# Theory Integration

theory Integration
imports Derivative Indicator_Function
```(*  Author:     John Harrison
Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light)
*)

header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}

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

lemma cSup_abs_le: (* TODO: is this really needed? *)
fixes S :: "real set"
shows "S ≠ {} ==> (∀x∈S. ¦x¦ ≤ a) ==> ¦Sup S¦ ≤ a"
by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2 bdd_aboveI)

lemma cInf_abs_ge: (* TODO: is this really needed? *)
fixes S :: "real set"
shows "S ≠ {} ==> (∀x∈S. ¦x¦ ≤ a) ==> ¦Inf S¦ ≤ a"
by (simp add: Inf_real_def) (insert cSup_abs_le [of "uminus ` S"], auto)

lemma cSup_asclose: (* TODO: is this really needed? *)
fixes S :: "real set"
assumes S: "S ≠ {}"
and b: "∀x∈S. ¦x - l¦ ≤ e"
shows "¦Sup S - l¦ ≤ e"
proof -
have th: "!!(x::real) 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: (* TODO: is this really needed? *)
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

lemma cSup_finite_ge_iff:
fixes S :: "real set"
shows "finite S ==> S ≠ {} ==> a ≤ Sup S <-> (∃x∈S. a ≤ x)"
by (metis cSup_eq_Max Max_ge_iff)

lemma cSup_finite_le_iff:
fixes S :: "real set"
shows "finite S ==> S ≠ {} ==> a ≥ Sup S <-> (∀x∈S. a ≥ x)"
by (metis cSup_eq_Max Max_le_iff)

lemma cInf_finite_ge_iff:
fixes S :: "real set"
shows "finite S ==> S ≠ {} ==> a ≤ Inf S <-> (∀x∈S. a ≤ x)"
by (metis cInf_eq_Min Min_ge_iff)

lemma cInf_finite_le_iff:
fixes S :: "real set"
shows "finite S ==> S ≠ {} ==> a ≥ Inf S <-> (∃x∈S. a ≥ x)"
by (metis cInf_eq_Min Min_le_iff)

(*declare not_less[simp] not_le[simp]*)

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
lemma conjunctD5: assumes "a ∧ b ∧ c ∧ d ∧ e" shows a b c d e 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 (rule assms)
apply assumption
apply assumption
using assms(2) apply auto
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
show ?thesis
apply (rule assms(2))
apply (rule Suc(1)[OF True])
using `?r` apply auto
done
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 (rule assms)
apply (rule assms,assumption,assumption)
using assms(3)
apply auto
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
apply -
apply (rule interior_maximal)
defer
apply (rule open_interior)
unfolding assms(1,2) interior_cbox
apply auto
done
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 (\<Union>f) = {}"
proof (rule ccontr, unfold ex_in_conv[symmetric])
case goal1
have lem1: "!!x e s U. ball x e ⊆ s ∩ interior U <-> ball x e ⊆ s ∩ U"
apply rule
defer
apply (rule_tac Int_greatest)
unfolding open_subset_interior[OF open_ball]
using interior_subset
apply auto
done
have lem2: "!!x s P. ∃x∈s. P x ==> ∃x∈insert x s. P x" by auto
have "!!f. finite f ==> ∀t∈f. ∃a b. t = cbox a b ==>
∃x. x ∈ s ∩ interior (\<Union>f) ==> ∃t∈f. ∃x. ∃e>0. ball x e ⊆ s ∩ t"
proof -
case goal1
then show ?case
proof (induct rule: finite_induct)
case empty
obtain x where "x ∈ s ∩ interior (\<Union>{})"
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 (\<Union>insert i f)"
using insert(5) ..
then obtain e where e: "0 < e ∧ ball x e ⊆ s ∩ interior (\<Union>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 (\<Union>f)"
using e unfolding lem1 unfolding  ball_min_Int by auto
then have "x ∈ s ∩ interior (\<Union>f)" using `d>0` e by auto
then have "∃t∈f. ∃x e. 0 < e ∧ ball x e ⊆ s ∩ t"
apply -
apply (rule insert(3))
using insert(4)
apply auto
done
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 ∩ (\<Union>f)"
proof (rule disjE)
let ?z = "x - (e/2) *⇩R k"
assume as: "x•k = a•k"
have "ball ?z (e / 2) ∩ i = {}"
apply (rule ccontr)
unfolding ex_in_conv[symmetric]
apply (erule exE)
proof -
fix y
assume "y ∈ ball ?z (e / 2) ∩ i"
then have "dist ?z y < e/2" and yi:"y∈i" 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[THEN conjunct1] 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 ∩ (\<Union>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 -
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
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
next
let ?z = "x + (e/2) *⇩R k"
assume as: "x•k = b•k"
have "ball ?z (e / 2) ∩ i = {}"
apply (rule ccontr)
unfolding ex_in_conv[symmetric]
apply (erule exE)
proof -
fix y
assume "y ∈ ball ?z (e / 2) ∩ i"
then have "dist ?z y < e/2" and yi: "y ∈ i"
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[THEN conjunct1] 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 ∩ (\<Union>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 -
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 ∩ \<Union>f" ..
then have "x ∈ s ∩ interior (\<Union>f)"
unfolding lem1[where U="\<Union>f", symmetric]
using centre_in_ball e[THEN conjunct1] by auto
then show ?thesis
apply -
apply (rule lem2, rule insert(3))
using insert(4)
apply auto
done
qed
qed
qed
qed
from this[OF assms(1,3) goal1]
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

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 simp: )

lemma content_cbox':
fixes a :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
shows "content (cbox a b) = (∏i∈Basis. b•i - a•i)"
apply (rule content_cbox)
using assms
unfolding box_ne_empty
apply assumption
done

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 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[intro]: "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)

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)"
apply rule
defer
apply (rule content_pos_lt, assumption)
proof -
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
qed

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

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
show ?thesis
unfolding content_def
unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
unfolding if_not_P[OF False] if_not_P[OF `cbox c d ≠ {}`]
apply (rule setprod_mono)
apply rule
proof
fix i :: 'a
assume i: "i ∈ Basis"
show "0 ≤ b • i - a • i"
using ab_ne[THEN bspec, OF i] i by auto
show "b • i - a • i ≤ d • i - c • i"
using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2),of i]
using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1),of i]
using i by auto
qed
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

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. \<Inter> {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) = {}) ∧
(\<Union>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 "\<Union>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 "\<Union>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
{
assume "s = {{a}}"
moreover fix k assume "k∈s"
ultimately have"∃x y. k = cbox x y"
apply (rule_tac x=a in exI)+
unfolding cbox_sing
apply auto
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 (\<Union>p)"
and "q ⊆ p"
shows "q division_of (\<Union>q)"
proof (rule division_ofI)
note * = division_ofD[OF assms(1)]
show "finite q"
apply (rule finite_subset)
using *(1) assms(2)
apply auto
done
{
fix k
assume "k ∈ q"
then have kp: "k ∈ p"
using assms(2) by auto
show "k ⊆ \<Union>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 (\<Union>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)]
apply (rule,rule,rule)
apply (drule division_ofD(2)[OF assms(2)])
apply (drule content_subset) unfolding assms(1)
proof -
case goal1
then show ?case using content_pos_le[of a b] by auto
qed

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. \<Union>{x∈s. x ≠ {}} = \<Union>s"
by auto
show "\<Union>?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 *)
defer
apply (rule_tac[1-4] interior_mono)
using division_ofD(5)[OF assms(1) k1(2) k2(2)]
using division_ofD(5)[OF assms(2) k1(3) k2(3)]
using 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)"
apply rule
apply (rule division_inter[OF assms])
done

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 (\<Inter> 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 \<Inter>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
apply -
unfolding Inter_insert
apply (rule elementary_inter)
apply assumption
apply assumption
done
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 "\<Union>(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 "\<Union>p = cbox a b"
by auto
qed
qed

lemma partial_division_extend_interval:
assumes "p division_of (\<Union>p)" "(\<Union>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
apply -
apply (rule that[of q])
unfolding True
apply auto
done
next
case False
note p = division_ofD[OF assms(1)]
have *: "∀k∈p. ∃q. q division_of cbox a b ∧ k ∈ q"
proof
case goal1
obtain c d where k: "k = cbox c d"
using p(4)[OF goal1] by blast
have *: "cbox c d ⊆ cbox a b" "cbox c d ≠ {}"
using p(2,3)[OF goal1, 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 ?case
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 *] by blast
have "!!x. x ∈ p ==> ∃d. d division_of \<Union>(q x - {x})"
apply rule
apply (rule_tac p="q x" in division_of_subset)
proof -
fix x
assume x: "x ∈ p"
show "q x division_of \<Union>q x"
apply -
apply (rule division_ofI)
using division_ofD[OF q(1)[OF x]]
apply auto
done
show "q x - {x} ⊆ q x"
by auto
qed
then have "∃d. d division_of \<Inter> ((λi. \<Union>(q i - {i})) ` p)"
apply -
apply (rule elementary_inters)
apply (rule finite_imageI[OF p(1)])
unfolding image_is_empty
apply (rule False)
apply auto
done
then obtain d where d: "d division_of \<Inter>((λi. \<Union>(q i - {i})) ` p)" ..
show ?thesis
apply (rule that[of "d ∪ p"])
proof -
have *: "!!s f t. s ≠ {} ==> ∀i∈s. f i ∪ i = t ==> t = \<Inter>(f ` s) ∪ \<Union>s" by auto
have *: "cbox a b = \<Inter>((λi. \<Union>(q i - {i})) ` p) ∪ \<Union>p"
apply (rule *[OF False])
proof
fix i
assume i: "i ∈ p"
show "\<Union>(q i - {i}) ∪ i = cbox a b"
using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
qed
show "d ∪ p division_of (cbox a b)"
unfolding *
apply (rule division_disjoint_union[OF d assms(1)])
apply (rule inter_interior_unions_intervals)
apply (rule p open_interior ballI)+
apply assumption
proof
fix k
assume k: "k ∈ p"
have *: "!!u t s. u ⊆ s ==> s ∩ t = {} ==> u ∩ t = {}"
by auto
show "interior (\<Inter> ((λi. \<Union>(q i - {i})) ` p)) ∩ interior k = {}"
apply (rule *[of _ "interior (\<Union>(q k - {k}))"])
defer
apply (subst Int_commute)
apply (rule inter_interior_unions_intervals)
proof -
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
have *: "!!x s. x ∈ s ==> \<Inter>s ⊆ x"
by auto
show "interior (\<Inter> ((λi. \<Union>(q i - {i})) ` p)) ⊆ interior (\<Union>(q k - {k}))"
apply (rule interior_mono *)+
using k
apply auto
done
qed
qed
qed auto
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
have *: "!!a b. {a, b} = {a} ∪ {b}" by auto
show ?thesis
apply (rule that[of "{cbox c d}"])
unfolding *
apply (rule division_disjoint_union)
using `cbox c d ≠ {}` True assms
using 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 *: "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 * False[unfolded uv]])
note p = this division_ofD[OF this(1)]
have *: "cbox a b ∪ cbox c d = cbox a b ∪ \<Union>(p - {cbox u v})" "!!x s. insert x s = {x} ∪ s"
using p(8) unfolding uv[symmetric] by auto
show ?thesis
apply (rule that[of "p - {cbox u v}"])
unfolding *(1)
apply (subst *(2))
apply (rule division_disjoint_union)
apply (rule, rule assms)
apply (rule division_of_subset[of p])
apply (rule division_of_union_self[OF p(1)])
defer
unfolding interior_inter[symmetric]
proof -
have *: "!!cd p uv ab. p ⊆ cd ==> ab ∩ cd = uv ==> ab ∩ p = uv ∩ p" by auto
have "interior (cbox a b ∩ \<Union>(p - {cbox u v})) = interior(cbox u v ∩ \<Union>(p - {cbox u v}))"
apply (rule arg_cong[of _ _ interior])
apply (rule *[OF _ uv])
using p(8)
apply auto
done
also have "… = {}"
unfolding interior_inter
apply (rule inter_interior_unions_intervals)
using p(6) p(7)[OF p(2)] p(3)
apply auto
done
finally show "interior (cbox a b ∩ \<Union>(p - {cbox u v})) = {}" .
qed auto
qed
qed

lemma division_of_unions:
assumes "finite f"
and "!!p. p ∈ f ==> p division_of (\<Union>p)"
and "!!k1 k2. k1 ∈ \<Union>f ==> k2 ∈ \<Union>f ==> k1 ≠ k2 ==> interior k1 ∩ interior k2 = {}"
shows "\<Union>f division_of \<Union>\<Union>f"
apply (rule division_ofI)
prefer 5
apply (rule assms(3)|assumption)+
apply (rule finite_Union assms(1))+
prefer 3
apply (erule UnionE)
apply (rule_tac s=X in division_ofD(3)[OF assms(2)])
using division_ofD[OF assms(2)]
apply auto
done

lemma elementary_union_interval:
fixes a b :: "'a::euclidean_space"
assumes "p division_of \<Union>p"
obtains q where "q division_of (cbox a b ∪ \<Union>p)"
proof -
note assm = division_ofD[OF assms]
have lem1: "!!f s. \<Union>\<Union>(f ` s) = \<Union>((λx. \<Union>(f x)) ` s)"
by auto
have lem2: "!!f s. f ≠ {} ==> \<Union>{s ∪ t |t. t ∈ f} = s ∪ \<Union>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
apply -
apply (rule that[of p])
unfolding as
apply auto
done
next
assume as: "cbox a b = {}"
show thesis
apply (rule that)
unfolding as
using assms
apply auto
done
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
case goal1
from assm(4)[OF this] obtain c d where "k = cbox c d" by blast
then show ?case
apply -
apply (rule division_union_intervals_exists[OF as(3), of c d])
apply auto
done
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 = "\<Union>{insert (cbox a b) (q k) | k. k ∈ p}"
show thesis
apply (rule that[of "?D"])
apply (rule division_ofI)
proof -
have *: "{insert (cbox a b) (q k) |k. k ∈ p} = (λk. insert (cbox a b) (q k)) ` p"
by auto
show "finite ?D"
apply (rule finite_Union)
unfolding *
apply (rule finite_imageI)
using assm(1) q(1)
apply auto
done
show "\<Union>?D = cbox a b ∪ \<Union>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 ∪ \<Union>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
apply(rule q(5))
using x x' k'
unfolding True
apply auto
done
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
apply (rule q(5))
using x' k'(2)
unfolding as'
apply auto
done
next
assume as': "k' = cbox a b"
show ?thesis
apply (rule q(5))
using x  k'(2)
unfolding as'
apply auto
done
}
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) = {}"
apply (rule q(5))
using x k'(2)
using as'
apply auto
done
then have "interior k ⊆ interior x"
apply -
apply (rule interior_subset_union_intervals[OF k _ as(2) q(2)[OF x(2,1)]])
apply auto
done
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) = {}"
apply (rule q(5))
using x' k'(2)
using as'
apply auto
done
then have "interior k' ⊆ interior x'"
apply -
apply (rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]])
apply auto
done
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 (\<Union>f)"
proof -
have "∃p. p division_of (\<Union>f)"
proof (induct_tac f rule:finite_subset_induct)
show "∃p. p division_of \<Union>{}" using elementary_empty by auto
next
fix x F
assume as: "finite F" "x ∉ F" "∃p. p division_of \<Union>F" "x∈f"
from this(3) obtain p where p: "p division_of \<Union>F" ..
from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
have *: "\<Union>F = \<Union>p"
using division_ofD[OF p] by auto
show "∃p. p division_of \<Union>insert x F"
using elementary_union_interval[OF p[unfolded *], of a b]
unfolding Union_insert x * by auto
qed (insert assms, auto)
then show ?thesis
apply -
apply (erule exE)
apply (rule that)
apply auto
done
qed

lemma elementary_union:
fixes s t :: "'a::euclidean_space set"
assumes "ps division_of s"
and "pt division_of t"
obtains p where "p division_of (s ∪ t)"
proof -
have "s ∪ t = \<Union>ps ∪ \<Union>pt"
using assms unfolding division_of_def by auto
then have *: "\<Union>(ps ∪ pt) = s ∪ t" by auto
show ?thesis
apply -
apply (rule elementary_unions_intervals[of "ps ∪ pt"])
unfolding *
prefer 3
apply (rule_tac p=p in that)
using assms[unfolded division_of_def]
apply auto
done
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)"
apply (rule partial_division_extend_interval)
apply (rule assms(1)[unfolded divp(6)[symmetric]])
apply (rule subset_trans)
apply (rule ab assms[unfolded divp(6)[symmetric]])+
apply assumption
done
note r1 = this division_ofD[OF this(2)]
obtain p' where "p' division_of \<Union>(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 (\<Union>(r1 - p)) ∩ (\<Union>q)"
apply -
apply (drule elementary_inter[OF _ assms(2)[unfolded divq(6)[symmetric]]])
apply auto
done
{
fix x
assume x: "x ∈ t" "x ∉ s"
then have "x∈\<Union>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∈\<Union>(r1 - p)" by auto
}
then have *: "t = \<Union>p ∪ (\<Union>(r1 - p) ∩ \<Union>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 (\<Union>(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 (\<Union>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 = {}"
apply (rule, rule r1(7))
using as
using r1
apply auto
done
qed
then show "interior s ∩ interior m = {}"
unfolding divp by auto
qed
qed
then show "interior s ∩ interior (\<Union>(r1-p) ∩ (\<Union>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 ∧ (\<Union>{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 = {}) ∧
(\<Union>{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 "(\<Union>{k. ∃x. (x,k) ∈ s} = i)"
shows "s tagged_division_of i"
unfolding tagged_division_of
apply rule
defer
apply rule
apply (rule allI impI conjI assms)+
apply assumption
apply rule
apply (rule assms)
apply assumption
apply (rule assms)
apply assumption
using assms(1,5-)
apply blast+
done

lemma tagged_division_ofD[dest]:
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 "(\<Union>{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 "\<Union>(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' = {}"
apply -
apply (rule assm(5))
apply (rule xk xk')+
using k'
apply auto
done
qed

lemma partial_division_of_tagged_division:
assumes "s tagged_partial_division_of i"
shows "(snd ` s) division_of \<Union>(snd ` s)"
proof (rule division_ofI)
note assm = tagged_partial_division_ofD[OF assms]
show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(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 ⊆ \<Union>(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' = {}"
apply -
apply (rule assm(5))
apply(rule xk xk')+
using k'
apply auto
done
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 o 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 pair_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 "\<Union>{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 (case_tac[!] "(x',k') ∈ p1")
apply (rule p1(5))
prefer 4
apply (rule *)
prefer 6
apply (subst Int_commute)
apply (rule *)
prefer 8
apply (rule p2(5))
using p1(3) p2(3)
using xk xk'
apply auto
done
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 "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF assms(2)[rule_format]]
show "finite (\<Union>(pfn ` iset))"
apply (rule finite_Union)
using assms
apply auto
done
have "\<Union>{k. ∃x. (x, k) ∈ \<Union>(pfn ` iset)} = \<Union>((λi. \<Union>{k. ∃x. (x, k) ∈ pfn i}) ` iset)"
by blast
also have "… = \<Union>iset"
using assm(6) by auto
finally show "\<Union>{k. ∃x. (x, k) ∈ \<Union>(pfn ` iset)} = \<Union>iset" .
fix x k
assume xk: "(x, k) ∈ \<Union>(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 ⊆ \<Union>iset"
using assm(2-4)[OF i] using i(1) by auto
fix x' k'
assume xk': "(x', k') ∈ \<Union>(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)[OF i _ xk'(2)] i'(2)
using assm(3)[OF i] assm(3)[OF i']
defer
apply -
apply (rule *)
apply auto
done
qed

lemma tagged_partial_division_of_union_self:
assumes "p tagged_partial_division_of s"
shows "p tagged_division_of (\<Union>(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 (\<Union>(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. \<Inter> {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 (\<Union>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 "\<Union>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="\<Union>(pfn ` iset)" in that)
unfolding assms(4)[symmetric]
apply (rule tagged_division_unions[OF assms(1) _ assms(3)])
defer
apply (rule fine_unions)
using pfn
apply auto
done
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∈f. P s ==>
∀s∈f. ∃a b. s = cbox a b ==>
∀s∈f.∀t∈f. s ≠ t --> interior s ∩ interior t = {} ==> P (\<Union>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])
apply rule
defer
apply rule
defer
apply (rule inter_interior_unions_intervals)
using insert
apply auto
done
qed
} note * = 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
apply -
apply (erule exE)+
apply (rule_tac c=x and d=xa in that)
apply auto
done
}
assume as: "∀c d. ?PP c d --> P (cbox c d)"
have "P (\<Union> ?A)"
apply (rule *)
apply (rule_tac[2-] ballI)
apply (rule_tac[4] ballI)
apply (rule_tac[4] impI)
proof -
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
case goal1
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
have *: "!!a b c d. a = c ==> b = d ==> cbox a b = cbox c d"
by auto
show "x ∈ ?B"
unfolding image_iff
apply (rule_tac x="{i. i∈Basis ∧ c•i = a•i}" in bexI)
unfolding x
apply (rule *)
apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
cong: ball_cong)
apply safe
proof -
fix i :: 'a
assume i: "i ∈ Basis"
then show "c • i = (if c • i = a • i then a • i else (a • i + b • i) / 2)"
and "d • i = (if c • i = a • i then (a • i + b • i) / 2 else b • i)"
using x(2)[of i] ab[OF i] by (auto simp add:field_simps)
qed
qed
then show "finite ?A"
by (rule finite_subset) auto
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])
proof -
case goal1
then show ?case
using s(2)[of i] using ab[OF `i ∈ Basis`] by auto
qed
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"
apply -
apply(erule_tac[!] disjE)
proof -
assume "c•i ≠ e•i"
then show "d•i ≠ f•i"
using s(2)[OF i'] t(2)[OF i'] by fastforce
next
assume "d•i ≠ f•i"
then show "c•i ≠ e•i"
using s(2)[OF i'] t(2)[OF i'] by fastforce
qed
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'
apply -
apply (erule_tac[!] x=i in ballE)+
apply auto
done
show False
using s(2)[OF i']
apply -
apply (erule_tac disjE)
apply (erule_tac[!] conjE)
proof -
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 "\<Union> ?A = cbox a b"
proof (rule set_eqI,rule)
fix x
assume "x ∈ \<Union>?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∈\<Union>?A"
unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
apply -
apply (erule exE)+
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))"
proof
case goal1
then show ?case
proof -
presume "¬ P (cbox (fst x) (snd x)) ==> ?thesis"
then show ?thesis by (cases "P (cbox (fst x) (snd x))") auto
next
assume as: "¬ P (cbox (fst x) (snd x))"
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
case goal3
note S = ab_def funpow.simps o_def id_apply
show ?case
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: "!!e. 0 < e ==> ∃n. ∀x∈cbox (A n) (B n). ∀y∈cbox (A n) (B n). dist x y < e"
proof -
case goal1
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 ?case
apply (rule_tac x=n in exI)
apply rule
apply rule
proof -
fix x y
assume xy: "x∈cbox (A n) (B n)" "y∈cbox (A n) (B n)"
have "dist x y ≤ setsum (λi. abs((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)
case goal1
then show ?case
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 goal1 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 goal1 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" ..
