
  by (auto simp: content_real)

lemma content_singleton: "content {a} = 0"
  by simp

lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1"
  by simp

lemma content_pos_le [iff]: "0 ≤ content X"
  by simp

corollary🍋‹tag unimportant› content_nonneg [simp]: "¬ content (cbox a b) < 0"
  using not_le by blast

lemma content_pos_lt: "∀i∈Basis. a∙i < b∙i ==> 0 < content (cbox a b)"
  by (auto simp: less_imp_le inner_diff box_eq_empty intro!: prod_pos)

lemma content_eq_0: "content (cbox a b) = 0 ⟷ (∃i∈Basis. b∙i ≤ a∙i)"
  by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl)

lemma content_eq_0_interior: "content (cbox a b) = 0 ⟷ interior(cbox a b) = {}"
  unfolding content_eq_0 interior_cbox box_eq_empty by auto

lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) ⟷ (∀i∈Basis. a∙i < b∙i)"
  by (auto simp: content_cbox_cases less_le prod_nonneg)

lemma content_empty [simp]: "content {} = 0"
  by simp

lemma content_real_if [simp]: "content {a..b} = (if a ≤ b then b - a else 0)"
  by (simp add: content_real)

lemma content_subset: "cbox a b ⊆ cbox c d ==> content (cbox a b) ≤ content (cbox c d)"
  unfolding measure_def
  by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq)

lemma content_lt_nz: "0 < content (cbox a b) ⟷ content (cbox a b) ≠ 0"
  by (fact zero_less_measure_iff)

lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
  unfolding measure_lborel_cbox_eq Basis_prod_def
  using Basis_zero 
  apply (simp add: prod.union_disjoint disjoint_iff image_iff ball_Un prod.reindex_nontrivial)
  apply (subst (1 2) prod.reindex_nontrivial)
  apply auto
  done

lemma content_cbox_pair_eq0_D:
   "content (cbox (a,c) (b,d)) = 0 ==> content (cbox a b) = 0 ∨ content (cbox c d) = 0"
  by (simp add: content_Pair)

lemma content_cbox_plus:
  fixes x :: "'a::euclidean_space"
  shows "content(cbox x (x + h *🪙R One)) = (if h ≥ 0 then h ^ DIM('a) else 0)"
  by (simp add: algebra_simps content_cbox_if box_eq_empty)

lemma content_0_subset: "content(cbox a b) = 0 ==> s ⊆ cbox a b ==> content s = 0"
  using emeasure_mono[of s "cbox a b" lborel]
  by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq)

lemma content_ball_pos:
  assumes "r > 0"
  shows   "content (ball c r) > 0"
proof -
  from rational_boxes[OF assms, of c] obtain a b where ab: "c ∈ box a b" "box a b ⊆ ball c r"
    by auto
  then have "0 < content (box a b)"
    by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
  have "emeasure lborel (box a b) ≤ emeasure lborel (ball c r)"
    using ab by (intro emeasure_mono) auto
  then show ?thesis
    using ‹content (box a b) > 0›
    by (metis Sigma_Algebra.measure_def emeasure_lborel_ball_finite enn2real_positive_iff
        infinity_ennreal_def le_zero_eq not_gr_zero)
qed

lemma content_cball_pos:
  assumes "r > 0"
  shows   "content (cball c r) > 0"
proof -
  from rational_boxes[OF assms, of c] obtain a b where ab: "c ∈ box a b" "box a b ⊆ ball c r"
    by auto
  then have "0 < content (box a b)"
    by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
  have "emeasure lborel (box a b) ≤ emeasure lborel (cball c r)"
    using ab by (intro emeasure_mono) auto
  also have "emeasure lborel (box a b) = ennreal (content (box a b))"
    using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto
  also have "emeasure lborel (cball c r) = ennreal (content (cball c r))"
    using emeasure_lborel_cball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto
  finally show ?thesis
    using ‹content (box a b) > 0› by simp
qed

lemma content_split:
  fixes a :: "'a::euclidean_space"
  assumes "k ∈ Basis"
  shows "content (cbox a b) = content(cbox a b ∩ {x. x∙k ≤ c}) + content(cbox a b ∩ {x. x∙k ≥ c})"
  🍋 ‹Prove using measure theory›
proof (cases "∀i∈Basis. a ∙ i ≤ b ∙ i")
  case True
  have 1: "∧X Y Z. (∏i∈Basis. Z i (if i = k then X else Y i)) = Z k X * (∏i∈Basis-{k}. Z i (Y i))"
    by (simp add: if_distrib prod.delta_remove assms)
  note simps = interval_split[OF assms] content_cbox_cases
  have 2: "(∏i∈Basis. b∙i - a∙i) = (∏i∈Basis-{k}. b∙i - a∙i) * (b∙k - a∙k)"
    by (metis (no_types, lifting) assms finite_Basis mult.commute prod.remove)
  have "∧x. min (b ∙ k) c = max (a ∙ k) c ==>
    x * (b∙k - a∙k) = x * (max (a ∙ k) c - a ∙ k) + x * (b ∙ k - max (a ∙ k) c)"
    by  (auto simp: field_simps)
  moreover
  have 3: "(∏i∈Basis. ((∑i∈Basis. (if i = k then min (b ∙ k) c else b ∙ i) *🪙R i) ∙ i - a ∙ i)) =
      (∏i∈Basis. (if i = k then min (b ∙ k) c else b ∙ i) - a ∙ i)"
    "(∏i∈Basis. b ∙ i - ((∑i∈Basis. (if i = k then max (a ∙ k) c else a ∙ i) *🪙R i) ∙ i)) =
      (∏i∈Basis. b ∙ i - (if i = k then max (a ∙ k) c else a ∙ i))"
    by (simp_all cong: prod.cong)
  have "¬ a ∙ k ≤ c ==> ¬ c ≤ b ∙ k ==> False"
    unfolding not_le using True assms by auto
  ultimately show ?thesis
    using assms unfolding simps 1[of "λi x. b∙i - x"] 1[of "λi x. x - a∙i"] 2 3
    by auto
next
  case False
  then show ?thesis
    using box_ne_empty(1) by force
qed

lemma division_of_content_0:
  assumes "content (cbox a b) = 0" "d division_of (cbox a b)" "K ∈ d"
  shows "content K = 0"
  unfolding forall_in_division[OF assms(2)]
  by (meson assms content_0_subset division_of_def)

lemma sum_content_null:
  assumes "content (cbox a b) = 0"
    and "p tagged_division_of (cbox a b)"
  shows "(∑(x,K)∈p. content K *🪙R f x) = (0::'a::real_normed_vector)"
proof (intro sum.neutral strip)
  fix y
  assume y: "y ∈ p"
  obtain x K where xk: "y = (x, K)"
    using surj_pair[of y] by blast
  then obtain c d where k: "K = cbox c d" "K ⊆ cbox a b"
    by (metis assms(2) tagged_division_ofD(3) tagged_division_ofD(4) y)
  have "(λ(x',K'). content K' *🪙R f x') y = content K *🪙R f x"
    unfolding xk by auto
  also have "… = 0"
    using assms(1) content_0_subset k(2) by auto
  finally show "(λ(x, k). content k *🪙R f x) y = 0" .
qed

global_interpretation sum_content: operative plus 0 content
  rewrites "comm_monoid_set.F plus 0 = sum"
proof -
  interpret operative plus 0 content
    by standard (auto simp: content_split [symmetric] content_eq_0_interior)
  show "operative plus 0 content"
    by standard
  show "comm_monoid_set.F plus 0 = sum"
    by (simp add: sum_def)
qed

lemma additive_content_division: "d division_of (cbox a b) ==> sum content d = content (cbox a b)"
  by (fact sum_content.division)

lemma additive_content_tagged_division:
  "d tagged_division_of (cbox a b) ==> sum (λ(x,l). content l) d = content (cbox a b)"
  by (fact sum_content.tagged_division)

lemma subadditive_content_division:
  assumes "D division_of S" "S ⊆ cbox a b"
  shows "sum content D ≤ content(cbox a b)"
proof -
  have "D division_of ∪D" "∪D ⊆ cbox a b"
    using assms by auto
  then obtain D' where "D ⊆ D'" "D' division_of cbox a b"
    using partial_division_extend_interval by metis
  then have "sum content D ≤ sum content D'"
    using sum_mono2 by blast
  also have "… ≤ content(cbox a b)"
    by (simp add: ‹D' division_of cbox a b› additive_content_division less_eq_real_def)
  finally show ?thesis .
qed

lemma content_real_eq_0: "content {a..b::real} = 0 ⟷ a ≥ b"
  by simp

lemma property_empty_interval: "∀a b. content (cbox a b) = 0 ⟶ P (cbox a b) ==> P {}"
  using content_empty unfolding empty_as_interval by auto

lemma interval_bounds_nz_content [simp]:
  assumes "content (cbox a b) ≠ 0"
  shows "interval_upperbound (cbox a b) = b"
    and "interval_lowerbound (cbox a b) = a"
  by (metis assms content_empty interval_bounds')+

subsection ‹Gauge integral›

text ‹Case distinction to define it first on compact intervals first, then use a limit. This is only
 much later unified. In Fremlin: Measure Theory, Volume 4I this is generalized using residual sets.›

definition has_integral :: "('n::euclidean_space ==> 'b::real_normed_vector) ==> 'b ==> 'n set ==> bool"
  (infixr ‹has'_integral› 46)
  where "(f has_integral I) s ⟷
    (if ∃a b. s = cbox a b
      then ((λp. ∑(x,k)∈p. content k *🪙R f x) ---> I) (division_filter s)
      else (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
        (∃z. ((λp. ∑(x,k)∈p. content k *🪙R (if x ∈ s then f x else 0)) ---> z) (division_filter (cbox a b)) ∧
          norm (z - I) < e)))"

lemma has_integral_cbox:
  "(f has_integral I) (cbox a b) ⟷ ((λp. ∑(x,k)∈p. content k *🪙R f x) ---> I) (division_filter (cbox a b))"
  by (auto simp: has_integral_def)

lemma has_integral:
  "(f has_integral y) (cbox a b) ⟷
    (∀e>0. ∃γ. gauge γ ∧
      (∀D. D tagged_division_of (cbox a b) ∧ γ fine D ⟶
        norm ((∑(x, k)∈D. content k *🪙R f x) - y) < e))"
  by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)

lemma has_integral_real:
  "(f has_integral y) {a..b::real} ⟷
    (∀e>0. ∃γ. gauge γ ∧
      (∀D. D tagged_division_of {a..b} ∧ γ fine D ⟶
        norm (sum (λ(x,k). content(k) *🪙R f x) D - y) < e))"
  unfolding box_real[symmetric] by (rule has_integral)

lemma has_integralD[dest]:
  assumes "(f has_integral y) (cbox a b)"
    and "e > 0"
  obtains γ
    where "gauge γ"
      and "∧D. D tagged_division_of (cbox a b) ==> γ fine D ==>
        norm ((∑(x,k)∈D. content k *🪙R f x) - y) < e"
  using assms unfolding has_integral by auto

lemma has_integral_alt:
  "(f has_integral y) i ⟷
    (if ∃a b. i = cbox a b
     then (f has_integral y) i
     else (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
      (∃z. ((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b) ∧ norm (z - y) < e)))"
  by (subst has_integral_def) (auto simp: has_integral_cbox)

lemma has_integral_altD:
  assumes "(f has_integral y) i"
    and "¬ (∃a b. i = cbox a b)"
    and "e>0"
  obtains B where "B > 0"
    and "∀a b. ball 0 B ⊆ cbox a b ⟶
      (∃z. ((λx. if x ∈ i then f(x) else 0) has_integral z) (cbox a b) ∧ norm(z - y) < e)"
  using assms has_integral_alt[of f y i] by auto

definition integrable_on (infixr ‹integrable'_on› 46)
  where "f integrable_on i ⟷ (∃y. (f has_integral y) i)"

definition "integral i f = (SOME y. (f has_integral y) i ∨ ¬ f integrable_on i ∧ y=0)"

lemma integrable_integral[intro]: "f integrable_on i ==> (f has_integral (integral i f)) i"
  unfolding integrable_on_def integral_def by (metis (mono_tags, lifting))

lemma not_integrable_integral: "¬ f integrable_on i ==> integral i f = 0"
  unfolding integrable_on_def integral_def by blast

lemma has_integral_integrable[dest]: "(f has_integral i) s ==> f integrable_on s"
  unfolding integrable_on_def by auto

lemma has_integral_integral: "f integrable_on s ⟷ (f has_integral (integral s f)) s"
  by auto

subsection ‹Basic theorems about integrals›

lemma has_integral_eq_rhs: "(f has_integral j) S ==> i = j ==> (f has_integral i) S"
  by (rule forw_subst)

lemma has_integral_unique_cbox:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  shows "(f has_integral k1) (cbox a b) ==> (f has_integral k2) (cbox a b) ==> k1 = k2"
  by (meson division_filter_not_empty has_integral_cbox tendsto_unique)

lemma has_integral_unique:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes "(f has_integral k1) i" "(f has_integral k2) i"
  shows "k1 = k2"
proof (rule ccontr)
  let ?e = "norm (k1 - k2)/2"
  let ?F = "(λx. if x ∈ i then f x else 0)"
  assume "k1 ≠ k2"
  then have e: "?e > 0"
    by auto
  have nonbox: "¬ (∃a b. i = cbox a b)"
    using ‹k1 ≠ k2› assms has_integral_unique_cbox by blast
  obtain B1 where B1:
      "0 < B1"
      "∧a b. ball 0 B1 ⊆ cbox a b ==>
        ∃z. (?F has_integral z) (cbox a b) ∧ norm (z - k1) < norm (k1 - k2)/2"
    by (rule has_integral_altD[OF assms(1) nonbox,OF e]) blast
  obtain B2 where B2:
      "0 < B2"
      "∧a b. ball 0 B2 ⊆ cbox a b ==>
        ∃z. (?F has_integral z) (cbox a b) ∧ norm (z - k2) < norm (k1 - k2)/2"
    by (rule has_integral_altD[OF assms(2) nonbox,OF e]) blast
  obtain a b :: 'n where ab: "ball 0 B1 ⊆ cbox a b" "ball 0 B2 ⊆ cbox a b"
    by (metis Un_subset_iff bounded_Un bounded_ball bounded_subset_cbox_symmetric)
  obtain w where w: "(?F has_integral w) (cbox a b)" "norm (w - k1) < norm (k1 - k2)/2"
    using B1(2)[OF ab(1)] by blast
  obtain z where z: "(?F has_integral z) (cbox a b)" "norm (z - k2) < norm (k1 - k2)/2"
    using B2(2)[OF ab(2)] by blast
  have "z = w"
    using has_integral_unique_cbox[OF w(1) z(1)] by auto
  then have "norm (k1 - k2) ≤ norm (z - k2) + norm (w - k1)"
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
    by (auto simp: norm_minus_commute)
  also have "… < norm (k1 - k2)/2 + norm (k1 - k2)/2"
    by (metis add_strict_mono z(2) w(2))
  finally show False by auto
qed

lemma integral_unique [intro]: "(f has_integral y) k ==> integral k f = y"
  unfolding integral_def
  by (rule some_equality) (auto intro: has_integral_unique)

lemma has_integral_iff: "(f has_integral i) S ⟷ (f integrable_on S ∧ integral S f = i)"
  by blast

lemma eq_integralD: "integral k f = y ==> (f has_integral y) k ∨ ¬ f integrable_on k ∧ y=0"
  unfolding integral_def integrable_on_def
  by (metis (mono_tags, lifting))

lemma has_integral_const [intro]:
  fixes a b :: "'a::euclidean_space"
  shows "((λx. c) has_integral (content (cbox a b) *🪙R c)) (cbox a b)"
  using eventually_division_filter_tagged_division[of "cbox a b"]
     additive_content_tagged_division[of _ a b]
  by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric]
           elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const])

lemma has_integral_const_real [intro]:
  fixes a b :: real
  shows "((λx. c) has_integral (content {a..b} *🪙R c)) {a..b}"
  by (metis box_real(2) has_integral_const)

lemma has_integral_integrable_integral: "(f has_integral i) s ⟷ f integrable_on s ∧ integral s f = i"
  by blast

lemma integral_const [simp]:
  fixes a b :: "'a::euclidean_space"
  shows "integral (cbox a b) (λx. c) = content (cbox a b) *🪙R c"
  by blast

lemma integral_const_real [simp]:
  fixes a b :: real
  shows "integral {a..b} (λx. c) = content {a..b} *🪙R c"
  by blast

lemma has_integral_is_0_cbox:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes "∧x. x ∈ cbox a b ==> f x = 0"
  shows "(f has_integral 0) (cbox a b)"
    unfolding has_integral_cbox
    using eventually_division_filter_tagged_division[of "cbox a b"] assms
    by (subst tendsto_cong[where g="λ_. 0"])
       (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval)

lemma has_integral_is_0:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes "∧x. x ∈ S ==> f x = 0"
  shows "(f has_integral 0) S"
proof (cases "(∃a b. S = cbox a b)")
  case True with assms has_integral_is_0_cbox show ?thesis
    by blast
next
  case False
  have *: "(λx. if x ∈ S then f x else 0) = (λx. 0)"
    by (auto simp: assms)
  show ?thesis
    using has_integral_is_0_cbox False
    by (subst has_integral_alt) (force simp: *)
qed

lemma has_integral_0[simp]: "((λx::'n::euclidean_space. 0) has_integral 0) S"
  by (rule has_integral_is_0) auto

lemma has_integral_0_eq[simp]: "((λx. 0) has_integral i) S ⟷ i = 0"
  using has_integral_unique[OF has_integral_0] by auto

lemma has_integral_linear_cbox:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes f: "(f has_integral y) (cbox a b)"
    and h: "bounded_linear h"
  shows "((h ∘ f) has_integral (h y)) (cbox a b)"
proof -
  interpret bounded_linear h using h .
  show ?thesis
    unfolding has_integral_cbox using tendsto [OF f [unfolded has_integral_cbox]]
    by (simp add: sum scaleR split_beta')
qed

lemma has_integral_linear:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes f: "(f has_integral y) S"
    and h: "bounded_linear h"
  shows "((h ∘ f) has_integral (h y)) S"
proof (cases "(∃a b. S = cbox a b)")
  case True with f h has_integral_linear_cbox show ?thesis 
    by blast
next
  case False
  interpret bounded_linear h using h .
  from pos_bounded obtain B where B: "0 < B" "∧x. norm (h x) ≤ norm x * B"
    by blast
  let ?S = "λf x. if x ∈ S then f x else 0"
  show ?thesis
  proof (subst has_integral_alt, clarsimp simp: False)
    fix e :: real
    assume e: "e > 0"
    have *: "0 < e/B" using e B(1) by simp
    obtain M where M:
      "M > 0"
      "∧a b. ball 0 M ⊆ cbox a b ==>
        ∃z. (?S f has_integral z) (cbox a b) ∧ norm (z - y) < e/B"
      using has_integral_altD[OF f False *] by blast
    show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
      (∃z. (?S(h ∘ f) has_integral z) (cbox a b) ∧ norm (z - h y) < e)"
    proof (rule exI, intro allI conjI impI)
      show "M > 0" using M by metis
    next
      fix a b::'n
      assume sb: "ball 0 M ⊆ cbox a b"
      obtain z where z: "(?S f has_integral z) (cbox a b)" "norm (z - y) < e/B"
        using M(2)[OF sb] by blast
      have *: "?S(h ∘ f) = h ∘ ?S f"
        using zero by auto
      show "∃z. (?S(h ∘ f) has_integral z) (cbox a b) ∧ norm (z - h y) < e"
      proof (intro exI conjI)
        show "(?S(h ∘ f) has_integral h z) (cbox a b)"
          by (simp add: * has_integral_linear_cbox[OF z(1) h])
        show "norm (h z - h y) < e"
          by (metis B diff le_less_trans pos_less_divide_eq z(2))
      qed
    qed
  qed
qed

lemma has_integral_of_real:
  "(f has_integral I) A ==>
     ((λx::'a::euclidean_space. of_real (f x) :: 'b :: {real_normed_vector,real_normed_algebra_1})
        has_integral of_real I) A"
  using has_integral_linear[of f I A "of_real :: _ ==> 'b"]
  by (simp add: o_def bounded_linear_of_real)

lemma has_integral_scaleR_left:
  "(f has_integral y) S ==> ((λx. f x *🪙R c) has_integral (y *🪙R c)) S"
  using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)

lemma integrable_on_scaleR_left:
  assumes "f integrable_on A"
  shows "(λx. f x *🪙R y) integrable_on A"
  using assms has_integral_scaleR_left unfolding integrable_on_def by blast

lemma has_integral_mult_left:
  fixes c :: "_ :: real_normed_algebra"
  shows "(f has_integral y) S ==> ((λx. f x * c) has_integral (y * c)) S"
  using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)

lemma integrable_on_mult_left:
  fixes c :: "'a :: real_normed_algebra"
  assumes "f integrable_on A"
  shows   "(λx. f x * c) integrable_on A"
  using assms has_integral_mult_left by blast

lemma has_integral_divide:
  fixes c :: "_ :: real_normed_div_algebra"
  shows "(f has_integral y) S ==> ((λx. f x / c) has_integral (y / c)) S"
  unfolding divide_inverse by (simp add: has_integral_mult_left)

lemma integrable_on_divide:
  fixes c :: "'a :: real_normed_div_algebra"
  assumes "f integrable_on A"
  shows   "(λx. f x / c) integrable_on A"
  using assms has_integral_divide by blast

text‹The case analysis eliminates the condition 🍋‹f integrable_on S› at the cost
     of the type class constraint ‹division_ring›\<close>
corollary integral_mult_left [simp]:
  fixes c:: "'a::{real_normed_algebra,division_ring}"
  shows "integral S (λx. f x * c) = integral S f * c"
proof (cases "f integrable_on S ∨ c = 0")
  case True then show ?thesis
    by (force intro: has_integral_mult_left)
next
  case False then have "¬ (λx. f x * c) integrable_on S"
    using has_integral_mult_left [of "(λx. f x * c)" _ S "inverse c"]
    by (auto simp: mult.assoc)
  with False show ?thesis by (simp add: not_integrable_integral)
qed

corollary integral_mult_right [simp]:
  fixes c:: "'a::{real_normed_field}"
  shows "integral S (λx. c * f x) = c * integral S f"
by (simp add: mult.commute [of c])

corollary integral_divide [simp]:
  fixes z :: "'a::real_normed_field"
  shows "integral S (λx. f x / z) = integral S (λx. f x) / z"
using integral_mult_left [of S f "inverse z"]
  by (simp add: divide_inverse_commute)

lemma has_integral_mult_right:
  fixes c :: "'a :: real_normed_algebra"
  shows "(f has_integral y) A ==> ((λx. c * f x) has_integral (c * y)) A"
  using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)

lemma integrable_on_mult_right:
  fixes c :: "'a :: real_normed_algebra"
  assumes "f integrable_on A"
  shows   "(λx. c * f x) integrable_on A"
  using assms has_integral_mult_right by blast

lemma has_integral_mult_right_iff:
  fixes c :: "'a :: real_normed_field"
  assumes "c ≠ 0"
  shows "((λx. c * f x) has_integral y) A ⟷ (f has_integral (y / c)) A"
  using has_integral_mult_right[of f "y / c" A c] 
        has_integral_mult_right[of "λx. c * f x" y A "1/c"] assms
  by auto

lemma integrable_on_mult_right_iff [simp]:
  fixes c :: "'a :: real_normed_field"
  assumes "c ≠ 0"
  shows   "(λx. c * f x) integrable_on A ⟷ f integrable_on A"
    using integrable_on_mult_right[of f A c]
          integrable_on_mult_right[of "λx. c * f x" A "inverse c"] assms
    by (auto simp: field_simps)

lemma integrable_on_mult_left_iff [simp]:
  fixes c :: "'a :: real_normed_field"
  assumes "c ≠ 0"
  shows   "(λx. f x * c) integrable_on A ⟷ f integrable_on A"
  using integrable_on_mult_right_iff[OF assms, of f A] by (simp add: mult.commute)

lemma integrable_on_div_iff [simp]:
  fixes c :: "'a :: real_normed_field"
  assumes "c ≠ 0"
  shows   "(λx. f x / c) integrable_on A ⟷ f integrable_on A"
  using integrable_on_mult_right_iff[of "inverse c" f A] assms by (simp add: field_simps)

lemma has_integral_cmul: "(f has_integral k) S ==> ((λx. c *🪙R f x) has_integral (c *🪙R k)) S"
  unfolding o_def[symmetric]
  by (metis has_integral_linear bounded_linear_scaleR_right)

lemma has_integral_cmult_real:
  fixes c :: real
  assumes "c ≠ 0 ==> (f has_integral x) A"
  shows "((λx. c * f x) has_integral c * x) A"
  by (metis assms has_integral_is_0 has_integral_mult_right lambda_zero)

lemma has_integral_neg: "(f has_integral k) S ==> ((λx. -(f x)) has_integral -k) S"
  by (drule_tac c="-1" in has_integral_cmul) auto

lemma has_integral_neg_iff: "((λx. - f x) has_integral k) S ⟷ (f has_integral - k) S"
  using has_integral_neg[of f "- k"] has_integral_neg[of "λx. - f x" k] by auto

lemma has_integral_add_cbox:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes "(f has_integral k) (cbox a b)" "(g has_integral l) (cbox a b)"
  shows "((λx. f x + g x) has_integral (k + l)) (cbox a b)"
  using assms
    unfolding has_integral_cbox
    by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)

lemma has_integral_add:
  fixes f :: "'n::euclidean_space ==> 'a::real_normed_vector"
  assumes f: "(f has_integral k) S" and g: "(g has_integral l) S"
  shows "((λx. f x + g x) has_integral (k + l)) S"
proof (cases "∃a b. S = cbox a b")
  case True with has_integral_add_cbox assms show ?thesis
    by blast 
next
  let ?S = "λf x. if x ∈ S then f x else 0"
  case False
  then show ?thesis
  proof (subst has_integral_alt, clarsimp, goal_cases)
    case (1 e)
    then have e2: "e/2 > 0"
      by auto
    obtain Bf where "0 < Bf"
      and Bf: "∧a b. ball 0 Bf ⊆ cbox a b ==>
                     ∃z. (?S f has_integral z) (cbox a b) ∧ norm (z - k) < e/2"
      using has_integral_altD[OF f False e2] by blast
    obtain Bg where "0 < Bg"
      and Bg: "∧a b. ball 0 Bg ⊆ (cbox a b) ==>
                     ∃z. (?S g has_integral z) (cbox a b) ∧ norm (z - l) < e/2"
      using has_integral_altD[OF g False e2] by blast
    show ?case
    proof (rule_tac x="max Bf Bg" in exI, clarsimp simp add: max.strict_coboundedI1 ‹0 🚫›)
      fix a b
      assume "ball 0 (max Bf Bg) ⊆ cbox a (b::'n)"
      then have fs: "ball 0 Bf ⊆ cbox a (b::'n)" and gs: "ball 0 Bg ⊆ cbox a (b::'n)"
        by auto
      obtain w where w: "(?S f has_integral w) (cbox a b)" "norm (w - k) < e/2"
        using Bf[OF fs] by blast
      obtain z where z: "(?S g has_integral z) (cbox a b)" "norm (z - l) < e/2"
        using Bg[OF gs] by blast
      have *: "∧x. (if x ∈ S then f x + g x else 0) = (?S f x) + (?S g x)"
        by auto
      show "∃z. (?S(λx. f x + g x) has_integral z) (cbox a b) ∧ norm (z - (k + l)) < e"
      proof (intro exI conjI)
        show "(?S(λx. f x + g x) has_integral (w + z)) (cbox a b)"
          by (simp add: has_integral_add_cbox[OF w(1) z(1), unfolded *[symmetric]])
        show "norm (w + z - (k + l)) < e"
          by (metis dist_norm dist_triangle_add_half w(2) z(2))
      qed
    qed
  qed
qed

lemma has_integral_diff:
  "(f has_integral k) S ==> (g has_integral l) S ==>
    ((λx. f x - g x) has_integral (k - l)) S"
  using has_integral_add[OF _ has_integral_neg, of f k S g l]
  by (auto simp: algebra_simps)

lemma integral_0 [simp]:
  "integral S (λx::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
  by auto

lemma integral_add: "f integrable_on S ==> g integrable_on S ==>
    integral S (λx. f x + g x) = integral S f + integral S g"
  by (rule integral_unique) (metis integrable_integral has_integral_add)

lemma integral_cmul [simp]: "integral S (λx. c *🪙R f x) = c *🪙R integral S f"
proof (cases "f integrable_on S ∨ c = 0")
  case True with has_integral_cmul integrable_integral show ?thesis
    by fastforce
next
  case False then have "¬ (λx. c *🪙R f x) integrable_on S"
    using has_integral_cmul [of "(λx. c *🪙R f x)" _ S "inverse c"] by auto
  with False show ?thesis by (simp add: not_integrable_integral)
qed

lemma integral_mult:
  fixes K::real
  shows "f integrable_on X ==> K * integral X f = integral X (λx. K * f x)"
  by simp

lemma integral_neg [simp]: "integral S (λx. - f x) = - integral S f"
  by (metis eq_integralD equation_minus_iff has_integral_iff has_integral_neg_iff neg_equal_0_iff_equal)

lemma integral_diff: "f integrable_on S ==> g integrable_on S ==>
    integral S (λx. f x - g x) = integral S f - integral S g"
  by (rule integral_unique) (metis integrable_integral has_integral_diff)

lemma integrable_0: "(λx. 0) integrable_on S"
  unfolding integrable_on_def using has_integral_0 by auto

lemma integrable_add: "f integrable_on S ==> g integrable_on S ==> (λx. f x + g x) integrable_on S"
  unfolding integrable_on_def by(auto intro: has_integral_add)

lemma integrable_cmul: "f integrable_on S ==> (λx. c *🪙R f(x)) integrable_on S"
  unfolding integrable_on_def by(auto intro: has_integral_cmul)

lemma integrable_on_scaleR_iff [simp]:
  fixes c :: real
  assumes "c ≠ 0"
  shows "(λx. c *🪙R f x) integrable_on S ⟷ f integrable_on S"
  using integrable_cmul[of "λx. c *🪙R f x" S "1 / c"] integrable_cmul[of f S c] ‹c ≠ 0›
  by auto

lemma integrable_on_cmult_iff [simp]:
  fixes c :: real
  assumes "c ≠ 0"
  shows "(λx. c * f x) integrable_on S ⟷ f integrable_on S"
  using integrable_on_scaleR_iff [of c f] assms by simp

lemma integrable_on_cmult_left:
  assumes "f integrable_on S"
  shows "(λx. of_real c * f x) integrable_on S"
    using integrable_cmul[of f S "of_real c"] assms
    by (simp add: scaleR_conv_of_real)

lemma integrable_neg: "f integrable_on S ==> (λx. -f(x)) integrable_on S"
  unfolding integrable_on_def by(auto intro: has_integral_neg)

lemma integrable_neg_iff: "(λx. -f(x)) integrable_on S ⟷ f integrable_on S"
  using integrable_neg by fastforce

lemma integrable_diff:
  "f integrable_on S ==> g integrable_on S ==> (λx. f x - g x) integrable_on S"
  unfolding integrable_on_def by(auto intro: has_integral_diff)

lemma integrable_linear:
  "f integrable_on S ==> bounded_linear h ==> (h ∘ f) integrable_on S"
  unfolding integrable_on_def by(auto intro: has_integral_linear)

lemma integral_linear:
  "f integrable_on S ==> bounded_linear h ==> integral S (h ∘ f) = h (integral S f)"
  by (meson has_integral_iff has_integral_linear)

lemma integrable_on_cnj_iff:
  "(λx. cnj (f x)) integrable_on A ⟷ f integrable_on A"
  using integrable_linear[OF _ bounded_linear_cnj, of f A]
        integrable_linear[OF _ bounded_linear_cnj, of "cnj ∘ f" A]
  by (auto simp: o_def)

lemma integral_cnj: "cnj (integral A f) = integral A (λx. cnj (f x))"
  by (cases "f integrable_on A")
     (simp_all add: integral_linear[OF _ bounded_linear_cnj, symmetric]
                    o_def integrable_on_cnj_iff not_integrable_integral)

lemma has_integral_cnj: "(cnj ∘ f has_integral (cnj I)) A = (f has_integral I) A"
  unfolding has_integral_iff comp_def
  by (metis integral_cnj complex_cnj_cancel_iff integrable_on_cnj_iff)

lemma integral_component_eq[simp]:
  fixes f :: "'n::euclidean_space ==> 'm::euclidean_space"
  assumes "f integrable_on S"
  shows "integral S (λx. f x ∙ k) = integral S f ∙ k"
  unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..

lemma integral_eq_iff_componentwise:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  assumes "f integrable_on A"
  shows "integral A f = I ⟷ (∀b∈Basis. integral A (λx. f x ∙ b) = I ∙ b)"
proof -
  have "integral A f = I ⟷ (∀b∈Basis. integral A f ∙ b = I ∙ b)"
    by (metis euclidean_eqI)
  also have "… ⟷ (∀b∈Basis. integral A (λx. f x ∙ b) = I ∙ b)"
    using assms by force
  finally show ?thesis .
qed

lemma has_integral_sum:
  assumes "finite T"
    and "∧a. a ∈ T ==> ((f a) has_integral (i a)) S"
  shows "((λx. sum (λa. f a x) T) has_integral (sum i T)) S"
  using ‹finite T› subset_refl[of T]
  by (induct rule: finite_subset_induct) (use assms in ‹auto simp: has_integral_add›)

lemma integral_sum:
  "[finite I; ∧a. a ∈ I ==> f a integrable_on S] ==>
   integral S (λx. ∑a∈I. f a x) = (∑a∈I. integral S (f a))"
  by (simp add: has_integral_sum integrable_integral integral_unique)

lemma integrable_sum:
  "[finite I; ∧a. a ∈ I ==> f a integrable_on S] ==> (λx. ∑a∈I. f a x) integrable_on S"
  unfolding integrable_on_def using has_integral_sum[of I] by metis

lemma has_integral_eq:
  assumes "∧x. x ∈ s ==> f x = g x"
    and f: "(f has_integral k) s"
  shows "(g has_integral k) s"
  using has_integral_diff[OF f, of "λx. f x - g x" 0]
  using has_integral_is_0[of s "λx. f x - g x"]
  using assms
  by auto

lemma integrable_eq: "[f integrable_on s; ∧x. x ∈ s ==> f x = g x] ==> g integrable_on s"
  unfolding integrable_on_def
  using has_integral_eq[of s f g] has_integral_eq by blast

lemma has_integral_cong:
  assumes "∧x. x ∈ s ==> f x = g x"
  shows "(f has_integral i) s = (g has_integral i) s"
  by (metis assms has_integral_eq)

lemma integrable_cong:
  assumes "∧x. x ∈ A ==> f x = g x"
  shows   "f integrable_on A ⟷ g integrable_on A"
  using has_integral_cong [OF assms] by fast

lemma integral_cong:
  assumes "∧x. x ∈ s ==> f x = g x"
  shows "integral s f = integral s g"
  unfolding integral_def
by (metis (full_types, opaque_lifting) assms has_integral_cong integrable_eq)

lemma integrable_on_cmult_left_iff [simp]:
  assumes "c ≠ 0"
  shows "(λx. of_real c * f x) integrable_on s ⟷ f integrable_on s"
        (is "?lhs = ?rhs")
proof
  assume ?lhs
  then have "(λx. of_real (1 / c) * (of_real c * f x)) integrable_on s"
    using integrable_cmul[of "λx. of_real c * f x" s "1 / of_real c"]
    by (simp add: scaleR_conv_of_real)
  then have "(λx. (of_real (1 / c) * of_real c * f x)) integrable_on s"
    by (simp add: algebra_simps)
  with ‹c ≠ 0› show ?rhs
    by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
qed (blast intro: integrable_on_cmult_left)

lemma integrable_on_cmult_right:
  fixes f :: "_ ==> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
  assumes "f integrable_on s"
  shows "(λx. f x * of_real c) integrable_on s"
  using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)

lemma integrable_on_cmult_right_iff [simp]:
  fixes f :: "_ ==> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
  assumes "c ≠ 0"
  shows "(λx. f x * of_real c) integrable_on s ⟷ f integrable_on s"
  using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)

lemma integrable_on_cdivide_iff [simp]:
  fixes f :: "_ ==> 'b :: real_normed_field"
  assumes "c ≠ 0"
  shows "(λx. f x / of_real c) integrable_on s ⟷ f integrable_on s"
  by (simp add: divide_inverse assms flip: of_real_inverse)

lemma has_integral_null [intro]: "content(cbox a b) = 0 ==> (f has_integral 0) (cbox a b)"
  unfolding has_integral_cbox
  using eventually_division_filter_tagged_division[of "cbox a b"]
  by (subst tendsto_cong[where g="λ_. 0"]) (auto elim: eventually_mono intro: sum_content_null)

lemma has_integral_null_real [intro]: "content {a..b::real} = 0 ==> (f has_integral 0) {a..b}"
  by (metis box_real(2) has_integral_null)

lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 ==> (f has_integral i) (cbox a b) ⟷ i = 0"
  by (auto simp: has_integral_null dest!: integral_unique)

lemma integral_null [simp]: "content (cbox a b) = 0 ==> integral (cbox a b) f = 0"
  by (metis has_integral_null integral_unique)

lemma integrable_on_null [intro]: "content (cbox a b) = 0 ==> f integrable_on (cbox a b)"
  by (simp add: has_integral_integrable)

lemma has_integral_empty[intro]: "(f has_integral 0) {}"
  by (meson ex_in_conv has_integral_is_0)

lemma has_integral_empty_eq[simp]: "(f has_integral i) {} ⟷ i = 0"
  by (auto simp: has_integral_empty has_integral_unique)

lemma integrable_on_empty[intro]: "f integrable_on {}"
  unfolding integrable_on_def by auto

lemma integral_empty[simp]: "integral {} f = 0"
  by blast

lemma has_integral_refl[intro]:
  fixes a :: "'a::euclidean_space"
  shows "(f has_integral 0) (cbox a a)"
    and "(f has_integral 0) {a}"
proof -
  show "(f has_integral 0) (cbox a a)"
     by (rule has_integral_null) simp
  then show "(f has_integral 0) {a}"
    by simp
qed

lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
  unfolding integrable_on_def by auto

lemma integral_refl [simp]: "integral (cbox a a) f = 0"
  by auto

lemma integral_singleton [simp]: "integral {a} f = 0"
  by auto

lemma integral_blinfun_apply:
  assumes "f integrable_on s"
  shows "integral s (λx. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
  using integral_linear[OF assms blinfun.bounded_linear_right]
  by (metis (no_types, lifting) ext comp_def)

lemma blinfun_apply_integral:
  assumes "f integrable_on s"
  shows "blinfun_apply (integral s f) x = integral s (λy. blinfun_apply (f y) x)"
  by (metis (no_types, lifting) ext assms blinfun.prod_left.rep_eq
      integral_blinfun_apply)

lemma has_integral_componentwise_iff:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  shows "(f has_integral y) A ⟷ (∀b∈Basis. ((λx. f x ∙ b) has_integral (y ∙ b)) A)"
proof (intro iffI strip)
  fix b :: 'b assume "(f has_integral y) A"
  from has_integral_linear[OF this(1) bounded_linear_inner_left, of b]
    show "((λx. f x ∙ b) has_integral (y ∙ b)) A" by (simp add: o_def)
next
  assume "(∀b∈Basis. ((λx. f x ∙ b) has_integral (y ∙ b)) A)"
  hence "∀b∈Basis. (((λx. x *🪙R b) ∘ (λx. f x ∙ b)) has_integral ((y ∙ b) *🪙R b)) A"
    using bounded_linear_scaleR_left has_integral_linear by blast
  hence "((λx. ∑b∈Basis. (f x ∙ b) *🪙R b) has_integral (∑b∈Basis. (y ∙ b) *🪙R b)) A"
    by (intro has_integral_sum) (simp_all add: o_def)
  thus "(f has_integral y) A" by (simp add: euclidean_representation)
qed

lemma has_integral_componentwise:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  shows "(∧b. b ∈ Basis ==> ((λx. f x ∙ b) has_integral (y ∙ b)) A) ==> (f has_integral y) A"
  by (subst has_integral_componentwise_iff) blast

lemma integrable_componentwise_iff:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  shows "f integrable_on A ⟷ (∀b∈Basis. (λx. f x ∙ b) integrable_on A)"
proof
  assume "f integrable_on A"
  then obtain y where "∀b∈Basis. ((λx. f x ∙ b) has_integral (y ∙ b)) A"
    using has_integral_componentwise_iff by blast
  thus "(∀b∈Basis. (λx. f x ∙ b) integrable_on A)" by (auto simp: integrable_on_def)
next
  assume "(∀b∈Basis. (λx. f x ∙ b) integrable_on A)"
  then obtain y where "∀b∈Basis. ((λx. f x ∙ b) has_integral y b) A"
    unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
  hence "∀b∈Basis. (((λx. x *🪙R b) ∘ (λx. f x ∙ b)) has_integral (y b *🪙R b)) A"
    by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
  hence "((λx. ∑b∈Basis. (f x ∙ b) *🪙R b) has_integral (∑b∈Basis. y b *🪙R b)) A"
    by (intro has_integral_sum) (simp_all add: o_def)
  thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
qed

lemma integrable_componentwise:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  shows "(∧b. b ∈ Basis ==> (λx. f x ∙ b) integrable_on A) ==> f integrable_on A"
  by (subst integrable_componentwise_iff) blast

lemma integral_componentwise:
  fixes f :: "'a :: euclidean_space ==> 'b :: euclidean_space"
  assumes "f integrable_on A"
  shows "integral A f = (∑b∈Basis. integral A (λx. (f x ∙ b) *🪙R b))"
proof -
  from assms have integrable: "∀b∈Basis. (λx. x *🪙R b) ∘ (λx. (f x ∙ b)) integrable_on A"
    using bounded_linear_scaleR_left integrable_componentwise_iff integrable_linear
    by blast
  have "integral A f = integral A (λx. ∑b∈Basis. (f x ∙ b) *🪙R b)"
    by (simp add: euclidean_representation)
  also from integrable have "… = (∑a∈Basis. integral A (λx. (f x ∙ a) *🪙R a))"
    by (subst integral_sum) (simp_all add: o_def)
  finally show ?thesis .
qed

lemma integrable_component:
  "f integrable_on A ==> (λx. f x ∙ (y :: 'b :: euclidean_space)) integrable_on A"
  by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def)

lemma
  assumes "(f has_integral I) A "
  shows has_integral_Re: "((λx. Re (f x)) has_integral (Re I)) A"
  and   has_integral_Im: "((λx. Im (f x)) has_integral (Im I)) A"
proof -
  have "((λx. Re (f x)) has_integral (Re I)) A ∧ ((λx. Im (f x)) has_integral (Im I)) A"
    using assms by (subst (asm) has_integral_componentwise_iff) (auto simp: Basis_complex_def)
  thus "((λx. Re (f x)) has_integral (Re I)) A" "((λx. Im (f x)) has_integral (Im I)) A"
    by blast+
qed


subsection ‹Cauchy-type criterion for integrability›

proposition integrable_Cauchy:
  fixes f :: "'n::euclidean_space ==> 'a::{real_normed_vector,complete_space}"
  shows "f integrable_on cbox a b ⟷
        (∀e>0. ∃γ. gauge γ ∧
          (∀D1 D2. D1 tagged_division_of (cbox a b) ∧ γ fine D1 ∧
            D2 tagged_division_of (cbox a b) ∧ γ fine D2 ⟶
            norm ((∑(x,K)∈D1. content K *🪙R f x) - (∑(x,K)∈D2. content K *🪙R f x)) < e))"
  (is "?l = (∀e>0. ∃γ. ?P e γ)")
proof (intro iffI allI impI)
  assume ?l
  then obtain y
    where y: "∧e. e > 0 ==>
        ∃γ. gauge γ ∧
            (∀D. D tagged_division_of cbox a b ∧ γ fine D ⟶
                 norm ((∑(x,K) ∈ D. content K *🪙R f x) - y) < e)"
    by (auto simp: integrable_on_def has_integral)
  show "∃γ. ?P e γ" if "e > 0" for e
  proof -
    have "e/2 > 0" using that by auto
    with y obtain γ where "gauge γ"
      and "∧D. D tagged_division_of cbox a b ∧ γ fine D ==>
                  norm ((∑(x,K)∈D. content K *🪙R f x) - y) < e/2"
      by meson
    then show ?thesis
      by (metis norm_triangle_half_l)
    qed
next
  assume "∀e>0. ∃γ. ?P e γ"
  then have "∀n::nat. ∃γ. ?P (1 / (n + 1)) γ"
    by auto
  then obtain γ :: "nat ==> 'n ==> 'n set" where γ:
    "∧m. gauge (γ m)"
    "∧m D1 D2. [D1 tagged_division_of cbox a b;
              γ m fine D1; D2 tagged_division_of cbox a b; γ m fine D2]
              ==> norm ((∑(x,K) ∈ D1. content K *🪙R f x) - (∑(x,K) ∈ D2. content K *🪙R f x))
                  < 1 / (m + 1)"
    by metis
  have "gauge (λx. ∩{γ i x |i. i ∈ {0..n}})" for n
    using γ by (intro gauge_Inter) auto
  then have "∀n. ∃p. p tagged_division_of (cbox a b) ∧ (λx. ∩{γ i x |i. i ∈ {0..n}}) fine p"
    by (meson fine_division_exists)
  then obtain p where p: "∧z. p z tagged_division_of cbox a b"
                         "∧z. (λx. ∩{γ i x |i. i ∈ {0..z}}) fine p z"
    by meson
  have dp: "∧i n. i≤n ==> γ i fine p n"
    using p unfolding fine_Inter
    using atLeastAtMost_iff by blast
  have "Cauchy (λn. sum (λ(x,K). content K *🪙R (f x)) (p n))"
  proof (rule CauchyI)
    fix e::real
    assume "0 < e"
    then obtain N where "N ≠ 0" and N: "inverse (real N) < e"
      using real_arch_inverse[of e] by blast
    show "∃M. ∀m≥M. ∀n≥M. norm ((∑(x,K) ∈ p m. content K *🪙R f x) - (∑(x,K) ∈ p n. content K *🪙R f x)) < e"
    proof (intro exI allI impI)
      fix m n
      assume mn: "N ≤ m" "N ≤ n"
      have "norm ((∑(x,K) ∈ p m. content K *🪙R f x)
                - (∑(x,K) ∈ p n. content K *🪙R f x)) < 1 / (real N + 1)"
        by (simp add: p(1) dp mn γ)
      also have "… < e"
        using  N ‹N ≠ 0› ‹0 🚫› by (auto simp: field_simps)
      finally show "norm ((∑(x,K) ∈ p m. content K *🪙R f x) - (∑(x,K) ∈ p n. content K *🪙R f x)) < e" .
    qed
  qed
  then obtain y where y: "∃no. ∀n≥no. norm ((∑(x,K) ∈ p n. content K *🪙R f x) - y) < r" if "r > 0" for r
    by (auto simp: convergent_eq_Cauchy[symmetric] dest: LIMSEQ_D)
  show ?l
    unfolding integrable_on_def has_integral
  proof (rule_tac x=y in exI, clarify)
    fix e :: real
    assume "e>0"
    then have e2: "e/2 > 0" by auto
    then obtain N1::nat where N1: "N1 ≠ 0" "inverse (real N1) < e/2"
      using real_arch_inverse by blast
    obtain N2::nat where N2: "∧n. n ≥ N2 ==> norm ((∑(x,K) ∈ p n. content K *🪙R f x) - y) < e/2"
      using y[OF e2] by metis
    show "∃γ. gauge γ ∧
              (∀D. D tagged_division_of (cbox a b) ∧ γ fine D ⟶
                norm ((∑(x,K) ∈ D. content K *🪙R f x) - y) < e)"
    proof (intro exI conjI allI impI)
      show "gauge (γ (N1+N2))"
        using γ by auto
      show "norm ((∑(x,K) ∈ q. content K *🪙R f x) - y) < e"
           if "q tagged_division_of cbox a b ∧ γ (N1+N2) fine q" for q
      proof (rule norm_triangle_half_r)
        have "norm ((∑(x,K) ∈ p (N1+N2). content K *🪙R f x)
                  - (∑(x,K) ∈ q. content K *🪙R f x))
               < 1 / (real (N1+N2) + 1)"
          by (rule γ; simp add: dp p that)
        also have "… < e/2"
          using N1 ‹0 🚫› by (auto simp: field_simps intro: less_le_trans)
        finally show "norm ((∑(x,K) ∈ p (N1+N2). content K *🪙R f x) - (∑(x,K) ∈ q. content K *🪙R f x)) < e/2" .
        show "norm ((∑(x,K) ∈ p (N1+N2). content K *🪙R f x) - y) < e/2"
          using N2 le_add_same_cancel2 by blast
      qed
    qed
  qed
qed


