# Theory Integration

theory Integration
imports Derivative
```(*  Author:     John Harrison
Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP
*)

section ‹Kurzweil-Henstock Gauge Integration in many dimensions.›

theory Integration
imports
Derivative
Uniform_Limit
"~~/src/HOL/Library/Indicator_Function"
begin

lemma cSup_abs_le: (* TODO: move to Conditionally_Complete_Lattices.thy? *)
fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
shows "S ≠ {} ⟹ (⋀x. x∈S ⟹ ¦x¦ ≤ a) ⟹ ¦Sup S¦ ≤ a"
apply (auto simp add: abs_le_iff intro: cSup_least)
by (metis bdd_aboveI cSup_upper neg_le_iff_le order_trans)

lemma cInf_abs_ge:
fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
assumes "S ≠ {}" and bdd: "⋀x. x∈S ⟹ ¦x¦ ≤ a"
shows "¦Inf S¦ ≤ a"
proof -
have "Sup (uminus ` S) = - (Inf S)"
proof (rule antisym)
show "- (Inf S) ≤ Sup(uminus ` S)"
apply (subst minus_le_iff)
apply (rule cInf_greatest [OF ‹S ≠ {}›])
apply (subst minus_le_iff)
apply (rule cSup_upper, force)
using bdd apply (force simp add: abs_le_iff bdd_above_def)
done
next
show "Sup (uminus ` S) ≤ - Inf S"
apply (rule cSup_least)
using ‹S ≠ {}› apply (force simp add: )
apply clarsimp
apply (rule cInf_lower)
apply assumption
using bdd apply (simp add: bdd_below_def)
apply (rule_tac x="-a" in exI)
apply force
done
qed
with cSup_abs_le [of "uminus ` S"] assms show ?thesis by fastforce
qed

lemma cSup_asclose:
fixes S :: "('a::{linordered_idom,conditionally_complete_linorder}) set"
assumes S: "S ≠ {}"
and b: "∀x∈S. ¦x - l¦ ≤ e"
shows "¦Sup S - l¦ ≤ e"
proof -
have th: "⋀(x::'a) l e. ¦x - l¦ ≤ e ⟷ l - e ≤ x ∧ x ≤ l + e"
by arith
have "bdd_above S"
using b by (auto intro!: bdd_aboveI[of _ "l + e"])
with S b show ?thesis
unfolding th by (auto intro!: cSup_upper2 cSup_least)
qed

lemma cInf_asclose:
fixes S :: "real set"
assumes S: "S ≠ {}"
and b: "∀x∈S. ¦x - l¦ ≤ e"
shows "¦Inf S - l¦ ≤ e"
proof -
have "¦- Sup (uminus ` S) - l¦ =  ¦Sup (uminus ` S) - (-l)¦"
by auto
also have "… ≤ e"
apply (rule cSup_asclose)
using abs_minus_add_cancel b by (auto simp add: S)
finally have "¦- Sup (uminus ` S) - l¦ ≤ e" .
then show ?thesis
by (simp add: Inf_real_def)
qed

lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one

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

subsection ‹Sundries›

lemma conjunctD2: assumes "a ∧ b" shows a b using assms by auto
lemma conjunctD3: assumes "a ∧ b ∧ c" shows a b c using assms by auto
lemma conjunctD4: assumes "a ∧ b ∧ c ∧ d" shows a b c d using assms by auto

declare norm_triangle_ineq4[intro]

lemma simple_image: "{f x |x . x ∈ s} = f ` s"
by blast

lemma linear_simps:
assumes "bounded_linear f"
shows
"f (a + b) = f a + f b"
"f (a - b) = f a - f b"
"f 0 = 0"
"f (- a) = - f a"
"f (s *⇩R v) = s *⇩R (f v)"
proof -
interpret f: bounded_linear f by fact
show "f (a + b) = f a + f b" by (rule f.add)
show "f (a - b) = f a - f b" by (rule f.diff)
show "f 0 = 0" by (rule f.zero)
show "f (- a) = - f a" by (rule f.minus)
show "f (s *⇩R v) = s *⇩R (f v)" by (rule f.scaleR)
qed

lemma bounded_linearI:
assumes "⋀x y. f (x + y) = f x + f y"
and "⋀r x. f (r *⇩R x) = r *⇩R f x"
and "⋀x. norm (f x) ≤ norm x * K"
shows "bounded_linear f"
using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)

lemma bounded_linear_component [intro]: "bounded_linear (λx::'a::euclidean_space. x ∙ k)"
by (rule bounded_linear_inner_left)

lemma transitive_stepwise_lt_eq:
assumes "(⋀x y z::nat. R x y ⟹ R y z ⟹ R x z)"
shows "((∀m. ∀n>m. R m n) ⟷ (∀n. R n (Suc n)))"
(is "?l = ?r")
proof safe
assume ?r
fix n m :: nat
assume "m < n"
then show "R m n"
proof (induct n arbitrary: m)
case 0
then show ?case by auto
next
case (Suc n)
show ?case
proof (cases "m < n")
case True
show ?thesis
apply (rule assms[OF Suc(1)[OF True]])
using ‹?r›
apply auto
done
next
case False
then have "m = n"
using Suc(2) by auto
then show ?thesis
using ‹?r› by auto
qed
qed
qed auto

lemma transitive_stepwise_gt:
assumes "⋀x y z. R x y ⟹ R y z ⟹ R x z" "⋀n. R n (Suc n)"
shows "∀n>m. R m n"
proof -
have "∀m. ∀n>m. R m n"
apply (subst transitive_stepwise_lt_eq)
apply (blast intro: assms)+
done
then show ?thesis by auto
qed

lemma transitive_stepwise_le_eq:
assumes "⋀x. R x x" "⋀x y z. R x y ⟹ R y z ⟹ R x z"
shows "(∀m. ∀n≥m. R m n) ⟷ (∀n. R n (Suc n))"
(is "?l = ?r")
proof safe
assume ?r
fix m n :: nat
assume "m ≤ n"
then show "R m n"
proof (induct n arbitrary: m)
case 0
with assms show ?case by auto
next
case (Suc n)
show ?case
proof (cases "m ≤ n")
case True
with Suc.hyps ‹∀n. R n (Suc n)› assms show ?thesis
by blast
next
case False
then have "m = Suc n"
using Suc(2) by auto
then show ?thesis
using assms(1) by auto
qed
qed
qed auto

lemma transitive_stepwise_le:
assumes "⋀x. R x x" "⋀x y z. R x y ⟹ R y z ⟹ R x z"
and "⋀n. R n (Suc n)"
shows "∀n≥m. R m n"
proof -
have "∀m. ∀n≥m. R m n"
apply (subst transitive_stepwise_le_eq)
apply (blast intro: assms)+
done
then show ?thesis by auto
qed

subsection ‹Some useful lemmas about intervals.›

lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
using nonempty_Basis
by (fastforce simp add: set_eq_iff mem_box)

lemma interior_subset_union_intervals:
assumes "i = cbox a b"
and "j = cbox c d"
and "interior j ≠ {}"
and "i ⊆ j ∪ s"
and "interior i ∩ interior j = {}"
shows "interior i ⊆ interior s"
proof -
have "box a b ∩ cbox c d = {}"
using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
unfolding assms(1,2) interior_cbox by auto
moreover
have "box a b ⊆ cbox c d ∪ s"
apply (rule order_trans,rule box_subset_cbox)
using assms(4) unfolding assms(1,2)
apply auto
done
ultimately
show ?thesis
unfolding assms interior_cbox
by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
qed

lemma inter_interior_unions_intervals:
fixes f::"('a::euclidean_space) set set"
assumes "finite f"
and "open s"
and "∀t∈f. ∃a b. t = cbox a b"
and "∀t∈f. s ∩ (interior t) = {}"
shows "s ∩ interior (⋃f) = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix x
assume x: "x ∈ s" "x ∈ interior (⋃f)"
have lem1: "⋀x e s U. ball x e ⊆ s ∩ interior U ⟷ ball x e ⊆ s ∩ U"
using interior_subset
by auto (meson Topology_Euclidean_Space.open_ball contra_subsetD interior_maximal mem_ball)
have "∃t∈f. ∃x. ∃e>0. ball x e ⊆ s ∩ t"
if "finite f" and "∀t∈f. ∃a b. t = cbox a b" and "∃x. x ∈ s ∩ interior (⋃f)" for f
using that
proof (induct rule: finite_induct)
case empty
obtain x where "x ∈ s ∩ interior (⋃{})"
using empty(2) ..
then have False
unfolding Union_empty interior_empty by auto
then show ?case by auto
next
case (insert i f)
obtain x where x: "x ∈ s ∩ interior (⋃insert i f)"
using insert(5) ..
then obtain e where e: "0 < e ∧ ball x e ⊆ s ∩ interior (⋃insert i f)"
unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
obtain a where "∃b. i = cbox a b"
using insert(4)[rule_format,OF insertI1] ..
then obtain b where ab: "i = cbox a b" ..
show ?case
proof (cases "x ∈ i")
case False
then have "x ∈ UNIV - cbox a b"
unfolding ab by auto
then obtain d where "0 < d ∧ ball x d ⊆ UNIV - cbox a b"
unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_cbox],rule_format] ..
then have "0 < d" "ball x (min d e) ⊆ UNIV - i"
unfolding ab ball_min_Int by auto
then have "ball x (min d e) ⊆ s ∩ interior (⋃f)"
using e unfolding lem1 unfolding  ball_min_Int by auto
then have "x ∈ s ∩ interior (⋃f)" using ‹d>0› e by auto
then have "∃t∈f. ∃x e. 0 < e ∧ ball x e ⊆ s ∩ t"
using insert.hyps(3) insert.prems(1) by blast
then show ?thesis by auto
next
case True show ?thesis
proof (cases "x∈box a b")
case True
then obtain d where "0 < d ∧ ball x d ⊆ box a b"
unfolding open_contains_ball_eq[OF open_box,rule_format] ..
then show ?thesis
apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
unfolding ab
using box_subset_cbox[of a b] and e
apply fastforce+
done
next
case False
then obtain k where "x∙k ≤ a∙k ∨ x∙k ≥ b∙k" and k: "k ∈ Basis"
unfolding mem_box by (auto simp add: not_less)
then have "x∙k = a∙k ∨ x∙k = b∙k"
using True unfolding ab and mem_box
apply (erule_tac x = k in ballE)
apply auto
done
then have "∃x. ball x (e/2) ⊆ s ∩ (⋃f)"
proof (rule disjE)
let ?z = "x - (e/2) *⇩R k"
assume as: "x∙k = a∙k"
have "ball ?z (e / 2) ∩ i = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix y
assume "y ∈ ball ?z (e / 2)" and yi: "y ∈ i"
then have "dist ?z y < e/2" by auto
then have "¦(?z - y) ∙ k¦ < e/2"
using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
then have "y∙k < a∙k"
using e k
by (auto simp add: field_simps abs_less_iff as inner_simps)
then have "y ∉ i"
unfolding ab mem_box by (auto intro!: bexI[OF _ k])
then show False using yi by auto
qed
moreover
have "ball ?z (e/2) ⊆ s ∩ (⋃insert i f)"
apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
proof
fix y
assume as: "y ∈ ball ?z (e/2)"
have "norm (x - y) ≤ ¦e¦ / 2 + norm (x - y - (e / 2) *⇩R k)"
apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *⇩R k"])
unfolding norm_scaleR norm_Basis[OF k]
apply auto
done
also have "… < ¦e¦ / 2 + ¦e¦ / 2"
apply (rule add_strict_left_mono)
using as e
apply (auto simp add: field_simps dist_norm)
done
finally show "y ∈ ball x e"
unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
apply (rule_tac x="?z" in exI)
unfolding Union_insert
apply auto
done
next
let ?z = "x + (e/2) *⇩R k"
assume as: "x∙k = b∙k"
have "ball ?z (e / 2) ∩ i = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix y
assume "y ∈ ball ?z (e / 2)" and yi: "y ∈ i"
then have "dist ?z y < e/2"
by auto
then have "¦(?z - y) ∙ k¦ < e/2"
using Basis_le_norm[OF k, of "?z - y"]
unfolding dist_norm by auto
then have "y∙k > b∙k"
using e k
by (auto simp add:field_simps inner_simps inner_Basis as)
then have "y ∉ i"
unfolding ab mem_box by (auto intro!: bexI[OF _ k])
then show False using yi by auto
qed
moreover
have "ball ?z (e/2) ⊆ s ∩ (⋃insert i f)"
apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
proof
fix y
assume as: "y∈ ball ?z (e/2)"
have "norm (x - y) ≤ ¦e¦ / 2 + norm (x - y + (e / 2) *⇩R k)"
apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *⇩R k"])
unfolding norm_scaleR
apply (auto simp: k)
done
also have "… < ¦e¦ / 2 + ¦e¦ / 2"
apply (rule add_strict_left_mono)
using as unfolding mem_ball dist_norm
using e apply (auto simp add: field_simps)
done
finally show "y ∈ ball x e"
unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
apply (rule_tac x="?z" in exI)
unfolding Union_insert
apply auto
done
qed
then obtain x where "ball x (e / 2) ⊆ s ∩ ⋃f" ..
then have "x ∈ s ∩ interior (⋃f)"
unfolding lem1[where U="⋃f", symmetric]
using centre_in_ball e by auto
then show ?thesis
using insert.hyps(3) insert.prems(1) by blast
qed
qed
qed
from this[OF assms(1,3)] x
obtain t x e where "t ∈ f" "0 < e" "ball x e ⊆ s ∩ t"
by blast
then have "x ∈ s" "x ∈ interior t"
using open_subset_interior[OF open_ball, of x e t]
by auto
then show False
using ‹t ∈ f› assms(4) by auto
qed

subsection ‹Bounds on intervals where they exist.›

definition interval_upperbound :: "('a::euclidean_space) set ⇒ 'a"
where "interval_upperbound s = (∑i∈Basis. (SUP x:s. x∙i) *⇩R i)"

definition interval_lowerbound :: "('a::euclidean_space) set ⇒ 'a"
where "interval_lowerbound s = (∑i∈Basis. (INF x:s. x∙i) *⇩R i)"

lemma interval_upperbound[simp]:
"∀i∈Basis. a∙i ≤ b∙i ⟹
interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
unfolding interval_upperbound_def euclidean_representation_setsum cbox_def SUP_def
by (safe intro!: cSup_eq) auto

lemma interval_lowerbound[simp]:
"∀i∈Basis. a∙i ≤ b∙i ⟹
interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def INF_def
by (safe intro!: cInf_eq) auto

lemmas interval_bounds = interval_upperbound interval_lowerbound

lemma
fixes X::"real set"
shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def)

lemma interval_bounds'[simp]:
assumes "cbox a b ≠ {}"
shows "interval_upperbound (cbox a b) = b"
and "interval_lowerbound (cbox a b) = a"
using assms unfolding box_ne_empty by auto

lemma interval_upperbound_Times:
assumes "A ≠ {}" and "B ≠ {}"
shows "interval_upperbound (A × B) = (interval_upperbound A, interval_upperbound B)"
proof-
from assms have fst_image_times': "A = fst ` (A × B)" by simp
have "(∑i∈Basis. (SUP x:A × B. x ∙ (i, 0)) *⇩R i) = (∑i∈Basis. (SUP x:A. x ∙ i) *⇩R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
moreover from assms have snd_image_times': "B = snd ` (A × B)" by simp
have "(∑i∈Basis. (SUP x:A × B. x ∙ (0, i)) *⇩R i) = (∑i∈Basis. (SUP x:B. x ∙ i) *⇩R i)"
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
ultimately show ?thesis unfolding interval_upperbound_def
by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
qed

lemma interval_lowerbound_Times:
assumes "A ≠ {}" and "B ≠ {}"
shows "interval_lowerbound (A × B) = (interval_lowerbound A, interval_lowerbound B)"
proof-
from assms have fst_image_times': "A = fst ` (A × B)" by simp
have "(∑i∈Basis. (INF x:A × B. x ∙ (i, 0)) *⇩R i) = (∑i∈Basis. (INF x:A. x ∙ i) *⇩R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
moreover from assms have snd_image_times': "B = snd ` (A × B)" by simp
have "(∑i∈Basis. (INF x:A × B. x ∙ (0, i)) *⇩R i) = (∑i∈Basis. (INF x:B. x ∙ i) *⇩R i)"
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
ultimately show ?thesis unfolding interval_lowerbound_def
by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
qed

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

definition "content (s::('a::euclidean_space) set) =
(if s = {} then 0 else (∏i∈Basis. (interval_upperbound s)∙i - (interval_lowerbound s)∙i))"

lemma interval_not_empty: "∀i∈Basis. a∙i ≤ b∙i ⟹ cbox a b ≠ {}"
unfolding box_eq_empty unfolding not_ex not_less by auto

lemma content_cbox:
fixes a :: "'a::euclidean_space"
assumes "∀i∈Basis. a∙i ≤ b∙i"
shows "content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
using interval_not_empty[OF assms]
unfolding content_def
by auto

lemma content_cbox':
fixes a :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
shows "content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
using assms box_ne_empty(1) content_cbox by blast

lemma content_real: "a ≤ b ⟹ content {a..b} = b - a"
by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def content_def)

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[simp]: "content {a} = 0"
proof -
have "content (cbox a a) = 0"
by (subst content_cbox) (auto simp: ex_in_conv)
then show ?thesis by (simp add: cbox_sing)
qed

lemma content_unit[iff]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
proof -
have *: "∀i∈Basis. (0::'a)∙i ≤ (One::'a)∙i"
by auto
have "0 ∈ cbox 0 (One::'a)"
unfolding mem_box by auto
then show ?thesis
unfolding content_def interval_bounds[OF *] using setprod.neutral_const by auto
qed

lemma content_pos_le[intro]:
fixes a::"'a::euclidean_space"
shows "0 ≤ content (cbox a b)"
proof (cases "cbox a b = {}")
case False
then have *: "∀i∈Basis. a ∙ i ≤ b ∙ i"
unfolding box_ne_empty .
have "0 ≤ (∏i∈Basis. interval_upperbound (cbox a b) ∙ i - interval_lowerbound (cbox a b) ∙ i)"
apply (rule setprod_nonneg)
unfolding interval_bounds[OF *]
using *
apply auto
done
also have "… = content (cbox a b)" using False by (simp add: content_def)
finally show ?thesis .
