-- Mathlib/Analysis/BoxIntegral/Basic.lean
theorem integrable_of_bounded_and_ae_continuousWithinAt [CompleteSpace E] {I : Box ι} {f : ℝⁿ → E}
    (hb : ∃ C : ℝ, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C) (μ : Measure ℝⁿ) [IsLocallyFiniteMeasure μ]
    (hc : ∀ᵐ x ∂(μ.restrict (Box.Icc I)), ContinuousWithinAt f (Box.Icc I) x) :
    Integrable I l f μ.toBoxAdditive.toSMul

Proof state:
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ⊤
t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J
B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes
B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
a✝ : J ∈ B'
⊢ ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)

Tactic:
rw [norm_smul, μ.toBoxAdditive_apply, Real.norm_of_nonneg toReal_nonneg, two_mul]

Result:
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ⊤
t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J
B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes
B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
J : Box ι
a✝ : J ∈ B'
⊢ (μ ↑J).toReal * ‖f (t₁ J) - f (t₂ J)‖ ≤ (μ ↑J).toReal * (C + C)

Proof state:
case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),
        l.MemBaseSet I c₁ (r c₁) π₁ →
          π₁.IsPartition →
            l.MemBaseSet I c₂ (r c₂) π₂ →
              π₂.IsPartition →
                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε

Tactic:
obtain ⟨U, UD, Uopen, hU⟩ := Set.exists_isOpen_lt_add D (show μ' D ≠ ⊤ by simp [μ'D]) ε₂0'

Result:
case intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < μ' D + ENNReal.ofReal ε₂
⊢ ∃ r,
    (∀ (c : ℝ≥0), l.RCond (r c)) ∧
      ∀ (c₁ c₂ : ℝ≥0) (π₁ π₂ : TaggedPrepartition I),
        l.MemBaseSet I c₁ (r c₁) π₁ →
          π₁.IsPartition →
            l.MemBaseSet I c₂ (r c₂) π₂ →
              π₂.IsPartition →
                dist (integralSum f μ.toBoxAdditive.toSMul π₁) (integralSum f μ.toBoxAdditive.toSMul π₂) ≤ ε

Proof state:
ι : Type u
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : Fintype ι
l : IntegrationParams
inst✝¹ : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hc : ∀ᵐ (x : ι → ℝ) ∂μ.restrict (Box.Icc I), ContinuousWithinAt f (Box.Icc I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : 2 * μ.toBoxAdditive I * ε₁ < ε
C : ℝ
hC : ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
ε₂0' : ENNReal.ofReal ε₂ ≠ 0
D : Set (ι → ℝ) := {x | x ∈ Box.Icc I ∧ ¬ContinuousWithinAt f (Box.Icc I) x}
μ' : Measure (ι → ℝ) := μ.restrict (Box.Icc I)
μ'D : μ' D = 0
U : Set (ι → ℝ)
UD : U ⊇ D
Uopen : IsOpen U
hU : μ' U < ENNReal.ofReal ε₂
comp : IsCompact (Box.Icc I \ U)
this✝¹ : ∀ x ∈ Box.Icc I \ U, oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁
r : ℝ
r0 : r > 0
hr : ∀ x ∈ Box.Icc I \ U, EMetric.diam (f '' (EMetric.ball x (ENNReal.ofReal r) ∩ Box.Icc I)) ≤ ENNReal.ofReal ε₁
c₁ c₂ : ℝ≥0
π₁ π₂ : TaggedPrepartition I
h₁ : l.MemBaseSet I c₁ ((fun x x => ⟨r / 2, ⋯⟩) c₁) π₁
h₁p : π₁.IsPartition
h₂ : l.MemBaseSet I c₂ ((fun x x => ⟨r / 2, ⋯⟩) c₂) π₂
h₂p : π₂.IsPartition
μI : μ ↑I < ⊤
t₁ : Box ι → ι → ℝ := fun J => (π₁.infPrepartition π₂.toPrepartition).tag J
t₂ : Box ι → ι → ℝ := fun J => (π₂.infPrepartition π₁.toPrepartition).tag J
B : Finset (Box ι) := (π₁.toPrepartition ⊓ π₂.toPrepartition).boxes
B' : Finset (Box ι) := Finset.filter (fun J => ↑J ⊆ U) B
hB' : B' ⊆ B
μJ_ne_top : ∀ J ∈ B, μ ↑J ≠ ⊤
un : ∀ S ⊆ B, ⋃ J ∈ S, ↑J ⊆ ↑I
this✝ : ∀ J ∈ B', ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * (2 * C)
this : (∑ J ∈ B', μ ↑J).toReal ≤ ε₂
⊢ (∑ a ∈ B', μ ↑a).toReal * (2 * C) ≤ ε / 2

Tactic:
linarith [mul_le_mul_of_nonneg_right this <| (mul_nonneg_iff_of_pos_left two_pos).2 C0]

Result:
no goals