awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (hypothesis)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
π : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c : ℝ≥0
ε : ℝ
inst_5 : CompleteSpace F
h : Integrable I l f vol
h0 : 0 < ε
hπ : l.MemBaseSet I c (h.convergenceR ε c) π
π₀ : Prepartition I
hU : π.iUnion = π₀.iUnion
δ : ℝ
δ0 : 0 < δ
δ' : ℝ := δ / HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
H0 : 0 < HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
δ'0 : 0 < δ'
C : ℝ≥0 := max π₀.distortion π₀.compl.distortion
πi : (J : Box ι) → TaggedPrepartition J
hπip : ∀ J ∈ π₀, (πi J).IsPartition
hπiδ' : ∀ J ∈ π₀, dist (integralSum f vol (πi J)) (integral J l f vol) ≤ δ'
hπiC : ∀ J ∈ π₀, l.MemBaseSet J C (h.convergenceR δ' C) (πi J)
this_1 : l.MemBaseSet I C (h.convergenceR δ' C) (π₀.biUnionTagged πi)
hU' : π.iUnion = (π₀.biUnionTagged πi).iUnion
this : dist (integralSum f vol π) (integralSum f vol (π₀.biUnionTagged πi)) ≤ ε + δ'
⊢ dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integral J l f vol) ≤ ε + δ
have this₁ : dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integralSum f vol (πi J)) ≤ ε + δ'
Tactic: rw [integralSum_biUnionTagged] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
π : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c : ℝ≥0
ε : ℝ
inst_5 : CompleteSpace F
h : Integrable I l f vol
h0 : 0 < ε
hπ : l.MemBaseSet I c (h.convergenceR ε c) π
π₀ : Prepartition I
hU : π.iUnion = π₀.iUnion
δ : ℝ
δ0 : 0 < δ
δ' : ℝ := δ / HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
H0 : 0 < HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
δ'0 : 0 < δ'
C : ℝ≥0 := max π₀.distortion π₀.compl.distortion
πi : (J : Box ι) → TaggedPrepartition J
hπip : ∀ J ∈ π₀, (πi J).IsPartition
hπiδ' : ∀ J ∈ π₀, dist (integralSum f vol (πi J)) (integral J l f vol) ≤ δ'
hπiC : ∀ J ∈ π₀, l.MemBaseSet J C (h.convergenceR δ' C) (πi J)
this_1 : l.MemBaseSet I C (h.convergenceR δ' C) (π₀.biUnionTagged πi)
hU' : π.iUnion = (π₀.biUnionTagged πi).iUnion
this : dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integralSum f vol (πi J)) ≤ ε + δ'
⊢ ε + δ' + ∑ _J ∈ π₀.boxes, δ' = ε + δ
have simp : ε * HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1 + δ + (↑(#π₀.boxes) : ℝ) * δ =
  HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1 * (ε + δ)
Tactic: simp [field, δ']
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
π : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
c : ℝ≥0
ε : ℝ
inst_5 : CompleteSpace F
h : Integrable I l f vol
h0 : 0 < ε
hπ : l.MemBaseSet I c (h.convergenceR ε c) π
π₀ : Prepartition I
hU : π.iUnion = π₀.iUnion
δ : ℝ
δ0 : 0 < δ
δ' : ℝ := δ / HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
H0 : 0 < HAdd.hAdd (α := ℝ) (↑(#π₀.boxes) : ℝ) 1
δ'0 : 0 < δ'
C : ℝ≥0 := max π₀.distortion π₀.compl.distortion
πi : (J : Box ι) → TaggedPrepartition J
hπip : ∀ J ∈ π₀, (πi J).IsPartition
hπiδ' : ∀ J ∈ π₀, dist (integralSum f vol (πi J)) (integral J l f vol) ≤ δ'
hπiC : ∀ J ∈ π₀, l.MemBaseSet J C (h.convergenceR δ' C) (πi J)
this_1 : l.MemBaseSet I C (h.convergenceR δ' C) (π₀.biUnionTagged πi)
hU' : π.iUnion = (π₀.biUnionTagged πi).iUnion
this : dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integralSum f vol (πi J)) ≤ ε + δ'