qed (simp add: content_def)

corollary content_nonneg [simp]:
fixes a::"'a::euclidean_space"
shows "~ content (cbox a b) < 0"
using not_le by blast

lemma content_pos_lt:
fixes a :: "'a::euclidean_space"
assumes "∀i∈Basis. a∙i < b∙i"
shows "0 < content (cbox a b)"
using assms
by (auto simp: content_def box_eq_empty intro!: setprod_pos)

lemma content_eq_0:
"content (cbox a b) = 0 ⟷ (∃i∈Basis. b∙i ≤ a∙i)"
by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)

lemma cond_cases: "(P ⟹ Q x) ⟹ (¬ P ⟹ Q y) ⟹ Q (if P then x else y)"
by auto

lemma content_cbox_cases:
"content (cbox a (b::'a::euclidean_space)) =
(if ∀i∈Basis. a∙i ≤ b∙i then setprod (λi. b∙i - a∙i) Basis else 0)"
by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)

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)"
proof (rule iffI)
assume "0 < content (cbox a b)"
then have "content (cbox a b) ≠ 0" by auto
then show "∀i∈Basis. a∙i < b∙i"
unfolding content_eq_0 not_ex not_le by fastforce
next
assume "∀i∈Basis. a ∙ i < b ∙ i"
then show "0 < content (cbox a b)"
by (metis content_pos_lt)
qed

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

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

lemma content_subset:
assumes "cbox a b ⊆ cbox c d"
shows "content (cbox a b) ≤ content (cbox c d)"
proof (cases "cbox a b = {}")
case True
then show ?thesis
using content_pos_le[of c d] by auto
next
case False
then have ab_ne: "∀i∈Basis. a ∙ i ≤ b ∙ i"
unfolding box_ne_empty by auto
then have ab_ab: "a∈cbox a b" "b∈cbox a b"
unfolding mem_box by auto
have "cbox c d ≠ {}" using assms False by auto
then have cd_ne: "∀i∈Basis. c ∙ i ≤ d ∙ i"
using assms unfolding box_ne_empty by auto
have "⋀i. i ∈ Basis ⟹ 0 ≤ b ∙ i - a ∙ i"
using ab_ne by auto
moreover
have "⋀i. i ∈ Basis ⟹ b ∙ i - a ∙ i ≤ d ∙ i - c ∙ i"
using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
by (metis diff_mono)
ultimately show ?thesis
unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF ‹cbox c d ≠ {}›])
qed

lemma content_lt_nz: "0 < content (cbox a b) ⟷ content (cbox a b) ≠ 0"
unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce

lemma content_times[simp]: "content (A × B) = content A * content B"
proof (cases "A × B = {}")
let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
assume nonempty: "A × B ≠ {}"
hence "content (A × B) = (∏i∈Basis. (?ub1 A, ?ub2 B) ∙ i - (?lb1 A, ?lb2 B) ∙ i)"
unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
also have "... = content A * content B" unfolding content_def using nonempty
apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
done
finally show ?thesis .
qed (auto simp: content_def)

lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
by (simp add: cbox_Pair_eq)

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_eq_0_gen:
fixes s :: "'a::euclidean_space set"
assumes "bounded s"
shows "content s = 0 ⟷ (∃i∈Basis. ∃v. ∀x ∈ s. x ∙ i = v)"  (is "_ = ?rhs")
proof safe
assume "content s = 0" then show ?rhs
apply (clarsimp simp: ex_in_conv content_def split: split_if_asm)
apply (rule_tac x=a in bexI)
apply (rule_tac x="interval_lowerbound s ∙ a" in exI)
apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
apply (drule cSUP_eq_cINF_D)
apply (auto simp: bounded_inner_imp_bdd_above [OF assms]  bounded_inner_imp_bdd_below [OF assms])
done
next
fix i a
assume "i ∈ Basis" "∀x∈s. x ∙ i = a"
then show "content s = 0"
apply (clarsimp simp: content_def)
apply (rule_tac x=i in bexI)
apply (auto simp: interval_upperbound_def interval_lowerbound_def)
done
qed

lemma content_0_subset_gen:
fixes a :: "'a::euclidean_space"
assumes "content t = 0" "s ⊆ t" "bounded t" shows "content s = 0"
proof -
have "bounded s"
using assms by (metis bounded_subset)
then show ?thesis
using assms
by (auto simp: content_eq_0_gen)
qed

lemma content_0_subset: "⟦content(cbox a b) = 0; s ⊆ cbox a b⟧ ⟹ content s = 0"
by (simp add: content_0_subset_gen bounded_cbox)

subsection ‹The notion of a gauge --- simply an open set containing the point.›

definition "gauge d ⟷ (∀x. x ∈ d x ∧ open (d x))"

lemma gaugeI:
assumes "⋀x. x ∈ g x"
and "⋀x. open (g x)"
shows "gauge g"
using assms unfolding gauge_def by auto

lemma gaugeD[dest]:
assumes "gauge d"
shows "x ∈ d x"
and "open (d x)"
using assms unfolding gauge_def by auto

lemma gauge_ball_dependent: "∀x. 0 < e x ⟹ gauge (λx. ball x (e x))"
unfolding gauge_def by auto

lemma gauge_ball[intro]: "0 < e ⟹ gauge (λx. ball x e)"
unfolding gauge_def by auto

lemma gauge_trivial[intro!]: "gauge (λx. ball x 1)"
by (rule gauge_ball) auto

lemma gauge_inter[intro]: "gauge d1 ⟹ gauge d2 ⟹ gauge (λx. d1 x ∩ d2 x)"
unfolding gauge_def by auto

lemma gauge_inters:
assumes "finite s"
and "∀d∈s. gauge (f d)"
shows "gauge (λx. ⋂{f d x | d. d ∈ s})"
proof -
have *: "⋀x. {f d x |d. d ∈ s} = (λd. f d x) ` s"
by auto
show ?thesis
unfolding gauge_def unfolding *
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
qed

lemma gauge_existence_lemma:
"(∀x. ∃d :: real. p x ⟶ 0 < d ∧ q d x) ⟷ (∀x. ∃d>0. p x ⟶ q d x)"
by (metis zero_less_one)

subsection ‹Divisions.›

definition division_of (infixl "division'_of" 40)
where
"s division_of i ⟷
finite s ∧
(∀k∈s. k ⊆ i ∧ k ≠ {} ∧ (∃a b. k = cbox a b)) ∧
(∀k1∈s. ∀k2∈s. k1 ≠ k2 ⟶ interior(k1) ∩ interior(k2) = {}) ∧
(⋃s = i)"

lemma division_ofD[dest]:
assumes "s division_of i"
shows "finite s"
and "⋀k. k ∈ s ⟹ k ⊆ i"
and "⋀k. k ∈ s ⟹ k ≠ {}"
and "⋀k. k ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀k1 k2. k1 ∈ s ⟹ k2 ∈ s ⟹ k1 ≠ k2 ⟹ interior(k1) ∩ interior(k2) = {}"
and "⋃s = i"
using assms unfolding division_of_def by auto

lemma division_ofI:
assumes "finite s"
and "⋀k. k ∈ s ⟹ k ⊆ i"
and "⋀k. k ∈ s ⟹ k ≠ {}"
and "⋀k. k ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀k1 k2. k1 ∈ s ⟹ k2 ∈ s ⟹ k1 ≠ k2 ⟹ interior k1 ∩ interior k2 = {}"
and "⋃s = i"
shows "s division_of i"
using assms unfolding division_of_def by auto

lemma division_of_finite: "s division_of i ⟹ finite s"
unfolding division_of_def by auto

lemma division_of_self[intro]: "cbox a b ≠ {} ⟹ {cbox a b} division_of (cbox a b)"
unfolding division_of_def by auto

lemma division_of_trivial[simp]: "s division_of {} ⟷ s = {}"
unfolding division_of_def by auto

lemma division_of_sing[simp]:
"s division_of cbox a (a::'a::euclidean_space) ⟷ s = {cbox a a}"
(is "?l = ?r")
proof
assume ?r
moreover
{ fix k
assume "s = {{a}}" "k∈s"
then have "∃x y. k = cbox x y"
apply (rule_tac x=a in exI)+
apply (force simp: cbox_sing)
done
}
ultimately show ?l
unfolding division_of_def cbox_sing by auto
next
assume ?l
note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
{
fix x
assume x: "x ∈ s" have "x = {a}"
using *(2)[rule_format,OF x] by auto
}
moreover have "s ≠ {}"
using *(4) by auto
ultimately show ?r
unfolding cbox_sing by auto
qed

lemma elementary_empty: obtains p where "p division_of {}"
unfolding division_of_trivial by auto

lemma elementary_interval: obtains p where "p division_of (cbox a b)"
by (metis division_of_trivial division_of_self)

lemma division_contains: "s division_of i ⟹ ∀x∈i. ∃k∈s. x ∈ k"
unfolding division_of_def by auto

lemma forall_in_division:
"d division_of i ⟹ (∀x∈d. P x) ⟷ (∀a b. cbox a b ∈ d ⟶ P (cbox a b))"
unfolding division_of_def by fastforce

lemma division_of_subset:
assumes "p division_of (⋃p)"
and "q ⊆ p"
shows "q division_of (⋃q)"
proof (rule division_ofI)
note * = division_ofD[OF assms(1)]
show "finite q"
using "*"(1) assms(2) infinite_super by auto
{
fix k
assume "k ∈ q"
then have kp: "k ∈ p"
using assms(2) by auto
show "k ⊆ ⋃q"
using ‹k ∈ q› by auto
show "∃a b. k = cbox a b"
using *(4)[OF kp] by auto
show "k ≠ {}"
using *(3)[OF kp] by auto
}
fix k1 k2
assume "k1 ∈ q" "k2 ∈ q" "k1 ≠ k2"
then have **: "k1 ∈ p" "k2 ∈ p" "k1 ≠ k2"
using assms(2) by auto
show "interior k1 ∩ interior k2 = {}"
using *(5)[OF **] by auto
qed auto

lemma division_of_union_self[intro]: "p division_of s ⟹ p division_of (⋃p)"
unfolding division_of_def by auto

lemma division_of_content_0:
assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
shows "∀k∈d. content k = 0"
unfolding forall_in_division[OF assms(2)]
by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))

lemma division_inter:
fixes s1 s2 :: "'a::euclidean_space set"
assumes "p1 division_of s1"
and "p2 division_of s2"
shows "{k1 ∩ k2 | k1 k2 .k1 ∈ p1 ∧ k2 ∈ p2 ∧ k1 ∩ k2 ≠ {}} division_of (s1 ∩ s2)"
(is "?A' division_of _")
proof -
let ?A = "{s. s ∈  (λ(k1,k2). k1 ∩ k2) ` (p1 × p2) ∧ s ≠ {}}"
have *: "?A' = ?A" by auto
show ?thesis
unfolding *
proof (rule division_ofI)
have "?A ⊆ (λ(x, y). x ∩ y) ` (p1 × p2)"
by auto
moreover have "finite (p1 × p2)"
using assms unfolding division_of_def by auto
ultimately show "finite ?A" by auto
have *: "⋀s. ⋃{x∈s. x ≠ {}} = ⋃s"
by auto
show "⋃?A = s1 ∩ s2"
apply (rule set_eqI)
unfolding * and Union_image_eq UN_iff
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
apply auto
done
{
fix k
assume "k ∈ ?A"
then obtain k1 k2 where k: "k = k1 ∩ k2" "k1 ∈ p1" "k2 ∈ p2" "k ≠ {}"
by auto
then show "k ≠ {}"
by auto
show "k ⊆ s1 ∩ s2"
using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
unfolding k by auto
obtain a1 b1 where k1: "k1 = cbox a1 b1"
using division_ofD(4)[OF assms(1) k(2)] by blast
obtain a2 b2 where k2: "k2 = cbox a2 b2"
using division_ofD(4)[OF assms(2) k(3)] by blast
show "∃a b. k = cbox a b"
unfolding k k1 k2 unfolding inter_interval by auto
}
fix k1 k2
assume "k1 ∈ ?A"
then obtain x1 y1 where k1: "k1 = x1 ∩ y1" "x1 ∈ p1" "y1 ∈ p2" "k1 ≠ {}"
by auto
assume "k2 ∈ ?A"
then obtain x2 y2 where k2: "k2 = x2 ∩ y2" "x2 ∈ p1" "y2 ∈ p2" "k2 ≠ {}"
by auto
assume "k1 ≠ k2"
then have th: "x1 ≠ x2 ∨ y1 ≠ y2"
unfolding k1 k2 by auto
have *: "interior x1 ∩ interior x2 = {} ∨ interior y1 ∩ interior y2 = {} ⟹
interior (x1 ∩ y1) ⊆ interior x1 ⟹ interior (x1 ∩ y1) ⊆ interior y1 ⟹
interior (x2 ∩ y2) ⊆ interior x2 ⟹ interior (x2 ∩ y2) ⊆ interior y2 ⟹
interior (x1 ∩ y1) ∩ interior (x2 ∩ y2) = {}" by auto
show "interior k1 ∩ interior k2 = {}"
unfolding k1 k2
apply (rule *)
using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
done
qed
qed

lemma division_inter_1:
assumes "d division_of i"
and "cbox a (b::'a::euclidean_space) ⊆ i"
shows "{cbox a b ∩ k | k. k ∈ d ∧ cbox a b ∩ k ≠ {}} division_of (cbox a b)"
proof (cases "cbox a b = {}")
case True
show ?thesis
unfolding True and division_of_trivial by auto
next
case False
have *: "cbox a b ∩ i = cbox a b" using assms(2) by auto
show ?thesis
using division_inter[OF division_of_self[OF False] assms(1)]
unfolding * by auto
qed

lemma elementary_inter:
fixes s t :: "'a::euclidean_space set"
assumes "p1 division_of s"
and "p2 division_of t"
shows "∃p. p division_of (s ∩ t)"
using assms division_inter by blast

lemma elementary_inters:
assumes "finite f"
and "f ≠ {}"
and "∀s∈f. ∃p. p division_of (s::('a::euclidean_space) set)"
shows "∃p. p division_of (⋂f)"
using assms
proof (induct f rule: finite_induct)
case (insert x f)
show ?case
proof (cases "f = {}")
case True
then show ?thesis
unfolding True using insert by auto
next
case False
obtain p where "p division_of ⋂f"
using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
moreover obtain px where "px division_of x"
using insert(5)[rule_format,OF insertI1] ..
ultimately show ?thesis
by (simp add: elementary_inter Inter_insert)
qed
qed auto

lemma division_disjoint_union:
assumes "p1 division_of s1"
and "p2 division_of s2"
and "interior s1 ∩ interior s2 = {}"
shows "(p1 ∪ p2) division_of (s1 ∪ s2)"
proof (rule division_ofI)
note d1 = division_ofD[OF assms(1)]
note d2 = division_ofD[OF assms(2)]
show "finite (p1 ∪ p2)"
using d1(1) d2(1) by auto
show "⋃(p1 ∪ p2) = s1 ∪ s2"
using d1(6) d2(6) by auto
{
fix k1 k2
assume as: "k1 ∈ p1 ∪ p2" "k2 ∈ p1 ∪ p2" "k1 ≠ k2"
moreover
let ?g="interior k1 ∩ interior k2 = {}"
{
assume as: "k1∈p1" "k2∈p2"
have ?g
using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
using assms(3) by blast
}
moreover
{
assume as: "k1∈p2" "k2∈p1"
have ?g
using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
using assms(3) by blast
}
ultimately show ?g
using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
}
fix k
assume k: "k ∈ p1 ∪ p2"
show "k ⊆ s1 ∪ s2"
using k d1(2) d2(2) by auto
show "k ≠ {}"
using k d1(3) d2(3) by auto
show "∃a b. k = cbox a b"
using k d1(4) d2(4) by auto
qed

lemma partial_division_extend_1:
fixes a b c d :: "'a::euclidean_space"
assumes incl: "cbox c d ⊆ cbox a b"
and nonempty: "cbox c d ≠ {}"
obtains p where "p division_of (cbox a b)" "cbox c d ∈ p"
proof
let ?B = "λf::'a⇒'a × 'a.
cbox (∑i∈Basis. (fst (f i) ∙ i) *⇩R i) (∑i∈Basis. (snd (f i) ∙ i) *⇩R i)"
def p ≡ "?B ` (Basis →⇩E {(a, c), (c, d), (d, b)})"

show "cbox c d ∈ p"
unfolding p_def
by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="λ(i::'a)∈Basis. (c, d)"])
{
fix i :: 'a
assume "i ∈ Basis"
with incl nonempty have "a ∙ i ≤ c ∙ i" "c ∙ i ≤ d ∙ i" "d ∙ i ≤ b ∙ i"
unfolding box_eq_empty subset_box by (auto simp: not_le)
}
note ord = this

show "p division_of (cbox a b)"
proof (rule division_ofI)
show "finite p"
unfolding p_def by (auto intro!: finite_PiE)
{
fix k
assume "k ∈ p"
then obtain f where f: "f ∈ Basis →⇩E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
by (auto simp: p_def)
then show "∃a b. k = cbox a b"
by auto
have "k ⊆ cbox a b ∧ k ≠ {}"
proof (simp add: k box_eq_empty subset_box not_less, safe)
fix i :: 'a
assume i: "i ∈ Basis"
with f have "f i = (a, c) ∨ f i = (c, d) ∨ f i = (d, b)"
by (auto simp: PiE_iff)
with i ord[of i]
show "a ∙ i ≤ fst (f i) ∙ i" "snd (f i) ∙ i ≤ b ∙ i" "fst (f i) ∙ i ≤ snd (f i) ∙ i"
by auto
qed
then show "k ≠ {}" "k ⊆ cbox a b"
by auto
{
fix l
assume "l ∈ p"
then obtain g where g: "g ∈ Basis →⇩E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
by (auto simp: p_def)
assume "l ≠ k"
have "∃i∈Basis. f i ≠ g i"
proof (rule ccontr)
assume "¬ ?thesis"
with f g have "f = g"
by (auto simp: PiE_iff extensional_def intro!: ext)
with ‹l ≠ k› show False
by (simp add: l k)
qed
then obtain i where *: "i ∈ Basis" "f i ≠ g i" ..