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 rule
apply (rule x0)
apply rule
defer
apply rule
proof -
show "¬ P (cbox (A n) (B n))"
apply (cases "0 < n")
using AB(3)[of "n - 1"] assms(3) AB(1-2)
apply auto
done
show "cbox (A n) (B n) ⊆ ball x0 e"
using n using x0[of n] by auto
show "cbox (A n) (B n) ⊆ cbox a b"
unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
qed
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",rule_format,OF _ _ as])
apply (rule_tac x="{}" in exI)
defer
apply (erule conjE exE)+
proof -
show "{} tagged_division_of {} ∧ g fine {}"
unfolding fine_def by auto
fix s t p p'
assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
"interior s ∩ interior t = {}"
then show "∃p. p tagged_division_of s ∪ t ∧ g fine p"
apply -
apply (rule_tac x="p ∪ p'" in exI)
apply rule
apply (rule tagged_division_union)
prefer 4
apply (rule fine_union)
apply auto
done
qed blast
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: "!!f::'n => 'a.  !!a b k1 k2.
(f has_integral k1) (cbox a b) ==> (f has_integral k2) (cbox a b) ==> k1 ≠ k2 ==> False"
proof -
case goal1
let ?e = "norm (k1 - k2) / 2"
from goal1(3) 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 goal1(1) 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 goal1(2) 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
apply -
apply (cases "∃a b. i = cbox a b")
using assms
apply (auto simp add:has_integral intro:lem[OF _ _ as])
done
}
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
apply rule
apply rule
proof -
fix a b e
fix f :: "'n => 'a"
assume as: "∀x∈cbox a b. f x = 0" "0 < (e::real)"
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)"
apply (rule_tac x="λx. ball x 1" in exI)
apply rule
apply (rule gaugeI)
unfolding centre_in_ball
defer
apply (rule open_ball)
apply rule
apply rule
apply (erule conjE)
proof -
case goal1
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 goal1(1), 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 ?case
using as by auto
qed auto
qed
{
presume "¬ (∃a b. s = cbox a b) ==> ?thesis"
then show ?thesis
apply -
apply (cases "∃a b. s = cbox a b")
using assms
apply (auto simp add:has_integral intro: lem)
done
}
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
apply (subst has_integral_alt)
unfolding if_not_P *
apply rule
apply rule
apply (rule_tac x=1 in exI)
apply rule
defer
apply rule
apply rule
apply rule
proof -
fix e :: real
fix a b
assume "e > 0"
then show "∃z. ((λx::'n. 0::'a) has_integral z) (cbox a b) ∧ norm (z - 0) < e"
apply (rule_tac x=0 in exI)
apply(rule,rule lem)
apply auto
done
qed auto
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 o 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 o f) has_integral h y) (cbox a b)"
apply (subst has_integral)
apply rule
apply rule
proof -
case goal1
from pos_bounded
obtain B where B: "0 < B" "!!x. norm (h x) ≤ norm x * B"
by blast
have *: "e / B > 0" using goal1(2) B by simp
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"
by (rule has_integralD[OF goal1(1) *]) blast
show ?case
apply (rule_tac x=g in exI)
apply rule
apply (rule g(1))
apply rule
apply rule
apply (erule conjE)
proof -
fix p
assume as: "p tagged_division_of (cbox a b)" "g fine p"
have *: "!!x k. h ((λ(x, k). content k *⇩R f x) x) = (λ(x, k). h (content k *⇩R f x)) x"
by auto
have "(∑(x, k)∈p. content k *⇩R (h o f) x) = setsum (h o (λ(x, k). content k *⇩R f x)) p"
unfolding o_def unfolding scaleR[symmetric] * 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 o f) x) = h (∑(x, k)∈p. content k *⇩R f x)" .
show "norm ((∑(x, k)∈p. content k *⇩R (h o f) x) - h y) < e"
unfolding * diff[symmetric]
apply (rule le_less_trans[OF B(2)])
using g(2)[OF as] B(1)
apply (auto simp add: field_simps)
done
qed
qed
{
presume "¬ (∃a b. s = cbox a b) ==> ?thesis"
then show ?thesis
apply -
apply (cases "∃a b. s = cbox a b")
using assms
apply (auto simp add:has_integral intro!:lem)
done
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
apply (subst has_integral_alt)
unfolding if_not_P
apply rule
apply rule
proof -
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 o f) x else 0) has_integral z) (cbox a b) ∧ norm (z - h y) < e)"
apply (rule_tac x=M in exI)
apply rule
apply (rule M(1))
apply rule
apply rule
apply rule
proof -
case goal1
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 goal1(1)] by blast
have *: "(λx. if x ∈ s then (h o f) x else 0) = h o (λx. if x ∈ s then f x else 0)"
unfolding o_def
apply (rule ext)
using zero
apply auto
done
show ?case
apply (rule_tac x="h z" in exI)
apply rule
unfolding *
apply (rule lem[OF z(1)])
unfolding diff[symmetric]
apply (rule le_less_trans[OF B(2)])
using B(1) z(2)
apply (auto simp add: field_simps)
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)

lemma has_integral_cmul: "(f has_integral k) s ==> ((λx. c *⇩R f x) has_integral (c *⇩R k)) s"
unfolding o_def[symmetric]
apply (rule has_integral_linear,assumption)
apply (rule bounded_linear_scaleR_right)
done

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"
apply (drule_tac c="-1" in has_integral_cmul)
apply auto
done

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:"!!(f:: 'n => 'a) g a b k l.
(f has_integral k) (cbox a b) ==>
(g has_integral l) (cbox a b) ==>
((λx. f x + g x) has_integral (k + l)) (cbox a b)"
proof -
case goal1
show ?case
unfolding has_integral
apply rule
apply rule
proof -
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 goal1(1) *] 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 goal1(2) *] 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)"
apply (rule_tac x="λx. (d1 x) ∩ (d2 x)" in exI)
apply rule
apply (rule gauge_inter[OF d1(1) d2(1)])
apply (rule,rule,erule conjE)
proof -
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
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
let ?res = "…"
from as have *: "d1 fine p" "d2 fine p"
unfolding fine_inter by auto
have "?res < e/2 + e/2"
apply (rule le_less_trans[OF norm_triangle_ineq])
apply (rule add_strict_mono)
using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)]
apply auto
done
finally show "norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) < e"
by auto
qed
qed
qed
{
presume "¬ (∃a b. s = cbox a b) ==> ?thesis"
then show ?thesis
apply -
apply (cases "∃a b. s = cbox a b")
using assms
apply (auto simp add:has_integral intro!:lem)
done
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
apply (subst has_integral_alt)
unfolding if_not_P
apply rule
apply rule
proof -
case goal1
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
apply (rule_tac x="max B1 B2" in exI)
apply rule
apply (rule max.strict_coboundedI1)
apply (rule B1)
apply rule
apply rule
apply rule
proof -
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 rule
apply (rule 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"
apply (rule integral_unique)
apply (drule integrable_integral)+
apply (rule has_integral_add)
apply assumption+
done

lemma integral_cmul: "f integrable_on s ==> integral s (λx. c *⇩R f x) = c *⇩R integral s f"
apply (rule integral_unique)
apply (drule integrable_integral)+
apply (rule has_integral_cmul)
apply assumption+
done

lemma integral_neg: "f integrable_on s ==> integral s (λx. - f x) = - integral s f"
apply (rule integral_unique)
apply (drule integrable_integral)+
apply (rule has_integral_neg)
apply assumption+
done

lemma integral_sub: "f integrable_on s ==> g integrable_on s ==>
integral s (λx. f x - g x) = integral s f - integral s g"
apply (rule integral_unique)
apply (drule integrable_integral)+
apply (rule has_integral_sub)
apply assumption+
done

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_sub:
"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 o 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 o f) = h (integral s f)"
apply (rule has_integral_unique)
defer
unfolding has_integral_integral
apply (drule (2) has_integral_linear)
unfolding has_integral_integral[symmetric]
apply (rule integrable_linear)
apply assumption+
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)
show ?case
unfolding setsum.insert[OF insert(1,3)]
apply (rule has_integral_add)
using insert assms
apply auto
done
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"
apply (rule integral_unique)
apply (rule has_integral_setsum)
using integrable_integral
apply auto
done

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∈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∈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]
by auto

lemma has_integral_eq_eq: "∀x∈s. f x = g x ==> (f has_integral k) s <-> (g has_integral k) s"
using has_integral_eq[of s f g] has_integral_eq[of s g f]
by auto

lemma has_integral_null[dest]:
assumes "content(cbox a b) = 0"
shows "(f has_integral 0) (cbox a b)"
unfolding has_integral
apply rule
apply rule
apply (rule_tac x="λx. ball x 1" in exI)
apply rule
defer
apply rule
apply rule
apply (erule conjE)
proof -
fix e :: real
assume e: "e > 0"
then show "gauge (λx. ball x 1)"
by auto
fix p
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 show "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
using e by auto
qed

lemma has_integral_null_real[dest]:
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"
apply rule
apply (rule has_integral_unique)
apply assumption
apply (drule (1) has_integral_null)
apply (drule has_integral_null)
apply auto
done

lemma integral_null[dest]: "content (cbox a b) = 0 ==> integral (cbox a b) f = 0"
apply (rule integral_unique)
apply (drule has_integral_null)
apply assumption
done

lemma integrable_on_null[dest]: "content (cbox a b) = 0 ==> f integrable_on (cbox a b)"
unfolding integrable_on_def
apply (drule has_integral_null)
apply auto
done

lemma has_integral_empty[intro]: "(f has_integral 0) {}"
unfolding empty_as_interval
apply (rule has_integral_null)
using content_empty
unfolding empty_as_interval
apply assumption
done

lemma has_integral_empty_eq[simp]: "(f has_integral i) {} <-> i = 0"
apply rule
apply (rule has_integral_unique)
apply assumption
apply auto
done

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: "integral (cbox a a) f = 0"
by (rule integral_unique) auto

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 (rule, rule)
case goal1
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
apply (rule_tac x=d in exI)
apply rule
apply (rule d)
apply rule
apply rule
apply rule
apply (erule conjE)+
proof -
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(real (n + 1))) d"
by auto
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
have "!!n. gauge (λx. \<Inter>{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. \<Inter>{d i x |i. i ∈ {0..n}}) fine p"
apply -
proof
case goal1
from this[of n]
show ?case
apply (drule_tac fine_division_exists)
apply auto
done
qed
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)
case goal1
then guess N unfolding real_arch_inv[of e] .. note N=this
show ?case
apply (rule_tac x=N in exI)
proof (rule, rule, rule, rule)
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 *)
apply(rule d(2))
using dp p(1)
using mn
apply auto
done
qed
qed
then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
show ?l
unfolding integrable_on_def has_integral
apply (rule_tac x=y in exI)
proof (rule, rule)
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
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)"
apply (rule_tac x="d (N1 + N2)" in exI)
apply rule
defer
proof (rule, rule, erule conjE)
show "gauge (d (N1 + N2))"
using d by auto
fix q
assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
have *: "inverse (real (N1 + N2 + 1)) < e / 2"
apply (rule less_trans)
using N1
apply auto
done
show "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)"])
defer
using N2[rule_format,of "N1+N2"]
using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"]
using p(1)[of "N1 + N2"]
using N1
apply auto
done
qed
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)]])
defer
apply (subst assms(5)[unfolded uv1 uv2])
unfolding uv1 uv2
apply auto
done
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)]])
defer
apply (subst assms(5)[unfolded uv1 uv2])
unfolding uv1 uv2
apply auto
done
qed

lemma tagged_division_split_left_inj:
fixes x1 :: "'a::euclidean_space"
assumes "d tagged_division_of i"
and "(x1, k1) ∈ d"
and "(x2, k2) ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x. x•k ≤ c} = k2 ∩ {x. x•k ≤ c}"
and k: "k ∈ Basis"
shows "content (k1 ∩ {x. x•k ≤ c}) = 0"
proof -
have *: "!!a b c. (a,b) ∈ c ==> b ∈ snd ` c"
unfolding image_iff
apply (rule_tac x="(a,b)" in bexI)
apply auto
done
show ?thesis
apply (rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])
apply (rule_tac[1-2] *)
using assms(2-)
apply auto
done
qed

lemma tagged_division_split_right_inj:
fixes x1 :: "'a::euclidean_space"
assumes "d tagged_division_of i"
and "(x1, k1) ∈ d"
and "(x2, k2) ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x. x•k ≥ c} = k2 ∩ {x. x•k ≥ c}"
and k: "k ∈ Basis"
shows "content (k1 ∩ {x. x•k ≥ c}) = 0"
proof -
have *: "!!a b c. (a,b) ∈ c ==> b ∈ snd ` c"
unfolding image_iff
apply (rule_tac x="(a,b)" in bexI)
apply auto
done
show ?thesis
apply (rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])
apply (rule_tac[1-2] *)
using assms(2-)
apply auto
done
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 "\<Union>?p1 = ?I1" "\<Union>?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" "k ≠ {}" "∃a b. k = cbox a b"
unfolding l
using p(2-3)[OF l(2)] l(3)
unfolding uv
apply -
prefer 3
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" "k ≠ {}" "∃a b. k = cbox a b"
unfolding l
using p(2-3)[OF l(2)] l(3)
unfolding uv
apply -
prefer 3
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 "(f has_integral i) (cbox a b ∩ {x. x•k ≤ c})"
and "(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)
case goal1
then have e: "e/2 > 0"
by auto
guess d1 using has_integralD[OF assms(1)[unfolded interval_split[OF k]] e] .
note d1=this[unfolded interval_split[symmetric,OF k]]
guess d2 using has_integralD[OF assms(2)[unfolded interval_split[OF k]] e] .
note d2=this[unfolded interval_split[symmetric,OF k]]
let ?d = "λx. if x•k = c then (d1 x ∩ d2 x) else ball x (abs(x•k - c)) ∩ d1 x ∩ d2 x"
show ?case
apply (rule_tac x="?d" in exI)
apply rule
defer
apply rule
apply rule
apply (elim conjE)
proof -
show "gauge ?d"
using d1(1) d2(1) unfolding gauge_def by auto
fix p
assume "p tagged_division_of (cbox a b)" "?d fine p"
note p = this tagged_division_ofD[OF this(1)]
have lem0:
"!!x kk. (x, kk) ∈ p ==> kk ∩ {x. x•k ≤ c} ≠ {} ==> x•k ≤ c"
"!!x kk. (x, kk) ∈ p ==> kk ∩ {x. x•k ≥ c} ≠ {} ==> x•k ≥ c"
proof -
fix x kk
assume as: "(x, kk) ∈ p"
{
assume *: "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 * have "∃y. y ∈ ball x ¦x • k - c¦ ∩ {x. x • k ≤ c}"
by blast
then guess y ..
then have "¦x • k - y • k¦ < ¦x • k - c¦" "y•k ≤ c"
apply -
apply (rule le_less_trans)
using Basis_le_norm[OF k, of "x - y"]
apply (auto simp add: dist_norm inner_diff_left)
done
then show False
using **[unfolded not_le] by (auto simp add: field_simps)
qed
next
assume *: "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 * have "∃y. y ∈ ball x ¦x • k - c¦ ∩ {x. x • k ≥ c}"
by blast
then guess y ..
then have "¦x • k - y • k¦ < ¦x • k - c¦" "y•k ≥ c"
apply -
apply (rule le_less_trans)
using Basis_le_norm[OF k, of "x - y"]
apply (auto simp add: dist_norm inner_diff_left)
done
then show False
using **[unfolded not_le] 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 lem2: "!!f s P f. finite s ==> finite {(x,f k) | x k. (x,k) ∈ s ∧ P x k}"
proof -
case goal1
then show ?case
apply -
apply (rule finite_subset[of _ "(λ(x,k). (x,f k)) ` s"])
apply auto
done
qed
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)"
apply (rule setsum.mono_neutral_left)
prefer 3
proof
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 show "(λ(x, k). content k *⇩R f x) i = 0"
unfolding xk split_conv by auto
qed auto
have lem4: "!!g. (λ(x,l). content (g l) *⇩R f x) = (λ(x,l). content l *⇩R f x) o (λ(x,l). (x,g l))"
by auto

let ?M1 = "{(x, kk ∩ {x. x•k ≤ c}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. x•k ≤ c} ≠ {}}"
have "norm ((∑(x, k)∈?M1. content k *⇩R f x) - i) < e/2"
apply (rule d1(2),rule tagged_division_ofI)
apply (rule lem2 p(3))+
prefer 6
apply (rule fineI)
proof -
show "\<Union>{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 Pair_eq by (elim exE conjE) note xl'=this
have "l' ⊆ d1 x'"
apply (rule order_trans[OF fineD[OF p(2) xl'(3)]])
apply auto
done
then show "l ⊆ d1 x"
unfolding xl' by auto
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x • k ≤ c}"
unfolding xl'
using p(4-6)[OF xl'(3)] using xl'(4)
using lem0(1)[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 Pair_eq 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 "norm ((∑(x, k)∈?M2. content k *⇩R f x) - j) < e/2"
apply (rule d2(2),rule tagged_division_ofI)
apply (rule lem2 p(3))+
prefer 6
apply (rule fineI)
proof -
show "\<Union>{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 Pair_eq by (elim exE conjE) note xl'=this
have "l' ⊆ d2 x'"
apply (rule order_trans[OF fineD[OF p(2) xl'(3)]])
apply auto
done
then show "l ⊆ d2 x"
unfolding xl' by auto
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x • k ≥ c}"
unfolding xl'
using p(4-6)[OF xl'(3)]
using xl'(4)
using lem0(2)[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 Pair_eq 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"
apply -
apply (rule norm_triangle_lt)
apply auto
done
also {
have *: "!!x y. x = (0::real) ==> x *⇩R (y::'b) = 0"
using scaleR_zero_left 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)]
apply (subst setsum.reindex_nontrivial[OF p(3)])
defer
apply (subst setsum.reindex_nontrivial[OF p(3)])
defer
unfolding lem4[symmetric]
apply (rule refl)
unfolding split_paired_all split_conv
apply (rule_tac[!] *)
proof -
case goal1
then show ?case
apply -
apply (rule tagged_division_split_left_inj [OF p(1), of a b aa ba])
using k
apply auto
done
next
case goal2
then show ?case
apply -
apply (rule tagged_division_split_right_inj[OF p(1), of a b aa ba])
using k
apply auto
done
qed
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"
unfolding split_paired_all split_conv
proof -
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
show ?thesis
apply (rule that[of d])
apply (rule d)
apply rule
apply rule
apply rule
apply (elim conjE)
proof -
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
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)"
apply (subst setsum.union_inter_neutral)
apply (rule p1 p2)+
apply rule
unfolding split_paired_all split_conv
proof -
fix a b
assume ab: "(a, b) ∈ p1 ∩ p2"
have "(a, b) ∈ p1"
using ab by auto
from p1(4)[OF this] guess u v by (elim exE) note uv = this
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))"
apply (rule setsum.cong)
apply (rule refl)
apply (subst *)
apply auto
done
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
apply -
apply (drule interior_mono)
apply auto
done
then show "content b *⇩R f a = 0"
by auto
qed auto
also have "… < e"
by (rule k d(2) p12 fine_union p1 p2)+
finally show "norm ((∑(x, k)∈p1. content k *⇩R f x) + (∑(x, k)∈p2. content k *⇩R f x) - i) < e" .