⊢ dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integral J l f vol) ≤ ε + δ
have dist : ε + δ' + ∑ _J ∈ π₀.boxes, δ' = ε + δ
Tactic: calc
    dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integral J l f vol) ≤
        dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integralSum f vol (πi J)) +
          dist (∑ J ∈ π₀.boxes, integralSum f vol (πi J)) (∑ J ∈ π₀.boxes, integral J l f vol) :=
      dist_triangle _ _ _
    _ ≤ ε + δ' + ∑ _J ∈ π₀.boxes, δ' := add_le_add this (dist_sum_sum_le_of_le _ hπiδ')
    _ = ε + δ := by simp [field, δ']; ring
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst_5 : CompleteSpace F
h : Integrable I l f vol
π₀ : Prepartition I
⊢ Tendsto (integralSum f vol) (toFilteriUnion (l := l) I π₀) (𝓝 (∑ J ∈ π₀.boxes, integral J l f vol))
have refine : ∀ (ε : ℝ) (ε0 : 0 < ε),
  ∃ (ia : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (c : ℝ≥0), l.RCond (ia c)) ∧
      ∀ (x : TaggedPrepartition I),
        Membership.mem (γ := Set (TaggedPrepartition I))
            {π : TaggedPrepartition I | ∃ (c : ℝ≥0), l.MemBaseSet I c (ia c) π ∧ π.iUnion = π₀.iUnion} x →
          Membership.mem (γ := Set F) (closedBall (∑ J ∈ π₀.boxes, integral J l f vol) ε) (integralSum f vol x)
Tactic: refine ((l.hasBasis_toFilteriUnion I π₀).tendsto_iff nhds_basis_closedBall).2 fun ε ε0 => ?_
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst_5 : CompleteSpace F
h : Integrable I l f vol
π₀ : Prepartition I
ε : ℝ
ε0 : 0 < ε
⊢ ∃ (ia : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (c : ℝ≥0), l.RCond (ia c)) ∧
      ∀ (x : TaggedPrepartition I),
        Membership.mem (γ := Set (TaggedPrepartition I))
            {π : TaggedPrepartition I | ∃ (c : ℝ≥0), l.MemBaseSet I c (ia c) π ∧ π.iUnion = π₀.iUnion} x →
          Membership.mem (γ := Set F) (closedBall (∑ J ∈ π₀.boxes, integral J l f vol) ε) (integralSum f vol x)
have refine : ∀ (x : TaggedPrepartition I),
  Membership.mem (γ := Set (TaggedPrepartition I))
      {π : TaggedPrepartition I | ∃ (c : ℝ≥0), l.MemBaseSet I c (h.convergenceR ε c) π ∧ π.iUnion = π₀.iUnion} x →
    Membership.mem (γ := Set F) (closedBall (∑ J ∈ π₀.boxes, integral J l f vol) ε) (integralSum f vol x)
Tactic: refine ⟨h.convergenceR ε, h.convergenceR_cond ε, ?_⟩
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst_5 : CompleteSpace F
h : Integrable I l f vol
π₁ π₂ : Prepartition I
hU : π₁.iUnion = π₂.iUnion
⊢ Eq.{w + 1} (α := F) (∑ J ∈ π₁.boxes, integral J l f vol) (∑ J ∈ π₂.boxes, integral J l f vol)
have refine : Tendsto (integralSum f vol) (toFilteriUnion (l := l) I π₁) (𝓝 (∑ J ∈ π₂.boxes, integral J l f vol))
Tactic: refine tendsto_nhds_unique (h.tendsto_integralSum_sum_integral π₁) ?_
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I : Box ι
inst_4 : Fintype ι
l : IntegrationParams
f : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
inst_5 : CompleteSpace F
h : Integrable I l f vol
π₁ π₂ : Prepartition I
hU : π₁.iUnion = π₂.iUnion
⊢ Tendsto (integralSum f vol) (toFilteriUnion (l := l) I π₁) (𝓝 (∑ J ∈ π₂.boxes, integral J l f vol))
have rw : Tendsto (integralSum f vol) (toFilteriUnion (l := l) I π₂) (𝓝 (∑ J ∈ π₂.boxes, integral J l f vol))
Tactic: rw [l.toFilteriUnion_congr _ hU]