then have "f i = (a, c) ∨ f i = (c, d) ∨ f i = (d, b)"
"g i = (a, c) ∨ g i = (c, d) ∨ g i = (d, b)"
using f g by (auto simp: PiE_iff)
with * ord[of i] show "interior l ∩ interior k = {}"
by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
}
note ‹k ⊆ cbox a b›
}
moreover
{
fix x assume x: "x ∈ cbox a b"
have "∀i∈Basis. ∃l. x ∙ i ∈ {fst l ∙ i .. snd l ∙ i} ∧ l ∈ {(a, c), (c, d), (d, b)}"
proof
fix i :: 'a
assume "i ∈ Basis"
with x ord[of i]
have "(a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ c ∙ i) ∨ (c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i) ∨
(d ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i)"
by (auto simp: cbox_def)
then show "∃l. x ∙ i ∈ {fst l ∙ i .. snd l ∙ i} ∧ l ∈ {(a, c), (c, d), (d, b)}"
by auto
qed
then obtain f where
f: "∀i∈Basis. x ∙ i ∈ {fst (f i) ∙ i..snd (f i) ∙ i} ∧ f i ∈ {(a, c), (c, d), (d, b)}"
unfolding bchoice_iff ..
moreover from f have "restrict f Basis ∈ Basis →⇩E {(a, c), (c, d), (d, b)}"
by auto
moreover from f have "x ∈ ?B (restrict f Basis)"
by (auto simp: mem_box)
ultimately have "∃k∈p. x ∈ k"
unfolding p_def by blast
}
ultimately show "⋃p = cbox a b"
by auto
qed
qed

lemma partial_division_extend_interval:
assumes "p division_of (⋃p)" "(⋃p) ⊆ cbox a b"
obtains q where "p ⊆ q" "q division_of cbox a (b::'a::euclidean_space)"
proof (cases "p = {}")
case True
obtain q where "q division_of (cbox a b)"
by (rule elementary_interval)
then show ?thesis
using True that by blast
next
case False
note p = division_ofD[OF assms(1)]
have div_cbox: "∀k∈p. ∃q. q division_of cbox a b ∧ k ∈ q"
proof
fix k
assume kp: "k ∈ p"
obtain c d where k: "k = cbox c d"
using p(4)[OF kp] by blast
have *: "cbox c d ⊆ cbox a b" "cbox c d ≠ {}"
using p(2,3)[OF kp, unfolded k] using assms(2)
by (blast intro: order.trans)+
obtain q where "q division_of cbox a b" "cbox c d ∈ q"
by (rule partial_division_extend_1[OF *])
then show "∃q. q division_of cbox a b ∧ k ∈ q"
unfolding k by auto
qed
obtain q where q: "⋀x. x ∈ p ⟹ q x division_of cbox a b" "⋀x. x ∈ p ⟹ x ∈ q x"
using bchoice[OF div_cbox] by blast
{ fix x
assume x: "x ∈ p"
have "q x division_of ⋃q x"
apply (rule division_ofI)
using division_ofD[OF q(1)[OF x]]
apply auto
done }
then have "⋀x. x ∈ p ⟹ ∃d. d division_of ⋃(q x - {x})"
by (meson Diff_subset division_of_subset)
then have "∃d. d division_of ⋂((λi. ⋃(q i - {i})) ` p)"
apply -
apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
done
then obtain d where d: "d division_of ⋂((λi. ⋃(q i - {i})) ` p)" ..
have "d ∪ p division_of cbox a b"
proof -
have te: "⋀s f t. s ≠ {} ⟹ ∀i∈s. f i ∪ i = t ⟹ t = ⋂(f ` s) ∪ ⋃s" by auto
have cbox_eq: "cbox a b = ⋂((λi. ⋃(q i - {i})) ` p) ∪ ⋃p"
proof (rule te[OF False], clarify)
fix i
assume i: "i ∈ p"
show "⋃(q i - {i}) ∪ i = cbox a b"
using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
qed
{ fix k
assume k: "k ∈ p"
have *: "⋀u t s. t ∩ s = {} ⟹ u ⊆ s ⟹ u ∩ t = {}"
by auto
have "interior (⋂i∈p. ⋃(q i - {i})) ∩ interior k = {}"
proof (rule *[OF inter_interior_unions_intervals])
note qk=division_ofD[OF q(1)[OF k]]
show "finite (q k - {k})" "open (interior k)" "∀t∈q k - {k}. ∃a b. t = cbox a b"
using qk by auto
show "∀t∈q k - {k}. interior k ∩ interior t = {}"
using qk(5) using q(2)[OF k] by auto
show "interior (⋂i∈p. ⋃(q i - {i})) ⊆ interior (⋃(q k - {k}))"
apply (rule interior_mono)+
using k
apply auto
done
qed } note [simp] = this
show "d ∪ p division_of (cbox a b)"
unfolding cbox_eq
apply (rule division_disjoint_union[OF d assms(1)])
apply (rule inter_interior_unions_intervals)
apply (rule p open_interior ballI)+
apply simp_all
done
qed
then show ?thesis
by (meson Un_upper2 that)
qed

lemma elementary_bounded[dest]:
fixes s :: "'a::euclidean_space set"
shows "p division_of s ⟹ bounded s"
unfolding division_of_def by (metis bounded_Union bounded_cbox)

lemma elementary_subset_cbox:
"p division_of s ⟹ ∃a b. s ⊆ cbox a (b::'a::euclidean_space)"
by (meson elementary_bounded bounded_subset_cbox)

lemma division_union_intervals_exists:
fixes a b :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
obtains p where "(insert (cbox a b) p) division_of (cbox a b ∪ cbox c d)"
proof (cases "cbox c d = {}")
case True
show ?thesis
apply (rule that[of "{}"])
unfolding True
using assms
apply auto
done
next
case False
show ?thesis
proof (cases "cbox a b ∩ cbox c d = {}")
case True
show ?thesis
apply (rule that[of "{cbox c d}"])
apply (subst insert_is_Un)
apply (rule division_disjoint_union)
using ‹cbox c d ≠ {}› True assms interior_subset
apply auto
done
next
case False
obtain u v where uv: "cbox a b ∩ cbox c d = cbox u v"
unfolding inter_interval by auto
have uv_sub: "cbox u v ⊆ cbox c d" using uv by auto
obtain p where "p division_of cbox c d" "cbox u v ∈ p"
by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
note p = this division_ofD[OF this(1)]
have "interior (cbox a b ∩ ⋃(p - {cbox u v})) = interior(cbox u v ∩ ⋃(p - {cbox u v}))"
apply (rule arg_cong[of _ _ interior])
using p(8) uv by auto
also have "… = {}"
unfolding interior_Int
apply (rule inter_interior_unions_intervals)
using p(6) p(7)[OF p(2)] p(3)
apply auto
done
finally have [simp]: "interior (cbox a b) ∩ interior (⋃(p - {cbox u v})) = {}" by simp
have cbe: "cbox a b ∪ cbox c d = cbox a b ∪ ⋃(p - {cbox u v})"
using p(8) unfolding uv[symmetric] by auto
show ?thesis
apply (rule that[of "p - {cbox u v}"])
apply (simp add: cbe)
apply (subst insert_is_Un)
apply (rule division_disjoint_union)
apply (simp_all add: assms division_of_self)
by (metis Diff_subset division_of_subset p(1) p(8))
qed
qed

lemma division_of_unions:
assumes "finite f"
and "⋀p. p ∈ f ⟹ p division_of (⋃p)"
and "⋀k1 k2. k1 ∈ ⋃f ⟹ k2 ∈ ⋃f ⟹ k1 ≠ k2 ⟹ interior k1 ∩ interior k2 = {}"
shows "⋃f division_of ⋃⋃f"
using assms
by (auto intro!: division_ofI)

lemma elementary_union_interval:
fixes a b :: "'a::euclidean_space"
assumes "p division_of ⋃p"
obtains q where "q division_of (cbox a b ∪ ⋃p)"
proof -
note assm = division_ofD[OF assms]
have lem1: "⋀f s. ⋃⋃(f ` s) = ⋃((λx. ⋃(f x)) ` s)"
by auto
have lem2: "⋀f s. f ≠ {} ⟹ ⋃{s ∪ t |t. t ∈ f} = s ∪ ⋃f"
by auto
{
presume "p = {} ⟹ thesis"
"cbox a b = {} ⟹ thesis"
"cbox a b ≠ {} ⟹ interior (cbox a b) = {} ⟹ thesis"
"p ≠ {} ⟹ interior (cbox a b)≠{} ⟹ cbox a b ≠ {} ⟹ thesis"
then show thesis by auto
next
assume as: "p = {}"
obtain p where "p division_of (cbox a b)"
by (rule elementary_interval)
then show thesis
using as that by auto
next
assume as: "cbox a b = {}"
show thesis
using as assms that by auto
next
assume as: "interior (cbox a b) = {}" "cbox a b ≠ {}"
show thesis
apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
unfolding finite_insert
apply (rule assm(1)) unfolding Union_insert
using assm(2-4) as
apply -
apply (fast dest: assm(5))+
done
next
assume as: "p ≠ {}" "interior (cbox a b) ≠ {}" "cbox a b ≠ {}"
have "∀k∈p. ∃q. (insert (cbox a b) q) division_of (cbox a b ∪ k)"
proof
fix k
assume kp: "k ∈ p"
from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast
then show "∃q. (insert (cbox a b) q) division_of (cbox a b ∪ k)"
by (meson as(3) division_union_intervals_exists)
qed
from bchoice[OF this] obtain q where "∀x∈p. insert (cbox a b) (q x) division_of (cbox a b) ∪ x" ..
note q = division_ofD[OF this[rule_format]]
let ?D = "⋃{insert (cbox a b) (q k) | k. k ∈ p}"
show thesis
proof (rule that[OF division_ofI])
have *: "{insert (cbox a b) (q k) |k. k ∈ p} = (λk. insert (cbox a b) (q k)) ` p"
by auto
show "finite ?D"
using "*" assm(1) q(1) by auto
show "⋃?D = cbox a b ∪ ⋃p"
unfolding * lem1
unfolding lem2[OF as(1), of "cbox a b", symmetric]
using q(6)
by auto
fix k
assume k: "k ∈ ?D"
then show "k ⊆ cbox a b ∪ ⋃p"
using q(2) by auto
show "k ≠ {}"
using q(3) k by auto
show "∃a b. k = cbox a b"
using q(4) k by auto
fix k'
assume k': "k' ∈ ?D" "k ≠ k'"
obtain x where x: "k ∈ insert (cbox a b) (q x)" "x∈p"
using k by auto
obtain x' where x': "k'∈insert (cbox a b) (q x')" "x'∈p"
using k' by auto
show "interior k ∩ interior k' = {}"
proof (cases "x = x'")
case True
show ?thesis
using True k' q(5) x' x by auto
next
case False
{
presume "k = cbox a b ⟹ ?thesis"
and "k' = cbox a b ⟹ ?thesis"
and "k ≠ cbox a b ⟹ k' ≠ cbox a b ⟹ ?thesis"
then show ?thesis by auto
next
assume as': "k  = cbox a b"
show ?thesis
using as' k' q(5) x' by auto
next
assume as': "k' = cbox a b"
show ?thesis
using as' k'(2) q(5) x by auto
}
assume as': "k ≠ cbox a b" "k' ≠ cbox a b"
obtain c d where k: "k = cbox c d"
using q(4)[OF x(2,1)] by blast
have "interior k ∩ interior (cbox a b) = {}"
using as' k'(2) q(5) x by auto
then have "interior k ⊆ interior x"
using interior_subset_union_intervals
by (metis as(2) k q(2) x interior_subset_union_intervals)
moreover
obtain c d where c_d: "k' = cbox c d"
using q(4)[OF x'(2,1)] by blast
have "interior k' ∩ interior (cbox a b) = {}"
using as'(2) q(5) x' by auto
then have "interior k' ⊆ interior x'"
by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
ultimately show ?thesis
using assm(5)[OF x(2) x'(2) False] by auto
qed
qed
}
qed

lemma elementary_unions_intervals:
assumes fin: "finite f"
and "⋀s. s ∈ f ⟹ ∃a b. s = cbox a (b::'a::euclidean_space)"
obtains p where "p division_of (⋃f)"
proof -
have "∃p. p division_of (⋃f)"
proof (induct_tac f rule:finite_subset_induct)
show "∃p. p division_of ⋃{}" using elementary_empty by auto
next
fix x F
assume as: "finite F" "x ∉ F" "∃p. p division_of ⋃F" "x∈f"
from this(3) obtain p where p: "p division_of ⋃F" ..
from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
have *: "⋃F = ⋃p"
using division_ofD[OF p] by auto
show "∃p. p division_of ⋃insert x F"
using elementary_union_interval[OF p[unfolded *], of a b]
unfolding Union_insert x * by metis
qed (insert assms, auto)
then show ?thesis
using that by auto
qed

lemma elementary_union:
fixes s t :: "'a::euclidean_space set"
assumes "ps division_of s" "pt division_of t"
obtains p where "p division_of (s ∪ t)"
proof -
have *: "s ∪ t = ⋃ps ∪ ⋃pt"
using assms unfolding division_of_def by auto
show ?thesis
apply (rule elementary_unions_intervals[of "ps ∪ pt"])
using assms apply auto
by (simp add: * that)
qed

lemma partial_division_extend:
fixes t :: "'a::euclidean_space set"
assumes "p division_of s"
and "q division_of t"
and "s ⊆ t"
obtains r where "p ⊆ r" and "r division_of t"
proof -
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
obtain a b where ab: "t ⊆ cbox a b"
using elementary_subset_cbox[OF assms(2)] by auto
obtain r1 where "p ⊆ r1" "r1 division_of (cbox a b)"
using assms
by (metis ab dual_order.trans partial_division_extend_interval divp(6))
note r1 = this division_ofD[OF this(2)]
obtain p' where "p' division_of ⋃(r1 - p)"
apply (rule elementary_unions_intervals[of "r1 - p"])
using r1(3,6)
apply auto
done
then obtain r2 where r2: "r2 division_of (⋃(r1 - p)) ∩ (⋃q)"
by (metis assms(2) divq(6) elementary_inter)
{
fix x
assume x: "x ∈ t" "x ∉ s"
then have "x∈⋃r1"
unfolding r1 using ab by auto
then obtain r where r: "r ∈ r1" "x ∈ r"
unfolding Union_iff ..