subsection ‹Additivity of integral on abutting intervals›

lemma tagged_division_split_left_inj_content:
  assumes D: "D tagged_division_of S"
    and "(x1, K1) ∈ D" "(x2, K2) ∈ D" "K1 ≠ K2" "K1 ∩ {x. x∙k ≤ c} = K2 ∩ {x. x∙k ≤ c}" "k ∈ Basis"
  shows "content (K1 ∩ {x. x∙k ≤ c}) = 0"
proof -
  from tagged_division_ofD(4)[OF D ‹(x1, K1) ∈ D›] obtain a b where K1: "K1 = cbox a b"
    by auto
  then have "interior (K1 ∩ {x. x ∙ k ≤ c}) = {}"
    by (metis tagged_division_split_left_inj assms)
  then show ?thesis
    unfolding K1 interval_split[OF ‹k ∈ Basis›] by (auto simp: content_eq_0_interior)
qed

lemma tagged_division_split_right_inj_content:
  assumes D: "D tagged_division_of S"
    and "(x1, K1) ∈ D" "(x2, K2) ∈ D" "K1 ≠ K2" "K1 ∩ {x. x∙k ≥ c} = K2 ∩ {x. x∙k ≥ c}" "k ∈ Basis"
  shows "content (K1 ∩ {x. x∙k ≥ c}) = 0"
proof -
  from tagged_division_ofD(4)[OF D ‹(x1, K1) ∈ D›] obtain a b where K1: "K1 = cbox a b"
    by auto
  then have "interior (K1 ∩ {x. c ≤ x ∙ k}) = {}"
    by (metis tagged_division_split_right_inj assms)
  then show ?thesis
    unfolding K1 interval_split[OF ‹k ∈ Basis›]
    by (auto simp: content_eq_0_interior)
qed


proposition has_integral_split:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes fi: "(f has_integral i) (cbox a b ∩ {x. x∙k ≤ c})"
      and fj: "(f has_integral j) (cbox a b ∩ {x. x∙k ≥ c})"
      and k: "k ∈ Basis"
shows "(f has_integral (i + j)) (cbox a b)"
  unfolding has_integral
proof clarify
  fix e::real
  assume "0 < e"
  then have e: "e/2 > 0"
    by auto
    obtain γ1 where γ1: "gauge γ1"
      and γ1norm:
        "∧D. [D tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c}; γ1 fine D]
             ==> norm ((∑(x,K) ∈ D. content K *🪙R f x) - i) < e/2"
      by (metis (no_types, lifting) ext e fi has_integral interval_split(1) k)
    obtain γ2 where γ2: "gauge γ2"
      and γ2norm:
        "∧D. [D tagged_division_of cbox a b ∩ {x. c ≤ x ∙ k}; γ2 fine D]
             ==> norm ((∑(x, k) ∈ D. content k *🪙R f x) - j) < e/2"
       apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
       apply (simp add: interval_split[symmetric] k)
       done
  let ?γ = "λx. if x∙k = c then (γ1 x ∩ γ2 x) else ball x ∣x∙k - c∣ ∩ γ1 x ∩ γ2 x"
  have "gauge ?γ"
    using γ1 γ2 unfolding gauge_def by auto
  then show "∃γ. gauge γ ∧
                 (∀D. D tagged_division_of cbox a b ∧ γ fine D ⟶
                      norm ((∑(x, k)∈D. content k *🪙R f x) - (i + j)) < e)"
  proof (rule_tac x="?γ" in exI, safe)
    fix p
    assume p: "p tagged_division_of (cbox a b)" and "?γ fine p"
    have ab_eqp: "cbox a b = ∪{K. ∃x. (x, K) ∈ p}"
      using p by blast
    have xk_le_c: "x∙k ≤ c" if as: "(x,K) ∈ p" and K: "K ∩ {x. x∙k ≤ c} ≠ {}" for x K
    proof (rule ccontr)
      assume **: "¬ x ∙ k ≤ c"
      then have "K ⊆ ball x ∣x ∙ k - c∣"
        using ‹?γ fine p› as by (fastforce simp: not_le algebra_simps)
      with K obtain y where y: "y ∈ ball x ∣x ∙ k - c∣" "y∙k ≤ c"
        by blast
      then have "∣x ∙ k - y ∙ k∣ < ∣x ∙ k - c∣"
        using Basis_le_norm[OF k, of "x - y"]
        by (auto simp: dist_norm inner_diff_left intro: le_less_trans)
      with y show False
        using ** by (auto simp: field_simps)
    qed
    have xk_ge_c: "x∙k ≥ c" if as: "(x,K) ∈ p" and K: "K ∩ {x. x∙k ≥ c} ≠ {}" for x K
    proof (rule ccontr)
      assume **: "¬ x ∙ k ≥ c"
      then have "K ⊆ ball x ∣x ∙ k - c∣"
        using ‹?γ fine p› as by (fastforce simp: not_le algebra_simps)
      with K obtain y where y: "y ∈ ball x ∣x ∙ k - c∣" "y∙k ≥ c"
        by blast
      then have "∣x ∙ k - y ∙ k∣ < ∣x ∙ k - c∣"
        using Basis_le_norm[OF k, of "x - y"]
        by (auto simp: dist_norm inner_diff_left intro: le_less_trans)
      with y show False
        using ** by (auto simp: field_simps)
    qed
    have fin_finite: "finite {(x,f K) | x K. (x,K) ∈ s ∧ P x K}"
      if "finite s" for s and f :: "'a set ==> 'a set" and P :: "'a ==> 'a set ==> bool"
    proof -
      from that have "finite ((λ(x,K). (x, f K)) ` s)"
        by auto
      then show ?thesis
        by (rule rev_finite_subset) auto
    qed
    { fix G :: "'a set ==> 'a set"
      fix i :: "'a × 'a set"
      assume "i ∈ (λ(x, k). (x, G k)) ` p - {(x, G k) |x k. (x, k) ∈ p ∧ G k ≠ {}}"
      then obtain x K where xk: "i = (x, G K)"  "(x,K) ∈ p"
                                "(x, G K) ∉ {(x, G K) |x K. (x,K) ∈ p ∧ G K ≠ {}}"
        by auto
      have "content (G K) = 0"
        using xk using content_empty by auto
      then have "(λ(x,K). content K *🪙R f x) i = 0"
        unfolding xk split_conv by auto
    } note [simp] = this
    have "finite p"
      using p by blast
    let ?M1 = "{(x, K ∩ {x. x∙k ≤ c}) |x K. (x,K) ∈ p ∧ K ∩ {x. x∙k ≤ c} ≠ {}}"
    have γ1_fine: "γ1 fine ?M1"
      using ‹?γ fine p› by (fastforce simp: fine_def split: if_split_asm)
    have "norm ((∑(x, k)∈?M1. content k *🪙R f x) - i) < e/2"
    proof (rule γ1norm [OF tagged_division_ofI γ1_fine])
      show "finite ?M1"
        by (rule fin_finite) (use p in blast)
      show "∪{k. ∃x. (x, k) ∈ ?M1} = cbox a b ∩ {x. x∙k ≤ c}"
        by (auto simp: ab_eqp)

      fix x L
      assume xL: "(x, L) ∈ ?M1"
      then obtain x' L' where xL': "x = x'" "L = L' ∩ {x. x ∙ k ≤ c}"
                                   "(x', L') ∈ p" "L' ∩ {x. x ∙ k ≤ c} ≠ {}"
        by blast
      then obtain a' b' where ab': "L' = cbox a' b'"
        using p by blast
      show "x ∈ L" "L ⊆ cbox a b ∩ {x. x ∙ k ≤ c}"
        using p xk_le_c xL' by auto
      show "∃a b. L = cbox a b"
        using ab' interval_split(1) k xL'(2) by blast

      fix y R
      assume yR: "(y, R) ∈ ?M1"
      then obtain y' R' where yR': "y = y'" "R = R' ∩ {x. x ∙ k ≤ c}"
                                   "(y', R') ∈ p" "R' ∩ {x. x ∙ k ≤ c} ≠ {}"
        by blast
      assume as: "(x, L) ≠ (y, R)"
      show "interior L ∩ interior R = {}"
      proof (cases "L' = R' ⟶ x' = y'")
        case False
        have "interior R' = {}"
          by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
        then show ?thesis
          using yR' by simp
      next
        case True
        then have "L' ≠ R'"
          using as unfolding xL' yR' by auto
        have "interior L' ∩ interior R' = {}"
          by (metis (no_types) Pair_inject ‹L' ≠ R'› p tagged_division_ofD(5) xL'(3) yR'(3))
        then show ?thesis
          using xL'(2) yR'(2) by auto
      qed
    qed
    moreover
    let ?M2 = "{(x,K ∩ {x. x∙k ≥ c}) |x K. (x,K) ∈ p ∧ K ∩ {x. x∙k ≥ c} ≠ {}}"
    have γ2_fine: "γ2 fine ?M2"
      using ‹?γ fine p› by (fastforce simp: fine_def split: if_split_asm)
    have "norm ((∑(x, k)∈?M2. content k *🪙R f x) - j) < e/2"
    proof (rule γ2norm [OF tagged_division_ofI γ2_fine])
      show "finite ?M2"
        by (rule fin_finite) (use p in blast)
      show "∪{k. ∃x. (x, k) ∈ ?M2} = cbox a b ∩ {x. x∙k ≥ c}"
        by (auto simp: ab_eqp)

      fix x L
      assume xL: "(x, L) ∈ ?M2"
      then obtain x' L' where xL': "x = x'" "L = L' ∩ {x. x ∙ k ≥ c}"
                                   "(x', L') ∈ p" "L' ∩ {x. x ∙ k ≥ c} ≠ {}"
        by blast
      then obtain a' b' where ab': "L' = cbox a' b'"
        using p by blast
      show "x ∈ L" "L ⊆ cbox a b ∩ {x. x ∙ k ≥ c}"
        using p xk_ge_c xL' by auto
      show "∃a b. L = cbox a b"
        using p xL' ab' by (auto simp: interval_split[OF k,where c=c])

      fix y R
      assume yR: "(y, R) ∈ ?M2"
      then obtain y' R' where yR': "y = y'" "R = R' ∩ {x. x ∙ k ≥ c}"
                                   "(y', R') ∈ p" "R' ∩ {x. x ∙ k ≥ c} ≠ {}"
        by blast
      assume as: "(x, L) ≠ (y, R)"
      show "interior L ∩ interior R = {}"
      proof (cases "L' = R' ⟶ x' = y'")
        case False
        have "interior R' = {}"
          by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
        then show ?thesis
          using yR' by simp
      next
        case True
        then have "L' ≠ R'"
          using as unfolding xL' yR' by auto
        have "interior L' ∩ interior R' = {}"
          by (metis (no_types) Pair_inject ‹L' ≠ R'› p tagged_division_ofD(5) xL'(3) yR'(3))
        then show ?thesis
          using xL'(2) yR'(2) by auto
      qed
    qed
    ultimately
    have "norm (((∑(x,K) ∈ ?M1. content K *🪙R f x) - i) + ((∑(x,K) ∈ ?M2. content K *🪙R f x) - j)) < e/2 + e/2"
      using norm_add_less by blast
    moreover have "((∑(x,K) ∈ ?M1. content K *🪙R f x) - i) +
                   ((∑(x,K) ∈ ?M2. content K *🪙R f x) - j) =
                   (∑(x, ka)∈p. content ka *🪙R f x) - (i + j)"
    proof -
      have eq0: "∧x y. x = (0::real) ==> x *🪙R (y::'b) = 0"
         by auto
      have cont_eq: "∧g. (λ(x,l). content l *🪙R f x) ∘ (λ(x,l). (x,g l)) = (λ(x,l). content (g l) *🪙R f x)"
        by auto
      have *: "∧G :: 'a set ==> 'a set.
                  (∑(x,K)∈{(x, G K) |x K. (x,K) ∈ p ∧ G K ≠ {}}. content K *🪙R f x) =
                  (∑(x,K)∈(λ(x,K). (x, G K)) ` p. content K *🪙R f x)"
        by (rule sum.mono_neutral_left) (auto simp: ‹finite p›)
      have "((∑(x, k)∈?M1. content k *🪙R f x) - i) + ((∑(x, k)∈?M2. content k *🪙R f x) - j) =
        (∑(x, k)∈?M1. content k *🪙R f x) + (∑(x, k)∈?M2. content k *🪙R f x) - (i + j)"
        by auto
      moreover have "… = (∑(x,K) ∈ p. content (K ∩ {x. x ∙ k ≤ c}) *🪙R f x) +
        (∑(x,K) ∈ p. content (K ∩ {x. c ≤ x ∙ k}) *🪙R f x) - (i + j)"
        unfolding *
        apply (subst (1 2) sum.reindex_nontrivial)
           apply (auto intro!: k p eq0 tagged_division_split_left_inj_content tagged_division_split_right_inj_content
                       simp: cont_eq ‹finite p›)
        done
      moreover have "∧x. x ∈ p ==> (λ(a,B). content (B ∩ {a. a ∙ k ≤ c}) *🪙R f a) x +
                                (λ(a,B). content (B ∩ {a. c ≤ a ∙ k}) *🪙R f a) x =
                                (λ(a,B). content B *🪙R f a) x"
      proof clarify
        fix a B
        assume "(a, B) ∈ p"
        with p obtain u v where uv: "B = cbox u v" by blast
        then show "content (B ∩ {x. x ∙ k ≤ c}) *🪙R f a + content (B ∩ {x. c ≤ x ∙ k}) *??R f a = content B *🪙R f a"
          by (auto simp: scaleR_left_distrib uv content_split[OF k,of u v c])
      qed
      ultimately show ?thesis
        by (auto simp: sum.distrib[symmetric])
      qed
    ultimately show "norm ((∑(x, k)∈p. content k *🪙R f x) - (i + j)) < e"
      by auto
  qed
qed


subsection ‹A sort of converse, integrability on subintervals›

lemma has_integral_separate_sides:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes f: "(f has_integral i) (cbox a b)"
    and "e > 0"
    and k: "k ∈ Basis"
  obtains d where "gauge d"
    "∀p1 p2. p1 tagged_division_of (cbox a b ∩ {x. x∙k ≤ c}) ∧ d fine p1 ∧
        p2 tagged_division_of (cbox a b ∩ {x. x∙k ≥ c}) ∧ d fine p2 ⟶
        norm ((sum (λ(x,k). content k *🪙R f x) p1 + sum (λ(x,k). content k *🪙R f x) p2) - i) < e"
proof -
  obtain γ where d: "gauge γ"
      "∧p. [p tagged_division_of cbox a b; γ fine p]
            ==> norm ((∑(x, k)∈p. content k *🪙R f x) - i) < e"
    using has_integralD[OF f ‹e > 0›] by metis
  { fix p1 p2
    assume tdiv1: "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" and "γ fine p1"
    note p1=tagged_division_ofD[OF this(1)] 
    assume tdiv2: "p2 tagged_division_of (cbox a b) ∩ {x. c ≤ x ∙ k}" and "γ fine p2"
    note p2=tagged_division_ofD[OF this(1)] 
    note tagged_division_Un_interval[OF tdiv1 tdiv2] 
    note p12 = tagged_division_ofD[OF this] this
    { fix a b
      assume ab: "(a, b) ∈ p1 ∩ p2"
      have "(a, b) ∈ p1"
        using ab by auto
      obtain u v where uv: "b = cbox u v"
        using ‹(a, b) ∈ p1› p1(4) by moura
      have "b ⊆ {x. x∙k = c}"
        using ab p1(3)[of a b] p2(3)[of a b] by fastforce
      moreover
      have "interior {x::'a. x ∙ k = c} = {}"
      proof (rule ccontr)
        assume "¬ ?thesis"
        then obtain x where x: "x ∈ interior {x::'a. x∙k = c}"
          by auto
        then obtain ε where "0 < ε" and ε: "ball x ε ⊆ {x. x ∙ k = c}"
          using mem_interior by metis
        have x: "x∙k = c"
          using x interior_subset by fastforce
        have "(∑i∈Basis. ∣(x - (x + (ε/2 ) *🪙R k)) ∙ i∣) =
              (∑i∈Basis. (if i = k then ε/2 else 0))"
          using ‹0 🚫ε› k by (intro sum.cong) (auto simp: inner_not_same_Basis)
        also have "… < ε"
          by (subst sum.delta) (use ‹0 🚫ε› in auto)
        finally have "x + (ε/2) *🪙R k ∈ ball x ε"
          unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
        then have "x + (ε/2) *🪙R k ∈ {x. x∙k = c}"
          using ε by auto
        then show False
          using ‹0 🚫ε› x k by (auto simp: inner_simps)
      qed
      ultimately have "content b = 0"
        unfolding uv content_eq_0_interior
        using interior_mono by blast
      then have "content b *🪙R f a = 0"
        by auto
    }
    then have "norm ((∑(x, k)∈p1. content k *🪙R f x) + (∑(x, k)∈p2. content k *🪙R f x) - i) =
               norm ((∑(x, k)∈p1 ∪ p2. content k *🪙R f x) - i)"
      by (subst sum.union_inter_neutral) (auto simp: p1 p2)
    also have "… < e"
      using d(2) p12 by (simp add: fine_Un k ‹γ fine p1› ‹γ fine p2›)
    finally have "norm ((∑(x, k)∈p1. content k *🪙R f x) + (∑(x, k)∈p2. content k *🪙R f x) - i) < e" .
   }
  then show ?thesis
    using d(1) that by auto
qed

lemma integrable_split [intro]:
  fixes f :: "'a::euclidean_space ==> 'b::{real_normed_vector,complete_space}"
  assumes f: "f integrable_on cbox a b"
      and k: "k ∈ Basis"
    shows "f integrable_on (cbox a b ∩ {x. x∙k ≤ c})"   (is ?thesis1)
    and   "f integrable_on (cbox a b ∩ {x. x∙k ≥ c})"   (is ?thesis2)
proof -
  obtain y where y: "(f has_integral y) (cbox a b)"
    using f by blast
  define a' where "a' = (∑i∈Basis. (if i = k then max (a∙k) c else a∙i)*🪙R i)"
  define b' where "b' = (∑i∈Basis. (if i = k then min (b∙k) c else b∙i)*🪙R i)"
  have "∃d. gauge d ∧
            (∀p1 p2. p1 tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c} ∧ d fine p1 ∧
                     p2 tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c} ∧ d fine p2 ⟶
                     norm ((∑(x,K) ∈ p1. content K *🪙R f x) - (∑(x,K) ∈ p2. content K *🪙R f x)) < e)"
    if "e > 0" for e
  proof -
    have "e/2 > 0" using that by auto
  with has_integral_separate_sides[OF y this k, of c]
  obtain d
    where "gauge d"
         and d: "∧p1 p2. [p1 tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c}; d fine p1;
                          p2 tagged_division_of cbox a b ∩ {x. c ≤ x ∙ k}; d fine p2]
                  ==> norm ((∑(x,K)∈p1. content K *🪙R f x) + (∑(x,K)∈p2. content K *🪙R f x) - y) < e/2"
    by metis
  show ?thesis
    proof (rule_tac x=d in exI, clarsimp simp add: ‹gauge d›)
      fix p1 p2
      assume as: "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p1"
                 "p2 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p2"
      show "norm ((∑(x, k)∈p1. content k *🪙R f x) - (∑(x, k)∈p2. content k *🪙R f x)) < e"
      proof (rule fine_division_exists[OF ‹gauge d›, of a' b])
        fix p
        assume "p tagged_division_of cbox a' b" "d fine p"
        then show ?thesis
          using as norm_triangle_half_l[OF d[of p1 p] d[of p2 p]]
          unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
          by (auto simp: algebra_simps)
      qed
    qed
  qed
  with f show ?thesis1
    by (simp add: interval_split[OF k] integrable_Cauchy)
  have "∃d. gauge d ∧
            (∀p1 p2. p1 tagged_division_of cbox a b ∩ {x. x ∙ k ≥ c} ∧ d fine p1 ∧
                     p2 tagged_division_of cbox a b ∩ {x. x ∙ k ≥ c} ∧ d fine p2 ⟶
                     norm ((∑(x,K) ∈ p1. content K *🪙R f x) - (∑(x,K) ∈ p2. content K *🪙R f x)) < e)"
    if "e > 0" for e
  proof -
    have "e/2 > 0" using that by auto
  with has_integral_separate_sides[OF y this k, of c]
  obtain d
    where "gauge d"
         and d: "∧p1 p2. [p1 tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c}; d fine p1;
                          p2 tagged_division_of cbox a b ∩ {x. c ≤ x ∙ k}; d fine p2]
                  ==> norm ((∑(x,K)∈p1. content K *🪙R f x) + (∑(x,K)∈p2. content K *🪙R f x) - y) < e/2"
    by metis
  show ?thesis
    proof (rule_tac x=d in exI, clarsimp simp add: ‹gauge d›)
      fix p1 p2
      assume as: "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≥ c}" "d fine p1"
                 "p2 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≥ c}" "d fine p2"
      show "norm ((∑(x, k)∈p1. content k *🪙R f x) - (∑(x, k)∈p2. content k *🪙R f x)) < e"
      proof (rule fine_division_exists[OF ‹gauge d›, of a b'])
        fix p
        assume "p tagged_division_of cbox a b'" "d fine p"
        then show ?thesis
          using as norm_triangle_half_l[OF d[of p p1] d[of p p2]]
          unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
          by (auto simp: algebra_simps)
      qed
    qed
  qed
  with f show ?thesis2
    by (simp add: interval_split[OF k] integrable_Cauchy)
qed

lemma operative_integralI:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  shows "operative (lift_option (+)) (Some 0)
    (λi. if f integrable_on i then Some (integral i f) else None)"
proof -
  interpret comm_monoid "lift_option plus" "Some (0::'b)"
    by (simp add: add.comm_monoid_axioms comm_monoid_lift_option)
  show ?thesis
  proof
    fix a b c
    fix k :: 'a
    assume k: "k ∈ Basis"
    show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
          lift_option (+) (if f integrable_on cbox a b ∩ {x. x ∙ k ≤ c} then Some (integral (cbox a b ∩ {x. x ∙ k ≤ c}) f) else None)
          (if f integrable_on cbox a b ∩ {x. c ≤ x ∙ k} then Some (integral (cbox a b ∩ {x. c ≤ x ∙ k}) f) else None)"
    proof (cases "f integrable_on cbox a b")
      case True
      with k show ?thesis
        by (auto simp: integrable_split intro: integral_unique [OF has_integral_split[OF _ _ k]])
    next
    case False
      have "¬ (f integrable_on cbox a b ∩ {x. x ∙ k ≤ c}) ∨ ¬ ( f integrable_on cbox a b ∩ {x. c ≤ x ∙ k})"
      proof (rule ccontr)
        assume "¬ ?thesis"
        then have "f integrable_on cbox a b"
          unfolding integrable_on_def using has_integral_split k by blast
        then show False
          using False by auto
      qed
      then show ?thesis
        using False by auto
    qed
  qed (auto simp: content_eq_0_interior)
qed

subsection ‹Bounds on the norm of Riemann sums and the integral itself›

lemma dsum_bound:
  assumes p: "p division_of (cbox a b)"
    and "norm c ≤ e"
  shows "norm (∑l∈p. content l *🪙R c) ≤ e * content(cbox a b)"
  by (metis abs_of_nonneg assms measure_nonneg mult.commute mult_right_mono norm_scaleR
      scaleR_left.sum sum_content.division)

lemma rsum_bound:
  assumes p: "p tagged_division_of (cbox a b)"
      and "∀x∈cbox a b. norm (f x) ≤ e"
    shows "norm (∑(x, k)∈p. content k *🪙R f x) ≤ e * content (cbox a b)"
proof (cases "cbox a b = {}")
  case True show ?thesis
    using p unfolding True tagged_division_of_trivial by auto
next
  case False
  then have e: "e ≥ 0"
    by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
  have sum_le: "sum (content ∘ snd) p ≤ content (cbox a b)"
    unfolding additive_content_tagged_division[OF p, symmetric] split_def
    by (auto intro: eq_refl)
  have con: "∧xk. xk ∈ p ==> 0 ≤ content (snd xk)"
    using tagged_division_ofD(4) [OF p] content_pos_le
    by force
  have "norm (sum (λ(x,k). content k *🪙R f x) p) ≤ (∑i∈p. norm (case i of (x, k) ==> content k *🪙R f x))"
    by (rule norm_sum)
  also have "… ≤ e * content (cbox a b)"
  proof -
    have "∧xk. xk ∈ p ==> norm (f (fst xk)) ≤ e"
      using assms(2) p tag_in_interval by force
    moreover have "(∑i∈p. ∣content (snd i)∣ * e) ≤ e * content (cbox a b)"
      unfolding sum_distrib_right[symmetric]
      using con sum_le by (auto simp: mult.commute intro: mult_left_mono [OF _ e])
    ultimately show ?thesis
      unfolding split_def norm_scaleR
      by (metis (no_types, lifting) mult_left_mono[OF _ abs_ge_zero]   order_trans[OF sum_mono])
  qed
  finally show ?thesis .
qed

lemma rsum_diff_bound:
  assumes "p tagged_division_of (cbox a b)"
    and "∀x∈cbox a b. norm (f x - g x) ≤ e"
  shows "norm (sum (λ(x,k). content k *🪙R f x) p - sum (λ(x,k). content k *🪙R g x) p) ≤
         e * content (cbox a b)"
  using order_trans[OF _ rsum_bound[OF assms]]
  by (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)

lemma has_integral_bound:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "0 ≤ B"
      and f: "(f has_integral i) (cbox a b)"
      and "∧x. x∈cbox a b ==> norm (f x) ≤ B"
    shows "norm i ≤ B * content (cbox a b)"
proof (rule ccontr)
  assume "¬ ?thesis"
  then have "norm i - B * content (cbox a b) > 0"
    by auto
  with f[unfolded has_integral]
  obtain γ where "gauge γ" and γ:
    "∧p. [p tagged_division_of cbox a b; γ fine p]
          ==> norm ((∑(x, K)∈p. content K *🪙R f x) - i) < norm i - B * content (cbox a b)"
    by metis
  then obtain p where p: "p tagged_division_of cbox a b" and "γ fine p"
    using fine_division_exists by blast
  have "∧s B. norm s ≤ B ==> ¬ norm (s - i) < norm i - B"
    unfolding not_less
    by (metis diff_left_mono dist_commute dist_norm norm_triangle_ineq2 order_trans)
  then show False
    using γ [OF p ‹γ fine p›] rsum_bound[OF p] assms by metis
qed

corollary integrable_bound:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "0 ≤ B"
    and "f integrable_on (cbox a b)"
    and "∧x. x∈cbox a b ==> norm (f x) ≤ B"
  shows "norm (integral (cbox a b) f) ≤ B * content (cbox a b)"
  by (metis integrable_integral has_integral_bound assms)


subsection ‹Similar theorems about relationship among components›

lemma rsum_component_le:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes p: "p tagged_division_of (cbox a b)"
      and "∧x. x ∈ cbox a b ==> (f x)∙i ≤ (g x)∙i"
    shows "(∑(x, K)∈p. content K *🪙R f x) ∙ i ≤ (∑(x, K)∈p. content K *🪙R g x) ∙ i"
proof -
have "∧a b. (a, b) ∈ p ==> content b *🪙R f a ∙ i ≤ content b *🪙R g a ∙ i"
  by (metis assms(2) inner_commute inner_scaleR_right mult_left_mono
      measure_nonneg p tag_in_interval)
  then show ?thesis
    by (simp add: inner_sum_left split_def sum_mono)
qed

lemma has_integral_component_le:
  fixes f g :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes k: "k ∈ Basis"
  assumes "(f has_integral i) S" "(g has_integral j) S"
    and f_le_g: "∧x. x ∈ S ==> (f x)∙k ≤ (g x)∙k"
  shows "i∙k ≤ j∙k"
proof -
  have ik_le_jk: "i∙k ≤ j∙k"
    if f_i: "(f has_integral i) (cbox a b)" and g_j: "(g has_integral j) (cbox a b)"
    and le: "∀x∈cbox a b. (f x)∙k ≤ (g x)∙k" 
    for a b i and j :: 'b and f g :: "'a ==> 'b"
  proof (rule ccontr)
    assume "¬ ?thesis"
    then have *: "0 < (i∙k - j∙k) / 3"
      by auto
    obtain γ1 where "gauge γ1" 
      and γ1: "∧p. [p tagged_division_of cbox a b; γ1 fine p]
                ==> norm ((∑(x, k)∈p. content k *🪙R f x) - i) < (i ∙ k - j ∙ k) / 3"
      by (metis "*" f_i has_integral)
    obtain γ2 where "gauge γ2" 
      and γ2: "∧p. [p tagged_division_of cbox a b; γ2 fine p]
                ==> norm ((∑(x, k)∈p. content k *🪙R g x) - j) < (i ∙ k - j ∙ k) / 3"
      by (metis "*" g_j has_integral)
    obtain p where p: "p tagged_division_of cbox a b" and "γ1 fine p" "γ2 fine p"
      using fine_division_exists[OF gauge_Int[OF ‹gauge γ1› ‹gauge γ2›]] 
      unfolding fine_Int
       by metis
    then have "∣((∑(x, k)∈p. content k *🪙R f x) - i) ∙ k∣ < (i ∙ k - j ∙ k) / 3"
         "∣((∑(x, k)∈p. content k *🪙R g x) - j) ∙ k∣ < (i ∙ k - j ∙ k) / 3"
      using le_less_trans[OF Basis_le_norm[OF k]] k γ1 γ2 by metis+ 
    then show False
      unfolding inner_simps
      using rsum_component_le[OF p] le
      by (fastforce simp: abs_real_def split: if_split_asm)
  qed
  show ?thesis
  proof (cases "∃a b. S = cbox a b")
    case True
    with ik_le_jk assms show ?thesis
      by auto
  next
    case False
    show ?thesis
    proof (rule ccontr)
      assume "¬ i∙k ≤ j∙k"
      then have ij: "(i∙k - j∙k) / 3 > 0"
        by auto
      obtain B1 where "0 < B1" 
           and B1: "∧a b. ball 0 B1 ⊆ cbox a b ==>
                    ∃z. ((λx. if x ∈ S then f x else 0) has_integral z) (cbox a b) ∧
                        norm (z - i) < (i ∙ k - j ∙ k) / 3"
        using has_integral_altD[OF _ False ij] assms by blast
      obtain B2 where "0 < B2" 
           and B2: "∧a b. ball 0 B2 ⊆ cbox a b ==>
                    ∃z. ((λx. if x ∈ S then g x else 0) has_integral z) (cbox a b) ∧
                        norm (z - j) < (i ∙ k - j ∙ k) / 3"
        using has_integral_altD[OF _ False ij] assms by blast
      have "bounded (ball 0 B1 ∪ ball (0::'a) B2)"
        unfolding bounded_Un by(rule conjI bounded_ball)+
      from bounded_subset_cbox_symmetric[OF this] 
      obtain a b::'a where ab: "ball 0 B1 ⊆ cbox a b" "ball 0 B2 ⊆ cbox a b"
        by (meson Un_subset_iff)
      then obtain w1 w2 where int_w1: "((λx. if x ∈ S then f x else 0) has_integral w1) (cbox a b)"
                        and norm_w1:  "norm (w1 - i) < (i ∙ k - j ∙ k) / 3"
                        and int_w2: "((λx. if x ∈ S then g x else 0) has_integral w2) (cbox a b)"
                        and norm_w2: "norm (w2 - j) < (i ∙ k - j ∙ k) / 3"
        using B1 B2 by blast
      have *: "∧w1 w2 j i::real .∣w1 - i∣ < (i - j) / 3 ==> ∣w2 - j∣ < (i - j) / 3 ==> w1 ≤ w2 ==> False"
        by (simp add: abs_real_def split: if_split_asm)
      have "∣(w1 - i) ∙ k∣ < (i ∙ k - j ∙ k) / 3"
           "∣(w2 - j) ∙ k∣ < (i ∙ k - j ∙ k) / 3"
        using Basis_le_norm k le_less_trans norm_w1 norm_w2 by blast+
      moreover
      have "w1∙k ≤ w2∙k"
        using ik_le_jk int_w1 int_w2 f_le_g by auto
      ultimately show False
        unfolding inner_simps by(rule *)
    qed
  qed
qed

lemma integral_component_le:
  fixes g f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "k ∈ Basis"
    and "f integrable_on S" "g integrable_on S"
    and "∧x. x ∈ S ==> (f x)∙k ≤ (g x)∙k"
  shows "(integral S f)∙k ≤ (integral S g)∙k"
  using has_integral_component_le assms by blast

lemma has_integral_component_nonneg:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "k ∈ Basis"
    and "(f has_integral i) S"
    and "∧x. x ∈ S ==> 0 ≤ (f x)∙k"
  shows "0 ≤ i∙k"
  by (metis (no_types, lifting) assms euclidean_all_zero_iff has_integral_0 has_integral_component_le)

lemma integral_component_nonneg:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "k ∈ Basis"
    and  "∧x. x ∈ S ==> 0 ≤ (f x)∙k"
  shows "0 ≤ (integral S f)∙k"
  by (metis assms order.refl has_integral_component_nonneg has_integral_integral
      inner_zero_left not_integrable_integral)

lemma has_integral_component_neg:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "k ∈ Basis"
    and "(f has_integral i) S"
    and "∧x. x ∈ S ==> (f x)∙k ≤ 0"
  shows "i∙k ≤ 0"
  by (metis (no_types, lifting) assms has_integral_0 has_integral_component_le inner_zero_left)

lemma has_integral_component_lbound:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "(f has_integral i) (cbox a b)"
    and "∀x∈cbox a b. B ≤ f(x)∙k"
    and "k ∈ Basis"
  shows "B * content (cbox a b) ≤ i∙k"
  using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(∑i∈Basis. B *🪙R i)::'b"] assms(2-)
  by (auto simp: field_simps)

lemma has_integral_component_ubound:
  fixes f::"'a::euclidean_space => 'b::euclidean_space"
  assumes "(f has_integral i) (cbox a b)"
    and "∀x∈cbox a b. f x∙k ≤ B"
    and "k ∈ Basis"
  shows "i∙k ≤ B * content (cbox a b)"
  using has_integral_component_le[OF assms(3,1) has_integral_const, of "∑i∈Basis. B *🪙R i"] assms(2-)
  by (auto simp: field_simps)

lemma integral_component_lbound:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "f integrable_on cbox a b"
    and "∀x∈cbox a b. B ≤ f(x)∙k"
    and "k ∈ Basis"
  shows "B * content (cbox a b) ≤ (integral(cbox a b) f)∙k"
  using assms has_integral_component_lbound by blast

lemma integral_component_lbound_real:
  assumes "f integrable_on {a ::real..b}"
    and "∀x∈{a..b}. B ≤ f(x)∙k"
    and "k ∈ Basis"
  shows "B * content {a..b} ≤ (integral {a..b} f)∙k"
  using assms by (metis box_real(2) integral_component_lbound)

lemma integral_component_ubound:
  fixes f :: "'a::euclidean_space ==> 'b::euclidean_space"
  assumes "f integrable_on cbox a b"
    and "∀x∈cbox a b. f x∙k ≤ B"
    and "k ∈ Basis"
  shows "(integral (cbox a b) f)∙k ≤ B * content (cbox a b)"
  using assms has_integral_component_ubound by blast

lemma integral_component_ubound_real:
  fixes f :: "real ==> 'a::euclidean_space"
  assumes "f integrable_on {a..b}"
    and "∀x∈{a..b}. f x∙k ≤ B"
    and "k ∈ Basis"
  shows "(integral {a..b} f)∙k ≤ B * content {a..b}"
  using assms by (metis box_real(2) integral_component_ubound)

subsection ‹Uniform limit of integrable functions is integrable›

lemma real_arch_invD:
  "0 < (e::real) ==> (∃n::nat. n ≠ 0 ∧ 0 < inverse (real n) ∧ inverse (real n) < e)"
  by (subst(asm) real_arch_inverse)


lemma integrable_uniform_limit:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "∧e. e > 0 ==> ∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
  shows "f integrable_on cbox a b"
proof (cases "content (cbox a b) > 0")
  case False then show ?thesis
    using has_integral_null by (simp add: content_lt_nz integrable_on_def)
next
  case True
  have "1 / (real n + 1) > 0" for n
    by auto
  then have "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ 1 / (real n + 1)) ∧ g integrable_on cbox a b" for n
    using assms by blast
  then obtain g where g_near_f: "∧n x. x ∈ cbox a b ==> norm (f x - g n x) ≤ 1 / (real n + 1)"
                  and int_g: "∧n. g n integrable_on cbox a b"
    by metis
  then obtain h where h: "∧n. (g n has_integral h n) (cbox a b)"
    unfolding integrable_on_def by metis
  have "Cauchy h"
    unfolding Cauchy_def
  proof clarify
    fix e :: real
    assume "e>0"
    then have "e/4 / content (cbox a b) > 0"
      using True by (auto simp: field_simps)
    then obtain M where "M ≠ 0" and M: "1 / (real M) < e/4 / content (cbox a b)"
      by (metis inverse_eq_divide real_arch_inverse)
    show "∃M. ∀m≥M. ∀n≥M. dist (h m) (h n) < e"
    proof (intro exI strip)
      fix m n
      assume m: "M ≤ m" and n: "M ≤ n"
      have "e/4>0" using ‹e>0› by auto
      then obtain gm gn where "gauge gm" "gauge gn"
              and gm: "∧D. D tagged_division_of cbox a b ∧ gm fine D
                            ==> norm ((∑(x,K) ∈ D. content K *🪙R g m x) - h m) < e/4"
              and gn: "∧D. D tagged_division_of cbox a b ∧ gn fine D ==>
                      norm ((∑(x,K) ∈ D. content K *🪙R g n x) - h n) < e/4"
        using h[unfolded has_integral] by meson
      then obtain D where D: "D tagged_division_of cbox a b" "(λx. gm x ∩ gn x) fine D"
        by (metis (full_types) fine_division_exists gauge_Int)
      have triangle3: "norm (i1 - i2) < e"
        if no: "norm(s2 - s1) ≤ e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4" for s1 s2 i1 and i2::'b
      proof -
        have "norm (i1 - i2) ≤ norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
          using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
          using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
          by (auto simp: algebra_simps)
        also have "… < e"
          using no by (auto simp: algebra_simps norm_minus_commute)
        finally show ?thesis .
      qed
      have finep: "gm fine D" "gn fine D"
        using fine_Int D  by auto
      have norm_le: "norm (g n x - g m x) ≤ 2 / real M" if x: "x ∈ cbox a b" for x
      proof -
        have "norm (f x - g n x) + norm (f x - g m x) ≤ 1 / (real n + 1) + 1 / (real m + 1)"          
          using g_near_f[OF x, of n] g_near_f[OF x, of m] by simp
        also have "… ≤ 1 / (real M) + 1 / (real M)"
          using ‹M ≠ 0› m n by (intro add_mono; force simp: field_split_simps)
        also have "… = 2 / real M"
          by auto
        finally show "norm (g n x - g m x) ≤ 2 / real M"
          using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
          by (auto simp: algebra_simps simp add: norm_minus_commute)
      qed
      have "norm ((∑(x,K) ∈ D. content K *🪙R g n x) - (∑(x,K) ∈ D. content K *🪙R g m x)) ≤ 2 / real M * content (cbox a b)"
        by (blast intro: norm_le rsum_diff_bound[OF D(1), where e="2 / real M"])
      also have "… ≤ e/2"
        using M True
        by (auto simp: field_simps)
      finally have le_e2: "norm ((∑(x,K) ∈ D. content K *🪙R g n x) - (∑(x,K) ∈ D. content K *🪙R g m x)) ≤ e/2" .
      then show "dist (h m) (h n) < e"
        unfolding dist_norm using gm gn D finep by (auto intro!: triangle3)
    qed
  qed
  then obtain s where s: "h <---- s"
    using convergent_eq_Cauchy[symmetric] by blast
  show ?thesis
    unfolding integrable_on_def has_integral
  proof (rule_tac x=s in exI, clarify)
    fix e::real
    assume e: "0 < e"
    then have "e/3 > 0" by auto
    then obtain N1 where N1: "∀n≥N1. norm (h n - s) < e/3"
      using LIMSEQ_D [OF s] by metis
    from e True have "e/3 / content (cbox a b) > 0"
      by (auto simp: field_simps)
    then obtain N2 :: nat
         where "N2 ≠ 0" and N2: "1 / (real N2) < e/3 / content (cbox a b)"
      by (metis inverse_eq_divide real_arch_inverse)
    obtain g' where "gauge g'"
            and g': "∧D. D tagged_division_of cbox a b ∧ g' fine D ==>
                    norm ((∑(x,K) ∈ D. content K *🪙R g (N1 + N2) x) - h (N1 + N2)) < e/3"
      by (metis h has_integral ‹e/3 > 0›)
    have *: "norm (sf - s) < e" 
        if no: "norm (sf - sg) ≤ e/3" "norm(h - s) < e/3" "norm (sg - h) < e/3" for sf sg h
    proof -
      have "norm (sf - s) ≤ norm (sf - sg) + norm (sg - h) + norm (h - s)"
        using norm_triangle_ineq[of "sf - sg" "sg - s"]
        using norm_triangle_ineq[of "sg - h" " h - s"]
        by (auto simp: algebra_simps)
      also have "… < e"
        using no by (auto simp: algebra_simps norm_minus_commute)
      finally show ?thesis .
    qed
    { fix D
      assume ptag: "D tagged_division_of (cbox a b)" and "g' fine D"
      then have norm_less: "norm ((∑(x,K) ∈ D. content K *🪙R g (N1 + N2) x) - h (N1 + N2)) < e/3"
        using g' by blast
      have "content (cbox a b) < e/3 * (of_nat N2)"
        using ‹N2 ≠ 0› N2 using True by (auto simp: field_split_simps)
      moreover have "e/3 * of_nat N2 ≤ e/3 * (of_nat (N1 + N2) + 1)"
        using ‹e>0› by auto
      ultimately have "content (cbox a b) < e/3 * (of_nat (N1 + N2) + 1)"
        by linarith
      then have le_e3: "1 / (real (N1 + N2) + 1) * content (cbox a b) ≤ e/3"
        unfolding inverse_eq_divide
        by (auto simp: field_simps)
      have ne3: "norm (h (N1 + N2) - s) < e/3"
        using N1 by auto
      have "norm ((∑(x,K) ∈ D. content K *🪙R f x) - (∑(x,K) ∈ D. content K *🪙R g (N1 + N2) x))
            ≤ 1 / (real (N1 + N2) + 1) * content (cbox a b)"
        by (blast intro: g_near_f rsum_diff_bound[OF ptag])
      then have "norm ((∑(x,K) ∈ D. content K *🪙R f x) - s) < e"
        by (rule *[OF order_trans [OF _ le_e3] ne3 norm_less])
    }
    then show "∃d. gauge d ∧
             (∀D. D tagged_division_of cbox a b ∧ d fine D ⟶ norm ((∑(x,K) ∈ D. content K *🪙R f x) - s) < e)"
      by (blast intro: g' ‹gauge g'›)
  qed
qed

lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]


subsection ‹Negligible sets›

definition "negligible (s:: 'a::euclidean_space set) ⟷
  (∀a b. ((indicator s :: 'a==>real) has_integral 0) (cbox a b))"


subsubsection ‹Negligibility of hyperplane›

lemma content_doublesplit:
  fixes a :: "'a::euclidean_space"
  assumes "0 < e"
    and k: "k ∈ Basis"
  obtains d where "0 < d" and "content (cbox a b ∩ {x. ∣x∙k - c∣ ≤ d}) < e"
proof cases
  assume *: "a ∙ k ≤ c ∧ c ≤ b ∙ k ∧ (∀j∈Basis. a ∙ j ≤ b ∙ j)"
  define a' where "a' d = (∑j∈Basis. (if j = k then max (a∙j) (c - d) else a∙j) *🪙R j)" for d
  define b' where "b' d = (∑j∈Basis. (if j = k then min (b∙j) (c + d) else b∙j) *🪙R j)" for d

  have "((λd. ∏j∈Basis. (b' d - a' d) ∙ j) ---> (∏j∈Basis. (b' 0 - a' 0) ∙ j)) (at_right 0)"
    by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros)
  also have "(∏j∈Basis. (b' 0 - a' 0) ∙ j) = 0"
    using k * unfolding b'_def a'_def
    by (auto simp: inner_diff intro!: prod_zero sum.cong)
  also have "((λd. ∏j∈Basis. (b' d - a' d) ∙ j) ---> 0) (at_right 0) =
    ((λd. content (cbox a b ∩ {x. ∣x∙k - c∣ ≤ d})) ---> 0) (at_right 0)"
  proof (intro tendsto_cong eventually_at_rightI)
    fix d :: real assume d: "d ∈ {0<..<1}"
    have "cbox a b ∩ {x. ∣x∙k - c∣ ≤ d} = cbox (a' d) (b' d)" for d
      using * d k by (auto simp: cbox_def set_eq_iff Int_def ball_conj_distrib abs_diff_le_iff a'_def b'_def)
    moreover have "j ∈ Basis ==> a' d ∙ j ≤ b' d ∙ j" for j
      using * d k by (auto simp: a'_def b'_def)
    ultimately show "(∏j∈Basis. (b' d - a' d) ∙ j) = content (cbox a b ∩ {x. ∣x∙k - c∣ ≤ d})"
      by simp
  qed simp
  finally have "((λd. content (cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ d})) ---> 0) (at_right 0)" .
  from order_tendstoD(2)[OF this ‹0🚫›]
  obtain d' where "0 < d'" and d': "∧y. y > 0 ==> y < d' ==> content (cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ y}) < e"
    by (subst (asm) eventually_at_right[of _ 1]) auto
  show ?thesis
    by (rule that[of "d'/2"], insert ‹0🚫› d'[of "d'/2"], auto)
next
  assume *: "¬ (a ∙ k ≤ c ∧ c ≤ b ∙ k ∧ (∀j∈Basis. a ∙ j ≤ b ∙ j))"
  then have "(∃j∈Basis. b ∙ j < a ∙ j) ∨ (c < a ∙ k ∨ b ∙ k < c)"
    by (auto simp: not_le)
  show thesis
  proof cases
    assume "∃j∈Basis. b ∙ j < a ∙ j"
    then have [simp]: "cbox a b = {}"
      using box_ne_empty(1)[of a b] by auto
    show ?thesis
      by (rule that[of 1]) (simp_all add: ‹0🚫›)
  next
    assume "¬ (∃j∈Basis. b ∙ j < a ∙ j)"
    with * have "c < a ∙ k ∨ b ∙ k < c"
      by auto
    then show thesis
    proof
      assume c: "c < a ∙ k"
      moreover have "x ∈ cbox a b ==> c ≤ x ∙ k" for x
        using k c by (auto simp: cbox_def)
      ultimately have "cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ (a ∙ k - c)/2} = {}"
        using k by (auto simp: cbox_def)
      with ‹0🚫› c that[of "(a ∙ k - c)/2"] show ?thesis
        by auto
    next
      assume c: "b ∙ k < c"
      moreover have "x ∈ cbox a b ==> x ∙ k ≤ c" for x
        using k c by (auto simp: cbox_def)
      ultimately have "cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ (c - b ∙ k)/2} = {}"
        using k by (auto simp: cbox_def)
      with ‹0🚫› c that[of "(c - b ∙ k)/2"] show ?thesis
        by auto
    qed
  qed
qed