qed
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}"
apply (rule_tac x=d in exI)
apply rule
apply (rule d)
apply rule
apply rule
apply rule
proof -
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 -
guess p using fine_division_exists[OF d(1), of a' b] . note p=this
show ?thesis
using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
using as unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
using p using assms
by (auto simp add: algebra_simps)
qed
qed
show "?P {x. x • k ≥ c}"
apply (rule_tac x=d in exI)
apply rule
apply (rule d)
apply rule
apply rule
apply rule
proof -
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 -
guess p using fine_division_exists[OF d(1), of a b'] . note p=this
show ?thesis
using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
using as
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
using p
using assms
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 operative_trivial: "operative opp f ==> content (cbox a b) = 0 ==> f (cbox a b) = neutral opp"
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. *}

lemma forall_option: "(∀x. P x) <-> P None ∧ (∀x. P (Some x))"
by (metis option.nchotomy)

lemma exists_option: "(∃x. P x) <-> P None ∨ (∃x. P (Some x))"
by (metis option.nchotomy)

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 "opp (neutral opp) a = a"
and "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 monoidal_simps[simp]:
assumes "monoidal opp"
shows "opp (neutral opp) a = a"
and "opp a (neutral opp) = a"
using monoidal_ac[OF assms] by auto

lemma neutral_lifted[cong]:
assumes "monoidal opp"
shows "neutral (lifted opp) = Some (neutral opp)"
apply (subst neutral_def)
apply (rule some_equality)
apply rule
apply (induct_tac y)
prefer 3
proof -
fix x
assume "∀y. lifted opp x y = y ∧ lifted opp y x = y"
then show "x = Some (neutral opp)"
apply (induct x)
defer
apply rule
apply (subst neutral_def)
apply (subst eq_commute)
apply(rule some_equality)
apply rule
apply (erule_tac x="Some y" in allE)
defer
apply (rename_tac x)
apply (erule_tac x="Some x" in allE)
apply auto
done
qed (auto simp add:monoidal_ac[OF assms])

lemma monoidal_lifted[intro]:
assumes "monoidal opp"
shows "monoidal (lifted opp)"
unfolding monoidal_def forall_option 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 o 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 have *: "insert x s = s"
by auto
show ?thesis unfolding iterate_def if_P[OF True] * by auto
next
case False
note x = this
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
show ?thesis
unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]
unfolding True monoidal_simps[OF assms(1)]
by auto
next
case False
show ?thesis
unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]
apply (subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
using `finite s`
unfolding support_def
using False x
apply auto
done
qed
qed

lemma iterate_some:
assumes "monoidal opp"
and "finite s"
shows "iterate (lifted opp) s (λx. Some(f x)) = Some (iterate opp s f)"
using assms(2)
proof (induct s)
case empty
then show ?case
using assms by auto
next
case (insert x F)
show ?case
apply (subst iterate_insert)
prefer 3
apply (subst if_not_P)
defer
unfolding insert(3) lifted.simps
apply rule
using assms insert
apply auto
done
qed

subsection {* Two key instances of additivity. *}

lemma neutral_add[simp]: "neutral op + = (0::'a::comm_monoid_add)"
unfolding neutral_def
apply (rule some_equality)
defer
apply (erule_tac x=0 in allE)
apply auto
done

lemma operative_content[intro]: "operative (op +) content"
unfolding operative_def neutral_add
apply safe
unfolding content_split[symmetric]
apply rule
done

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
unfolding neutral_lifted[OF monoidal_monoid] neutral_add
apply rule
apply rule
apply rule
apply rule
defer
apply (rule allI ballI)+
proof -
fix a b c
fix k :: 'a
assume k: "k ∈ Basis"
show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
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
show ?thesis
unfolding if_P[OF True]
using k
apply -
unfolding if_P[OF integrable_split(1)[OF True]]
unfolding if_P[OF integrable_split(2)[OF True]]
unfolding lifted.simps option.inject
apply (rule integral_unique)
apply (rule has_integral_split[OF _ _ k])
apply (rule_tac[!] integrable_integral integrable_split)+
using True k
apply auto
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"
apply -
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 (rule_tac[!] integrable_integral)
apply auto
done
then show False
using False by auto
qed
then show ?thesis
using False by auto
qed
next
fix a b :: 'a
assume as: "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"
unfolding if_P[OF integrable_on_null[OF as]]
using has_integral_null_eq[OF as]
by auto
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 = \<Union>(?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
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:
"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
unfolding subset_eq
apply -
defer
apply (erule_tac x=u in ballE)
apply (erule_tac x=v in ballE)
unfolding mem_box
apply auto
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 interval_split[OF k]
apply (subst interval_bounds)
prefer 3
apply (subst interval_bounds)
unfolding l interval_bounds[OF uv(1)]
using uv[rule_format,of k] ab k
apply auto
done
have "∃x. x ∈ ?D - ?D1"
using assms(2-)
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)•k)" in exI)
defer
apply (rule_tac x="(k,(interval_upperbound l)•k)" in exI)
unfolding division_points_def
unfolding interval_bounds[OF ab]
apply (auto simp add:*)
done
then show "?D1 ⊂ ?D"
apply -
apply rule
apply (rule division_points_subset[OF assms(1-4)])
using k
apply auto
done

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 interval_split[OF k]
apply (subst interval_bounds)
prefer 3
apply (subst interval_bounds)
unfolding l interval_bounds[OF uv(1)]
using uv[rule_format,of k] ab k
apply auto
done
have "∃x. x ∈ ?D - ?D2"
using assms(2-)
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)•k)" in exI)
defer
apply (rule_tac x="(k,(interval_upperbound l)•k)" in exI)
unfolding division_points_def
unfolding interval_bounds[OF ab]
apply (auto simp add:*)
done
then show "?D2 ⊂ ?D"
apply -
apply rule
apply (rule division_points_subset[OF assms(1-4) k])
apply auto
done
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)"
apply cases
apply (subst if_P, assumption)
unfolding iterate_def support_support fold'_def
apply auto
done

lemma iterate_image:
assumes "monoidal opp"
and "inj_on f s"
shows "iterate opp (f ` s) g = iterate opp s (g o f)"
proof -
have *: "!!s. finite s ==>  ∀x∈s. ∀y∈s. f x = f y --> x = y ==>
iterate opp (f ` s) g = iterate opp s (g o f)"
proof -
case goal1
then show ?case
proof (induct s)
case empty
then show ?case
using assms(1) by auto
next
case (insert x s)
show ?case
unfolding iterate_insert[OF assms(1) insert(1)]
unfolding if_not_P[OF insert(2)]
apply (subst insert(3)[symmetric])
unfolding image_insert
defer
apply (subst iterate_insert[OF assms(1)])
apply (rule finite_imageI insert)+
apply (subst if_not_P)
unfolding image_iff o_def
using insert(2,4)
apply auto
done
qed
qed
show ?thesis
apply (cases "finite (support opp g (f ` s))")
apply (subst (1) iterate_support[symmetric],subst (2) iterate_support[symmetric])
unfolding support_clauses
apply (rule *)
apply (rule finite_imageD,assumption)
unfolding inj_on_def[symmetric]
apply (rule subset_inj_on[OF assms(2) support_subset])+
apply (subst iterate_expand_cases)
unfolding support_clauses
apply (simp only: if_False)
apply (subst iterate_expand_cases)
apply (subst if_not_P)
apply auto
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 o f)"
proof -
have *: "{f x |x. x ∈ s ∧ f x ≠ a} = f ` {x. x ∈ s ∧ f x ≠ a}"
by auto
have **: "support opp (g o f) {x ∈ s. f x ≠ a} = support opp (g o f) s"
unfolding support_def using assms(3) by auto
show ?thesis
unfolding *
apply (subst iterate_support[symmetric])
unfolding support_clauses
apply (subst iterate_image[OF assms(1)])
defer
apply (subst(2) iterate_support[symmetric])
apply (subst **)
unfolding inj_on_def
using assms(3,4)
unfolding support_def
apply auto
done
qed

lemma iterate_eq_neutral:
assumes "monoidal opp"
and "∀x ∈ s. f x = neutral opp"
shows "iterate opp s f = neutral opp"
proof -
have *: "support opp f s = {}"
unfolding support_def using assms(2) by auto
show ?thesis
apply (subst iterate_support[symmetric])
unfolding *
using assms(1)
apply auto
done
qed

lemma iterate_op:
assumes "monoidal opp"
and "finite s"
shows "iterate opp s (λx. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)"
using assms(2)
proof (induct s)
case empty
then show ?case
unfolding iterate_insert[OF assms(1)] using assms(1) by auto
next
case (insert x F)
show ?case
unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)
by (simp add: monoidal_ac[OF assms(1)])
qed

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
apply (subst iterate_expand_cases)
apply (subst(2) iterate_expand_cases)
unfolding *
apply auto
done
next
def su ≡ "support opp f s"
case True note support_subset[of opp f s]
then show ?thesis
apply -
apply (subst iterate_support[symmetric])
apply (subst(2) iterate_support[symmetric])
unfolding *
using True
unfolding su_def[symmetric]
proof (induct su)
case empty
show ?case by auto
next
case (insert x s)
show ?case
unfolding iterate_insert[OF assms(1) insert(1)]
unfolding if_not_P[OF insert(2)]
apply (subst insert(3))
defer
apply (subst assms(2)[of x])
using insert
apply auto
done
qed
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 goal1
{ presume *: "content (cbox a b) ≠ 0 ==> ?case"
then show ?case
apply -
apply cases
defer
apply assumption
proof -
assume as: "content (cbox a b) = 0"
show ?case
unfolding operativeD(1)[OF assms(2) as]
apply(rule iterate_eq_neutral[OF goal1(2)])
proof
fix x
assume x: "x ∈ d"
then guess u v
apply (drule_tac division_ofD(4)[OF goal1(4)])
apply (elim exE)
done
then show "f x = neutral opp"
using division_of_content_0[OF as goal1(4)]
using operativeD(1)[OF assms(2)] x
by auto
qed
qed
}
assume "content (cbox a b) ≠ 0" 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 ?case
proof (cases "division_points (cbox a b) d = {}")
case True
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 goal1(4)]
apply rule
apply rule
apply rule
apply (rule_tac x=a in exI)
apply (rule_tac x=b in exI)
apply rule
apply (rule refl)
proof
fix u v
fix j :: 'a
assume j: "j ∈ Basis"
assume as: "cbox u v ∈ d"
note division_ofD(3)[OF goal1(4) this]
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 goal1(4) as]
unfolding subset_eq
apply -
apply (erule_tac x=u in ballE,erule_tac[3] x=v in ballE)
unfolding box_ne_empty mem_box
using j
apply auto
done
ultimately show "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 auto
qed
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 goal1(4),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 -
{ presume "i = cbox a b" then show ?thesis using i by auto }
{ presume "u = a" "v = b" then show "i = cbox a b" using uv by auto }
show "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
qed
then have *: "d = insert (cbox a b) (d - {cbox a b})"
by auto
have "iterate opp (d - {cbox a b}) f = neutral opp"
apply (rule iterate_eq_neutral[OF goal1(2)])
proof
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
apply (rule_tac x=j in bexI)
using j
apply auto
done
then show "f x = neutral opp"
unfolding uv(1) by (rule operativeD(1)[OF goal1(3)])
qed
then show "iterate opp d f = f (cbox a b)"
apply -
apply (subst *)
apply (subst iterate_insert[OF goal1(2)])
using goal1(2,4)
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
unfolding 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 goal1(4) 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[!] goal1(1)[rule_format])
using division_split[OF goal1(4), where k=k and c=c]
unfolding interval_split[OF kc(4)] d1_def[symmetric] d2_def[symmetric]
unfolding goal1(2) Suc_le_mono
using goal1(2-3)
using division_points_finite[OF goal1(4)]
using kc(4)
apply auto
done
have "f (cbox a b) = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
unfolding *
apply (rule operativeD(2))
using goal1(3)
using kc(4)
apply auto
done
also have "iterate opp d1 f = iterate opp d (λl. f(l ∩ {x. x•k ≤ c}))"
unfolding d1_def
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
unfolding empty_as_interval
apply (rule goal1 division_of_finite operativeD[OF goal1(3)])+
unfolding empty_as_interval[symmetric]
apply (rule content_empty)
proof (rule, rule, rule, erule conjE)
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 goal1(4) this(1)] guess u v by (elim exE) note l=this
show "f (l ∩ {x. x • k ≤ c}) = neutral opp"
unfolding l interval_split[OF kc(4)]
apply (rule operativeD(1) goal1)+
unfolding interval_split[symmetric,OF kc(4)]
apply (rule division_split_left_inj)
apply (rule goal1)
unfolding l[symmetric]
apply (rule as(1), rule as(2))
apply (rule kc(4) as)+
done
qed
also have "iterate opp d2 f = iterate opp d (λl. f(l ∩ {x. x•k ≥ c}))"
unfolding d2_def
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
unfolding empty_as_interval
apply (rule goal1 division_of_finite operativeD[OF goal1(3)])+
unfolding empty_as_interval[symmetric]
apply (rule content_empty)
proof (rule, rule, rule, erule conjE)
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 goal1(4) this(1)] guess u v by (elim exE) note l=this
show "f (l ∩ {x. x • k ≥ c}) = neutral opp"
unfolding l interval_split[OF kc(4)]
apply (rule operativeD(1) goal1)+
unfolding interval_split[symmetric,OF kc(4)]
apply (rule division_split_right_inj)
apply (rule goal1)
unfolding l[symmetric]
apply (rule as(1))
apply (rule as(2))
apply (rule as kc(4))+
done
qed also have *: "∀x∈d. f x = opp (f (x ∩ {x. x • k ≤ c})) (f (x ∩ {x. c ≤ x • k}))"
unfolding forall_in_division[OF goal1(4)]
apply (rule, rule, rule, rule operativeD(2))
using goal1(3) kc
by auto
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]])
prefer 3
apply (rule iterate_op[symmetric])
using goal1
apply auto
done
finally show ?thesis by auto
qed
qed
qed

lemma iterate_image_nonzero:
assumes "monoidal opp"
and "finite s"
and "∀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 o f)"
using assms
proof (induct rule: finite_subset_induct[OF assms(2) subset_refl])
case goal1
show ?case
using assms(1) by auto
next
case goal2
have *: "!!x y. y = neutral opp ==> x = opp y x"
using assms(1) by auto
show ?case
unfolding image_insert
apply (subst iterate_insert[OF assms(1)])
apply (rule finite_imageI goal2)+
apply (cases "f a ∈ f ` F")
unfolding if_P if_not_P
apply (subst goal2(4)[OF assms(1) goal2(1)])
defer
apply (subst iterate_insert[OF assms(1) goal2(1)])
defer
apply (subst iterate_insert[OF assms(1) goal2(1)])
unfolding if_not_P[OF goal2(3)]
defer unfolding image_iff
defer
apply (erule bexE)
apply (rule *)
unfolding o_def
apply (rule_tac y=x in goal2(7)[rule_format])
using goal2
unfolding o_def
apply auto
done
qed

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 o snd"
unfolding o_def by rule auto note assm = tagged_division_ofD[OF assms(3)]
have "iterate opp d (λ(x,l). f l) = iterate opp (snd ` d) f"
unfolding *
apply (rule iterate_image_nonzero[symmetric,OF assms(1)])
apply (rule tagged_division_of_finite assms)+
unfolding Ball_def split_paired_All snd_conv
apply (rule, rule, rule, rule, rule, rule, rule, erule conjE)
proof -
fix a b aa ba
assume as: "(a, b) ∈ d" "(aa, ba) ∈ d" "(a, b) ≠ (aa, ba)" "b = ba"
guess u v using assm(4)[OF as(1)] by (elim exE) note uv=this
show "f b = neutral opp"
unfolding uv
apply (rule operativeD(1)[OF assms(2)])
unfolding content_eq_0_interior
using tagged_division_ofD(5)[OF assms(3) as(1-3)]
unfolding as(4)[symmetric] uv
apply auto
done
qed
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)"
apply (rule setsum.mono_neutral_right)
unfolding support_def neutral_add
using assms
apply auto
done
then show ?thesis unfolding * iterate_def fold'_def setsum.eq_fold
unfolding neutral_add by (simp add: comp_def)
qed

lemma additive_content_division:
assumes "d division_of (cbox a b)"
shows "setsum content d = content (cbox a b)"
unfolding operative_division[OF monoidal_monoid operative_content assms,symmetric]
apply (subst setsum_iterate)
using assms
apply auto
done

lemma additive_content_tagged_division:
assumes "d tagged_division_of (cbox a b)"
shows "setsum (λ(x,l). content l) d = content (cbox a b)"
unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,symmetric]
apply (subst setsum_iterate)
using assms
apply auto
done

subsection {* Finally, the integral of a constant *}

lemma has_integral_const[intro]:
fixes a b :: "'a::euclidean_space"
shows "((λx. c) has_integral (content (cbox a b) *⇩R c)) (cbox a b)"
unfolding has_integral
apply rule
apply rule
apply (rule_tac x="λx. ball x 1" in exI)
apply rule
apply (rule gauge_trivial)
apply rule
apply rule
apply (erule conjE)
unfolding split_def
apply (subst scaleR_left.setsum[symmetric, unfolded o_def])
defer
apply (subst additive_content_tagged_division[unfolded split_def])
apply assumption
apply auto
done

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

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

lemma integral_const_real[simp]:
fixes a b :: real
shows "integral {a .. b} (λx. c) = content {a .. b} *⇩R c"
by (metis box_real(2) integral_const)

subsection {* Bounds on the norm of Riemann sums and the integral itself. *}

lemma dsum_bound:
assumes "p division_of (cbox a b)"
and "norm c ≤ e"
shows "norm (setsum (λl. content l *⇩R c) p) ≤ e * content(cbox a b)"
apply (rule order_trans)
apply (rule norm_setsum)
unfolding norm_scaleR setsum_left_distrib[symmetric]
apply (rule order_trans[OF mult_left_mono])
apply (rule assms)
apply (rule setsum_abs_ge_zero)
apply (subst mult.commute)
apply (rule mult_left_mono)
apply (rule order_trans[of _ "setsum content p"])
apply (rule eq_refl)
apply (rule setsum.cong)
apply (rule refl)
apply (subst abs_of_nonneg)
unfolding additive_content_division[OF assms(1)]
proof -
from order_trans[OF norm_ge_zero[of c] assms(2)]
show "0 ≤ e" .
fix x assume "x ∈ p"
from division_ofD(4)[OF assms(1) this] guess u v by (elim exE)
then show "0 ≤ content x"
using content_pos_le by auto
qed (insert assms, auto)

lemma rsum_bound:
assumes "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. norm (f x) ≤ e"
shows "norm (setsum (λ(x,k). content k *⇩R f x) p) ≤ e * content (cbox a b)"
proof (cases "cbox a b = {}")
case True
show ?thesis
using assms(1) unfolding True tagged_division_of_trivial by auto
next
case False
show ?thesis
apply (rule order_trans)
apply (rule norm_setsum)
unfolding split_def norm_scaleR
apply (rule order_trans[OF setsum_mono])
apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
defer
unfolding setsum_left_distrib[symmetric]
apply (subst mult.commute)
apply (rule mult_left_mono)
apply (rule order_trans[of _ "setsum (content o snd) p"])
apply (rule eq_refl)
apply (rule setsum.cong)
apply (rule refl)
apply (subst o_def)
apply (rule abs_of_nonneg)
proof -
show "setsum (content o snd) p ≤ content (cbox a b)"
apply (rule eq_refl)
unfolding additive_content_tagged_division[OF assms(1),symmetric] split_def
apply auto
done
guess w using nonempty_witness[OF False] .
then show "e ≥ 0"
apply -
apply (rule order_trans)
defer
apply (rule assms(2)[rule_format])
apply assumption
apply auto
done
fix xk
assume *: "xk ∈ p"
guess x k using surj_pair[of xk] by (elim exE) note xk = this *[unfolded this]
from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v by (elim exE) note uv=this
show "0 ≤ content (snd xk)"
unfolding xk snd_conv uv by(rule content_pos_le)
show "norm (f (fst xk)) ≤ e"
unfolding xk fst_conv using tagged_division_ofD(2,3)[OF assms(1) xk(2)] assms(2) by auto
qed
qed

lemma rsum_diff_bound:
assumes "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. norm (f x - g x) ≤ e"
shows "norm (setsum (λ(x,k). content k *⇩R f x) p - setsum (λ(x,k). content k *⇩R g x) p) ≤
e * content (cbox a b)"
apply (rule order_trans[OF _ rsum_bound[OF assms]])
apply (rule eq_refl)
apply (rule arg_cong[where f=norm])
unfolding setsum_subtractf[symmetric]
apply (rule setsum.cong)
unfolding scaleR_diff_right
apply auto
done

lemma has_integral_bound:
fixes f :: "'a::euclidean_space => 'b::real_normed_vector"
assumes "0 ≤ B"
and "(f has_integral i) (cbox a b)"
and "∀x∈cbox a b. norm (f x) ≤ B"
shows "norm i ≤ B * content (cbox a b)"
proof -
let ?P = "content (cbox a b) > 0"
{
presume "?P ==> ?thesis"
then show ?thesis
proof (cases ?P)
case False
then have *: "content (cbox a b) = 0"
using content_lt_nz by auto
then have **: "i = 0"
using assms(2)
apply (subst has_integral_null_eq[symmetric])
apply auto
done
show ?thesis
unfolding * ** using assms(1) by auto
qed auto
}
assume ab: ?P
{ presume "¬ ?thesis ==> False" then show ?thesis by auto }
assume "¬ ?thesis"
then have *: "norm i - B * content (cbox a b) > 0"
by auto
from assms(2)[unfolded has_integral,rule_format,OF *]
guess d by (elim exE conjE) note d=this[rule_format]
from fine_division_exists[OF this(1), of a b] guess p . note p=this
have *: "!!s B. norm s ≤ B ==> ¬ norm (s - i) < norm i - B"
proof -
case goal1
then show ?case
unfolding not_less
using norm_triangle_sub[of i s]
unfolding norm_minus_commute
by auto
qed
show False
using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
qed

lemma has_integral_bound_real:
fixes f :: "real => 'b::real_normed_vector"
assumes "0 ≤ B"
and "(f has_integral i) {a .. b}"
and "∀x∈{a .. b}. norm (f x) ≤ B"
shows "norm i ≤ B * content {a .. b}"
by (metis assms(1) assms(2) assms(3) box_real(2) has_integral_bound)

subsection {* Similar theorems about relationship among components. *}

lemma rsum_component_le:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. (f x)•i ≤ (g x)•i"
shows "(setsum (λ(x,k). content k *⇩R f x) p)•i ≤ (setsum (λ(x,k). content k *⇩R g x) p)•i"
unfolding inner_setsum_left
apply (rule setsum_mono)
apply safe
proof -
fix a b
assume ab: "(a, b) ∈ p"
note assm = tagged_division_ofD(2-4)[OF assms(1) ab]
from this(3) guess u v by (elim exE) note b=this
show "(content b *⇩R f a) • i ≤ (content b *⇩R g a) • i"
unfolding b
unfolding inner_simps real_scaleR_def
apply (rule mult_left_mono)
defer
apply (rule content_pos_le,rule assms(2)[rule_format])
using assm
apply auto
done
qed

lemma has_integral_component_le:
fixes f g :: "'a::euclidean_space => 'b::euclidean_space"
assumes k: "k ∈ Basis"
assumes "(f has_integral i) s" "(g has_integral j) s"
and "∀x∈s. (f x)•k ≤ (g x)•k"
shows "i•k ≤ j•k"
proof -
have lem: "!!a b i j::'b. !!g f::'a => 'b. (f has_integral i) (cbox a b) ==>
(g has_integral j) (cbox a b) ==> ∀x∈cbox a b. (f x)•k ≤ (g x)•k ==> i•k ≤ j•k"
proof (rule ccontr)
case goal1
then have *: "0 < (i•k - j•k) / 3"
by auto
guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .
note p = this(1) conjunctD2[OF this(2)]
note le_less_trans[OF Basis_le_norm[OF k]]
note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]
then show False
unfolding inner_simps
using rsum_component_le[OF p(1) goal1(3)]
by (simp add: abs_real_def split: split_if_asm)
qed
let ?P = "∃a b. s = cbox a b"
{
presume "¬ ?P ==> ?thesis"
then show ?thesis
proof (cases ?P)
case True
then guess a b by (elim exE) note s=this
show ?thesis
apply (rule lem)
using assms[unfolded s]
apply auto
done
qed auto
}
assume as: "¬ ?P"
{ presume "¬ ?thesis ==> False" then show ?thesis by auto }
assume "¬ i•k ≤ j•k"
then have ij: "(i•k - j•k) / 3 > 0"
by auto
note has_integral_altD[OF _ as this]
from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
have "bounded (ball 0 B1 ∪ ball (0::'a) B2)"
unfolding bounded_Un by(rule conjI bounded_ball)+
from bounded_subset_cbox[OF this] guess a b by (elim exE)
note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
have *: "!!w1 w2 j i::real .¦w1 - i¦ < (i - j) / 3 ==> ¦w2 - j¦ < (i - j) / 3 ==> w1 ≤ w2 ==> False"
by (simp add: abs_real_def split: split_if_asm)
note le_less_trans[OF Basis_le_norm[OF k]]
note this[OF w1(2)] this[OF w2(2)]
moreover
have "w1•k ≤ w2•k"
apply (rule lem[OF w1(1) w2(1)])
using assms
apply auto
done
ultimately show False
unfolding inner_simps by(rule *)
qed

lemma integral_component_le:
fixes g f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "k ∈ Basis"
and "f integrable_on s" "g integrable_on s"
and "∀x∈s. (f x)•k ≤ (g x)•k"
shows "(integral s f)•k ≤ (integral s g)•k"
apply (rule has_integral_component_le)
using integrable_integral assms
apply auto
done

lemma has_integral_component_nonneg:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "k ∈ Basis"
and "(f has_integral i) s"
and "∀x∈s. 0 ≤ (f x)•k"
shows "0 ≤ i•k"
using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
using assms(3-)
by auto

lemma integral_component_nonneg:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "k ∈ Basis"
and "f integrable_on s" "∀x∈s. 0 ≤ (f x)•k"
shows "0 ≤ (integral s f)•k"
apply (rule has_integral_component_nonneg)
using assms
apply auto
done

lemma has_integral_component_neg:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "k ∈ Basis"
and "(f has_integral i) s"
and "∀x∈s. (f x)•k ≤ 0"
shows "i•k ≤ 0"
using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
by auto

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

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

lemma integral_component_lbound:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "f integrable_on cbox a b"
and "∀x∈cbox a b. B ≤ f(x)•k"
and "k ∈ Basis"
shows "B * content (cbox a b) ≤ (integral(cbox a b) f)•k"
apply (rule has_integral_component_lbound)
using assms
unfolding has_integral_integral
apply auto
done

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

lemma integral_component_ubound:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes "f integrable_on cbox a b"
and "∀x∈cbox a b. f x•k ≤ B"
and "k ∈ Basis"
shows "(integral (cbox a b) f)•k ≤ B * content (cbox a b)"
apply (rule has_integral_component_ubound)
using assms
unfolding has_integral_integral
apply auto
done

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

subsection {* Uniform limit of integrable functions is integrable. *}

lemma integrable_uniform_limit:
fixes f :: "'a::euclidean_space => 'b::banach"
assumes "∀e>0. ∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
shows "f integrable_on cbox a b"
proof -
{
presume *: "content (cbox a b) > 0 ==> ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
unfolding content_lt_nz integrable_on_def
using has_integral_null
apply auto
done
}
assume as: "content (cbox a b) > 0"
have *: "!!P. ∀e>(0::real). P e ==> ∀n::nat. P (inverse (real n + 1))"
by auto
from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "λx. x"]] guess i .. note i=this[rule_format]

have "Cauchy i"
unfolding Cauchy_def
proof (rule, rule)
fix e :: real
assume "e>0"
then have "e / 4 / content (cbox a b) > 0"
using as by (auto simp add: field_simps)
then guess M
apply -
apply (subst(asm) real_arch_inv)
apply (elim exE conjE)
done
note M=this
show "∃M. ∀m≥M. ∀n≥M. dist (i m) (i n) < e"
apply (rule_tac x=M in exI,rule,rule,rule,rule)
proof -
case goal1
have "e/4>0" using `e>0` by auto
note * = i[unfolded has_integral,rule_format,OF this]
from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this
have lem2: "!!s1 s2 i1 i2. norm(s2 - s1) ≤ e/2 ==> norm (s1 - i1) < e / 4 ==>
norm (s2 - i2) < e / 4 ==> norm (i1 - i2) < e"
proof -
case goal1
have "norm (i1 - i2) ≤ norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
by (auto simp add: algebra_simps)
also have "… < e"
using goal1
unfolding norm_minus_commute
by (auto simp add: algebra_simps)
finally show ?case .