moreover
have "r ∉ p"
proof
assume "r ∈ p"
then have "x ∈ s" using divp(2) r by auto
then show False using x by auto
qed
ultimately have "x∈⋃(r1 - p)" by auto
}
then have *: "t = ⋃p ∪ (⋃(r1 - p) ∩ ⋃q)"
unfolding divp divq using assms(3) by auto
show ?thesis
apply (rule that[of "p ∪ r2"])
unfolding *
defer
apply (rule division_disjoint_union)
unfolding divp(6)
apply(rule assms r2)+
proof -
have "interior s ∩ interior (⋃(r1-p)) = {}"
proof (rule inter_interior_unions_intervals)
show "finite (r1 - p)" and "open (interior s)" and "∀t∈r1-p. ∃a b. t = cbox a b"
using r1 by auto
have *: "⋀s. (⋀x. x ∈ s ⟹ False) ⟹ s = {}"
by auto
show "∀t∈r1-p. interior s ∩ interior t = {}"
proof
fix m x
assume as: "m ∈ r1 - p"
have "interior m ∩ interior (⋃p) = {}"
proof (rule inter_interior_unions_intervals)
show "finite p" and "open (interior m)" and "∀t∈p. ∃a b. t = cbox a b"
using divp by auto
show "∀t∈p. interior m ∩ interior t = {}"
by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
qed
then show "interior s ∩ interior m = {}"
unfolding divp by auto
qed
qed
then show "interior s ∩ interior (⋃(r1-p) ∩ (⋃q)) = {}"
using interior_subset by auto
qed auto
qed

subsection ‹Tagged (partial) divisions.›

definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
where "s tagged_partial_division_of i ⟷
finite s ∧
(∀x k. (x, k) ∈ s ⟶ x ∈ k ∧ k ⊆ i ∧ (∃a b. k = cbox a b)) ∧
(∀x1 k1 x2 k2. (x1, k1) ∈ s ∧ (x2, k2) ∈ s ∧ (x1, k1) ≠ (x2, k2) ⟶
interior k1 ∩ interior k2 = {})"

lemma tagged_partial_division_ofD[dest]:
assumes "s tagged_partial_division_of i"
shows "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1,k1) ∈ s ⟹
(x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹ interior k1 ∩ interior k2 = {}"
using assms unfolding tagged_partial_division_of_def by blast+

definition tagged_division_of (infixr "tagged'_division'_of" 40)
where "s tagged_division_of i ⟷ s tagged_partial_division_of i ∧ (⋃{k. ∃x. (x,k) ∈ s} = i)"

lemma tagged_division_of_finite: "s tagged_division_of i ⟹ finite s"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto

lemma tagged_division_of:
"s tagged_division_of i ⟷
finite s ∧
(∀x k. (x, k) ∈ s ⟶ x ∈ k ∧ k ⊆ i ∧ (∃a b. k = cbox a b)) ∧
(∀x1 k1 x2 k2. (x1, k1) ∈ s ∧ (x2, k2) ∈ s ∧ (x1, k1) ≠ (x2, k2) ⟶
interior k1 ∩ interior k2 = {}) ∧
(⋃{k. ∃x. (x,k) ∈ s} = i)"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto

lemma tagged_division_ofI:
assumes "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1,k1) ∈ s ⟹ (x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹
interior k1 ∩ interior k2 = {}"
and "(⋃{k. ∃x. (x,k) ∈ s} = i)"
shows "s tagged_division_of i"
unfolding tagged_division_of
using assms
apply auto
apply fastforce+
done

lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
assumes "s tagged_division_of i"
shows "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1, k1) ∈ s ⟹ (x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹
interior k1 ∩ interior k2 = {}"
and "(⋃{k. ∃x. (x,k) ∈ s} = i)"
using assms unfolding tagged_division_of by blast+

lemma division_of_tagged_division:
assumes "s tagged_division_of i"
shows "(snd ` s) division_of i"
proof (rule division_ofI)
note assm = tagged_division_ofD[OF assms]
show "⋃(snd ` s) = i" "finite (snd ` s)"
using assm by auto
fix k
assume k: "k ∈ snd ` s"
then obtain xk where xk: "(xk, k) ∈ s"
by auto
then show "k ⊆ i" "k ≠ {}" "∃a b. k = cbox a b"
using assm by fastforce+
fix k'
assume k': "k' ∈ snd ` s" "k ≠ k'"
from this(1) obtain xk' where xk': "(xk', k') ∈ s"
by auto
then show "interior k ∩ interior k' = {}"
using assm(5) k'(2) xk by blast
qed

lemma partial_division_of_tagged_division:
assumes "s tagged_partial_division_of i"
shows "(snd ` s) division_of ⋃(snd ` s)"
proof (rule division_ofI)
note assm = tagged_partial_division_ofD[OF assms]
show "finite (snd ` s)" "⋃(snd ` s) = ⋃(snd ` s)"
using assm by auto
fix k
assume k: "k ∈ snd ` s"
then obtain xk where xk: "(xk, k) ∈ s"
by auto
then show "k ≠ {}" "∃a b. k = cbox a b" "k ⊆ ⋃(snd ` s)"
using assm by auto
fix k'
assume k': "k' ∈ snd ` s" "k ≠ k'"
from this(1) obtain xk' where xk': "(xk', k') ∈ s"
by auto
then show "interior k ∩ interior k' = {}"
using assm(5) k'(2) xk by auto
qed

lemma tagged_partial_division_subset:
assumes "s tagged_partial_division_of i"
and "t ⊆ s"
shows "t tagged_partial_division_of i"
using assms
unfolding tagged_partial_division_of_def
using finite_subset[OF assms(2)]
by blast

lemma setsum_over_tagged_division_lemma:
assumes "p tagged_division_of i"
and "⋀u v. cbox u v ≠ {} ⟹ content (cbox u v) = 0 ⟹ d (cbox u v) = 0"
shows "setsum (λ(x,k). d k) p = setsum d (snd ` p)"
proof -
have *: "(λ(x,k). d k) = d ∘ snd"
unfolding o_def by (rule ext) auto
note assm = tagged_division_ofD[OF assms(1)]
show ?thesis
unfolding *
proof (rule setsum.reindex_nontrivial[symmetric])
show "finite p"
using assm by auto
fix x y
assume "x∈p" "y∈p" "x≠y" "snd x = snd y"
obtain a b where ab: "snd x = cbox a b"
using assm(4)[of "fst x" "snd x"] ‹x∈p› by auto
have "(fst x, snd y) ∈ p" "(fst x, snd y) ≠ y"
by (metis prod.collapse ‹x∈p› ‹snd x = snd y› ‹x ≠ y›)+
with ‹x∈p› ‹y∈p› have "interior (snd x) ∩ interior (snd y) = {}"
by (intro assm(5)[of "fst x" _ "fst y"]) auto
then have "content (cbox a b) = 0"
unfolding ‹snd x = snd y›[symmetric] ab content_eq_0_interior by auto
then have "d (cbox a b) = 0"
using assm(2)[of "fst x" "snd x"] ‹x∈p› ab[symmetric] by (intro assms(2)) auto
then show "d (snd x) = 0"
unfolding ab by auto
qed
qed

lemma tag_in_interval: "p tagged_division_of i ⟹ (x, k) ∈ p ⟹ x ∈ i"
by auto

lemma tagged_division_of_empty: "{} tagged_division_of {}"
unfolding tagged_division_of by auto

lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} ⟷ p = {}"
unfolding tagged_partial_division_of_def by auto

lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} ⟷ p = {}"
unfolding tagged_division_of by auto

lemma tagged_division_of_self: "x ∈ cbox a b ⟹ {(x,cbox a b)} tagged_division_of (cbox a b)"
by (rule tagged_division_ofI) auto

lemma tagged_division_of_self_real: "x ∈ {a .. b::real} ⟹ {(x,{a .. b})} tagged_division_of {a .. b}"
unfolding box_real[symmetric]
by (rule tagged_division_of_self)

lemma tagged_division_union:
assumes "p1 tagged_division_of s1"
and "p2 tagged_division_of s2"
and "interior s1 ∩ interior s2 = {}"
shows "(p1 ∪ p2) tagged_division_of (s1 ∪ s2)"
proof (rule tagged_division_ofI)
note p1 = tagged_division_ofD[OF assms(1)]
note p2 = tagged_division_ofD[OF assms(2)]
show "finite (p1 ∪ p2)"
using p1(1) p2(1) by auto
show "⋃{k. ∃x. (x, k) ∈ p1 ∪ p2} = s1 ∪ s2"
using p1(6) p2(6) by blast
fix x k
assume xk: "(x, k) ∈ p1 ∪ p2"
show "x ∈ k" "∃a b. k = cbox a b"
using xk p1(2,4) p2(2,4) by auto
show "k ⊆ s1 ∪ s2"
using xk p1(3) p2(3) by blast
fix x' k'
assume xk': "(x', k') ∈ p1 ∪ p2" "(x, k) ≠ (x', k')"
have *: "⋀a b. a ⊆ s1 ⟹ b ⊆ s2 ⟹ interior a ∩ interior b = {}"
using assms(3) interior_mono by blast
show "interior k ∩ interior k' = {}"
apply (cases "(x, k) ∈ p1")
apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
qed

lemma tagged_division_unions:
assumes "finite iset"
and "∀i∈iset. pfn i tagged_division_of i"
and "∀i1∈iset. ∀i2∈iset. i1 ≠ i2 ⟶ interior(i1) ∩ interior(i2) = {}"
shows "⋃(pfn ` iset) tagged_division_of (⋃iset)"
proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF assms(2)[rule_format]]
show "finite (⋃(pfn ` iset))"
apply (rule finite_Union)
using assms
apply auto
done
have "⋃{k. ∃x. (x, k) ∈ ⋃(pfn ` iset)} = ⋃((λi. ⋃{k. ∃x. (x, k) ∈ pfn i}) ` iset)"
by blast
also have "… = ⋃iset"
using assm(6) by auto
finally show "⋃{k. ∃x. (x, k) ∈ ⋃(pfn ` iset)} = ⋃iset" .
fix x k
assume xk: "(x, k) ∈ ⋃(pfn ` iset)"
then obtain i where i: "i ∈ iset" "(x, k) ∈ pfn i"
by auto
show "x ∈ k" "∃a b. k = cbox a b" "k ⊆ ⋃iset"
using assm(2-4)[OF i] using i(1) by auto
fix x' k'
assume xk': "(x', k') ∈ ⋃(pfn ` iset)" "(x, k) ≠ (x', k')"
then obtain i' where i': "i' ∈ iset" "(x', k') ∈ pfn i'"
by auto
have *: "⋀a b. i ≠ i' ⟹ a ⊆ i ⟹ b ⊆ i' ⟹ interior a ∩ interior b = {}"
using i(1) i'(1)
using assms(3)[rule_format] interior_mono
by blast
show "interior k ∩ interior k' = {}"
apply (cases "i = i'")
using assm(5) i' i(2) xk'(2) apply blast
using "*" assm(3) i' i by auto
qed

lemma tagged_partial_division_of_union_self:
assumes "p tagged_partial_division_of s"
shows "p tagged_division_of (⋃(snd ` p))"
apply (rule tagged_division_ofI)
using tagged_partial_division_ofD[OF assms]
apply auto
done

lemma tagged_division_of_union_self:
assumes "p tagged_division_of s"
shows "p tagged_division_of (⋃(snd ` p))"
apply (rule tagged_division_ofI)
using tagged_division_ofD[OF assms]
apply auto
done

subsection ‹Fine-ness of a partition w.r.t. a gauge.›

definition fine  (infixr "fine" 46)
where "d fine s ⟷ (∀(x,k) ∈ s. k ⊆ d x)"

lemma fineI:
assumes "⋀x k. (x, k) ∈ s ⟹ k ⊆ d x"
shows "d fine s"
using assms unfolding fine_def by auto

lemma fineD[dest]:
assumes "d fine s"
shows "⋀x k. (x,k) ∈ s ⟹ k ⊆ d x"
using assms unfolding fine_def by auto

lemma fine_inter: "(λx. d1 x ∩ d2 x) fine p ⟷ d1 fine p ∧ d2 fine p"
unfolding fine_def by auto

lemma fine_inters:
"(λx. ⋂{f d x | d.  d ∈ s}) fine p ⟷ (∀d∈s. (f d) fine p)"
unfolding fine_def by blast

lemma fine_union: "d fine p1 ⟹ d fine p2 ⟹ d fine (p1 ∪ p2)"
unfolding fine_def by blast

lemma fine_unions: "(⋀p. p ∈ ps ⟹ d fine p) ⟹ d fine (⋃ps)"
unfolding fine_def by auto

lemma fine_subset: "p ⊆ q ⟹ d fine q ⟹ d fine p"
unfolding fine_def by blast

subsection ‹Gauge integral. Define on compact intervals first, then use a limit.›

definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
where "(f has_integral_compact_interval y) i ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of i ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - y) < e))"

definition has_integral ::
"('n::euclidean_space ⇒ 'b::real_normed_vector) ⇒ 'b ⇒ 'n set ⇒ bool"
(infixr "has'_integral" 46)
where "(f has_integral y) i ⟷
(if ∃a b. i = cbox a b
then (f has_integral_compact_interval 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_compact_interval z) (cbox a b) ∧
norm (z - y) < e)))"

lemma has_integral:
"(f has_integral y) (cbox a b) ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm (setsum (λ(x,k). content(k) *⇩R f x) p - y) < e))"
unfolding has_integral_def has_integral_compact_interval_def
by auto

lemma has_integral_real:
"(f has_integral y) {a .. b::real} ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of {a .. b} ∧ d fine p ⟶
norm (setsum (λ(x,k). content(k) *⇩R f x) p - 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 d where "gauge d"
and "⋀p. p tagged_division_of (cbox a b) ⟹ d fine p ⟹
norm (setsum (λ(x,k). content(k) *⇩R f(x)) p - 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)))"
unfolding has_integral
unfolding has_integral_compact_interval_def has_integral_def
by auto

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
unfolding has_integral
unfolding has_integral_compact_interval_def has_integral_def
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)"

lemma integrable_integral[dest]: "f integrable_on i ⟹ (f has_integral (integral i f)) i"
unfolding integrable_on_def integral_def by (rule someI_ex)

lemma has_integral_integrable[intro]: "(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

lemma setsum_content_null:
assumes "content (cbox a b) = 0"
and "p tagged_division_of (cbox a b)"
shows "setsum (λ(x,k). content k *⇩R f x) p = (0::'a::real_normed_vector)"
proof (rule setsum.neutral, rule)
fix y
assume y: "y ∈ p"
obtain x k where xk: "y = (x, k)"
using surj_pair[of y] by blast
note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
from this(2) obtain c d where k: "k = cbox c d" by blast
have "(λ(x, k). content k *⇩R f x) y = content k *⇩R f x"
unfolding xk by auto
also have "… = 0"
using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
unfolding assms(1) k
by auto
finally show "(λ(x, k). content k *⇩R f x) y = 0" .
qed

subsection ‹Some basic combining lemmas.›

lemma tagged_division_unions_exists:
assumes "finite iset"
and "∀i∈iset. ∃p. p tagged_division_of i ∧ d fine p"
and "∀i1∈iset. ∀i2∈iset. i1 ≠ i2 ⟶ interior i1 ∩ interior i2 = {}"
and "⋃iset = i"
obtains p where "p tagged_division_of i" and "d fine p"
proof -
obtain pfn where pfn:
"⋀x. x ∈ iset ⟹ pfn x tagged_division_of x"
"⋀x. x ∈ iset ⟹ d fine pfn x"
using bchoice[OF assms(2)] by auto
show thesis
apply (rule_tac p="⋃(pfn ` iset)" in that)
using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
qed

subsection ‹The set we're concerned with must be closed.›

lemma division_of_closed:
fixes i :: "'n::euclidean_space set"
shows "s division_of i ⟹ closed i"
unfolding division_of_def by fastforce

subsection ‹General bisection principle for intervals; might be useful elsewhere.›

lemma interval_bisection_step:
fixes type :: "'a::euclidean_space"
assumes "P {}"
and "∀s t. P s ∧ P t ∧ interior(s) ∩ interior(t) = {} ⟶ P (s ∪ t)"
and "¬ P (cbox a (b::'a))"
obtains c d where "¬ P (cbox c d)"
and "∀i∈Basis. a∙i ≤ c∙i ∧ c∙i ≤ d∙i ∧ d∙i ≤ b∙i ∧ 2 * (d∙i - c∙i) ≤ b∙i - a∙i"
proof -
have "cbox a b ≠ {}"
using assms(1,3) by metis
then have ab: "⋀i. i∈Basis ⟹ a ∙ i ≤ b ∙ i"
by (force simp: mem_box)
{ fix f
have "⟦finite f;
⋀s. s∈f ⟹ P s;
⋀s. s∈f ⟹ ∃a b. s = cbox a b;
⋀s t. s∈f ⟹ t∈f ⟹ s ≠ t ⟹ interior s ∩ interior t = {}⟧ ⟹ P (⋃f)"
proof (induct f rule: finite_induct)
case empty
show ?case
using assms(1) by auto
next
case (insert x f)
show ?case
unfolding Union_insert
apply (rule assms(2)[rule_format])
using inter_interior_unions_intervals [of f "interior x"]
apply (auto simp: insert)
by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
qed
} note UN_cases = this
let ?A = "{cbox c d | c d::'a. ∀i∈Basis. (c∙i = a∙i) ∧ (d∙i = (a∙i + b∙i) / 2) ∨
(c∙i = (a∙i + b∙i) / 2) ∧ (d∙i = b∙i)}"
let ?PP = "λc d. ∀i∈Basis. a∙i ≤ c∙i ∧ c∙i ≤ d∙i ∧ d∙i ≤ b∙i ∧ 2 * (d∙i - c∙i) ≤ b∙i - a∙i"
{
presume "∀c d. ?PP c d ⟶ P (cbox c d) ⟹ False"
then show thesis
unfolding atomize_not not_all
by (blast intro: that)
}
assume as: "∀c d. ?PP c d ⟶ P (cbox c d)"
have "P (⋃?A)"
proof (rule UN_cases)
let ?B = "(λs. cbox (∑i∈Basis. (if i ∈ s then a∙i else (a∙i + b∙i) / 2) *⇩R i::'a)
(∑i∈Basis. (if i ∈ s then (a∙i + b∙i) / 2 else b∙i) *⇩R i)) ` {s. s ⊆ Basis}"
have "?A ⊆ ?B"
proof
fix x
assume "x ∈ ?A"
then obtain c d
where x:  "x = cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i" by blast
show "x ∈ ?B"
unfolding image_iff x
apply (rule_tac x="{i. i∈Basis ∧ c∙i = a∙i}" in bexI)
apply (rule arg_cong2 [where f = cbox])
using x(2) ab
apply (auto simp add: euclidean_eq_iff[where 'a='a])
by fastforce
qed
then show "finite ?A"
by (rule finite_subset) auto
next
fix s
assume "s ∈ ?A"
then obtain c d
where s: "s = cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i"
by blast
show "P s"
unfolding s
apply (rule as[rule_format])
using ab s(2) by force
show "∃a b. s = cbox a b"
unfolding s by auto
fix t
assume "t ∈ ?A"
then obtain e f where t:
"t = cbox e f"
"⋀i. i ∈ Basis ⟹
e ∙ i = a ∙ i ∧ f ∙ i = (a ∙ i + b ∙ i) / 2 ∨
e ∙ i = (a ∙ i + b ∙ i) / 2 ∧ f ∙ i = b ∙ i"
by blast
assume "s ≠ t"
then have "¬ (c = e ∧ d = f)"
unfolding s t by auto
then obtain i where "c∙i ≠ e∙i ∨ d∙i ≠ f∙i" and i': "i ∈ Basis"
unfolding euclidean_eq_iff[where 'a='a] by auto
then have i: "c∙i ≠ e∙i" "d∙i ≠ f∙i"
using s(2) t(2) apply fastforce
using t(2)[OF i'] ‹c ∙ i ≠ e ∙ i ∨ d ∙ i ≠ f ∙ i› i' s(2) t(2) by fastforce
have *: "⋀s t. (⋀a. a ∈ s ⟹ a ∈ t ⟹ False) ⟹ s ∩ t = {}"
by auto
show "interior s ∩ interior t = {}"
unfolding s t interior_cbox
proof (rule *)
fix x
assume "x ∈ box c d" "x ∈ box e f"
then have x: "c∙i < d∙i" "e∙i < f∙i" "c∙i < f∙i" "e∙i < d∙i"
unfolding mem_box using i'
by force+
show False  using s(2)[OF i']
proof safe
assume as: "c ∙ i = a ∙ i" "d ∙ i = (a ∙ i + b ∙ i) / 2"
show False
using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
next
assume as: "c ∙ i = (a ∙ i + b ∙ i) / 2" "d ∙ i = b ∙ i"
show False
using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
qed
qed
qed
also have "⋃?A = cbox a b"
proof (rule set_eqI,rule)
fix x
assume "x ∈ ⋃?A"
then obtain c d where x:
"x ∈ cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i"
by blast
show "x∈cbox a b"
unfolding mem_box
proof safe
fix i :: 'a
assume i: "i ∈ Basis"
then show "a ∙ i ≤ x ∙ i" "x ∙ i ≤ b ∙ i"
using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
qed
next
fix x
assume x: "x ∈ cbox a b"
have "∀i∈Basis.