proposition negligible_standard_hyperplane[intro]:
  fixes k :: "'a::euclidean_space"
  assumes k: "k ∈ Basis"
  shows "negligible {x. x∙k = c}"
  unfolding negligible_def has_integral
proof clarsimp
  fix a b and e::real assume "e > 0"
  with k obtain d where "0 < d" and d: "content (cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ d}) < e"
    by (metis content_doublesplit)
  let ?i = "indicator {x::'a. x∙k = c} :: 'a==>real"
  show "∃γ. gauge γ ∧
           (∀D. D tagged_division_of cbox a b ∧ γ fine D ⟶
                 ∣∑(x,K) ∈ D. content K * ?i x∣ < e)"
  proof (intro exI, safe)
    show "gauge (λx. ball x d)"
      using ‹0 🚫› by blast
  next
    fix D
    assume p: "D tagged_division_of (cbox a b)" "(λx. ball x d) fine D"
    have "content L = content (L ∩ {x. ∣x ∙ k - c∣ ≤ d})" 
      if "(x, L) ∈ D" "?i x ≠ 0" for x L
    proof -
      have xk: "x∙k = c"
        using that by (simp add: indicator_def split: if_split_asm)
      have "L ⊆ {x. ∣x ∙ k - c∣ ≤ d}"
      proof 
        fix y
        assume y: "y ∈ L"
        have "L ⊆ ball x d"
          using p(2) that(1) by auto
        then have "norm (x - y) < d"
          by (simp add: dist_norm subset_iff y)
        then have "∣(x - y) ∙ k∣ < d"
          using k norm_bound_Basis_lt by blast
        then show "y ∈ {x. ∣x ∙ k - c∣ ≤ d}"
          unfolding inner_simps xk by auto
      qed 
      then show "content L = content (L ∩ {x. ∣x ∙ k - c∣ ≤ d})"
        by (metis inf.orderE)
    qed
    then have *: "(∑(x,K)∈D. content K * ?i x) = (∑(x,K)∈D. content (K ∩ {x. ∣x∙k - c∣ ≤ d}) *🪙R ?i x)"
      by (force simp: split_paired_all intro!: sum.cong [OF refl])
    note p'= tagged_division_ofD[OF p(1)] and p''=division_of_tagged_division[OF p(1)]
    have "(∑(x,K)∈D. content (K ∩ {x. ∣x ∙ k - c∣ ≤ d}) * indicator {x. x ∙ k = c} x) < e"
    proof -
      have "(∑(x,K)∈D. content (K ∩ {x. ∣x ∙ k - c∣ ≤ d}) * ?i x) ≤ (∑(x,K)∈D. content (K ∩ {x. ∣x ∙ k - c∣ ≤ d}))"
        by (force simp: indicator_def intro!: sum_mono)
      also have "… < e"
      proof (subst sum.over_tagged_division_lemma[OF p(1)])
        fix u v::'a
        assume "box u v = {}"
        then have *: "content (cbox u v) = 0"
          unfolding content_eq_0_interior by simp
        have "cbox u v ∩ {x. ∣x ∙ k - c∣ ≤ d} ⊆ cbox u v"
          by auto
        then have "content (cbox u v ∩ {x. ∣x ∙ k - c∣ ≤ d}) ≤ content (cbox u v)"
          unfolding interval_doublesplit[OF k] by (rule content_subset)
        then show "content (cbox u v ∩ {x. ∣x ∙ k - c∣ ≤ d}) = 0"
          unfolding * interval_doublesplit[OF k]
          by (blast intro: antisym)
      next
        have "(∑l∈snd ` D. content (l ∩ {x. ∣x ∙ k - c∣ ≤ d})) =
          sum content ((λl. l ∩ {x. ∣x ∙ k - c∣ ≤ d})`{l∈snd ` D. l ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}})"
        proof (subst (2) sum.reindex_nontrivial)
          fix x y assume "x ∈ {l ∈ snd ` D. l ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}}" "y ∈ {l ∈ snd ` D. l ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}}"
            "x ≠ y" and eq: "x ∩ {x. ∣x ∙ k - c∣ ≤ d} = y ∩ {x. ∣x ∙ k - c∣ ≤ d}"
          then obtain x' y' where "(x', x) ∈ D" "x ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}" "(y', y) ∈ D" "y ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}"
            by (auto)
          from p'(5)[OF ‹(x', x) ∈ D› ‹(y', y) ∈ D›] ‹x ≠ y› have "interior (x ∩ y) = {}"
            by auto
          moreover have "interior ((x ∩ {x. ∣x ∙ k - c∣ ≤ d}) ∩ (y ∩ {x. ∣x ∙ k - c∣ ≤ d})) ⊆ interior (x ∩ y)"
            by (auto intro: interior_mono)
          ultimately have "interior (x ∩ {x. ∣x ∙ k - c∣ ≤ d}) = {}"
            by (auto simp: eq)
          then show "content (x ∩ {x. ∣x ∙ k - c∣ ≤ d}) = 0"
            using p'(4)[OF ‹(x', x) ∈ D›] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int)
        qed (insert p'(1), auto intro!: sum.mono_neutral_right)
        also have "… ≤ norm (∑l∈(λl. l ∩ {x. ∣x ∙ k - c∣ ≤ d})`{l∈snd ` D. l ∩ {x. ∣x ∙ k - c∣ ≤ d} ≠ {}}. content l *🪙R 1::real)"
          by simp
        also have "… ≤ 1 * content (cbox a b ∩ {x. ∣x ∙ k - c∣ ≤ d})"
          using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
          unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
        also have "… < e"
          using d by simp
        finally show "(∑K∈snd ` D. content (K ∩ {x. ∣x ∙ k - c∣ ≤ d})) < e" .
      qed
      finally show "(∑(x, K)∈D. content (K ∩ {x. ∣x ∙ k - c∣ ≤ d}) * ?i x) < e" .
    qed
    then show "∣∑(x, K)∈D. content K * ?i x∣ < e"
      unfolding *  by (simp add: sum_nonneg split: prod.split)
  qed
qed

corollary negligible_standard_hyperplane_cart:
  fixes k :: "'a::finite"
  shows "negligible {x. x$k = (0::real)}"
  by (simp add: cart_eq_inner_axis negligible_standard_hyperplane)


subsubsection ‹Hence the main theorem about negligible sets›


lemma has_integral_negligible_cbox:
  fixes f :: "'b::euclidean_space ==> 'a::real_normed_vector"
  assumes negs: "negligible S"
    and 0: "∧x. x ∉ S ==> f x = 0"
  shows "(f has_integral 0) (cbox a b)"
  unfolding has_integral
proof clarify
  fix e::real
  assume "e > 0"
  then have nn_gt0: "e/2 / ((real n+1) * (2 ^ n)) > 0" for n
    by simp
  then have "∃γ. gauge γ ∧
                 (∀D. D tagged_division_of cbox a b ∧ γ fine D ⟶
                      ∣∑(x,K) ∈ D. content K *🪙R indicator S x∣
                      < e/2 / ((real n + 1) * 2 ^ n))" for n
    using negs [unfolded negligible_def has_integral] by auto
  then obtain γ where 
    gd: "∧n. gauge (γ n)"
    and γ: "∧n D. [D tagged_division_of cbox a b; γ n fine D]
                  ==> ∣∑(x,K) ∈ D. content K *🪙R indicator S x∣ < e/2 / ((real n + 1) * 2 ^ n)"
    by metis
  show "∃γ. gauge γ ∧
             (∀D. D tagged_division_of cbox a b ∧ γ fine D ⟶
                  norm ((∑(x,K) ∈ D. content K *🪙R f x) - 0) < e)"
  proof (intro exI, safe)
    show "gauge (λx. γ (nat ⌊norm (f x)⌋) x)"
      using gd by (auto simp: gauge_def)

    show "norm ((∑(x,K) ∈ D. content K *🪙R f x) - 0) < e"
      if "D tagged_division_of (cbox a b)" "(λx. γ (nat ⌊norm (f x)⌋) x) fine D" for D
    proof (cases "D = {}")
      case True with ‹0 🚫› show ?thesis by simp
    next
      case False
      obtain N where "Max ((λ(x, K). norm (f x)) ` D) ≤ real N"
        using real_arch_simple by blast
      then have N: "∧x. x ∈ (λ(x, K). norm (f x)) ` D ==> x ≤ real N"
        by (meson Max_ge that(1) dual_order.trans finite_imageI tagged_division_of_finite)
      have "∀i. ∃q. q tagged_division_of (cbox a b) ∧ (γ i) fine q ∧ (∀(x,K) ∈ D. K ⊆ (γ i) x ⟶ (x, K) ∈ q)"
        by (auto intro: tagged_division_finer[OF that(1) gd])
      from choice[OF this] 
      obtain q where q: "∧n. q n tagged_division_of cbox a b"
                        "∧n. γ n fine q n"
                        "∧n x K. [(x, K) ∈ D; K ⊆ γ n x] ==> (x, K) ∈ q n"
        by fastforce
      have "finite D"
        using that(1) by blast
      then have sum_le_inc: "[finite T; ∧x y. (x,y) ∈ T ==> (0::real) ≤ g(x,y);
                      ∧y. y∈D ==> ∃x. (x,y) ∈ T ∧ f(y) ≤ g(x,y)] ==> sum f D ≤ sum g T" for f g T
        by (rule sum_le_included[of D T g snd f]; force)
      have "norm (∑(x,K) ∈ D. content K *🪙R f x) ≤ (∑(x,K) ∈ D. norm (content K *🪙R f x))"
        unfolding split_def by (rule norm_sum)
      also have "… ≤ (∑(i, j) ∈ Sigma {..N + 1} q.
                          (real i + 1) * (case j of (x, K) ==> content K *🪙R indicator S x))"
      proof (rule sum_le_inc, safe)
        show "finite (Sigma {..N+1} q)"
          by (meson finite_SigmaI finite_atMost tagged_division_of_finite q(1)) 
      next
        fix x K
        assume xk: "(x, K) ∈ D"
        define n where "n = nat ⌊norm (f x)⌋"
        have *: "norm (f x) ∈ (λ(x, K). norm (f x)) ` D"
          using xk by auto
        have nfx: "real n ≤ norm (f x)" "norm (f x) ≤ real n + 1"
          unfolding n_def by auto
        then have "n ∈ {0..N + 1}"
          using N[OF *] by auto
        moreover have "(x, K) ∈ q n"
          using n_def q(3) that(2) xk by fastforce
        moreover
        have "norm (content K *🪙R f x) ≤ (real n + 1) * (content K * indicator S x)"
        proof (cases "x ∈ S")
          case False
          then show ?thesis by (simp add: 0)
        next
          case True
          have *: "content K ≥ 0"
            using tagged_division_ofD(4)[OF that(1) xk] by auto
          moreover have "content K * norm (f x) ≤ content K * (real n + 1)"
            by (simp add: mult_left_mono nfx(2))
          ultimately show ?thesis
            using nfx True by (auto simp: field_simps)
        qed
        ultimately show "∃y. (y, x, K) ∈ (Sigma {..N + 1} q) ∧ norm (content K *🪙R f x) ≤
          (real y + 1) * (content K *🪙R indicator S x)"
          by force
      qed auto
      also have "… = (∑i≤N + 1. ∑j∈q i. (real i + 1) * (case j of (x, K) ==> content K *🪙R indicator S x))"
        using q(1) by (intro sum_Sigma_product [symmetric]) auto
      also have "… ≤ (∑i≤N + 1. (real i + 1) * ∣∑(x,K) ∈ q i. content K *🪙R indicator S x∣)"
        by (rule sum_mono) (simp flip: sum_distrib_left)
      also have "… ≤ (∑i≤N + 1. e/2/2 ^ i)"
      proof (rule sum_mono)
        show "(real i + 1) * ∣∑(x,K) ∈ q i. content K *🪙R indicator S x∣ ≤ e/2/2 ^ i"
          if "i ∈ {..N + 1}" for i
          using γ[of "q i" i] q by (simp add: divide_simps mult.left_commute)
      qed
      also have "… = e/2 * (∑i≤N + 1. (1/2) ^ i)"
        unfolding sum_distrib_left by (metis divide_inverse inverse_eq_divide power_one_over)
      also have "… < e/2 * 2"
      proof (rule mult_strict_left_mono)
        have "sum (power (1/2)) {..N + 1} = sum (power (1/2::real)) {..
          using lessThan_Suc_atMost by auto
        also have "… < 2"
          by (auto simp: geometric_sum)
        finally show "sum (power (1/2::real)) {..N + 1} < 2" .
      qed (use ‹0 🚫› in auto)
      finally show ?thesis by simp
    qed
  qed
qed


proposition has_integral_negligible:
  fixes f :: "'b::euclidean_space ==> 'a::real_normed_vector"
  assumes negs: "negligible S"
    and "∧x. x ∈ (T - S) ==> f x = 0"
  shows "(f has_integral 0) T"
proof (cases "∃a b. T = cbox a b")
  case True
  then have "((λx. if x ∈ T then f x else 0) has_integral 0) T"
    using assms by (auto intro!: has_integral_negligible_cbox)
  then show ?thesis
    by (rule has_integral_eq [rotated]) auto
next
  case False
  let ?f = "(λx. if x ∈ T then f x else 0)"
  have "((λx. if x ∈ T then f x else 0) has_integral 0) T"
    apply (auto simp: False has_integral_alt [of ?f])
    apply (rule_tac x=1 in exI, auto)
    apply (rule_tac x=0 in exI, simp add: has_integral_negligible_cbox [OF negs] assms)
    done
  then show ?thesis
    by (rule_tac f="?f" in has_integral_eq) auto
qed

lemma
  assumes "negligible S"
  shows integrable_negligible: "f integrable_on S" and integral_negligible: "integral S f = 0"
  using has_integral_negligible [OF assms]
  by (auto simp: has_integral_iff)

lemma has_integral_spike:
  fixes f :: "'b::euclidean_space ==> 'a::real_normed_vector"
  assumes "negligible S"
    and gf: "∧x. x ∈ T - S ==> g x = f x"
    and fint: "(f has_integral y) T"
  shows "(g has_integral y) T"
proof -
  have *: "(g has_integral y) (cbox a b)"
       if "(f has_integral y) (cbox a b)" "∀x ∈ cbox a b - S. g x = f x" for a b f and g:: "'b ==> 'a" and y
  proof -
    have "((λx. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
      using that by (intro has_integral_add has_integral_negligible) (auto intro!: ‹negligible S›)
    then show ?thesis
      by auto
  qed
  have 🍋: "∧a b z. [∧x. x ∈ T ∧ x ∉ S ==> g x = f x;
                     ((λx. if x ∈ T then f x else 0) has_integral z) (cbox a b)]
                    ==> ((λx. if x ∈ T then g x else 0) has_integral z) (cbox a b)"
      by (auto dest!: *[where f="λx. if x∈T then f x else 0" and g="λx. if x ∈ T then g x else 0"])
  show ?thesis
    using fint gf
    apply (subst has_integral_alt)
    apply (subst (asm) has_integral_alt)
    apply (auto split: if_split_asm)
     apply (blast dest: *)
    using 🍋 by meson
qed

lemma has_integral_spike_eq:
  assumes "negligible S" and "∧x. x ∈ T - S ==> g x = f x"
  shows "(f has_integral y) T ⟷ (g has_integral y) T"
  by (metis assms has_integral_spike)

lemma integrable_spike:
  assumes "f integrable_on T" "negligible S" "∧x. x ∈ T - S ==> g x = f x"
    shows "g integrable_on T"
  using assms unfolding integrable_on_def by (blast intro: has_integral_spike)

lemma integral_spike:
  assumes "negligible S" and "∧x. x ∈ T - S ==> g x = f x"
  shows "integral T f = integral T g"
  using has_integral_spike_eq[OF assms]
    by (auto simp: integral_def integrable_on_def)


subsection ‹Some other trivialities about negligible sets›

lemma negligible_subset:
  assumes "negligible S" "T ⊆ S"
  shows "negligible T"
  unfolding negligible_def
    by (metis (no_types) Diff_iff assms contra_subsetD has_integral_negligible indicator_simps(2))

lemma negligible_diff[intro?]:
  assumes "negligible S"
  shows "negligible (S - T)"
  using assms by (meson Diff_subset negligible_subset)

lemma negligible_Int:
  assumes "negligible S ∨ negligible T"
  shows "negligible (S ∩ T)"
  using assms negligible_subset by force

lemma negligible_Un:
  assumes "negligible S" and T: "negligible T"
  shows "negligible (S ∪ T)"
proof -
  have "(indicat_real (S ∪ T) has_integral 0) (cbox a b)"
    if S0: "(indicat_real S has_integral 0) (cbox a b)" 
      and  "(indicat_real T has_integral 0) (cbox a b)" for a b
  proof (subst has_integral_spike_eq[OF T])
    show "indicat_real S x = indicat_real (S ∪ T) x" if "x ∈ cbox a b - T" for x
      using that by (simp add: indicator_def)
    show "(indicat_real S has_integral 0) (cbox a b)"
      by (simp add: S0)
  qed
  with assms show ?thesis
    unfolding negligible_def by blast
qed

lemma negligible_Un_eq[simp]: "negligible (S ∪ T) ⟷ negligible S ∧ negligible T"
  using negligible_Un negligible_subset by blast

lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
  using negligible_standard_hyperplane[OF SOME_Basis, of "a ∙ (SOME i. i ∈ Basis)"] negligible_subset by blast

lemma negligible_insert[simp]: "negligible (insert a S) ⟷ negligible S"
  by (metis insert_is_Un negligible_Un_eq negligible_sing)

lemma negligible_empty[iff]: "negligible {}"
  using negligible_insert by blast

text‹Useful in this form for backchaining›
lemma empty_imp_negligible: "S = {} ==> negligible S"
  by simp

lemma negligible_finite[intro]:
  assumes "finite S"
  shows "negligible S"
  using assms by (induct S) auto

lemma negligible_Union[intro]:
  assumes "finite T"
    and "∧T. T ∈ T ==> negligible T"
  shows "negligible(∪T)"
  using assms by induct auto

lemma negligible: "negligible S ⟷ (∀T. (indicat_real S has_integral 0) T)"
  by (meson DiffD2 has_integral_negligible indicator_simps(2) negligible_def)

subsection ‹Finite case of the spike theorem is quite commonly needed›

lemma has_integral_spike_finite:
  assumes "finite S"
    and "∧x. x ∈ T - S ==> g x = f x"
    and "(f has_integral y) T"
  shows "(g has_integral y) T"
  using assms has_integral_spike negligible_finite by blast

lemma has_integral_spike_finite_eq:
  assumes "finite S"
    and "∧x. x ∈ T - S ==> g x = f x"
  shows "((f has_integral y) T ⟷ (g has_integral y) T)"
  by (metis assms has_integral_spike_finite)

lemma integrable_spike_finite:
  assumes "finite S"
    and "∧x. x ∈ T - S ==> g x = f x"
    and "f integrable_on T"
  shows "g integrable_on T"
  using assms has_integral_spike_finite by blast

lemma integrable_spike_finite_eq:
  assumes "finite S"
    and "∧x. x ∈ T - S ==> f x = g x"
  shows "f integrable_on T ⟷ g integrable_on T"
  by (metis assms integrable_spike_finite)

lemma has_integral_bound_spike_finite:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "0 ≤ B" "finite S"
      and f: "(f has_integral i) (cbox a b)"
      and leB: "∧x. x ∈ cbox a b - S ==> norm (f x) ≤ B"
    shows "norm i ≤ B * content (cbox a b)"
proof -
  define g where "g ≡ (λx. if x ∈ S then 0 else f x)"
  then have "∧x. x ∈ cbox a b - S ==> norm (g x) ≤ B"
    using leB by simp
  moreover have "(g has_integral i) (cbox a b)"
    using has_integral_spike_finite [OF ‹finite S› _ f]
    by (simp add: g_def)
  ultimately show ?thesis
    by (simp add: ‹0 ≤ B› g_def has_integral_bound)
qed

corollary has_integral_bound_real:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "0 ≤ B" "finite S"
      and "(f has_integral i) {a..b}"
      and "∧x. x ∈ {a..b} - S ==> norm (f x) ≤ B"
    shows "norm i ≤ B * content {a..b}"
  by (metis assms box_real(2) has_integral_bound_spike_finite)


subsection ‹In particular, the boundary of an interval is negligible›

lemma negligible_frontier_interval: "negligible(cbox a b - box a b)"
proof -
  let ?A = "∪((λk. {x. x∙k = a∙k} ∪ {x::'a. x∙k = b∙k}) ` Basis)"
  have "negligible ?A"
    by (force simp: negligible_Union[OF finite_imageI])
  moreover have "cbox a b - box a b ⊆ ?A"
    by (force simp: mem_box)
  ultimately show ?thesis
    by (rule negligible_subset)
qed

lemma has_integral_spike_interior:
  assumes f: "(f has_integral y) (cbox a b)" and gf: "∧x. x ∈ box a b ==> g x = f x"
  shows "(g has_integral y) (cbox a b)"
  by (meson Diff_iff gf has_integral_spike[OF negligible_frontier_interval _ f])
  
lemma has_integral_spike_interior_eq:
  assumes "∧x. x ∈ box a b ==> g x = f x"
  shows "(f has_integral y) (cbox a b) ⟷ (g has_integral y) (cbox a b)"
  by (metis assms has_integral_spike_interior)

lemma integrable_spike_interior:
  assumes "∧x. x ∈ box a b ==> g x = f x"
    and "f integrable_on cbox a b"
  shows "g integrable_on cbox a b"
  using assms has_integral_spike_interior_eq by blast


subsection ‹Integrability of continuous functions›

lemma operative_approximableI:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes "0 ≤ e"
  shows "operative conj True (λi. ∃g. (∀x∈i. norm (f x - g (x::'b)) ≤ e) ∧ g integrable_on i)"
proof -
  interpret comm_monoid conj True
    by standard auto
  show ?thesis
  proof (standard, safe)
    fix a b :: 'b
    show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
      if "box a b = {}" for a b
      using assms that
      by (metis content_eq_0_interior integrable_on_null interior_cbox norm_zero right_minus_eq)
    {
      fix c g and k :: 'b
      assume fg: "∀x∈cbox a b. norm (f x - g x) ≤ e" and g: "g integrable_on cbox a b"
      assume k: "k ∈ Basis"
      show "∃g. (∀x∈cbox a b ∩ {x. x ∙ k ≤ c}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. x ∙ k ≤ c}"
           "∃g. (∀x∈cbox a b ∩ {x. c ≤ x ∙ k}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. c ≤ x ∙ k}"
        using fg g k by auto
    }
    show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
      if fg1: "∀x∈cbox a b ∩ {x. x ∙ k ≤ c}. norm (f x - g1 x) ≤ e" 
        and g1: "g1 integrable_on cbox a b ∩ {x. x ∙ k ≤ c}"
        and fg2: "∀x∈cbox a b ∩ {x. c ≤ x ∙ k}. norm (f x - g2 x) ≤ e" 
        and g2: "g2 integrable_on cbox a b ∩ {x. c ≤ x ∙ k}" 
        and k: "k ∈ Basis"
      for c k g1 g2
    proof -
      let ?g = "λx. if x∙k = c then f x else if x∙k ≤ c then g1 x else g2 x"
      show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
      proof (intro exI conjI ballI)
        show "norm (f x - ?g x) ≤ e" if "x ∈ cbox a b" for x
          by (auto simp: that assms fg1 fg2)
        show "?g integrable_on cbox a b"
        proof -
          have "?g integrable_on cbox a b ∩ {x. x ∙ k ≤ c}" "?g integrable_on cbox a b ∩ {x. x ∙ k ≥ c}"
            by(rule integrable_spike[OF _ negligible_standard_hyperplane[of k c]], use k g1 g2 in auto)+
          with has_integral_split[OF _ _ k] show ?thesis
            unfolding integrable_on_def by blast
        qed
      qed
    qed
  qed
qed

lemma comm_monoid_set_F_and: "comm_monoid_set.F (∧) True f s ⟷ (finite s ⟶ (∀x∈s. f x))"
proof -
  interpret bool: comm_monoid_set ‹(∧)› True ..
  show ?thesis
    by (induction s rule: infinite_finite_induct) auto
qed

lemma approximable_on_division:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes "0 ≤ e"
    and d: "d division_of (cbox a b)"
    and f: "∀i∈d. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
  obtains g where "∀x∈cbox a b. norm (f x - g x) ≤ e" "g integrable_on cbox a b"
proof -
  interpret operative conj True "λi. ∃g. (∀x∈i. norm (f x - g (x::'b)) ≤ e) ∧ g integrable_on i"
    using ‹0 ≤ e› by (rule operative_approximableI)
  from f local.division [OF d] that show thesis
    by auto
qed

lemma integrable_continuous:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes "continuous_on (cbox a b) f"
  shows "f integrable_on cbox a b"
proof (rule integrable_uniform_limit)
  fix e :: real
  assume e: "e > 0"
  then obtain d where "0 < d" and d: "∧x x'. [x ∈ cbox a b; x' ∈ cbox a b; dist x' x < d] ==> dist (f x') (f x) < e"
    using compact_uniformly_continuous[OF assms compact_cbox] unfolding uniformly_continuous_on_def by metis
  obtain p where ptag: "p tagged_division_of cbox a b" and finep: "(λx. ball x d) fine p"
    using fine_division_exists[OF gauge_ball[OF ‹0 🚫›], of a b] .
  have *: "∀i∈snd ` p. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
  proof clarsimp 
    fix x l
    assume as: "(x, l) ∈ p"
    obtain a b where l: "l = cbox a b"
      using as ptag by blast
    then have x: "x ∈ cbox a b"
      using as ptag by auto
    show "∃g. (∀x∈l. norm (f x - g x) ≤ e) ∧ g integrable_on l"
    proof (intro exI conjI strip)
      show "(λy. f x) integrable_on l"
        unfolding integrable_on_def l by blast
    next
      fix y
      assume y: "y ∈ l"
      then have "y ∈ ball x d"
        using as finep by fastforce
      then show "norm (f y - f x) ≤ e"
        using d x y as l
        by (metis dist_commute dist_norm less_imp_le mem_ball ptag subsetCE tagged_division_ofD(3))
    qed
  qed
  from e have "e ≥ 0"
    by auto
  from approximable_on_division[OF this division_of_tagged_division[OF ptag] *]
  show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
    by metis
qed

lemma integrable_continuous_interval:
  fixes f :: "'b::ordered_euclidean_space ==> 'a::banach"
  assumes "continuous_on {a..b} f"
  shows "f integrable_on {a..b}"
  by (metis assms integrable_continuous interval_cbox)

lemmas integrable_continuous_real = integrable_continuous_interval[where 'b=real]

lemma integrable_continuous_closed_segment:
  fixes f :: "real ==> 'a::banach"
  assumes "continuous_on (closed_segment a b) f"
  shows "f integrable_on (closed_segment a b)"
  by (metis assms closed_segment_eq_real_ivl integrable_continuous_interval)


subsection ‹Specialization of additivity to one dimension›


subsection ‹A useful lemma allowing us to factor out the content size›

lemma has_integral_factor_content:
  "(f has_integral i) (cbox a b) ⟷
    (∀e>0. ∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
      norm (sum (λ(x,k). content k *🪙R f x) p - i) ≤ e * content (cbox a b)))"
proof (cases "content (cbox a b) = 0")
  case True
  have "∧e p. p tagged_division_of cbox a b ==> norm ((∑(x, k)∈p. content k *🪙R f x)) ≤ e * content (cbox a b)"
    unfolding sum_content_null[OF True] True by force
  moreover have "i = 0" 
    if "∧e. e > 0 ==> ∃d. gauge d ∧
              (∀p. p tagged_division_of cbox a b ∧
                   d fine p ⟶
                   norm ((∑(x, k)∈p. content k *🪙R f x) - i) ≤ e * content (cbox a b))"
    using that [of 1]
    by (force simp: True sum_content_null[OF True] intro: fine_division_exists[of _ a b])
  ultimately show ?thesis
    unfolding has_integral_null_eq[OF True]
    by force
next
  case False
  then have F: "0 < content (cbox a b)"
    using zero_less_measure_iff by blast
  let ?P = "λe opp. ∃d. gauge d ∧
    (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶ opp (norm ((∑(x, k)∈p. content k *🪙R f x) - i)) e)"
  show ?thesis
  proof (subst has_integral, safe)
    fix e :: real
    assume e: "e > 0"
    show "?P (e * content (cbox a b)) (≤)" if 🍋[rule_format]: "∀ε>0. ?P ε (<)"
      using 🍋 [of "e * content (cbox a b)"]
      by (meson F e less_imp_le mult_pos_pos)
    show "?P e (<)" if 🍋[rule_format]:  "∀ε>0. ?P (ε * content (cbox a b)) (≤)"
      using 🍋 [of "e/2 / content (cbox a b)"]
        using F e by (force simp: algebra_simps)
  qed
qed

lemma has_integral_factor_content_real:
  "(f has_integral i) {a..b::real} ⟷
    (∀e>0. ∃d. gauge d ∧ (∀p. p tagged_division_of {a..b} ∧ d fine p ⟶
      norm (sum (λ(x,k). content k *🪙R f x) p - i) ≤ e * content {a..b} ))"
  unfolding box_real[symmetric]
  by (rule has_integral_factor_content)


subsection ‹Fundamental theorem of calculus›

lemma interval_bounds_real:
  fixes q b :: real
  assumes "a ≤ b"
  shows "Sup {a..b} = b"
    and "Inf {a..b} = a"
  using assms by auto

theorem fundamental_theorem_of_calculus:
  fixes f :: "real ==> 'a::banach"
  assumes "a ≤ b" 
      and vecd: "∧x. x ∈ {a..b} ==> (f has_vector_derivative f' x) (at x within {a..b})"
  shows "(f' has_integral (f b - f a)) {a..b}"
  unfolding has_integral_factor_content box_real[symmetric]
proof safe
  fix e :: real
  assume "e > 0"
  then have "∀x. ∃d>0. x ∈ {a..b} ⟶
         (∀y∈{a..b}. norm (y-x) < d ⟶ norm (f y - f x - (y-x) *🪙R f' x) ≤ e * norm (y-x))"
    using vecd[unfolded has_vector_derivative_def has_derivative_within_alt] by blast
  then obtain d where d: "∧x. 0 < d x"
                 "∧x y. [x ∈ {a..b}; y ∈ {a..b}; norm (y-x) < d x]
                        ==> norm (f y - f x - (y-x) *🪙R f' x) ≤ e * norm (y-x)"
    by metis  
  show "∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
    norm ((∑(x, k)∈p. content k *🪙R f' x) - (f b - f a)) ≤ e * content (cbox a b))"
  proof (rule exI, safe)
    show "gauge (λx. ball x (d x))"
      using d(1) gauge_ball_dependent by blast
  next
    fix p
    assume ptag: "p tagged_division_of cbox a b" and finep: "(λx. ball x (d x)) fine p"
    have ba: "b - a = (∑(x,K)∈p. Sup K - Inf K)" "f b - f a = (∑(x,K)∈p. f(Sup K) - f(Inf K))"
      using additive_tagged_division_1[where f= "λx. x"] additive_tagged_division_1[where f= f]
            ‹a ≤ b› ptag by auto
    have "norm (∑(x, K) ∈ p. (content K *🪙R f' x) - (f (Sup K) - f (Inf K)))
          ≤ (∑n∈p. e * (case n of (x, k) ==> Sup k - Inf k))"
    proof (rule sum_norm_le,safe)
      fix x K
      assume "(x, K) ∈ p"
      then have "x ∈ K" and kab: "K ⊆ cbox a b"
        using ptag by blast+
      then obtain u v where k: "K = cbox u v" and "x ∈ K" and kab: "K ⊆ cbox a b"
        using ptag ‹(x, K) ∈ p› by auto 
      have "u ≤ v"
        using ‹x ∈ K› unfolding k by auto
      have ball: "∀y∈K. y ∈ ball x (d x)"
        using finep ‹(x, K) ∈ p› by blast
      have "u ∈ K" "v ∈ K"
        by (simp_all add: ‹u ≤ v› k)
      have "norm ((v - u) *🪙R f' x - (f v - f u)) = norm (f u - f x - (u - x) *🪙R f' x - (f v - f x - (v - x) *🪙R f' x))"
        by (auto simp: algebra_simps)
      also have "… ≤ norm (f u - f x - (u - x) *🪙R f' x) + norm (f v - f x - (v - x) *??R f' x)"
        by (rule norm_triangle_ineq4)
      finally have "norm ((v - u) *🪙R f' x - (f v - f u)) ≤
        norm (f u - f x - (u - x) *🪙R f' x) + norm (f v - f x - (v - x) *🪙R f' x)" .
      also have "… ≤ e * norm (u - x) + e * norm (v - x)"
      proof (rule add_mono)
        show "norm (f u - f x - (u - x) *🪙R f' x) ≤ e * norm (u - x)"
        proof (rule d)
          show "norm (u - x) < d x"
            using ‹u ∈ K› ball by (auto simp: dist_real_def)
        qed (use ‹x ∈ K› ‹u ∈ K› kab in auto)
        show "norm (f v - f x - (v - x) *🪙R f' x) ≤ e * norm (v - x)"
        proof (rule d)
          show "norm (v - x) < d x"
            using ‹v ∈ K› ball by (auto simp: dist_real_def)
        qed (use ‹x ∈ K› ‹v ∈ K› kab in auto)
      qed
      also have "… ≤ e * (Sup K - Inf K)"
        using ‹x ∈ K› by (auto simp: k interval_bounds_real[OF ‹u ≤ v›] field_simps)
      finally show "norm (content K *🪙R f' x - (f (Sup K) - f (Inf K))) ≤ e * (Sup K - Inf K)"
        using ‹u ≤ v› by (simp add: k)
    qed
    with ‹a ≤ b› show "norm ((∑(x, K)∈p. content K *🪙R f' x) - (f b - f a)) ≤ e * content (cbox a b)"
      by (auto simp: ba split_def sum_subtractf [symmetric] sum_distrib_left)
  qed
qed

lemma has_complex_derivative_imp_has_vector_derivative:
  fixes f :: "complex ==> complex"
  assumes "(f has_field_derivative f') (at (of_real a) within (cbox (of_real x) (of_real y)))"
  shows "((f o of_real) has_vector_derivative f') (at a within {x..y})"
  using has_derivative_in_compose[of of_real of_real a "{x..y}" f "(*) f'"] assms
  by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def o_def cbox_complex_of_real)

lemma ident_has_integral:
  fixes a::real
  assumes "a ≤ b"
  shows "((λx. x) has_integral (b🪙2 - a🪙2)/2) {a..b}"
proof -
  have "((λx. x) has_integral inverse 2 * b🪙2 - inverse 2 * a🪙2) {a..b}"
    unfolding power2_eq_square
    by (rule fundamental_theorem_of_calculus [OF assms] derivative_eq_intros | simp)+
  then show ?thesis
    by (simp add: field_simps)
qed

lemma integral_ident [simp]:
  fixes a::real
  assumes "a ≤ b"
  shows "integral {a..b} (λx. x) = (if a ≤ b then (b🪙2 - a🪙2)/2 else 0)"
  by (metis assms ident_has_integral integral_unique)

lemma ident_integrable_on:
  fixes a::real
  shows "(λx. x) integrable_on {a..b}"
  using continuous_on_id integrable_continuous_real by blast

lemma integral_sin [simp]:
  fixes a::real
  assumes "a ≤ b" shows "integral {a..b} sin = cos a - cos b"
proof -
  have "(sin has_integral (- cos b - - cos a)) {a..b}"
  proof (rule fundamental_theorem_of_calculus)
    show "((λa. - cos a) has_vector_derivative sin x) (at x within {a..b})" for x
      unfolding has_real_derivative_iff_has_vector_derivative [symmetric]
      by (rule derivative_eq_intros | force)+
  qed (use assms in auto)
  then show ?thesis
    by (simp add: integral_unique)
qed

lemma integral_cos [simp]:
  fixes a::real
  assumes "a ≤ b" shows "integral {a..b} cos = sin b - sin a"
proof -
  have "(cos has_integral (sin b - sin a)) {a..b}"
  proof (rule fundamental_theorem_of_calculus)
    show "(sin has_vector_derivative cos x) (at x within {a..b})" for x
      unfolding has_real_derivative_iff_has_vector_derivative [symmetric]
      by (rule derivative_eq_intros | force)+
  qed (use assms in auto)
  then show ?thesis
    by (simp add: integral_unique)
qed

lemma integral_exp [simp]:
  fixes a::real
  assumes "a ≤ b" shows "integral {a..b} exp = exp b - exp a"
  by (meson DERIV_exp assms fundamental_theorem_of_calculus has_real_derivative_iff_has_vector_derivative
       has_vector_derivative_at_within integral_unique)

lemma has_integral_sin_nx: "((λx. sin(real_of_int n * x)) has_integral 0) {-pi..pi}"
proof (cases "n = 0")
  case False
  have "((λx. sin (n * x)) has_integral (- cos (n * pi)/n - - cos (n * - pi)/n)) {-pi..pi}"
  proof (rule fundamental_theorem_of_calculus)
    show "((λx. - cos (n * x) / n) has_vector_derivative sin (n * a)) (at a within {-pi..pi})"
      if "a ∈ {-pi..pi}" for a :: real
      using that False
      unfolding has_vector_derivative_def
      by (intro derivative_eq_intros | force)+
  qed auto
  then show ?thesis
    by simp
qed auto

lemma integral_sin_nx:
   "integral {-pi..pi} (λx. sin(x * real_of_int n)) = 0"
  using has_integral_sin_nx [of n] by (force simp: mult.commute)

lemma has_integral_cos_nx:
  "((λx. cos(real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {-pi..pi}"
proof (cases "n = 0")
  case True
  then show ?thesis
    using has_integral_const_real [of "1::real" "-pi" pi] by auto
next
  case False
  have "((λx. cos (n * x)) has_integral (sin (n * pi)/n - sin (n * - pi)/n)) {-pi..pi}"
  proof (rule fundamental_theorem_of_calculus)
    show "((λx. sin (n * x) / n) has_vector_derivative cos (n * x)) (at x within {-pi..pi})"
      if "x ∈ {-pi..pi}"
      for x :: real
      using that False
      unfolding has_vector_derivative_def
      by (intro derivative_eq_intros | force)+
  qed auto
  with False show ?thesis
    by (simp add: mult.commute)
qed

lemma integral_cos_nx:
   "integral {-pi..pi} (λx. cos(x * real_of_int n)) = (if n = 0 then 2 * pi else 0)"
  using has_integral_cos_nx [of n] by (force simp: mult.commute)


subsection ‹Taylor series expansion›

lemma mvt_integral:
  fixes f::"'a::real_normed_vector==>'b::banach"
  assumes f'[derivative_intros]:
    "∧x. x ∈ S ==> (f has_derivative f' x) (at x within S)"
  assumes line_in: "∧t. t ∈ {0..1} ==> x + t *🪙R y ∈ S"
  shows "f (x + y) - f x = integral {0..1} (λt. f' (x + t *🪙R y) y)"
proof -
  from assms have subset: "(λxa. x + xa *🪙R y) ` {0..1} ⊆ S" by auto
  note [derivative_intros] =
    has_derivative_subset[OF _ subset]
    has_derivative_in_compose[where f="(λu. x + u *🪙R y)" and g = f]
  have "∧t. t ∈ {0..1} ==>
    ((λt. f (x + t *🪙R y)) has_vector_derivative f' (x + t *🪙R y) y)
    (at t within {0..1})"
    using assms
    by (auto simp: has_vector_derivative_def
        linear_cmul[OF has_derivative_linear[OF f'], symmetric]
      intro!: derivative_eq_intros)
  from fundamental_theorem_of_calculus[OF _ this]
  show ?thesis
    by (simp add: integral_unique)
qed

lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative:
  assumes "p>0"
  and f0: "Df 0 = f"
  and Df: "∧m t. m < p ==> a ≤ t ==> t ≤ b ==>
    (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
  and g0: "Dg 0 = g"
  and Dg: "∧m t. m < p ==> a ≤ t ==> t ≤ b ==>
    (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
  and ivl: "a ≤ t" "t ≤ b"
  shows "((λt. ∑i<p. (-1)^i *🪙R prod (Df i t) (Dg (p - Suc i) t))
    has_vector_derivative
      prod (f t) (Dg p t) - (-1)^p *🪙R prod (Df p t) (g t))
    (at t within {a..b})"
  using assms
proof cases
  assume p: "p ≠ 1"
  define p' where "p' = p - 2"
  from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
    by (auto simp: p'_def)
  have *: "∧i. i ≤ p' ==> Suc (Suc p' - i) = (Suc (Suc p') - i)"
    by auto
  let ?f = "λi. (-1) ^ i *🪙R (prod (Df i t) (Dg ((p - i)) t))"
  have "(∑i<p. (-1) ^ i *🪙R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
    prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
    (∑i≤(Suc p'). ?f i - ?f (Suc i))"
    by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
  also note sum_telescope
  finally
  have "(∑i<p. (-1) ^ i *🪙R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
    prod (Df (Suc i) t) (Dg (p - Suc i) t)))
    = prod (f t) (Dg p t) - (- 1) ^ p *🪙R prod (Df p t) (g t)"
    unfolding p'[symmetric]
    by (simp add: assms)
  thus ?thesis
    using assms
    by (auto intro!: derivative_eq_intros has_vector_derivative)
qed (auto intro!: derivative_eq_intros has_vector_derivative)

lemma
  fixes f::"real==>'a::banach"
  assumes "p>0"
  and f0: "Df 0 = f"
  and Df: "∧m t. m < p ==> a ≤ t ==> t ≤ b ==>
    (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
  and ivl: "a ≤ b"
  defines "i ≡ λx. ((b - x) ^ (p - 1) / fact (p - 1)) *🪙R Df p x"
  shows Taylor_has_integral:
    "(i has_integral f b - (∑i<p. ((b-a) ^ i / fact i) *🪙R Df i a)) {a..b}"
  and Taylor_integral:
    "f b = (∑i<p. ((b-a) ^ i / fact i) *🪙R Df i a) + integral {a..b} i"
  and Taylor_integrable:
    "i integrable_on {a..b}"
proof goal_cases
  case 1
  interpret bounded_bilinear "scaleR::real==>'a==>'a"
    by (rule bounded_bilinear_scaleR)
  define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
  define Dg where [abs_def]:
    "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
  have g0: "Dg 0 = g"
    using ‹p > 0›
    by (auto simp: Dg_def field_split_simps g_def split: if_split_asm)
  have fact_eq: "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
    if "p > Suc m" for m
    by (metis Suc_diff_Suc fact_Suc that)
  have Dg: "∧m t. m < p ==> a ≤ t ==> t ≤ b ==>
    (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
    unfolding Dg_def has_vector_derivative_def
    by (auto intro!: derivative_eq_intros simp: fact_eq divide_simps simp del: of_nat_diff)
  let ?sum = "λt. ∑i<p. (- 1) ^ i *🪙R Dg i t *🪙R Df (p - Suc i) t"
  from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
      OF ‹p > 0› g0 Dg f0 Df]
  have deriv: "∧t. a ≤ t ==> t ≤ b ==>
    (?sum has_vector_derivative
      g t *🪙R Df p t - (- 1) ^ p *🪙R Dg p t *🪙R f t) (at t within {a..b})"
    by auto
  from fundamental_theorem_of_calculus[OF ‹a ≤ b› deriv]
  have "(i has_integral ?sum b - ?sum a) {a..b}"
    using Dg_def g_def i_def by fastforce
  also
  have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)" "{..<p} ∩ {i. p = Suc i} = {p - 1}" for p'
    using ‹p > 0› by (auto simp: power_mult_distrib)
  then have "?sum b = f b"
    using Suc_pred'[OF ‹p > 0›]
    by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
        if_distribR sum.If_cases f0)
  also
  have "?sum a = (∑i<p. ((b-a) ^ i / fact i) *🪙R Df i a)"
  proof (rule sum.reindex_cong)
    have "∧i. i < p ==> ∃j<p. i = p - Suc j"
      by (metis Suc_diff_Suc ‹p>0› diff_Suc_less diff_diff_cancel less_or_eq_imp_le)
    then show "{..<p} = (λx. p - x - 1) ` {..<p}"
      by force
  qed (auto simp: inj_on_def Dg_def one)
  finally show c: ?case .
  case 2 show ?case
    by (metis (lifting) add_diff_cancel_left' add_diff_eq c integral_unique)
  case 3 show ?case
    using c by force
qed