qed
show ?case
unfolding dist_norm
apply (rule lem2)
defer
apply (rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])
using conjunctD2[OF p(2)[unfolded fine_inter]]
apply -
apply assumption+
apply (rule order_trans)
apply (rule rsum_diff_bound[OF p(1), where e="2 / real M"])
proof
show "2 / real M * content (cbox a b) ≤ e / 2"
unfolding divide_inverse
using M as
by (auto simp add: field_simps)
fix x
assume x: "x ∈ cbox a b"
have "norm (f x - g n x) + norm (f x - g m x) ≤ inverse (real n + 1) + inverse (real m + 1)"
using g(1)[OF x, of n] g(1)[OF x, of m] by auto
also have "… ≤ inverse (real M) + inverse (real M)"
apply (rule add_mono)
apply (rule_tac[!] le_imp_inverse_le)
using goal1 M
apply auto
done
also have "… = 2 / real M"
unfolding divide_inverse by auto
finally show "norm (g n x - g m x) ≤ 2 / real M"
using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
by (auto simp add: algebra_simps simp add: norm_minus_commute)
qed
qed
qed
from this[unfolded convergent_eq_cauchy[symmetric]] guess s .. note s=this

show ?thesis
unfolding integrable_on_def
apply (rule_tac x=s in exI)
unfolding has_integral
proof (rule, rule)
case goal1
then have *: "e/3 > 0" by auto
from LIMSEQ_D [OF s this] guess N1 .. note N1=this
from goal1 as have "e / 3 / content (cbox a b) > 0"
by (auto simp add: field_simps)
from real_arch_invD[OF this] guess N2 by (elim exE conjE) note N2=this
from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
have lem: "!!sf sg i. norm (sf - sg) ≤ e / 3 ==>
norm(i - s) < e / 3 ==> norm (sg - i) < e / 3 ==> norm (sf - s) < e"
proof -
case goal1
have "norm (sf - s) ≤ norm (sf - sg) + norm (sg - i) + norm (i - s)"
using norm_triangle_ineq[of "sf - sg" "sg - s"]
using norm_triangle_ineq[of "sg -  i" " i - s"]
by (auto simp add: algebra_simps)
also have "… < e"
using goal1
unfolding norm_minus_commute
by (auto simp add: algebra_simps)
finally show ?case .
qed
show ?case
apply (rule_tac x=g' in exI)
apply rule
apply (rule g')
proof (rule, rule)
fix p
assume p: "p tagged_division_of (cbox a b) ∧ g' fine p"
note * = g'(2)[OF this]
show "norm ((∑(x, k)∈p. content k *⇩R f x) - s) < e"
apply -
apply (rule lem[OF _ _ *])
apply (rule order_trans)
apply (rule rsum_diff_bound[OF p[THEN conjunct1]])
apply rule
apply (rule g)
apply assumption
proof -
have "content (cbox a b) < e / 3 * (real N2)"
using N2 unfolding inverse_eq_divide using as by (auto simp add: field_simps)
then have "content (cbox a b) < e / 3 * (real (N1 + N2) + 1)"
apply -
apply (rule less_le_trans,assumption)
using `e>0`
apply auto
done
then show "inverse (real (N1 + N2) + 1) * content (cbox a b) ≤ e / 3"
unfolding inverse_eq_divide
by (auto simp add: field_simps)
show "norm (i (N1 + N2) - s) < e / 3"
by (rule N1[rule_format]) auto
qed
qed
qed
qed

subsection {* Negligible sets. *}

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

subsection {* Negligibility of hyperplane. *}

lemma vsum_nonzero_image_lemma:
assumes "finite s"
and "g a = 0"
and "∀x∈s. ∀y∈s. f x = f y ∧ x ≠ y --> g (f x) = 0"
shows "setsum g {f x |x. x ∈ s ∧ f x ≠ a} = setsum (g o f) s"
unfolding setsum_iterate[OF assms(1)]
apply (subst setsum_iterate)
defer
apply (rule iterate_nonzero_image_lemma)
apply (rule assms monoidal_monoid)+
unfolding assms
unfolding neutral_add
using assms
apply auto
done

lemma interval_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "k ∈ Basis"
shows "cbox a b ∩ {x . abs(x•k - c) ≤ (e::real)} =
cbox (∑i∈Basis. (if i = k then max (a•k) (c - e) else a•i) *⇩R i)
(∑i∈Basis. (if i = k then min (b•k) (c + e) else b•i) *⇩R i)"
proof -
have *: "!!x c e::real. abs(x - c) ≤ e <-> x ≥ c - e ∧ x ≤ c + e"
by auto
have **: "!!s P Q. s ∩ {x. P x ∧ Q x} = (s ∩ {x. Q x}) ∩ {x. P x}"
by blast
show ?thesis
unfolding * ** interval_split[OF assms] by (rule refl)
qed

lemma division_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k ∈ Basis"
shows "{l ∩ {x. abs(x•k - c) ≤ e} |l. l ∈ p ∧ l ∩ {x. abs(x•k - c) ≤ e} ≠ {}} division_of (cbox a b ∩ {x. abs(x•k - c) ≤ e})"
proof -
have *: "!!x c. abs (x - c) ≤ e <-> x ≥ c - e ∧ x ≤ c + e"
by auto
have **: "!!p q p' q'. p division_of q ==> p = p' ==> q = q' ==> p' division_of q'"
by auto
note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
note division_split(2)[OF this, where c="c-e" and k=k,OF k]
then show ?thesis
apply (rule **)
using k
apply -
unfolding interval_doublesplit
unfolding *
unfolding interval_split interval_doublesplit
apply (rule set_eqI)
unfolding mem_Collect_eq
apply rule
apply (erule conjE exE)+
apply (rule_tac x=la in exI)
defer
apply (erule conjE exE)+
apply (rule_tac x="l ∩ {x. c + e ≥ x • k}" in exI)
apply rule
defer
apply rule
apply (rule_tac x=l in exI)
apply blast+
done
qed

lemma content_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "0 < e"
and k: "k ∈ Basis"
obtains d where "0 < d" and "content (cbox a b ∩ {x. abs(x•k - c) ≤ d}) < e"
proof (cases "content (cbox a b) = 0")
case True
show ?thesis
apply (rule that[of 1])
defer
unfolding interval_doublesplit[OF k]
apply (rule le_less_trans[OF content_subset])
defer
apply (subst True)
unfolding interval_doublesplit[symmetric,OF k]
using assms
apply auto
done
next
case False
def d ≡ "e / 3 / setprod (λi. b•i - a•i) (Basis - {k})"
note False[unfolded content_eq_0 not_ex not_le, rule_format]
then have "!!x. x ∈ Basis ==> b•x > a•x"
by (auto simp add:not_le)
then have prod0: "0 < setprod (λi. b•i - a•i) (Basis - {k})"
apply -
apply (rule setprod_pos)
apply (auto simp add: field_simps)
done
then have "d > 0"
unfolding d_def
using assms
by (auto simp add:field_simps)
then show ?thesis
proof (rule that[of d])
have *: "Basis = insert k (Basis - {k})"
using k by auto
have **: "cbox a b ∩ {x. ¦x • k - c¦ ≤ d} ≠ {} ==>
(∏i∈Basis - {k}. interval_upperbound (cbox a b ∩ {x. ¦x • k - c¦ ≤ d}) • i -
interval_lowerbound (cbox a b ∩ {x. ¦x • k - c¦ ≤ d}) • i) =
(∏i∈Basis - {k}. b•i - a•i)"
apply (rule setprod.cong)
apply (rule refl)
unfolding interval_doublesplit[OF k]
apply (subst interval_bounds)
defer
apply (subst interval_bounds)
unfolding box_eq_empty not_ex not_less
apply auto
done
show "content (cbox a b ∩ {x. ¦x • k - c¦ ≤ d}) < e"
apply cases
unfolding content_def
apply (subst if_P)
apply assumption
apply (rule assms)
unfolding if_not_P
apply (subst *)
apply (subst setprod.insert)
unfolding **
unfolding interval_doublesplit[OF k] box_eq_empty not_ex not_less
prefer 3
apply (subst interval_bounds)
defer
apply (subst interval_bounds)
apply (simp_all only: k inner_setsum_left_Basis simp_thms if_P cong: bex_cong ball_cong)
proof -
have "(min (b • k) (c + d) - max (a • k) (c - d)) ≤ 2 * d"
by auto
also have "… < e / (∏i∈Basis - {k}. b • i - a • i)"
unfolding d_def
using assms prod0
by (auto simp add: field_simps)
finally show "(min (b • k) (c + d) - max (a • k) (c - d)) * (∏i∈Basis - {k}. b • i - a • i) < e"
unfolding pos_less_divide_eq[OF prod0] .
qed auto
qed
qed

lemma negligible_standard_hyperplane[intro]:
fixes k :: "'a::euclidean_space"
assumes k: "k ∈ Basis"
shows "negligible {x. x•k = c}"
unfolding negligible_def has_integral
apply (rule, rule, rule, rule)
proof -
case goal1
from content_doublesplit[OF this k,of a b c] guess d . note d=this
let ?i = "indicator {x::'a. x•k = c} :: 'a=>real"
show ?case
apply (rule_tac x="λx. ball x d" in exI)
apply rule
apply (rule gauge_ball)
apply (rule d)
proof (rule, rule)
fix p
assume p: "p tagged_division_of (cbox a b) ∧ (λx. ball x d) fine p"
have *: "(∑(x, ka)∈p. content ka *⇩R ?i x) =
(∑(x, ka)∈p. content (ka ∩ {x. abs(x•k - c) ≤ d}) *⇩R ?i x)"
apply (rule setsum.cong)
apply (rule refl)
unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
apply cases
apply (rule disjI1)
apply assumption
apply (rule disjI2)
proof -
fix x l
assume as: "(x, l) ∈ p" "?i x ≠ 0"
then have xk: "x•k = c"
unfolding indicator_def
apply -
apply (rule ccontr)
apply auto
done
show "content l = content (l ∩ {x. ¦x • k - c¦ ≤ d})"
apply (rule arg_cong[where f=content])
apply (rule set_eqI)
apply rule
apply rule
unfolding mem_Collect_eq
proof -
fix y
assume y: "y ∈ l"
note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y]
note le_less_trans[OF Basis_le_norm[OF k] this]
then show "¦y • k - c¦ ≤ d"
unfolding inner_simps xk by auto
qed auto
qed
note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
show "norm ((∑(x, ka)∈p. content ka *⇩R ?i x) - 0) < e"
unfolding diff_0_right *
unfolding real_scaleR_def real_norm_def
apply (subst abs_of_nonneg)
apply (rule setsum_nonneg)
apply rule
unfolding split_paired_all split_conv
apply (rule mult_nonneg_nonneg)
apply (drule p'(4))
apply (erule exE)+
apply(rule_tac b=b in back_subst)
prefer 2
apply (subst(asm) eq_commute)
apply assumption
apply (subst interval_doublesplit[OF k])
apply (rule content_pos_le)
apply (rule indicator_pos_le)
proof -
have "(∑(x, ka)∈p. content (ka ∩ {x. ¦x • k - c¦ ≤ d}) * ?i x) ≤
(∑(x, ka)∈p. content (ka ∩ {x. ¦x • k - c¦ ≤ d}))"
apply (rule setsum_mono)
unfolding split_paired_all split_conv
apply (rule mult_right_le_one_le)
apply (drule p'(4))
apply (auto simp add:interval_doublesplit[OF k])
done
also have "… < e"
apply (subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])
proof -
case goal1
have "content (cbox u v ∩ {x. ¦x • k - c¦ ≤ d}) ≤ content (cbox u v)"
unfolding interval_doublesplit[OF k]
apply (rule content_subset)
unfolding interval_doublesplit[symmetric,OF k]
apply auto
done
then show ?case
unfolding goal1
unfolding interval_doublesplit[OF k]
by (blast intro: antisym)
next
have *: "setsum content {l ∩ {x. ¦x • k - c¦ ≤ d} |l. l ∈ snd ` p ∧ l ∩ {x. ¦x • k - c¦ ≤ d} ≠ {}} ≥ 0"
apply (rule setsum_nonneg)
apply rule
unfolding mem_Collect_eq image_iff
apply (erule exE bexE conjE)+
unfolding split_paired_all
proof -
fix x l a b
assume as: "x = l ∩ {x. ¦x • k - c¦ ≤ d}" "(a, b) ∈ p" "l = snd (a, b)"
guess u v using p'(4)[OF as(2)] by (elim exE) note * = this
show "content x ≥ 0"
unfolding as snd_conv * interval_doublesplit[OF k]
by (rule content_pos_le)
qed
have **: "norm (1::real) ≤ 1"
by auto
note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
note dsum_bound[OF this **,unfolded interval_doublesplit[symmetric,OF k]]
note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d]
note le_less_trans[OF this d(2)]
from this[unfolded abs_of_nonneg[OF *]]
show "(∑ka∈snd ` p. content (ka ∩ {x. ¦x • k - c¦ ≤ d})) < e"
apply (subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,symmetric])
apply (rule finite_imageI p' content_empty)+
unfolding forall_in_division[OF p'']
proof (rule,rule,rule,rule,rule,rule,rule,erule conjE)
fix m n u v
assume as:
"cbox m n ∈ snd ` p" "cbox u v ∈ snd ` p"
"cbox m n ≠ cbox u v"
"cbox m n ∩ {x. ¦x • k - c¦ ≤ d} = cbox u v ∩ {x. ¦x • k - c¦ ≤ d}"
have "(cbox m n ∩ {x. ¦x • k - c¦ ≤ d}) ∩ (cbox u v ∩ {x. ¦x • k - c¦ ≤ d}) ⊆ cbox m n ∩ cbox u v"
by blast
note interior_mono[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "cbox m n"]]
then have "interior (cbox m n ∩ {x. ¦x • k - c¦ ≤ d}) = {}"
unfolding as Int_absorb by auto
then show "content (cbox m n ∩ {x. ¦x • k - c¦ ≤ d}) = 0"
unfolding interval_doublesplit[OF k] content_eq_0_interior[symmetric] .
qed
qed
finally show "(∑(x, ka)∈p. content (ka ∩ {x. ¦x • k - c¦ ≤ d}) * ?i x) < e" .
qed
qed
qed

subsection {* A technical lemma about "refinement" of division. *}

lemma tagged_division_finer:
fixes p :: "('a::euclidean_space × ('a::euclidean_space set)) set"
assumes "p tagged_division_of (cbox a b)"
and "gauge d"
obtains q where "q tagged_division_of (cbox a b)"
and "d fine q"
and "∀(x,k) ∈ p. k ⊆ d(x) --> (x,k) ∈ q"
proof -
let ?P = "λp. p tagged_partial_division_of (cbox a b) --> gauge d -->
(∃q. q tagged_division_of (\<Union>{k. ∃x. (x,k) ∈ p}) ∧ d fine q ∧
(∀(x,k) ∈ p. k ⊆ d(x) --> (x,k) ∈ q))"
{
have *: "finite p" "p tagged_partial_division_of (cbox a b)"
using assms(1)
unfolding tagged_division_of_def
by auto
presume "!!p. finite p ==> ?P p"
from this[rule_format,OF * assms(2)] guess q .. note q=this
then show ?thesis
apply -
apply (rule that[of q])
unfolding tagged_division_ofD[OF assms(1)]
apply auto
done
}
fix p :: "('a::euclidean_space × ('a::euclidean_space set)) set"
assume as: "finite p"
show "?P p"
apply rule
apply rule
using as
proof (induct p)
case empty
show ?case
apply (rule_tac x="{}" in exI)
unfolding fine_def
apply auto
done
next
case (insert xk p)
guess x k using surj_pair[of xk] by (elim exE) note xk=this
note tagged_partial_division_subset[OF insert(4) subset_insertI]
from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
have *: "\<Union>{l. ∃y. (y,l) ∈ insert xk p} = k ∪ \<Union>{l. ∃y. (y,l) ∈ p}"
unfolding xk by auto
note p = tagged_partial_division_ofD[OF insert(4)]
from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this

have "finite {k. ∃x. (x, k) ∈ p}"
apply (rule finite_subset[of _ "snd ` p"],rule)
unfolding subset_eq image_iff mem_Collect_eq
apply (erule exE)
apply (rule_tac x="(xa,x)" in bexI)
using p
apply auto
done
then have int: "interior (cbox u v) ∩ interior (\<Union>{k. ∃x. (x, k) ∈ p}) = {}"
apply (rule inter_interior_unions_intervals)
apply (rule open_interior)
apply (rule_tac[!] ballI)
unfolding mem_Collect_eq
apply (erule_tac[!] exE)
apply (drule p(4)[OF insertI2])
apply assumption
apply (rule p(5))
unfolding uv xk
apply (rule insertI1)
apply (rule insertI2)
apply assumption
using insert(2)
unfolding uv xk
apply auto
done
show ?case
proof (cases "cbox u v ⊆ d x")
case True
then show ?thesis
apply (rule_tac x="{(x,cbox u v)} ∪ q1" in exI)
apply rule
unfolding * uv
apply (rule tagged_division_union)
apply (rule tagged_division_of_self)
apply (rule p[unfolded xk uv] insertI1)+
apply (rule q1)
apply (rule int)
apply rule
apply (rule fine_union)
apply (subst fine_def)
defer
apply (rule q1)
unfolding Ball_def split_paired_All split_conv
apply rule
apply rule
apply rule
apply rule
apply (erule insertE)
defer
apply (rule UnI2)
apply (drule q1(3)[rule_format])
unfolding xk uv
apply auto
done
next
case False
from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
show ?thesis
apply (rule_tac x="q2 ∪ q1" in exI)
apply rule
unfolding * uv
apply (rule tagged_division_union q2 q1 int fine_union)+
unfolding Ball_def split_paired_All split_conv
apply rule
apply (rule fine_union)
apply (rule q1 q2)+
apply rule
apply rule
apply rule
apply rule
apply (erule insertE)
apply (rule UnI2)
defer
apply (drule q1(3)[rule_format])
using False
unfolding xk uv
apply auto
done
qed
qed
qed

subsection {* Hence the main theorem about negligible sets. *}

lemma finite_product_dependent:
assumes "finite s"
and "!!x. x ∈ s ==> finite (t x)"
shows "finite {(i, j) |i j. i ∈ s ∧ j ∈ t i}"
using assms
proof induct
case (insert x s)
have *: "{(i, j) |i j. i ∈ insert x s ∧ j ∈ t i} =
(λy. (x,y)) ` (t x) ∪ {(i, j) |i j. i ∈ s ∧ j ∈ t i}" by auto
show ?case
unfolding *
apply (rule finite_UnI)
using insert
apply auto
done
qed auto

lemma sum_sum_product:
assumes "finite s"
and "∀i∈s. finite (t i)"
shows "setsum (λi. setsum (x i) (t i)::real) s =
setsum (λ(i,j). x i j) {(i,j) | i j. i ∈ s ∧ j ∈ t i}"
using assms
proof induct
case (insert a s)
have *: "{(i, j) |i j. i ∈ insert a s ∧ j ∈ t i} =
(λy. (a,y)) ` (t a) ∪ {(i, j) |i j. i ∈ s ∧ j ∈ t i}" by auto
show ?case
unfolding *
apply (subst setsum.union_disjoint)
unfolding setsum.insert[OF insert(1-2)]
prefer 4
apply (subst insert(3))
unfolding add_right_cancel
proof -
show "setsum (x a) (t a) = (∑(xa, y)∈ Pair a ` t a. x xa y)"
apply (subst setsum.reindex)
unfolding inj_on_def
apply auto
done
show "finite {(i, j) |i j. i ∈ s ∧ j ∈ t i}"
apply (rule finite_product_dependent)
using insert
apply auto
done
qed (insert insert, auto)
qed auto

lemma has_integral_negligible:
fixes f :: "'b::euclidean_space => 'a::real_normed_vector"
assumes "negligible s"
and "∀x∈(t - s). f x = 0"
shows "(f has_integral 0) t"
proof -
presume P: "!!f::'b::euclidean_space => 'a.