∃c d. (c = a∙i ∧ d = (a∙i + b∙i) / 2 ∨ c = (a∙i + b∙i) / 2 ∧ d = b∙i) ∧ c≤x∙i ∧ x∙i ≤ d"
(is "∀i∈Basis. ∃c d. ?P i c d")
unfolding mem_box
proof
fix i :: 'a
assume i: "i ∈ Basis"
have "?P i (a∙i) ((a ∙ i + b ∙ i) / 2) ∨ ?P i ((a ∙ i + b ∙ i) / 2) (b∙i)"
using x[unfolded mem_box,THEN bspec, OF i] by auto
then show "∃c d. ?P i c d"
by blast
qed
then show "x∈⋃?A"
unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
apply auto
apply (rule_tac x="cbox xa xaa" in exI)
unfolding mem_box
apply auto
done
qed
finally show False
using assms by auto
qed

lemma interval_bisection:
fixes type :: "'a::euclidean_space"
assumes "P {}"
and "(∀s t. P s ∧ P t ∧ interior(s) ∩ interior(t) = {} ⟶ P(s ∪ t))"
and "¬ P (cbox a (b::'a))"
obtains x where "x ∈ cbox a b"
and "∀e>0. ∃c d. x ∈ cbox c d ∧ cbox c d ⊆ ball x e ∧ cbox c d ⊆ cbox a b ∧ ¬ P (cbox c d)"
proof -
have "∀x. ∃y. ¬ P (cbox (fst x) (snd x)) ⟶ (¬ P (cbox (fst y) (snd y)) ∧
(∀i∈Basis. fst x∙i ≤ fst y∙i ∧ fst y∙i ≤ snd y∙i ∧ snd y∙i ≤ snd x∙i ∧
2 * (snd y∙i - fst y∙i) ≤ snd x∙i - fst x∙i))" (is "∀x. ?P x")
proof
show "?P x" for x
proof (cases "P (cbox (fst x) (snd x))")
case True
then show ?thesis by auto
next
case as: False
obtain c d where "¬ P (cbox c d)"
"∀i∈Basis.
fst x ∙ i ≤ c ∙ i ∧
c ∙ i ≤ d ∙ i ∧
d ∙ i ≤ snd x ∙ i ∧
2 * (d ∙ i - c ∙ i) ≤ snd x ∙ i - fst x ∙ i"
by (rule interval_bisection_step[of P, OF assms(1-2) as])
then show ?thesis
apply -
apply (rule_tac x="(c,d)" in exI)
apply auto
done
qed
qed
then obtain f where f:
"∀x.
¬ P (cbox (fst x) (snd x)) ⟶
¬ P (cbox (fst (f x)) (snd (f x))) ∧
(∀i∈Basis.
fst x ∙ i ≤ fst (f x) ∙ i ∧
fst (f x) ∙ i ≤ snd (f x) ∙ i ∧
snd (f x) ∙ i ≤ snd x ∙ i ∧
2 * (snd (f x) ∙ i - fst (f x) ∙ i) ≤ snd x ∙ i - fst x ∙ i)"
apply -
apply (drule choice)
apply blast
done
def AB ≡ "λn. (f ^^ n) (a,b)"
def A ≡ "λn. fst(AB n)"
def B ≡ "λn. snd(AB n)"
note ab_def = A_def B_def AB_def
have "A 0 = a" "B 0 = b" "⋀n. ¬ P (cbox (A(Suc n)) (B(Suc n))) ∧
(∀i∈Basis. A(n)∙i ≤ A(Suc n)∙i ∧ A(Suc n)∙i ≤ B(Suc n)∙i ∧ B(Suc n)∙i ≤ B(n)∙i ∧
2 * (B(Suc n)∙i - A(Suc n)∙i) ≤ B(n)∙i - A(n)∙i)" (is "⋀n. ?P n")
proof -
show "A 0 = a" "B 0 = b"
unfolding ab_def by auto
note S = ab_def funpow.simps o_def id_apply
show "?P n" for n
proof (induct n)
case 0
then show ?case
unfolding S
apply (rule f[rule_format]) using assms(3)
apply auto
done
next
case (Suc n)
show ?case
unfolding S
apply (rule f[rule_format])
using Suc
unfolding S
apply auto
done
qed
qed
note AB = this(1-2) conjunctD2[OF this(3),rule_format]

have interv: "∃n. ∀x∈cbox (A n) (B n). ∀y∈cbox (A n) (B n). dist x y < e"
if e: "0 < e" for e
proof -
obtain n where n: "(∑i∈Basis. b ∙ i - a ∙ i) / e < 2 ^ n"
using real_arch_pow2[of "(setsum (λi. b∙i - a∙i) Basis) / e"] ..
show ?thesis
proof (rule exI [where x=n], clarify)
fix x y
assume xy: "x∈cbox (A n) (B n)" "y∈cbox (A n) (B n)"
have "dist x y ≤ setsum (λi. ¦(x - y)∙i¦) Basis"
unfolding dist_norm by(rule norm_le_l1)
also have "… ≤ setsum (λi. B n∙i - A n∙i) Basis"
proof (rule setsum_mono)
fix i :: 'a
assume i: "i ∈ Basis"
show "¦(x - y) ∙ i¦ ≤ B n ∙ i - A n ∙ i"
using xy[unfolded mem_box,THEN bspec, OF i]
by (auto simp: inner_diff_left)
qed
also have "… ≤ setsum (λi. b∙i - a∙i) Basis / 2^n"
unfolding setsum_divide_distrib
proof (rule setsum_mono)
show "B n ∙ i - A n ∙ i ≤ (b ∙ i - a ∙ i) / 2 ^ n" if i: "i ∈ Basis" for i
proof (induct n)
case 0
then show ?case
unfolding AB by auto
next
case (Suc n)
have "B (Suc n) ∙ i - A (Suc n) ∙ i ≤ (B n ∙ i - A n ∙ i) / 2"
using AB(4)[of i n] using i by auto
also have "… ≤ (b ∙ i - a ∙ i) / 2 ^ Suc n"
using Suc by (auto simp add: field_simps)
finally show ?case .
qed
qed
also have "… < e"
using n using e by (auto simp add: field_simps)
finally show "dist x y < e" .
qed
qed
{
fix n m :: nat
assume "m ≤ n" then have "cbox (A n) (B n) ⊆ cbox (A m) (B m)"
proof (induction rule: inc_induct)
case (step i)
show ?case
using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
qed simp
} note ABsubset = this
have "∃a. ∀n. a∈ cbox (A n) (B n)"
by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
(metis nat.exhaust AB(1-3) assms(1,3))
then obtain x0 where x0: "⋀n. x0 ∈ cbox (A n) (B n)"
by blast
show thesis
proof (rule that[rule_format, of x0])
show "x0∈cbox a b"
using x0[of 0] unfolding AB .
fix e :: real
assume "e > 0"
from interv[OF this] obtain n
where n: "∀x∈cbox (A n) (B n). ∀y∈cbox (A n) (B n). dist x y < e" ..
have "¬ P (cbox (A n) (B n))"
apply (cases "0 < n")
using AB(3)[of "n - 1"] assms(3) AB(1-2)
apply auto
done
moreover have "cbox (A n) (B n) ⊆ ball x0 e"
using n using x0[of n] by auto
moreover have "cbox (A n) (B n) ⊆ cbox a b"
unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
ultimately show "∃c d. x0 ∈ cbox c d ∧ cbox c d ⊆ ball x0 e ∧ cbox c d ⊆ cbox a b ∧ ¬ P (cbox c d)"
apply (rule_tac x="A n" in exI)
apply (rule_tac x="B n" in exI)
apply (auto simp: x0)
done
qed
qed

subsection ‹Cousin's lemma.›

lemma fine_division_exists:
fixes a b :: "'a::euclidean_space"
assumes "gauge g"
obtains p where "p tagged_division_of (cbox a b)" "g fine p"
proof -
presume "¬ (∃p. p tagged_division_of (cbox a b) ∧ g fine p) ⟹ False"
then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
by blast
then show thesis ..
next
assume as: "¬ (∃p. p tagged_division_of (cbox a b) ∧ g fine p)"
obtain x where x:
"x ∈ (cbox a b)"
"⋀e. 0 < e ⟹
∃c d.
x ∈ cbox c d ∧
cbox c d ⊆ ball x e ∧
cbox c d ⊆ (cbox a b) ∧
¬ (∃p. p tagged_division_of cbox c d ∧ g fine p)"
apply (rule interval_bisection[of "λs. ∃p. p tagged_division_of s ∧ g fine p", OF _ _ as])
apply (simp add: fine_def)
apply (metis tagged_division_union fine_union)
apply (auto simp: )
done
obtain e where e: "e > 0" "ball x e ⊆ g x"
using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
from x(2)[OF e(1)]
obtain c d where c_d: "x ∈ cbox c d"
"cbox c d ⊆ ball x e"
"cbox c d ⊆ cbox a b"
"¬ (∃p. p tagged_division_of cbox c d ∧ g fine p)"
by blast
have "g fine {(x, cbox c d)}"
unfolding fine_def using e using c_d(2) by auto
then show False
using tagged_division_of_self[OF c_d(1)] using c_d by auto
qed

lemma fine_division_exists_real:
fixes a b :: real
assumes "gauge g"
obtains p where "p tagged_division_of {a .. b}" "g fine p"
by (metis assms box_real(2) fine_division_exists)

subsection ‹Basic theorems about integrals.›

lemma has_integral_unique:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral k1) i"
and "(f has_integral k2) i"
shows "k1 = k2"
proof (rule ccontr)
let ?e = "norm (k1 - k2) / 2"
assume as: "k1 ≠ k2"
then have e: "?e > 0"
by auto
have lem: False
if f_k1: "(f has_integral k1) (cbox a b)"
and f_k2: "(f has_integral k2) (cbox a b)"
and "k1 ≠ k2"
for f :: "'n ⇒ 'a" and a b k1 k2
proof -
let ?e = "norm (k1 - k2) / 2"
from ‹k1 ≠ k2› have e: "?e > 0" by auto
obtain d1 where d1:
"gauge d1"
"⋀p. p tagged_division_of cbox a b ⟹
d1 fine p ⟹ norm ((∑(x, k)∈p. content k *⇩R f x) - k1) < norm (k1 - k2) / 2"
by (rule has_integralD[OF f_k1 e]) blast
obtain d2 where d2:
"gauge d2"
"⋀p. p tagged_division_of cbox a b ⟹
d2 fine p ⟹ norm ((∑(x, k)∈p. content k *⇩R f x) - k2) < norm (k1 - k2) / 2"
by (rule has_integralD[OF f_k2 e]) blast
obtain p where p:
"p tagged_division_of cbox a b"
"(λx. d1 x ∩ d2 x) fine p"
by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
let ?c = "(∑(x, k)∈p. content k *⇩R f x)"
have "norm (k1 - k2) ≤ norm (?c - k2) + norm (?c - k1)"
using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
by (auto simp add:algebra_simps norm_minus_commute)
also have "… < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
apply (rule add_strict_mono)
apply (rule_tac[!] d2(2) d1(2))
using p unfolding fine_def
apply auto
done
finally show False by auto
qed
{
presume "¬ (∃a b. i = cbox a b) ⟹ False"
then show False
using as assms lem by blast
}
assume as: "¬ (∃a b. i = cbox a b)"
obtain B1 where B1:
"0 < B1"
"⋀a b. ball 0 B1 ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b) ∧
norm (z - k1) < norm (k1 - k2) / 2"
by (rule has_integral_altD[OF assms(1) as,OF e]) blast
obtain B2 where B2:
"0 < B2"
"⋀a b. ball 0 B2 ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b) ∧
norm (z - k2) < norm (k1 - k2) / 2"
by (rule has_integral_altD[OF assms(2) as,OF e]) blast
have "∃a b::'n. ball 0 B1 ∪ ball 0 B2 ⊆ cbox a b"
apply (rule bounded_subset_cbox)
using bounded_Un bounded_ball
apply auto
done
then obtain a b :: 'n where ab: "ball 0 B1 ⊆ cbox a b" "ball 0 B2 ⊆ cbox a b"
by blast
obtain w where w:
"((λx. if x ∈ i then f x else 0) 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:
"((λx. if x ∈ i then f x else 0) 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 lem[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 add: norm_minus_commute)
also have "… < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
apply (rule add_strict_mono)
apply (rule_tac[!] z(2) w(2))
done
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_is_0:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "∀x∈s. f x = 0"
shows "(f has_integral 0) s"
proof -
have lem: "⋀a b. ⋀f::'n ⇒ 'a.
(∀x∈cbox a b. f(x) = 0) ⟹ (f has_integral 0) (cbox a b)"
unfolding has_integral
proof clarify
fix a b e
fix f :: "'n ⇒ 'a"
assume as: "∀x∈cbox a b. f x = 0" "0 < (e::real)"
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
if p: "p tagged_division_of cbox a b" for p
proof -
have "(∑(x, k)∈p. content k *⇩R f x) = 0"
proof (rule setsum.neutral, rule)
fix x
assume x: "x ∈ p"
have "f (fst x) = 0"
using tagged_division_ofD(2-3)[OF p, of "fst x" "snd x"] using as x by auto
then show "(λ(x, k). content k *⇩R f x) x = 0"
apply (subst surjective_pairing[of x])
unfolding split_conv
apply auto
done
qed
then show ?thesis
using as by auto
qed
then show "∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶ norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e)"
by auto
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
with assms lem show ?thesis
by blast
}
have *: "(λx. if x ∈ s then f x else 0) = (λx. 0)"
apply (rule ext)
using assms
apply auto
done
assume "¬ (∃a b. s = cbox a b)"
then show ?thesis
using lem
by (subst has_integral_alt) (force simp add: *)
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:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral y) s"
and "bounded_linear h"
shows "((h ∘ f) has_integral ((h y))) s"
proof -
interpret bounded_linear h
using assms(2) .
from pos_bounded obtain B where B: "0 < B" "⋀x. norm (h x) ≤ norm x * B"
by blast
have lem: "⋀(f :: 'n ⇒ 'a) y a b.
(f has_integral y) (cbox a b) ⟹ ((h ∘ f) has_integral h y) (cbox a b)"
unfolding has_integral
proof (clarify, goal_cases)
case prems: (1 f y a b e)
from pos_bounded
obtain B where B: "0 < B" "⋀x. norm (h x) ≤ norm x * B"
by blast
have "e / B > 0" using prems(2) B by simp
then obtain g
where g: "gauge g"
"⋀p. p tagged_division_of (cbox a b) ⟹ g fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R f x) - y) < e / B"
using prems(1) by auto
{
fix p
assume as: "p tagged_division_of (cbox a b)" "g fine p"
have hc: "⋀x k. h ((λ(x, k). content k *⇩R f x) x) = (λ(x, k). h (content k *⇩R f x)) x"
by auto
then have "(∑(x, k)∈p. content k *⇩R (h ∘ f) x) = setsum (h ∘ (λ(x, k). content k *⇩R f x)) p"
unfolding o_def unfolding scaleR[symmetric] hc by simp
also have "… = h (∑(x, k)∈p. content k *⇩R f x)"
using setsum[of "λ(x,k). content k *⇩R f x" p] using as by auto
finally have "(∑(x, k)∈p. content k *⇩R (h ∘ f) x) = h (∑(x, k)∈p. content k *⇩R f x)" .
then have "norm ((∑(x, k)∈p. content k *⇩R (h ∘ f) x) - h y) < e"
apply (simp add: diff[symmetric])
apply (rule le_less_trans[OF B(2)])
using g(2)[OF as] B(1)
apply (auto simp add: field_simps)
done
}
with g show ?case
by (rule_tac x=g in exI) auto
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
then show ?thesis
using assms(1) lem by blast
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
proof (subst has_integral_alt, clarsimp)
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. ((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b) ∧ norm (z - y) < e / B"
using has_integral_altD[OF assms(1) as *] by blast
show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ s then (h ∘ f) x else 0) has_integral z) (cbox a b) ∧ norm (z - h y) < e)"
proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
case prems: (1 a b)
obtain z where z:
"((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b)"
"norm (z - y) < e / B"
using M(2)[OF prems(1)] by blast
have *: "(λx. if x ∈ s then (h ∘ f) x else 0) = h ∘ (λx. if x ∈ s then f x else 0)"
using zero by auto
show ?case
apply (rule_tac x="h z" in exI)
apply (simp add: * lem z(1))
apply (metis B diff le_less_trans pos_less_divide_eq z(2))
done
qed
qed
qed

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 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)

corollary integral_mult_left:
fixes c:: "'a::real_normed_algebra"
shows "f integrable_on s ⟹ integral s (λx. f x * c) = integral s f * c"
by (blast intro:  has_integral_mult_left)

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

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"
proof (cases "c = 0")
case True
then show ?thesis by simp
next
case False
from has_integral_cmul[OF assms[OF this], of c] show ?thesis
unfolding real_scaleR_def .
qed

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_add:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral k) s"
and "(g has_integral l) s"
shows "((λx. f x + g x) has_integral (k + l)) s"
proof -
have lem: "((λx. f x + g x) has_integral (k + l)) (cbox a b)"
if f_k: "(f has_integral k) (cbox a b)"
and g_l: "(g has_integral l) (cbox a b)"
for f :: "'n ⇒ 'a" and g a b k l
unfolding has_integral
proof clarify
fix e :: real
assume e: "e > 0"
then have *: "e / 2 > 0"
by auto
obtain d1 where d1:
"gauge d1"
"⋀p. p tagged_division_of (cbox a b) ⟹ d1 fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R f x) - k) < e / 2"
using has_integralD[OF f_k *] by blast
obtain d2 where d2:
"gauge d2"
"⋀p. p tagged_division_of (cbox a b) ⟹ d2 fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R g x) - l) < e / 2"
using has_integralD[OF g_l *] by blast
show "∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) < e)"
proof (rule exI [where x="λx. (d1 x) ∩ (d2 x)"], clarsimp simp add: gauge_inter[OF d1(1) d2(1)])
fix p
assume as: "p tagged_division_of (cbox a b)" "(λx. d1 x ∩ d2 x) fine p"
have *: "(∑(x, k)∈p. content k *⇩R (f x + g x)) =
(∑(x, k)∈p. content k *⇩R f x) + (∑(x, k)∈p. content k *⇩R g x)"
unfolding scaleR_right_distrib setsum.distrib[of "λ(x,k). content k *⇩R f x" "λ(x,k). content k *⇩R g x" p,symmetric]
by (rule setsum.cong) auto
from as have fine: "d1 fine p" "d2 fine p"
unfolding fine_inter by auto
have "norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) =
norm (((∑(x, k)∈p. content k *⇩R f x) - k) + ((∑(x, k)∈p. content k *⇩R g x) - l))"
unfolding * by (auto simp add: algebra_simps)
also have "… < e/2 + e/2"
apply (rule le_less_trans[OF norm_triangle_ineq])
using as d1 d2 fine
apply (blast intro: add_strict_mono)
done
finally show "norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) < e"
by auto
qed
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
then show ?thesis
using assms lem by force
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
proof (subst has_integral_alt, clarsimp, goal_cases)
case (1 e)
then have *: "e / 2 > 0"
by auto
from has_integral_altD[OF assms(1) as *]
obtain B1 where B1:
"0 < 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 - k) < e / 2"
by blast
from has_integral_altD[OF assms(2) as *]
obtain B2 where B2:
"0 < 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 - l) < e / 2"
by blast
show ?case
proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
fix a b
assume "ball 0 (max B1 B2) ⊆ cbox a (b::'n)"
then have *: "ball 0 B1 ⊆ cbox a (b::'n)" "ball 0 B2 ⊆ cbox a (b::'n)"
by auto
obtain w where w:
"((λx. if x ∈ s then f x else 0) has_integral w) (cbox a b)"
"norm (w - k) < e / 2"
using B1(2)[OF *(1)] by blast
obtain z where z:
"((λx. if x ∈ s then g x else 0) has_integral z) (cbox a b)"
"norm (z - l) < e / 2"
using B2(2)[OF *(2)] by blast
have *: "⋀x. (if x ∈ s then f x + g x else 0) =
(if x ∈ s then f x else 0) + (if x ∈ s then g x else 0)"
by auto
show "∃z. ((λx. if x ∈ s then f x + g x else 0) has_integral z) (cbox a b) ∧ norm (z - (k + l)) < e"
apply (rule_tac x="w + z" in exI)
apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
apply (auto simp add: field_simps)
done
qed
qed
qed

lemma has_integral_sub:
"(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]
unfolding algebra_simps
by auto

lemma integral_0:
"integral s (λx::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
by (rule integral_unique has_integral_0)+

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: "f integrable_on s ⟹ integral s (λx. c *⇩R f x) = c *⇩R integral s f"
by (rule integral_unique) (metis integrable_integral has_integral_cmul)

lemma integral_neg: "f integrable_on s ⟹ integral s (λx. - f x) = - integral s f"
by (rule integral_unique) (metis integrable_integral has_integral_neg)

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_sub)

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_cmult_iff:
fixes c :: real
assumes "c ≠ 0"
shows "(λx. c * f x) integrable_on s ⟷ f integrable_on s"
using integrable_cmul[of "λx. c * f x" s "1 / c"] integrable_cmul[of f s c] ‹c ≠ 0›
by auto

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_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_sub)

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)"
apply (rule has_integral_unique [where i=s and f = "h ∘ f"])
apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
done

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_component,unfolded o_def] ..