subsection ‹Only need trivial subintervals if the interval itself is trivial›

proposition division_of_nontrivial:
  fixes D :: "'a::euclidean_space set set"
  assumes sdiv: "D division_of (cbox a b)"
     and cont0: "content (cbox a b) ≠ 0"
  shows "{k. k ∈ D ∧ content k ≠ 0} division_of (cbox a b)"
  using sdiv
proof (induction "card D" arbitrary: D rule: less_induct)
  case less
  note D = division_ofD[OF less.prems]
  show ?case
  proof (cases "{k ∈ D. content k ≠ 0} = D")
    case False
    then obtain K c d where "K ∈ D" and contk: "content K = 0" and keq: "K = cbox c d"
      using D(4) by blast
    then have "card D > 0"
      unfolding card_gt_0_iff using less by auto
    then have card: "card (D - {K}) < card D"
      using less ‹K ∈ D› by (simp add: D(1))
    have closed: "closed (∪(D - {K}))"
      using less.prems by auto
    have "x islimpt ∪(D - {K})" if "x ∈ K" for x
      unfolding islimpt_approachable
    proof (intro allI impI)
      fix e::real
      assume "e > 0"
      obtain i where i: "c∙i = d∙i" "i∈Basis"
        using contk D(3) [OF ‹K ∈ D›] unfolding box_ne_empty keq
        by (meson content_eq_0 dual_order.antisym)
      then have xi: "x∙i = d∙i"
        using ‹x ∈ K› unfolding keq mem_box by (metis antisym)
      define y where "y = (∑j∈Basis. (if j = i then if c∙i ≤ (a∙i + b∙i)/2 then c∙i +
      min e (b∙i - c∙i)/2 else c∙i - min e (c∙i - a∙i)/2 else x∙j) *🪙R j)"
      show "∃x'∈∪(D - {K}). x' ≠ x ∧ dist x' x < e"
      proof (intro bexI conjI)
        have "d ∈ cbox c d"
          using D(3)[OF ‹K ∈ D›] by (simp add: box_ne_empty(1) keq mem_box(2))
        then have "d ∈ cbox a b"
          using D(2)[OF ‹K ∈ D›] by (auto simp: keq)
        then have di: "a ∙ i ≤ d ∙ i ∧ d ∙ i ≤ b ∙ i"
          using ‹i ∈ Basis› mem_box(2) by blast
        then have xyi: "y∙i ≠ x∙i"
          unfolding y_def i xi using ‹e > 0› cont0 ‹i ∈ Basis›
          by (auto simp: content_eq_0 elim!: ballE[of _ _ i])
        then show "y ≠ x"
          unfolding euclidean_eq_iff[where 'a='a] using i by auto
        have "norm (y-x) ≤ (∑b∈Basis. ∣(y - x) ∙ b∣)"
          by (rule norm_le_l1)
        also have "… = ∣(y - x) ∙ i∣ + (∑b ∈ Basis - {i}. ∣(y - x) ∙ b∣)"
          by (meson finite_Basis i(2) sum.remove)
        also have "… <  e + sum (λi. 0) Basis"
        proof (rule add_less_le_mono)
          show "∣(y-x) ∙ i∣ < e"
            using di ‹e > 0› y_def i xi by (auto simp: inner_simps)
          show "(∑i∈Basis - {i}. ∣(y-x) ∙ i∣) ≤ (∑i∈Basis. 0)"
            unfolding y_def by (auto simp: inner_simps)
        qed
        finally have "norm (y-x) < e + sum (λi. 0) Basis" .
        then show "dist y x < e"
          unfolding dist_norm by auto
        have "y ∉ K"
          unfolding keq mem_box using i(1) i(2) xi xyi by fastforce
        moreover have "y ∈ ∪D"
          using subsetD[OF D(2)[OF ‹K ∈ D›] ‹x ∈ K›] ‹e > 0› di i
          by (auto simp: D mem_box y_def field_simps elim!: ballE[of _ _ i])
        ultimately show "y ∈ ∪(D - {K})" by auto
      qed
    qed
    then have "K ⊆ ∪(D - {K})"
      using closed closed_limpt by blast
    then have "∪(D - {K}) = cbox a b"
      unfolding D(6)[symmetric] by auto
    then have "D - {K} division_of cbox a b"
      by (metis Diff_subset less.prems division_of_subset D(6))
    then have "{ka ∈ D - {K}. content ka ≠ 0} division_of (cbox a b)"
      using card less.hyps by blast
    moreover have "{ka ∈ D - {K}. content ka ≠ 0} = {K ∈ D. content K ≠ 0}"
      using contk by auto
    ultimately show ?thesis by auto
  qed (use less.prems in auto)
qed

subsection ‹Integrability on subintervals›

lemma operative_integrableI:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes "0 ≤ e"
  shows "operative conj True (λi. f integrable_on i)"
proof -
  interpret comm_monoid conj True
  proof qed
  show ?thesis
  proof
    show "∧a b. box a b = {} ==> (f integrable_on cbox a b) = True"
      by (simp add: content_eq_0_interior integrable_on_null)
    show "∧a b c k.
             k ∈ Basis ==>
             (f integrable_on cbox a b) ⟷
             (f integrable_on cbox a b ∩ {x. x ∙ k ≤ c} ∧ f integrable_on cbox a b ∩ {x. c ≤ x ∙ k})"
      unfolding integrable_on_def by (auto intro!: has_integral_split)
  qed
qed

lemma integrable_subinterval:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes f: "f integrable_on cbox a b"
    and cd: "cbox c d ⊆ cbox a b"
  shows "f integrable_on cbox c d"
proof -
  interpret operative conj True "λi. f integrable_on i"
    using order_refl by (rule operative_integrableI)
  show ?thesis
    by (metis cd division division_of_finite empty f partial_division_extend_1 remove)
qed

lemma integrable_subinterval_real:
  fixes f :: "real ==> 'a::banach"
  assumes "f integrable_on {a..b}"
    and "{c..d} ⊆ {a..b}"
  shows "f integrable_on {c..d}"
  by (metis assms box_real(2) integrable_subinterval)

subsection ‹Combining adjacent intervals in 1 dimension›

lemma has_integral_combine:
  fixes a b c :: real and j :: "'a::banach"
  assumes "a ≤ c"
      and "c ≤ b"
      and ac: "(f has_integral i) {a..c}"
      and cb: "(f has_integral j) {c..b}"
  shows "(f has_integral (i + j)) {a..b}"
proof -
  interpret operative_real "lift_option plus" "Some 0"
    "λi. if f integrable_on i then Some (integral i f) else None"
    using operative_integralI by (rule operative_realI)
  from ‹a ≤ c› ‹c ≤ b› ac cb coalesce_less_eq
  have *: "lift_option (+)
             (if f integrable_on {a..c} then Some (integral {a..c} f) else None)
             (if f integrable_on {c..b} then Some (integral {c..b} f) else None) =
            (if f integrable_on {a..b} then Some (integral {a..b} f) else None)"
    by (auto simp: split: if_split_asm)
  then have "f integrable_on cbox a b"
    using ac cb by (auto split: if_split_asm)
  with * show ?thesis
    using ac cb by (auto simp: integrable_on_def integral_unique split: if_split_asm)
qed

lemma integral_combine:
  fixes f :: "real ==> 'a::banach"
  assumes "a ≤ c"
    and "c ≤ b"
    and ab: "f integrable_on {a..b}"
  shows "integral {a..c} f + integral {c..b} f = integral {a..b} f"
proof -
  have "(f has_integral integral {a..c} f) {a..c}" "(f has_integral integral {c..b} f) {c..b}"
    using ab ‹c ≤ b› ‹a ≤ c› integrable_subinterval_real by fastforce+
  then show ?thesis
    by (smt (verit, best) assms has_integral_combine integral_unique)
qed

lemma integrable_combine:
  fixes f :: "real ==> 'a::banach"
  assumes "a ≤ c"
    and "c ≤ b"
    and "f integrable_on {a..c}"
    and "f integrable_on {c..b}"
  shows "f integrable_on {a..b}"
  using assms has_integral_combine by blast

lemma integral_minus_sets:
  fixes f::"real ==> 'a::banach"
  shows "c ≤ a ==> c ≤ b ==> f integrable_on {c .. max a b} ==>
    integral {c .. a} f - integral {c .. b} f =
    (if a ≤ b then - integral {a .. b} f else integral {b .. a} f)"
  using integral_combine[of c a b f] integral_combine[of c b a f]
  by (auto simp: algebra_simps max_def)

lemma integral_minus_sets':
  fixes f::"real ==> 'a::banach"
  shows "c ≥ a ==> c ≥ b ==> f integrable_on {min a b .. c} ==>
    integral {a .. c} f - integral {b .. c} f =
    (if a ≤ b then integral {a .. b} f else - integral {b .. a} f)"
  using integral_combine[of b a c f] integral_combine[of a b c f]
  by (auto simp: algebra_simps min_def)

subsection ‹Reduce integrability to "local" integrability›

lemma integrable_on_little_subintervals:
  fixes f :: "'b::euclidean_space ==> 'a::banach"
  assumes "∀x∈cbox a b. ∃d>0. ∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b ⟶
    f integrable_on cbox u v"
  shows "f integrable_on cbox a b"
proof -
  interpret operative conj True "λi. f integrable_on i"
    using order_refl by (rule operative_integrableI)
  have "∀x. ∃d>0. x∈cbox a b ⟶ (∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b ⟶
    f integrable_on cbox u v)"
    using assms by (metis zero_less_one)
  then obtain d where d: "∧x. 0 < d x"
     "∧x u v. [x ∈ cbox a b; x ∈ cbox u v; cbox u v ⊆ ball x (d x); cbox u v ⊆ cbox a b] 
               ==> f integrable_on cbox u v"
    by metis
  obtain p where p: "p tagged_division_of cbox a b" "(λx. ball x (d x)) fine p"
    using fine_division_exists[OF gauge_ball_dependent,of d a b] d(1) by blast
  then have sndp: "snd ` p division_of cbox a b"
    by (metis division_of_tagged_division)
  have "f integrable_on k" if "(x, k) ∈ p" for x k
    using tagged_division_ofD(2-4)[OF p(1) that] fineD[OF p(2) that] d by blast
  then show ?thesis
    unfolding division [symmetric, OF sndp] comm_monoid_set_F_and
    by auto
qed


subsection ‹Second FTC or existence of antiderivative›

lemma integrable_const[intro]: "(λx. c) integrable_on cbox a b"
  unfolding integrable_on_def by blast

lemma integral_has_vector_derivative_continuous_at:
  fixes f :: "real ==> 'a::banach"
  assumes f: "f integrable_on {a..b}"
     and x: "x ∈ {a..b} - S"
     and "finite S"
     and fx: "continuous (at x within ({a..b} - S)) f"
 shows "((λu. integral {a..u} f) has_vector_derivative f x) (at x within ({a..b} - S))"
proof -
  let ?I = "λa b. integral {a..b} f"
  { fix e::real
    assume "e > 0"
    obtain d where "d>0" and d: "∧x'. [x' ∈ {a..b} - S; ∣x' - x∣ < d] ==> norm(f x' - f x) ≤ e"
      using ‹e>0› fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
    have "norm (integral {a..y} f - integral {a..x} f - (y-x) *🪙R f x) ≤ e * ∣y - x∣" (is "?lhs ≤ ?rhs")
           if y: "y ∈ {a..b} - S" and yx: "∣y - x∣ < d" for y
    proof (cases "y < x")
      case False
      have "f integrable_on {a..y}"
        using f y by (simp add: integrable_subinterval_real)
      then have Idiff: "?I a y - ?I a x = ?I x y"
        using False x by (simp add: algebra_simps integral_combine)
      have fux_int: "((λu. f u - f x) has_integral integral {x..y} f - (y-x) *🪙R f x) {x..y}"
      proof (rule has_integral_diff)
        show "(f has_integral integral {x..y} f) {x..y}"
          using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
        show "((λu. f x) has_integral (y - x) *🪙R f x) {x..y}"
          using has_integral_const_real [of "f x" x y] False by simp
      qed
      have "?lhs ≤ e * content {x..y}"
        using yx False d x y ‹e>0› assms
        by (intro has_integral_bound_real[where f="(λu. f u - f x)"]) (auto simp: Idiff fux_int)
      also have "… ≤ ?rhs"
        using False by auto
      finally show ?thesis .
    next
      case True
      have "f integrable_on {a..x}"
        using f x by (simp add: integrable_subinterval_real)
      then have Idiff: "?I a x - ?I a y = ?I y x"
        using True x y by (simp add: algebra_simps integral_combine)
      have fux_int: "((λu. f u - f x) has_integral integral {y..x} f - (x - y) *🪙R f x) {y..x}"
      proof (rule has_integral_diff)
        show "(f has_integral integral {y..x} f) {y..x}"
          using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
        show "((λu. f x) has_integral (x - y) *🪙R f x) {y..x}"
          using has_integral_const_real [of "f x" y x] True by simp
      qed
      have "norm (integral {a..x} f - integral {a..y} f - (x - y) *🪙R f x) ≤ e * content {y..x}"
        using yx True d x y ‹e>0› assms
        by (intro has_integral_bound_real[where f="(λu. f u - f x)"]) (auto simp: Idiff fux_int)
      also have "… ≤ e * ∣y - x∣"
        using True by auto
      finally have "norm (integral {a..x} f - integral {a..y} f - (x - y) *🪙R f x) ≤ e * ∣y - x∣" .
      then show ?thesis
        by (simp add: algebra_simps norm_minus_commute)
    qed
    then have "∃d>0. ∀y∈{a..b} - S. ∣y - x∣ < d ⟶ norm (integral {a..y} f - integral {a..x} f - (y-x) *🪙R f x) ≤ e * ∣y - x∣"
      using ‹d>0› by blast
  }
  then show ?thesis
    by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
qed


lemma integral_has_vector_derivative:
  fixes f :: "real ==> 'a::banach"
  assumes "continuous_on {a..b} f"
    and "x ∈ {a..b}"
  shows "((λu. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
  using assms integral_has_vector_derivative_continuous_at [OF integrable_continuous_real]
  by (fastforce simp: continuous_on_eq_continuous_within)

lemma integral_has_real_derivative:
  assumes "continuous_on {a..b} g"
  assumes "t ∈ {a..b}"
  shows "((λx. integral {a..x} g) has_real_derivative g t) (at t within {a..b})"
  using integral_has_vector_derivative[of a b g t] assms
  by (auto simp: has_real_derivative_iff_has_vector_derivative)

lemma antiderivative_continuous:
  fixes q b :: real
  assumes "continuous_on {a..b} f"
  obtains g where "∧x. x ∈ {a..b} ==> (g has_vector_derivative (f x::_::banach)) (at x within {a..b})"
  using integral_has_vector_derivative[OF assms] by auto

subsection ‹Combined fundamental theorem of calculus›

lemma antiderivative_integral_continuous:
  fixes f :: "real ==> 'a::banach"
  assumes "continuous_on {a..b} f"
  obtains g where "∀u∈{a..b}. ∀v ∈ {a..b}. u ≤ v ⟶ (f has_integral (g v - g u)) {u..v}"
proof -
  obtain g
    where g: "∧x. x ∈ {a..b} ==> (g has_vector_derivative f x) (at x within {a..b})"
    using antiderivative_continuous[OF assms] by metis
  have "(f has_integral g v - g u) {u..v}" if "u ∈ {a..b}" "v ∈ {a..b}" "u ≤ v" for u v
  proof -
    have "∧x. x ∈ cbox u v ==> (g has_vector_derivative f x) (at x within cbox u v)"
      by (metis atLeastAtMost_iff atLeastatMost_subset_iff box_real(2) g
          has_vector_derivative_within_subset subsetCE that)
    then show ?thesis
      by (metis box_real(2) ‹u ≤ v› fundamental_theorem_of_calculus)
  qed
  then show ?thesis
    using that by blast
qed


subsection ‹General "twiddling" for interval-to-interval function image›

lemma has_integral_twiddle:
  assumes "0 < r"
    and hg: "∧x. h(g x) = x"
    and gh: "∧x. g(h x) = x"
    and contg: "∧x. continuous (at x) g"
    and g: "∧u v. ∃w z. g ` cbox u v = cbox w z"
    and h: "∧u v. ∃w z. h ` cbox u v = cbox w z"
    and r: "∧u v. content(g ` cbox u v) = r * content (cbox u v)"
    and intfi: "(f has_integral i) (cbox a b)"
  shows "((λx. f(g x)) has_integral (1 / r) *🪙R i) (h ` cbox a b)"
proof (cases "cbox a b = {}")
  case False
  obtain w z where wz: "h ` cbox a b = cbox w z"
    using h by blast
  have inj: "inj g" "inj h"
    using hg gh injI by metis+
  from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast
  have "∃d. gauge d ∧ (∀p. p tagged_division_of h ` cbox a b ∧ d fine p 
              ⟶ norm ((∑(x, k)∈p. content k *🪙R f (g x)) - (1 / r) *🪙R i) < e)"
    if "e > 0" for e
  proof -
    have "e * r > 0" using that ‹0 < r› by simp
    with intfi[unfolded has_integral]
    obtain d where "gauge d"
               and d: "∧p. p tagged_division_of cbox a b ∧ d fine p 
                        ==> norm ((∑(x, k)∈p. content k *🪙R f x) - i) < e * r"
      by metis
    define d' where "d' x = g -` d (g x)" for x
    show ?thesis
    proof (rule_tac x=d' in exI, safe)
      show "gauge d'"
        using ‹gauge d› continuous_open_vimage[OF _ contg] by (auto simp: gauge_def d'_def)
    next
      fix p
      assume ptag: "p tagged_division_of h ` cbox a b" and finep: "d' fine p"
      note p = tagged_division_ofD[OF ptag]
      have gab: "g y ∈ cbox a b" if "y ∈ K" "(x, K) ∈ p" for x y K
        by (metis hg inj(2) inj_image_mem_iff p(3) subsetCE that that)
      have gimp: "(λ(x,K). (g x, g ` K)) ` p tagged_division_of (cbox a b) ∧ 
                  d fine (λ(x, k). (g x, g ` k)) ` p"
        unfolding tagged_division_of
      proof safe
        show "finite ((λ(x, k). (g x, g ` k)) ` p)"
          using ptag by auto
        show "d fine (λ(x, k). (g x, g ` k)) ` p"
          using finep unfolding fine_def d'_def by auto
      next
        fix x K
        assume xk: "(x, K) ∈ p"
        show "g x ∈ g ` K"
          using p(2)[OF xk] by auto
        show "∃u v. g ` K = cbox u v"
          using p(4)[OF xk] using assms(5-6) by auto
        fix x' K' u
        assume xk': "(x', K') ∈ p" and u: "u ∈ interior (g ` K)" "u ∈ interior (g ` K')"
        have "interior K ∩ interior K' ≠ {}"
        proof
          assume "interior K ∩ interior K' = {}"
          moreover have "u ∈ g ` (interior K ∩ interior K')"
            using interior_image_subset[OF ‹inj g› contg] u
            unfolding image_Int[OF inj(1)] by blast
          ultimately show False by blast
        qed
        then have same: "(x, K) = (x', K')"
          using ptag xk' xk by blast
        then show "g x = g x'"
          by auto
        show "g u ∈ g ` K'"if "u ∈ K" for u
          using that same by auto
        show "g u ∈ g ` K"if "u ∈ K'" for u
          using that same by auto
      next
        fix x
        assume "x ∈ cbox a b"
        then have "h x ∈  ∪{k. ∃x. (x, k) ∈ p}"
          using p(6) by auto
        then obtain X y where "h x ∈ X" "(y, X) ∈ p" by blast
        then show "x ∈ ∪{k. ∃x. (x, k) ∈ (λ(x, k). (g x, g ` k)) ` p}"
          by clarsimp (metis (no_types, lifting) gh image_eqI pair_imageI)
      qed (use gab in auto)
      have *: "inj_on (λ(x, k). (g x, g ` k)) p"
        using inj(1) unfolding inj_on_def by fastforce
      have "(∑(x,K)∈(λ(y,L). (g y, g ` L)) ` p. content K *🪙R f x) 
          = (∑u∈p. case case u of (x,K) ==> (g x, g ` K) of (y,L) ==> content L *🪙R f y)"
        by (metis (mono_tags, lifting) "*" sum.reindex_cong)
      also have "… = (∑(x,K)∈p. r *🪙R content K *🪙R f (g x))"
        using r by (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
      finally
      have "(∑(x, K)∈(λ(x,K). (g x, g ` K)) ` p. content K *🪙R f x) - i = r *🪙R (∑(x,K)∈p. content K *🪙R f (g x)) - i"
        by (simp add: scaleR_right.sum split_def)
      also have "… = r *🪙R ((∑(x,K)∈p. content K *🪙R f (g x)) - (1 / r) *🪙R i)"
        using ‹0 < r› by (auto simp: scaleR_diff_right)
      finally show "norm ((∑(x,K)∈p. content K *🪙R f (g x)) - (1 / r) *🪙R i) < e"
        using d[OF gimp] ‹0 < r› by auto
    qed
  qed
  then show ?thesis
    by (auto simp: h_eq has_integral)
qed (use intfi in auto)


subsection ‹Special case of a basic affine transformation›

lemma AE_lborel_inner_neq:
  assumes k: "k ∈ Basis"
  shows "AE x in lborel. x ∙ k ≠ c"
proof -
  interpret finite_product_sigma_finite "λ_. lborel" Basis
    proof qed simp
    have "emeasure lborel {x∈space lborel. x ∙ k = c} 
        = emeasure (Π🪙M j::'a∈Basis. lborel) (Π🪙E j∈Basis. if j = k then {c} else UNIV)"
    using k
    by (auto simp: lborel_eq[where 'a='a] emeasure_distr intro!: arg_cong2[where f=emeasure])
       (auto simp: space_PiM PiE_iff extensional_def split: if_split_asm)
  also have "… = (∏j∈Basis. emeasure lborel (if j = k then {c} else UNIV))"
    by (intro measure_times) auto
  also have "… = 0"
    by (intro prod_zero bexI[OF _ k]) auto
  finally show ?thesis
    by (subst AE_iff_measurable[OF _ refl]) auto
qed

lemma content_image_stretch_interval:
  fixes m :: "'a::euclidean_space ==> real"
  defines "s f x ≡ (∑k::'a∈Basis. (f k * (x∙k)) *🪙R k)"
  shows "content (s m ` cbox a b) = ∣∏k∈Basis. m k∣ * content (cbox a b)"
proof cases
  have s[measurable]: "s f ∈ borel →🪙M borel" for f
    by (auto simp: s_def[abs_def])
  assume m: "∀k∈Basis. m k ≠ 0"
  then have s_comp_s: "s (λk. 1 / m k) ∘ s m = id" "s m ∘ s (λk. 1 / m k) = id"
    by (auto simp: s_def[abs_def] fun_eq_iff euclidean_representation)
  then have "inv (s (λk. 1 / m k)) = s m" "bij (s (λk. 1 / m k))"
    by (auto intro: inv_unique_comp o_bij)
  then have eq: "s m ` cbox a b = s (λk. 1 / m k) -` cbox a b"
    using bij_vimage_eq_inv_image[OF ‹bij (s (λk. 1 / m k))›, of "cbox a b"] by auto
  show ?thesis
    using m unfolding eq measure_def
    by (subst lborel_affine_euclidean[where c=m and t=0])
       (simp_all add: emeasure_density measurable_sets_borel[OF s] abs_prod nn_integral_cmult
                      s_def[symmetric] emeasure_distr vimage_comp s_comp_s enn2real_mult prod_nonneg)
next
  assume "¬ (∀k∈Basis. m k ≠ 0)"
  then obtain k where k: "k ∈ Basis" "m k = 0" by auto
  then have [simp]: "(∏k∈Basis. m k) = 0"
    by (intro prod_zero) auto
  have "emeasure lborel {x∈space lborel. x ∈ s m ` cbox a b} = 0"
  proof (rule emeasure_eq_0_AE)
    show "AE x in lborel. x ∉ s m ` cbox a b"
      using AE_lborel_inner_neq[OF ‹k∈Basis›]
    proof eventually_elim
      show "x ∙ k ≠ 0 ==> x ∉ s m ` cbox a b " for x
        using k by (auto simp: s_def[abs_def] cbox_def)
    qed
  qed
  then show ?thesis
    by (simp add: measure_def)
qed

lemma interval_image_affinity_interval:
  "∃u v. (λx. m *🪙R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v"
  unfolding image_affinity_cbox
  by auto

lemma content_image_affinity_cbox:
  "content((λx::'a::euclidean_space. m *🪙R x + c) ` cbox a b) =
    ∣m∣ ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
proof (cases "cbox a b = {}")
  case True then show ?thesis by simp
next
  case False
  show ?thesis
  proof (cases "m ≥ 0")
    case True
    with ‹cbox a b ≠ {}› have "cbox (m *🪙R a + c) (m *🪙R b + c) ≠ {}"
      by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
    moreover from True have *: "∧i. (m *🪙R b + c) ∙ i - (m *🪙R a + c) ∙ i = m *🪙R (b-a) ∙ i"
      by (simp add: inner_simps field_simps)
    ultimately show ?thesis
      by (simp add: image_affinity_cbox True content_cbox' prod.distrib inner_diff_left)
  next
    case False
    with ‹cbox a b ≠ {}› have "cbox (m *🪙R b + c) (m *🪙R a + c) ≠ {}"
      by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
    moreover from False have *: "∧i. (m *🪙R a + c) ∙ i - (m *🪙R b + c) ∙ i = (-m) *🪙R (b-a) ∙ i"
      by (simp add: inner_simps field_simps)
    ultimately show ?thesis using False
      by (simp add: image_affinity_cbox content_cbox' inner_diff_left flip: prod_constant prod.distrib)
  qed
qed

lemma has_integral_affinity:
  fixes a :: "'a::euclidean_space"
  assumes "(f has_integral i) (cbox a b)"
      and "m ≠ 0"
  shows "((λx. f(m *🪙R x + c)) has_integral (1 / (∣m∣ ^ DIM('a))) *🪙R i) ((λx. (1 / m) *🪙R x + -((1 / m) *🪙R c)) ` cbox a b)"
proof (rule has_integral_twiddle)
  show "∃w z. (λx. (1 / m) *🪙R x + - ((1 / m) *🪙R c)) ` cbox u v = cbox w z"
       "∃w z. (λx. m *🪙R x + c) ` cbox u v = cbox w z" for u v
    using interval_image_affinity_interval by blast+
  show "content ((λx. m *🪙R x + c) ` cbox u v) = ∣m∣ ^ DIM('a) * content (cbox u v)" for u v
    using content_image_affinity_cbox by blast
qed (use assms zero_less_power in ‹auto simp: field_simps›)

lemma integrable_affinity:
  assumes "f integrable_on cbox a b"
    and "m ≠ 0"
  shows "(λx. f(m *🪙R x + c)) integrable_on ((λx. (1 / m) *🪙R x + -((1/m) *🪙R c)) ` cbox a b)"
  using has_integral_affinity assms
  unfolding integrable_on_def by blast

lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified]

lemma integrable_on_affinity:
  assumes "m ≠ 0" "f integrable_on (cbox a b)"
  shows "(λx. f (m *🪙R x + c)) integrable_on ((λx. (1 / m) *🪙R x - ((1 / m) *🪙R c)) ` cbox a b)"
proof -
  from assms obtain I where "(f has_integral I) (cbox a b)"
    by (auto simp: integrable_on_def)
  from has_integral_affinity[OF this assms(1), of c] show ?thesis
    by (auto simp: integrable_on_def)
qed

lemma has_integral_cmul_iff:
  assumes "c ≠ 0"
  shows "((λx. c *🪙R f x) has_integral (c *🪙R I)) A ⟷ (f has_integral I) A"
  using assms has_integral_cmul[of f I A c]
        has_integral_cmul[of "λx. c *🪙R f x" "c *🪙R I" A "inverse c"]
  by (auto simp: field_simps)

lemma has_integral_cmul_iff':
  assumes "c ≠ 0"
  shows "((λx. c *🪙R f x) has_integral I) A ⟷ (f has_integral I /🪙R c) A"
  using assms by (metis divideR_right has_integral_cmul_iff)

lemma has_integral_affinity':
  fixes a :: "'a::euclidean_space"
  assumes "(f has_integral i) (cbox a b)" and "m > 0"
  shows "((λx. f(m *🪙R x + c)) has_integral (i /🪙R m ^ DIM('a)))
           (cbox ((a - c) /🪙R m) ((b - c) /🪙R m))"
proof (cases "cbox a b = {}")
  case True
  hence "(cbox ((a - c) /🪙R m) ((b - c) /🪙R m)) = {}"
    using ‹m > 0› unfolding box_eq_empty by (auto simp: algebra_simps)
  with True and assms show ?thesis by simp
next
  case False
  have "((λx. f (m *🪙R x + c)) has_integral (1 / ∣m∣ ^ DIM('a)) *🪙R i)
          ((λx. (1 / m) *🪙R x + - ((1 / m) *🪙R c)) ` cbox a b)"
    using assms by (intro has_integral_affinity) auto
  also have "((λx. (1 / m) *🪙R x + - ((1 / m) *🪙R c)) ` cbox a b) =
               ((λx.  - ((1 / m) *🪙R c) + x) ` (λx. (1 / m) *🪙R x) ` cbox a b)"
    by (simp add: image_image algebra_simps)
  also have "(λx. (1 / m) *🪙R x) ` cbox a b = cbox ((1 / m) *🪙R a) ((1 / m) *🪙R b)" using ‹m > 0› False
    by (subst image_smult_cbox) simp_all
  also have "(λx. - ((1 / m) *🪙R c) + x) ` … = cbox ((a - c) /🪙R m) ((b - c) /🪙R m)"
    by (subst cbox_translation [symmetric]) (simp add: field_simps vector_add_divide_simps)
  finally show ?thesis using ‹m > 0›
    by (simp add: field_simps)
qed

lemma has_integral_affinity_iff:
  fixes f :: "'a :: euclidean_space ==> 'b :: real_normed_vector"
  assumes "m > 0"
  shows "((λx. f (m *🪙R x + c)) has_integral (I /🪙R m ^ DIM('a)))
               (cbox ((a - c) /🪙R m) ((b - c) /🪙R m)) ⟷
           (f has_integral I) (cbox a b)" (is "?lhs = ?rhs")
proof
  assume ?lhs
  from has_integral_affinity'[OF this, of "1 / m" "-c /🪙R m"] and ‹m > 0›
  show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps)
next
  assume ?rhs
  from has_integral_affinity'[OF this, of m c] and ‹m > 0›
  show ?lhs by simp
qed

lemma integrable_on_shift_cbox:
  "(f ∘ ((+) c)) integrable_on cbox a b ⟷ f integrable_on (cbox (a + c) (b + c))"
proof
  assume *: "(f ∘ ((+) c)) integrable_on cbox a b"
  show "f integrable_on (cbox (a+c) (b+c))"
    using integrable_on_affinity[OF _ *, of 1 "-c"] cbox_translation[of c a b]
    by (simp add: add_ac)
next
  assume *: "f integrable_on (cbox (a+c) (b+c))"
  show "(f ∘ ((+) c)) integrable_on cbox a b"
    using integrable_on_affinity[OF _ *, of 1 "c"] cbox_translation[of "-c" "a+c" "b+c"]
    by (simp add: o_def add_ac)
qed

lemma integrable_on_shift_Icc_real:
  "(f ∘ ((+) c)) integrable_on {a..b::real} ⟷ f integrable_on {a+c..b+c}"
  using integrable_on_shift_cbox[of f c a b] by simp

lemma has_integral_shift_cbox_iff:
  "((f ∘ ((+) c)) has_integral I) (cbox a b) ⟷
   (f has_integral I) (cbox (a + c) (b + c))"
proof
  assume *: "((f ∘ ((+) c)) has_integral I) (cbox a b)"
  show "(f has_integral I) (cbox (a + c) (b + c))"
    using has_integral_affinity[OF *, of 1 "-c"] cbox_translation[of c a b]
    by (simp add: add_ac)
next
  assume *: "(f has_integral I) (cbox (a + c) (b + c))"
  show "((f ∘ ((+) c)) has_integral I) (cbox a b)"
    using has_integral_affinity[OF *, of 1 "c"] cbox_translation[of "-c" "a+c" "b+c"]
    by (simp add: o_def add_ac)
qed

lemma has_integral_shift_Icc_real:
  "((f ∘ ((+) c)) has_integral I) {a..b::real} ⟷ (f has_integral I) {a+c..b+c}"
  using has_integral_shift_cbox_iff[of f c I a b] by simp

lemma integral_shift_cbox_plus:
  "integral (cbox a b) (f ∘ ((+) c)) = integral (cbox (a+c) (b+c)) f"
  using has_integral_shift_cbox_iff[of f c _ a b] integrable_on_shift_cbox[of f c a b]
  by (metis integrable_integral integral_unique not_integrable_integral)

lemma integral_shift_Icc_real:
  "integral {a..b::real} (f ∘ ((+) c)) = integral {a+c..b+c} f"
  using integral_shift_cbox_plus[of a b f c] by simp

subsection ‹Special case of stretching coordinate axes separately›

lemma has_integral_stretch:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "(f has_integral i) (cbox a b)" and "∀k∈Basis. m k ≠ 0"
  shows "((λx. f (∑k∈Basis. (m k * (x∙k))*🪙R k)) has_integral
         ((1/ ∣prod m Basis∣) *🪙R i)) ((λx. (∑k∈Basis. (1 / m k * (x∙k))*🪙R k)) ` cbox a b)"
  apply (rule has_integral_twiddle[where f=f])
  unfolding zero_less_abs_iff content_image_stretch_interval
  unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
  using assms
  by auto

lemma integrable_stretch:
  fixes f :: "'a::euclidean_space ==> 'b::real_normed_vector"
  assumes "f integrable_on cbox a b" and "∀k∈Basis. m k ≠ 0"
  shows "(λx::'a. f (∑k∈Basis. (m k * (x∙k))*🪙R k)) integrable_on
    ((λx. ∑k∈Basis. (1 / m k * (x∙k))*🪙R k) ` cbox a b)"
  using assms unfolding integrable_on_def
  by (force dest: has_integral_stretch)

lemma has_integral_stretch_real:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "(f has_integral i) {a..b}" and "m ≠ 0"
  shows "((λx. f (m * x)) has_integral (1 / ∣m∣) *🪙R i) ((λx. x / m) ` {a..b})"
  using has_integral_stretch [of f i a b "λb. m"] assms by simp

lemma integrable_stretch_real:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "f integrable_on {a..b}" and "m ≠ 0"
  shows "(λx. f (m * x)) integrable_on ((λx. x / m) ` {a..b})"
proof -
  from assms obtain I where "(f has_integral I) {a..b}"
    by (auto simp: integrable_on_def)
  from has_integral_stretch_real[OF this assms(2)] show ?thesis
    by (auto simp: integrable_on_def)
qed

lemma integrable_stretch_real_iff:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "m ≠ 0"
  shows "(λx. f (m * x)) integrable_on ((λx. x / m) ` {a..b}) ⟷ f integrable_on {a..b}"
proof
  assume "f integrable_on {a..b}"
  thus "(λx. f (m * x)) integrable_on ((λx. x / m) ` {a..b})"
    using assms by (intro integrable_stretch_real) auto
next
  assume *: "(λx. f (m * x)) integrable_on ((λx. x / m) ` {a..b})"
  define a' where "a' = (if m > 0 then a / m else b / m)"
  define b' where "b' = (if m > 0 then b / m else a / m)"
  have "bij_betw (λx. x / m) {a..b} {a'..b'}"
    by (rule bij_betwI[of _ _ _ "λx. x * m"]) (use assms in ‹auto simp: field_simps a'_def b'_def›)
  hence eq: "(λx. x / m) ` {a..b} = {a'..b'}"
    by (simp add: bij_betw_def)
  from * have "(λx. f (m * x)) integrable_on {a'..b'}"
    unfolding eq .
  hence "(λx. f (m * (1 / m * x))) integrable_on (λx. x / (1 / m)) ` {a'..b'}"
    using assms by (intro integrable_stretch_real) auto
  also have "(λx. f (m * (1 / m * x))) = f"
    using assms by simp
  also have "bij_betw (λx. x / (1 / m)) {a'..b'} {a..b}"
    by (rule bij_betwI[of _ _ _ "λx. x / m"]) (use assms in ‹auto simp: field_simps a'_def b'_def›)
  hence "(λx. x / (1 / m)) ` {a'..b'} = {a..b}"
    by (simp add: bij_betw_def)
  finally show "f integrable_on {a..b}" .
qed

lemma integral_stretch_real:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "m ≠ 0"
  shows "integral ((λx. x / m) ` {a..b}) (λx. f (m * x)) = (1 / ∣m∣) *🪙R integral {a..b} f"
proof (cases "f integrable_on {a..b}")
  case True
  hence "(f has_integral integral {a..b} f) {a..b}"
    by blast
  from has_integral_stretch_real[OF this assms] show ?thesis
    by (simp add: has_integral_iff)
next
  case False
  hence "¬(λx. f (m * x)) integrable_on ((λx. x / m) ` {a..b})"
    using assms by (subst integrable_stretch_real_iff)
  with False show ?thesis
    by (simp add: not_integrable_integral)
qed

lemma has_integral_stretch_real_iff:
  fixes f :: "real ==> 'b::real_normed_vector"
  assumes "m ≠ 0"
  shows "((λx. f (m * x)) has_integral I) ((λx. x / m) ` {a..b}) ⟷ 
             (f has_integral (∣m∣ *🪙R I)) {a..b}"
  using integral_stretch_real[of m a b f] integrable_stretch_real_iff[of m f a b] assms
  by (auto simp: has_integral_iff)

lemma has_integral_shift_cbox:
  fixes f :: "'a :: euclidean_space ==> 'b :: real_normed_vector"
  assumes "(f has_integral I) (cbox a b)"
  shows "((λx. f (x + c)) has_integral I) (cbox (a - c) (b - c))"
proof -
  have "((λx. f (x + c)) has_integral (1 / 1) *🪙R I) ((λx. x - c) ` cbox a b)"
    by (rule has_integral_twiddle)
       (use assms in ‹auto simp: cbox_shift'' cbox_shift' content_cbox_if
  algebra_simps box_eq_empty›)
  thus ?thesis
    by (simp add: cbox_shift'')
qed

lemma integrable_shift_cbox:
  fixes f :: "'a :: euclidean_space ==> 'b :: real_normed_vector"
  assumes "f integrable_on cbox a b"
  shows "(λx. f (x + c)) integrable_on (cbox (a - c) (b - c))"
  using has_integral_shift_cbox[of f _ a b c] assms
  by (auto simp: integrable_on_def)

lemma integrable_shift_cbox_iff:
  fixes f :: "'a :: euclidean_space ==> 'b :: real_normed_vector"
  shows "(λx. f (x + c)) integrable_on (cbox (a - c) (b - c)) ⟷ f integrable_on cbox a b"
  using integrable_shift_cbox[of f a b c]
        integrable_shift_cbox[of "λx. f (x + c)" "a - c" "b - c" "-c"] by auto

lemma integral_shift_cbox:
  fixes f :: "'a :: euclidean_space ==> 'b :: real_normed_vector"
  shows "integral (cbox (a - c) (b - c)) (λx. f (x + c)) = integral (cbox a b) f"
  by (metis eq_integralD has_integral_integral has_integral_shift_cbox
      integrable_shift_cbox_iff integral_unique)

lemma has_integral_shift_real_ivl:
  fixes f :: "real ==> 'b :: real_normed_vector"
  assumes "(f has_integral I) {a..b}"
  shows "((λx. f (x + c)) has_integral I) {a-c..b-c}"
  using has_integral_shift_cbox[of f I a b c] assms by simp

lemma has_integral_shift_real_ivl_iff:
  fixes f :: "real ==> 'b :: real_normed_vector"
  shows "(f has_integral I) {a..b} ⟷ ((λx. f (x + c)) has_integral I) {a-c..b-c}"
  using has_integral_shift_real_ivl[of f I a b c]
        has_integral_shift_real_ivl[of "λx. f (x + c)" I "a-c" "b-c" "-c"] by auto

lemma integrable_shift_real_ivl:
  fixes f :: "real ==> 'b :: real_normed_vector"
  assumes "f integrable_on {a..b}"
  shows "(λx. f (x + c)) integrable_on {a-c..b-c}"
  using integrable_shift_cbox[of f a b c] assms by simp

lemma integrable_shift_real_ivl_iff:
  fixes f :: "real ==> 'b :: real_normed_vector"
  shows "(λx. f (x + c)) integrable_on {a-c..b-c} ⟷ f integrable_on {a..b}"
  using integrable_shift_cbox_iff[of f c a b] by simp

lemma integral_shift_real_ivl:
  fixes f :: "real ==> 'b :: real_normed_vector"
  shows "integral {a-c..b-c} (λx. f (x + c)) = integral {a..b} f"
  using integral_shift_cbox[of a c b f] by simp

lemma vec_lambda_eq_sum:
     "(χ k. f k (x $ k)) = (∑k∈Basis. (f (axis_index k) (x ∙ k)) *🪙R k)" (is "?lhs = ?rhs")
proof -
  have "?lhs = (χ k. f k (x ∙ axis k 1))"
    by (simp add: cart_eq_inner_axis)
  also have "… = (∑u∈UNIV. f u (x ∙ axis u 1) *🪙R axis u 1)"
    by (simp add: vec_eq_iff axis_def if_distrib cong: if_cong)
  also have "… = ?rhs"
    by (simp add: Basis_vec_def UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
  finally show ?thesis .
qed

lemma has_integral_stretch_cart:
  fixes m :: "'n::finite ==> real"
  assumes f: "(f has_integral i) (cbox a b)" and m: "∧k. m k ≠ 0"
  shows "((λx. f(χ k. m k * x$k)) has_integral i /🪙R ∣prod m UNIV∣)
            ((λx. χ k. x$k / m k) ` (cbox a b))"
proof -
  have *: "∀k:: real^'n ∈ Basis. m (axis_index k) ≠ 0"
    using axis_index by (simp add: m)
  have eqp: "(∏k:: real^'n ∈ Basis. m (axis_index k)) = prod m UNIV"
    by (simp add: Basis_vec_def UNION_singleton_eq_range prod.reindex axis_eq_axis inj_on_def)
  show ?thesis
    using has_integral_stretch [OF f *] vec_lambda_eq_sum [where f="λi x. m i * x"] vec_lambda_eq_sum [where f="λi x. x / m i"]
    by (simp add: field_simps eqp)
qed

lemma image_stretch_interval_cart:
  fixes m :: "'n::finite ==> real"
  shows "(λx. χ k. m k * x$k) ` cbox a b =
            (if cbox a b = {} then {}
            else cbox (χ k. min (m k * a$k) (m k * b$k)) (χ k. max (m k * a$k) (m k * b$k)))"
proof -
  have *: "(∑k∈Basis. min (m (axis_index k) * (a ∙ k)) (m (axis_index k) * (b ∙ k)) *🪙R k)
           = (χ k. min (m k * a $ k) (m k * b $ k))"
          "(∑k∈Basis. max (m (axis_index k) * (a ∙ k)) (m (axis_index k) * (b ∙ k)) *🪙R k)
           = (χ k. max (m k * a $ k) (m k * b $ k))"
    apply (simp_all add: Basis_vec_def cart_eq_inner_axis UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
    apply (simp_all add: vec_eq_iff axis_def if_distrib cong: if_cong)
    done
  show ?thesis
    by (simp add: vec_lambda_eq_sum [where f="λi x. m i * x"] image_stretch_interval eq_cbox *)
qed


subsection ‹even more special cases›

lemma uminus_interval_vector[simp]:
  fixes a b :: "'a::euclidean_space"
  shows "uminus ` cbox a b = cbox (-b) (-a)"
proof -
  have "x ∈ uminus ` cbox a b" if "x ∈ cbox (- b) (- a)" for x
    by (smt (verit) add.inverse_inverse image_iff inner_minus_left mem_box(2) that)
  then show ?thesis
    by (auto simp: mem_box)
qed

lemma has_integral_reflect_lemma[intro]:
  assumes "(f has_integral i) (cbox a b)"
  shows "((λx. f(-x)) has_integral i) (cbox (-b) (-a))"
  using has_integral_affinity[OF assms, of "-1" 0]
  by auto

lemma has_integral_reflect_lemma_real[intro]:
  assumes "(f has_integral i) {a..b::real}"
  shows "((λx. f(-x)) has_integral i) {-b .. -a}"
  by (metis has_integral_reflect_lemma interval_cbox assms)

lemma has_integral_reflect[simp]:
  "((λx. f (-x)) has_integral i) (cbox (-b) (-a)) ⟷ (f has_integral i) (cbox a b)"
  by (auto dest: has_integral_reflect_lemma)

lemma has_integral_reflect_real[simp]:
  fixes a b::real
  shows "((λx. f (-x)) has_integral i) {-b..-a} ⟷ (f has_integral i) {a..b}"
  by (metis has_integral_reflect interval_cbox)

lemma integrable_reflect[simp]: "(λx. f(-x)) integrable_on cbox (-b) (-a) ⟷ f integrable_on cbox a b"
  unfolding integrable_on_def by auto

lemma integrable_reflect_real[simp]: "(λx. f(-x)) integrable_on {-b .. -a} ⟷ f integrable_on {a..b::real}"
  unfolding box_real[symmetric]
  by (rule integrable_reflect)

lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (λx. f (-x)) = integral (cbox a b) f"
  unfolding integral_def by auto

lemma integral_reflect_real[simp]: "integral {-b .. -a} (λx. f (-x)) = integral {a..b::real} f"
  unfolding box_real[symmetric]
  by (rule integral_reflect)

subsection ‹Stronger form of FCT; quite a tedious proof›

lemma split_minus[simp]: "(λ(x, k). f x k) x - (λ(x, k). g x k) x = (λ(x, k). f x k - g x k) x"
  by (simp add: split_def)