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (hypothesis)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I J_1 : Box ι
π_1 : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst_5 : CompleteSpace F
h : Integrable I l f vol
J : Box ι
hJ : (↑J : WithTop (Box ι)) ≤ (↑I : WithTop (Box ι))
π : Prepartition J
hπ : π.IsPartition
⊢ ∑ Ji ∈ π.boxes, integral Ji l f vol = integral J l f vol
have hπ₁ : π.iUnion = (↑J : Set (ι → ℝ))
Tactic: replace hπ := hπ.iUnion_eq
Cleared hypotheses: hπ
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (hypothesis)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I J_1 : Box ι
π_1 : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst_5 : CompleteSpace F
h : Integrable I l f vol
J : Box ι
hJ : (↑J : WithTop (Box ι)) ≤ (↑I : WithTop (Box ι))
π : Prepartition J
hπ : π.iUnion = (↑J : Set (ι → ℝ))
⊢ ∑ Ji ∈ π.boxes, integral Ji l f vol = integral J l f vol
have hπ₁ : π.iUnion = Prepartition.iUnion (I := J) ⊤
Tactic: rw [← Prepartition.iUnion_top] at hπ
Cleared hypotheses: hπ
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I J_1 : Box ι
π_1 : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst_5 : CompleteSpace F
h : Integrable I l f vol
J : Box ι
hJ : (↑J : WithTop (Box ι)) ≤ (↑I : WithTop (Box ι))
π : Prepartition J
hπ : π.iUnion = Prepartition.iUnion (I := J) ⊤
⊢ ∑ Ji ∈ π.boxes, integral Ji l f vol = integral J l f vol
have rw : ∑ J ∈ Prepartition.boxes (I := J) ⊤, integral J l f vol = integral J l f vol
Tactic: rw [(h.to_subbox (WithTop.coe_le_coe.1 hJ)).sum_integral_congr hπ, Prepartition.top_boxes,
      sum_singleton]
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I J_1 : Box ι
π_1 : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst_5 : CompleteSpace F
h : Integrable I l f vol
J : Box ι
hJ : (↑J : WithTop (Box ι)) ≤ (↑I : WithTop (Box ι))
π : Prepartition J
hπ : π.iUnion = Prepartition.iUnion (I := J) ⊤
rw : ∑ J ∈ Prepartition.boxes (I := J) ⊤, integral J l f vol = integral J l f vol
⊢ ∑ Ji ∈ π.boxes, integral Ji l f vol = integral J l f vol
have rw₁ : ∑ J ∈ {J}, integral J l f vol = integral J l f vol
Tactic: rw [(h.to_subbox (WithTop.coe_le_coe.1 hJ)).sum_integral_congr hπ, Prepartition.top_boxes,
      sum_singleton]
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
F : Type w
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : NormedAddCommGroup F
inst_3 : NormedSpace ℝ F
I J_1 : Box ι
π_1 : TaggedPrepartition I
inst_4 : Fintype ι
l : IntegrationParams
f g : (ι → ℝ) → E
vol : ι →ᵇᵃ[⊤] E →L[ℝ] F
y y' : F
c c₁ c₂ : ℝ≥0
ε ε₁ ε₂ : ℝ
π₁ π₂ : TaggedPrepartition I
inst_5 : CompleteSpace F
h : Integrable I l f vol
J : Box ι
hJ : (↑J : WithTop (Box ι)) ≤ (↑I : WithTop (Box ι))
π : Prepartition J
hπ : π.iUnion = Prepartition.iUnion (I := J) ⊤
rw : ∑ J ∈ Prepartition.boxes (I := J) ⊤, integral J l f vol = integral J l f vol
rw₁ : ∑ J ∈ {J}, integral J l f vol = integral J l f vol
⊢ ∑ Ji ∈ π.boxes, integral Ji l f vol = integral J l f vol
have rw₂ : integral J l f vol = integral J l f vol
Tactic: rw [(h.to_subbox (WithTop.coe_le_coe.1 hJ)).sum_integral_congr hπ, Prepartition.top_boxes,
      sum_singleton]
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hb :
  ∃ (C : ℝ),
    ∀ (x : ι → ℝ),
      Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