!!a b. ∀x. x ∉ s --> f x = 0 ==> (f has_integral 0) (cbox a b)"
let ?f = "(λx. if x ∈ t then f x else 0)"
show ?thesis
apply (rule_tac f="?f" in has_integral_eq)
apply rule
unfolding if_P
apply (rule refl)
apply (subst has_integral_alt)
apply cases
apply (subst if_P, assumption)
unfolding if_not_P
proof -
assume "∃a b. t = cbox a b"
then guess a b apply - by (erule exE)+ note t = this
show "(?f has_integral 0) t"
unfolding t
apply (rule P)
using assms(2)
unfolding t
apply auto
done
next
show "∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b -->
(∃z. ((λx. if x ∈ t then ?f x else 0) has_integral z) (cbox a b) ∧ norm (z - 0) < e)"
apply safe
apply (rule_tac x=1 in exI)
apply rule
apply (rule zero_less_one)
apply safe
apply (rule_tac x=0 in exI)
apply rule
apply (rule P)
using assms(2)
apply auto
done
qed
next
fix f :: "'b => 'a"
fix a b :: 'b
assume assm: "∀x. x ∉ s --> f x = 0"
show "(f has_integral 0) (cbox a b)"
unfolding has_integral
proof safe
case goal1
then have "!!n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
apply -
apply (rule divide_pos_pos)
defer
apply (rule mult_pos_pos)
apply (auto simp add:field_simps)
done
note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b]
note allI[OF this,of "λx. x"]
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
show ?case
apply (rule_tac x="λx. d (nat ⌊norm (f x)⌋) x" in exI)
proof safe
show "gauge (λx. d (nat ⌊norm (f x)⌋) x)"
using d(1) unfolding gauge_def by auto
fix p
assume as: "p tagged_division_of (cbox a b)" "(λx. d (nat ⌊norm (f x)⌋) x) fine p"
let ?goal = "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
{
presume "p ≠ {} ==> ?goal"
then show ?goal
apply (cases "p = {}")
using goal1
apply auto
done
}
assume as': "p ≠ {}"
from real_arch_simple[of "Sup((λ(x,k). norm(f x)) ` p)"] guess N ..
then have N: "∀x∈(λ(x, k). norm (f x)) ` p. x ≤ real N"
apply (subst(asm) cSup_finite_le_iff)
using as as'
apply auto
done
have "∀i. ∃q. q tagged_division_of (cbox a b) ∧ (d i) fine q ∧ (∀(x, k)∈p. k ⊆ (d i) x --> (x, k) ∈ q)"
apply rule
apply (rule tagged_division_finer[OF as(1) d(1)])
apply auto
done
from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
have *: "!!i. (∑(x, k)∈q i. content k *⇩R indicator s x) ≥ (0::real)"
apply (rule setsum_nonneg)
apply safe
unfolding real_scaleR_def
apply (drule tagged_division_ofD(4)[OF q(1)])
apply (auto intro: mult_nonneg_nonneg)
done
have **: "!!f g s t. finite s ==> finite t ==> (∀(x,y) ∈ t. (0::real) ≤ g(x,y)) ==>
(∀y∈s. ∃x. (x,y) ∈ t ∧ f(y) ≤ g(x,y)) ==> setsum f s ≤ setsum g t"
proof -
case goal1
then show ?case
apply -
apply (rule setsum_le_included[of s t g snd f])
prefer 4
apply safe
apply (erule_tac x=x in ballE)
apply (erule exE)
apply (rule_tac x="(xa,x)" in bexI)
apply auto
done
qed
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) ≤ setsum (λi. (real i + 1) *
norm (setsum (λ(x,k). content k *⇩R indicator s x :: real) (q i))) {..N+1}"
unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
apply (rule order_trans)
apply (rule norm_setsum)
apply (subst sum_sum_product)
prefer 3
proof (rule **, safe)
show "finite {(i, j) |i j. i ∈ {..N + 1} ∧ j ∈ q i}"
apply (rule finite_product_dependent)
using q
apply auto
done
fix i a b
assume as'': "(a, b) ∈ q i"
show "0 ≤ (real i + 1) * (content b *⇩R indicator s a)"
unfolding real_scaleR_def
using tagged_division_ofD(4)[OF q(1) as'']
by (auto intro!: mult_nonneg_nonneg)
next
fix i :: nat
show "finite (q i)"
using q by auto
next
fix x k
assume xk: "(x, k) ∈ p"
def n ≡ "nat ⌊norm (f x)⌋"
have *: "norm (f x) ∈ (λ(x, k). norm (f x)) ` p"
using xk by auto
have nfx: "real n ≤ norm (f x)" "norm (f x) ≤ real n + 1"
unfolding n_def by auto
then have "n ∈ {0..N + 1}"
using N[rule_format,OF *] by auto
moreover
note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this]
note this[unfolded n_def[symmetric]]
moreover
have "norm (content k *⇩R f x) ≤ (real n + 1) * (content k * indicator s x)"
proof (cases "x ∈ s")
case False
then show ?thesis
using assm by auto
next
case True
have *: "content k ≥ 0"
using tagged_division_ofD(4)[OF as(1) xk] by auto
moreover
have "content k * norm (f x) ≤ content k * (real n + 1)"
apply (rule mult_mono)
using nfx *
apply auto
done
ultimately
show ?thesis
unfolding abs_mult
using nfx True
by (auto simp add: field_simps)
qed
ultimately show "∃y. (y, x, k) ∈ {(i, j) |i j. i ∈ {..N + 1} ∧ j ∈ q i} ∧ norm (content k *⇩R f x) ≤
(real y + 1) * (content k *⇩R indicator s x)"
apply (rule_tac x=n in exI)
apply safe
apply (rule_tac x=n in exI)
apply (rule_tac x="(x,k)" in exI)
apply safe
apply auto
done
qed (insert as, auto)
also have "… ≤ setsum (λi. e / 2 / 2 ^ i) {..N+1}"
apply (rule setsum_mono)
proof -
case goal1
then show ?case
apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
using d(2)[rule_format,of "q i" i]
using q[rule_format]
apply (auto simp add: field_simps)
done
qed
also have "… < e * inverse 2 * 2"
unfolding divide_inverse setsum_right_distrib[symmetric]
apply (rule mult_strict_left_mono)
unfolding power_inverse lessThan_Suc_atMost[symmetric]
apply (subst geometric_sum)
using goal1
apply auto
done
finally show "?goal" by auto
qed
qed
qed

lemma has_integral_spike:
fixes f :: "'b::euclidean_space => 'a::real_normed_vector"
assumes "negligible s"
and "(∀x∈(t - s). g x = f x)"
and "(f has_integral y) t"
shows "(g has_integral y) t"
proof -
{
fix a b :: 'b
fix f g :: "'b => 'a"
fix y :: 'a
assume as: "∀x ∈ cbox a b - s. g x = f x" "(f has_integral y) (cbox a b)"
have "((λx. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
apply (rule has_integral_add[OF as(2)])
apply (rule has_integral_negligible[OF assms(1)])
using as
apply auto
done
then have "(g has_integral y) (cbox a b)"
by auto
} note * = this
show ?thesis
apply (subst has_integral_alt)
using assms(2-)
apply -
apply (rule cond_cases)
apply safe
apply (rule *)
apply assumption+
apply (subst(asm) has_integral_alt)
unfolding if_not_P
apply (erule_tac x=e in allE)
apply safe
apply (rule_tac x=B in exI)
apply safe
apply (erule_tac x=a in allE)
apply (erule_tac x=b in allE)
apply safe
apply (rule_tac x=z in exI)
apply safe
apply (rule *[where fa2="λx. if x∈t then f x else 0"])
apply auto
done
qed

lemma has_integral_spike_eq:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
shows "((f has_integral y) t <-> (g has_integral y) t)"
apply rule
apply (rule_tac[!] has_integral_spike[OF assms(1)])
using assms(2)
apply auto
done

lemma integrable_spike:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
and "f integrable_on t"
shows "g integrable_on  t"
using assms
unfolding integrable_on_def
apply -
apply (erule exE)
apply rule
apply (rule has_integral_spike)
apply fastforce+
done

lemma integral_spike:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
shows "integral t f = integral t g"
unfolding integral_def
using has_integral_spike_eq[OF assms]
by auto

subsection {* Some other trivialities about negligible sets. *}

lemma negligible_subset[intro]:
assumes "negligible s"
and "t ⊆ s"
shows "negligible t"
unfolding negligible_def
proof safe
case goal1
show ?case
using assms(1)[unfolded negligible_def,rule_format,of a b]
apply -
apply (rule has_integral_spike[OF assms(1)])
defer
apply assumption
using assms(2)
unfolding indicator_def
apply auto
done
qed

lemma negligible_diff[intro?]:
assumes "negligible s"
shows "negligible (s - t)"
using assms by auto

lemma negligible_inter:
assumes "negligible s ∨ negligible t"
shows "negligible (s ∩ t)"
using assms by auto

lemma negligible_union:
assumes "negligible s"
and "negligible t"
shows "negligible (s ∪ t)"
unfolding negligible_def
proof safe
case goal1
note assm = assms[unfolded negligible_def,rule_format,of a b]
then show ?case
apply (subst has_integral_spike_eq[OF assms(2)])
defer
apply assumption
unfolding indicator_def
apply auto
done
qed

lemma negligible_union_eq[simp]: "negligible (s ∪ t) <-> negligible s ∧ negligible t"
using negligible_union by auto

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

lemma negligible_insert[simp]: "negligible (insert a s) <-> negligible s"
apply (subst insert_is_Un)
unfolding negligible_union_eq
apply auto
done

lemma negligible_empty[intro]: "negligible {}"
by auto

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

lemma negligible_unions[intro]:
assumes "finite s"
and "∀t∈s. negligible t"
shows "negligible(\<Union>s)"
using assms by induct auto

lemma negligible:
"negligible s <-> (∀t::('a::euclidean_space) set. ((indicator s::'a=>real) has_integral 0) t)"
apply safe
defer
apply (subst negligible_def)
proof -
fix t :: "'a set"
assume as: "negligible s"
have *: "(λx. if x ∈ s ∩ t then 1 else 0) = (λx. if x∈t then if x∈s then 1 else 0 else 0)"
by auto
show "((indicator s::'a=>real) has_integral 0) t"
apply (subst has_integral_alt)
apply cases
apply (subst if_P,assumption)
unfolding if_not_P
apply safe
apply (rule as[unfolded negligible_def,rule_format])
apply (rule_tac x=1 in exI)
apply safe
apply (rule zero_less_one)
apply (rule_tac x=0 in exI)
using negligible_subset[OF as,of "s ∩ t"]
unfolding negligible_def indicator_def [abs_def]
unfolding *
apply auto
done
qed auto

subsection {* Finite case of the spike theorem is quite commonly needed. *}

lemma has_integral_spike_finite:
assumes "finite s"
and "∀x∈t-s. g x = f x"
and "(f has_integral y) t"
shows "(g has_integral y) t"
apply (rule has_integral_spike)
using assms
apply auto
done

lemma has_integral_spike_finite_eq:
assumes "finite s"
and "∀x∈t-s. g x = f x"
shows "((f has_integral y) t <-> (g has_integral y) t)"
apply rule
apply (rule_tac[!] has_integral_spike_finite)
using assms
apply auto
done

lemma integrable_spike_finite:
assumes "finite s"
and "∀x∈t-s. g x = f x"
and "f integrable_on t"
shows "g integrable_on  t"
using assms
unfolding integrable_on_def
apply safe
apply (rule_tac x=y in exI)
apply (rule has_integral_spike_finite)
apply auto
done

subsection {* In particular, the boundary of an interval is negligible. *}

lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
proof -
let ?A = "\<Union>((λk. {x. x•k = a•k} ∪ {x::'a. x•k = b•k}) ` Basis)"
have "cbox a b - box a b ⊆ ?A"
apply rule unfolding Diff_iff mem_box
apply simp
apply(erule conjE bexE)+
apply(rule_tac x=i in bexI)
apply auto
done
then show ?thesis
apply -
apply (rule negligible_subset[of ?A])
apply (rule negligible_unions[OF finite_imageI])
apply auto
done
qed

lemma has_integral_spike_interior:
assumes "∀x∈box a b. g x = f x"
and "(f has_integral y) (cbox a b)"
shows "(g has_integral y) (cbox a b)"
apply (rule has_integral_spike[OF negligible_frontier_interval _ assms(2)])
using assms(1)
apply auto
done

lemma has_integral_spike_interior_eq:
assumes "∀x∈box a b. g x = f x"
shows "(f has_integral y) (cbox a b) <-> (g has_integral y) (cbox a b)"
apply rule
apply (rule_tac[!] has_integral_spike_interior)
using assms
apply auto
done

lemma integrable_spike_interior:
assumes "∀x∈box a b. g x = f x"
and "f integrable_on cbox a b"
shows "g integrable_on cbox a b"
using assms
unfolding integrable_on_def
using has_integral_spike_interior[OF assms(1)]
by auto

subsection {* Integrability of continuous functions. *}

lemma neutral_and[simp]: "neutral op ∧ = True"
unfolding neutral_def by (rule some_equality) auto

lemma monoidal_and[intro]: "monoidal op ∧"
unfolding monoidal_def by auto

lemma iterate_and[simp]:
assumes "finite s"
shows "(iterate op ∧) s p <-> (∀x∈s. p x)"
using assms
apply induct
unfolding iterate_insert[OF monoidal_and]
apply auto
done

lemma operative_division_and:
assumes "operative op ∧ P"
and "d division_of (cbox a b)"
shows "(∀i∈d. P i) <-> P (cbox a b)"
using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)]
by auto

lemma operative_approximable:
fixes f::"'b::euclidean_space => 'a::banach"
assumes "0 ≤ e"
shows "operative op ∧ (λi. ∃g. (∀x∈i. norm (f x - g (x::'b)) ≤ e) ∧ g integrable_on i)"
unfolding operative_def neutral_and
proof safe
fix a b :: 'b
{
assume "content (cbox a b) = 0"
then show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
apply (rule_tac x=f in exI)
using assms
apply (auto intro!:integrable_on_null)
done
}
{
fix c g
fix k :: 'b
assume as: "∀x∈cbox a b. norm (f x - g x) ≤ e" "g integrable_on cbox a b"
assume k: "k ∈ Basis"
show "∃g. (∀x∈cbox a b ∩ {x. x • k ≤ c}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. x • k ≤ c}"
"∃g. (∀x∈cbox a b ∩ {x. c ≤ x • k}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. c ≤ x • k}"
apply (rule_tac[!] x=g in exI)
using as(1) integrable_split[OF as(2) k]
apply auto
done
}
fix c k g1 g2
assume as: "∀x∈cbox a b ∩ {x. x • k ≤ c}. norm (f x - g1 x) ≤ e" "g1 integrable_on cbox a b ∩ {x. x • k ≤ c}"
"∀x∈cbox a b ∩ {x. c ≤ x • k}. norm (f x - g2 x) ≤ e" "g2 integrable_on cbox a b ∩ {x. c ≤ x • k}"
assume k: "k ∈ Basis"
let ?g = "λx. if x•k = c then f x else if x•k ≤ c then g1 x else g2 x"
show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
apply (rule_tac x="?g" in exI)
proof safe
case goal1
then show ?case
apply -
apply (cases "x•k=c")
apply (case_tac "x•k < c")
using as assms
apply auto
done
next
case goal2
presume "?g integrable_on cbox a b ∩ {x. x • k ≤ c}"
and "?g integrable_on cbox a b ∩ {x. x • k ≥ c}"
then guess h1 h2 unfolding integrable_on_def by auto
from has_integral_split[OF this k] show ?case
unfolding integrable_on_def by auto
next
show "?g integrable_on cbox a b ∩ {x. x • k ≤ c}" "?g integrable_on cbox a b ∩ {x. x • k ≥ c}"
apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
using k as(2,4)
apply auto
done
qed
qed

lemma approximable_on_division:
fixes f :: "'b::euclidean_space => 'a::banach"
assumes "0 ≤ e"
and "d division_of (cbox a b)"
and "∀i∈d. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
obtains g where "∀x∈cbox a b. norm (f x - g x) ≤ e" "g integrable_on cbox a b"
proof -
note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]]
from assms(3)[unfolded this[of f]] guess g ..
then show thesis
apply -
apply (rule that[of g])
apply auto
done
qed

lemma integrable_continuous:
fixes f :: "'b::euclidean_space => 'a::banach"
assumes "continuous_on (cbox a b) f"
shows "f integrable_on cbox a b"
proof (rule integrable_uniform_limit, safe)
fix e :: real
assume e: "e > 0"
from compact_uniformly_continuous[OF assms compact_cbox,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
note d=conjunctD2[OF this,rule_format]
from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
note p' = tagged_division_ofD[OF p(1)]
have *: "∀i∈snd ` p. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
proof (safe, unfold snd_conv)
fix x l
assume as: "(x, l) ∈ p"
from p'(4)[OF this] guess a b by (elim exE) note l=this
show "∃g. (∀x∈l. norm (f x - g x) ≤ e) ∧ g integrable_on l"
apply (rule_tac x="λy. f x" in exI)
proof safe
show "(λy. f x) integrable_on l"
unfolding integrable_on_def l
apply rule
apply (rule has_integral_const)
done
fix y
assume y: "y ∈ l"
note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
note d(2)[OF _ _ this[unfolded mem_ball]]
then show "norm (f y - f x) ≤ e"
using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce
qed
qed
from e have "e ≥ 0"
by auto
from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
then show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
by auto
qed

lemma integrable_continuous_real:
fixes f :: "real => 'a::banach"
assumes "continuous_on {a .. b} f"
shows "f integrable_on {a .. b}"
by (metis assms box_real(2) integrable_continuous)

subsection {* Specialization of additivity to one dimension. *}

lemma
shows real_inner_1_left: "inner 1 x = x"
and real_inner_1_right: "inner x 1 = x"
by simp_all

lemma content_real_eq_0: "content {a .. b::real} = 0 <-> a ≥ b"
by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)

lemma interval_real_split:
"{a .. b::real} ∩ {x. x ≤ c} = {a .. min b c}"
"{a .. b} ∩ {x. c ≤ x} = {max a c .. b}"
apply (metis Int_atLeastAtMostL1 atMost_def)
apply (metis Int_atLeastAtMostL2 atLeast_def)
done

lemma operative_1_lt:
assumes "monoidal opp"
shows "operative opp f <-> ((∀a b. b ≤ a --> f {a .. b::real} = neutral opp) ∧
(∀a b c. a < c ∧ c < b --> opp (f {a .. c}) (f {c .. b}) = f {a .. b}))"
apply (simp add: operative_def content_real_eq_0)
proof safe
fix a b c :: real
assume as:
"∀a b c. f {a..b} = opp (f ({a..b} ∩ {x. x ≤ c})) (f ({a..b} ∩ Collect (op ≤ c)))"
"a < c"
"c < b"
from this(2-) have "cbox a b ∩ {x. x ≤ c} = cbox a c" "cbox a b ∩ {x. x ≥ c} = cbox c b"
by (auto simp: mem_box)
then show "opp (f {a..c}) (f {c..b}) = f {a..b}"
unfolding as(1)[rule_format,of a b "c"] by auto
next
fix a b c :: real
assume as: "∀a b. b ≤ a --> f {a..b} = neutral opp"
"∀a b c. a < c ∧ c < b --> opp (f {a..c}) (f {c..b}) = f {a..b}"
show " f {a..b} = opp (f ({a..b} ∩ {x. x ≤ c})) (f ({a..b} ∩ Collect (op ≤ c)))"
proof (cases "c ∈ {a..b}")
case False
then have "c < a ∨ c > b" by auto
then show ?thesis
proof
assume "c < a"
then have *: "{a..b} ∩ {x. x ≤ c} = {1..0}" "{a..b} ∩ {x. c ≤ x} = {a..b}"
by auto
show ?thesis
unfolding *
apply (subst as(1)[rule_format,of 0 1])
using assms
apply auto
done
next
assume "b < c"
then have *: "{a..b} ∩ {x. x ≤ c} = {a..b}" "{a..b} ∩ {x. c ≤ x} = {1 .. 0}"
by auto
show ?thesis
unfolding *
apply (subst as(1)[rule_format,of 0 1])
using assms
apply auto
done
qed
next
case True
then have *: "min (b) c = c" "max a c = c"
by auto
have **: "(1::real) ∈ Basis"
by simp
have ***: "!!P Q. (∑i∈Basis. (if i = 1 then P i else Q i) *⇩R i) = (P 1::real)"
by simp
show ?thesis
unfolding interval_real_split unfolding *
proof (cases "c = a ∨ c = b")
case False
then show "f {a..b} = opp (f {a..c}) (f {c..b})"
apply -
apply (subst as(2)[rule_format])
using True
apply auto
done
next
case True
then show "f {a..b} = opp (f {a..c}) (f {c..b})"
proof
assume *: "c = a"
then have "f {a .. c} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
next
assume *: "c = b"
then have "f {c .. b} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
qed
qed
qed
qed

lemma operative_1_le:
assumes "monoidal opp"
shows "operative opp f <-> ((∀a b. b ≤ a --> f {a .. b::real} = neutral opp) ∧
(∀a b c. a ≤ c ∧ c ≤ b --> opp (f {a .. c}) (f {c .. b}) = f {a .. b}))"
unfolding operative_1_lt[OF assms]
proof safe
fix a b c :: real
assume as:
"∀a b c. a ≤ c ∧ c ≤ b --> opp (f {a..c}) (f {c..b}) = f {a..b}"
"a < c"
"c < b"
show "opp (f {a..c}) (f {c..b}) = f {a..b}"
apply (rule as(1)[rule_format])
using as(2-)
apply auto
done
next
fix a b c :: real
assume "∀a b. b ≤ a --> f {a .. b} = neutral opp"
and "∀a b c. a < c ∧ c < b --> opp (f {a..c}) (f {c..b}) = f {a..b}"
and "a ≤ c"
and "c ≤ b"
note as = this[rule_format]
show "opp (f {a..c}) (f {c..b}) = f {a..b}"
proof (cases "c = a ∨ c = b")
case False
then show ?thesis
apply -
apply (subst as(2))
using as(3-)
apply auto
done
next
case True
then show ?thesis
proof
assume *: "c = a"
then have "f {a .. c} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
next
assume *: "c = b"
then have "f {c .. b} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
qed
qed
qed

subsection {* Special case of additivity we need for the FCT. *}

lemma additive_tagged_division_1:
fixes f :: "real => 'a::real_normed_vector"
assumes "a ≤ b"
and "p tagged_division_of {a..b}"
shows "setsum (λ(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
proof -
let ?f = "(λk::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
have ***: "∀i∈Basis. a • i ≤ b • i"
using assms by auto
have *: "operative op + ?f"
unfolding operative_1_lt[OF monoidal_monoid] box_eq_empty
by auto
have **: "cbox a b ≠ {}"
using assms(1) by auto
note operative_tagged_division[OF monoidal_monoid * assms(2)[simplified box_real[symmetric]]]
note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
show ?thesis
unfolding *
apply (subst setsum_iterate[symmetric])
defer
apply (rule setsum.cong)
unfolding split_paired_all split_conv
using assms(2)
apply auto
done
qed

subsection {* A useful lemma allowing us to factor out the content size. *}

lemma has_integral_factor_content:
"(f has_integral i) (cbox a b) <->
(∀e>0. ∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p -->
norm (setsum (λ(x,k). content k *⇩R f x) p - i) ≤ e * content (cbox a b)))"
proof (cases "content (cbox a b) = 0")
case True
show ?thesis
unfolding has_integral_null_eq[OF True]
apply safe
apply (rule, rule, rule gauge_trivial, safe)
unfolding setsum_content_null[OF True] True
defer
apply (erule_tac x=1 in allE)
apply safe
defer
apply (rule fine_division_exists[of _ a b])
apply assumption
apply (erule_tac x=p in allE)
unfolding setsum_content_null[OF True]
apply auto
done
next
case False
note F = this[unfolded content_lt_nz[symmetric]]
let ?P = "λe opp. ∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p --> opp (norm ((∑(x, k)∈p. content k *⇩R f x) - i)) e)"
show ?thesis
apply (subst has_integral)
proof safe
fix e :: real
assume e: "e > 0"
{
assume "∀e>0. ?P e op <"
then show "?P (e * content (cbox a b)) op ≤"
apply (erule_tac x="e * content (cbox a b)" in allE)
apply (erule impE)
defer
apply (erule exE,rule_tac x=d in exI)
using F e
apply (auto simp add:field_simps)
done
}
{
assume "∀e>0. ?P (e * content (cbox a b)) op ≤"
then show "?P e op <"
apply (erule_tac x="e / 2 / content (cbox a b)" in allE)
apply (erule impE)
defer
apply (erule exE,rule_tac x=d in exI)
using F e
apply (auto simp add: field_simps)
done
}
qed
qed

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

subsection {* Fundamental theorem of calculus. *}

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

lemma fundamental_theorem_of_calculus:
fixes f :: "real => 'a::banach"
assumes "a ≤ b"
and "∀x∈{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
shows "(f' has_integral (f b - f a)) {a .. b}"
unfolding has_integral_factor_content box_real[symmetric]
proof safe
fix e :: real
assume e: "e > 0"
note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
have *: "!!P Q. ∀x∈{a .. b}. P x ∧ (∀e>0. ∃d>0. Q x e d) ==> ∀x. ∃(d::real)>0. x∈{a .. b} --> Q x e d"
using e by blast
note this[OF assm,unfolded gauge_existence_lemma]
from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
note d=conjunctD2[OF this[rule_format],rule_format]
show "∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p -->
norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content (cbox a b))"
apply (rule_tac x="λx. ball x (d x)" in exI)
apply safe
apply (rule gauge_ball_dependent)
apply rule
apply (rule d(1))
proof -
fix p
assume as: "p tagged_division_of cbox a b" "(λx. ball x (d x)) fine p"
show "norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content (cbox a b)"
unfolding content_real[OF assms(1), simplified box_real[symmetric]] additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of f,symmetric]
unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "λx. x",symmetric]
unfolding setsum_right_distrib
defer
unfolding setsum_subtractf[symmetric]
proof (rule setsum_norm_le,safe)
fix x k
assume "(x, k) ∈ p"
note xk = tagged_division_ofD(2-4)[OF as(1) this]
from this(3) guess u v by (elim exE) note k=this
have *: "u ≤ v"
using xk unfolding k by auto
have ball: "∀xa∈k. xa ∈ ball x (d x)"
using as(2)[unfolded fine_def,rule_format,OF `(x,k)∈p`,unfolded split_conv subset_eq] .