theorem fundamental_theorem_of_calculus_interior:
  fixes f :: "real ==> 'a::real_normed_vector"
  assumes "a ≤ b"
    and contf: "continuous_on {a..b} f"
    and derf: "∧x. x ∈ {a <..< b} ==> (f has_vector_derivative f' x) (at x)"
  shows "(f' has_integral (f b - f a)) {a..b}"
proof (cases "a = b")
  case True
  then have *: "cbox a b = {b}" "f b - f a = 0"
    by (auto simp: order_antisym)
  with True show ?thesis by auto
next
  case False
  with ‹a ≤ b› have ab: "a < b" by arith
  show ?thesis
    unfolding has_integral_factor_content_real
  proof (intro allI impI)
    fix e :: real
    assume e: "e > 0"
    then have eba8: "(e * (b-a)) / 8 > 0"
      using ab by (auto simp: field_simps)
    note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt, THEN conjunct2, rule_format]
    have bounded: "∧x. x ∈ {a<..<b} ==> bounded_linear (λu. u *🪙R f' x)"
      by (simp add: bounded_linear_scaleR_left)
    have "∀x ∈ box a b. ∃d>0. ∀y. norm (y-x) < d ⟶ norm (f y - f x - (y-x) *🪙R f' x) ≤ e/2 * norm (y-x)"
      (is "∀x ∈ box a b. ?Q x") 🍋‹The explicit quantifier is required by the following step›
        using e derf_exp [of _ "e/2"] by auto
    then obtain d where d: "∧x. 0 < d x"
      "∧x y. [x ∈ box a b; norm (y-x) < d x] ==> norm (f y - f x - (y-x) *🪙R f' x) ≤ e/2 * norm (y-x)"
      unfolding bgauge_existence_lemma by metis
    have "bounded (f ` cbox a b)"
      using compact_cbox assms by (auto simp: compact_imp_bounded compact_continuous_image)
    then obtain B
      where "0 < B" and B: "∧x. x ∈ f ` cbox a b ==> norm x ≤ B"
      unfolding bounded_pos by metis
    obtain da where "0 < da"
      and da: "∧c. [a ≤ c; {a..c} ⊆ {a..b}; {a..c} ⊆ ball a da]
                          ==> norm (content {a..c} *🪙R f' a - (f c - f a)) ≤ (e * (b-a)) / 4"
    proof -
      have "continuous (at a within {a..b}) f"
        using contf continuous_on_eq_continuous_within by force
      with eba8 obtain k where "0 < k"
        and k: "∧x. [x ∈ {a..b}; 0 < norm (x-a); norm (x-a) < k] ==> norm (f x - f a) < e * (b-a) / 8"
        unfolding continuous_within Lim_within dist_norm by metis
      obtain l where l: "0 < l" "norm (l *🪙R f' a) ≤ e * (b-a) / 8"
      proof (cases "f' a = 0")
        case True with ab e that show ?thesis by auto
      next
        case False
        show ?thesis
        proof
          show "norm ((e * (b - a) / 8 / norm (f' a)) *🪙R f' a) ≤ e * (b - a) / 8"
               "0 < e * (b - a) / 8 / norm (f' a)"
            using False ab e by (auto simp: field_simps)
        qed
      qed
      have "norm (content {a..c} *🪙R f' a - (f c - f a)) ≤ e * (b-a) / 4"
        if "a ≤ c" "{a..c} ⊆ {a..b}" and bmin: "{a..c} ⊆ ball a (min k l)" for c
      proof -
        have minkl: "∣a - x∣ < min k l" if "x ∈ {a..c}" for x
          using bmin dist_real_def that by auto
        then have lel: "∣c - a∣ ≤ ∣l∣"
          using that by force
        have "norm ((c - a) *🪙R f' a - (f c - f a)) ≤ norm ((c - a) *🪙R f' a) + norm (f c - f a)"
          by (rule norm_triangle_ineq4)
        also have "… ≤ e * (b-a) / 8 + e * (b-a) / 8"
        proof (rule add_mono)
          have "norm ((c - a) *🪙R f' a) ≤ norm (l *🪙R f' a)"
            by (auto intro: mult_right_mono [OF lel])
          with l show "norm ((c - a) *🪙R f' a) ≤ e * (b-a) / 8"
            by linarith
        next
          have "norm (f c - f a) < e * (b-a) / 8"
          proof (cases "a = c")
            case True then show ?thesis
              using eba8 by auto
          next
            case False show ?thesis
              by (rule k) (use minkl ‹a ≤ c› that False in auto)
          qed
          then show "norm (f c - f a) ≤ e * (b-a) / 8" by simp
        qed
        finally show "norm (content {a..c} *🪙R f' a - (f c - f a)) ≤ e * (b-a) / 4"
          unfolding content_real[OF ‹a ≤ c›] by auto
      qed
      then show ?thesis
        by (rule_tac da="min k l" in that) (auto simp: l ‹0 < k›)
    qed
    obtain db where "0 < db"
      and db: "∧c. [c ≤ b; {c..b} ⊆ {a..b}; {c..b} ⊆ ball b db]
                   ==> norm (content {c..b} *🪙R f' b - (f b - f c)) ≤ (e * (b-a)) / 4"
    proof -
      have "continuous (at b within {a..b}) f"
        using contf continuous_on_eq_continuous_within by force
      with eba8 obtain k
        where "0 < k"
          and k: "∧x. [x ∈ {a..b}; 0 < norm(x-b); norm(x-b) < k]
                     ==> norm (f b - f x) < e * (b-a) / 8"
        unfolding continuous_within Lim_within dist_norm norm_minus_commute by metis
      obtain l where l: "0 < l" "norm (l *🪙R f' b) ≤ (e * (b-a)) / 8"
      proof (cases "f' b = 0")
        case True thus ?thesis
          using ab e that by auto
      next
        case False show ?thesis
        proof
          show "norm ((e * (b - a) / 8 / norm (f' b)) *🪙R f' b) ≤ e * (b - a) / 8"
               "0 < e * (b - a) / 8 / norm (f' b)"
            using False ab e by (auto simp: field_simps)
        qed
      qed
      have "norm (content {c..b} *🪙R f' b - (f b - f c)) ≤ e * (b-a) / 4"
        if "c ≤ b" "{c..b} ⊆ {a..b}" and bmin: "{c..b} ⊆ ball b (min k l)" for c
      proof -
        have minkl: "∣b - x∣ < min k l" if "x ∈ {c..b}" for x
          using bmin dist_real_def that by auto
        then have lel: "∣b - c∣ ≤ ∣l∣"
          using that by force
        have "norm ((b - c) *🪙R f' b - (f b - f c)) ≤ norm ((b - c) *🪙R f' b) + norm (f b - f c)"
          by (rule norm_triangle_ineq4)
        also have "… ≤ e * (b-a) / 8 + e * (b-a) / 8"
        proof (rule add_mono)
          have "norm ((b - c) *🪙R f' b) ≤ norm (l *🪙R f' b)"
            by (auto intro: mult_right_mono [OF lel])
          also have "… ≤ e * (b-a) / 8"
            by (rule l)
          finally show "norm ((b - c) *🪙R f' b) ≤ e * (b-a) / 8" .
        next
          have "norm (f b - f c) < e * (b-a) / 8"
          proof (cases "b = c")
            case True with eba8 show ?thesis
              by auto
          next
            case False show ?thesis
              by (rule k) (use minkl ‹c ≤ b› that False in auto)
          qed
          then show "norm (f b - f c) ≤ e * (b-a) / 8" by simp
        qed
        finally show "norm (content {c..b} *🪙R f' b - (f b - f c)) ≤ e * (b-a) / 4"
          unfolding content_real[OF ‹c ≤ b›] by auto
      qed
      then show ?thesis
        by (rule_tac db="min k l" in that) (auto simp: l ‹0 < k›)
    qed
    let ?d = "(λx. ball x (if x=a then da else if x=b then db else d x))"
    show "∃d. gauge d ∧ (∀p. p tagged_division_of {a..b} ∧ d fine p ⟶
              norm ((∑(x,K)∈p. content K *🪙R f' x) - (f b - f a)) ≤ e * content {a..b})"
    proof (rule exI, safe)
      show "gauge ?d"
        using ab ‹db > 0› ‹da > 0› d(1) by (auto intro: gauge_ball_dependent)
    next
      fix p
      assume ptag: "p tagged_division_of {a..b}" and fine: "?d fine p"
      let ?A = "{t. fst t ∈ {a, b}}"
      note p = tagged_division_ofD[OF ptag]
      have pA: "p = (p ∩ ?A) ∪ (p - ?A)" "finite (p ∩ ?A)" "finite (p - ?A)" "(p ∩ ?A) ∩ (p - ?A) = {}"
        using ptag fine by auto
      have le_xz: "∧w x y z::real. y ≤ z/2 ==> w - x ≤ z/2 ==> w + y ≤ x + z"
        by arith
      have non: False if xk: "(x,K) ∈ p" and "x ≠ a" "x ≠ b"
        and less: "e * (Sup K - Inf K)/2 < norm (content K *🪙R f' x - (f (Sup K) - f (Inf K)))"
      for x K
      proof -
        obtain u v where k: "K = cbox u v"
          using p(4) xk by blast
        then have "u ≤ v" and uv: "{u, v} ⊆ cbox u v"
          using p(2)[OF xk] by auto
        then have result: "e * (v - u)/2 < norm ((v - u) *🪙R f' x - (f v - f u))"
          using less[unfolded k box_real interval_bounds_real content_real] by auto
        then have "x ∈ box a b"
          using p(2) p(3) ‹x ≠ a› ‹x ≠ b› xk by fastforce
        with d have *: "∧y. norm (y-x) < d x 
                ==> norm (f y - f x - (y-x) *🪙R f' x) ≤ e/2 * norm (y-x)"
          by metis
        have xd: "norm (u - x) < d x" "norm (v - x) < d x"
          using fineD[OF fine xk] ‹x ≠ a› ‹x ≠ b› uv
          by (auto simp: k subset_eq dist_commute dist_real_def)
        have "norm ((v - u) *🪙R f' x - (f v - f u)) =
              norm ((f u - f x - (u - x) *🪙R f' x) - (f v - f x - (v - x) *🪙R f' x))"
          by (rule arg_cong[where f=norm]) (auto simp: scaleR_left.diff)
        also have "… ≤ e/2 * norm (u - x) + e/2 * norm (v - x)"
          by (metis norm_triangle_le_diff add_mono * xd)
        also have "… ≤ e/2 * norm (v - u)"
          using p(2)[OF xk] by (auto simp: field_simps k)
        also have "… < norm ((v - u) *🪙R f' x - (f v - f u))"
          using result by (simp add: ‹u ≤ v›)
        finally have "e * (v - u)/2 < e * (v - u)/2"
          using uv by auto
        then show False by auto
      qed
      have "norm (∑(x, K)∈p - ?A. content K *🪙R f' x - (f (Sup K) - f (Inf K)))
          ≤ (∑(x, K)∈p - ?A. norm (content K *🪙R f' x - (f (Sup K) - f (Inf K))))"
        by (auto intro: sum_norm_le)
      also have "… ≤ (∑n∈p - ?A. e * (case n of (x, k) ==> Sup k - Inf k)/2)"
        using non by (fastforce intro: sum_mono)
      finally have I: "norm (∑(x, k)∈p - ?A. content k *🪙R f' x - (f (Sup k) - f (Inf k)))
                     ≤ (∑n∈p - ?A. e * (case n of (x, k) ==> Sup k - Inf k))/2"
        by (simp add: sum_divide_distrib)
      have II: "norm (∑(x, k)∈p ∩ ?A. content k *🪙R f' x - (f (Sup k) - f (Inf k))) -
             (∑n∈p ∩ ?A. e * (case n of (x, k) ==> Sup k - Inf k))
             ≤ (∑n∈p - ?A. e * (case n of (x, k) ==> Sup k - Inf k))/2"
      proof -
        have ge0: "0 ≤ e * (Sup k - Inf k)" if xkp: "(x, k) ∈ p ∩ ?A" for x k
        proof -
          obtain u v where uv: "k = cbox u v"
            by (meson Int_iff xkp p(4))
          with p that have "cbox u v ≠ {}"
            by blast
          then show "0 ≤ e * ((Sup k) - (Inf k))"
            unfolding uv using e by (auto simp: field_simps)
        qed
        let ?B = "λx. {t ∈ p. fst t = x ∧ content (snd t) ≠ 0}"
        let ?C = "{t ∈ p. fst t ∈ {a, b} ∧ content (snd t) ≠ 0}"
        have "norm (∑(x, k)∈p ∩ {t. fst t ∈ {a, b}}. content k *🪙R f' x - (f (Sup k) - f (Inf k))) ≤ e * (b-a)/2"
        proof -
          have *: "∧S f e. sum f S = sum f (p ∩ ?C) ==> norm (sum f (p ∩ ?C)) ≤ e ==> norm (sum f S) ≤ e"
            by auto
          have 1: "content K *🪙R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
            if "(x,K) ∈ p ∩ {t. fst t ∈ {a, b}} - p ∩ ?C" for x K
          proof -
            have xk: "(x,K) ∈ p" and k0: "content K = 0"
              using that by auto
            then obtain u v where uv: "K = cbox u v" "u = v"
              using xk k0 p by fastforce
            then show "content K *🪙R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
              using xk unfolding uv by auto
          qed
          have 2: "norm(∑(x,K)∈p ∩ ?C. content K *🪙R f' x - (f (Sup K) - f (Inf K)))  ≤ e * (b-a)/2"
          proof -
            have norm_le: "norm (sum f S) ≤ e"
              if 🍋: "∧x y. [x ∈ S; y ∈ S] ==> x = y" "∧x. x ∈ S ==> norm (f x) ≤ e" "e > 0"
              for S f and e :: real
            proof (cases "S = {}")
              case True
              with that show ?thesis by auto
            next
              case False then obtain x where "S = {x}"
                using 🍋 by blast
              then show ?thesis
                by (simp add: that(2))
            qed
            have *: "p ∩ ?C = ?B a ∪ ?B b"
              by blast
            then have "norm (∑(x,K)∈p ∩ ?C. content K *🪙R f' x - (f (Sup K) - f (Inf K))) =
                       norm (∑(x,K) ∈ ?B a ∪ ?B b. content K *🪙R f' x - (f (Sup K) - f (Inf K)))"
              by simp
            also have "… = norm ((∑(x,K) ∈ ?B a. content K *🪙R f' x - (f (Sup K) - f (Inf K))) + 
                                   (∑(x,K) ∈ ?B b. content K *🪙R f' x - (f (Sup K) - f (Inf K))))"
              using p(1) ab e by (subst sum.union_disjoint) auto
            also have "… ≤ e * (b - a) / 4 + e * (b - a) / 4"
            proof (rule norm_triangle_le [OF add_mono])
              have pa: "∃v. k = cbox a v ∧ a ≤ v" if "(a, k) ∈ p" for k
                using p that by fastforce
              show "norm (∑(x,K) ∈ ?B a. content K *🪙R f' x - (f (Sup K) - f (Inf K))) ≤ e * (b - a) / 4"
              proof (intro norm_le; clarsimp)
                fix K K'
                assume K: "(a, K) ∈ p" "(a, K') ∈ p" and ne0: "content K ≠ 0" "content K' ≠ 0"
                with pa obtain v v' where v: "K = cbox a v" "a ≤ v" and v': "K' = cbox a v'" "a ≤ v'"
                  by blast
                let ?v = "min v v'"
                have "box a ?v ⊆ K ∩ K'"
                  unfolding v v' by (auto simp: mem_box)
                then have "interior (box a (min v v')) ⊆ interior K ∩ interior K'"
                  using interior_Int interior_mono by blast
                moreover have "(a + ?v)/2 ∈ box a ?v"
                  using ne0 unfolding v v' content_eq_0 not_le
                  by (auto simp: mem_box)
                ultimately have "(a + ?v)/2 ∈ interior K ∩ interior K'"
                  unfolding interior_open[OF open_box] by auto
                then show "K = K'"
                  using p(5)[OF K] by auto
              next
                fix K
                assume K: "(a, K) ∈ p" and ne0: "content K ≠ 0"
                show "norm (content c *🪙R f' a - (f (Sup c) - f (Inf c))) * 4 ≤ e * (b-a)"
                  if "(a, c) ∈ p" and ne0: "content c ≠ 0" for c
                proof -
                  obtain v where v: "c = cbox a v" and "a ≤ v"
                    using pa[OF ‹(a, c) ∈ p›] by metis
                  then have "a ∈ {a..v}" "v ≤ b"
                    using p(3)[OF ‹(a, c) ∈ p›] by auto
                  moreover have "{a..v} ⊆ ball a da"
                    using fineD[OF ‹?d fine p› ‹(a, c) ∈ p›] by (simp add: v split: if_split_asm)
                  ultimately show ?thesis
                    unfolding v interval_bounds_real[OF ‹a ≤ v›] box_real
                    using da ‹a ≤ v› by auto
                qed
              qed (use ab e in auto)
            next
              have pb: "∃v. k = cbox v b ∧ b ≥ v" if "(b, k) ∈ p" for k
                using p that by fastforce
              show "norm (∑(x,K) ∈ ?B b. content K *🪙R f' x - (f (Sup K) - f (Inf K))) ≤ e * (b - a) / 4"
              proof (intro norm_le; clarsimp)
                fix K K'
                assume K: "(b, K) ∈ p" "(b, K') ∈ p" and ne0: "content K ≠ 0" "content K' ≠ 0"
                with pb obtain v v' where v: "K = cbox v b" "v ≤ b" and v': "K' = cbox v' b" "v' ≤ b"
                  by blast
                let ?v = "max v v'"
                have "box ?v b ⊆ K ∩ K'"
                  unfolding v v' by (auto simp: mem_box)
                then have "interior (box (max v v') b) ⊆ interior K ∩ interior K'"
                  using interior_Int interior_mono by blast
                moreover have "((b + ?v)/2) ∈ box ?v b"
                  using ne0 unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
                ultimately have "((b + ?v)/2) ∈ interior K ∩ interior K'"
                  unfolding interior_open[OF open_box] by auto
                then show "K = K'"
                  using p(5)[OF K] by auto
              next
                fix K
                assume K: "(b, K) ∈ p" and ne0: "content K ≠ 0"
                show "norm (content c *🪙R f' b - (f (Sup c) - f (Inf c))) * 4 ≤ e * (b-a)"
                  if "(b, c) ∈ p" and ne0: "content c ≠ 0" for c
                proof -
                obtain v where v: "c = cbox v b" and "v ≤ b"
                  using ‹(b, c) ∈ p› pb by blast
                then have "v ≥ a""b ∈ {v.. b}"
                  using p(3)[OF ‹(b, c) ∈ p›] by auto
                moreover have "{v..b} ⊆ ball b db"
                  using fineD[OF ‹?d fine p› ‹(b, c) ∈ p›] box_real(2) v False by force
                ultimately show ?thesis
                  using db v by auto
                qed
              qed (use ab e in auto)
            qed
            also have "… = e * (b-a)/2"
              by simp
            finally show "norm (∑(x,k)∈p ∩ ?C.
                        content k *🪙R f' x - (f (Sup k) - f (Inf k))) ≤ e * (b-a)/2" .
          qed
          show "norm (∑(x, k)∈p ∩ ?A. content k *🪙R f' x - (f ((Sup k)) - f ((Inf k)))) ≤ e * (b-a)/2"
            apply (rule * [OF sum.mono_neutral_right[OF pA(2)]])
            using 1 2 by (auto simp: split_paired_all)
        qed
        also have "… = (∑n∈p. e * (case n of (x, k) ==> Sup k - Inf k))/2"
          unfolding sum_distrib_left[symmetric]
          by (subst additive_tagged_division_1[OF ‹a ≤ b› ptag]) auto
        finally have norm_le: "norm (∑(x,K)∈p ∩ {t. fst t ∈ {a, b}}. content K *🪙R f' x - (f (Sup K) - f (Inf K)))
                ≤ (∑n∈p. e * (case n of (x, K) ==> Sup K - Inf K))/2" .
        have le2: "∧x s1 s2::real. 0 ≤ s1 ==> x ≤ (s1 + s2)/2 ==> x - s1 ≤ s2/2"
          by auto
        show ?thesis
          apply (rule le2 [OF sum_nonneg])
          using ge0 apply force
          by (metis (no_types, lifting) Diff_Diff_Int Diff_subset norm_le p(1) sum.subset_diff)
      qed
      note * = additive_tagged_division_1[OF assms(1) ptag, symmetric]
      have "norm (∑(x,K)∈p ∩ ?A ∪ (p - ?A). content K *🪙R f' x - (f (Sup K) - f (Inf K)))
               ≤ e * (∑(x,K)∈p ∩ ?A ∪ (p - ?A). Sup K - Inf K)"
        unfolding sum_distrib_left
        unfolding sum.union_disjoint[OF pA(2-)]
        using le_xz norm_triangle_le I II by blast
      then
      show "norm ((∑(x,K)∈p. content K *🪙R f' x) - (f b - f a)) ≤ e * content {a..b}"
        by (simp only: content_real[OF ‹a ≤ b›] *[of "λx. x"] *[of f] sum_subtractf[symmetric] split_minus pA(1) [symmetric])
    qed
  qed
qed


subsection ‹Stronger form with finite number of exceptional points›

lemma fundamental_theorem_of_calculus_interior_strong:
 fixes f :: "real ==> 'a::banach"
 assumes "finite S"
   and "a ≤ b" "∧x. x ∈ {a <..< b} - S ==> (f has_vector_derivative f'(x)) (at x)"
   and "continuous_on {a .. b} f"
 shows "(f' has_integral (f b - f a)) {a .. b}"
  using assms
proof (induction arbitrary: a b)
case empty
  then show ?case
    using fundamental_theorem_of_calculus_interior by force
next
case (insert x S)
  show ?case
  proof (cases "x ∈ {a<..<b}")
    case False then show ?thesis
      using insert by blast
  next
    case True then have "a < x" "x < b"
      by auto
    have "(f' has_integral f x - f a) {a..x}" "(f' has_integral f b - f x) {x..b}"
      using ‹continuous_on {a..b} f› ‹a < x› ‹x < b› continuous_on_subset by (force simp: intro!: insert)+
    then have "(f' has_integral f x - f a + (f b - f x)) {a..b}"
      using ‹a < x› ‹x < b› has_integral_combine less_imp_le by blast
    then show ?thesis
      by simp
  qed
qed

corollary fundamental_theorem_of_calculus_strong:
  fixes f :: "real ==> 'a::banach"
  assumes "finite S"
    and "a ≤ b"
    and vec: "∧x. x ∈ {a..b} - S ==> (f has_vector_derivative f'(x)) (at x)"
    and "continuous_on {a..b} f"
  shows "(f' has_integral (f b - f a)) {a..b}"
  by (rule fundamental_theorem_of_calculus_interior_strong [OF ‹finite S›]) (force simp: assms)+

proposition indefinite_integral_continuous_left:
  fixes f:: "real ==> 'a::banach"
  assumes intf: "f integrable_on {a..b}" and "a < c" "c ≤ b" "e > 0"
  obtains d where "d > 0"
    and "∀t. c - d < t ∧ t ≤ c ⟶ norm (integral {a..c} f - integral {a..t} f) < e"
proof -
  obtain w where "w > 0" and w: "∧t. [c - w < t; t < c] ==> norm (f c) * norm(c - t) < e/3"
  proof (cases "f c = 0")
    case False
    hence e3: "0 < e/3 / norm (f c)" using ‹e>0› by simp
    moreover have "norm (f c) * norm (c - t) < e/3"
      if "t < c" and "c - e/3 / norm (f c) < t" for t
      unfolding real_norm_def
      by (smt (verit) False divide_right_mono nonzero_mult_div_cancel_left norm_eq_zero norm_ge_zero that)
    ultimately show ?thesis
      using that by auto
  qed (use ‹e > 0› in auto)

  let ?SUM = "λp. (∑(x,K) ∈ p. content K *🪙R f x)"
  have e3: "e/3 > 0"
    using ‹e > 0› by auto
  have "f integrable_on {a..c}"
    using ‹a < c› ‹c ≤ b› by (auto intro: integrable_subinterval_real[OF intf])
  then obtain d1 where "gauge d1" and d1:
    "∧p. [p tagged_division_of {a..c}; d1 fine p] ==> norm (?SUM p - integral {a..c} f) < e/3"
    using integrable_integral has_integral_real e3 by metis
  define d where [abs_def]: "d x = ball x w ∩ d1 x" for x
  have "gauge d"
    unfolding d_def using ‹w > 0› ‹gauge d1› by auto
  then obtain k where "0 < k" and k: "ball c k ⊆ d c"
    by (meson gauge_def open_contains_ball)

  let ?d = "min k (c - a)/2"
  show thesis
  proof (intro that[of ?d] allI impI, safe)
    show "?d > 0"
      using ‹0 < k› ‹a < c› by auto
  next
    fix t
    assume t: "c - ?d < t" "t ≤ c"
    show "norm (integral ({a..c}) f - integral ({a..t}) f) < e"
    proof (cases "t < c")
      case False with ‹t ≤ c› show ?thesis
        by (simp add: ‹e > 0›)
    next
      case True
      have "f integrable_on {a..t}"
        using ‹t < c› ‹c ≤ b› by (auto intro: integrable_subinterval_real[OF intf])
      then obtain d2 where d2: "gauge d2"
        "∧p. p tagged_division_of {a..t} ∧ d2 fine p ==> norm (?SUM p - integral {a..t} f) < e/3"
        using integrable_integral has_integral_real e3 by metis
      define d3 where "d3 x = (if x ≤ t then d1 x ∩ d2 x else d1 x)" for x
      have "gauge d3"
        using ‹gauge d1› ‹gauge d2› unfolding d3_def gauge_def by auto
      then obtain p where ptag: "p tagged_division_of {a..t}" and pfine: "d3 fine p"
        by (metis box_real(2) fine_division_exists)
      note p' = tagged_division_ofD[OF ptag]
      have pt: "(x,K)∈p ==> x ≤ t" for x K
        by (meson atLeastAtMost_iff p'(2) p'(3) subsetCE)
      with pfine have "d2 fine p"
        unfolding fine_def d3_def by fastforce
      then have d2_fin: "norm (?SUM p - integral {a..t} f) < e/3"
        using d2(2) ptag by auto
      have eqs: "{a..c} ∩ {x. x ≤ t} = {a..t}" "{a..c} ∩ {x. x ≥ t} = {t..c}"
        using t by (auto simp: field_simps)
      have "p ∪ {(c, {t..c})} tagged_division_of {a..c}"
        using ‹t ≤ c›
        by (intro tagged_division_Un_interval_real [of _ _ _ 1 t])
           (auto simp: eqs tagged_division_of_self_real eqs ptag)
      moreover
      have "d1 fine p ∪ {(c, {t..c})}"
        unfolding fine_def
      proof safe
        fix x K y
        assume "(x,K) ∈ p" and "y ∈ K" then show "y ∈ d1 x"
          by (metis Int_iff d3_def subsetD fineD pfine)
      next
        fix x assume "x ∈ {t..c}"
        then have "dist c x < k"
          using t(1) by (auto simp: field_simps dist_real_def)
        with k show "x ∈ d1 c"
          unfolding d_def by auto
      qed
      ultimately have d1_fin: "norm (?SUM(p ∪ {(c, {t..c})}) - integral {a..c} f) < e/3"
        using d1 by metis
      have SUMEQ: "?SUM(p ∪ {(c, {t..c})}) = (c - t) *🪙R f c + ?SUM p"
      proof -
        have "?SUM(p ∪ {(c, {t..c})}) = (content{t..c} *🪙R f c) + ?SUM p"
        proof (subst sum.union_disjoint)
          show "p ∩ {(c, {t..c})} = {}"
            using ‹t < c› pt by force
        qed (use p'(1) in auto)
        also have "… = (c - t) *🪙R f c + ?SUM p"
          using ‹t ≤ c› by auto
        finally show ?thesis .
      qed
      have "c - k < t"
        using ‹k>0› t(1) by (auto simp: field_simps)
      moreover have "k ≤ w"
      proof (rule ccontr)
        assume "¬ k ≤ w"
        then have "c + (k + w) / 2 ∉ d c"
          by (auto simp: field_simps not_le not_less dist_real_def d_def)
        then have "c + (k + w) / 2 ∉ ball c k"
          using k by blast
        then show False
          using ‹0 < w› ‹¬ k ≤ w› dist_real_def by auto
      qed
      ultimately have cwt: "c - w < t"
        by (auto simp: field_simps)
      have eq: "integral {a..c} f - integral {a..t} f = -(((c - t) *🪙R f c + ?SUM p) -
                integral {a..c} f) + (?SUM p - integral {a..t} f) + (c - t) *🪙R f c"
        by auto
      have "norm (integral {a..c} f - integral {a..t} f) < e/3 + e/3 + e/3"
        unfolding eq
      proof (intro norm_triangle_lt add_strict_mono)
        show "norm (- ((c - t) *🪙R f c + ?SUM p - integral {a..c} f)) < e/3"
          by (metis SUMEQ d1_fin norm_minus_cancel)
        show "norm (?SUM p - integral {a..t} f) < e/3"
          using d2_fin by blast
        show "norm ((c - t) *🪙R f c) < e/3"
          using w cwt ‹t < c› by simp (simp add: field_simps)
      qed
      then show ?thesis by simp
    qed
  qed
qed

lemma indefinite_integral_continuous_right:
  fixes f :: "real ==> 'a::banach"
  assumes "f integrable_on {a..b}"
    and "a ≤ c"
    and "c < b"
    and "e > 0"
  obtains d where "0 < d"
    and "∀t. c ≤ t ∧ t < c + d ⟶ norm (integral {a..c} f - integral {a..t} f) < e"
proof -
  have intm: "(λx. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c ≤ - a"
    using assms by auto
  from indefinite_integral_continuous_left[OF intm ‹e>0›]
  obtain d where "0 < d"
    and d: "∧t. [- c - d < t; t ≤ -c] 
             ==> norm (integral {- b..- c} (λx. f (-x)) - integral {- b..t} (λx. f (-x))) < e"
    by metis
  let ?d = "min d (b - c)"
  show ?thesis
  proof (intro that[of "?d"] allI impI)
    show "0 < ?d"
      using ‹0 < d› ‹c < b› by auto
    fix t :: real
    assume t: "c ≤ t ∧ t < c + ?d"
    have *: "integral {a..c} f = integral {a..b} f - integral {c..b} f"
            "integral {a..t} f = integral {a..b} f - integral {t..b} f"
      using assms t by (auto simp: algebra_simps integral_combine)
    have "(- c) - d < (- t)" "- t ≤ - c"
      using t by auto
    from d[OF this] show "norm (integral {a..c} f - integral {a..t} f) < e"
      by (auto simp: algebra_simps norm_minus_commute *)
  qed
qed

lemma indefinite_integral_continuous_1:
  fixes f :: "real ==> 'a::banach"
  assumes "f integrable_on {a..b}"
  shows "continuous_on {a..b} (λx. integral {a..x} f)"
proof -
  have "∃d>0. ∀x'∈{a..b}. dist x' x < d ⟶ dist (integral {a..x'} f) (integral {a..x} f) < e"
       if x: "x ∈ {a..b}" and "e > 0" for x e :: real
  proof (cases "a = b")
    case True
    with that show ?thesis by force
  next
    case False
    with x have "a < b" by force
    with x consider "x = a" | "x = b" | "a < x" "x < b"
      by force
    then show ?thesis
    proof cases
      case 1 then show ?thesis
        by (force simp: dist_norm algebra_simps intro: indefinite_integral_continuous_right [OF assms _ ‹a < b› ‹e > 0›])
    next
      case 2 then show ?thesis
        by (force simp: dist_norm norm_minus_commute algebra_simps intro: indefinite_integral_continuous_left [OF assms ‹a < b› _ ‹e > 0›])
    next
      case 3
      obtain d1 where "0 < d1"
        and d1: "∧t. [x - d1 < t; t ≤ x] ==> norm (integral {a..x} f - integral {a..t} f) < e"
        using 3 by (auto intro: indefinite_integral_continuous_left [OF assms ‹a < x› _ ‹e > 0›])
      obtain d2 where "0 < d2"
        and d2: "∧t. [x ≤ t; t < x + d2] ==> norm (integral {a..x} f - integral {a..t} f) < e"
        using 3 by (auto intro: indefinite_integral_continuous_right [OF assms _ ‹x < b› ‹e > 0›])
      show ?thesis
      proof (intro exI ballI conjI impI)
        show "0 < min d1 d2"
          using ‹0 < d1› ‹0 < d2› by simp
        show "dist (integral {a..y} f) (integral {a..x} f) < e"
          if "dist y x < min d1 d2" for y
          by (smt (verit) d1 d2 dist_commute dist_norm dist_real_def that)
      qed
    qed
  qed
  then show ?thesis
    by (auto simp: continuous_on_iff)
qed

lemma indefinite_integral_continuous_1':
  fixes f::"real ==> 'a::banach"
  assumes "f integrable_on {a..b}"
  shows "continuous_on {a..b} (λx. integral {x..b} f)"
proof -
  have "integral {a..b} f - integral {a..x} f = integral {x..b} f" if "x ∈ {a..b}" for x
    using integral_combine[OF _ _ assms, of x] that
    by (auto simp: algebra_simps)
  with _ show ?thesis
    by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous_1 assms)
qed

theorem integral_has_vector_derivative':
  fixes f :: "real ==> 'b::banach"
  assumes "continuous_on {a..b} f"
    and "x ∈ {a..b}"
  shows "((λu. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})"
proof -
  have *: "integral {x..b} f = integral {a .. b} f - integral {a .. x} f" if "a ≤ x" "x ≤ b" for x
    using integral_combine[of a x b for x, OF that integrable_continuous_real[OF assms(1)]]
    by (simp add: algebra_simps)
  show ?thesis
    using ‹x ∈ _› *
    by (rule has_vector_derivative_transform)
      (auto intro!: derivative_eq_intros assms integral_has_vector_derivative)
qed

lemma integral_has_real_derivative':
  assumes "continuous_on {a..b} g"
  assumes "t ∈ {a..b}"
  shows "((λx. integral {x..b} g) has_real_derivative -g t) (at t within {a..b})"
  using integral_has_vector_derivative'[OF assms]
  by (auto simp: has_real_derivative_iff_has_vector_derivative)


subsection ‹This doesn't directly involve integration, but that gives an easy proof›

lemma has_derivative_zero_unique_strong_interval:
  fixes f :: "real ==> 'a::banach"
  assumes "finite k"
    and contf: "continuous_on {a..b} f"
    and "f a = y"
    and fder: "∧x. x ∈ {a..b} - k ==> (f has_derivative (λh. 0)) (at x within {a..b})"
    and x: "x ∈ {a..b}"
  shows "f x = y"
proof -
  have "a ≤ b" "a ≤ x"
    using assms by auto
  have "((λx. 0::'a) has_integral f x - f a) {a..x}"
  proof (rule fundamental_theorem_of_calculus_interior_strong[OF ‹finite k› ‹a ≤ x›]; clarify?)
    have "{a..x} ⊆ {a..b}"
      using x by auto
    then show "continuous_on {a..x} f"
      by (rule continuous_on_subset[OF contf])
    show "(f has_vector_derivative 0) (at z)" if z: "z ∈ {a<..<x}" and notin: "z ∉ k" for z
      unfolding has_vector_derivative_def
    proof (simp add: at_within_open[OF z, symmetric])
      show "(f has_derivative (λx. 0)) (at z within {a<..<x})"
        by (rule has_derivative_subset [OF fder]) (use x z notin in auto)
    qed
  qed
  from has_integral_unique[OF has_integral_0 this]
  show ?thesis
    unfolding assms by auto
qed


subsection ‹Generalize a bit to any convex set›

lemma has_derivative_zero_unique_strong_convex:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "convex S" "finite K"
    and contf: "continuous_on S f"
    and "c ∈ S" "f c = y"
    and derf: "∧x. x ∈ S - K ==> (f has_derivative (λh. 0)) (at x within S)"
    and "x ∈ S"
  shows "f x = y"
proof (cases "x = c")
  case True with ‹f c = y› show ?thesis
    by blast
next
  case False
  let ?φ = "λu. (1 - u) *🪙R c + u *🪙R x"
  have contf': "continuous_on {0 ..1} (f ∘ ?φ)"
  proof (rule continuous_intros continuous_on_subset[OF contf])+
    show "(λu. (1 - u) *🪙R c + u *🪙R x) ` {0..1} ⊆ S"
      using ‹convex S› ‹x ∈ S› ‹c ∈ S› by (auto simp: convex_alt algebra_simps)
  qed
  have "t = u" if "?φ t = ?φ u" for t u
  proof -
    from that have "(t - u) *🪙R x = (t - u) *🪙R c"
      by (auto simp: algebra_simps)
    then show ?thesis
      using ‹x ≠ c› by auto
  qed
  then have eq: "(SOME t. ?φ t = ?φ u) = u" for u
    by blast
  then have "(?φ -` K) ⊆ (λz. SOME t. ?φ t = z) ` K"
    by (clarsimp simp: image_iff) (metis (no_types) eq)
  then have fin: "finite (?φ -` K)"
    by (rule finite_surj[OF ‹finite K›])

  have derf': "((λu. f (?φ u)) has_derivative (λh. 0)) (at t within {0..1})"
               if "t ∈ {0..1} - {t. ?φ t ∈ K}" for t
  proof -
    have df: "(f has_derivative (λh. 0)) (at (?φ t) within ?φ ` {0..1})"
      using ‹convex S› ‹x ∈ S› ‹c ∈ S› that
      by (auto simp: convex_alt algebra_simps intro: has_derivative_subset [OF derf])
    have "(f ∘ ?φ has_derivative (λx. 0) ∘ (λz. (0 - z *🪙R c) + z *🪙R x)) (at t within {0..1})"
      by (rule derivative_eq_intros df | simp)+
    then show ?thesis
      unfolding o_def .
  qed
  have "(f ∘ ?φ) 1 = y"
    apply (rule has_derivative_zero_unique_strong_interval[OF fin contf'])
    unfolding o_def using ‹f c = y› derf' by auto
  then show ?thesis
    by auto
qed


text ‹Also to any open connected set with finite set of exceptions. Could
  generalize to locally convex set with limpt-free set of exceptions.›

lemma has_derivative_zero_unique_strong_connected:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "connected S"
    and "open S"
    and "finite K"
    and contf: "continuous_on S f"
    and "c ∈ S"
    and "f c = y"
    and derf: "∧x. x ∈ S - K ==> (f has_derivative (λh. 0)) (at x within S)"
    and "x ∈ S"
  shows "f x = y"
proof -
  have "∃e>0. ball x e ⊆ (S ∩ f -` {f x})" if "x ∈ S" for x
  proof -
    obtain e where "0 < e" and e: "ball x e ⊆ S"
      using ‹x ∈ S› ‹open S› open_contains_ball by blast
    have "ball x e ⊆ {u ∈ S. f u ∈ {f x}}"
    proof safe
      fix y
      assume y: "y ∈ ball x e"
      then show "y ∈ S"
        using e by auto
      show "f y = f x"
      proof (rule has_derivative_zero_unique_strong_convex[OF convex_ball ‹finite K›])
        show "continuous_on (ball x e) f"
          using contf continuous_on_subset e by blast
        show "(f has_derivative (λh. 0)) (at u within ball x e)"
             if "u ∈ ball x e - K" for u
          by (metis Diff_iff contra_subsetD derf e has_derivative_subset that)
      qed (use y e ‹0 < e› in auto)
    qed
    then show "∃e>0. ball x e ⊆ (S ∩ f -` {f x})"
      using ‹0 < e› by blast
  qed
  then have "openin (top_of_set S) (S ∩ f -` {y})"
    by (auto intro!: open_openin_trans[OF ‹open S›] simp: open_contains_ball)
  moreover have "closedin (top_of_set S) (S ∩ f -` {y})"
    by (force intro!: continuous_closedin_preimage [OF contf])
  ultimately have "(S ∩ f -` {y}) = {} ∨ (S ∩ f -` {y}) = S"
    using ‹connected S› by (simp add: connected_clopen)
  then show ?thesis
    using ‹x ∈ S› ‹f c = y› ‹c ∈ S› by auto
qed

lemma has_derivative_zero_connected_constant_on:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "connected S" "open S" "finite K" "continuous_on S f"
      and "∀x∈S-K. (f has_derivative (λh. 0)) (at x within S)"
  shows "f constant_on S"
  by (smt (verit, best) assms constant_on_def has_derivative_zero_unique_strong_connected)

lemma DERIV_zero_connected_constant_on:
  fixes f :: "'a::{real_normed_field,euclidean_space} ==> 'a"
  assumes *: "connected S" "open S" "finite K" "continuous_on S f"
      and 0: "∀x∈S-K. DERIV f x :> 0"
  shows "f constant_on S"
  using has_derivative_zero_connected_constant_on [OF *] 0
  by (metis has_derivative_at_withinI has_field_derivative_def lambda_zero)

lemma DERIV_zero_connected_constant:
  fixes f :: "'a::{real_normed_field,euclidean_space} ==> 'a"
  assumes "connected S" and "open S" and "finite K" and "continuous_on S f"
      and "∀x∈S-K. DERIV f x :> 0"
    obtains c where "∧x. x ∈ S ==> f(x) = c"
  by (metis DERIV_zero_connected_constant_on [OF assms] constant_on_def)

lemma has_field_derivative_0_imp_constant_on:
  fixes f :: "'a::{real_normed_field,euclidean_space} ==> 'a"
  assumes "∧z. z ∈ S ==> (f has_field_derivative 0) (at z)" and S: "connected S" "open S"
  shows "f constant_on S"
  using DERIV_zero_connected_constant_on [where K="Basis"]
  by (metis DERIV_isCont Diff_iff assms continuous_at_imp_continuous_on eucl.finite_Basis)


subsection ‹Integrating characteristic function of an interval›

lemma has_integral_restrict_open_subinterval:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes intf: "(f has_integral i) (cbox c d)"
    and cb: "cbox c d ⊆ cbox a b"
  shows "((λx. if x ∈ box c d then f x else 0) has_integral i) (cbox a b)"
proof (cases "cbox c d = {}")
  case True
  then have "box c d = {}"
    by (metis bot.extremum_uniqueI box_subset_cbox)
  then show ?thesis
    using True intf by auto
next
  case False
  then obtain p where pdiv: "p division_of cbox a b" and inp: "cbox c d ∈ p"
    using cb partial_division_extend_1 by blast
  define g where [abs_def]: "g x = (if x ∈box c d then f x else 0)" for x
  interpret operative "lift_option plus" "Some (0 :: 'b)"
    "λi. if g integrable_on i then Some (integral i g) else None"
    by (fact operative_integralI)
  note operat = division [OF pdiv, symmetric]
  show ?thesis
  proof (cases "(g has_integral i) (cbox a b)")
    case True then show ?thesis
      by (simp add: g_def)
  next
    case False
    have iterate: "F (λi. if g integrable_on i then Some (integral i g) else None) (p - {cbox c d}) = Some 0"
    proof (intro neutral ballI)
      fix x
      assume x: "x ∈ p - {cbox c d}"
      then have "x ∈ p"
        by auto
      then obtain u v where uv: "x = cbox u v"
        using pdiv by blast
      have "interior x ∩ interior (cbox c d) = {}"
        using pdiv inp x by blast
      then have "(g has_integral 0) x"
        unfolding uv using has_integral_spike_interior[where f="λx. 0"]
        by (metis (no_types, lifting) disjoint_iff_not_equal g_def has_integral_0_eq interior_cbox)
      then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
        by auto
    qed
    interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)"
      by (intro comm_monoid_set.intro comm_monoid_lift_option add.comm_monoid_axioms)
    have intg: "g integrable_on cbox c d"
      using integrable_spike_interior[where f=f]
      by (meson g_def has_integral_integrable intf)
    moreover have "integral (cbox c d) g = i"
      by (meson g_def has_integral_iff has_integral_spike_interior intf)
    ultimately have "F (λA. if g integrable_on A then Some (integral A g) else None) p = Some i"
      by (metis (full_types, lifting) division_of_finite inp iterate pdiv remove right_neutral)
    with False show ?thesis
      by (metis integrable_integral not_None_eq operat option.inject)
  qed
qed


lemma has_integral_restrict_closed_subinterval:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "(f has_integral i) (cbox c d)"
    and "cbox c d ⊆ cbox a b"
  shows "((λx. if x ∈ cbox c d then f x else 0) has_integral i) (cbox a b)"
proof -
  note has_integral_restrict_open_subinterval[OF assms]
  note * = has_integral_spike[OF negligible_frontier_interval _ this]
  show ?thesis
    by (rule *[of c d]) (use box_subset_cbox[of c d] in auto)
qed

lemma has_integral_restrict_closed_subintervals_eq:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "cbox c d ⊆ cbox a b"
  shows "((λx. if x ∈ cbox c d then f x else 0) has_integral i) (cbox a b) ⟷ (f has_integral i) (cbox c d)"
  (is "?l = ?r")
proof (cases "cbox c d = {}")
  case False
  let ?g = "λx. if x ∈ cbox c d then f x else 0"
  show ?thesis
  proof
    assume ?l
    then have "?g integrable_on cbox c d"
      using assms has_integral_integrable integrable_subinterval by blast
    then have f: "f integrable_on cbox c d"
      by (rule integrable_eq) auto
    moreover have "i = integral (cbox c d) f"
      using f ‹(?g has_integral i) (cbox a b)›
      by (simp add: assms has_integral_restrict_closed_subinterval has_integral_unique
          integrable_integral)
    ultimately show ?r by auto
  next
    assume ?r then show ?l
      by (rule has_integral_restrict_closed_subinterval[OF _ assms])
  qed
qed auto


text ‹Hence we can apply the limit process uniformly to all integrals.›

lemma has_integral':
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "(f has_integral i) S ⟷
    (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
      (∃z. ((λx. if x ∈ S then f(x) else 0) has_integral z) (cbox a b) ∧ norm(z - i) < e))"
  (is "?l ⟷ (∀e>0. ?r e)")
proof (cases "∃a b. S = cbox a b")
  case False then show ?thesis
    by (simp add: has_integral_alt)
next
  case True
  then obtain a b where S: "S = cbox a b"
    by blast
  obtain B where " 0 < B" and B: "∧x. x ∈ cbox a b ==> norm x ≤ B"
    using bounded_cbox[unfolded bounded_pos] by blast
  show ?thesis
  proof safe
    fix e :: real
    assume ?l and "e > 0"
    have "((λx. if x ∈ S then f x else 0) has_integral i) (cbox c d)"
      if "ball 0 (B+1) ⊆ cbox c d" for c d
        unfolding S using B that
        by (force intro: ‹?l›[unfolded S] has_integral_restrict_closed_subinterval)
    then show "?r e"
      by (meson ‹0 < B› ‹0 < e› add_pos_pos le_less_trans zero_less_one norm_pths(2))
  next
    assume as: "∀e>0. ?r e"
    then obtain C
      where C: "∧a b. ball 0 C ⊆ cbox a b ==>
                ∃z. ((λx. if x ∈ S then f x else 0) has_integral z) (cbox a b)"
      by (meson zero_less_one)
    define c :: 'n where "c = (∑i∈Basis. (- max B C) *🪙R i)"
    define d :: 'n where "d = (∑i∈Basis. max B C *🪙R i)"
    have "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i" if "norm x ≤ B" "i ∈ Basis" for x i
      using that and Basis_le_norm[OF ‹i∈Basis›, of x]
      by (auto simp: field_simps sum_negf c_def d_def)
    then have c_d: "cbox a b ⊆ cbox c d"
      by (meson B mem_box(2) subsetI)
    have "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i"
      if x: "norm (0 - x) < C" and i: "i ∈ Basis" for x i
        using Basis_le_norm[OF i, of x] x i by (auto simp: sum_negf c_def d_def)
      then have "ball 0 C ⊆ cbox c d"
        by (auto simp: mem_box dist_norm)
    with C obtain y where y: "(f has_integral y) (cbox a b)"
      using c_d has_integral_restrict_closed_subintervals_eq S by blast
    have "y = i"
    proof (rule ccontr)
      assume "y ≠ i"
      then have "0 < norm (y - i)"
        by auto
      from as[rule_format,OF this]
      obtain C where C: "∧a b. ball 0 C ⊆ cbox a b ==> 
           ∃z. ((λx. if x ∈ S then f x else 0) has_integral z) (cbox a b) ∧ norm (z-i) < norm (y-i)"
        by auto
      define c :: 'n where "c = (∑i∈Basis. (- max B C) *🪙R i)"
      define d :: 'n where "d = (∑i∈Basis. max B C *🪙R i)"
      have "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i"
        if "norm x ≤ B" and "i ∈ Basis" for x i
          using that Basis_le_norm[of i x] by (auto simp: field_simps sum_negf c_def d_def)
        then have c_d: "cbox a b ⊆ cbox c d"
        by (simp add: B mem_box(2) subset_eq)
      have "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i" if "norm (0 - x) < C" and "i ∈ Basis" for x i
        using Basis_le_norm[of i x] that by (auto simp: sum_negf c_def d_def)
      then have "ball 0 C ⊆ cbox c d"
        by (auto simp: mem_box dist_norm)
      with C obtain z where z: "(f has_integral z) (cbox a b)" "norm (z-i) < norm (y-i)"
        using has_integral_restrict_closed_subintervals_eq[OF c_d] S by blast
      moreover have "z = y"
        using z by (blast intro: has_integral_unique[OF _ y])
      ultimately show False
        by auto
    qed
    then show ?l
      using y by (auto simp: S)
  qed
qed

lemma has_integral_le:
  fixes f :: "'n::euclidean_space ==> real"
  assumes fg: "(f has_integral i) S" "(g has_integral j) S"
    and le: "∧x. x ∈ S ==> f x ≤ g x"
  shows "i ≤ j"
  using has_integral_component_le[OF _ fg, of 1] le by auto