lemma has_integral_setsum:
assumes "finite t"
and "∀a∈t. ((f a) has_integral (i a)) s"
shows "((λx. setsum (λa. f a x) t) has_integral (setsum i t)) s"
using assms(1) subset_refl[of t]
proof (induct rule: finite_subset_induct)
case empty
then show ?case by auto
next
case (insert x F)
with assms show ?case
by (simp add: has_integral_add)
qed

lemma integral_setsum:
"⟦finite t;  ∀a∈t. (f a) integrable_on s⟧ ⟹
integral s (λx. setsum (λa. f a x) t) = setsum (λa. integral s (f a)) t"
by (auto intro: has_integral_setsum integrable_integral)

lemma integrable_setsum:
"⟦finite t;  ∀a∈t. (f a) integrable_on s⟧ ⟹ (λx. setsum (λa. f a x) t) integrable_on s"
unfolding integrable_on_def
apply (drule bchoice)
using has_integral_setsum[of t]
apply auto
done

lemma has_integral_eq:
assumes "⋀x. x ∈ s ⟹ f x = g x"
and "(f has_integral k) s"
shows "(g has_integral k) s"
using has_integral_sub[OF assms(2), of "λx. f x - g x" 0]
using has_integral_is_0[of s "λx. f x - g x"]
using assms(1)
by auto

lemma integrable_eq: "(⋀x. x ∈ s ⟹ f x = g x) ⟹ f integrable_on s ⟹ 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"
using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
by auto

lemma integral_cong:
assumes "⋀x. x ∈ s ⟹ f x = g x"
shows "integral s f = integral s g"
unfolding integral_def
by (metis assms has_integral_cong)

lemma has_integral_null [intro]:
assumes "content(cbox a b) = 0"
shows "(f has_integral 0) (cbox a b)"
proof -
have "gauge (λx. ball x 1)"
by auto
moreover
{
fix e :: real
fix p
assume e: "e > 0"
assume p: "p tagged_division_of (cbox a b)"
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) = 0"
unfolding norm_eq_zero diff_0_right
using setsum_content_null[OF assms(1) p, of f] .
then have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
using e by auto
}
ultimately show ?thesis
by (auto simp: has_integral)
qed

lemma has_integral_null_real [intro]:
assumes "content {a .. b::real} = 0"
shows "(f has_integral 0) {a .. b}"
by (metis assms 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 add: 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 (simp add: has_integral_is_0)

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

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 -
have *: "{a} = cbox a a"
apply (rule set_eqI)
unfolding mem_box singleton_iff euclidean_eq_iff[where 'a='a]
apply safe
prefer 3
apply (erule_tac x=b in ballE)
apply (auto simp add: field_simps)
done
show "(f has_integral 0) (cbox a a)" "(f has_integral 0) {a}"
unfolding *
apply (rule_tac[!] has_integral_null)
unfolding content_eq_0_interior
unfolding interior_cbox
using box_sing
apply auto
done
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 (rule integral_unique) 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)"
by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_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) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)

subsection ‹Cauchy-type criterion for integrability.›

(* XXXXXXX *)
lemma integrable_cauchy:
fixes f :: "'n::euclidean_space ⇒ 'a::{real_normed_vector,complete_space}"
shows "f integrable_on cbox a b ⟷
(∀e>0.∃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 ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p1 -
setsum (λ(x,k). content k *⇩R f x) p2) < e))"
(is "?l = (∀e>0. ∃d. ?P e d)")
proof
assume ?l
then guess y unfolding integrable_on_def has_integral .. note y=this
show "∀e>0. ∃d. ?P e d"
proof (clarify, goal_cases)
case (1 e)
then have "e/2 > 0" by auto
then guess d
apply -
apply (drule y[rule_format])
apply (elim exE conjE)
done
note d=this[rule_format]
show ?case
proof (rule_tac x=d in exI, clarsimp simp: d)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
"p2 tagged_division_of (cbox a b)" "d fine p2"
show "norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e"
apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
qed
qed
next
assume "∀e>0. ∃d. ?P e d"
then have "∀n::nat. ∃d. ?P (inverse(of_nat (n + 1))) d"
by auto
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
have "⋀n. gauge (λx. ⋂{d i x |i. i ∈ {0..n}})"
apply (rule gauge_inters)
using d(1)
apply auto
done
then have "∀n. ∃p. p tagged_division_of (cbox a b) ∧ (λx. ⋂{d i x |i. i ∈ {0..n}}) fine p"
by (meson fine_division_exists)
from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
have dp: "⋀i n. i≤n ⟹ d i fine p n"
using p(2) unfolding fine_inters by auto
have "Cauchy (λn. setsum (λ(x,k). content k *⇩R (f x)) (p n))"
proof (rule CauchyI, goal_cases)
case (1 e)
then guess N unfolding real_arch_inv[of e] .. note N=this
show ?case
apply (rule_tac x=N in exI)
proof clarify
fix m n
assume mn: "N ≤ m" "N ≤ n"
have *: "N = (N - 1) + 1" using N by auto
show "norm ((∑(x, k)∈p m. content k *⇩R f x) - (∑(x, k)∈p n. content k *⇩R f x)) < e"
apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
apply(subst *)
using dp p(1) mn d(2) by auto
qed
qed
then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN 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 *:"e/2 > 0" by auto
then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this
then have N1': "N1 = N1 - 1 + 1"
by auto
guess N2 using y[OF *] .. note N2=this
have "gauge (d (N1 + N2))"
using d by auto
moreover
{
fix q
assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
apply (rule less_trans)
using N1
apply auto
done
have "norm ((∑(x, k)∈q. content k *⇩R f x) - y) < e"
apply (rule norm_triangle_half_r)
apply (rule less_trans[OF _ *])
apply (subst N1', rule d(2)[of "p (N1+N2)"])
using N1' as(1) as(2) dp
apply (simp add: ‹∀x. p x tagged_division_of cbox a b ∧ (λxa. ⋂{d i xa |i. i ∈ {0..x}}) fine p x›)
using N2 le_add2 by blast
}
ultimately show "∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R f x) - y) < e)"
by (rule_tac x="d (N1 + N2)" in exI) auto
qed
qed

subsection ‹Additivity of integral on abutting intervals.›

lemma interval_split:
fixes a :: "'a::euclidean_space"
assumes "k ∈ Basis"
shows
"cbox a b ∩ {x. x∙k ≤ c} = cbox a (∑i∈Basis. (if i = k then min (b∙k) c else b∙i) *⇩R i)"
"cbox a b ∩ {x. x∙k ≥ c} = cbox (∑i∈Basis. (if i = k then max (a∙k) c else a∙i) *⇩R i) b"
apply (rule_tac[!] set_eqI)
unfolding Int_iff mem_box mem_Collect_eq
using assms
apply auto
done

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})"
proof cases
note simps = interval_split[OF assms] content_cbox_cases
have *: "Basis = insert k (Basis - {k})" "⋀x. finite (Basis-{x})" "⋀x. x∉Basis-{x}"
using assms by auto
have *: "⋀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))"
"(∏i∈Basis. b∙i - a∙i) = (∏i∈Basis-{k}. b∙i - a∙i) * (b∙k - a∙k)"
apply (subst *(1))
defer
apply (subst *(1))
unfolding setprod.insert[OF *(2-)]
apply auto
done
assume as: "∀i∈Basis. a ∙ i ≤ b ∙ i"
moreover
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 add: field_simps)
moreover
have **: "(∏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 (auto intro!: setprod.cong)
have "¬ a ∙ k ≤ c ⟹ ¬ c ≤ b ∙ k ⟹ False"
unfolding not_le
using as[unfolded ,rule_format,of k] assms
by auto
ultimately show ?thesis
using assms
unfolding simps **
unfolding *(1)[of "λi x. b∙i - x"] *(1)[of "λi x. x - a∙i"]
unfolding *(2)
by auto
next
assume "¬ (∀i∈Basis. a ∙ i ≤ b ∙ i)"
then have "cbox a b = {}"
unfolding box_eq_empty by (auto simp: not_le)
then show ?thesis
by (auto simp: not_le)
qed

lemma division_split_left_inj:
fixes type :: "'a::euclidean_space"
assumes "d division_of i"
and "k1 ∈ d"
and "k2 ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x::'a. x∙k ≤ c} = k2 ∩ {x. x∙k ≤ c}"
and k: "k∈Basis"
shows "content(k1 ∩ {x. x∙k ≤ c}) = 0"
proof -
note d=division_ofD[OF assms(1)]
have *: "⋀(a::'a) b c. content (cbox a b ∩ {x. x∙k ≤ c}) = 0 ⟷
interior(cbox a b ∩ {x. x∙k ≤ c}) = {}"
unfolding  interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
have **: "⋀s t u. s ∩ t = {} ⟹ u ⊆ s ⟹ u ⊆ t ⟹ u = {}"
by auto
show ?thesis
unfolding uv1 uv2 *
apply (rule **[OF d(5)[OF assms(2-4)]])
apply (simp add: uv1)
using assms(5) uv1 by auto
qed

lemma division_split_right_inj:
fixes type :: "'a::euclidean_space"
assumes "d division_of i"
and "k1 ∈ d"
and "k2 ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x::'a. x∙k ≥ c} = k2 ∩ {x. x∙k ≥ c}"
and k: "k ∈ Basis"
shows "content (k1 ∩ {x. x∙k ≥ c}) = 0"
proof -
note d=division_ofD[OF assms(1)]
have *: "⋀a b::'a. ⋀c. content(cbox a b ∩ {x. x∙k ≥ c}) = 0 ⟷
interior(cbox a b ∩ {x. x∙k ≥ c}) = {}"
unfolding interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
have **: "⋀s t u. s ∩ t = {} ⟹ u ⊆ s ⟹ u ⊆ t ⟹ u = {}"
by auto
show ?thesis
unfolding uv1 uv2 *
apply (rule **[OF d(5)[OF assms(2-4)]])
apply (simp add: uv1)
using assms(5) uv1 by auto
qed

lemma tagged_division_split_left_inj:
fixes x1 :: "'a::euclidean_space"
assumes d: "d tagged_division_of i"
and k12: "(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 -
have *: "⋀a b c. (a,b) ∈ c ⟹ b ∈ snd ` c"
by force
show ?thesis
using k12
by (fastforce intro!:  division_split_left_inj[OF division_of_tagged_division[OF d]] *)
qed

lemma tagged_division_split_right_inj:
fixes x1 :: "'a::euclidean_space"
assumes d: "d tagged_division_of i"
and k12: "(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 -
have *: "⋀a b c. (a,b) ∈ c ⟹ b ∈ snd ` c"
by force
show ?thesis
using k12
by (fastforce intro!:  division_split_right_inj[OF division_of_tagged_division[OF d]] *)
qed

lemma division_split:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k∈Basis"
shows "{l ∩ {x. x∙k ≤ c} | l. l ∈ p ∧ l ∩ {x. x∙k ≤ c} ≠ {}} division_of(cbox a b ∩ {x. x∙k ≤ c})"
(is "?p1 division_of ?I1")
and "{l ∩ {x. x∙k ≥ c} | l. l ∈ p ∧ l ∩ {x. x∙k ≥ c} ≠ {}} division_of (cbox a b ∩ {x. x∙k ≥ c})"
(is "?p2 division_of ?I2")
proof (rule_tac[!] division_ofI)
note p = division_ofD[OF assms(1)]
show "finite ?p1" "finite ?p2"
using p(1) by auto
show "⋃?p1 = ?I1" "⋃?p2 = ?I2"
unfolding p(6)[symmetric] by auto
{
fix k
assume "k ∈ ?p1"
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
show "k ⊆ ?I1"
using l p(2) uv by force
show  "k ≠ {}"
by (simp add: l)
show  "∃a b. k = cbox a b"
apply (simp add: l uv p(2-3)[OF l(2)])
apply (subst interval_split[OF k])
apply (auto intro: order.trans)
done
fix k'
assume "k' ∈ ?p1"
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
assume "k ≠ k'"
then show "interior k ∩ interior k' = {}"
unfolding l l' using p(5)[OF l(2) l'(2)] by auto
}
{
fix k
assume "k ∈ ?p2"
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
show "k ⊆ ?I2"
using l p(2) uv by force
show  "k ≠ {}"
by (simp add: l)
show  "∃a b. k = cbox a b"
apply (simp add: l uv p(2-3)[OF l(2)])
apply (subst interval_split[OF k])
apply (auto intro: order.trans)
done
fix k'
assume "k' ∈ ?p2"
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
assume "k ≠ k'"
then show "interior k ∩ interior k' = {}"
unfolding l l' using p(5)[OF l(2) l'(2)] by auto
}
qed

lemma 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)"
proof (unfold has_integral, rule, rule, goal_cases)
case (1 e)
then have e: "e/2 > 0"
by auto
obtain d1
where d1: "gauge d1"
and d1norm:
"⋀p. ⟦p tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c};
d1 fine p⟧ ⟹ norm ((∑(x, k) ∈ p. content k *⇩R f x) - i) < e / 2"
apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
obtain d2
where d2: "gauge d2"
and d2norm:
"⋀p. ⟦p tagged_division_of cbox a b ∩ {x. c ≤ x ∙ k};
d2 fine p⟧ ⟹ norm ((∑(x, k) ∈ p. 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 ?d = "λx. if x∙k = c then (d1 x ∩ d2 x) else ball x ¦x∙k - c¦ ∩ d1 x ∩ d2 x"
have "gauge ?d"
using d1 d2 unfolding gauge_def by auto
then show ?case
proof (rule_tac x="?d" in exI, safe)
fix p
assume "p tagged_division_of (cbox a b)" "?d fine p"
note p = this tagged_division_ofD[OF this(1)]
have xk_le_c: "⋀x kk. (x, kk) ∈ p ⟹ kk ∩ {x. x∙k ≤ c} ≠ {} ⟹ x∙k ≤ c"
proof -
fix x kk
assume as: "(x, kk) ∈ p" and kk: "kk ∩ {x. x∙k ≤ c} ≠ {}"
show "x∙k ≤ c"
proof (rule ccontr)
assume **: "¬ ?thesis"
from this[unfolded not_le]
have "kk ⊆ ball x ¦x ∙ k - c¦"
using p(2)[unfolded fine_def, rule_format,OF as] by auto
with kk 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 add: dist_norm inner_diff_left intro: le_less_trans)
with y show False
using ** by (auto simp add: field_simps)
qed
qed
have xk_ge_c: "⋀x kk. (x, kk) ∈ p ⟹ kk ∩ {x. x∙k ≥ c} ≠ {} ⟹ x∙k ≥ c"
proof -
fix x kk
assume as: "(x, kk) ∈ p" and kk: "kk ∩ {x. x∙k ≥ c} ≠ {}"
show "x∙k ≥ c"
proof (rule ccontr)
assume **: "¬ ?thesis"
from this[unfolded not_le] have "kk ⊆ ball x ¦x ∙ k - c¦"
using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
with kk 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 add: dist_norm inner_diff_left intro: le_less_trans)
with y show False
using ** by (auto simp add: field_simps)
qed
qed

have lem1: "⋀f P Q. (∀x k. (x, k) ∈ {(x, f k) | x k. P x k} ⟶ Q x k) ⟷
(∀x k. P x k ⟶ Q x (f k))"
by auto
have fin_finite: "finite {(x,f k) | x k. (x,k) ∈ s ∧ P x k}" if "finite s" for f s P
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 lem3: "⋀g :: 'a set ⇒ 'a set. finite p ⟹
setsum (λ(x, k). content k *⇩R f x) {(x,g k) |x k. (x,k) ∈ p ∧ g k ≠ {}} =
setsum (λ(x, k). content k *⇩R f x) ((λ(x, k). (x, g k)) ` p)"
by (rule setsum.mono_neutral_left) auto
let ?M1 = "{(x, kk ∩ {x. x∙k ≤ c}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. x∙k ≤ c} ≠ {}}"
have d1_fine: "d1 fine ?M1"
by (force intro: fineI dest: fineD[OF p(2)] simp add: split: split_if_asm)
have "norm ((∑(x, k)∈?M1. content k *⇩R f x) - i) < e/2"
proof (rule d1norm [OF tagged_division_ofI d1_fine])
show "finite ?M1"
by (rule fin_finite p(3))+
show "⋃{k. ∃x. (x, k) ∈ ?M1} = cbox a b ∩ {x. x∙k ≤ c}"
unfolding p(8)[symmetric] by auto
fix x l
assume xl: "(x, l) ∈ ?M1"
then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x ∙ k ≤ c}"
unfolding xl'
using p(4-6)[OF xl'(3)] using xl'(4)
using xk_le_c[OF xl'(3-4)] by auto
show "∃a b. l = cbox a b"
unfolding xl'
using p(6)[OF xl'(3)]
by (fastforce simp add: interval_split[OF k,where c=c])
fix y r
let ?goal = "interior l ∩ interior r = {}"
assume yr: "(y, r) ∈ ?M1"
then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
assume as: "(x, l) ≠ (y, r)"
show "interior l ∩ interior r = {}"
proof (cases "l' = r' ⟶ x' = y'")
case False
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next
case True
then have "l' ≠ r'"
using as unfolding xl' yr' by auto
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
qed
qed
moreover
let ?M2 = "{(x,kk ∩ {x. x∙k ≥ c}) |x kk. (x,kk) ∈ p ∧ kk ∩ {x. x∙k ≥ c} ≠ {}}"
have d2_fine: "d2 fine ?M2"
by (force intro: fineI dest: fineD[OF p(2)] simp add: split: split_if_asm)
have "norm ((∑(x, k)∈?M2. content k *⇩R f x) - j) < e/2"
proof (rule d2norm [OF tagged_division_ofI d2_fine])