have "norm ((v - u) *⇩R f' x - (f v - f u)) ≤
norm (f u - f x - (u - x) *⇩R f' x) + norm (f v - f x - (v - x) *⇩R f' x)"
apply (rule order_trans[OF _ norm_triangle_ineq4])
apply (rule eq_refl)
apply (rule arg_cong[where f=norm])
unfolding scaleR_diff_left
apply (auto simp add:algebra_simps)
done
also have "… ≤ e * norm (u - x) + e * norm (v - x)"
apply (rule add_mono)
apply (rule d(2)[of "x" "u",unfolded o_def])
prefer 4
apply (rule d(2)[of "x" "v",unfolded o_def])
using ball[rule_format,of u] ball[rule_format,of v]
using xk(1-2)
unfolding k subset_eq
apply (auto simp add:dist_real_def)
done
also have "… ≤ e * (Sup k - Inf k)"
unfolding k interval_bounds_real[OF *]
using xk(1)
unfolding k
by (auto simp add: dist_real_def field_simps)
finally show "norm (content k *⇩R f' x - (f (Sup k) - f (Inf k))) ≤
e * (Sup k - Inf k)"
unfolding box_real k interval_bounds_real[OF *] content_real[OF *]
interval_upperbound_real interval_lowerbound_real
.
qed
qed
qed

subsection {* Attempt a systematic general set of "offset" results for components. *}

lemma gauge_modify:
assumes "(∀s. open s --> open {x. f(x) ∈ s})" "gauge d"
shows "gauge (λx. {y. f y ∈ d (f x)})"
using assms
unfolding gauge_def
apply safe
defer
apply (erule_tac x="f x" in allE)
apply (erule_tac x="d (f x)" in allE)
apply auto
done

subsection {* Only need trivial subintervals if the interval itself is trivial. *}

lemma division_of_nontrivial:
fixes s :: "'a::euclidean_space set set"
assumes "s division_of (cbox a b)"
and "content (cbox a b) ≠ 0"
shows "{k. k ∈ s ∧ content k ≠ 0} division_of (cbox a b)"
using assms(1)
apply -
proof (induct "card s" arbitrary: s rule: nat_less_induct)
fix s::"'a set set"
assume assm: "s division_of (cbox a b)"
"∀m<card s. ∀x. m = card x -->
x division_of (cbox a b) --> {k ∈ x. content k ≠ 0} division_of (cbox a b)"
note s = division_ofD[OF assm(1)]
let ?thesis = "{k ∈ s. content k ≠ 0} division_of (cbox a b)"
{
presume *: "{k ∈ s. content k ≠ 0} ≠ s ==> ?thesis"
show ?thesis
apply cases
defer
apply (rule *)
apply assumption
using assm(1)
apply auto
done
}
assume noteq: "{k ∈ s. content k ≠ 0} ≠ s"
then obtain k where k: "k ∈ s" "content k = 0"
by auto
from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
from k have "card s > 0"
unfolding card_gt_0_iff using assm(1) by auto
then have card: "card (s - {k}) < card s"
using assm(1) k(1)
apply (subst card_Diff_singleton_if)
apply auto
done
have *: "closed (\<Union>(s - {k}))"
apply (rule closed_Union)
defer
apply rule
apply (drule DiffD1,drule s(4))
using assm(1)
apply auto
done
have "k ⊆ \<Union>(s - {k})"
apply safe
apply (rule *[unfolded closed_limpt,rule_format])
unfolding islimpt_approachable
proof safe
fix x
fix e :: real
assume as: "x ∈ k" "e > 0"
from k(2)[unfolded k content_eq_0] guess i ..
then have i:"c•i = d•i" "i∈Basis"
using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
then have xi: "x•i = d•i"
using as unfolding k mem_box by (metis antisym)
def y ≡ "∑j∈Basis. (if j = i then if c•i ≤ (a•i + b•i) / 2 then c•i +
min e (b•i - c•i) / 2 else c•i - min e (c•i - a•i) / 2 else x•j) *⇩R j"
show "∃x'∈\<Union>(s - {k}). x' ≠ x ∧ dist x' x < e"
apply (rule_tac x=y in bexI)
proof
have "d ∈ cbox c d"
using s(3)[OF k(1)]
unfolding k box_eq_empty mem_box
by (fastforce simp add: not_less)
then have "d ∈ cbox a b"
using s(2)[OF k(1)]
unfolding k
by auto
note di = this[unfolded mem_box,THEN bspec[where x=i]]
then have xyi: "y•i ≠ x•i"
unfolding y_def i xi
using as(2) assms(2)[unfolded content_eq_0] i(2)
by (auto elim!: ballE[of _ _ i])
then show "y ≠ x"
unfolding euclidean_eq_iff[where 'a='a] using i by auto
have *: "Basis = insert i (Basis - {i})"
using i by auto
have "norm (y - x) < e + setsum (λi. 0) Basis"
apply (rule le_less_trans[OF norm_le_l1])
apply (subst *)
apply (subst setsum.insert)
prefer 3
apply (rule add_less_le_mono)
proof -
show "¦(y - x) • i¦ < e"
using di as(2) y_def i xi by (auto simp: inner_simps)
show "(∑i∈Basis - {i}. ¦(y - x) • i¦) ≤ (∑i∈Basis. 0)"
unfolding y_def by (auto simp: inner_simps)
qed auto
then show "dist y x < e"
unfolding dist_norm by auto
have "y ∉ k"
unfolding k mem_box
apply rule
apply (erule_tac x=i in ballE)
using xyi k i xi
apply auto
done
moreover
have "y ∈ \<Union>s"
using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i
unfolding s mem_box y_def
by (auto simp: field_simps elim!: ballE[of _ _ i])
ultimately
show "y ∈ \<Union>(s - {k})" by auto
qed
qed
then have "\<Union>(s - {k}) = cbox a b"
unfolding s(6)[symmetric] by auto
then have  "{ka ∈ s - {k}. content ka ≠ 0} division_of (cbox a b)"
apply -
apply (rule assm(2)[rule_format,OF card refl])
apply (rule division_ofI)
defer
apply (rule_tac[1-4] s)
using assm(1)
apply auto
done
moreover
have "{ka ∈ s - {k}. content ka ≠ 0} = {k ∈ s. content k ≠ 0}"
using k by auto
ultimately show ?thesis by auto
qed

subsection {* Integrability on subintervals. *}

lemma operative_integrable:
fixes f :: "'b::euclidean_space => 'a::banach"
shows "operative op ∧ (λi. f integrable_on i)"
unfolding operative_def neutral_and
apply safe
apply (subst integrable_on_def)
unfolding has_integral_null_eq
apply (rule, rule refl)
apply (rule, assumption, assumption)+
unfolding integrable_on_def
by (auto intro!: has_integral_split)

lemma integrable_subinterval:
fixes f :: "'b::euclidean_space => 'a::banach"
assumes "f integrable_on cbox a b"
and "cbox c d ⊆ cbox a b"
shows "f integrable_on cbox c d"
apply (cases "cbox c d = {}")
defer
apply (rule partial_division_extend_1[OF assms(2)],assumption)
using operative_division_and[OF operative_integrable,symmetric,of _ _ _ f] assms(1)
apply auto
done

lemma integrable_subinterval_real:
fixes f :: "real => 'a::banach"
assumes "f integrable_on {a .. b}"
and "{c .. d} ⊆ {a .. b}"
shows "f integrable_on {c .. d}"
by (metis assms(1) assms(2) box_real(2) integrable_subinterval)

subsection {* Combining adjacent intervals in 1 dimension. *}

lemma has_integral_combine:
fixes a b c :: real
assumes "a ≤ c"
and "c ≤ b"
and "(f has_integral i) {a .. c}"
and "(f has_integral (j::'a::banach)) {c .. b}"
shows "(f has_integral (i + j)) {a .. b}"
proof -
note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
note conjunctD2[OF this,rule_format]
note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
then have "f integrable_on cbox a b"
apply -
apply (rule ccontr)
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
using assms(3-)
apply auto
done
with *
show ?thesis
apply -
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
unfolding lifted.simps
using assms(3-)
apply (auto simp add: integrable_on_def integral_unique)
done
qed

lemma integral_combine:
fixes f :: "real => 'a::banach"
assumes "a ≤ c"
and "c ≤ b"
and "f integrable_on {a .. b}"
shows "integral {a .. c} f + integral {c .. b} f = integral {a .. b} f"
apply (rule integral_unique[symmetric])
apply (rule has_integral_combine[OF assms(1-2)])
apply (metis assms(2) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel2 monoid_add_class.add.left_neutral)
by (metis assms(1) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel1 monoid_add_class.add.right_neutral)

lemma integrable_combine:
fixes f :: "real => 'a::banach"
assumes "a ≤ c"
and "c ≤ b"
and "f integrable_on {a .. c}"
and "f integrable_on {c .. b}"
shows "f integrable_on {a .. b}"
using assms
unfolding integrable_on_def
by (fastforce intro!:has_integral_combine)

subsection {* Reduce integrability to "local" integrability. *}

lemma integrable_on_little_subintervals:
fixes f :: "'b::euclidean_space => 'a::banach"
assumes "∀x∈cbox a b. ∃d>0. ∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b -->
f integrable_on cbox u v"
shows "f integrable_on cbox a b"
proof -
have "∀x. ∃d. x∈cbox a b --> d>0 ∧ (∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b -->
f integrable_on cbox u v)"
using assms by auto
note this[unfolded gauge_existence_lemma]
from choice[OF this] guess d .. note d=this[rule_format]
guess p
apply (rule fine_division_exists[OF gauge_ball_dependent,of d a b])
using d
by auto
note p=this(1-2)
note division_of_tagged_division[OF this(1)]
note * = operative_division_and[OF operative_integrable,OF this,symmetric,of f]
show ?thesis
unfolding *
apply safe
unfolding snd_conv
proof -
fix x k
assume "(x, k) ∈ p"
note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
then show "f integrable_on k"
apply safe
apply (rule d[THEN conjunct2,rule_format,of x])
apply (auto intro: order.trans)
done
qed
qed

subsection {* Second FCT or existence of antiderivative. *}

lemma integrable_const[intro]: "(λx. c) integrable_on cbox a b"
unfolding integrable_on_def
apply rule
apply (rule has_integral_const)
done

lemma integral_has_vector_derivative:
fixes f :: "real => 'a::banach"
assumes "continuous_on {a .. b} f"
and "x ∈ {a .. b}"
shows "((λu. integral {a .. u} f) has_vector_derivative f(x)) (at x within {a .. b})"
unfolding has_vector_derivative_def has_derivative_within_alt
apply safe
apply (rule bounded_linear_scaleR_left)
proof -
fix e :: real
assume e: "e > 0"
note compact_uniformly_continuous[OF assms(1) compact_Icc,unfolded uniformly_continuous_on_def]
from this[rule_format,OF e] guess d by (elim conjE exE) note d=this[rule_format]
let ?I = "λa b. integral {a .. b} f"
show "∃d>0. ∀y∈{a .. b}. norm (y - x) < d -->
norm (?I a y - ?I a x - (y - x) *⇩R f x) ≤ e * norm (y - x)"
proof (rule, rule, rule d, safe)
case goal1
show ?case
proof (cases "y < x")
case False
have "f integrable_on {a .. y}"
apply (rule integrable_subinterval_real,rule integrable_continuous_real)
apply (rule assms)
unfolding not_less
using assms(2) goal1
apply auto
done
then have *: "?I a y - ?I a x = ?I x y"
unfolding algebra_simps
apply (subst eq_commute)
apply (rule integral_combine)
using False
unfolding not_less
using assms(2) goal1
apply auto
done
have **: "norm (y - x) = content {x .. y}"
using False by (auto simp: content_real)
show ?thesis
unfolding **
apply (rule has_integral_bound_real[where f="(λu. f u - f x)"])
unfolding *
defer
apply (rule has_integral_sub)
apply (rule integrable_integral)
apply (rule integrable_subinterval_real)
apply (rule integrable_continuous_real)
apply (rule assms)+
proof -
show "{x .. y} ⊆ {a .. b}"
using goal1 assms(2) by auto
have *: "y - x = norm (y - x)"
using False by auto
show "((λxa. f x) has_integral (y - x) *⇩R f x) {x .. y}"
apply (subst *)
unfolding **
by auto
show "∀xa∈{x .. y}. norm (f xa - f x) ≤ e"
apply safe
apply (rule less_imp_le)
apply (rule d(2)[unfolded dist_norm])
using assms(2)
using goal1
apply auto
done
qed (insert e, auto)
next
case True
have "f integrable_on cbox a x"
apply (rule integrable_subinterval,rule integrable_continuous)
unfolding box_real
apply (rule assms)+
unfolding not_less
using assms(2) goal1
apply auto
done
then have *: "?I a x - ?I a y = ?I y x"
unfolding algebra_simps
apply (subst eq_commute)
apply (rule integral_combine)
using True using assms(2) goal1
apply auto
done
have **: "norm (y - x) = content {y .. x}"
apply (subst content_real)
using True
unfolding not_less
apply auto
done
have ***: "!!fy fx c::'a. fx - fy - (y - x) *⇩R c = -(fy - fx - (x - y) *⇩R c)"
unfolding scaleR_left.diff by auto
show ?thesis
apply (subst ***)
unfolding norm_minus_cancel **
apply (rule has_integral_bound_real[where f="(λu. f u - f x)"])
unfolding *
unfolding o_def
defer
apply (rule has_integral_sub)
apply (subst minus_minus[symmetric])
unfolding minus_minus
apply (rule integrable_integral)
apply (rule integrable_subinterval_real,rule integrable_continuous_real)
apply (rule assms)+
proof -
show "{y .. x} ⊆ {a .. b}"
using goal1 assms(2) by auto
have *: "x - y = norm (y - x)"
using True by auto
show "((λxa. f x) has_integral (x - y) *⇩R f x) {y .. x}"
apply (subst *)
unfolding **
apply auto
done
show "∀xa∈{y .. x}. norm (f xa - f x) ≤ e"
apply safe
apply (rule less_imp_le)
apply (rule d(2)[unfolded dist_norm])
using assms(2)
using goal1
apply auto
done
qed (insert e, auto)
qed
qed
qed

lemma antiderivative_continuous:
fixes q b :: real
assumes "continuous_on {a .. b} f"
obtains g where "∀x∈{a .. b}. (g has_vector_derivative (f x::_::banach)) (at x within {a .. b})"
apply (rule that)
apply rule
using integral_has_vector_derivative[OF assms]
apply auto
done

subsection {* Combined fundamental theorem of calculus. *}

lemma antiderivative_integral_continuous:
fixes f :: "real => 'a::banach"
assumes "continuous_on {a .. b} f"
obtains g where "∀u∈{a .. b}. ∀v ∈ {a .. b}. u ≤ v --> (f has_integral (g v - g u)) {u .. v}"
proof -
from antiderivative_continuous[OF assms] guess g . note g=this
show ?thesis
apply (rule that[of g])
proof safe
case goal1
have "∀x∈cbox u v. (g has_vector_derivative f x) (at x within cbox u v)"
apply rule
apply (rule has_vector_derivative_within_subset)
apply (rule g[rule_format])
using goal1(1-2)
apply auto
done
then show ?case
using fundamental_theorem_of_calculus[OF goal1(3),of "g" "f"] by auto
qed
qed

subsection {* General "twiddling" for interval-to-interval function image. *}

lemma has_integral_twiddle:
assumes "0 < r"
and "∀x. h(g x) = x"
and "∀x. g(h x) = x"
and "∀x. continuous (at x) g"
and "∀u v. ∃w z. g ` cbox u v = cbox w z"
and "∀u v. ∃w z. h ` cbox u v = cbox w z"
and "∀u v. content(g ` cbox u v) = r * content (cbox u v)"
and "(f has_integral i) (cbox a b)"
shows "((λx. f(g x)) has_integral (1 / r) *⇩R i) (h ` cbox a b)"
proof -
{
presume *: "cbox a b ≠ {} ==> ?thesis"
show ?thesis
apply cases
defer
apply (rule *)
apply assumption
proof -
case goal1
then show ?thesis
unfolding goal1 assms(8)[unfolded goal1 has_integral_empty_eq] by auto qed
}
assume "cbox a b ≠ {}"
from assms(6)[rule_format,of a b] guess w z by (elim exE) note wz=this
have inj: "inj g" "inj h"
unfolding inj_on_def
apply safe
apply(rule_tac[!] ccontr)
using assms(2)
apply(erule_tac x=x in allE)
using assms(2)
apply(erule_tac x=y in allE)
defer
using assms(3)
apply (erule_tac x=x in allE)
using assms(3)
apply(erule_tac x=y in allE)
apply auto
done
show ?thesis
unfolding has_integral_def has_integral_compact_interval_def
apply (subst if_P)
apply rule
apply rule
apply (rule wz)
proof safe
fix e :: real
assume e: "e > 0"
with assms(1) have "e * r > 0" by simp
from assms(8)[unfolded has_integral,rule_format,OF this] guess d by (elim exE conjE) note d=this[rule_format]
def d' ≡ "λx. {y. g y ∈ d (g x)}"
have d': "!!x. d' x = {y. g y ∈ (d (g x))}"
unfolding d'_def ..
show "∃d. gauge d ∧ (∀p. p tagged_division_of h ` cbox a b ∧ d fine p --> norm ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i) < e)"
proof (rule_tac x=d' in exI, safe)
show "gauge d'"
using d(1)
unfolding gauge_def d'
using continuous_open_preimage_univ[OF assms(4)]
by auto
fix p
assume as: "p tagged_division_of h ` cbox a b" "d' fine p"
note p = tagged_division_ofD[OF as(1)]
have "(λ(x, k). (g x, g ` k)) ` p tagged_division_of (cbox a b) ∧ d fine (λ(x, k). (g x, g ` k)) ` p"
unfolding tagged_division_of
proof safe
show "finite ((λ(x, k). (g x, g ` k)) ` p)"
using as by auto
show "d fine (λ(x, k). (g x, g ` k)) ` p"
using as(2) unfolding fine_def d' by auto
fix x k
assume xk[intro]: "(x, k) ∈ p"
show "g x ∈ g ` k"
using p(2)[OF xk] by auto
show "∃u v. g ` k = cbox u v"
using p(4)[OF xk] using assms(5-6) by auto
{
fix y
assume "y ∈ k"
then show "g y ∈ cbox a b" "g y ∈ cbox a b"
using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
using assms(2)[rule_format,of y]
unfolding inj_image_mem_iff[OF inj(2)]
by auto
}
fix x' k'
assume xk': "(x', k') ∈ p"
fix z
assume "z ∈ interior (g ` k)" and "z ∈ interior (g ` k')"
then have *: "interior (g ` k) ∩ interior (g ` k') ≠ {}"
by auto
have same: "(x, k) = (x', k')"
apply -
apply (rule ccontr)
apply (drule p(5)[OF xk xk'])
proof -
assume as: "interior k ∩ interior k' = {}"
from nonempty_witness[OF *] guess z .
then have "z ∈ g ` (interior k ∩ interior k')"
using interior_image_subset[OF assms(4) inj(1)]
unfolding image_Int[OF inj(1)]
by auto
then show False
using as by blast
qed
then show "g x = g x'"
by auto
{
fix z
assume "z ∈ k"
then show "g z ∈ g ` k'"
using same by auto
}
{
fix z
assume "z ∈ k'"
then show "g z ∈ g ` k"
using same by auto
}
next
fix x
assume "x ∈ cbox a b"
then have "h x ∈  \<Union>{k. ∃x. (x, k) ∈ p}"
using p(6) by auto
then guess X unfolding Union_iff .. note X=this
from this(1) guess y unfolding mem_Collect_eq ..
then show "x ∈ \<Union>{k. ∃x. (x, k) ∈ (λ(x, k). (g x, g ` k)) ` p}"
apply -
apply (rule_tac X="g ` X" in UnionI)
defer
apply (rule_tac x="h x" in image_eqI)
using X(2) assms(3)[rule_format,of x]
apply auto
done
qed
note ** = d(2)[OF this]
have *: "inj_on (λ(x, k). (g x, g ` k)) p"
using inj(1) unfolding inj_on_def by fastforce
have "(∑(x, k)∈(λ(x, k). (g x, g ` k)) ` p. content k *⇩R f x) - i = r *⇩R (∑(x, k)∈p. content k *⇩R f (g x)) - i" (is "?l = _")
using assms(7)
unfolding algebra_simps add_left_cancel scaleR_right.setsum
by (subst setsum.reindex_bij_betw[symmetric, where h="λ(x, k). (g x, g ` k)" and S=p])
(auto intro!: * setsum.cong simp: bij_betw_def dest!: p(4))
also have "… = r *⇩R ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i)" (is "_ = ?r")
unfolding scaleR_diff_right scaleR_scaleR
using assms(1)
by auto
finally have *: "?l = ?r" .