lemma integral_le:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "f integrable_on S"
    and "g integrable_on S"
    and "∧x. x ∈ S ==> f x ≤ g x"
  shows "integral S f ≤ integral S g"
  by (meson assms has_integral_le integrable_integral)

lemma has_integral_nonneg:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "(f has_integral i) S" and "∧x. x ∈ S ==> 0 ≤ f x"
  shows "0 ≤ i"
  using assms has_integral_0 has_integral_le by blast

lemma integral_nonneg:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "f integrable_on S" and "∧x. x ∈ S ==> 0 ≤ f x"
  shows "0 ≤ integral S f"
  using assms has_integral_nonneg by blast


text ‹Hence a general restriction property.›

lemma has_integral_restrict [simp]:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  assumes "S ⊆ T"
  shows "((λx. if x ∈ S then f x else 0) has_integral i) T ⟷ (f has_integral i) S"
proof -
  have *: "∧x. (if x ∈ T then if x ∈ S then f x else 0 else 0) =  (if x∈S then f x else 0)"
    using assms by auto
  show ?thesis
    apply (subst(2) has_integral')
    apply (subst has_integral')
      apply (simp add: *)
    done
qed

corollary has_integral_restrict_UNIV:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "((λx. if x ∈ s then f x else 0) has_integral i) UNIV ⟷ (f has_integral i) s"
  by auto

lemma has_integral_restrict_Int:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "((λx. if x ∈ S then f x else 0) has_integral i) T ⟷ (f has_integral i) (S ∩ T)"
proof -
  have "((λx. if x ∈ T then if x ∈ S then f x else 0 else 0) has_integral i) UNIV =
        ((λx. if x ∈ S ∩ T then f x else 0) has_integral i) UNIV"
    by (rule has_integral_cong) auto
  then show ?thesis
    using has_integral_restrict_UNIV by fastforce
qed

lemma integral_restrict_Int:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "integral T (λx. if x ∈ S then f x else 0) = integral (S ∩ T) f"
  by (metis (no_types, lifting) has_integral_cong has_integral_restrict_Int integrable_integral integral_unique not_integrable_integral)

lemma integrable_restrict_Int:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "(λx. if x ∈ S then f x else 0) integrable_on T ⟷ f integrable_on (S ∩ T)"
  using has_integral_restrict_Int by fastforce

lemma has_integral_on_superset:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "(f has_integral i) S"
      and "∧x. x ∉ S ==> f x = 0"
      and "S ⊆ T"
    shows "(f has_integral i) T"
  by (smt (verit, ccfv_SIG) assms has_integral_cong has_integral_restrict)

lemma integrable_on_superset:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "f integrable_on S" and "∧x. x ∉ S ==> f x = 0" and "S ⊆ t"
  shows "f integrable_on t"
  by (meson assms has_integral_on_superset integrable_integral integrable_on_def)

lemma integral_subset_negligible:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  assumes "S ⊆ T" "negligible (T - S)"
  shows "integral S f = integral T f"
proof -
  have "integral T f = integral T (λx. if x ∈ S then f x else 0)"
    by (rule integral_spike[of "T - S"]) (use assms in auto)
  also have "… = integral (S ∩ T) f"
    by (subst integral_restrict_Int) auto
  also have "S ∩ T = S" using assms by auto
  finally show ?thesis ..
qed

lemma integral_restrict_UNIV:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "integral UNIV (λx. if x ∈ S then f x else 0) = integral S f"
  by (simp add: integral_restrict_Int)

lemma integrable_restrict_UNIV:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "(λx. if x ∈ s then f x else 0) integrable_on UNIV ⟷ f integrable_on s"
  unfolding integrable_on_def
  by auto

lemma has_integral_subset_component_le:
  fixes f :: "'n::euclidean_space ==> 'm::euclidean_space"
  assumes k: "k ∈ Basis"
    and "S ⊆ T" "(f has_integral i) S" "(f has_integral j) T" "∧x. x∈T ==> 0 ≤ f(x)∙k"
  shows "i∙k ≤ j∙k"
proof -
  have 🍋: "((λx. if x ∈ S then f x else 0) has_integral i) UNIV"
          "((λx. if x ∈ T then f x else 0) has_integral j) UNIV"
    by (simp_all add: assms)
  show ?thesis
    using assms by (force intro!: has_integral_component_le[OF k 🍋])
qed

subsection‹Integrals on set differences›

lemma has_integral_setdiff:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes S: "(f has_integral i) S" and T: "(f has_integral j) T"
    and neg: "negligible (T - S)"
  shows "(f has_integral (i - j)) (S - T)"
proof -
  show ?thesis
    unfolding has_integral_restrict_UNIV [symmetric, of f]
  proof (rule has_integral_spike [OF neg])
    have eq: "(λx. (if x ∈ S then f x else 0) - (if x ∈ T then f x else 0)) =
            (λx. if x ∈ T - S then - f x else if x ∈ S - T then f x else 0)"
      by force
    have "((λx. if x ∈ S then f x else 0) has_integral i) UNIV"
      "((λx. if x ∈ T then f x else 0) has_integral j) UNIV"
      using S T has_integral_restrict_UNIV by auto
    from has_integral_diff [OF this]
    show "((λx. if x ∈ T - S then - f x else if x ∈ S - T then f x else 0)
                   has_integral i-j) UNIV"
      by (simp add: eq)
  qed force
qed

lemma integral_setdiff:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "f integrable_on S" "f integrable_on T" "negligible(T - S)"
 shows "integral (S - T) f = integral S f - integral T f"
  by (rule integral_unique) (simp add: assms has_integral_setdiff integrable_integral)

lemma integrable_setdiff:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  assumes "(f has_integral i) S" "(f has_integral j) T" "negligible (T - S)"
  shows "f integrable_on (S - T)"
  using has_integral_setdiff [OF assms]
  by (simp add: has_integral_iff)

lemma negligible_setdiff [simp]: "T ⊆ S ==> negligible (T - S)"
  by (metis Diff_eq_empty_iff negligible_empty)

lemma negligible_on_intervals: "negligible s ⟷ (∀a b. negligible(s ∩ cbox a b))" (is "?l ⟷ ?r")
proof
  assume R: ?r
  show ?l
    unfolding negligible_def
    by (metis Diff_iff Int_iff R has_integral_negligible indicator_simps(2))
qed (simp add: negligible_Int)

lemma negligible_translation:
  assumes "negligible S"
    shows "negligible ((+) c ` S)"
proof -
  have inj: "inj ((+) c)"
    by simp
  show ?thesis
  using assms
  proof (clarsimp simp: negligible_def)
    fix a b
    assume "∀x y. (indicator S has_integral 0) (cbox x y)"
    then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))"
      by (meson Diff_iff assms has_integral_negligible indicator_simps(2))
    have eq: "indicator ((+) c ` S) = (λx. indicator S (x - c))"
      by (force simp: indicator_def)
    show "(indicator ((+) c ` S) has_integral 0) (cbox a b)"
      using has_integral_affinity [OF *, of 1 "-c"]
            cbox_translation [of "c" "-c+a" "-c+b"]
      by (simp add: eq) (simp add: ac_simps)
  qed
qed

lemma negligible_translation_rev:
  assumes "negligible ((+) c ` S)"
  shows "negligible S"
  by (metis negligible_translation [OF assms, of "-c"] translation_galois)

lemma negligible_atLeastAtMostI: "b ≤ a ==> negligible {a..(b::real)}"
  using negligible_insert by fastforce

lemma has_integral_spike_set_eq:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "negligible {x ∈ S - T. f x ≠ 0}" "negligible {x ∈ T - S. f x ≠ 0}"
  shows "(f has_integral y) S ⟷ (f has_integral y) T"
proof -
  have "((λx. if x ∈ S then f x else 0) has_integral y) UNIV =
        ((λx. if x ∈ T then f x else 0) has_integral y) UNIV"
  proof (rule has_integral_spike_eq)
    show "negligible ({x ∈ S - T. f x ≠ 0} ∪ {x ∈ T - S. f x ≠ 0})"
      by (rule negligible_Un [OF assms])
  qed auto
  then show ?thesis
    by (simp add: has_integral_restrict_UNIV)
qed

corollary integral_spike_set:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "negligible {x ∈ S - T. f x ≠ 0}" "negligible {x ∈ T - S. f x ≠ 0}"
  shows "integral S f = integral T f"
  using has_integral_spike_set_eq [OF assms]
  by (metis eq_integralD integral_unique)

lemma integrable_spike_set:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes f: "f integrable_on S" and neg: "negligible {x ∈ S - T. f x ≠ 0}" "negligible {x ∈ T - S. f x ≠ 0}"
  shows "f integrable_on T"
  using has_integral_spike_set_eq [OF neg] f by blast

lemma integrable_spike_set_eq:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "negligible ((S - T) ∪ (T - S))"
  shows "f integrable_on S ⟷ f integrable_on T"
  by (blast intro: integrable_spike_set assms negligible_subset)

lemma integrable_on_insert_iff: "f integrable_on (insert x X) ⟷ f integrable_on X"
  for f::"_ ==> 'a::banach"
  by (rule integrable_spike_set_eq) (auto simp: insert_Diff_if)

lemma has_integral_interior:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "negligible(frontier S) ==> (f has_integral y) (interior S) ⟷ (f has_integral y) S"
proof (rule has_integral_spike_set_eq [OF empty_imp_negligible negligible_subset])
qed (use interior_subset in ‹auto simp: frontier_def closure_def›)

lemma has_integral_closure:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "negligible(frontier S) ==> (f has_integral y) (closure S) ⟷ (f has_integral y) S"
proof (rule has_integral_spike_set_eq [OF negligible_subset empty_imp_negligible])
qed (auto simp: closure_Un_frontier )

lemma has_integral_open_interval:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "(f has_integral y) (box a b) ⟷ (f has_integral y) (cbox a b)"
  unfolding interior_cbox [symmetric]
  by (metis frontier_cbox has_integral_interior negligible_frontier_interval)

lemma integrable_on_open_interval:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "f integrable_on box a b ⟷ f integrable_on cbox a b"
  by (simp add: has_integral_open_interval integrable_on_def)

lemma integral_open_interval:
  fixes f :: "'a :: euclidean_space ==> 'b :: banach"
  shows "integral(box a b) f = integral(cbox a b) f"
  by (metis has_integral_integrable_integral has_integral_open_interval not_integrable_integral)

lemma integrable_on_open_interval_real:
  fixes f :: "real ==> 'b :: banach"
  shows "f integrable_on {a<..<b} ⟷ f integrable_on {a..b}"
  using integrable_on_open_interval[of f a b] by simp

lemma integral_open_interval_real:
  "integral {a..b} (f :: real ==> 'a :: banach) = integral {a<..<(b::real)} f"
  using integral_open_interval[of a b f] by simp

lemma has_integral_Icc_iff_Ioo:
  fixes f :: "real ==> 'a :: banach"
  shows "(f has_integral I) {a..b} ⟷ (f has_integral I) {a<..<b}"
  by (metis box_real(1) cbox_interval has_integral_open_interval)

lemma integrable_on_Icc_iff_Ioo:
  fixes f :: "real ==> 'a :: banach"
  shows "f integrable_on {a..b} ⟷ f integrable_on {a<..<b}"
  using has_integral_Icc_iff_Ioo by blast


subsection ‹More lemmas that are useful later›

lemma has_integral_subset_le:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "s ⊆ t"
    and "(f has_integral i) s"
    and "(f has_integral j) t"
    and "∀x∈t. 0 ≤ f x"
  shows "i ≤ j"
  using assms has_integral_subset_component_le[OF _ assms(1), of 1 f i j]
  by auto

lemma integral_subset_component_le:
  fixes f :: "'n::euclidean_space ==> 'm::euclidean_space"
  assumes "k ∈ Basis"
    and "s ⊆ t"
    and "f integrable_on s"
    and "f integrable_on t"
    and "∀x ∈ t. 0 ≤ f x ∙ k"
  shows "(integral s f)∙k ≤ (integral t f)∙k"
  by (meson assms has_integral_subset_component_le integrable_integral)

lemma integral_subset_le:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "s ⊆ t"
    and "f integrable_on s"
    and "f integrable_on t"
    and "∀x ∈ t. 0 ≤ f x"
  shows "integral s f ≤ integral t f"
  using assms has_integral_subset_le by blast

lemma has_integral_alt':
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "(f has_integral i) s ⟷ 
          (∀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b) ∧
          (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
            norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - i) < e)"
  (is "?l = ?r")
proof
  assume rhs: ?r
  show ?l
  proof (subst has_integral', intro allI impI)
    fix e::real
    assume "e > 0"
    from rhs[THEN conjunct2,rule_format,OF this]
    show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
                   (∃z. ((λx. if x ∈ s then f x else 0) has_integral z)
                         (cbox a b) ∧ norm (z - i) < e)"
      by (simp add: has_integral_iff rhs)
  qed
next
  let ?Φ = "λe a b. ∃z. ((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b) ∧ norm (z - i) < e"
  assume ?l
  then have lhs: "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶ ?Φ e a b" if "e > 0" for e
    using that has_integral'[of f] by auto
  let ?f = "λx. if x ∈ s then f x else 0"
  show ?r
  proof (intro conjI allI impI)
    fix a b :: 'n
    from lhs[OF zero_less_one]
    obtain B where "0 < B" and B: "∧a b. ball 0 B ⊆ cbox a b ==> ?Φ 1 a b"
      by blast
    let ?a = "∑i∈Basis. min (a∙i) (-B) *🪙R i::'n"
    let ?b = "∑i∈Basis. max (b∙i) B *🪙R i::'n"
    show "?f integrable_on cbox a b"
    proof (rule integrable_subinterval[of _ ?a ?b])
      have "?a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ ?b ∙ i" if "norm (0 - x) < B" "i ∈ Basis" for x i
        using Basis_le_norm[of i x] that by (auto simp:field_simps)
      then have "ball 0 B ⊆ cbox ?a ?b"
        by (auto simp: mem_box dist_norm)
      then show "?f integrable_on cbox ?a ?b"
        unfolding integrable_on_def using B by blast
      show "cbox a b ⊆ cbox ?a ?b"
        by (force simp: mem_box)
    qed
    fix e :: real
    assume "e > 0"
    with lhs show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
      norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - i) < e"
      by (metis (no_types, lifting) has_integral_integrable_integral)
  qed
qed


subsection ‹Continuity of the integral (for a 1-dimensional interval)›

lemma integrable_alt:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  shows "f integrable_on s ⟷
    (∀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b) ∧
    (∀e>0. ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
    norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) -
      integral (cbox c d)  (λx. if x ∈ s then f x else 0)) < e)"
  (is "?l = ?r")
proof
  let ?F = "λx. if x ∈ s then f x else 0"
  assume ?l
  then obtain y where intF: "∧a b. ?F integrable_on cbox a b"
          and y: "∧e. 0 < e ==>
              ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶ norm (integral (cbox a b) ?F - y) < e"
    unfolding integrable_on_def has_integral_alt'[of f] by auto
  show ?r
  proof (intro conjI allI impI intF)
    fix e::real
    assume "e > 0"
    then have "e/2 > 0"
      by auto
    obtain B where "0 < B"
       and B: "∧a b. ball 0 B ⊆ cbox a b ==> norm (integral (cbox a b) ?F - y) < e/2"
      using ‹0 < e/2› y by blast
    show "∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
                  norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
    proof (intro conjI exI impI allI, rule ‹0 < B›)
      fix a b c d::'n
      assume sub: "ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d"
      show "norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
        using sub by (auto intro: norm_triangle_half_l dest: B)
    qed
  qed
next
  let ?F = "λx. if x ∈ s then f x else 0"
  assume rhs: ?r
  let ?cube = "λn. cbox (∑i∈Basis. - real n *🪙R i::'n) (∑i∈Basis. real n *🪙R i)"
  have "Cauchy (λn. integral (?cube n) ?F)"
    unfolding Cauchy_def
  proof (intro allI impI)
    fix e::real
    assume "e > 0"
    with rhs obtain B where "0 < B"
      and B: "∧a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d 
                        ==> norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
      by blast
    obtain N where N: "B ≤ real N"
      using real_arch_simple by blast
    have "ball 0 B ⊆ ?cube n" if n: "n ≥ N" for n
    proof -
      have "sum ((*\<^sub>R) (- real n)) Basis \<bullet> i \<le> x \<bullet> i \<and>
  x ∙ i ≤ sum ((*🪙R) (real n)) Basis ∙ i"
  if "norm x 🚫" "i ∈ Basis" for x i::'n
  using Basis_le_norm[of i x] n N that by (auto simp: field_simps sum_negf)
  then show ?thesis
  by (auto simp: mem_box dist_norm)
  qed
  then show "∃M. ∀m≥M. ∀n≥M. dist (integral (?cube m) ?F) (integral (?cube n) ?F) 🚫"
  by (fastforce simp: dist_norm intro!: B)
  qed
  then obtain i where i: "(λn. integral (?cube n) ?F) <---- i"
  using convergent_eq_Cauchy by blast
  have "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶ norm (integral (cbox a b) ?F - i) 🚫"
  if "e > 0" for e
  proof -
  have *: "e/2 > 0" using that by auto
  then obtain N where N: "∧n. N ≤ n ==> norm (i - integral (?cube n) ?F) 🚫/2"
  using i[THEN LIMSEQ_D, simplified norm_minus_commute] by meson
  obtain B where "0 🚫"
  and B: "∧a b c d. [ball 0 B ⊆ cbox a b; ball 0 B ⊆ cbox c d] ==>
  norm (integral (cbox a b) ?F - integral (cbox c d) ?F) 🚫/2"
  using rhs * by meson
  let ?B = "max (real N) B"
  show ?thesis
  proof (intro exI conjI allI impI)
  show "0 🚫B"
  using ‹B > 0› by auto
  fix a b :: 'n
  assume "ball 0 ?B ⊆ cbox a b"
  moreover obtain n where n: "max (real N) B ≤ real n"
  using real_arch_simple by blast
  moreover have "ball 0 B ⊆ ?cube n"
  proof
  fix x :: 'n
  assume x: "x ∈ ball 0 B"
  have "[norm (0 - x) 🚫; i ∈ Basis]
  ==> sum ((*🪙R) (-n)) Basis ∙ i≤ x ∙ i ∧ x ∙ i ≤ sum ((*🪙R) n) Basis ∙ i" for i
  using Basis_le_norm[of i x] n by (auto simp: field_simps sum_negf)
  then show "x ∈ ?cube n"
  using x by (auto simp: mem_box dist_norm)
  qed
  ultimately show "norm (integral (cbox a b) ?F - i) 🚫"
  using norm_triangle_half_l [OF B N] by force
  qed
  qed
  then show ?l unfolding integrable_on_def has_integral_alt'[of f]
  using rhs by blast
 qed
 
 lemma integrable_altD:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "f integrable_on s"
  shows "∧a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b"
  and "∧e. e > 0 ==> ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
  norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - integral (cbox c d) (λx. if x ∈ s then f x else 0)) 🚫"
  using assms[unfolded integrable_alt[of f]] by auto
 
 lemma integrable_alt_subset:
  fixes f :: "'a::euclidean_space ==> 'b::banach"
  shows
  "f integrable_on S ⟷
  (∀a b. (λx. if x ∈ S then f x else 0) integrable_on cbox a b) ∧
  (∀e>0. ∃B>0. ∀a b c d.
  ball 0 B ⊆ cbox a b ∧ cbox a b ⊆ cbox c d
  ⟶ norm(integral (cbox a b) (λx. if x ∈ S then f x else 0) -
  integral (cbox c d) (λx. if x ∈ S then f x else 0)) 🚫)"
  (is "_ = ?rhs")
 proof -
  let ?g = "λx. if x ∈ S then f x else 0"
  have "f integrable_on S ⟷
  (∀a b. ?g integrable_on cbox a b) ∧
  (∀e>0. ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
  norm (integral (cbox a b) ?g - integral (cbox c d) ?g) 🚫)"
  by (rule integrable_alt)
  also have "… = ?rhs"
  proof -
  { fix e :: "real"
  assume e: "∧e. e>0 ==> ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ cbox a b ⊆ cbox c d ⟶
  norm (integral (cbox a b) ?g - integral (cbox c d) ?g) 🚫"
  and "e > 0"
  obtain B where "B > 0"
  and B: "∧a b c d. [ball 0 B ⊆ cbox a b; cbox a b ⊆ cbox c d] ==>
  norm (integral (cbox a b) ?g - integral (cbox c d) ?g) 🚫/2"
  using ‹e > 0› e [of "e/2"] by force
  have "∃B>0. ∀a b c d.
  ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
  norm (integral (cbox a b) ?g - integral (cbox c d) ?g) 🚫"
  proof (intro exI allI conjI impI)
  fix a b c d :: "'a"
  let ?α = "∑i∈Basis. max (a ∙ i) (c ∙ i) *🪙R i"
  let ?β = "∑i∈Basis. min (b ∙ i) (d ∙ i) *🪙R i"
  show "norm (integral (cbox a b) ?g - integral (cbox c d) ?g) 🚫"
  if ball: "ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d"
  proof -
  have B': "norm (integral (cbox a b ∩ cbox c d) ?g - integral (cbox x y) ?g) 🚫/2"
  if "cbox a b ∩ cbox c d ⊆ cbox x y" for x y
  using B [of ?α ?β x y] ball that by (simp add: Int_interval [symmetric])
  show ?thesis
  using B' [of a b] B' [of c d] norm_triangle_half_r by blast
  qed
  qed (use ‹B > 0› in auto)}
  then show ?thesis
  by force
  qed
  finally show ?thesis .
 qed
 
 lemma integrable_on_subcbox:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes intf: "f integrable_on S"
  and sub: "cbox a b ⊆ S"
  shows "f integrable_on cbox a b"
 proof -
  have "(λx. if x ∈ S then f x else 0) integrable_on cbox a b"
  by (simp add: intf integrable_altD(1))
 then show ?thesis
  by (simp add: inf.absorb2 integrable_restrict_Int sub)
 qed
 
 
 subsection ‹A straddling criterion for integrability›
 
 lemma integrable_straddle_interval:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "∧e. e>0 ==> ∃g h i j. (g has_integral i) (cbox a b) ∧ (h has_integral j) (cbox a b) ∧
  ∣i - j∣ 🚫 ∧ (∀x∈cbox a b. (g x) ≤ f x ∧ f x ≤ h x)"
  shows "f integrable_on cbox a b"
 proof -
  have "∃d. gauge d ∧
  (∀p1 p2. p1 tagged_division_of cbox a b ∧ d fine p1 ∧
  p2 tagged_division_of cbox a b ∧ d fine p2 ⟶
  ∣(∑(x,K)∈p1. content K *🪙R f x) - (∑(x,K)∈p2. content K *??R f x)∣ 🚫)"
  if "e > 0" for e
  proof -
  have e: "e/3 > 0"
  using that by auto
  then obtain g h i j where ij: "∣i - j∣ 🚫/3"
  and "(g has_integral i) (cbox a b)"
  and "(h has_integral j) (cbox a b)"
  and fgh: "∧x. x ∈ cbox a b ==> g x ≤ f x ∧ f x ≤ h x"
  using assms real_norm_def by metis
  then obtain d1 d2 where "gauge d1" "gauge d2"
  and d1: "∧p. [p tagged_division_of cbox a b; d1 fine p] ==>
  ∣(∑(x,K)∈p. content K *🪙R g x) - i∣ 🚫/3"
  and d2: "∧p. [p tagged_division_of cbox a b; d2 fine p] ==>
  ∣(∑(x,K) ∈ p. content K *🪙R h x) - j∣ 🚫/3"
  by (metis e has_integral real_norm_def)
  have "∣(∑(x,K) ∈ p1. content K *🪙R f x) - (∑(x,K) ∈ p2. content K *🪙R f x)∣ 🚫"
  if p1: "p1 tagged_division_of cbox a b" and 11: "d1 fine p1" and 21: "d2 fine p1"
  and p2: "p2 tagged_division_of cbox a b" and 12: "d1 fine p2" and 22: "d2 fine p2" for p1 p2
  proof -
  have *: "∧g1 g2 h1 h2 f1 f2.
  [∣g2 - i∣ 🚫/3; ∣g1 - i∣ 🚫/3; ∣h2 - j∣ 🚫/3; ∣h1 - j∣ 🚫/3;
  g1 - h2 ≤ f1 - f2; f1 - f2 ≤ h1 - g2]
  ==> ∣f1 - f2∣ 🚫"
  using ‹e > 0› ij by arith
  have 0: "(∑(x, k)∈p1. content k *🪙R f x) - (∑(x, k)∈p1. content k *🪙R g x) ≥ 0"
  "0 ≤ (∑(x, k)∈p2. content k *🪙R h x) - (∑(x, k)∈p2. content k *🪙R f x)"
  "(∑(x, k)∈p2. content k *🪙R f x) - (∑(x, k)∈p2. content k *🪙R g x) ≥ 0"
  "0 ≤ (∑(x, k)∈p1. content k *🪙R h x) - (∑(x, k)∈p1. content k *🪙R f x)"
  unfolding sum_subtractf[symmetric]
  apply (auto intro!: sum_nonneg)
  apply (meson fgh measure_nonneg mult_left_mono tag_in_interval that sum_nonneg)+
  done
  show ?thesis
      proof (rule *)
        show "∣(∑(x,K) ∈ p2. content K *🪙R g x) - i∣ < e/3"
          by (rule d1[OF p2 12])
        show "∣(∑(x,K) ∈ p1. content K *🪙R g x) - i∣ < e/3"
          by (rule d1[OF p1 11])
        show "∣(∑(x,K) ∈ p2. content K *🪙R h x) - j∣ < e/3"
          by (rule d2[OF p2 22])
        show "∣(∑(x,K) ∈ p1. content K *🪙R h x) - j∣ < e/3"
          by (rule d2[OF p1 21])
      qed (use 0 in auto)
    qed
    then show ?thesis
      by (rule_tac x="λx. d1 x ∩ d2 x" in exI)
        (auto simp: fine_Int intro: ‹gauge d1› ‹gauge d2› d1 d2)
  qed
  then show ?thesis
    by (simp add: integrable_Cauchy)
qed

lemma integrable_straddle:
  fixes f :: "'n::euclidean_space ==> real"
  assumes "∧e. e>0 ==> ∃g h i j. (g has_integral i) s ∧ (h has_integral j) s ∧
                     ∣i - j∣ < e ∧ (∀x∈s. g x ≤ f x ∧ f x ≤ h x)"
  shows "f integrable_on s"
proof -
  let ?fs = "(λx. if x ∈ s then f x else 0)"
  have "?fs integrable_on cbox a b" for a b
  proof (rule integrable_straddle_interval)
    fix e::real
    assume "e > 0"
    then have *: "e/4 > 0"
      by auto
    with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                 and ij: "∣i - j∣ < e/4"
                 and fgh: "∧x. x ∈ s ==> g x ≤ f x ∧ f x ≤ h x"
      by metis
    let ?gs = "(λx. if x ∈ s then g x else 0)"
    let ?hs = "(λx. if x ∈ s then h x else 0)"
    obtain Bg where Bg: "∧a b. ball 0 Bg ⊆ cbox a b ==> ∣integral (cbox a b) ?gs - i∣ < e/4"
              and int_g: "∧a b. ?gs integrable_on cbox a b"
      using g * unfolding has_integral_alt' real_norm_def by meson
    obtain Bh where
          Bh: "∧a b. ball 0 Bh ⊆ cbox a b ==> ∣integral (cbox a b) ?hs - j∣ < e/4"
         and int_h: "∧a b. ?hs integrable_on cbox a b"
      using h * unfolding has_integral_alt' real_norm_def by meson
    define c where "c = (∑i∈Basis. min (a∙i) (- (max Bg Bh)) *🪙R i)"
    define d where "d = (∑i∈Basis. max (b∙i) (max Bg Bh) *🪙R i)"
    have "[norm (0 - x) < Bg; i ∈ Basis] ==> c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i" for x i
      using Basis_le_norm[of i x] unfolding c_def d_def by auto
    then have ballBg: "ball 0 Bg ⊆ cbox c d"
      by (auto simp: mem_box dist_norm)
    have "[norm (0 - x) < Bh; i ∈ Basis] ==> c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i" for x i
      using Basis_le_norm[of i x] unfolding c_def d_def by auto
    then have ballBh: "ball 0 Bh ⊆ cbox c d"
      by (auto simp: mem_box dist_norm)
    have ab_cd: "cbox a b ⊆ cbox c d"
      by (auto simp: c_def d_def subset_box_imp)
    have **: "∧ch cg ag ah::real. [∣ah - ag∣ ≤ ∣ch - cg∣; ∣cg - i∣ < e/4; ∣ch - j∣ < e/4]
       ==> ∣ag - ah∣ < e"
      using ij by arith
    show "∃g h i j. (g has_integral i) (cbox a b) ∧ (h has_integral j) (cbox a b) ∧ ∣i - j∣ < e ∧
          (∀x∈cbox a b. g x ≤ (if x ∈ s then f x else 0) ∧
                        (if x ∈ s then f x else 0) ≤ h x)"
    proof (intro exI ballI conjI)
      have eq: "∧x f g. (if x ∈ s then f x else 0) - (if x ∈ s then g x else 0) =
                       (if x ∈ s then f x - g x else (0::real))"
        by auto
      have int_hg: "(λx. if x ∈ s then h x - g x else 0) integrable_on cbox a b"
                   "(λx. if x ∈ s then h x - g x else 0) integrable_on cbox c d"
        by (metis (no_types) integrable_diff g h has_integral_integrable integrable_altD(1))+
      show "(?gs has_integral integral (cbox a b) ?gs) (cbox a b)"
           "(?hs has_integral integral (cbox a b) ?hs) (cbox a b)"
        by (intro integrable_integral int_g int_h)+
      then have "integral (cbox a b) ?gs ≤ integral (cbox a b) ?hs"
        using fgh by (force intro: has_integral_le)
      then have "0 ≤ integral (cbox a b) ?hs - integral (cbox a b) ?gs"
        by simp
      then have "∣integral (cbox a b) ?hs - integral (cbox a b) ?gs∣
              ≤ ∣integral (cbox c d) ?hs - integral (cbox c d) ?gs∣"
        apply (simp add: integral_diff [symmetric] int_g int_h)
        apply (subst abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF int_h int_g]])
        using fgh apply (force simp: eq intro!: integral_subset_le [OF ab_cd int_hg])+
        done
      then show "∣integral (cbox a b) ?gs - integral (cbox a b) ?hs∣ < e"
        using ** Bg ballBg Bh ballBh by blast
      show "∧x. x ∈ cbox a b ==> ?gs x ≤ ?fs x" "∧x. x ∈ cbox a b ==> ?fs x ≤ ?hs x"
        using fgh by auto
    qed
  qed
  then have int_f: "?fs integrable_on cbox a b" for a b
    by simp
  have "∃B>0. ∀a b c d.
                  ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
                  abs (integral (cbox a b) ?fs - integral (cbox c d) ?fs) < e"
      if "0 < e" for e
  proof -
    have *: "e/3 > 0"
      using that by auto
    with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                 and ij: "∣i - j∣ < e/3"
                 and fgh: "∧x. x ∈ s ==> g x ≤ f x ∧ f x ≤ h x"
      by metis
    let ?gs = "(λx. if x ∈ s then g x else 0)"
    let ?hs = "(λx. if x ∈ s then h x else 0)"
    obtain Bg where "Bg > 0"
              and Bg: "∧a b. ball 0 Bg ⊆ cbox a b ==> ∣integral (cbox a b) ?gs - i∣ < e/3"
              and int_g: "∧a b. ?gs integrable_on cbox a b"
      using g * unfolding has_integral_alt' real_norm_def by meson
    obtain Bh where "Bh > 0"
              and Bh: "∧a b. ball 0 Bh ⊆ cbox a b ==> ∣integral (cbox a b) ?hs - j∣ < e/3"
              and int_h: "∧a b. ?hs integrable_on cbox a b"
      using h * unfolding has_integral_alt' real_norm_def by meson
    { fix a b c d :: 'n
      assume as: "ball 0 (max Bg Bh) ⊆ cbox a b" "ball 0 (max Bg Bh) ⊆ cbox c d"
      have **: "ball 0 Bg ⊆ ball (0::'n) (max Bg Bh)" "ball 0 Bh ⊆ ball (0::'n) (max Bg Bh)"
        by auto
      have *: "∧ga gc ha hc fa fc. [∣ga - i∣ < e/3; ∣gc - i∣ < e/3; ∣ha - j∣ < e/3;
                     ∣hc - j∣ < e/3; ga ≤ fa; fa ≤ ha; gc ≤ fc; fc ≤ hc] ==>
        ∣fa - fc∣ < e"
        using ij by arith
      have "abs (integral (cbox a b) (λx. if x ∈ s then f x else 0) - integral (cbox c d)
        (λx. if x ∈ s then f x else 0)) < e"
      proof (rule *)
        show "∣integral (cbox a b) ?gs - i∣ < e/3"
          using "**" Bg as by blast
        show "∣integral (cbox c d) ?gs - i∣ < e/3"
          using "**" Bg as by blast
        show "∣integral (cbox a b) ?hs - j∣ < e/3"
          using "**" Bh as by blast
        show "∣integral (cbox c d) ?hs - j∣ < e/3"
          using "**" Bh as by blast
      qed (use int_f int_g int_h fgh in ‹simp_all add: integral_le›)
    }
    then show ?thesis
      by (meson ‹0 🚫› linorder_not_less max.bounded_iff)
  qed
  then show ?thesis
    unfolding integrable_alt[of f] real_norm_def by (blast intro: int_f)
qed


subsection ‹Adding integrals over several sets›

lemma has_integral_Un:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes f: "(f has_integral i) S" "(f has_integral j) T"
    and neg: "negligible (S ∩ T)"
  shows "(f has_integral (i + j)) (S ∪ T)"
  unfolding has_integral_restrict_UNIV[symmetric, of f]
proof (rule has_integral_spike[OF neg])
  let ?f = "λx. (if x ∈ S then f x else 0) + (if x ∈ T then f x else 0)"
  show "(?f has_integral i + j) UNIV"
    by (simp add: f has_integral_add)
qed auto

lemma integral_Un [simp]:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "f integrable_on S" "f integrable_on T" "negligible (S ∩ T)"
  shows "integral (S ∪ T) f = integral S f + integral T f"
  by (simp add: has_integral_Un assms integrable_integral integral_unique)

lemma integrable_Un:
  fixes f :: "'a::euclidean_space ==> 'b :: banach"
  assumes "negligible (A ∩ B)" "f integrable_on A" "f integrable_on B"
  shows   "f integrable_on (A ∪ B)"
proof -
  from assms obtain y z where "(f has_integral y) A" "(f has_integral z) B"
     by (auto simp: integrable_on_def)
  from has_integral_Un[OF this assms(1)] show ?thesis by (auto simp: integrable_on_def)
qed

lemma integrable_Un':
  fixes f :: "'a::euclidean_space ==> 'b :: banach"
  assumes "f integrable_on A" "f integrable_on B" "negligible (A ∩ B)" "C = A ∪ B"
  shows   "f integrable_on C"
  by (simp add: assms integrable_Un set_zero)

lemma has_integral_UN:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "finite I"
    and int: "∧i. i ∈ I ==> (f has_integral (g i)) (T i)"
    and neg: "pairwise (λi i'. negligible (T i ∩ T i')) I"
  shows "(f has_integral (sum g I)) (∪i∈I. T i)"
proof -
  let ?U = "((λ(a,b). T a ∩ T b) ` {(a,b). a ∈ I ∧ b ∈ I-{a}})"
  have "((λx. if x ∈ (∪i∈I. T i) then f x else 0) has_integral sum g I) UNIV"
  proof (rule has_integral_spike)
    show "negligible (∪?U)"
    proof (rule negligible_Union)
      have "finite (I × I)"
        by (simp add: ‹finite I›)
      moreover have "{(a,b). a ∈ I ∧ b ∈ I-{a}} ⊆ I × I"
        by auto
      ultimately show "finite ?U"
        by (simp add: finite_subset)
      show "∧t. t ∈ ?U ==> negligible t"
        using neg unfolding pairwise_def by auto
    qed
  next
    show "(if x ∈ (∪i∈I. T i) then f x else 0) = (∑i∈I. if x ∈ T i then f x else 0)"
      if "x ∈ UNIV - (∪?U)" for x
    proof clarsimp
      fix i assume i: "i ∈ I" "x ∈ T i"
      then have "∀j∈I. x ∈ T j ⟷ j = i"
        using that by blast
      with i show "f x = (∑i∈I. if x ∈ T i then f x else 0)"
        by (simp add: sum.delta[OF ‹finite I›])
    qed
  next
    show "((λx. (∑i∈I. if x ∈ T i then f x else 0)) has_integral sum g I) UNIV"
      using int by (simp add: has_integral_restrict_UNIV has_integral_sum [OF ‹finite I›])
  qed
  then show ?thesis
    using has_integral_restrict_UNIV by blast
qed

lemma has_integral_Union:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "finite T"
    and "∧S. S ∈ T ==> (f has_integral (i S)) S"
    and "pairwise (λS S'. negligible (S ∩ S')) T"
  shows "(f has_integral (sum i T)) (∪T)"
proof -
  have "(f has_integral (sum i T)) (∪S∈T. S)"
    by (intro has_integral_UN assms)
  then show ?thesis
    by force
qed

text ‹In particular adding integrals over a division, maybe not of an interval.›

lemma has_integral_combine_division:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "D division_of S"
    and "∧k. k ∈ D ==> (f has_integral (i k)) k"
  shows "(f has_integral (sum i D)) S"
proof -
  note D = division_ofD[OF assms(1)]
  have neg: "negligible (S ∩ s')" if "S ∈ D" "s' ∈ D" "S ≠ s'" for S s'
  proof -
    obtain a c b D where obt: "S = cbox a b" "s' = cbox c D"
      by (meson ‹S ∈ D› ‹s' ∈ D› D(4))
    from D(5)[OF that] show ?thesis
      unfolding obt interior_cbox
      by (metis (lifting) Diff_empty Int_interval box_Int_box negligible_frontier_interval)
  qed
  show ?thesis
    unfolding D(6)[symmetric]
    by (auto intro: D neg assms has_integral_Union pairwiseI)
qed

lemma integral_combine_division_bottomup:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "D division_of S" "∧k. k ∈ D ==> f integrable_on k"
  shows "integral S f = sum (λi. integral i f) D"
  by (meson assms integral_unique has_integral_combine_division has_integral_integrable_integral)

lemma has_integral_combine_division_topdown:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes f: "f integrable_on S"
    and D: "D division_of K"
    and "K ⊆ S"
  shows "(f has_integral (sum (λi. integral i f) D)) K"
proof -
  have "f integrable_on L" if "L ∈ D" for L
    by (smt (verit, best) assms cbox_division_memE f integrable_on_subcbox subset_trans that)
  then show ?thesis
    by (meson D has_integral_combine_division has_integral_integrable_integral)
qed

lemma integral_combine_division_topdown:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "f integrable_on S"
    and "D division_of S"
  shows "integral S f = sum (λi. integral i f) D"
  using assms has_integral_combine_division_topdown by blast

lemma integrable_combine_division:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes D: "D division_of S"
    and f: "∧i. i ∈ D ==> f integrable_on i"
  shows "f integrable_on S"
  using f unfolding integrable_on_def by (metis has_integral_combine_division[OF D])

lemma integrable_on_subdivision:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes D: "D division_of i"
    and f: "f integrable_on S"
    and "i ⊆ S"
  shows "f integrable_on i"
proof -
  have "f integrable_on j" if "j ∈ D" for j
    using assms integrable_on_def that
    by (metis cbox_division_memE elementary_interval has_integral_combine_division_topdown)
  then show ?thesis
    using D integrable_combine_division by blast
qed


subsection ‹Also tagged divisions›

lemma has_integral_combine_tagged_division:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "p tagged_division_of S"
    and "∧x k. (x,k) ∈ p ==> (f has_integral (i k)) k"
  shows "(f has_integral (∑(x,k)∈p. i k)) S"
proof -
  have *: "(f has_integral (∑k∈snd`p. integral k f)) S"
    by (smt (verit, del_insts) assms division_of_tagged_division has_integral_combine_division has_integral_iff imageE prod.collapse)
  also have "(∑k∈snd`p. integral k f) = (∑(x, k)∈p. integral k f)"
    by (metis assms(1) eq_integralD has_integral_empty_eq has_integral_open_interval
        sum.over_tagged_division_lemma)
  finally show ?thesis
    using assms by (auto simp: has_integral_iff intro!: sum.cong)
qed

lemma integral_combine_tagged_division_bottomup:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes p: "p tagged_division_of (cbox a b)"
    and f: "∧x k. (x,k)∈p ==> f integrable_on k"
  shows "integral (cbox a b) f = sum (λ(x,k). integral k f) p"
  by (simp add: has_integral_combine_tagged_division[OF p] integral_unique f integrable_integral)

lemma has_integral_combine_tagged_division_topdown:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes f: "f integrable_on cbox a b"
    and p: "p tagged_division_of (cbox a b)"
  shows "(f has_integral (sum (λ(x,K). integral K f) p)) (cbox a b)"
proof -
  have "(f has_integral integral K f) K" if "(x,K) ∈ p" for x K
    by (metis assms integrable_integral integrable_on_subcbox tagged_division_ofD(3,4) that)
  then show ?thesis
    by (simp add: has_integral_combine_tagged_division p)
qed

lemma integral_combine_tagged_division_topdown:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes "f integrable_on cbox a b"
    and "p tagged_division_of (cbox a b)"
  shows "integral (cbox a b) f = (∑(x, k)∈p. integral k f)"
  using assms by (auto intro: integral_unique [OF has_integral_combine_tagged_division_topdown])


subsection ‹Henstock's lemma›

lemma Henstock_lemma_part1:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes intf: "f integrable_on cbox a b"
    and "e > 0"
    and "gauge d"
    and less_e: "∧p. [p tagged_division_of (cbox a b); d fine p] ==>
                     norm (sum (λ(x,K). content K *🪙R f x) p - integral(cbox a b) f) < e"
    and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
  shows "norm (sum (λ(x,K). content K *🪙R f x - integral K f) p) ≤ e" (is "?lhs ≤ e")
proof (rule field_le_epsilon)
  fix k :: real
  assume "k > 0"
  let ?SUM = "λp. (∑(x,K) ∈ p. content K *🪙R f x)"
  note p' = tagged_partial_division_ofD[OF p(1)]
  have "∪(snd ` p) ⊆ cbox a b"
    using p'(3) by fastforce
  then obtain q where q: "snd ` p ⊆ q" and qdiv: "q division_of cbox a b"
    by (meson p(1) partial_division_extend_interval partial_division_of_tagged_division)
  note q' = division_ofD[OF qdiv]
  define r where "r = q - snd ` p"
  have "snd ` p ∩ r = {}"
    unfolding r_def by auto
  have "finite r"
    using q' unfolding r_def by auto
  have "∃p. p tagged_division_of i ∧ d fine p ∧
        norm (?SUM p - integral i f) < k / (real (card r) + 1)"
    if "i∈r" for i
  proof -
    have gt0: "k / (real (card r) + 1) > 0" using ‹k > 0› by simp
    have i: "i ∈ q"
      using that unfolding r_def by auto
    then obtain u v where uv: "i = cbox u v"
      using q'(4) by blast
    then have "cbox u v ⊆ cbox a b"
      using i q'(2) by auto  
    then have "f integrable_on cbox u v"
      by (rule integrable_subinterval[OF intf])
    with integrable_integral[OF this, unfolded has_integral[of f]]
    obtain dd where "gauge dd" and dd:
      "∧D. [D tagged_division_of cbox u v; dd fine D] ==>
    norm (?SUM D - integral (cbox u v) f) < k / (real (card r) + 1)"
      using gt0 by auto
    with gauge_Int[OF ‹gauge d› ‹gauge dd›]
    obtain qq where qq: "qq tagged_division_of cbox u v" "(λx. d x ∩ dd x) fine qq"
      using fine_division_exists by blast
    with dd[of qq]  show ?thesis
      by (auto simp: fine_Int uv)
  qed
  then obtain qq where qq: "∧i. i ∈ r ==> qq i tagged_division_of i ∧
      d fine qq i ∧ norm (?SUM (qq i) - integral i f) < k / (real (card r) + 1)"
    by metis

  let ?p = "p ∪ ∪(qq ` r)"
  have "norm (?SUM ?p - integral (cbox a b) f) < e"
  proof (rule less_e)
    show "d fine ?p"
      by (metis (mono_tags, opaque_lifting) qq fine_Un fine_Union imageE p(2))
    note ptag = tagged_partial_division_of_Union_self[OF p(1)]
    have "p ∪ ∪(qq ` r) tagged_division_of ∪(snd ` p) ∪ ∪r"
    proof (rule tagged_division_Un[OF ptag tagged_division_Union [OF ‹finite r›]])
      show "∧i. i ∈ r ==> qq i tagged_division_of i"
        using qq by auto
      show "∧i1 i2. [i1 ∈ r; i2 ∈ r; i1 ≠ i2] ==> interior i1 ∩ interior i2 = {}"
        by (simp add: q'(5) r_def)
      show "interior (∪(snd ` p)) ∩ interior (∪r) = {}"
      proof (rule Int_interior_Union_intervals [OF ‹finite r›])
        show "open (interior (∪(snd ` p)))"
          by blast
        show "∧T. T ∈ r ==> ∃a b. T = cbox a b"
          by (simp add: q'(4) r_def)
        have "interior T ∩ interior (∪(snd ` p)) = {}" if "T ∈ r" for T
        proof (rule Int_interior_Union_intervals)
          show "∧U. U ∈ snd ` p ==> ∃a b. U = cbox a b"
            using q q'(4) by blast
          show "∧U. U ∈ snd ` p ==> interior T ∩ interior U = {}"
            by (metis DiffE q q'(5) r_def subsetD that)
        qed (use p' in auto)
        then show "∧T. T ∈ r ==> interior (∪(snd ` p)) ∩ interior T = {}"
          by (metis Int_commute)
      qed
    qed
    moreover have "∪(snd ` p) ∪ ∪r = cbox a b" and "{qq i |i. i ∈ r} = qq ` r"
      using qdiv q unfolding Union_Un_distrib[symmetric] r_def by auto
    ultimately show "?p tagged_division_of (cbox a b)"
      by fastforce
  qed
  then have "norm (?SUM p + (?SUM (∪(qq ` r))) - integral (cbox a b) f) < e"
  proof (subst sum.union_inter_neutral[symmetric, OF ‹finite p›], safe)
    show "content L *🪙R f x = 0" if "(x, L) ∈ p" "(x, L) ∈ qq K" "K ∈ r" for x K L 
    proof -
      obtain u v where uv: "L = cbox u v"
        using ‹(x,L) ∈ p› p'(4) by blast
      have "L ⊆ K"
        using  qq[OF that(3)] tagged_division_ofD(3) ‹(x,L) ∈ qq K› by metis
      have "L ∈ snd ` p" 
        using ‹(x,L) ∈ p› image_iff by fastforce 
      then have "L ∈ q" "K ∈ q" "L ≠ K"
        using that q(1) unfolding r_def by auto
      with q'(5) show "content L *🪙R f x = 0"
        by (metis ‹L ⊆ K› content_eq_0_interior inf.orderE interior_Int scaleR_eq_0_iff uv)
    qed
    show "finite (∪(qq ` r))"
      by (meson finite_UN qq ‹finite r› tagged_division_of_finite)
  qed
  moreover have "content M *🪙R f x = 0" 
      if x: "(x,M) ∈ qq K" "(x,M) ∈ qq L" and KL: "qq K ≠ qq L" and r: "K ∈ r" "L ∈ r"
    for x M K L
  proof -
    note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
    obtain u v where uv: "M = cbox u v"
      using ‹(x, M) ∈ qq L› ‹L ∈ r› kl(2) by blast
    have empty: "interior (K ∩ L) = {}"
      by (metis DiffD1 interior_Int q'(5) r_def KL r)
    with that kl show "content M *🪙R f x = 0"
      by (metis content_eq_0_interior dual_order.refl inf.orderE inf_mono interior_mono 
                scaleR_eq_0_iff subset_empty uv x)
  qed 
  ultimately have "norm (?SUM p + sum ?SUM (qq ` r) - integral (cbox a b) f) < e"
    apply (subst (asm) sum.Union_comp)
    using qq by (force simp: split_paired_all)+
  moreover have "content M *🪙R f x = 0" 
       if "K ∈ r" "L ∈ r" "K ≠ L" "qq K = qq L" "(x, M) ∈ qq K" for K L x M
    using tagged_division_ofD(6) qq that by (metis (no_types, lifting)) 
  ultimately have less_e: "norm (?SUM p + sum (?SUM ∘ qq) r - integral (cbox a b) f) < e"
  proof (subst (asm) sum.reindex_nontrivial [OF ‹finite r›])
    qed (auto simp: split_paired_all sum.neutral)
  have norm_le: "norm (cp - ip) ≤ e + k"
                  if "norm ((cp + cr) - i) < e" "norm (cr - ir) < k" "ip + ir = i"
                  for ir ip i cr cp::'a
    using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"] that
    unfolding that(3)[symmetric] norm_minus_cancel
    by (auto simp: algebra_simps)