show "finite ?M2"
by (rule fin_finite p(3))+
show "⋃{k. ∃x. (x, k) ∈ ?M2} = cbox a b ∩ {x. x∙k ≥ c}"
unfolding p(8)[symmetric] by auto
fix x l
assume xl: "(x, l) ∈ ?M2"
then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x ∙ k ≥ c}"
unfolding xl'
using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
by auto
show "∃a b. l = cbox a b"
unfolding xl'
using p(6)[OF xl'(3)]
by (fastforce simp add: interval_split[OF k, where c=c])
fix y r
let ?goal = "interior l ∩ interior r = {}"
assume yr: "(y, r) ∈ ?M2"
then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
assume as: "(x, l) ≠ (y, r)"
show "interior l ∩ interior r = {}"
proof (cases "l' = r' ⟶ x' = y'")
case False
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next
case True
then have "l' ≠ r'"
using as unfolding xl' yr' by auto
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' 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
also {
have eq0: "⋀x y. x = (0::real) ⟹ x *⇩R (y::'b) = 0"
using scaleR_zero_left 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 "((∑(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
also have "… = (∑(x, ka)∈p. content (ka ∩ {x. x ∙ k ≤ c}) *⇩R f x) +
(∑(x, ka)∈p. content (ka ∩ {x. c ≤ x ∙ k}) *⇩R f x) - (i + j)"
unfolding lem3[OF p(3)]
by (subst setsum.reindex_nontrivial[OF p(3)], auto intro!: k eq0 tagged_division_split_left_inj[OF p(1)] tagged_division_split_right_inj[OF p(1)]
simp: cont_eq)+
also note setsum.distrib[symmetric]
also have "⋀x. x ∈ p ⟹
(λ(x,ka). content (ka ∩ {x. x ∙ k ≤ c}) *⇩R f x) x +
(λ(x,ka). content (ka ∩ {x. c ≤ x ∙ k}) *⇩R f x) x =
(λ(x,ka). content ka *⇩R f x) x"
proof clarify
fix a b
assume "(a, b) ∈ p"
from p(6)[OF this] guess u v by (elim exE) note uv=this
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"
unfolding scaleR_left_distrib[symmetric]
unfolding uv content_split[OF k,of u v c]
by auto
qed
note setsum.cong [OF _ this]
finally have "(∑(x, k)∈{(x, kk ∩ {x. x ∙ k ≤ c}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. x ∙ k ≤ c} ≠ {}}. content k *⇩R f x) - i +
((∑(x, k)∈{(x, kk ∩ {x. c ≤ x ∙ k}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. c ≤ x ∙ k} ≠ {}}. content k *⇩R f x) - j) =
(∑(x, ka)∈p. content ka *⇩R f x) - (i + j)"
by auto
}
finally 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 tagged_division_union_interval:
fixes a :: "'a::euclidean_space"
assumes "p1 tagged_division_of (cbox a b ∩ {x. x∙k ≤ (c::real)})"
and "p2 tagged_division_of (cbox a b ∩ {x. x∙k ≥ c})"
and k: "k ∈ Basis"
shows "(p1 ∪ p2) tagged_division_of (cbox a b)"
proof -
have *: "cbox a b = (cbox a b ∩ {x. x∙k ≤ c}) ∪ (cbox a b ∩ {x. x∙k ≥ c})"
by auto
show ?thesis
apply (subst *)
apply (rule tagged_division_union[OF assms(1-2)])
unfolding interval_split[OF k] interior_cbox
using k
apply (auto simp add: box_def elim!: ballE[where x=k])
done
qed

lemma tagged_division_union_interval_real:
fixes a :: real
assumes "p1 tagged_division_of ({a .. b} ∩ {x. x∙k ≤ (c::real)})"
and "p2 tagged_division_of ({a .. b} ∩ {x. x∙k ≥ c})"
and k: "k ∈ Basis"
shows "(p1 ∪ p2) tagged_division_of {a .. b}"
using assms
unfolding box_real[symmetric]
by (rule tagged_division_union_interval)

lemma has_integral_separate_sides:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "(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 ((setsum (λ(x,k). content k *⇩R f x) p1 + setsum (λ(x,k). content k *⇩R f x) p2) - i) < e"
proof -
guess d using has_integralD[OF assms(1-2)] . note d=this
{ fix p1 p2
assume "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p1"
note p1=tagged_division_ofD[OF this(1)] this
assume "p2 tagged_division_of (cbox a b) ∩ {x. c ≤ x ∙ k}" "d fine p2"
note p2=tagged_division_ofD[OF this(1)] this
note tagged_division_union_interval[OF p1(7) p2(7)] 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
with p1 obtain u v where uv: "b = cbox u v" by auto
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 guess e unfolding mem_interior .. note e=this
have x: "x∙k = c"
using x interior_subset by fastforce
have *: "⋀i. i ∈ Basis ⟹ ¦(x - (x + (e / 2) *⇩R k)) ∙ i¦ = (if i = k then e/2 else 0)"
using e k by (auto simp: inner_simps inner_not_same_Basis)
have "(∑i∈Basis. ¦(x - (x + (e / 2 ) *⇩R k)) ∙ i¦) =
(∑i∈Basis. (if i = k then e / 2 else 0))"
using "*" by (blast intro: setsum.cong)
also have "… < e"
apply (subst setsum.delta)
using e
apply auto
done
finally have "x + (e/2) *⇩R k ∈ ball x e"
unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
then have "x + (e/2) *⇩R k ∈ {x. x∙k = c}"
using e by auto
then show False
unfolding mem_Collect_eq using e 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 setsum.union_inter_neutral) (auto simp: p1 p2)
also have "… < e"
by (rule k d(2) p12 fine_union p1 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
by (auto intro: that[of d] d elim: )
qed

lemma integrable_split[intro]:
fixes f :: "'a::euclidean_space ⇒ 'b::{real_normed_vector,complete_space}"
assumes "f integrable_on cbox a b"
and k: "k ∈ Basis"
shows "f integrable_on (cbox a b ∩ {x. x∙k ≤ c})" (is ?t1)
and "f integrable_on (cbox a b ∩ {x. x∙k ≥ c})" (is ?t2)
proof -
guess y using assms(1) unfolding integrable_on_def .. note y=this
def b' ≡ "∑i∈Basis. (if i = k then min (b∙k) c else b∙i)*⇩R i::'a"
def a' ≡ "∑i∈Basis. (if i = k then max (a∙k) c else a∙i)*⇩R i::'a"
show ?t1 ?t2
unfolding interval_split[OF k] integrable_cauchy
unfolding interval_split[symmetric,OF k]
proof (rule_tac[!] allI impI)+
fix e :: real
assume "e > 0"
then have "e/2>0"
by auto
from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
let ?P = "λA. ∃d. gauge d ∧ (∀p1 p2. p1 tagged_division_of (cbox a b) ∩ A ∧ d fine p1 ∧
p2 tagged_division_of (cbox a b) ∩ A ∧ d fine p2 ⟶
norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e)"
show "?P {x. x ∙ k ≤ c}"
proof (rule_tac x=d in exI, clarsimp simp add: 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 d(1), 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(2)[of p1 p] d(2)[of p2 p]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
by (auto simp add: algebra_simps)
qed
qed
show "?P {x. x ∙ k ≥ c}"
proof (rule_tac x=d in exI, clarsimp simp add: 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 d(1), 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(2)[of p p1] d(2)[of p p2]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
by (auto simp add: algebra_simps)
qed
qed
qed
qed

subsection ‹Generalized notion of additivity.›

definition "neutral opp = (SOME x. ∀y. opp x y = y ∧ opp y x = y)"

definition operative :: "('a ⇒ 'a ⇒ 'a) ⇒ (('b::euclidean_space) set ⇒ 'a) ⇒ bool"
where "operative opp f ⟷
(∀a b. content (cbox a b) = 0 ⟶ f (cbox a b) = neutral opp) ∧
(∀a b c. ∀k∈Basis. f (cbox a b) = opp (f(cbox a b ∩ {x. x∙k ≤ c})) (f (cbox a b ∩ {x. x∙k ≥ c})))"

lemma operativeD[dest]:
fixes type :: "'a::euclidean_space"
assumes "operative opp f"
shows "⋀a b::'a. content (cbox a b) = 0 ⟹ f (cbox a b) = neutral opp"
and "⋀a b c k. k ∈ Basis ⟹ f (cbox a b) =
opp (f (cbox a b ∩ {x. x∙k ≤ c})) (f (cbox a b ∩ {x. x∙k ≥ c}))"
using assms unfolding operative_def by auto

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 operative_empty: "operative opp f ⟹ f {} = neutral opp"
unfolding operative_def by (rule property_empty_interval) auto

subsection ‹Using additivity of lifted function to encode definedness.›

fun lifted where
"lifted (opp :: 'a ⇒ 'a ⇒ 'b) (Some x) (Some y) = Some (opp x y)"
| "lifted opp None _ = (None::'b option)"
| "lifted opp _ None = None"

lemma lifted_simp_1[simp]: "lifted opp v None = None"
by (induct v) auto

definition "monoidal opp ⟷
(∀x y. opp x y = opp y x) ∧
(∀x y z. opp x (opp y z) = opp (opp x y) z) ∧
(∀x. opp (neutral opp) x = x)"

lemma monoidalI:
assumes "⋀x y. opp x y = opp y x"
and "⋀x y z. opp x (opp y z) = opp (opp x y) z"
and "⋀x. opp (neutral opp) x = x"
shows "monoidal opp"
unfolding monoidal_def using assms by fastforce

lemma monoidal_ac:
assumes "monoidal opp"
shows [simp]: "opp (neutral opp) a = a"
and [simp]: "opp a (neutral opp) = a"
and "opp a b = opp b a"
and "opp (opp a b) c = opp a (opp b c)"
and "opp a (opp b c) = opp b (opp a c)"
using assms unfolding monoidal_def by metis+

lemma neutral_lifted [cong]:
assumes "monoidal opp"
shows "neutral (lifted opp) = Some (neutral opp)"
proof -
{ fix x
assume "∀y. lifted opp x y = y ∧ lifted opp y x = y"
then have "Some (neutral opp) = x"
apply (induct x)
apply force
by (metis assms lifted.simps(1) monoidal_ac(2) option.inject) }
note [simp] = this
show ?thesis
apply (subst neutral_def)
apply (intro some_equality allI)
apply (induct_tac y)
apply (auto simp add:monoidal_ac[OF assms])
done
qed

lemma monoidal_lifted[intro]:
assumes "monoidal opp"
shows "monoidal (lifted opp)"
unfolding monoidal_def split_option_all neutral_lifted[OF assms]
using monoidal_ac[OF assms]
by auto

definition "support opp f s = {x. x∈s ∧ f x ≠ neutral opp}"
definition "fold' opp e s = (if finite s then Finite_Set.fold opp e s else e)"
definition "iterate opp s f = fold' (λx a. opp (f x) a) (neutral opp) (support opp f s)"

lemma support_subset[intro]: "support opp f s ⊆ s"
unfolding support_def by auto

lemma support_empty[simp]: "support opp f {} = {}"
using support_subset[of opp f "{}"] by auto

lemma comp_fun_commute_monoidal[intro]:
assumes "monoidal opp"
shows "comp_fun_commute opp"
unfolding comp_fun_commute_def
using monoidal_ac[OF assms]
by auto

lemma support_clauses:
"⋀f g s. support opp f {} = {}"
"⋀f g s. support opp f (insert x s) =
(if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
"⋀f g s. support opp f (s - {x}) = (support opp f s) - {x}"
"⋀f g s. support opp f (s ∪ t) = (support opp f s) ∪ (support opp f t)"
"⋀f g s. support opp f (s ∩ t) = (support opp f s) ∩ (support opp f t)"
"⋀f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
"⋀f g s. support opp g (f ` s) = f ` (support opp (g ∘ f) s)"
unfolding support_def by auto

lemma finite_support[intro]: "finite s ⟹ finite (support opp f s)"
unfolding support_def by auto

lemma iterate_empty[simp]: "iterate opp {} f = neutral opp"
unfolding iterate_def fold'_def by auto

lemma iterate_insert[simp]:
assumes "monoidal opp"
and "finite s"
shows "iterate opp (insert x s) f =
(if x ∈ s then iterate opp s f else opp (f x) (iterate opp s f))"
proof (cases "x ∈ s")
case True
then show ?thesis by (auto simp: insert_absorb iterate_def)
next
case False
note * = comp_fun_commute.comp_comp_fun_commute [OF comp_fun_commute_monoidal[OF assms(1)]]
show ?thesis
proof (cases "f x = neutral opp")
case True
then show ?thesis
using assms ‹x ∉ s›
by (auto simp: iterate_def support_clauses)
next
case False
with ‹x ∉ s› ‹finite s› support_subset show ?thesis
apply (simp add: iterate_def fold'_def support_clauses)
apply (subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
apply (force simp add: )+
done
qed
qed

lemma iterate_some:
"⟦monoidal opp; finite s⟧ ⟹ iterate (lifted opp) s (λx. Some(f x)) = Some (iterate opp s f)"
by (erule finite_induct) (auto simp: monoidal_lifted)

subsection ‹Two key instances of additivity.›

lemma neutral_add[simp]: "neutral op + = (0::'a::comm_monoid_add)"
unfolding neutral_def
by (force elim: allE [where x=0])

lemma operative_content[intro]: "operative (op +) content"
by (force simp add: operative_def content_split[symmetric])

lemma monoidal_monoid[intro]: "monoidal ((op +)::('a::comm_monoid_add) ⇒ 'a ⇒ 'a)"
unfolding monoidal_def neutral_add
by (auto simp add: algebra_simps)

lemma operative_integral:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
shows "operative (lifted(op +)) (λi. if f integrable_on i then Some(integral i f) else None)"
unfolding operative_def neutral_lifted[OF monoidal_monoid] neutral_add
proof safe
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) =
lifted op + (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
apply (simp add: integrable_split)
apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
apply (auto intro: integrable_integral)
done
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
apply (rule_tac x="integral (cbox a b ∩ {x. x ∙ k ≤ c}) f + integral (cbox a b ∩ {x. x ∙ k ≥ c}) f" in exI)
apply (rule has_integral_split[OF _ _ k])
apply (auto intro: integrable_integral)
done
then show False
using False by auto
qed
then show ?thesis
using False by auto
qed
next
fix a b :: 'a
assume "content (cbox a b) = 0"
then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
using has_integral_null_eq
by (auto simp: integrable_on_null)
qed

subsection ‹Points of division of a partition.›

definition "division_points (k::('a::euclidean_space) set) d =
{(j,x). j ∈ Basis ∧ (interval_lowerbound k)∙j < x ∧ x < (interval_upperbound k)∙j ∧
(∃i∈d. (interval_lowerbound i)∙j = x ∨ (interval_upperbound i)∙j = x)}"

lemma division_points_finite:
fixes i :: "'a::euclidean_space set"
assumes "d division_of i"
shows "finite (division_points i d)"
proof -
note assm = division_ofD[OF assms]
let ?M = "λj. {(j,x)|x. (interval_lowerbound i)∙j < x ∧ x < (interval_upperbound i)∙j ∧
(∃i∈d. (interval_lowerbound i)∙j = x ∨ (interval_upperbound i)∙j = x)}"
have *: "division_points i d = ⋃(?M ` Basis)"
unfolding division_points_def by auto
show ?thesis
unfolding * using assm by auto
qed

lemma division_points_subset:
fixes a :: "'a::euclidean_space"
assumes "d division_of (cbox a b)"
and "∀i∈Basis. a∙i < b∙i"  "a∙k < c" "c < b∙k"
and k: "k ∈ Basis"
shows "division_points (cbox a b ∩ {x. x∙k ≤ c}) {l ∩ {x. x∙k ≤ c} | l . l ∈ d ∧ l ∩ {x. x∙k ≤ c} ≠ {}} ⊆
division_points (cbox a b) d" (is ?t1)
and "division_points (cbox a b ∩ {x. x∙k ≥ c}) {l ∩ {x. x∙k ≥ c} | l . l ∈ d ∧ ~(l ∩ {x. x∙k ≥ c} = {})} ⊆
division_points (cbox a b) d" (is ?t2)
proof -
note assm = division_ofD[OF assms(1)]
have *: "∀i∈Basis. a∙i ≤ b∙i"
"∀i∈Basis. a∙i ≤ (∑i∈Basis. (if i = k then min (b ∙ k) c else  b ∙ i) *⇩R i) ∙ i"
"∀i∈Basis. (∑i∈Basis. (if i = k then max (a ∙ k) c else a ∙ i) *⇩R i) ∙ i ≤ b∙i"
"min (b ∙ k) c = c" "max (a ∙ k) c = c"
using assms using less_imp_le by auto
show ?t1 (*FIXME a horrible mess*)
unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
unfolding *
apply (rule subsetI)
unfolding mem_Collect_eq split_beta
apply (erule bexE conjE)+
apply (simp add: )
apply (erule exE conjE)+
proof
fix i l x
assume as:
"a ∙ fst x < snd x" "snd x < (if fst x = k then c else b ∙ fst x)"
"interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
"i = l ∩ {x. x ∙ k ≤ c}" "l ∈ d" "l ∩ {x. x ∙ k ≤ c} ≠ {}"
and fstx: "fst x ∈ Basis"
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *: "∀i∈Basis. u ∙ i ≤ (∑i∈Basis. (if i = k then min (v ∙ k) c else v ∙ i) *⇩R i) ∙ i"
using as(6) unfolding l interval_split[OF k] box_ne_empty as .