show "norm ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i) < e"
using **
unfolding *
unfolding norm_scaleR
using assms(1)
by (auto simp add:field_simps)
qed
qed
qed

subsection {* Special case of a basic affine transformation. *}

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

lemma content_image_affinity_cbox:
"content((λx::'a::euclidean_space. m *⇩R x + c) ` cbox a b) =
abs m ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
proof -
{
presume *: "cbox a b ≠ {} ==> ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
unfolding not_not
using content_empty
apply auto
done
}
assume as: "cbox a b ≠ {}"
show ?thesis
proof (cases "m ≥ 0")
case True
with as have "cbox (m *⇩R a + c) (m *⇩R b + c) ≠ {}"
unfolding box_ne_empty
apply (intro ballI)
apply (erule_tac x=i in ballE)
apply (auto simp: inner_simps intro!: mult_left_mono)
done
moreover from True have *: "!!i. (m *⇩R b + c) • i - (m *⇩R a + c) • i = m *⇩R (b - a) • i"
by (simp add: inner_simps field_simps)
ultimately show ?thesis
by (simp add: image_affinity_cbox True content_cbox'
setprod.distrib setprod_constant inner_diff_left)
next
case False
with as have "cbox (m *⇩R b + c) (m *⇩R a + c) ≠ {}"
unfolding box_ne_empty
apply (intro ballI)
apply (erule_tac x=i in ballE)
apply (auto simp: inner_simps intro!: mult_left_mono)
done
moreover from False have *: "!!i. (m *⇩R a + c) • i - (m *⇩R b + c) • i = (-m) *⇩R (b - a) • i"
by (simp add: inner_simps field_simps)
ultimately show ?thesis using False
by (simp add: image_affinity_cbox content_cbox'
setprod.distrib[symmetric] setprod_constant[symmetric] inner_diff_left)
qed
qed

lemma has_integral_affinity:
fixes a :: "'a::euclidean_space"
assumes "(f has_integral i) (cbox a b)"
and "m ≠ 0"
shows "((λx. f(m *⇩R x + c)) has_integral ((1 / (abs(m) ^ DIM('a))) *⇩R i)) ((λx. (1 / m) *⇩R x + -((1 / m) *⇩R c)) ` cbox a b)"
apply (rule has_integral_twiddle)
apply safe
apply (rule zero_less_power)
unfolding euclidean_eq_iff[where 'a='a]
unfolding scaleR_right_distrib inner_simps scaleR_scaleR
defer
apply (insert assms(2))
apply (simp add: field_simps)
apply (insert assms(2))
apply (simp add: field_simps)
apply (rule continuous_intros)+
apply (rule interval_image_affinity_interval)+
apply (rule content_image_affinity_cbox)
using assms
apply auto
done

lemma integrable_affinity:
assumes "f integrable_on cbox a b"
and "m ≠ 0"
shows "(λx. f(m *⇩R x + c)) integrable_on ((λx. (1 / m) *⇩R x + -((1/m) *⇩R c)) ` cbox a b)"
using assms
unfolding integrable_on_def
apply safe
apply (drule has_integral_affinity)
apply auto
done

subsection {* Special case of stretching coordinate axes separately. *}

lemma image_stretch_interval:
"(λx. ∑k∈Basis. (m k * (x•k)) *⇩R k) ` cbox a (b::'a::euclidean_space) =
(if (cbox a b) = {} then {} else
cbox (∑k∈Basis. (min (m k * (a•k)) (m k * (b•k))) *⇩R k::'a)
(∑k∈Basis. (max (m k * (a•k)) (m k * (b•k))) *⇩R k))"
proof cases
assume *: "cbox a b ≠ {}"
show ?thesis
unfolding box_ne_empty if_not_P[OF *]
apply (simp add: cbox_def image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
apply (subst choice_Basis_iff[symmetric])
proof (intro allI ball_cong refl)
fix x i :: 'a assume "i ∈ Basis"
with * have a_le_b: "a • i ≤ b • i"
unfolding box_ne_empty by auto
show "(∃xa. x • i = m i * xa ∧ a • i ≤ xa ∧ xa ≤ b • i) <->
min (m i * (a • i)) (m i * (b • i)) ≤ x • i ∧ x • i ≤ max (m i * (a • i)) (m i * (b • i))"
proof (cases "m i = 0")
case True
with a_le_b show ?thesis by auto
next
case False
then have *: "!!a b. a = m i * b <-> b = a / m i"
by (auto simp add: field_simps)
from False have
"min (m i * (a • i)) (m i * (b • i)) = (if 0 < m i then m i * (a • i) else m i * (b • i))"
"max (m i * (a • i)) (m i * (b • i)) = (if 0 < m i then m i * (b • i) else m i * (a • i))"
using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
with False show ?thesis using a_le_b
unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps)
qed
qed
qed simp

lemma interval_image_stretch_interval:
"∃u v. (λx. ∑k∈Basis. (m k * (x•k))*⇩R k) ` cbox a (b::'a::euclidean_space) = cbox u (v::'a::euclidean_space)"
unfolding image_stretch_interval by auto

lemma content_image_stretch_interval:
"content ((λx::'a::euclidean_space. (∑k∈Basis. (m k * (x•k))*⇩R k)::'a) ` cbox a b) =
abs (setprod m Basis) * content (cbox a b)"
proof (cases "cbox a b = {}")
case True
then show ?thesis
unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
next
case False
then have "(λx. (∑k∈Basis. (m k * (x•k))*⇩R k)) ` cbox a b ≠ {}"
by auto
then show ?thesis
using False
unfolding content_def image_stretch_interval
apply -
unfolding interval_bounds' if_not_P
unfolding abs_setprod setprod.distrib[symmetric]
apply (rule setprod.cong)
apply (rule refl)
unfolding lessThan_iff
apply (simp only: inner_setsum_left_Basis)
proof -
fix i :: 'a
assume i: "i ∈ Basis"
have "(m i < 0 ∨ m i > 0) ∨ m i = 0"
by auto
then show "max (m i * (a • i)) (m i * (b • i)) - min (m i * (a • i)) (m i * (b • i)) =
¦m i¦ * (b • i - a • i)"
apply -
apply (erule disjE)+
unfolding min_def max_def
using False[unfolded box_ne_empty,rule_format,of i] i
apply (auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos)
done
qed
qed

lemma has_integral_stretch:
fixes f :: "'a::euclidean_space => 'b::real_normed_vector"
assumes "(f has_integral i) (cbox a b)"
and "∀k∈Basis. m k ≠ 0"
shows "((λx. f (∑k∈Basis. (m k * (x•k))*⇩R k)) has_integral
((1/(abs(setprod m Basis))) *⇩R i)) ((λx. (∑k∈Basis. (1 / m k * (x•k))*⇩R k)) ` cbox a b)"
apply (rule has_integral_twiddle[where f=f])
unfolding zero_less_abs_iff content_image_stretch_interval
unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
using assms
proof -
show "∀y::'a. continuous (at y) (λx. (∑k∈Basis. (m k * (x•k))*⇩R k))"
apply rule
apply (rule linear_continuous_at)
unfolding linear_linear
unfolding linear_iff inner_simps euclidean_eq_iff[where 'a='a]
apply (auto simp add: field_simps)
done
qed auto

lemma integrable_stretch:
fixes f :: "'a::euclidean_space => 'b::real_normed_vector"
assumes "f integrable_on cbox a b"
and "∀k∈Basis. m k ≠ 0"
shows "(λx::'a. f (∑k∈Basis. (m k * (x•k))*⇩R k)) integrable_on
((λx. ∑k∈Basis. (1 / m k * (x•k))*⇩R k) ` cbox a b)"
using assms
unfolding integrable_on_def
apply -
apply (erule exE)
apply (drule has_integral_stretch)
apply assumption
apply auto
done

subsection {* even more special cases. *}

lemma uminus_interval_vector[simp]:
fixes a b :: "'a::euclidean_space"
shows "uminus ` cbox a b = cbox (-b) (-a)"
apply (rule set_eqI)
apply rule
defer
unfolding image_iff
apply (rule_tac x="-x" in bexI)
apply (auto simp add:minus_le_iff le_minus_iff mem_box)
done

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

lemma has_integral_reflect_lemma_real[intro]:
assumes "(f has_integral i) {a .. b::real}"
shows "((λx. f(-x)) has_integral i) {-b .. -a}"
using assms
unfolding box_real[symmetric]
by (rule has_integral_reflect_lemma)

lemma has_integral_reflect[simp]:
"((λx. f (-x)) has_integral i) (cbox (-b) (-a)) <-> (f has_integral i) (cbox a b)"
apply rule
apply (drule_tac[!] has_integral_reflect_lemma)
apply auto
done

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

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

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

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

subsection {* Stronger form of FCT; quite a tedious proof. *}

lemma bgauge_existence_lemma: "(∀x∈s. ∃d::real. 0 < d ∧ q d x) <-> (∀x. ∃d>0. x∈s --> q d x)"
by (meson zero_less_one)

lemma additive_tagged_division_1':
fixes f :: "real => 'a::real_normed_vector"
assumes "a ≤ b"
and "p tagged_division_of {a..b}"
shows "setsum (λ(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
using additive_tagged_division_1[OF _ assms(2), of f]
using assms(1)
by auto

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

lemma norm_triangle_le_sub: "norm x + norm y ≤ e ==> norm (x - y) ≤ e"
apply (subst(asm)(2) norm_minus_cancel[symmetric])
apply (drule norm_triangle_le)
apply (auto simp add: algebra_simps)
done

lemma fundamental_theorem_of_calculus_interior:
fixes f :: "real => 'a::real_normed_vector"
assumes "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a <..< b}. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
proof -
{
presume *: "a < b ==> ?thesis"
show ?thesis
proof (cases "a < b")
case True
then show ?thesis by (rule *)
next
case False
then have "a = b"
using assms(1) by auto
then have *: "cbox a b = {b}" "f b - f a = 0"
by (auto simp add:  order_antisym)
show ?thesis
unfolding *(2)
unfolding content_eq_0
using * `a = b`
by (auto simp: ex_in_conv)
qed
}
assume ab: "a < b"
let ?P = "λe. ∃d. gauge d ∧ (∀p. p tagged_division_of {a .. b} ∧ d fine p -->
norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content {a .. b})"
{ presume "!!e. e > 0 ==> ?P e" then show ?thesis unfolding has_integral_factor_content_real by auto }
fix e :: real
assume e: "e > 0"
note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
note conjunctD2[OF this]
note bounded=this(1) and this(2)
from this(2) have "∀x∈box a b. ∃d>0. ∀y. norm (y - x) < d -->
norm (f y - f x - (y - x) *⇩R f' x) ≤ e/2 * norm (y - x)"
apply -
apply safe
apply (erule_tac x=x in ballE)
apply (erule_tac x="e/2" in allE)
using e
apply auto
done
note this[unfolded bgauge_existence_lemma]
from choice[OF this] guess d ..
note conjunctD2[OF this[rule_format]]
note d = this[rule_format]
have "bounded (f ` cbox a b)"
apply (rule compact_imp_bounded compact_continuous_image)+
using compact_cbox assms
apply auto
done
from this[unfolded bounded_pos] guess B .. note B = this[rule_format]

have "∃da. 0 < da ∧ (∀c. a ≤ c ∧ {a .. c} ⊆ {a .. b} ∧ {a .. c} ⊆ ball a da -->
norm (content {a .. c} *⇩R f' a - (f c - f a)) ≤ (e * (b - a)) / 4)"
proof -
have "a ∈ {a .. b}"
using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format]
have "(e * (b - a)) / 8 > 0"
using e ab by (auto simp add: field_simps)
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "∃l. 0 < l ∧ norm(l *⇩R f' a) ≤ (e * (b - a)) / 8"
proof (cases "f' a = 0")
case True
thus ?thesis using ab e by auto
next
case False
then show ?thesis
apply (rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
using ab e
apply (auto simp add: field_simps)
done
qed
then guess l .. note l = conjunctD2[OF this]
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
unfolding min_less_iff_conj
apply rule
apply (rule l k)+
proof -
fix c
assume as: "a ≤ c" "{a .. c} ⊆ {a .. b}" "{a .. c} ⊆ ball a (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
have "norm ((c - a) *⇩R f' a - (f c - f a)) ≤ norm ((c - a) *⇩R f' a) + norm (f c - f a)"
by (rule norm_triangle_ineq4)
also have "… ≤ e * (b - a) / 8 + e * (b - a) / 8"
proof (rule add_mono)
case goal1
have "¦c - a¦ ≤ ¦l¦"
using as' by auto
then show ?case
apply -
apply (rule order_trans[OF _ l(2)])
unfolding norm_scaleR
apply (rule mult_right_mono)
apply auto
done
next
case goal2
show ?case
apply (rule less_imp_le)
apply (cases "a = c")
defer
apply (rule k(2)[unfolded dist_norm])
using as' e ab
apply (auto simp add: field_simps)
done
qed
finally show "norm (content {a .. c} *⇩R f' a - (f c - f a)) ≤ e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed
qed
then guess da .. note da=conjunctD2[OF this,rule_format]

have "∃db>0. ∀c≤b. {c .. b} ⊆ {a .. b} ∧ {c .. b} ⊆ ball b db -->
norm (content {c .. b} *⇩R f' b - (f b - f c)) ≤ (e * (b - a)) / 4"
proof -
have "b ∈ {a .. b}"
using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
using e ab by (auto simp add: field_simps)
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "∃l. 0 < l ∧ norm (l *⇩R f' b) ≤ (e * (b - a)) / 8"
proof (cases "f' b = 0")
case True
thus ?thesis using ab e by auto
next
case False
then show ?thesis
apply (rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
using ab e
apply (auto simp add: field_simps)
done
qed
then guess l .. note l = conjunctD2[OF this]
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
unfolding min_less_iff_conj
apply rule
apply (rule l k)+
proof -
fix c
assume as: "c ≤ b" "{c..b} ⊆ {a..b}" "{c..b} ⊆ ball b (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
have "norm ((b - c) *⇩R f' b - (f b - f c)) ≤ norm ((b - c) *⇩R f' b) + norm (f b - f c)"
by (rule norm_triangle_ineq4)
also have "… ≤ e * (b - a) / 8 + e * (b - a) / 8"
proof (rule add_mono)
case goal1
have "¦c - b¦ ≤ ¦l¦"
using as' by auto
then show ?case
apply -
apply (rule order_trans[OF _ l(2)])
unfolding norm_scaleR
apply (rule mult_right_mono)
apply auto
done
next
case goal2
show ?case
apply (rule less_imp_le)
apply (cases "b = c")
defer
apply (subst norm_minus_commute)
apply (rule k(2)[unfolded dist_norm])
using as' e ab
apply (auto simp add: field_simps)
done
qed
finally show "norm (content {c .. b} *⇩R f' b - (f b - f c)) ≤ e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed
qed
then guess db .. note db=conjunctD2[OF this,rule_format]

let ?d = "(λx. ball x (if x=a then da else if x=b then db else d x))"
show "?P e"
apply (rule_tac x="?d" in exI)
proof safe
case goal1
show ?case
apply (rule gauge_ball_dependent)
using ab db(1) da(1) d(1)
apply auto
done
next
case goal2
note as=this
let ?A = "{t. fst t ∈ {a, b}}"
note p = tagged_division_ofD[OF goal2(1)]
have pA: "p = (p ∩ ?A) ∪ (p - ?A)" "finite (p ∩ ?A)" "finite (p - ?A)" "(p ∩ ?A) ∩ (p - ?A) = {}"
using goal2 by auto
note * = additive_tagged_division_1'[OF assms(1) goal2(1), symmetric]
have **: "!!n1 s1 n2 s2::real. n2 ≤ s2 / 2 ==> n1 - s1 ≤ s2 / 2 ==> n1 + n2 ≤ s1 + s2"
by arith
show ?case
unfolding content_real[OF assms(1)] and *[of "λx. x"] *[of f] setsum_subtractf[symmetric] split_minus
unfolding setsum_right_distrib
apply (subst(2) pA)
apply (subst pA)
unfolding setsum.union_disjoint[OF pA(2-)]
proof (rule norm_triangle_le, rule **)
case goal1
show ?case
apply (rule order_trans)
apply (rule setsum_norm_le)
defer
apply (subst setsum_divide_distrib)
apply (rule order_refl)
apply safe
apply (unfold not_le o_def split_conv fst_conv)
proof (rule ccontr)
fix x k
assume as: "(x, k) ∈ p"
"e * (Sup k -  Inf k) / 2 <
norm (content k *⇩R f' x - (f (Sup k) - f (Inf k)))"
from p(4)[OF this(1)] guess u v by (elim exE) note k=this
then have "u ≤ v" and uv: "{u, v} ⊆ cbox u v"
using p(2)[OF as(1)] by auto
note result = as(2)[unfolded k box_real interval_bounds_real[OF this(1)] content_real[OF this(1)]]

assume as': "x ≠ a" "x ≠ b"
then have "x ∈ box a b"
using p(2-3)[OF as(1)] by (auto simp: mem_box)
note  * = d(2)[OF this]
have "norm ((v - u) *⇩R f' (x) - (f (v) - f (u))) =
norm ((f (u) - f (x) - (u - x) *⇩R f' (x)) - (f (v) - f (x) - (v - x) *⇩R f' (x)))"
apply (rule arg_cong[of _ _ norm])
unfolding scaleR_left.diff
apply auto
done
also have "… ≤ e / 2 * norm (u - x) + e / 2 * norm (v - x)"
apply (rule norm_triangle_le_sub)
apply (rule add_mono)
apply (rule_tac[!] *)
using fineD[OF goal2(2) as(1)] as'
unfolding k subset_eq
apply -
apply (erule_tac x=u in ballE)
apply (erule_tac[3] x=v in ballE)
using uv
apply (auto simp:dist_real_def)
done
also have "… ≤ e / 2 * norm (v - u)"
using p(2)[OF as(1)]
unfolding k
by (auto simp add: field_simps)
finally have "e * (v - u) / 2 < e * (v - u) / 2"
apply -
apply (rule less_le_trans[OF result])
using uv
apply auto
done
then show False by auto
qed
next
have *: "!!x s1 s2::real. 0 ≤ s1 ==> x ≤ (s1 + s2) / 2 ==> x - s1 ≤ s2 / 2"
by auto
case goal2
show ?case
apply (rule *)
apply (rule setsum_nonneg)
apply rule
apply (unfold split_paired_all split_conv)
defer
unfolding setsum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
unfolding setsum_right_distrib[symmetric]
thm additive_tagged_division_1
apply (subst additive_tagged_division_1[OF _ as(1)])
apply (rule assms)
proof -
fix x k
assume "(x, k) ∈ p ∩ {t. fst t ∈ {a, b}}"
note xk=IntD1[OF this]
from p(4)[OF this] guess u v by (elim exE) note uv=this
with p(2)[OF xk] have "cbox u v ≠ {}"
by auto
then show "0 ≤ e * ((Sup k) - (Inf k))"
unfolding uv using e by (auto simp add: field_simps)
next
have *: "!!s f t e. setsum f s = setsum f t ==> norm (setsum f t) ≤ e ==> norm (setsum f s) ≤ e"
by auto
show "norm (∑(x, k)∈p ∩ ?A. content k *⇩R f' x -
(f ((Sup k)) - f ((Inf k)))) ≤ e * (b - a) / 2"
apply (rule *[where t="p ∩ {t. fst t ∈ {a, b} ∧ content(snd t) ≠ 0}"])
apply (rule setsum.mono_neutral_right[OF pA(2)])
defer
apply rule
unfolding split_paired_all split_conv o_def
proof -
fix x k
assume "(x, k) ∈ p ∩ {t. fst t ∈ {a, b}} - p ∩ {t. fst t ∈ {a, b} ∧ content (snd t) ≠ 0}"
then have xk: "(x, k) ∈ p" "content k = 0"
by auto
from p(4)[OF xk(1)] guess u v by (elim exE) note uv=this
have "k ≠ {}"
using p(2)[OF xk(1)] by auto
then have *: "u = v"
using xk
unfolding uv content_eq_0 box_eq_empty
by auto
then show "content k *⇩R (f' (x)) - (f ((Sup k)) - f ((Inf k))) = 0"
using xk unfolding uv by auto
next
have *: "p ∩ {t. fst t ∈ {a, b} ∧ content(snd t) ≠ 0} =
{t. t∈p ∧ fst t = a ∧ content(snd t) ≠ 0} ∪ {t. t∈p ∧ fst t = b ∧ content(snd t) ≠ 0}"
by blast
have **: "!!s f. !!e::real. (∀x y. x ∈ s ∧ y ∈ s --> x = y) ==>
(∀x. x ∈ s --> norm (f x) ≤ e) ==> e > 0 ==> norm (setsum f s) ≤ e"
proof (case_tac "s = {}")
case goal2
then obtain x where "x ∈ s"
by auto
then have *: "s = {x}"
using goal2(1) by auto
then show ?case
using `x ∈ s` goal2(2) by auto
qed auto
case goal2
show ?case
apply (subst *)
apply (subst setsum.union_disjoint)
prefer 4
apply (rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
apply (rule norm_triangle_le,rule add_mono)
apply (rule_tac[1-2] **)
proof -
let ?B = "λx. {t ∈ p. fst t = x ∧ content (snd t) ≠ 0}"
have pa: "!!k. (a, k) ∈ p ==> ∃v. k = cbox a v ∧ a ≤ v"
proof -
case goal1
guess u v using p(4)[OF goal1] by (elim exE) note uv=this
have *: "u ≤ v"
using p(2)[OF goal1] unfolding uv by auto
have u: "u = a"
proof (rule ccontr)
have "u ∈ cbox u v"
using p(2-3)[OF goal1(1)] unfolding uv by auto
have "u ≥ a"
using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto
moreover assume "¬ ?thesis"
ultimately have "u > a" by auto
then show False
using p(2)[OF goal1(1)] unfolding uv by (auto simp add:)
qed
then show ?case
apply (rule_tac x=v in exI)
unfolding uv
using *
apply auto
done
qed
have pb: "!!k. (b, k) ∈ p ==> ∃v. k = cbox v b ∧ b ≥ v"
proof -
case goal1
guess u v using p(4)[OF goal1] by (elim exE) note uv=this
have *: "u ≤ v"
using p(2)[OF goal1] unfolding uv by auto
have u: "v =  b"
proof (rule ccontr)
have "u ∈ cbox u v"
using p(2-3)[OF goal1(1)] unfolding uv by auto
have "v ≤ b"
using p(2-3)[OF goal1(1)] unfolding uv subset_eq by auto
moreover assume "¬ ?thesis"
ultimately have "v < b" by auto
then show False
using p(2)[OF goal1(1)] unfolding uv by (auto simp add:)
qed
then show ?case
apply (rule_tac x=u in exI)
unfolding uv
using *
apply auto
done
qed
show "∀x y. x ∈ ?B a ∧ y ∈ ?B a --> x = y"
apply (rule,rule,rule,unfold split_paired_all)
unfolding mem_Collect_eq fst_conv snd_conv
apply safe
proof -
fix x k k'
assume k: "(a, k) ∈ p" "(a, k') ∈ p" "content k ≠ 0" "content k' ≠ 0"
guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "min v v'"
have "box a ?v ⊆ k ∩ k'"
unfolding v v' by (auto simp add: mem_box)
note interior_mono[OF this,unfolded interior_inter]
moreover have "(a + ?v)/2 ∈ box a ?v"
using k(3-)
unfolding v v' content_eq_0 not_le
by (auto simp add: mem_box)
ultimately have "(a + ?v)/2 ∈ interior k ∩ interior k'"