  have "?lhs = norm (?SUM p - (∑(x, k)∈p. integral k f))"
    unfolding split_def sum_subtractf ..
  also have "… ≤ e + k"
  proof (rule norm_le[OF less_e])
    have lessk: "k * real (card r) / (1 + real (card r)) < k"
      using ‹k>0› by (auto simp: field_simps)
    have "norm (sum (?SUM ∘ qq) r - (∑k∈r. integral k f)) ≤ (∑x∈r. k / (real (card r) + 1))"
      unfolding sum_subtractf[symmetric] by (force dest: qq intro!: sum_norm_le)
    also have "… < k"
      by (simp add: lessk add.commute mult.commute)
    finally show "norm (sum (?SUM ∘ qq) r - (∑k∈r. integral k f)) < k" .
  next
    from q(1) have [simp]: "snd ` p ∪ q = q" by auto
    have "integral l f = 0"
      if inp: "(x, l) ∈ p" "(y, m) ∈ p" and ne: "(x, l) ≠ (y, m)" and "l = m" for x l y m
    proof -
      obtain u v where uv: "l = cbox u v"
        using inp p'(4) by blast
      then show ?thesis
        using uv that p
        by (metis content_eq_0_interior inf.orderE integral_null p'(5) subset_refl)
    qed
    then have "(∑(x, K)∈p. integral K f) = (∑K∈snd ` p. integral K f)"
      apply (subst sum.reindex_nontrivial [OF ‹finite p›])
      unfolding split_paired_all split_def by auto
    then show "(∑(x, k)∈p. integral k f) + (∑k∈r. integral k f) = integral (cbox a b) f"
      unfolding integral_combine_division_topdown[OF intf qdiv] r_def
      by (metis add.commute q q'(1) sum.subset_diff)
  qed
  finally show "?lhs ≤ e + k" .
qed

lemma Henstock_lemma_part2:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes fed: "f integrable_on cbox a b" "e > 0" "gauge d"
    and less_e: "∧D. [D tagged_division_of (cbox a b); d fine D] ==>
                     norm (sum (λ(x,k). content k *🪙R f x) D - integral (cbox a b) f) < e"
    and tag: "p tagged_partial_division_of (cbox a b)"
    and "d fine p"
  shows "sum (λ(x,k). norm (content k *🪙R f x - integral k f)) p ≤ 2 * real (DIM('n)) * e"
proof -
  have "finite p"
    using tag tagged_partial_division_ofD by blast
  then show ?thesis
    unfolding split_def
  proof (rule sum_norm_allsubsets_bound)
    fix Q
    assume Q: "Q ⊆ p"
    then have fine: "d fine Q"
      by (simp add: ‹d fine p› fine_subset)
    show "norm (∑x∈Q. content (snd x) *🪙R f (fst x) - integral (snd x) f) ≤ e"
      apply (rule Henstock_lemma_part1[OF fed less_e, unfolded split_def])
      using Q tag tagged_partial_division_subset by (force simp: fine)+
  qed
qed

lemma Henstock_lemma:
  fixes f :: "'m::euclidean_space ==> 'n::euclidean_space"
  assumes intf: "f integrable_on cbox a b"
    and "e > 0"
  obtains γ where "gauge γ"
    and "∧p. [p tagged_partial_division_of (cbox a b); γ fine p] ==>
             sum (λ(x,k). norm(content k *🪙R f x - integral k f)) p < e"
proof -
  have *: "e/(2 * (real DIM('n) + 1)) > 0" using ‹e > 0› by simp
  with integrable_integral[OF intf, unfolded has_integral]
  obtain γ where "gauge γ"
    and γ: "∧D. [D tagged_division_of cbox a b; γ fine D] ==>
         norm ((∑(x,K)∈D. content K *🪙R f x) - integral (cbox a b) f)
         < e/(2 * (real DIM('n) + 1))"
    by metis
  show thesis
  proof (rule that [OF ‹gauge γ›])
    fix p
    assume p: "p tagged_partial_division_of cbox a b" "γ fine p"
    have "(∑(x,K)∈p. norm (content K *🪙R f x - integral K f))
          ≤ 2 * real DIM('n) * (e/(2 * (real DIM('n) + 1)))"
      using Henstock_lemma_part2[OF intf * ‹gauge γ› γ p] by metis
    also have "… < e"
      using ‹e > 0› by (auto simp: field_simps)
    finally
    show "(∑(x,K)∈p. norm (content K *🪙R f x - integral K f)) < e" .
  qed
qed


subsection ‹Monotone convergence (bounded interval first)›

lemma bounded_increasing_convergent:
  fixes f :: "nat ==> real"
  shows "[bounded (range f); ∧n. f n ≤ f (Suc n)] ==> ∃l. f <---- l"
  using Bseq_mono_convergent[of f] incseq_Suc_iff[of f]
  by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

lemma monotone_convergence_interval:
  fixes f :: "nat ==> 'n::euclidean_space ==> real"
  assumes intf: "∧k. (f k) integrable_on cbox a b"
    and le: "∧k x. x ∈ cbox a b ==> (f k x) ≤ f (Suc k) x"
    and fg: "∧x. x ∈ cbox a b ==> ((λk. f k x) ---> g x) sequentially"
    and bou: "bounded (range (λk. integral (cbox a b) (f k)))"
  shows "g integrable_on cbox a b ∧ ((λk. integral (cbox a b) (f k)) ---> integral (cbox a b) g) sequentially"
proof (cases "content (cbox a b) = 0")
  case True then show ?thesis
    by auto
next
  case False
  have fg1: "(f k x) ≤ (g x)" if x: "x ∈ cbox a b" for x k
  proof -
    have "∀🪙F j in sequentially. f k x ≤ f j x"
      by (metis eventually_sequentiallyI le lift_Suc_mono_le x)
    then show "f k x ≤ g x"
      using tendsto_lowerbound [OF fg] x trivial_limit_sequentially by blast
  qed
  have int_inc: "∧n. integral (cbox a b) (f n) ≤ integral (cbox a b) (f (Suc n))"
    by (metis integral_le intf le)
  then obtain i where i: "(λk. integral (cbox a b) (f k)) <---- i"
    using bounded_increasing_convergent bou by blast
  have "∧k. ∀🪙F x in sequentially. integral (cbox a b) (f k) ≤ integral (cbox a b) (f x)"
    unfolding eventually_sequentially
    by (force intro: transitive_stepwise_le int_inc)
  then have i': "∧k. (integral(cbox a b) (f k)) ≤ i"
    using tendsto_le [OF trivial_limit_sequentially i] by blast
  have "(g has_integral i) (cbox a b)"
    unfolding has_integral real_norm_def
  proof clarify
    fix e::real
    assume e: "e > 0"
    have "∧k. (∃γ. gauge γ ∧ (∀D. D tagged_division_of (cbox a b) ∧ γ fine D ⟶
      abs ((∑(x,K)∈D. content K *🪙R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))"
      using intf e by (auto simp: has_integral_integral has_integral)
    then obtain c where c: "∧x. gauge (c x)"
          "∧x D. [D tagged_division_of cbox a b; c x fine D] ==>
              abs ((∑(u,K)∈D. content K *🪙R f x u) - integral (cbox a b) (f x))
              < e/2 ^ (x + 2)"
      by metis

    have "∃r. ∀k≥r. 0 ≤ i - (integral (cbox a b) (f k)) ∧ i - (integral (cbox a b) (f k)) < e/4"
    proof -
      have "e/4 > 0"
        using e by auto
      show ?thesis
        using LIMSEQ_D [OF i ‹e/4 > 0›] i' by auto
    qed
    then obtain r where r: "∧k. r ≤ k ==> 0 ≤ i - integral (cbox a b) (f k)"
                       "∧k. r ≤ k ==> i - integral (cbox a b) (f k) < e/4" 
      by metis
    have "∃n≥r. ∀k≥n. 0 ≤ (g x) - (f k x) ∧ (g x) - (f k x) < e/(4 * content(cbox a b))"
      if "x ∈ cbox a b" for x
    proof -
      have "e/(4 * content (cbox a b)) > 0"
        by (simp add: False content_lt_nz e)
      with fg that LIMSEQ_D
      obtain N where "∀n≥N. norm (f n x - g x) < e/(4 * content (cbox a b))"
        by metis
      with fg1[OF that] show "∃n≥r. ∀k≥n. 0 ≤ g x - f k x ∧ g x - f k x < e/(4 * content (cbox a b))"
        by (rule_tac x="N + r" in exI) (auto simp: field_simps)
    qed
    then obtain m where r_le_m: "∧x. x ∈ cbox a b ==> r ≤ m x"
       and m: "∧x k. [x ∈ cbox a b; m x ≤ k]
                     ==> 0 ≤ g x - f k x ∧ g x - f k x < e/(4 * content (cbox a b))"
      by metis
    define d where "d x = c (m x) x" for x
    show "∃γ. gauge γ ∧
             (∀D. D tagged_division_of cbox a b ∧
                  γ fine D ⟶ abs ((∑(x,K)∈D. content K *🪙R g x) - i) < e)"
    proof (rule exI, safe)
      show "gauge d"
        using c(1) unfolding gauge_def d_def by auto
    next
      fix D
      assume ptag: "D tagged_division_of (cbox a b)" and "d fine D"
      note p'=tagged_division_ofD[OF ptag]
      obtain s where s: "∧x. x ∈ D ==> m (fst x) ≤ s"
        by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
      have *: "∣a - d∣ < e" if "∣a - b∣ ≤ e/4" "∣b - c∣ < e/2" "∣c - d∣ < e/4" for a b c d
        using that norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
          norm_triangle_lt[of "a - b + (b - c)" "c - d" e]
        by (auto simp: algebra_simps)
      show "∣(∑(x, k)∈D. content k *🪙R g x) - i∣ < e"
      proof (rule *)
        have "∣(∑(x,K)∈D. content K *🪙R g x) - (∑(x,K)∈D. content K *🪙R f (m x) x)∣
              ≤ (∑i∈D. ∣(case i of (x, K) ==> content K *🪙R g x) - (case i of (x, K) ==> content K *🪙R f (m x) x)∣)"
          by (metis (mono_tags) sum_subtractf sum_abs) 
        also have "… ≤ (∑(x, k)∈D. content k * (e/(4 * content (cbox a b))))"
        proof (rule sum_mono, simp add: split_paired_all)
          fix x K
          assume xk: "(x,K) ∈ D"
          with ptag have x: "x ∈ cbox a b"
            by blast
          then have "abs (content K * (g x - f (m x) x)) ≤ content K * (e/(4 * content (cbox a b)))"
            by (metis m[OF x] mult_nonneg_nonneg abs_of_nonneg less_eq_real_def measure_nonneg mult_left_mono order_refl)
          then show "∣content K * g x - content K * f (m x) x∣ ≤ content K * e/(4 * content (cbox a b))"
            by (simp add: algebra_simps)
        qed
        also have "… = (e/(4 * content (cbox a b))) * (∑(x, k)∈D. content k)"
          by (simp add: sum_distrib_left sum_divide_distrib split_def mult.commute)
        also have "… ≤ e/4"
          by (metis False additive_content_tagged_division [OF ptag] nonzero_mult_divide_mult_cancel_right order_refl times_divide_eq_left)
        finally show "∣(∑(x,K)∈D. content K *🪙R g x) - (∑(x,K)∈D. content K *🪙R f (m x) x)∣ ≤ e/4" .

      next
        have "norm ((∑(x,K)∈D. content K *🪙R f (m x) x) - (∑(x,K)∈D. integral K (f (m x))))
              ≤ norm (∑j = 0..s. ∑(x,K)∈{xk ∈ D. m (fst xk) = j}. content K *🪙R f (m x) x - integral K (f (m x)))"
          using s by (subst sum.group) (auto simp: sum_subtractf split_def p')
        also have "… < e/2"
        proof -
          have "norm (∑j = 0..s. ∑(x, k)∈{xk ∈ D. m (fst xk) = j}. content k *🪙R f (m x) x - integral k (f (m x)))
                ≤ (∑i = 0..s. e/2 ^ (i + 2))"
          proof (rule sum_norm_le)
            fix t
            assume "t ∈ {0..s}"
            have "norm (∑(x,k)∈{xk ∈ D. m (fst xk) = t}. content k *🪙R f (m x) x - integral k (f (m x))) =
                  norm (∑(x,k)∈{xk ∈ D. m (fst xk) = t}. content k *🪙R f t x - integral k (f t))"
              by (force intro!: sum.cong arg_cong[where f=norm])
            also have "… ≤ e/2 ^ (t + 2)"
            proof (rule Henstock_lemma_part1 [OF intf])
              show "{xk ∈ D. m (fst xk) = t} tagged_partial_division_of cbox a b"
              proof (rule tagged_partial_division_subset[of D])
                show "D tagged_partial_division_of cbox a b"
                  using ptag tagged_division_of_def by blast
              qed auto
              show "c t fine {xk ∈ D. m (fst xk) = t}"
                using ‹d fine D› by (auto simp: fine_def d_def)
            qed (use c e in auto)
            finally show "norm (∑(x,K)∈{xk ∈ D. m (fst xk) = t}. content K *🪙R f (m x) x -
                                integral K (f (m x))) ≤ e/2 ^ (t + 2)" .
          qed
          also have "… = (e/2/2) * (∑i = 0..s. (1/2) ^ i)"
            by (simp add: sum_distrib_left field_simps)
          also have "… < e/2"
            by (simp add: sum_gp mult_strict_left_mono[OF _ e])
          finally show "norm (∑j = 0..s. ∑(x, k)∈{xk ∈ D.
            m (fst xk) = j}. content k *🪙R f (m x) x - integral k (f (m x))) < e/2" .
        qed 
        finally show "∣(∑(x,K)∈D. content K *🪙R f (m x) x) - (∑(x,K)∈D. integral K (f (m x)))∣ < e/2"
          by simp
      next
        have comb: "integral (cbox a b) (f y) = (∑(x, k)∈D. integral k (f y))" for y
          using integral_combine_tagged_division_topdown[OF intf ptag] by metis
        have f_le: "∧y m n. [y ∈ cbox a b; n≥m] ==> f m y ≤ f n y"
          using le by (auto intro: transitive_stepwise_le)        
        have "(∑(x, k)∈D. integral k (f r)) ≤ (∑(x, K)∈D. integral K (f (m x)))"
        proof (rule sum_mono, simp add: split_paired_all)
          fix x K
          assume xK: "(x, K) ∈ D"
          show "integral K (f r) ≤ integral K (f (m x))"
          proof (rule integral_le)
            show "f r integrable_on K"
              by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
            show "f (m x) integrable_on K"
              by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
            show "f r y ≤ f (m x) y" if "y ∈ K" for y
              using that r_le_m[of x] p'(2-3)[OF xK] f_le by auto
          qed
        qed
        moreover have "(∑(x, K)∈D. integral K (f (m x))) ≤ (∑(x, k)∈D. integral k (f s))"
        proof (rule sum_mono, simp add: split_paired_all)
          fix x K
          assume xK: "(x, K) ∈ D"
          show "integral K (f (m x)) ≤ integral K (f s)"
          proof (rule integral_le)
            show "f (m x) integrable_on K"
              by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
            show "f s integrable_on K"
              by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
            show "f (m x) y ≤ f s y" if "y ∈ K" for y
              using that s xK f_le p'(3) by fastforce
          qed
        qed
        moreover have "0 ≤ i - integral (cbox a b) (f r)" "i - integral (cbox a b) (f r) < e/4"
          using r by auto
        ultimately show "∣(∑(x,K)∈D. integral K (f (m x))) - i∣ < e/4"
          using comb i'[of s] by auto
      qed
    qed
  qed 
  with i integral_unique show ?thesis
    by blast
qed

lemma monotone_convergence_increasing:
  fixes f :: "nat ==> 'n::euclidean_space ==> real"
  assumes int_f: "∧k. (f k) integrable_on S"
    and "∧k x. x ∈ S ==> (f k x) ≤ (f (Suc k) x)"
    and fg: "∧x. x ∈ S ==> ((λk. f k x) ---> g x) sequentially"
    and bou: "bounded (range (λk. integral S (f k)))"
  shows "g integrable_on S ∧ ((λk. integral S (f k)) ---> integral S g) sequentially"
proof -
  have lem: "g integrable_on S ∧ ((λk. integral S (f k)) ---> integral S g) sequentially"
    if f0: "∧k x. x ∈ S ==> 0 ≤ f k x"
    and int_f: "∧k. (f k) integrable_on S"
    and le: "∧k x. x ∈ S ==> f k x ≤ f (Suc k) x"
    and lim: "∧x. x ∈ S ==> ((λk. f k x) ---> g x) sequentially"
    and bou: "bounded (range(λk. integral S (f k)))"
    for f :: "nat ==> 'n::euclidean_space ==> real" and g S
  proof -
    have fg: "(f k x) ≤ (g x)" if "x ∈ S" for x k
    proof -
      have "∧xa. k ≤ xa ==> f k x ≤ f xa x"
        using le  by (force intro: transitive_stepwise_le that)
      then show ?thesis
        using tendsto_lowerbound [OF lim [OF that]] eventually_sequentiallyI by force
    qed
    obtain i where i: "(λk. integral S (f k)) <---- i"
      using bounded_increasing_convergent [OF bou] le int_f integral_le by blast
    have i': "(integral S (f k)) ≤ i" for k
    proof -
      have "∧k. ∧x. x ∈ S ==> ∀n≥k. f k x ≤ f n x"
        using le  by (force intro: transitive_stepwise_le)
      then show ?thesis
        using tendsto_lowerbound [OF i eventually_sequentiallyI trivial_limit_sequentially]
        by (meson int_f integral_le)
    qed
    let ?f = "(λk x. if x ∈ S then f k x else 0)"
    let ?g = "(λx. if x ∈ S then g x else 0)"
    have int: "?f k integrable_on cbox a b" for a b k
      by (simp add: int_f integrable_altD(1))
    have int': "∧k a b. f k integrable_on cbox a b ∩ S"
      using int by (simp add: Int_commute integrable_restrict_Int) 
    have g: "?g integrable_on cbox a b ∧
             (λk. integral (cbox a b) (?f k)) <---- integral (cbox a b) ?g" for a b
    proof (rule monotone_convergence_interval)
      have "norm (integral (cbox a b) (?f k)) ≤ norm (integral S (f k))" for k
      proof -
        have "0 ≤ integral (cbox a b) (?f k)"
          by (metis (no_types) integral_nonneg Int_iff f0 inf_commute integral_restrict_Int int')
        moreover have "0 ≤ integral S (f k)"
          by (simp add: integral_nonneg f0 int_f)
        moreover have "integral (S ∩ cbox a b) (f k) ≤ integral S (f k)"
          by (metis f0 inf_commute int' int_f integral_subset_le le_inf_iff order_refl)
        ultimately show ?thesis
          by (simp add: integral_restrict_Int)
      qed
      moreover obtain B where "∧x. x ∈ range (λk. integral S (f k)) ==> norm x ≤ B"
        using bou unfolding bounded_iff by blast
      ultimately show "bounded (range (λk. integral (cbox a b) (?f k)))"
        unfolding bounded_iff by (blast intro: order_trans)
    qed (use int le lim in auto)
    moreover have "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶ norm (integral (cbox a b) ?g - i) < e"
      if "0 < e" for e
    proof -
      have "e/4>0"
        using that by auto
      with LIMSEQ_D [OF i] obtain N where N: "∧n. n ≥ N ==> norm (integral S (f n) - i) < e/4"
        by metis
      obtain B where "0 < B" and B: 
        "∧a b. ball 0 B ⊆ cbox a b ==> norm (integral (cbox a b) (?f N) - integral S (f N)) < e/4"
        by (metis (lifting) ext ‹0 🚫/4› has_integral_alt' int_f integrable_integral)
      have "norm (integral (cbox a b) ?g - i) < e" if  ab: "ball 0 B ⊆ cbox a b" for a b
      proof -
        obtain M where M: "∧n. n ≥ M ==> abs (integral (cbox a b) (?f n) - integral (cbox a b) ?g) < e/2"
          using ‹e > 0› g by (fastforce simp: dest!: LIMSEQ_D [where r = "e/2"])
        have *: "∧α β g. [∣α - i∣ < e/2; ∣β - g∣ < e/2; α ≤ β; β ≤ i] ==> ∣g - i∣ < e"
          unfolding real_inner_1_right by arith
        show "norm (integral (cbox a b) ?g - i) < e"
          unfolding real_norm_def
        proof (rule *)
          show "∣integral (cbox a b) (?f N) - i∣ < e/2"
          proof (rule abs_triangle_half_l)
            show "∣integral (cbox a b) (?f N) - integral S (f N)∣ < e/2/2"
              using B[OF ab] by simp
            show "abs (i - integral S (f N)) < e/2/2"
              using N by (simp add: abs_minus_commute)
          qed
          show "∣integral (cbox a b) (?f (M + N)) - integral (cbox a b) ?g∣ < e/2"
            by (metis le_add1 M[of "M + N"])
          show "integral (cbox a b) (?f N) ≤ integral (cbox a b) (?f (M + N))"
          proof (intro ballI integral_le[OF int int])
            fix x assume "x ∈ cbox a b"
            have "(f m x) ≤ (f n x)" if "x ∈ S" "n ≥ m" for m n
            proof (rule transitive_stepwise_le [OF ‹n ≥ m› order_refl])
              show "∧u y z. [f u x ≤ f y x; f y x ≤ f z x] ==> f u x ≤ f z x"
                using dual_order.trans by blast
            qed (simp add: le ‹x ∈ S›)
            then show "(?f N)x ≤ (?f (M+N))x"
              by auto
          qed
          have "integral (cbox a b ∩ S) (f (M + N)) ≤ integral S (f (M + N))"
            by (metis Int_lower1 f0 inf_commute int' int_f integral_subset_le)
          then have "integral (cbox a b) (?f (M + N)) ≤ integral S (f (M + N))"
            by (metis (no_types) inf_commute integral_restrict_Int)
          also have "… ≤ i"
            using i'[of "M + N"] by auto
          finally show "integral (cbox a b) (?f (M + N)) ≤ i" .
        qed
      qed
      then show ?thesis
        using ‹0 🚫› by blast
    qed
    ultimately have "(g has_integral i) S"
      unfolding has_integral_alt' by auto
    then show ?thesis
      using has_integral_integrable_integral i integral_unique by metis
  qed

  have sub: "∧k. integral S (λx. f k x - f 0 x) = integral S (f k) - integral S (f 0)"
    by (simp add: integral_diff int_f)
  have *: "∧x m n. x ∈ S ==> n≥m ==> f m x ≤ f n x"
    using assms(2) by (force intro: transitive_stepwise_le)
  have gf: "(λx. g x - f 0 x) integrable_on S ∧ ((λk. integral S (λx. f (Suc k) x - f 0 x)) --->
    integral S (λx. g x - f 0 x)) sequentially"
  proof (rule lem)
    show "∧k. (λx. f (Suc k) x - f 0 x) integrable_on S"
      by (simp add: integrable_diff int_f)
    show "(λk. f (Suc k) x - f 0 x) <---- g x - f 0 x" if "x ∈ S" for x
      using LIMSEQ_ignore_initial_segment[OF fg[OF ‹x ∈ S›], of 1]
      by (simp add: tendsto_diff)
    show "bounded (range (λk. integral S (λx. f (Suc k) x - f 0 x)))"
    proof -
      obtain B where B: "∧k. norm (integral S (f k)) ≤ B"
        using  bou by (auto simp: bounded_iff)
      then have "norm (integral S (λx. f (Suc k) x - f 0 x))
              ≤ B + norm (integral S (f 0))" for k
        unfolding sub by (meson add_le_cancel_right norm_triangle_le_diff)
      then show ?thesis
        unfolding bounded_iff by blast
    qed
  qed (use * in auto)
  then have "(λx. integral S (λxa. f (Suc x) xa - f 0 xa) + integral S (f 0))
             <---- integral S (λx. g x - f 0 x) + integral S (f 0)"
    by (auto simp: tendsto_add)
  moreover have "(λx. g x - f 0 x + f 0 x) integrable_on S"
    using gf integrable_add int_f [of 0] by metis
  ultimately show ?thesis
    by (simp add: integral_diff int_f LIMSEQ_imp_Suc sub)
qed

lemma has_integral_monotone_convergence_increasing:
  fixes f :: "nat ==> 'a::euclidean_space ==> real"
  assumes f: "∧k. (f k has_integral x k) s"
  assumes "∧k x. x ∈ s ==> f k x ≤ f (Suc k) x"
  assumes "∧x. x ∈ s ==> (λk. f k x) <---- g x"
  assumes "x <---- x'"
  shows "(g has_integral x') s"
proof -
  have x_eq: "x = (λi. integral s (f i))"
    by (simp add: integral_unique[OF f])
  then have x: "range(λk. integral s (f k)) = range x"
    by auto
  have *: "g integrable_on s ∧ (λk. integral s (f k)) <---- integral s g"
  proof (intro monotone_convergence_increasing allI ballI assms)
    show "bounded (range(λk. integral s (f k)))"
      using x convergent_imp_bounded assms by metis
  qed (use f in auto)
  then have "integral s g = x'"
    by (intro LIMSEQ_unique[OF _ ‹x <---- x'›]) (simp add: x_eq)
  with * show ?thesis
    by (simp add: has_integral_integral)
qed

lemma monotone_convergence_decreasing:
  fixes f :: "nat ==> 'n::euclidean_space ==> real"
  assumes intf: "∧k. (f k) integrable_on S"
    and le: "∧k x. x ∈ S ==> f (Suc k) x ≤ f k x"
    and fg: "∧x. x ∈ S ==> ((λk. f k x) ---> g x) sequentially"
    and bou: "bounded (range(λk. integral S (f k)))"
  shows "g integrable_on S ∧ (λk. integral S (f k)) <---- integral S g"
proof -
  have *: "range(λk. integral S (λx. - f k x)) = (*🪙R) (- 1) ` (range(λk. integral S (f k)))"
    by force
  have "(λx. - g x) integrable_on S ∧ (λk. integral S (λx. - f k x)) <---- integral S (λx. - g x)"
  proof (rule monotone_convergence_increasing)
    show "∧k. (λx. - f k x) integrable_on S"
      by (blast intro: integrable_neg intf)
    show "∧k x. x ∈ S ==> - f k x ≤ - f (Suc k) x"
      by (simp add: le)
    show "∧x. x ∈ S ==> (λk. - f k x) <---- - g x"
      by (simp add: fg tendsto_minus)
    show "bounded (range(λk. integral S (λx. - f k x)))"
      using "*" bou bounded_scaling by auto
  qed
  then show ?thesis
    by (force dest: integrable_neg tendsto_minus)
qed

lemma integral_norm_bound_integral:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  assumes int_f: "f integrable_on S"
    and int_g: "g integrable_on S"
    and le_g: "∧x. x ∈ S ==> norm (f x) ≤ g x"
  shows "norm (integral S f) ≤ integral S g"
proof -
  have norm: "norm 🪙 ≤ y + e"
    if "norm ζ ≤ x" and "∣x - y∣ < e/2" and "norm (ζ - 🪙) < e/2"
    for e x y and ζ 🪙 :: 'a
  proof -
    have "norm (🪙 - ζ) < e/2"
      by (metis norm_minus_commute that(3))
    moreover have "x ≤ y + e/2"
      using that(2) by linarith
    ultimately show ?thesis
      using that(1) le_less_trans[OF norm_triangle_sub[of 🪙 ζ]] by (auto simp: less_imp_le)
  qed
  have lem: "norm (integral(cbox a b) f) ≤ integral (cbox a b) g"
    if f: "f integrable_on cbox a b"
    and g: "g integrable_on cbox a b"
    and nle: "∧x. x ∈ cbox a b ==> norm (f x) ≤ g x"
    for f :: "'n ==> 'a" and g a b
  proof (rule field_le_epsilon)
    fix e :: real
    assume "e > 0"
    then have e: "e/2 > 0"
      by auto
    with integrable_integral[OF f,unfolded has_integral[of f]]
    obtain γ where γ: "gauge γ"
              "∧D. D tagged_division_of cbox a b ∧ γ fine D
           ==> norm ((∑(x, k)∈D. content k *🪙R f x) - integral (cbox a b) f) < e/2"
      by meson 
    moreover
    from integrable_integral[OF g,unfolded has_integral[of g]] e
    obtain δ where δ: "gauge δ"
              "∧D. D tagged_division_of cbox a b ∧ δ fine D
           ==> norm ((∑(x, k)∈D. content k *🪙R g x) - integral (cbox a b) g) < e/2"
      by meson
    ultimately have "gauge (λx. γ x ∩ δ x)"
      using gauge_Int by blast
    with fine_division_exists obtain D 
      where p: "D tagged_division_of cbox a b" "(λx. γ x ∩ δ x) fine D" 
      by metis
    have "γ fine D" "δ fine D"
      using fine_Int p(2) by blast+
    show "norm (integral (cbox a b) f) ≤ integral (cbox a b) g + e"
    proof (rule norm)
      have "norm (content K *🪙R f x) ≤ content K *🪙R g x" if  "(x, K) ∈ D" for x K
      proof-
        have K: "x ∈ K" "K ⊆ cbox a b"
          using ‹(x, K) ∈ D› p(1) by blast+
        obtain u v where  "K = cbox u v"
          using ‹(x, K) ∈ D› p(1) by blast
        moreover have "content K * norm (f x) ≤ content K * g x"
          by (meson K(1) K(2) content_pos_le mult_left_mono nle subsetD)
        then show ?thesis
          by simp
      qed
      then show "norm (∑(x, k)∈D. content k *🪙R f x) ≤ (∑(x, k)∈D. content k *🪙R g x)"
        by (simp add: sum_norm_le split_def)
      show "norm ((∑(x, k)∈D. content k *🪙R f x) - integral (cbox a b) f) < e/2"
        using ‹γ fine D› γ p(1) by simp
      show "∣(∑(x, k)∈D. content k *🪙R g x) - integral (cbox a b) g∣ < e/2"
        using ‹δ fine D› δ p(1) by simp
    qed
  qed
  show ?thesis
  proof (rule field_le_epsilon)
    fix e :: real
    assume "e > 0"
    then have e: "e/2 > 0"
      by auto
    let ?f = "(λx. if x ∈ S then f x else 0)"
    let ?g = "(λx. if x ∈ S then g x else 0)"
    have f: "?f integrable_on cbox a b" and g: "?g integrable_on cbox a b" for a b
      using int_f int_g integrable_altD by auto
    obtain Bf where "0 < Bf"
      and Bf: "∧a b. ball 0 Bf ⊆ cbox a b ==>
        ∃z. (?f has_integral z) (cbox a b) ∧ norm (z - integral S f) < e/2"
      using integrable_integral [OF int_f,unfolded has_integral'[of f]] e that by blast
    obtain Bg where "0 < Bg"
      and Bg: "∧a b. ball 0 Bg ⊆ cbox a b ==>
        ∃z. (?g has_integral z) (cbox a b) ∧ norm (z - integral S g) < e/2"
      using integrable_integral [OF int_g,unfolded has_integral'[of g]] e that by blast
    obtain a b::'n where ab: "ball 0 Bf ∪ ball 0 Bg ⊆ cbox a b"
      using ball_max_Un  by (metis bounded_ball bounded_subset_cbox_symmetric)
    have "ball 0 Bf ⊆ cbox a b"
      using ab by auto
    with Bf obtain z where int_fz: "(?f has_integral z) (cbox a b)" and z: "norm (z - integral S f) < e/2"
      by meson
    have "ball 0 Bg ⊆ cbox a b"
      using ab by auto
    with Bg obtain w where int_gw: "(?g has_integral w) (cbox a b)" and w: "norm (w - integral S g) < e/2"
      by meson
    show "norm (integral S f) ≤ integral S g + e"
    proof (rule norm)
      show "norm (integral (cbox a b) ?f) ≤ integral (cbox a b) ?g"
        by (simp add: le_g lem[OF f g, of a b])
      show "∣integral (cbox a b) ?g - integral S g∣ < e/2"
        using int_gw integral_unique w by auto
      show "norm (integral (cbox a b) ?f - integral S f) < e/2"
        using int_fz integral_unique z by blast
    qed
  qed
qed

lemma continuous_on_imp_absolutely_integrable_on:
  fixes f::"real ==> 'a::banach"
  shows "continuous_on {a..b} f ==>
    norm (integral {a..b} f) ≤ integral {a..b} (λx. norm (f x))"
  by (intro integral_norm_bound_integral integrable_continuous_real continuous_on_norm) auto

lemma integral_bound:
  fixes f::"real ==> 'a::banach"
  assumes "a ≤ b"
  assumes "continuous_on {a .. b} f"
  assumes "∧t. t ∈ {a .. b} ==> norm (f t) ≤ B"
  shows "norm (integral {a .. b} f) ≤ B * (b - a)"
proof -
  note continuous_on_imp_absolutely_integrable_on[OF assms(2)]
  also have "integral {a..b} (λx. norm (f x)) ≤ integral {a..b} (λ_. B)"
    by (rule integral_le)
      (auto intro!: integrable_continuous_real continuous_intros assms)
  also have "… = B * (b - a)" using assms by simp
  finally show ?thesis .
qed

lemma integral_norm_bound_integral_component:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  fixes g :: "'n ==> 'b::euclidean_space"
  assumes f: "f integrable_on S" and g: "g integrable_on S"
    and fg: "∧x. x ∈ S ==> norm(f x) ≤ (g x)∙k"
  shows "norm (integral S f) ≤ (integral S g)∙k"
proof -
  have "norm (integral S f) ≤ integral S ((λx. x ∙ k) ∘ g)"
    using integral_norm_bound_integral[OF f integrable_linear[OF g]]
    by (simp add: bounded_linear_inner_left fg)
  then show ?thesis
    unfolding o_def integral_component_eq[OF g] .
qed

lemma has_integral_norm_bound_integral_component:
  fixes f :: "'n::euclidean_space ==> 'a::banach"
  fixes g :: "'n ==> 'b::euclidean_space"
  assumes "(f has_integral i) S" and "(g has_integral j) S"
    and "∧x. x ∈ S ==> norm (f x) ≤ (g x)∙k"
  shows "norm i ≤ j∙k"
  by (metis assms has_integral_integrable integral_norm_bound_integral_component integral_unique)


lemma uniformly_convergent_improper_integral:
  fixes f :: "'b ==> real ==> 'a :: {banach}"
  assumes deriv: "∧x. x ≥ a ==> (G has_field_derivative g x) (at x within {a..})"
  assumes integrable: "∧a' b x. x ∈ A ==> a' ≥ a ==> b ≥ a' ==> f x integrable_on {a'..b}"
  assumes G: "convergent G"
  assumes le: "∧y x. y ∈ A ==> x ≥ a ==> norm (f y x) ≤ g x"
  shows   "uniformly_convergent_on A (λb x. integral {a..b} (f x))"
proof (intro Cauchy_uniformly_convergent uniformly_Cauchy_onI', goal_cases)
  case (1 ε)
  from G have "Cauchy G"
    by (auto intro!: convergent_Cauchy)
  with 1 obtain M where M: "dist (G (real m)) (G (real n)) < ε" if "m ≥ M" "n ≥ M" for m n
    by (force simp: Cauchy_def)
  define M' where "M' = max (nat ⌈a⌉) M"

  show ?case
  proof (rule exI[of _ M'], safe, goal_cases)
    case (1 x m n)
    have M': "M' ≥ a" "M' ≥ M" unfolding M'_def by linarith+
    have int_g: "(g has_integral (G (real n) - G (real m))) {real m..real n}"
      using 1 M' by (intro fundamental_theorem_of_calculus) 
                    (auto simp: has_real_derivative_iff_has_vector_derivative [symmetric] 
                          intro!: DERIV_subset[OF deriv])
    have int_f: "f x integrable_on {a'..real n}" if "a' ≥ a" for a'
      using that 1 by (cases "a' ≤ real n") (auto intro: integrable)

    have "dist (integral {a..real m} (f x)) (integral {a..real n} (f x)) =
            norm (integral {a..real n} (f x) - integral {a..real m} (f x))"
      by (simp add: dist_norm norm_minus_commute)
    also have "integral {a..real m} (f x) + integral {real m..real n} (f x) =
                 integral {a..real n} (f x)"
      using M' and 1 by (intro integral_combine int_f) auto
    hence "integral {a..real n} (f x) - integral {a..real m} (f x) =
             integral {real m..real n} (f x)"
      by (simp add: algebra_simps)
    also have "norm … ≤ integral {real m..real n} g"
      using le 1 M' int_f int_g by (intro integral_norm_bound_integral) auto 
    also from int_g have "integral {real m..real n} g = G (real n) - G (real m)"
      by (simp add: has_integral_iff)
    also have "… ≤ dist (G m) (G n)" 
      by (simp add: dist_norm)
    also from 1 and M' have "… < ε"
      by (intro M) auto
    finally show ?case .
  qed
qed

lemma uniformly_convergent_improper_integral':
  fixes f :: "'b ==> real ==> 'a :: {banach, real_normed_algebra}"
  assumes deriv: "∧x. x ≥ a ==> (G has_field_derivative g x) (at x within {a..})"
  assumes integrable: "∧a' b x. x ∈ A ==> a' ≥ a ==> b ≥ a' ==> f x integrable_on {a'..b}"
  assumes G: "convergent G"
  assumes le: "eventually (λx. ∀y∈A. norm (f y x) ≤ g x) at_top"
  shows   "uniformly_convergent_on A (λb x. integral {a..b} (f x))"
proof -
  from le obtain a'' where le: "∧y x. y ∈ A ==> x ≥ a'' ==> norm (f y x) ≤ g x"
    by (auto simp: eventually_at_top_linorder)
  define a' where "a' = max a a''"

  have "uniformly_convergent_on A (λb x. integral {a'..real b} (f x))"
  proof (rule uniformly_convergent_improper_integral)
    fix t assume t: "t ≥ a'"
    hence "(G has_field_derivative g t) (at t within {a..})"
      by (intro deriv) (auto simp: a'_def)
    moreover have "{a'..} ⊆ {a..}" unfolding a'_def by auto
    ultimately show "(G has_field_derivative g t) (at t within {a'..})"
      by (rule DERIV_subset)
  qed (insert le, auto simp: a'_def intro: integrable G)
  hence "uniformly_convergent_on A (λb x. integral {a..a'} (f x) + integral {a'..real b} (f x))" 
    (is ?P) by (intro uniformly_convergent_add) auto
  also have "eventually (λx. ∀y∈A. integral {a..a'} (f y) + integral {a'..x} (f y) =
                   integral {a..x} (f y)) sequentially"
    by (intro eventually_mono [OF eventually_ge_at_top[of "nat ⌈a'⌉"]] ballI integral_combine)
       (auto simp: a'_def intro: integrable)
  hence "?P ⟷ ?thesis"
    by (intro uniformly_convergent_cong) simp_all
  finally show ?thesis .
qed

subsection ‹differentiation under the integral sign›

lemma integral_continuous_on_param:
  fixes f::"'a::topological_space ==> 'b::euclidean_space ==> 'c::banach"
  assumes cont_fx: "continuous_on (U × cbox a b) (λ(x, t). f x t)"
  shows "continuous_on U (λx. integral (cbox a b) (f x))"
proof cases
  assume "content (cbox a b) ≠ 0"
  then have ne: "cbox a b ≠ {}" by auto

  note [continuous_intros] =
    continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
      unfolded split_beta fst_conv snd_conv]

  show ?thesis
    unfolding continuous_on_def
  proof (intro strip tendstoI)
    fix e'::real and x
    assume "e' > 0"
    define e where "e = e' / (content (cbox a b) + 1)"
    have "e > 0" using ‹e' > 0› by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
    assume "x ∈ U"
    from continuous_on_prod_compactE[OF cont_fx compact_cbox ‹x ∈ U› ‹0 🚫›]
    obtain X0 where X0: "x ∈ X0" "open X0"
      and fx_bound: "∧y t. y ∈ X0 ∩ U ==> t ∈ cbox a b ==> norm (f y t - f x t) ≤ e"
      unfolding split_beta fst_conv snd_conv dist_norm
      by metis
    have "∀🪙F y in at x within U. y ∈ X0 ∩ U"
      using X0(1) X0(2) eventually_at_topological by auto
    then show "∀🪙F y in at x within U. dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'"
    proof eventually_elim
      case (elim y)
      have "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) =
        norm (integral (cbox a b) (λt. f y t - f x t))"
        using elim ‹x ∈ U›
        unfolding dist_norm
        proof (subst integral_diff)
        qed (auto intro!: integrable_continuous continuous_intros)
      also have "… ≤ e * content (cbox a b)"
        using elim ‹x ∈ U›
        by (intro integrable_bound)
           (auto intro!: fx_bound ‹x ∈ U › less_imp_le[OF ‹0 🚫›]
              integrable_continuous continuous_intros)
      also have "… < e'"
        using ‹0 🚫'› ‹e > 0›
        by (auto simp: e_def field_split_simps)
      finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" .
    qed
  qed
qed (auto intro!: continuous_on_const)

lemma leibniz_rule:
  fixes f::"'a::banach ==> 'b::euclidean_space ==> 'c::banach"
  assumes fx: "∧x t. x ∈ U ==> t ∈ cbox a b ==>
    ((λx. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"
  assumes integrable_f2: "∧x. x ∈ U ==> f x integrable_on cbox a b"
  assumes cont_fx: "continuous_on (U × (cbox a b)) (λ(x, t). fx x t)"
  assumes [intro]: "x0 ∈ U"
  assumes "convex U"
  shows
    "((λx. integral (cbox a b) (f x)) has_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
    (is "(?F has_derivative ?dF) _")
proof cases
  assume "content (cbox a b) ≠ 0"
  then have ne: "cbox a b ≠ {}" by auto
  note [continuous_intros] =
    continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
      unfolded split_beta fst_conv snd_conv]
  show ?thesis
  proof (intro has_derivativeI bounded_linear_scaleR_left tendstoI, fold norm_conv_dist)
    have cont_f1: "∧t. t ∈ cbox a b ==> continuous_on U (λx. f x t)"
      by (auto simp: continuous_on_eq_continuous_within intro!: has_derivative_continuous fx)
    note [continuous_intros] = continuous_on_compose2[OF cont_f1]
    fix e'::real
    assume "e' > 0"
    define e where "e = e' / (content (cbox a b) + 1)"
    have "e > 0" using ‹e' > 0› by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
    from continuous_on_prod_compactE[OF cont_fx compact_cbox ‹x0 ∈ U› ‹e > 0›]
    obtain X0 where X0: "x0 ∈ X0" "open X0"
      and fx_bound: "∧x t. x ∈ X0 ∩ U ==> t ∈ cbox a b ==> norm (fx x t - fx x0 t) ≤ e"
      unfolding split_beta fst_conv snd_conv
      by (metis dist_norm)

    note eventually_closed_segment[OF ‹open X0› ‹x0 ∈ X0›, of U]
    moreover
    have "∀🪙F x in at x0 within U. x ∈ X0"
      using ‹open X0› ‹x0 ∈ X0› eventually_at_topological by blast
    moreover have "∀🪙F x in at x0 within U. x ≠ x0" "∀🪙F x in at x0 within U. x ∈ U"
      by (auto simp: eventually_at_filter)
    ultimately
    show "∀🪙F x in at x0 within U. norm ((?F x - ?F x0 - ?dF (x - x0)) /🪙R norm (x - x0)) < e'"
    proof eventually_elim
      case (elim x)
      from elim have "0 < norm (x - x0)" by simp
      have "closed_segment x0 x ⊆ U"
        by (simp add: assms closed_segment_subset elim(4))
      from elim have [intro]: "x ∈ U" by auto
      have "?F x - ?F x0 - ?dF (x - x0) =
        integral (cbox a b) (λy. f x y - f x0 y - fx x0 y (x - x0))"
        (is "_ = ?id")
        using ‹x ≠ x0›
        by (subst blinfun_apply_integral integral_diff,
            auto intro!: integrable_diff integrable_f2 continuous_intros
              intro: integrable_continuous)+
      also
      have "norm ?id ≤ integral (cbox a b) (λ_. e * norm (x - x0))"
      proof (intro integral_norm_bound_integral)
        fix t assume t: "t ∈ (cbox a b)"
        then have deriv:
          "((λx. f x t) has_derivative (fx y t)) (at y within X0 ∩ U)"
          if "y ∈ X0 ∩ U" for y
          using fx has_derivative_subset that by fastforce
        have seg: "∧t. t ∈ {0..1} ==> x0 + t *🪙R (x - x0) ∈ X0 ∩ U"
          using ‹closed_segment x0 x ⊆ U› ‹closed_segment x0 x ⊆ X0›
          by (force simp: closed_segment_def algebra_simps)
        have "∧x. x ∈ X0 ∩ U ==> onorm (blinfun_apply (fx x t) - (fx x0 t)) ≤ e"
          using fx_bound t
          by (auto simp: norm_blinfun_def fun_diff_def blinfun.bilinear_simps[symmetric])
        from differentiable_bound_linearization[OF seg deriv this] X0
        show "norm (f x t - f x0 t - fx x0 t (x - x0)) ≤ e * norm (x - x0)"
          by (auto simp: ac_simps)
      qed (force intro: continuous_intros integrable_diff integrable_f2 integrable_continuous)+
      also have "… = content (cbox a b) * e * norm (x - x0)"
        by simp
      also have "… < e' * norm (x - x0)"
      proof (intro mult_strict_right_mono[OF _ ‹0 🚫 (x - x0)›])
        show "content (cbox a b) * e < e'"
          using ‹e' > 0› by (simp add: divide_simps e_def not_less)
      qed
      finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .
      then show ?case
        by (auto simp: divide_simps)
    qed
  qed (rule blinfun.bounded_linear_right)
qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps)

