l3lab/ntp-mathlib huggingface dataset:
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  

Proof state:
case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

Next tactic:
have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    

Proof state:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ ↑↑(Measure.restrict μ ↑I) U < ↑(δ n) / ↑n

Next tactic:
refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    

Proof state:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {n}) < ↑(δ n) / ↑n

Next tactic:
cases' n with n
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · 

Proof state:
case zero
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {Nat.zero}) < ↑(δ Nat.zero) / ↑Nat.zero

Next tactic:
simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · 

Proof state:
case succ
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ ↑↑(Measure.restrict μ ↑I) (N ⁻¹' {Nat.succ n}) < ↑(δ (Nat.succ n)) / ↑(Nat.succ n)

Next tactic:
refine' (measure_mono_null _ hf).le.trans_lt _
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · 

Proof state:
case succ.refine'_1
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ N ⁻¹' {Nat.succ n} ⊆ {x | f x ≠ 0}

Next tactic:
exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · 

Proof state:
case succ.refine'_2
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
n : ℕ
⊢ 0 < ↑(δ (Nat.succ n)) / ↑(Nat.succ n)

Next tactic:
simp [(δ0 _).ne']
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
  

Proof state:
case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this✝ : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
this : ∀ (n : ℕ), ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ ↑↑(Measure.restrict μ ↑I) U < ↑(δ n) / ↑n
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

Next tactic:
choose U hNU hUo hμU using this
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
  choose U hNU hUo hμU using this
  

Proof state:
case intro.intro.intro.intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
⊢ ∃ r,
    (∀ (c : ℝ≥0), IntegrationParams.RCond l (r c)) ∧
      ∀ (c : ℝ≥0) (π : TaggedPrepartition I),
        IntegrationParams.MemBaseSet l I c (r c) π →
          TaggedPrepartition.IsPartition π →
            dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive μ)) π) 0 ≤ ↑ε

Next tactic:
have : ∀ x, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U (N x) := fun x => by
    obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
    exact ⟨⟨r, hr₀⟩, hr⟩
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
  choose U hNU hUo hμU using this
  have : ∀ x, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U (N x) := fun x => by
    

Proof state:
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
x : ι → ℝ
⊢ ∃ r, closedBall x ↑r ⊆ U (N x)

Next tactic:
obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
/-- If `f` is a.e. equal to zero on a rectangular box, then it has McShane integral zero on this
box. -/
theorem HasIntegral.of_aeEq_zero {l : IntegrationParams} {I : Box ι} {f : (ι → ℝ) → E}
    {μ : Measure (ι → ℝ)} [IsLocallyFiniteMeasure μ] (hf : f =ᵐ[μ.restrict I] 0)
    (hl : l.bRiemann = false) : HasIntegral.{u, v, v} I l f μ.toBoxAdditive.toSMul 0 := by
  /- Each set `{x | n < ‖f x‖ ≤ n + 1}`, `n : ℕ`, has measure zero. We cover it by an open set of
    measure less than `ε / 2 ^ n / (n + 1)`. Then the norm of the integral sum is less than `ε`. -/
  refine' hasIntegral_iff.2 fun ε ε0 => _
  lift ε to ℝ≥0 using ε0.lt.le; rw [gt_iff_lt, NNReal.coe_pos] at ε0
  rcases NNReal.exists_pos_sum_of_countable ε0.ne' ℕ with ⟨δ, δ0, c, hδc, hcε⟩
  haveI := Fact.mk (I.measure_coe_lt_top μ)
  change μ.restrict I {x | f x ≠ 0} = 0 at hf
  set N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
  have N0 : ∀ {x}, N x = 0 ↔ f x = 0 := by simp
  have : ∀ n, ∃ U, N ⁻¹' {n} ⊆ U ∧ IsOpen U ∧ μ.restrict I U < δ n / n := fun n ↦ by
    refine (N ⁻¹' {n}).exists_isOpen_lt_of_lt _ ?_
    cases' n with n
    · simpa [ENNReal.div_zero (ENNReal.coe_pos.2 (δ0 _)).ne'] using measure_lt_top (μ.restrict I) _
    · refine' (measure_mono_null _ hf).le.trans_lt _
      · exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
      · simp [(δ0 _).ne']
  choose U hNU hUo hμU using this
  have : ∀ x, ∃ r : Ioi (0 : ℝ), closedBall x r ⊆ U (N x) := fun x => by
    obtain ⟨r, hr₀, hr⟩ := nhds_basis_closedBall.mem_iff.1 ((hUo _).mem_nhds (hNU _ rfl))
    

Proof state:
case intro.intro
ι : Type u
E : Type v
inst✝³ : Fintype ι
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
l : IntegrationParams
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst✝ : IsLocallyFiniteMeasure μ
hl : l.bRiemann = false
ε : ℝ≥0
ε0 : 0 < ε
δ : ℕ → ℝ≥0
δ0 : ∀ (i : ℕ), 0 < δ i
c : ℝ≥0
hδc : HasSum δ c
hcε : c < ε
this : Fact (↑↑μ ↑I < ⊤)
hf : ↑↑(Measure.restrict μ ↑I) {x | f x ≠ 0} = 0
N : (ι → ℝ) → ℕ := fun x => ⌈‖f x‖⌉₊
N0 : ∀ {x : ι → ℝ}, N x = 0 ↔ f x = 0
U : ℕ → Set (ι → ℝ)
hNU : ∀ (n : ℕ), N ⁻¹' {n} ⊆ U n
hUo : ∀ (n : ℕ), IsOpen (U n)
hμU : ∀ (n : ℕ), ↑↑(Measure.restrict μ ↑I) (U n) < ↑(δ n) / ↑n
x : ι → ℝ
r : ℝ
hr₀ : 0 < r
hr : closedBall ?m.317652 r ⊆ U (N ?m.317652)
⊢ ∃ r, closedBall x ↑r ⊆ U (N x)

Next tactic:
exact ⟨⟨r, hr₀⟩, hr⟩