unfolding interior_open[OF open_box] by auto
then have *: "k = k'"
apply -
apply (rule ccontr)
using p(5)[OF k(1-2)]
apply auto
done
{ assume "x ∈ k" then show "x ∈ k'" unfolding * . }
{ assume "x ∈ k'" then show "x ∈ k" unfolding * . }
qed
show "∀x y. x ∈ ?B b ∧ y ∈ ?B b --> x = y"
apply rule
apply rule
apply rule
apply (unfold split_paired_all)
unfolding mem_Collect_eq fst_conv snd_conv
apply safe
proof -
fix x k k'
assume k: "(b, k) ∈ p" "(b, k') ∈ p" "content k ≠ 0" "content k' ≠ 0"
guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this]
let ?v = "max v v'"
have "box ?v b ⊆ k ∩ k'"
unfolding v v' by (auto simp: mem_box)
note interior_mono[OF this,unfolded interior_inter]
moreover have " ((b + ?v)/2) ∈ box ?v b"
using k(3-) unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
ultimately have " ((b + ?v)/2) ∈ interior k ∩ interior k'"
unfolding interior_open[OF open_box] by auto
then have *: "k = k'"
apply -
apply (rule ccontr)
using p(5)[OF k(1-2)]
apply auto
done
{ assume "x ∈ k" then show "x ∈ k'" unfolding * . }
{ assume "x ∈ k'" then show "x∈k" unfolding * . }
qed

let ?a = a and ?b = b (* a is something else while proofing the next theorem. *)
show "∀x. x ∈ ?B a --> norm ((λ(x, k). content k *⇩R f' x - (f (Sup k) -
f (Inf k))) x) ≤ e * (b - a) / 4"
apply rule
apply rule
unfolding mem_Collect_eq
unfolding split_paired_all fst_conv snd_conv
proof safe
case goal1
guess v using pa[OF goal1(1)] .. note v = conjunctD2[OF this]
have "?a ∈ {?a..v}"
using v(2) by auto
then have "v ≤ ?b"
using p(3)[OF goal1(1)] unfolding subset_eq v by auto
moreover have "{?a..v} ⊆ ball ?a da"
using fineD[OF as(2) goal1(1)]
apply -
apply (subst(asm) if_P)
apply (rule refl)
unfolding subset_eq
apply safe
apply (erule_tac x=" x" in ballE)
apply (auto simp add:subset_eq dist_real_def v)
done
ultimately show ?case
unfolding v interval_bounds_real[OF v(2)] box_real
apply -
apply(rule da(2)[of "v"])
using goal1 fineD[OF as(2) goal1(1)]
unfolding v content_eq_0
apply auto
done
qed
show "∀x. x ∈ ?B b --> norm ((λ(x, k). content k *⇩R f' x -
(f (Sup k) - f (Inf k))) x) ≤ e * (b - a) / 4"
apply rule
apply rule
unfolding mem_Collect_eq
unfolding split_paired_all fst_conv snd_conv
proof safe
case goal1 guess v using pb[OF goal1(1)] .. note v = conjunctD2[OF this]
have "?b ∈ {v.. ?b}"
using v(2) by auto
then have "v ≥ ?a" using p(3)[OF goal1(1)]
unfolding subset_eq v by auto
moreover have "{v..?b} ⊆ ball ?b db"
using fineD[OF as(2) goal1(1)]
apply -
apply (subst(asm) if_P, rule refl)
unfolding subset_eq
apply safe
apply (erule_tac x=" x" in ballE)
using ab
apply (auto simp add:subset_eq v dist_real_def)
done
ultimately show ?case
unfolding v
unfolding interval_bounds_real[OF v(2)] box_real
apply -
apply(rule db(2)[of "v"])
using goal1 fineD[OF as(2) goal1(1)]
unfolding v content_eq_0
apply auto
done
qed
qed (insert p(1) ab e, auto simp add: field_simps)
qed auto
qed
qed
qed
qed

subsection {* Stronger form with finite number of exceptional points. *}

lemma fundamental_theorem_of_calculus_interior_strong:
fixes f :: "real => 'a::banach"
assumes "finite s"
and "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a <..< b} - s. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
using assms
proof (induct "card s" arbitrary: s a b)
case 0
show ?case
apply (rule fundamental_theorem_of_calculus_interior)
using 0
apply auto
done
next
case (Suc n)
from this(2) guess c s'
apply -
apply (subst(asm) eq_commute)
unfolding card_Suc_eq
apply (subst(asm)(2) eq_commute)
apply (elim exE conjE)
done
note cs = this[rule_format]
show ?case
proof (cases "c ∈ box a b")
case False
then show ?thesis
apply -
apply (rule Suc(1)[OF cs(3) _ Suc(4,5)])
apply safe
defer
apply (rule Suc(6)[rule_format])
using Suc(3)
unfolding cs
apply auto
done
next
have *: "f b - f a = (f c - f a) + (f b - f c)"
by auto
case True
then have "a ≤ c" "c ≤ b"
by (auto simp: mem_box)
then show ?thesis
apply (subst *)
apply (rule has_integral_combine)
apply assumption+
apply (rule_tac[!] Suc(1)[OF cs(3)])
using Suc(3)
unfolding cs
proof -
show "continuous_on {a .. c} f" "continuous_on {c .. b} f"
apply (rule_tac[!] continuous_on_subset[OF Suc(5)])
using True
apply (auto simp: mem_box)
done
let ?P = "λi j. ∀x∈{i <..< j} - s'. (f has_vector_derivative f' x) (at x)"
show "?P a c" "?P c b"
apply safe
apply (rule_tac[!] Suc(6)[rule_format])
using True
unfolding cs
apply (auto simp: mem_box)
done
qed auto
qed
qed

lemma fundamental_theorem_of_calculus_strong:
fixes f :: "real => 'a::banach"
assumes "finite s"
and "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a .. b} - s. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
using assms(4)
apply (auto simp: mem_box)
done

lemma indefinite_integral_continuous_left:
fixes f:: "real => 'a::banach"
assumes "f integrable_on {a .. b}"
and "a < c"
and "c ≤ b"
and "e > 0"
obtains d where "d > 0"
and "∀t. c - d < t ∧ t ≤ c --> norm (integral {a .. c} f - integral {a .. t} f) < e"
proof -
have "∃w>0. ∀t. c - w < t ∧ t < c --> norm (f c) * norm(c - t) < e / 3"
proof (cases "f c = 0")
case False
hence "0 < e / 3 / norm (f c)" using `e>0` by simp
then show ?thesis
apply -
apply rule
apply rule
apply assumption
apply safe
proof -
fix t
assume as: "t < c" and "c - e / 3 / norm (f c) < t"
then have "c - t < e / 3 / norm (f c)"
by auto
then have "norm (c - t) < e / 3 / norm (f c)"
using as by auto
then show "norm (f c) * norm (c - t) < e / 3"
using False
apply -
apply (subst mult.commute)
apply (subst pos_less_divide_eq[symmetric])
apply auto
done
qed
next
case True
show ?thesis
apply (rule_tac x=1 in exI)
unfolding True
using `e > 0`
apply auto
done
qed
then guess w .. note w = conjunctD2[OF this,rule_format]

have *: "e / 3 > 0"
using assms by auto
have "f integrable_on {a .. c}"
apply (rule integrable_subinterval_real[OF assms(1)])
using assms(2-3)
apply auto
done
from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d1 ..
note d1 = conjunctD2[OF this,rule_format]
def d ≡ "λx. ball x w ∩ d1 x"
have "gauge d"
unfolding d_def using w(1) d1 by auto
note this[unfolded gauge_def,rule_format,of c]
note conjunctD2[OF this]
from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k ..
note k=conjunctD2[OF this]

let ?d = "min k (c - a) / 2"
show ?thesis
apply (rule that[of ?d])
apply safe
proof -
show "?d > 0"
using k(1) using assms(2) by auto
fix t
assume as: "c - ?d < t" "t ≤ c"
let ?thesis = "norm (integral ({a .. c}) f - integral ({a .. t}) f) < e"
{
presume *: "t < c ==> ?thesis"
show ?thesis
apply (cases "t = c")
defer
apply (rule *)
apply (subst less_le)
using `e > 0` as(2)
apply auto
done
}
assume "t < c"

have "f integrable_on {a .. t}"
apply (rule integrable_subinterval_real[OF assms(1)])
using assms(2-3) as(2)
apply auto
done
from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d2 ..
note d2 = conjunctD2[OF this,rule_format]
def d3 ≡ "λx. if x ≤ t then d1 x ∩ d2 x else d1 x"
have "gauge d3"
using d2(1) d1(1) unfolding d3_def gauge_def by auto
from fine_division_exists_real[OF this, of a t] guess p . note p=this
note p'=tagged_division_ofD[OF this(1)]
have pt: "∀(x,k)∈p. x ≤ t"
proof safe
case goal1
from p'(2,3)[OF this] show ?case
by auto
qed
with p(2) have "d2 fine p"
unfolding fine_def d3_def
apply safe
apply (erule_tac x="(a,b)" in ballE)+
apply auto
done
note d2_fin = d2(2)[OF conjI[OF p(1) this]]

have *: "{a .. c} ∩ {x. x • 1 ≤ t} = {a .. t}" "{a .. c} ∩ {x. x • 1 ≥ t} = {t .. c}"
using assms(2-3) as by (auto simp add: field_simps)
have "p ∪ {(c, {t .. c})} tagged_division_of {a .. c} ∧ d1 fine p ∪ {(c, {t .. c})}"
apply rule
apply (rule tagged_division_union_interval_real[of _ _ _ 1 "t"])
unfolding *
apply (rule p)
apply (rule tagged_division_of_self_real)
unfolding fine_def
apply safe
proof -
fix x k y
assume "(x,k) ∈ p" and "y ∈ k"
then show "y ∈ d1 x"
using p(2) pt
unfolding fine_def d3_def
apply -
apply (erule_tac x="(x,k)" in ballE)+
apply auto
done
next
fix x assume "x ∈ {t..c}"
then have "dist c x < k"
unfolding dist_real_def
using as(1)
by (auto simp add: field_simps)
then show "x ∈ d1 c"
using k(2)
unfolding d_def
by auto
qed (insert as(2), auto) note d1_fin = d1(2)[OF this]

have *: "integral {a .. c} f - integral {a .. t} f = -(((c - t) *⇩R f c + (∑(x, k)∈p. content k *⇩R f x)) -
integral {a .. c} f) + ((∑(x, k)∈p. content k *⇩R f x) - integral {a .. t} f) + (c - t) *⇩R f c"
"e = (e/3 + e/3) + e/3"
by auto
have **: "(∑(x, k)∈p ∪ {(c, {t .. c})}. content k *⇩R f x) =
(c - t) *⇩R f c + (∑(x, k)∈p. content k *⇩R f x)"
proof -
have **: "!!x F. F ∪ {x} = insert x F"
by auto
have "(c, cbox t c) ∉ p"
proof safe
case goal1
from p'(2-3)[OF this] have "c ∈ cbox a t"
by auto
then show False using `t < c`
by auto
qed
then show ?thesis
unfolding ** box_real
apply -
apply (subst setsum.insert)
apply (rule p')
unfolding split_conv
defer
apply (subst content_real)
using as(2)
apply auto
done
qed
have ***: "c - w < t ∧ t < c"
proof -
have "c - k < t"
using `k>0` as(1) by (auto simp add: field_simps)
moreover have "k ≤ w"
apply (rule ccontr)
using k(2)
unfolding subset_eq
apply (erule_tac x="c + ((k + w)/2)" in ballE)
unfolding d_def
using `k > 0` `w > 0`
apply (auto simp add: field_simps not_le not_less dist_real_def)
done
ultimately show ?thesis using `t < c`
by (auto simp add: field_simps)
qed
show ?thesis
unfolding *(1)
apply (subst *(2))
apply (rule norm_triangle_lt add_strict_mono)+
unfolding norm_minus_cancel
apply (rule d1_fin[unfolded **])
apply (rule d2_fin)
using w(2)[OF ***]
unfolding norm_scaleR
apply (auto simp add: field_simps)
done
qed
qed

lemma indefinite_integral_continuous_right:
fixes f :: "real => 'a::banach"
assumes "f integrable_on {a .. b}"
and "a ≤ c"
and "c < b"
and "e > 0"
obtains d where "0 < d"
and "∀t. c ≤ t ∧ t < c + d --> norm (integral {a .. c} f - integral {a .. t} f) < e"
proof -
have *: "(λx. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c ≤ - a"
using assms by auto
from indefinite_integral_continuous_left[OF * `e>0`] guess d . note d = this
let ?d = "min d (b - c)"
show ?thesis
apply (rule that[of "?d"])
apply safe
proof -
show "0 < ?d"
using d(1) assms(3) by auto
fix t :: real
assume as: "c ≤ t" "t < c + ?d"
have *: "integral {a .. c} f = integral {a .. b} f - integral {c .. b} f"
"integral {a .. t} f = integral {a .. b} f - integral {t .. b} f"
unfolding algebra_simps
apply (rule_tac[!] integral_combine)
using assms as
apply auto
done
have "(- c) - d < (- t) ∧ - t ≤ - c"
using as by auto note d(2)[rule_format,OF this]
then show "norm (integral {a .. c} f - integral {a .. t} f) < e"
unfolding *
unfolding integral_reflect
apply (subst norm_minus_commute)
apply (auto simp add: algebra_simps)
done
qed
qed

lemma indefinite_integral_continuous:
fixes f :: "real => 'a::banach"
assumes "f integrable_on {a .. b}"
shows "continuous_on {a .. b} (λx. integral {a .. x} f)"
proof (unfold continuous_on_iff, safe)
fix x e :: real
assume as: "x ∈ {a .. b}" "e > 0"
let ?thesis = "∃d>0. ∀x'∈{a .. b}. dist x' x < d --> dist (integral {a .. x'} f) (integral {a .. x} f) < e"
{
presume *: "a < b ==> ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
proof -
case goal1
then have "cbox a b = {x}"
using as(1)
apply -
apply (rule set_eqI)
apply auto
done
then show ?case using `e > 0` by auto
qed
}
assume "a < b"
have "(x = a ∨ x = b) ∨ (a < x ∧ x < b)"
using as(1) by auto
then show ?thesis
apply (elim disjE)
proof -
assume "x = a"
have "a ≤ a" ..
from indefinite_integral_continuous_right[OF assms(1) this `a<b` `e>0`] guess d . note d=this
show ?thesis
apply rule
apply rule
apply (rule d)
apply safe
apply (subst dist_commute)
unfolding `x = a` dist_norm
apply (rule d(2)[rule_format])
apply auto
done
next
assume "x = b"
have "b ≤ b" ..
from indefinite_integral_continuous_left[OF assms(1) `a<b` this `e>0`] guess d . note d=this
show ?thesis
apply rule
apply rule
apply (rule d)
apply safe
apply (subst dist_commute)
unfolding `x = b` dist_norm
apply (rule d(2)[rule_format])
apply auto
done
next
assume "a < x ∧ x < b"
then have xl: "a < x" "x ≤ b" and xr: "a ≤ x" "x < b"
by auto
from indefinite_integral_continuous_left [OF assms(1) xl `e>0`] guess d1 . note d1=this
from indefinite_integral_continuous_right[OF assms(1) xr `e>0`] guess d2 . note d2=this
show ?thesis
apply (rule_tac x="min d1 d2" in exI)
proof safe
show "0 < min d1 d2"
using d1 d2 by auto
fix y
assume "y ∈ {a .. b}" and "dist y x < min d1 d2"
then show "dist (integral {a .. y} f) (integral {a .. x} f) < e"
apply (subst dist_commute)
apply (cases "y < x")
unfolding dist_norm
apply (rule d1(2)[rule_format])
defer
apply (rule d2(2)[rule_format])
unfolding not_less
apply (auto simp add: field_simps)
done
qed
qed
qed

subsection {* This doesn't directly involve integration, but that gives an easy proof. *}

lemma has_derivative_zero_unique_strong_interval:
fixes f :: "real => 'a::banach"
assumes "finite k"
and "continuous_on {a .. b} f"
and "f a = y"
and "∀x∈({a .. b} - k). (f has_derivative (λh. 0)) (at x within {a .. b})" "x ∈ {a .. b}"
shows "f x = y"
proof -
have ab: "a ≤ b"
using assms by auto
have *: "a ≤ x"
using assms(5) by auto
have "((λx. 0::'a) has_integral f x - f a) {a .. x}"
apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
apply (rule continuous_on_subset[OF assms(2)])
defer
apply safe
unfolding has_vector_derivative_def
apply (subst has_derivative_within_open[symmetric])
apply assumption
apply (rule open_greaterThanLessThan)
apply (rule has_derivative_within_subset[where s="{a .. b}"])
using assms(4) assms(5)
apply (auto simp: mem_box)
done
note this[unfolded *]
note has_integral_unique[OF has_integral_0 this]
then show ?thesis
unfolding assms by auto
qed

subsection {* Generalize a bit to any convex set. *}

lemma has_derivative_zero_unique_strong_convex:
fixes f :: "'a::euclidean_space => 'b::banach"
assumes "convex s"
and "finite k"
and "continuous_on s f"
and "c ∈ s"
and "f c = y"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
and "x ∈ s"
shows "f x = y"
proof -
{
presume *: "x ≠ c ==> ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
unfolding assms(5)[symmetric]
apply auto
done
}
assume "x ≠ c"
note conv = assms(1)[unfolded convex_alt,rule_format]
have as1: "continuous_on {0 ..1} (f o (λt. (1 - t) *⇩R c + t *⇩R x))"
apply (rule continuous_intros)+
apply (rule continuous_on_subset[OF assms(3)])
apply safe
apply (rule conv)
using assms(4,7)
apply auto
done
have *: "!!t xa. (1 - t) *⇩R c + t *⇩R x = (1 - xa) *⇩R c + xa *⇩R x ==> t = xa"
proof -
case goal1
then have "(t - xa) *⇩R x = (t - xa) *⇩R c"
unfolding scaleR_simps by (auto simp add: algebra_simps)
then show ?case
using `x ≠ c` by auto
qed
have as2: "finite {t. ((1 - t) *⇩R c + t *⇩R x) ∈ k}"
using assms(2)
apply (rule finite_surj[where f="λz. SOME t. (1-t) *⇩R c + t *⇩R x = z"])
apply safe
unfolding image_iff
apply rule
defer
apply assumption
apply (rule sym)
apply (rule some_equality)
defer
apply (drule *)
apply auto
done
have "(f o (λt. (1 - t) *⇩R c + t *⇩R x)) 1 = y"
apply (rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
unfolding o_def
using assms(5)
defer
apply -
apply rule
proof -
fix t
assume as: "t ∈ {0 .. 1} - {t. (1 - t) *⇩R c + t *⇩R x ∈ k}"
have *: "c - t *⇩R c + t *⇩R x ∈ s - k"
apply safe
apply (rule conv[unfolded scaleR_simps])
using `x ∈ s` `c ∈ s` as
by (auto simp add: algebra_simps)
have "(f o (λt. (1 - t) *⇩R c + t *⇩R x) has_derivative (λx. 0) o (λz. (0 - z *⇩R c) + z *⇩R x))
(at t within {0 .. 1})"
apply (intro derivative_eq_intros)
apply simp_all
apply (simp add: field_simps)
unfolding scaleR_simps
apply (rule has_derivative_within_subset,rule assms(6)[rule_format])
apply (rule *)
apply safe
apply (rule conv[unfolded scaleR_simps])
using `x ∈ s` `c ∈ s`
apply auto
done
then show "((λxa. f ((1 - xa) *⇩R c + xa *⇩R x)) has_derivative (λh. 0)) (at t within {0 .. 1})"
unfolding o_def .
qed auto
then show ?thesis
by auto
qed

text {* Also to any open connected set with finite set of exceptions. Could
generalize to locally convex set with limpt-free set of exceptions. *}

lemma has_derivative_zero_unique_strong_connected:
fixes f :: "'a::euclidean_space => 'b::banach"
assumes "connected s"
and "open s"
and "finite k"
and "continuous_on s f"
and "c ∈ s"
and "f c = y"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
and "x∈s"
shows "f x = y"
proof -
have "{x ∈ s. f x ∈ {y}} = {} ∨ {x ∈ s. f x ∈ {y}} = s"
apply (rule assms(1)[unfolded connected_clopen,rule_format])
apply rule
defer
apply (rule continuous_closed_in_preimage[OF assms(4) closed_singleton])
apply (rule open_openin_trans[OF assms(2)])
unfolding open_contains_ball
proof safe
fix x
assume "x ∈ s"
from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
show "∃e>0. ball x e ⊆ {xa ∈ s. f xa ∈ {f x}}"
apply rule
apply rule
apply (rule e)
proof safe
fix y
assume y: "y ∈ ball x e"
then show "y ∈ s"
using e by auto
show "f y = f x"
apply (rule has_derivative_zero_unique_strong_convex[OF convex_ball])
apply (rule assms)
apply (rule continuous_on_subset)
apply (rule assms)
apply (rule e)+
apply (subst centre_in_ball)
apply (rule e)
apply rule
apply safe
apply (rule has_derivative_within_subset)
apply (rule assms(7)[rule_format])
using y e
apply auto
done
qed
qed
then show ?thesis
using `x ∈ s` `f c = y` `c ∈ s` by auto
qed

lemma has_derivative_zero_connected_constant:
fixes f :: "'a::euclidean_space => 'b::banach"
assumes "connected s"
and "open s"
and "finite k"
and "continuous_on s f"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
obtains c where "!!x. x ∈ s ==> f(x) = c"
proof (cases "s = {}")
case True
then show ?thesis
by (metis empty_iff that)
next
case False
then obtain c where "c ∈ s"
by (metis equals0I)
then show ?thesis
by (metis has_derivative_zero_unique_strong_connected assms that)
qed

subsection {* Integrating characteristic function of an interval *}

lemma has_integral_restrict_open_subinterval:
fixes f :: "'a::euclidean_space => 'b::banach"
assumes "(f has_integral i) (cbox c d)"
and "cbox c d ⊆ cbox a b"
shows "((λx. if x ∈ box c d then f x else 0) has_integral i) (cbox a b)"
proof -
def g ≡ "λx. if x ∈box c d then f x else 0"
{
presume *: "cbox c d ≠ {} ==> ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
proof -
case goal1
then have *: "box c d = {}"
by (metis bot.extremum_uniqueI box_subset_cbox)
show ?thesis
using assms(1)
unfolding *
using goal1
by auto
qed
}
assume "cbox c d ≠ {}"
from partial_division_extend_1[OF assms(2) this] guess p . note p=this
note mon = monoidal_lifted[OF monoidal_monoid]
note operat = operative_division[OF this operative_integral p(1), symmetric]
let ?P = "(if g integrable_on cbox a b then Some (integral (cbox a b) g) else None) = Some i"
{
presume "?P"
then have "g integrable_on cbox a b ∧ integral (cbox a b) g = i"
apply -
apply cases
apply (subst(asm) if_P)
apply assumption
apply auto
done
then show ?thesis
using integrable_integral
unfolding g_def
by auto
}

note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
note * = this[unfolded neutral_add]
have iterate:"iterate (lifted op +) (p - {cbox c d})
(λi. if g integrable_on i then Some (integral i g) else None) = Some 0"
proof (rule *, rule)
case goal1
then have "x ∈ p"
by auto
note div = division_ofD(2-5)[OF p(1) this]
from div(3) guess u v by (elim exE) n```