lemma has_vector_derivative_eq_has_derivative_blinfun:
  "(f has_vector_derivative f') (at x within U) ⟷
    (f has_derivative blinfun_scaleR_left f') (at x within U)"
  by (simp add: has_vector_derivative_def)

lemma leibniz_rule_vector_derivative:
  fixes f::"real ==> 'b::euclidean_space ==> 'c::banach"
  assumes fx: "∧x t. x ∈ U ==> t ∈ cbox a b ==>
      ((λx. f x t) has_vector_derivative (fx x t)) (at x within U)"
  assumes integrable_f2: "∧x. x ∈ U ==> (f x) integrable_on cbox a b"
  assumes cont_fx: "continuous_on (U × cbox a b) (λ(x, t). fx x t)"
  assumes U: "x0 ∈ U" "convex U"
  shows "((λx. integral (cbox a b) (f x)) has_vector_derivative integral (cbox a b) (fx x0))
      (at x0 within U)"
proof -
  note [continuous_intros] =
    continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
      unfolded split_beta fst_conv snd_conv]
  show ?thesis
    unfolding has_vector_derivative_eq_has_derivative_blinfun
  proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U]])
    show "continuous_on (U × cbox a b) (λ(x, t). blinfun_scaleR_left (fx x t))"
      using cont_fx by (auto simp: split_beta intro!: continuous_intros)
    show "blinfun_apply (integral (cbox a b) (λt. blinfun_scaleR_left (fx x0 t))) =
          blinfun_apply (blinfun_scaleR_left (integral (cbox a b) (fx x0)))"
    by (subst integral_linear[symmetric])
       (auto simp: has_vector_derivative_def o_def
         intro!: integrable_continuous U continuous_intros bounded_linear_intros)
    qed (use fx in ‹auto simp: has_vector_derivative_def›)
qed

lemma has_field_derivative_eq_has_derivative_blinfun:
  "(f has_field_derivative f') (at x within U) ⟷ (f has_derivative blinfun_mult_right f') (at x within U)"
  by (simp add: has_field_derivative_def)

lemma leibniz_rule_field_derivative:
  fixes f::"'a::{real_normed_field, banach} ==> 'b::euclidean_space ==> 'a"
  assumes fx: "∧x t. x ∈ U ==> t ∈ cbox a b ==> ((λx. f x t) has_field_derivative fx x t) (at x within U)"
  assumes integrable_f2: "∧x. x ∈ U ==> (f x) integrable_on cbox a b"
  assumes cont_fx: "continuous_on (U × (cbox a b)) (λ(x, t). fx x t)"
  assumes U: "x0 ∈ U" "convex U"
  shows "((λx. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
proof -
  note [continuous_intros] =
    continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
      unfolded split_beta fst_conv snd_conv]
  have *: "blinfun_mult_right (integral (cbox a b) (fx x0)) =
    integral (cbox a b) (λt. blinfun_mult_right (fx x0 t))"
    by (subst integral_linear[symmetric])
      (auto simp: has_vector_derivative_def o_def
        intro!: integrable_continuous U continuous_intros bounded_linear_intros)
  show ?thesis
    unfolding has_field_derivative_eq_has_derivative_blinfun
  proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U, where fx="λx t. blinfun_mult_right (fx x t)"]])
    show "continuous_on (U × cbox a b) (λ(x, t). blinfun_mult_right (fx x t))"
      using cont_fx by (auto simp: split_beta intro!: continuous_intros)
    show "blinfun_apply (integral (cbox a b) (λt. blinfun_mult_right (fx x0 t))) =
          blinfun_apply (blinfun_mult_right (integral (cbox a b) (fx x0)))"
      by (subst integral_linear[symmetric])
        (auto simp: has_vector_derivative_def o_def
          intro!: integrable_continuous U continuous_intros bounded_linear_intros)
  qed (use fx in ‹auto simp: has_field_derivative_def›)
qed

lemma leibniz_rule_field_differentiable:
  fixes f::"'a::{real_normed_field, banach} ==> 'b::euclidean_space ==> 'a"
  assumes "∧x t. x ∈ U ==> t ∈ cbox a b ==> ((λx. f x t) has_field_derivative fx x t) (at x within U)"
  assumes "∧x. x ∈ U ==> (f x) integrable_on cbox a b"
  assumes "continuous_on (U × (cbox a b)) (λ(x, t). fx x t)"
  assumes "x0 ∈ U" "convex U"
  shows "(λx. integral (cbox a b) (f x)) field_differentiable at x0 within U"
  using leibniz_rule_field_derivative[OF assms] by (auto simp: field_differentiable_def)


subsection ‹Exchange uniform limit and integral›

lemma uniform_limit_integral_cbox:
  fixes f::"'a ==> 'b::euclidean_space ==> 'c::banach"
  assumes u: "uniform_limit (cbox a b) f g F"
  assumes c: "∧n. continuous_on (cbox a b) (f n)"
  assumes [simp]: "F ≠ bot"
  obtains I J where
    "∧n. (f n has_integral I n) (cbox a b)"
    "(g has_integral J) (cbox a b)"
    "(I ---> J) F"
proof -
  have fi[simp]: "f n integrable_on (cbox a b)" for n
    by (auto intro!: integrable_continuous assms)
  then obtain I where I: "∧n. (f n has_integral I n) (cbox a b)"
    unfolding integrable_on_def by metis

  moreover
  have gi[simp]: "g integrable_on (cbox a b)"
    by (auto intro!: integrable_continuous uniform_limit_theorem[OF _ u] eventuallyI c)
  then obtain J where J: "(g has_integral J) (cbox a b)"
    by blast

  moreover
  have "(I ---> J) F"
  proof cases
    assume "content (cbox a b) = 0"
    hence "I = (λ_. 0)" "J = 0"
      by (auto intro!: has_integral_unique I J)
    thus ?thesis by simp
  next
    assume content_nonzero: "content (cbox a b) ≠ 0"
    show ?thesis
    proof (rule tendstoI)
      fix e::real
      assume "e > 0"
      define e' where "e' = e/2"
      with ‹e > 0› have "e' > 0" by simp
      then have "∀🪙F n in F. ∀x∈cbox a b. norm (f n x - g x) < e' / content (cbox a b)"
        using u content_nonzero by (auto simp: uniform_limit_iff dist_norm zero_less_measure_iff)
      then show "∀🪙F n in F. dist (I n) J < e"
      proof eventually_elim
        case (elim n)
        have "I n = integral (cbox a b) (f n)" "J = integral (cbox a b) g"
          using I[of n] J by (simp_all add: integral_unique)
        then have "dist (I n) J = norm (integral (cbox a b) (λx. f n x - g x))"
          by (simp add: integral_diff dist_norm)
        also have "… ≤ integral (cbox a b) (λx. (e' / content (cbox a b)))"
          using elim
          by (intro integral_norm_bound_integral) (auto intro!: integrable_diff)
        also have "… < e"
          using ‹0 🚫› by (simp add: e'_def)
        finally show ?case .
      qed
    qed
  qed
  ultimately show ?thesis ..
qed

lemma uniform_limit_integral:
  fixes f::"'a ==> 'b::ordered_euclidean_space ==> 'c::banach"
  assumes u: "uniform_limit {a..b} f g F"
  assumes c: "∧n. continuous_on {a..b} (f n)"
  assumes [simp]: "F ≠ bot"
  obtains I J where
    "∧n. (f n has_integral I n) {a..b}"
    "(g has_integral J) {a..b}"
    "(I ---> J) F"
  by (metis interval_cbox assms uniform_limit_integral_cbox)


subsection ‹Integration by parts›

lemma integration_by_parts_interior_strong:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes bilinear: "bounded_bilinear (prod)"
  assumes s: "finite s" and le: "a ≤ b"
  assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes deriv: "∧x. x∈{a<..==> (f has_vector_derivative f' x) (at x)"
                 "∧x. x∈{a<..==> (g has_vector_derivative g' x) (at x)"
  assumes int: "((λx. prod (f x) (g' x)) has_integral
                  (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
  shows   "((λx. prod (f' x) (g x)) has_integral y) {a..b}"
proof -
  interpret bounded_bilinear prod by fact
  have "((λx. prod (f x) (g' x) + prod (f' x) (g x)) has_integral
          (prod (f b) (g b) - prod (f a) (g a))) {a..b}"
    using deriv by (intro fundamental_theorem_of_calculus_interior_strong[OF s le])
                   (auto intro!: continuous_intros continuous_on has_vector_derivative)
  from has_integral_diff[OF this int] show ?thesis by (simp add: algebra_simps)
qed

lemma integration_by_parts_interior:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes "bounded_bilinear (prod)" "a ≤ b"
          "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes "∧x. x∈{a<..==> (f has_vector_derivative f' x) (at x)"
          "∧x. x∈{a<..==> (g has_vector_derivative g' x) (at x)"
  assumes "((λx. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
  shows   "((λx. prod (f' x) (g x)) has_integral y) {a..b}"
  by (rule integration_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (use assms in simp_all)

lemma integration_by_parts:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes "bounded_bilinear (prod)" "a ≤ b"
          "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes "∧x. x∈{a..b} ==> (f has_vector_derivative f' x) (at x)"
          "∧x. x∈{a..b} ==> (g has_vector_derivative g' x) (at x)"
  assumes "((λx. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
  shows   "((λx. prod (f' x) (g x)) has_integral y) {a..b}"
  by (rule integration_by_parts_interior[of _ _ _ f g f' g']) (use assms in simp_all)

lemma integrable_by_parts_interior_strong:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes bilinear: "bounded_bilinear (prod)"
  assumes s: "finite s" and le: "a ≤ b"
  assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes deriv: "∧x. x∈{a<..==> (f has_vector_derivative f' x) (at x)"
                 "∧x. x∈{a<..==> (g has_vector_derivative g' x) (at x)"
  assumes int: "(λx. prod (f x) (g' x)) integrable_on {a..b}"
  shows   "(λx. prod (f' x) (g x)) integrable_on {a..b}"
proof -
  from int obtain I where "((λx. prod (f x) (g' x)) has_integral I) {a..b}"
    unfolding integrable_on_def by blast
  hence "((λx. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) -
           (prod (f b) (g b) - prod (f a) (g a) - I))) {a..b}" by simp
  from integration_by_parts_interior_strong[OF assms(1-7) this]
    show ?thesis unfolding integrable_on_def by blast
qed

lemma integrable_by_parts_interior:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes "bounded_bilinear (prod)" "a ≤ b"
          "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes "∧x. x∈{a<..==> (f has_vector_derivative f' x) (at x)"
          "∧x. x∈{a<..==> (g has_vector_derivative g' x) (at x)"
  assumes "(λx. prod (f x) (g' x)) integrable_on {a..b}"
  shows   "(λx. prod (f' x) (g x)) integrable_on {a..b}"
  by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (use assms in simp_all)

lemma integrable_by_parts:
  fixes prod :: "_ ==> _ ==> 'b :: banach"
  assumes "bounded_bilinear (prod)" "a ≤ b"
          "continuous_on {a..b} f" "continuous_on {a..b} g"
  assumes "∧x. x∈{a..b} ==> (f has_vector_derivative f' x) (at x)"
          "∧x. x∈{a..b} ==> (g has_vector_derivative g' x) (at x)"
  assumes "(λx. prod (f x) (g' x)) integrable_on {a..b}"
  shows   "(λx. prod (f' x) (g x)) integrable_on {a..b}"
  by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (use assms in simp_all)


subsection ‹Integration by substitution›

lemma has_integral_substitution_general:
  fixes f :: "real ==> 'a::euclidean_space" and g :: "real ==> real"
  assumes s: "finite s" and le: "a ≤ b"
      and subset: "g ` {a..b} ⊆ {c..d}"
      and f [continuous_intros]: "continuous_on {c..d} f"
      and g [continuous_intros]: "continuous_on {a..b} g"
      and deriv [derivative_intros]:
              "∧x. x ∈ {a..b} - s ==> (g has_field_derivative g' x) (at x within {a..b})"
    shows "((λx. g' x *🪙R f (g x)) has_integral (integral {g a..g b} f - integral {g b..g a} f)) {a..b}"
proof -
  let ?F = "λx. integral {c..g x} f"
  have cont_int: "continuous_on {a..b} ?F"
    by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous_1
          f integrable_continuous_real)+
  have deriv: "(((λx. integral {c..x} f) ∘ g) has_vector_derivative g' x *🪙R f (g x))
                 (at x within {a..b})" if "x ∈ {a..b} - s" for x
  proof (rule has_vector_derivative_eq_rhs [OF vector_diff_chain_within refl])
    show "(g has_vector_derivative g' x) (at x within {a..b})"
      using deriv has_real_derivative_iff_has_vector_derivative that by blast
    show "((λx. integral {c..x} f) has_vector_derivative f (g x))
          (at (g x) within g ` {a..b})"
      using that le subset
      by (blast intro: has_vector_derivative_within_subset integral_has_vector_derivative f)
  qed
  have deriv: "(?F has_vector_derivative g' x *🪙R f (g x))
                  (at x)" if "x ∈ {a..b} - (s ∪ {a,b})" for x
    using deriv[of x] that by (simp add: at_within_Icc_at o_def)
  have "((λx. g' x *🪙R f (g x)) has_integral (?F b - ?F a)) {a..b}"
    using le cont_int s deriv cont_int
    by (intro fundamental_theorem_of_calculus_interior_strong[of "s ∪ {a,b}"]) simp_all
  also
  from subset have "g x ∈ {c..d}" if "x ∈ {a..b}" for x using that by blast
  from this[of a] this[of b] le have cd: "c ≤ g a" "g b ≤ d" "c ≤ g b" "g a ≤ d" by auto
  have "integral {c..g b} f - integral {c..g a} f = integral {g a..g b} f - integral {g b..g a} f"
  proof cases
    assume "g a ≤ g b"
    note le = le this
    from cd have "integral {c..g a} f + integral {g a..g b} f = integral {c..g b} f"
      by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
    with le show ?thesis
      by (cases "g a = g b") (simp_all add: algebra_simps)
  next
    assume less: "¬g a ≤ g b"
    then have "g a ≥ g b" by simp
    note le = le this
    from cd have "integral {c..g b} f + integral {g b..g a} f = integral {c..g a} f"
      by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
    with less show ?thesis
      by (simp_all add: algebra_simps)
  qed
  finally show ?thesis .
qed

lemma has_integral_substitution_strong:
  fixes f :: "real ==> 'a::euclidean_space" and g :: "real ==> real"
  assumes s: "finite s" and le: "a ≤ b" "g a ≤ g b"
    and subset: "g ` {a..b} ⊆ {c..d}"
    and f [continuous_intros]: "continuous_on {c..d} f"
    and g [continuous_intros]: "continuous_on {a..b} g"
    and deriv [derivative_intros]:
    "∧x. x ∈ {a..b} - s ==> (g has_field_derivative g' x) (at x within {a..b})"
  shows "((λx. g' x *🪙R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
  using has_integral_substitution_general[OF s le(1) subset f g deriv] le(2)
  by (cases "g a = g b") auto

lemma has_integral_substitution:
  fixes f :: "real ==> 'a::euclidean_space" and g :: "real ==> real"
  assumes "a ≤ b" "g a ≤ g b" "g ` {a..b} ⊆ {c..d}"
      and "continuous_on {c..d} f"
      and "∧x. x ∈ {a..b} ==> (g has_field_derivative g' x) (at x within {a..b})"
    shows "((λx. g' x *🪙R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
proof (intro has_integral_substitution_strong[of "{}" a b g c d] assms)
qed (auto intro: DERIV_continuous_on assms)

lemma integral_shift:
  fixes f :: "real ==> 'a::euclidean_space"
  assumes cont: "continuous_on {a + c..b + c} f"
  shows "integral {a..b} (f ∘ (λx. x + c)) = integral {a + c..b + c} f"
proof (cases "a ≤ b")
  case True
  have "((λx. 1 *🪙R f (x + c)) has_integral integral {a+c..b+c} f) {a..b}"
    using True cont
    by (intro has_integral_substitution[where c = "a + c" and d = "b + c"])
       (auto intro!: derivative_eq_intros)
  thus ?thesis by (simp add: has_integral_iff o_def)
qed auto


subsection ‹Compute a double integral using iterated integrals and switching the order of integration›

lemma continuous_on_imp_integrable_on_Pair1:
  fixes f :: "_ ==> 'b::banach"
  assumes con: "continuous_on (cbox (a,c) (b,d)) f" and x: "x ∈ cbox a b"
  shows "(λy. f (x, y)) integrable_on (cbox c d)"
proof -
  have "f ∘ (λy. (x, y)) integrable_on (cbox c d)"
  proof (intro integrable_continuous continuous_on_compose [OF _ continuous_on_subset [OF con]])
    show "continuous_on (cbox c d) (Pair x)"
      by (simp add: continuous_on_Pair)
    show "Pair x ` cbox c d ⊆ cbox (a,c) (b,d)"
      using x by blast
  qed
  then show ?thesis
    by (simp add: o_def)
qed

lemma integral_integrable_2dim:
  fixes f :: "('a::euclidean_space * 'b::euclidean_space) ==> 'c::banach"
  assumes "continuous_on (cbox (a,c) (b,d)) f"
    shows "(λx. integral (cbox c d) (λy. f (x,y))) integrable_on cbox a b"
proof (cases "content(cbox c d) = 0")
case True
  then show ?thesis
    by (simp add: True integrable_const)
next
  case False
  have uc: "uniformly_continuous_on (cbox (a,c) (b,d)) f"
    by (simp add: assms compact_cbox compact_uniformly_continuous)
  { fix x::'a and e::real
    assume x: "x ∈ cbox a b" and e: "0 < e"
    then have e2_gt: "0 < e/2 / content (cbox c d)" and e2_less: "e/2 / content (cbox c d) * content (cbox c d) < e"
      by (auto simp: False content_lt_nz e)
    then obtain dd
    where dd: "∧x x'. [x∈cbox (a, c) (b, d); x'∈cbox (a, c) (b, d); norm (x' - x) < dd]
                       ==> norm (f x' - f x) ≤ e/(2 * content (cbox c d))"  "dd>0"
      using uc [unfolded uniformly_continuous_on_def, THEN spec, of "e/(2 * content (cbox c d))"]
      by (auto simp: dist_norm intro: less_imp_le)
    have "∃delta>0. ∀x'∈cbox a b. norm (x' - x) < delta ⟶ norm (integral (cbox c d) (λu. f (x', u) - f (x, u))) < e"
      using dd e2_gt assms x
      apply (rule_tac x=dd in exI)
      apply clarify
      apply (rule le_less_trans [OF integrable_bound e2_less])
      apply (auto intro: integrable_diff continuous_on_imp_integrable_on_Pair1)
      done
  } note * = this
  show ?thesis
  proof (rule integrable_continuous)
    show "continuous_on (cbox a b) (λx. integral (cbox c d) (λy. f (x, y)))"
      by (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms])
  qed
qed

lemma integral_split:
  fixes f :: "'a::euclidean_space ==> 'b::{real_normed_vector,complete_space}"
  assumes f: "f integrable_on (cbox a b)"
      and k: "k ∈ Basis"
  shows "integral (cbox a b) f =
           integral (cbox a b ∩ {x. x∙k ≤ c}) f +
           integral (cbox a b ∩ {x. x∙k ≥ c}) f"
  using k f
  by (auto simp: has_integral_integral intro: integral_unique [OF has_integral_split])

lemma integral_swap_operativeI:
  fixes f :: "('a::euclidean_space * 'b::euclidean_space) ==> 'c::banach"
  assumes f: "continuous_on s f" and e: "0 < e"
    shows "operative conj True
           (λk. ∀a b c d.
                cbox (a,c) (b,d) ⊆ k ∧ cbox (a,c) (b,d) ⊆ s
                ⟶ norm(integral (cbox (a,c) (b,d)) f -
                         integral (cbox a b) (λx. integral (cbox c d) (λy. f((x,y)))))
                    ≤ e * content (cbox (a,c) (b,d)))"
proof (standard, auto)
  fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b
  assume *: "box (a, c) (b, d) = {}"
     and cb1: "cbox (u, w) (v, z) ⊆ cbox (a, c) (b, d)"
     and cb2: "cbox (u, w) (v, z) ⊆ s"
  then have c0: "content (cbox (a, c) (b, d)) = 0"
    using * unfolding content_eq_0_interior by simp
  have c0': "content (cbox (u, w) (v, z)) = 0"
    by (fact content_0_subset [OF c0 cb1])
  show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))))
          ≤ e * content (cbox (u,w) (v,z))"
    using content_cbox_pair_eq0_D [OF c0']
    by (force simp: c0')
next
  fix a::'a and c::'b and b::'a and d::'b
  and M::real and i::'a and j::'b
  and u::'a and v::'a and w::'b and z::'b
  assume ij: "(i,j) ∈ Basis"
     and n1: "∀a' b' c' d'.
                cbox (a',c') (b',d') ⊆ cbox (a,c) (b,d) ∧
                cbox (a',c') (b',d') ⊆ {x. x ∙ (i,j) ≤ M} ∧ cbox (a',c') (b',d') ⊆ s ⟶
                norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (λx. integral (cbox c' d') (λy. f (x,y))))
                ≤ e * content (cbox (a',c') (b',d'))"
     and n2: "∀a' b' c' d'.
                cbox (a',c') (b',d') ⊆ cbox (a,c) (b,d) ∧
                cbox (a',c') (b',d') ⊆ {x. M ≤ x ∙ (i,j)} ∧ cbox (a',c') (b',d') ⊆ s ⟶
                norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (λx. integral (cbox c' d') (λy. f (x,y))))
                ≤ e * content (cbox (a',c') (b',d'))"
     and subs: "cbox (u,w) (v,z) ⊆ cbox (a,c) (b,d)"  "cbox (u,w) (v,z) ⊆ s"
  have fcont: "continuous_on (cbox (u, w) (v, z)) f"
    using assms(1) continuous_on_subset  subs(2) by blast
  then have fint: "f integrable_on cbox (u, w) (v, z)"
    by (metis integrable_continuous)
  consider "i ∈ Basis" "j=0" | "j ∈ Basis" "i=0"  using ij
    by (auto simp: Euclidean_Space.Basis_prod_def)
  then show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))))
             ≤ e * content (cbox (u,w) (v,z))" (is ?normle)
  proof cases
    case 1
    then have e: "e * content (cbox (u, w) (v, z)) =
                  e * (content (cbox u v ∩ {x. x ∙ i ≤ M}) * content (cbox w z)) +
                  e * (content (cbox u v ∩ {x. M ≤ x ∙ i}) * content (cbox w z))"
      by (simp add: content_split [where c=M] content_Pair algebra_simps)
    have *: "integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))) =
                integral (cbox u v ∩ {x. x ∙ i ≤ M}) (λx. integral (cbox w z) (λy. f (x, y))) +
                integral (cbox u v ∩ {x. M ≤ x ∙ i}) (λx. integral (cbox w z) (λy. f (x, y)))"
      using 1 f subs integral_integrable_2dim continuous_on_subset
      by (blast intro: integral_split)
    show ?normle
      apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
      using 1 subs
      apply (simp_all add: cbox_Pair_eq setcomp_dot1 [of "λu. M≤u"] setcomp_dot1 [of "λu. u≤M"] Sigma_Int_Paircomp1)
      apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
      apply (force simp flip: interval_split intro!: n1 [rule_format])
      apply (force simp flip: interval_split intro!: n2 [rule_format])
      done
  next
    case 2
    then have e: "e * content (cbox (u, w) (v, z)) =
                  e * (content (cbox u v) * content (cbox w z ∩ {x. x ∙ j ≤ M})) +
                  e * (content (cbox u v) * content (cbox w z ∩ {x. M ≤ x ∙ j}))"
      by (simp add: content_split [where c=M] content_Pair algebra_simps)
    have "(λx. integral (cbox w z ∩ {x. x ∙ j ≤ M}) (λy. f (x, y))) integrable_on cbox u v"
         "(λx. integral (cbox w z ∩ {x. M ≤ x ∙ j}) (λy. f (x, y))) integrable_on cbox u v"
      using 2 subs
      apply (simp_all add: interval_split)
      apply (rule integral_integrable_2dim [OF continuous_on_subset [OF f]]; auto simp: cbox_Pair_eq interval_split [symmetric])+
      done
    with 2 have *: "integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))) =
                   integral (cbox u v) (λx. integral (cbox w z ∩ {x. x ∙ j ≤ M}) (λy. f (x, y))) +
                   integral (cbox u v) (λx. integral (cbox w z ∩ {x. M ≤ x ∙ j}) (λy. f (x, y)))"
      by (simp add: integral_add [symmetric] integral_split [symmetric]
                    continuous_on_imp_integrable_on_Pair1 [OF fcont] cong: integral_cong)
    show ?normle
      apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
      using 2 subs
      apply (simp_all add: cbox_Pair_eq setcomp_dot2 [of "λu. M≤u"] setcomp_dot2 [of "λu. u≤M"] Sigma_Int_Paircomp2)
      apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
      apply (force simp flip: interval_split intro!: n1 [rule_format])
      apply (force simp flip: interval_split intro!: n2 [rule_format])
      done
  qed
qed

lemma integral_Pair_const:
    "integral (cbox (a,c) (b,d)) (λx. k) =
     integral (cbox a b) (λx. integral (cbox c d) (λy. k))"
  by (simp add: content_Pair)

lemma integral_prod_continuous:
  fixes f :: "('a::euclidean_space * 'b::euclidean_space) ==> 'c::banach"
  assumes "continuous_on (cbox (a, c) (b, d)) f" (is "continuous_on ?CBOX f")
    shows "integral (cbox (a, c) (b, d)) f = integral (cbox a b) (λx. integral (cbox c d) (λy. f (x, y)))"
proof (cases "content ?CBOX = 0")
  case True
  then show ?thesis
    by (auto simp: content_Pair)
next
  case False
  then have cbp: "content ?CBOX > 0"
    using content_lt_nz by blast
  have "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) = 0"
  proof (rule dense_eq0_I, simp)
    fix e :: real 
    assume "0 < e"
    with ‹content ?CBOX > 0› have "0 < e/content ?CBOX"
      by simp
    have f_int_acbd: "f integrable_on ?CBOX"
      by (rule integrable_continuous [OF assms])
    { fix p
      assume p: "p division_of ?CBOX"
      then have "finite p"
        by blast
      define e' where "e' = e/content ?CBOX"
      with ‹0 🚫› ‹0 🚫/content ?CBOX›
      have "0 < e'"
        by simp
      interpret operative conj True
           "λk. ∀a' b' c' d'.
                cbox (a', c') (b', d') ⊆ k ∧ cbox (a', c') (b', d') ⊆ ?CBOX
                ⟶ norm (integral (cbox (a', c') (b', d')) f -
                         integral (cbox a' b') (λx. integral (cbox c' d') (λy. f ((x, y)))))
                    ≤ e' * content (cbox (a', c') (b', d'))"
        using assms ‹0 🚫'› by (rule integral_swap_operativeI)
      have "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x, y))))
          ≤ e' * content ?CBOX"
        if "∧t u v w z. t ∈ p ==> cbox (u, w) (v, z) ⊆ t ==> cbox (u, w) (v, z) ⊆ ?CBOX
          ==> norm (integral (cbox (u, w) (v, z)) f -
              integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))))
              ≤ e' * content (cbox (u, w) (v, z))"
        using that division [of p "(a, c)" "(b, d)"] p ‹finite p› by (auto simp: comm_monoid_set_F_and)
      with False have "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x, y))))
          ≤ e"
        if "∧t u v w z. t ∈ p ==> cbox (u, w) (v, z) ⊆ t ==> cbox (u, w) (v, z) ⊆ ?CBOX
          ==> norm (integral (cbox (u, w) (v, z)) f -
              integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))))
              ≤ e * content (cbox (u, w) (v, z)) / content ?CBOX"
        using that by (simp add: e'_def)
    } note op_acbd = this
    { fix k::real and D and u::'a and v w and z::'b and t1 t2 l
      let ?CBUZ = "cbox (u,w) (v,z)"
      assume k: "0 < k"
         and nf: "∧x y u v.
                  [x ∈ cbox a b; y ∈ cbox c d; u ∈ cbox a b; v∈cbox c d; norm (u-x, v-y) < k]
                  ==> norm (f(u,v) - f(x,y)) < e/(2 * (content ?CBOX))"
         and p_acbd: "D tagged_division_of cbox (a,c) (b,d)"
         and fine: "(λx. ball x k) fine D"  "((t1,t2), l) ∈ D"
         and uwvz_sub: "?CBUZ ⊆ l" "?CBUZ ⊆ cbox (a,c) (b,d)"
      have t: "t1 ∈ cbox a b" "t2 ∈ cbox c d"
        by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+
      have ls: "l ⊆ ball (t1,t2) k"
        using fine by (simp add: fine_def Ball_def)
      { fix x1 x2
        assume xuvwz: "x1 ∈ cbox u v" "x2 ∈ cbox w z"
        then have x: "x1 ∈ cbox a b" "x2 ∈ cbox c d"
          using uwvz_sub by auto
        have "norm (x1 - t1, x2 - t2) = norm (t1 - x1, t2 - x2)"
          by (simp add: norm_Pair norm_minus_commute)
        also have "norm (t1 - x1, t2 - x2) < k"
          using xuvwz ls uwvz_sub unfolding ball_def
          by (force simp: cbox_Pair_eq dist_norm )
        finally have "norm (f (x1,x2) - f (t1,t2)) ≤ e/content ?CBOX/2"
          using nf [OF t x]  by simp
      } note nf' = this
      have f_int_uwvz: "f integrable_on ?CBUZ"
        using f_int_acbd uwvz_sub integrable_on_subcbox by blast
      have f_int_uv: "∧x. x ∈ cbox u v ==> (λy. f (x,y)) integrable_on cbox w z"
        using assms continuous_on_subset uwvz_sub
        by (blast intro: continuous_on_imp_integrable_on_Pair1)
      have 1: "norm (integral ?CBUZ f - integral ?CBUZ (λx. f (t1,t2)))
             ≤ e * content ?CBUZ / content ?CBOX/2"
        using cbp ‹0 🚫/content ?CBOX› nf'
        apply (simp only: integral_diff [symmetric] f_int_uwvz integrable_const)
        apply (auto simp: integrable_diff f_int_uwvz integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
        done
      have int_integrable: "(λx. integral (cbox w z) (λy. f (x, y))) integrable_on cbox u v"
        using assms integral_integrable_2dim continuous_on_subset uwvz_sub(2) by blast
      have normint_wz:
         "∧x. x ∈ cbox u v ==>
               norm (integral (cbox w z) (λy. f (x, y)) - integral (cbox w z) (λy. f (t1, t2)))
               ≤ e * content (cbox w z) / content (cbox (a, c) (b, d))/2"
        using cbp ‹0 🚫/content ?CBOX› nf'
        apply (simp only: integral_diff [symmetric] f_int_uv integrable_const)
        apply (auto simp: integrable_diff f_int_uv integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
        done
      have "norm (integral (cbox u v)
               (λx. integral (cbox w z) (λy. f (x,y)) - integral (cbox w z) (λy. f (t1,t2))))
            ≤ e * content (cbox w z) / content ?CBOX/2 * content (cbox u v)"
        using cbp ‹0 🚫/content ?CBOX›
        apply (intro integrable_bound [OF _ _ normint_wz])
        apply (auto simp: field_split_simps integrable_diff int_integrable integrable_const)
        done
      also have "… ≤ e * content ?CBUZ / content ?CBOX/2"
        by (simp add: content_Pair field_split_simps)
      finally have 2: "norm (integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))) -
                      integral (cbox u v) (λx. integral (cbox w z) (λy. f (t1,t2))))
                ≤ e * content ?CBUZ / content ?CBOX/2"
        by (simp only: integral_diff [symmetric] int_integrable integrable_const)
      have norm_xx: "[x' = y'; norm(x - x') ≤ e/2; norm(y - y') ≤ e/2] ==> norm(x - y) ≤ e" for x::'c and y x' y' e
        using norm_triangle_mono [of "x-y'" "e/2" "y'-y" "e/2"] field_sum_of_halves
        by (simp add: norm_minus_commute)
      have "norm (integral ?CBUZ f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))))
            ≤ e * content ?CBUZ / content ?CBOX"
        by (rule norm_xx [OF integral_Pair_const 1 2])
    } note * = this
    have "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) ≤ e" 
      if "∀x∈?CBOX. ∀x'∈?CBOX. norm (x' - x) < k ⟶ norm (f x' - f x) < e /(2 * content (?CBOX))" "0 < k" for k
    proof -
      obtain p where ptag: "p tagged_division_of cbox (a, c) (b, d)" 
                 and fine: "(λx. ball x k) fine p"
        using fine_division_exists ‹0 🚫› by blast
      show ?thesis
        using that fine ptag ‹0 🚫›
        by (auto simp: * intro: op_acbd [OF division_of_tagged_division [OF ptag]])
    qed
    then show "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) ≤ e"
      using compact_uniformly_continuous [OF assms compact_cbox]
      apply (simp add: uniformly_continuous_on_def dist_norm)
      apply (drule_tac x="e/2 / content?CBOX" in spec)
      using cbp ‹0 🚫› by (auto simp: zero_less_mult_iff)
  qed
  then show ?thesis
    by simp
qed

lemma integral_swap_2dim:
  fixes f :: "['a::euclidean_space, 'b::euclidean_space] ==> 'c::banach"
  assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
    shows "integral (cbox (a, c) (b, d)) (λ(x, y). f x y) = integral (cbox (c, a) (d, b)) (λ(x, y). f y x)"
proof -
  have "((λ(x, y). f x y) has_integral integral (cbox (c, a) (d, b)) (λ(x, y). f y x)) (prod.swap ` (cbox (c, a) (d, b)))"
  proof (rule has_integral_twiddle [of 1 prod.swap prod.swap "λ(x,y). f y x" "integral (cbox (c, a) (d, b)) (λ(x, y). f y x)", simplified])
    show "∧u v. content (prod.swap ` cbox u v) = content (cbox u v)"
      by (metis content_Pair mult.commute old.prod.exhaust swap_cbox_Pair)
    show "((λ(x, y). f y x) has_integral integral (cbox (c, a) (d, b)) (λ(x, y). f y x)) (cbox (c, a) (d, b))"
      by (simp add: assms integrable_continuous integrable_integral swap_continuous)
  qed (use isCont_swap in ‹fastforce+›)
 then show ?thesis
   by force
qed

theorem integral_swap_continuous:
  fixes f :: "['a::euclidean_space, 'b::euclidean_space] ==> 'c::banach"
  assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
    shows "integral (cbox a b) (λx. integral (cbox c d) (f x)) =
           integral (cbox c d) (λy. integral (cbox a b) (λx. f x y))"
proof -
  have "integral (cbox a b) (λx. integral (cbox c d) (f x)) = integral (cbox (a, c) (b, d)) (λ(x, y). f x y)"
    using integral_prod_continuous [OF assms] by auto
  also have "… = integral (cbox (c, a) (d, b)) (λ(x, y). f y x)"
    by (rule integral_swap_2dim [OF assms])
  also have "… = integral (cbox c d) (λy. integral (cbox a b) (λx. f x y))"
    using integral_prod_continuous [OF swap_continuous] assms
    by auto
  finally show ?thesis .
qed


subsection ‹Definite integrals for exponential and power function›

text ‹This lemma packages up a reference to @{thm[source]monotone_convergence_increasing}›
lemma has_integral_to_inf:
  fixes h ::"real ==> real"
  assumes int: "∧y::real. h integrable_on {a..y}"
    and lim: "((λy. integral {a..y} h) ---> l) at_top"
    and nonneg: "∧y. y ≥ a ==> h y ≥ 0"
  shows "(h has_integral l) {a..}"
proof -
  have ge: "integral {a..y} h ≥ 0" for y
    by (meson Henstock_Kurzweil_Integration.integral_nonneg atLeastAtMost_iff int nonneg)
  then have "l ≥ 0"
    using tendsto_lowerbound [OF lim] by simp 
  have "mono (λy. integral {a..y} h)"
    by (simp add: int integral_subset_le monoI nonneg)
  then have int_le_l: "integral {a..y} h ≤ l" for y
    using order_tendstoD [OF lim, of "integral {a..y} h"]
    by (smt (verit) eventually_at_top_linorder monotoneD nle_le)
  define f where "f ≡ λn x. if x ∈ {a..of_nat n} then h x else 0"
  have has_integral_f: "∧n. (f n has_integral (integral {a..of_nat n} h)) {a..}"
    unfolding f_def
    by (metis (no_types, lifting) ext Icc_subset_Ici_iff order.refl
        has_integral_restrict int integrable_integral)

  have integral_f: "integral {a..} (f n) = (if n ≥ a then integral {a..of_nat n} h else 0)" for n
    by (meson atLeastAtMost_iff f_def has_integral_f has_integral_iff has_integral_is_0 order_trans)
  have *: "h integrable_on {a..} ∧ (λn. integral {a..} (f n)) <---- integral {a..} h"
  proof (intro monotone_convergence_increasing allI ballI)
    fix n
    show "f n integrable_on {a..}"
      using has_integral_f by blast
  next
    fix n x
    show "f n x ≤ f (Suc n) x" using nonneg by (auto simp: f_def)
  next
    fix x :: real assume x: "x ∈ {a..}"
    have "eventually (λn. real n ≥ x) at_top"
      by (meson eventually_sequentiallyI nat_ceiling_le_eq)
    with x have "eventually (λn. f n x = h x) at_top"
      by (simp add: eventually_mono f_def)
    thus "(λn. f n x) <---- h x" by (simp add: tendsto_eventually)
  next
    have "norm (integral {a..} (f n)) ≤ l" for n
      by (simp add: ‹0 ≤ l› ge int_le_l integral_f)
    thus "bounded (range(λk. integral {a..} (f k)))"
      by (metis (no_types, lifting) boundedI rangeE)
  qed
  have "eventually (λn. integral {a..n} h = integral {a..} (f n)) at_top"
    by (metis (mono_tags, lifting) eventuallyI has_integral_f integral_unique)
  moreover have "((λy. integral {a..y} h) ∘ real) <---- l"
    unfolding tendsto_compose_filtermap
    using filterlim_def filterlim_real_sequentially lim tendsto_mono by blast
  ultimately have "(λn. integral {a..} (f n)) <---- l"
    by (force intro: Lim_transform_eventually)
  then show ?thesis
    using "*" LIMSEQ_unique by blast
qed

lemma has_integral_exp_minus_to_infinity:
  assumes "a > 0"
  shows   "((λx::real. exp (-a*x)) has_integral exp (-a*c)/a) {c..}"
proof (intro has_integral_to_inf integrable_continuous_interval continuous_intros)
  have "((λx. exp (-a*x)) has_integral (-exp (-a*y)/a - (-exp (-a*c)/a))) {c..y}"
    if "y ≥ c" for y
    using that ‹a > 0›
      by (intro fundamental_theorem_of_calculus)
         (auto intro!: derivative_eq_intros
               simp flip: has_real_derivative_iff_has_vector_derivative)    
  then have "∀🪙F y in at_top. integral {c..y} (λx. exp (-a*x)) = -exp (-a*y)/a + exp (-a*c)/a"
    using eventually_at_top_linorder[of
        "λy. integral {c..y} (λx. exp (-a*x)) = -exp (-a*y)/a + exp (-a*c)/a"]
    by auto
  moreover have "((λy::real. -exp (-a*y)/a + exp (-a*c)/a) ---> exp (-a*c)/a) at_top"
    using ‹a > 0› by real_asymp
  ultimately show "((λy. integral {c..y} (λx. exp (-a*x))) ---> exp (-a*c)/a) at_top"
    by (simp add: filterlim_cong)
qed auto

lemma integrable_on_exp_minus_to_infinity: "a > 0 ==> (λx. exp (-a*x) :: real) integrable_on {c..}"
  using has_integral_exp_minus_to_infinity[of a c] unfolding integrable_on_def by blast

lemma has_integral_powr_from_0:
  assumes a: "a > (-1::real)" and c: "c ≥ 0"
  shows   "((λx. x powr a) has_integral (c powr (a+1) / (a+1))) {0..c}"
proof (cases "c = 0")
  case False
  define f where "f = (λk x. if x ∈ {inverse (of_nat (Suc k))..c} then x powr a else 0)"
  define F where "F = (λk. if inverse (of_nat (Suc k)) ≤ c then
                             c powr (a+1)/(a+1) - inverse (real (Suc k)) powr (a+1)/(a+1) else 0)"
  have has_integral_f: "(f k has_integral F k) {0..c}" for k::nat
  proof (cases "inverse (of_nat (Suc k)) ≤ c")
    case True
    have x: "x > 0" if "x ≥ inverse (1 + real k)" for x
      by (metis inverse_Suc of_nat_Suc order_less_le_trans that)
    hence "((λx. x powr a) has_integral c powr (a + 1) / (a + 1) -
               inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}"
      using True a 
    proof (intro fundamental_theorem_of_calculus)
    qed (auto intro!: derivative_eq_intros continuous_on_powr' continuous_on_const
          simp flip: has_real_derivative_iff_has_vector_derivative)
    with True show ?thesis unfolding f_def F_def by (subst has_integral_restrict) simp_all
  next
    case False
    thus ?thesis unfolding f_def F_def
      by force 
  qed
  then have integral_f: "integral {0..c} (f k) = F k" for k
    by blast

  have A: "(λx. x powr a) integrable_on {0..c} ∧
           (λk. integral {0..c} (f k)) <---- integral {0..c} (λx. x powr a)"
  proof (intro monotone_convergence_increasing ballI allI)
    fix k from has_integral_f[of k] show "f k integrable_on {0..c}"
      by (auto simp: integrable_on_def)
  next
    fix k :: nat and x :: real
    {
      assume x: "inverse (real (Suc k)) ≤ x"
      then have "inverse (real (Suc (Suc k))) ≤ x"
        using dual_order.trans by fastforce 
    }
    thus "f k x ≤ f (Suc k) x" by (auto simp: f_def simp del: of_nat_Suc)
  next
    fix x assume x: "x ∈ {0..c}"
    show "(λk. f k x) <---- x powr a"
    proof (cases "x = 0")
      case False
      with x have "x > 0" by simp
      from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
        have "eventually (λk. x powr a = f k x) sequentially"
        by eventually_elim (insert x, simp add: f_def)
      moreover have "(λ_. x powr a) <---- x powr a" by simp
      ultimately show ?thesis by (blast intro: Lim_transform_eventually)
    qed (simp_all add: f_def)
  next
    {
      fix k
      from a have "F k ≤ c powr (a + 1) / (a + 1)"
        by (auto simp: F_def divide_simps)
      also from a have "F k ≥ 0"
        by (auto simp: F_def divide_simps simp del: of_nat_Suc intro!: powr_mono2)
      hence "F k = abs (F k)" by simp
      finally have "abs (F k) ≤ c powr (a + 1) / (a + 1)" .
    }
    thus "bounded (range(λk. integral {0..c} (f k)))"
      by (intro boundedI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f)
  qed

  from False c have "c > 0" by simp
  from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
    have "eventually (λk. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a+1) / (a+1) =
            integral {0..c} (f k)) sequentially"
    by eventually_elim (simp add: integral_f F_def)
  moreover have "(λk. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
                   <---- c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)"
    using a by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) auto
  hence "(λk. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
          <---- c powr (a + 1) / (a + 1)" by simp
  ultimately have "(λk. integral {0..c} (f k)) <---- c powr (a+1) / (a+1)"
    by (blast intro: Lim_transform_eventually)
  with A have "integral {0..c} (λx. x powr a) = c powr (a+1) / (a+1)"
    by (blast intro: LIMSEQ_unique)
  with A show ?thesis by (simp add: has_integral_integral)
qed (simp_all add: has_integral_refl)

lemma integrable_on_powr_from_0:
  assumes a: "a > (-1::real)" and c: "c ≥ 0"
  shows   "(λx. x powr a) integrable_on {0..c}"
  using has_integral_powr_from_0[OF assms] unfolding integrable_on_def by blast
lemma has_integral_powr_to_inf:
  fixes a e :: real
  assumes "e < -1" "a > 0"
  shows   "((λx. x powr e) has_integral -(a powr (e+1)) / (e+1)) {a..}"
proof (intro has_integral_to_inf integrable_continuous_interval continuous_intros)
  define F where "F ≡ λx. x powr (e+1) / (e+1)"
  have "((λx. x powr e) has_integral (F y - F a)) {a..y}" if "y ≥ a" for y
    unfolding F_def using assms
    by (intro fundamental_theorem_of_calculus that)
       (auto intro!: derivative_eq_intros
               simp flip: has_real_derivative_iff_has_vector_derivative)
  then have "∀🪙F y in at_top. integral {a..y} (λx. x powr e) = F y - F a"
    by (meson eventually_at_top_linorderI integral_unique)
  moreover have "((λy::real. F y - F a) ---> - F a) at_top"
    using assms unfolding F_def by real_asymp
  ultimately 
  show "((λy. integral {a..y} (λx. x powr e)) ---> - (a powr (e+1)) / (e+1)) at_top"
    by (simp add: F_def filterlim_cong)
qed (use assms in auto)

lemma has_integral_inverse_power_to_inf:
  fixes a :: real and n :: nat
  assumes "n > 1" "a > 0"
  shows   "((λx. 1 / x ^ n) has_integral 1 / (real (n - 1) * a ^ (n - 1))) {a..}"
proof -
  from assms have "real_of_int (-int n) < -1" by simp
  note has_integral_powr_to_inf[OF this ‹a > 0›]
  also have "- (a powr (real_of_int (- int n) + 1)) / (real_of_int (- int n) + 1) =
                 1 / (real (n - 1) * a powr (real (n - 1)))" using assms
    by (simp add: field_split_simps powr_add [symmetric] of_nat_diff)
  also from assms have "a powr (real (n - 1)) = a ^ (n - 1)"
    by (intro powr_realpow)
  finally show ?thesis
  proof (rule has_integral_eq [rotated])
  qed (insert assms, simp_all add: powr_minus powr_realpow field_split_simps)
qed

subsection ‹Adaption to ordered Euclidean spaces and the Cartesian Euclidean space›

lemma integral_component_eq_cart[simp]:
  fixes f :: "'n::euclidean_space ==> real^'m"
  assumes "f integrable_on s"
  shows "integral s (λx. f x $ k) = integral s f $ k"
  using integral_linear[OF assms(1) bounded_linear_vec_nth,unfolded o_def] .

lemma content_closed_interval:
  fixes a :: "'a::ordered_euclidean_space"
  assumes "a ≤ b"
  shows "content {a..b} = (∏i∈Basis. b∙i - a∙i)"
  using content_cbox[of a b] assms by (simp add: cbox_interval eucl_le[where 'a='a])

lemma integrable_const_ivl[intro]:
  fixes a::"'a::ordered_euclidean_space"
  shows "(λx. c) integrable_on {a..b}"
  unfolding cbox_interval[symmetric] by (rule integrable_const)

lemma integrable_on_subinterval:
  fixes f :: "'n::ordered_euclidean_space ==> 'a::banach"
  assumes "f integrable_on S" "{a..b} ⊆ S"
  shows "f integrable_on {a..b}"
  using integrable_on_subcbox[of f S a b] assms by (simp add: cbox_interval)

end