-- 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
hJ : J ∈ B \ B'
⊢ ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * ε₁

Tactic:
rw [mem_sdiff, B.mem_filter, not_and] at hJ

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
hJ : J ∈ B ∧ (J ∈ B → ¬↑J ⊆ U)
⊢ ‖μ.toBoxAdditive J • (f (t₁ J) - f (t₂ J))‖ ≤ μ.toBoxAdditive J * ε₁

Proof state:
case 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
x : ι → ℝ
hx : x ∈ ↑I
⊢ 0 ≤ C

Tactic:
exact le_trans (norm_nonneg (f x)) <| hC x (I.coe_subset_Icc hx)

Result:
no goals

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)
x : ι → ℝ
hx : x ∈ Box.Icc I \ U
⊢ oscillationWithin f (Box.Icc I) x < ENNReal.ofReal ε₁

Tactic:
suffices oscillationWithin f (Box.Icc I) x = 0 by rw [this]; exact ofReal_pos.2 ε₁0

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)
x : ι → ℝ
hx : x ∈ Box.Icc I \ U
⊢ oscillationWithin f (Box.Icc I) x = 0

Proof state:
case 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)
V : Set (ι → ℝ)
ae : V ∈ MeasureTheory.ae (μ.restrict (Box.Icc I))
hV : ∀ y ∈ V, ContinuousWithinAt f (Box.Icc I) y
⊢ μ' D = 0

Tactic:
exact eq_of_le_of_not_lt (mem_ae_iff.1 ae ▸ (μ'.mono <| fun x h xV ↦ h.2 (hV x xV))) not_lt_zero

Result:
no goals

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
⊢ μ' D ≠ ⊤

Tactic:
simp [μ'D]

Result:
no goals