have **: "∀i∈Basis. u∙i ≤ v∙i"
using l using as(6) unfolding box_ne_empty[symmetric] by auto
show "∃i∈d. interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
apply (rule bexI[OF _ ‹l ∈ d›])
using as(1-3,5) fstx
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
apply (auto split: split_if_asm)
done
show "snd x < b ∙ fst x"
using as(2) ‹c < b∙k› by (auto split: split_if_asm)
qed
show ?t2
unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
unfolding *
unfolding subset_eq
apply rule
unfolding mem_Collect_eq split_beta
apply (erule bexE conjE)+
apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
apply (erule exE conjE)+
proof
fix i l x
assume as:
"(if fst x = k then c else a ∙ fst x) < snd x" "snd x < b ∙ fst x"
"interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
"i = l ∩ {x. c ≤ x ∙ k}" "l ∈ d" "l ∩ {x. c ≤ x ∙ k} ≠ {}"
and fstx: "fst x ∈ Basis"
from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
have *: "∀i∈Basis. (∑i∈Basis. (if i = k then max (u ∙ k) c else u ∙ i) *⇩R i) ∙ i ≤ v ∙ i"
using as(6) unfolding l interval_split[OF k] box_ne_empty as .
have **: "∀i∈Basis. u∙i ≤ v∙i"
using l using as(6) unfolding box_ne_empty[symmetric] by auto
show "∃i∈d. interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
apply (rule bexI[OF _ ‹l ∈ d›])
using as(1-3,5) fstx
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
apply (auto split: split_if_asm)
done
show "a ∙ fst x < snd x"
using as(1) ‹a∙k < c› by (auto split: split_if_asm)
qed
qed

lemma division_points_psubset:
fixes a :: "'a::euclidean_space"
assumes "d division_of (cbox a b)"
and "∀i∈Basis. a∙i < b∙i"  "a∙k < c" "c < b∙k"
and "l ∈ d"
and "interval_lowerbound l∙k = c ∨ interval_upperbound l∙k = c"
and k: "k ∈ Basis"
shows "division_points (cbox a b ∩ {x. x∙k ≤ c}) {l ∩ {x. x∙k ≤ c} | l. l∈d ∧ l ∩ {x. x∙k ≤ c} ≠ {}} ⊂
division_points (cbox a b) d" (is "?D1 ⊂ ?D")
and "division_points (cbox a b ∩ {x. x∙k ≥ c}) {l ∩ {x. x∙k ≥ c} | l. l∈d ∧ l ∩ {x. x∙k ≥ c} ≠ {}} ⊂
division_points (cbox a b) d" (is "?D2 ⊂ ?D")
proof -
have ab: "∀i∈Basis. a∙i ≤ b∙i"
using assms(2) by (auto intro!:less_imp_le)
guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
have uv: "∀i∈Basis. u∙i ≤ v∙i" "∀i∈Basis. a∙i ≤ u∙i ∧ v∙i ≤ b∙i"
using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
using subset_box(1)
apply auto
apply blast+
done
have *: "interval_upperbound (cbox a b ∩ {x. x ∙ k ≤ interval_upperbound l ∙ k}) ∙ k = interval_upperbound l ∙ k"
"interval_upperbound (cbox a b ∩ {x. x ∙ k ≤ interval_lowerbound l ∙ k}) ∙ k = interval_lowerbound l ∙ k"
unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
using uv[rule_format, of k] ab k
by auto
have "∃x. x ∈ ?D - ?D1"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)∙k)" in exI, force simp add: *)
apply (rule_tac x="(k,(interval_upperbound l)∙k)" in exI, force simp add: *)
done
moreover have "?D1 ⊆ ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D1 ⊂ ?D"
by blast
have *: "interval_lowerbound (cbox a b ∩ {x. x ∙ k ≥ interval_lowerbound l ∙ k}) ∙ k = interval_lowerbound l ∙ k"
"interval_lowerbound (cbox a b ∩ {x. x ∙ k ≥ interval_upperbound l ∙ k}) ∙ k = interval_upperbound l ∙ k"
unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
using uv[rule_format, of k] ab k
by auto
have "∃x. x ∈ ?D - ?D2"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)∙k)" in exI, force simp add: *)
apply (rule_tac x="(k,(interval_upperbound l)∙k)" in exI, force simp add: *)
done
moreover have "?D2 ⊆ ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D2 ⊂ ?D"
by blast
qed

subsection ‹Preservation by divisions and tagged divisions.›

lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
unfolding support_def by auto

lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
unfolding iterate_def support_support by auto

lemma iterate_expand_cases:
"iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
by (simp add: iterate_def fold'_def)

lemma iterate_image:
assumes "monoidal opp"
and "inj_on f s"
shows "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
proof -
have *: "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
if "finite s" "∀x∈s. ∀y∈s. f x = f y ⟶ x = y" for s
using that
proof (induct s)
case empty
then show ?case by simp
next
case insert
with assms(1) show ?case by auto
qed
show ?thesis
apply (cases "finite (support opp g (f ` s))")
prefer 2
apply (metis finite_imageI iterate_expand_cases support_clauses(7))
apply (subst (1) iterate_support[symmetric], subst (2) iterate_support[symmetric])
unfolding support_clauses
apply (rule *)
apply (meson assms(2) finite_imageD subset_inj_on support_subset)
apply (meson assms(2) inj_on_contraD rev_subsetD support_subset)
done
qed

(* This lemma about iterations comes up in a few places. *)
lemma iterate_nonzero_image_lemma:
assumes "monoidal opp"
and "finite s" "g(a) = neutral opp"
and "∀x∈s. ∀y∈s. f x = f y ∧ x ≠ y ⟶ g(f x) = neutral opp"
shows "iterate opp {f x | x. x ∈ s ∧ f x ≠ a} g = iterate opp s (g ∘ f)"
proof -
have *: "{f x |x. x ∈ s ∧ f x ≠ a} = f ` {x. x ∈ s ∧ f x ≠ a}"
by auto
have **: "support opp (g ∘ f) {x ∈ s. f x ≠ a} = support opp (g ∘ f) s"
unfolding support_def using assms(3) by auto
have inj: "inj_on f (support opp (g ∘ f) {x ∈ s. f x ≠ a})"
apply (simp add: inj_on_def)
apply (metis (mono_tags, lifting) assms(4) comp_def mem_Collect_eq support_def)
done
show ?thesis
apply (subst iterate_support[symmetric])
apply (simp add: * support_clauses iterate_image[OF assms(1) inj])
apply (simp add: iterate_def **)
done
qed

lemma iterate_eq_neutral:
assumes "monoidal opp"
and "⋀x. x ∈ s ⟹ f x = neutral opp"
shows "iterate opp s f = neutral opp"
proof -
have [simp]: "support opp f s = {}"
unfolding support_def using assms(2) by auto
show ?thesis
by (subst iterate_support[symmetric]) simp
qed

lemma iterate_op:
"⟦monoidal opp; finite s⟧
⟹ iterate opp s (λx. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)"
by (erule finite_induct) (auto simp: monoidal_ac(4) monoidal_ac(5))

lemma iterate_eq:
assumes "monoidal opp"
and "⋀x. x ∈ s ⟹ f x = g x"
shows "iterate opp s f = iterate opp s g"
proof -
have *: "support opp g s = support opp f s"
unfolding support_def using assms(2) by auto
show ?thesis
proof (cases "finite (support opp f s)")
case False
then show ?thesis
by (simp add: "*" iterate_expand_cases)
next
case True
def su ≡ "support opp f s"
have fsu: "finite su"
using True by (simp add: su_def)
moreover
{ assume "finite su" "su ⊆ s"
then have "iterate opp su f = iterate opp su g"
by (induct su) (auto simp: assms)
}
ultimately have "iterate opp (support opp f s) f = iterate opp (support opp g s) g"
by (simp add: "*" su_def support_subset)
then show ?thesis
by simp
qed
qed

lemma nonempty_witness:
assumes "s ≠ {}"
obtains x where "x ∈ s"
using assms by auto

lemma operative_division:
fixes f :: "'a::euclidean_space set ⇒ 'b"
assumes "monoidal opp"
and "operative opp f"
and "d division_of (cbox a b)"
shows "iterate opp d f = f (cbox a b)"
proof -
def C ≡ "card (division_points (cbox a b) d)"
then show ?thesis
using assms
proof (induct C arbitrary: a b d rule: full_nat_induct)
case (1 a b d)
show ?case
proof (cases "content (cbox a b) = 0")
case True
show "iterate opp d f = f (cbox a b)"
unfolding operativeD(1)[OF assms(2) True]
proof (rule iterate_eq_neutral[OF ‹monoidal opp›])
fix x
assume x: "x ∈ d"
then show "f x = neutral opp"
by (metis division_ofD(4) 1(4) division_of_content_0[OF True] operativeD(1)[OF assms(2)] x)
qed
next
case False
note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
then have ab': "∀i∈Basis. a∙i ≤ b∙i"
by (auto intro!: less_imp_le)
show "iterate opp d f = f (cbox a b)"
proof (cases "division_points (cbox a b) d = {}")
case True
{ fix u v and j :: 'a
assume j: "j ∈ Basis" and as: "cbox u v ∈ d"
then have "cbox u v ≠ {}"
using "1.prems"(3) by blast
then have uv: "∀i∈Basis. u∙i ≤ v∙i" "u∙j ≤ v∙j"
using j unfolding box_ne_empty by auto
have *: "⋀p r Q. ¬ j∈Basis ∨ p ∨ r ∨ (∀x∈d. Q x) ⟹ p ∨ r ∨ Q (cbox u v)"
using as j by auto
have "(j, u∙j) ∉ division_points (cbox a b) d"
"(j, v∙j) ∉ division_points (cbox a b) d" using True by auto
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
moreover
have "a∙j ≤ u∙j" "v∙j ≤ b∙j"
using division_ofD(2,2,3)[OF ‹d division_of cbox a b› as]
apply (metis j subset_box(1) uv(1))
by (metis ‹cbox u v ⊆ cbox a b› j subset_box(1) uv(1))
ultimately have "u∙j = a∙j ∧ v∙j = a∙j ∨ u∙j = b∙j ∧ v∙j = b∙j ∨ u∙j = a∙j ∧ v∙j = b∙j"
unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force }
then have d': "∀i∈d. ∃u v. i = cbox u v ∧
(∀j∈Basis. u∙j = a∙j ∧ v∙j = a∙j ∨ u∙j = b∙j ∧ v∙j = b∙j ∨ u∙j = a∙j ∧ v∙j = b∙j)"
unfolding forall_in_division[OF 1(4)]
by blast
have "(1/2) *⇩R (a+b) ∈ cbox a b"
unfolding mem_box using ab by(auto intro!: less_imp_le simp: inner_simps)
note this[unfolded division_ofD(6)[OF ‹d division_of cbox a b›,symmetric] Union_iff]
then guess i .. note i=this
guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
have "cbox a b ∈ d"
proof -
have "u = a" "v = b"
unfolding euclidean_eq_iff[where 'a='a]
proof safe
fix j :: 'a
assume j: "j ∈ Basis"
note i(2)[unfolded uv mem_box,rule_format,of j]
then show "u ∙ j = a ∙ j" and "v ∙ j = b ∙ j"
using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
qed
then have "i = cbox a b" using uv by auto
then show ?thesis using i by auto
qed
then have deq: "d = insert (cbox a b) (d - {cbox a b})"
by auto
have "iterate opp (d - {cbox a b}) f = neutral opp"
proof (rule iterate_eq_neutral[OF 1(2)])
fix x
assume x: "x ∈ d - {cbox a b}"
then have "x∈d"
by auto note d'[rule_format,OF this]
then guess u v by (elim exE conjE) note uv=this
have "u ≠ a ∨ v ≠ b"
using x[unfolded uv] by auto
then obtain j where "u∙j ≠ a∙j ∨ v∙j ≠ b∙j" and j: "j ∈ Basis"
unfolding euclidean_eq_iff[where 'a='a] by auto
then have "u∙j = v∙j"
using uv(2)[rule_format,OF j] by auto
then have "content (cbox u v) = 0"
unfolding content_eq_0 using j
by force
then show "f x = neutral opp"
unfolding uv(1) by (rule operativeD(1)[OF 1(3)])
qed
then show "iterate opp d f = f (cbox a b)"
apply (subst deq)
apply (subst iterate_insert[OF 1(2)])
using 1
apply auto
done
next
case False
then have "∃x. x ∈ division_points (cbox a b) d"
by auto
then guess k c
unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
apply (elim exE conjE)
done
note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
from this(3) guess j .. note j=this
def d1 ≡ "{l ∩ {x. x∙k ≤ c} | l. l ∈ d ∧ l ∩ {x. x∙k ≤ c} ≠ {}}"
def d2 ≡ "{l ∩ {x. x∙k ≥ c} | l. l ∈ d ∧ l ∩ {x. x∙k ≥ c} ≠ {}}"
def cb ≡ "(∑i∈Basis. (if i = k then c else b∙i) *⇩R i)::'a"
def ca ≡ "(∑i∈Basis. (if i = k then c else a∙i) *⇩R i)::'a"
note division_points_psubset[OF ‹d division_of cbox a b› ab kc(1-2) j]
note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
then have *: "(iterate opp d1 f) = f (cbox a b ∩ {x. x∙k ≤ c})"
"(iterate opp d2 f) = f (cbox a b ∩ {x. x∙k ≥ c})"
unfolding interval_split[OF kc(4)]
apply (rule_tac[!] "1.hyps"[rule_format])
using division_split[OF ‹d division_of cbox a b›, where k=k and c=c]
apply (simp_all add: interval_split 1 kc d1_def d2_def division_points_finite[OF ‹d division_of cbox a b›])
done
{ fix l y
assume as: "l ∈ d" "y ∈ d" "l ∩ {x. x ∙ k ≤ c} = y ∩ {x. x ∙ k ≤ c}" "l ≠ y"
from division_ofD(4)[OF ‹d division_of cbox a b› this(1)] guess u v by (elim exE) note leq=this
have "f (l ∩ {x. x ∙ k ≤ c}) = neutral opp"
unfolding leq interval_split[OF kc(4)]
apply (rule operativeD(1) 1)+
unfolding interval_split[symmetric,OF kc(4)]
using division_split_left_inj 1 as kc leq by blast
} note fxk_le = this
{ fix l y
assume as: "l ∈ d" "y ∈ d" "l ∩ {x. c ≤ x ∙ k} = y ∩ {x. c ≤ x ∙ k}" "l ≠ y"
from division_ofD(4)[OF ‹d division_of cbox a b› this(1)] guess u v by (elim exE) note leq=this
have "f (l ∩ {x. x ∙ k ≥ c}) = neutral opp"
unfolding leq interval_split[OF kc(4)]
apply (rule operativeD(1) 1)+
unfolding interval_split[symmetric,OF kc(4)]
using division_split_right_inj 1 leq as kc by blast
} note fxk_ge = this
have "f (cbox a b) = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
unfolding *
using assms(2) kc(4) by blast
also have "iterate opp d1 f = iterate opp d (λl. f(l ∩ {x. x∙k ≤ c}))"
unfolding d1_def empty_as_interval
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
apply (rule 1 division_of_finite operativeD[OF 1(3)])+
apply (force simp add: empty_as_interval[symmetric] fxk_le)+
done
also have "iterate opp d2 f = iterate opp d (λl. f(l ∩ {x. x∙k ≥ c}))"
unfolding d2_def empty_as_interval
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
apply (rule 1 division_of_finite operativeD[OF 1(3)])+
apply (force simp add: empty_as_interval[symmetric] fxk_ge)+
done
also have *: "∀x∈d. f x = opp (f (x ∩ {x. x ∙ k ≤ c})) (f (x ∩ {x. c ≤ x ∙ k}))"
unfolding forall_in_division[OF ‹d division_of cbox a b›]
using assms(2) kc(4) by blast
have "opp (iterate opp d (λl. f (l ∩ {x. x ∙ k ≤ c}))) (iterate opp d (λl. f (l ∩ {x. c ≤ x ∙ k}))) =
iterate opp d f"
apply (subst(3) iterate_eq[OF _ *[rule_format]])
using 1
apply (auto simp: iterate_op[symmetric])
done
finally show ?thesis by auto
qed
qed
qed
qed

lemma iterate_image_nonzero:
assumes "monoidal opp"
and "finite s"
and "⋀x y. ∀x∈s. ∀y∈s. x ≠ y ∧ f x = f y ⟶ g (f x) = neutral opp"
shows "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
using assms
by (induct rule: finite_subset_induct[OF assms(2) subset_refl]) auto

lemma operative_tagged_division:
assumes "monoidal opp"
and "operative opp f"
and "d tagged_division_of (cbox a b)"
shows "iterate opp d (λ(x,l). f l) = f (cbox a b)"
proof -
have *: "(λ(x,l). f l) = f ∘ snd"
unfolding o_def by rule auto note tagged = tagged_division_ofD[OF assms(3)]
{ fix a b a'
assume as: "(a, b) ∈ d" "(a', b) ∈ d" "(a, b) ≠ (a', b)"
have "f b = neutral opp"
using tagged(4)[OF as(1)]
apply clarify
apply (rule operativeD(1)[OF assms(2)])
by (metis content_eq_0_interior inf.idem tagged_division_ofD(5)[OF assms(3) as(1-3)])
}
then have "iterate opp d (λ(x,l). f l) = iterate opp (snd ` d) f"
unfolding *
by (force intro!: assms iterate_image_nonzero[symmetric, OF _ tagged_division_of_finite])
also have "… = f (cbox a b)"
using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
finally show ?thesis .
qed

subsection ‹Additivity of content.›

lemma setsum_iterate:
assumes "finite s"
shows "setsum f s = iterate op + s f"
proof -
have "setsum f s = setsum f (support op + f s)"
using assms
by (auto simp: support_def intro: setsum.mono_neutral_right)
then show ?thesis unfolding iterate_def fold'_def setsum.eq_fold
by (simp```