⊢ Integrable I l f μ.toBoxAdditive.toSMul
have refine : ∀ (ε : ℝ) (ε0 : ε > 0),
  ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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: refine integrable_iff_cauchy_basis.2 fun ε ε0 ↦ ?_
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hb :
  ∃ (C : ℝ),
    ∀ (x : ι → ℝ),
      Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
⊢ ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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
have rcases : ∀ (ε₁ : ℝ) (ε₁0 : 0 < ε₁)
  (hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε),
  ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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: rcases exists_pos_mul_lt ε0 (2 * μ.toBoxAdditive I) with ⟨ε₁, ε₁0, hε₁⟩
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
hb :
  ∃ (C : ℝ),
    ∀ (x : ι → ℝ),
      Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
⊢ ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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
have rcases : ∀ (C : ℝ)
  (hC :
    ∀ (x : ι → ℝ),
      Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C),
  ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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: rcases hb with ⟨C, hC⟩
Cleared hypotheses: hb
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
C : ℝ
hC :
  ∀ (x : ι → ℝ), Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
⊢ 0 ≤ C
have obtain : ∀ (x : ι → ℝ) (hx : x ∈ (↑I : Set (ι → ℝ))), 0 ≤ C
Tactic: obtain ⟨x, hx⟩ := BoxIntegral.Box.nonempty_coe I
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (hypothesis)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
C : ℝ
hC :
  ∀ (x : ι → ℝ), Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
⊢ ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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
have C0 : 0 ≤ C
Tactic: have C0 : 0 ≤ C := by
    obtain ⟨x, hx⟩ := BoxIntegral.Box.nonempty_coe I
    exact le_trans (norm_nonneg (f x)) <| hC x (I.coe_subset_Icc hx)
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
C : ℝ
hC :
  ∀ (x : ι → ℝ), Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
C0 : 0 ≤ C
⊢ ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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
have rcases : ∀ (ε₂ : ℝ) (ε₂0 : 0 < ε₂) (hε₂ : 4 * C * ε₂ < ε),
  ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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: rcases exists_pos_mul_lt ε0 (4 * C) with ⟨ε₂, ε₂0, hε₂⟩
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (hypothesis)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
C : ℝ
hC :
  ∀ (x : ι → ℝ), Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖f x‖ ≤ C
C0 : 0 ≤ C
ε₂ : ℝ
ε₂0 : 0 < ε₂
hε₂ : 4 * C * ε₂ < ε
⊢ ∃ (r : ℝ≥0 → (ι → ℝ) → (↑(Set.Ioi 0) : Type)),
    (∀ (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
have ε₂0' : ENNReal.ofReal ε₂ ≠ 0
Tactic: have ε₂0' : ENNReal.ofReal ε₂ ≠ 0 := ne_of_gt <| ofReal_pos.2 ε₂0
-- Lean 4 proof step from Mathlib.Analysis.BoxIntegral.Basic (goal)
Goal:
ι : Type u
E : Type v
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : Fintype ι
l : IntegrationParams
inst_3 : CompleteSpace E
I : Box ι
f : (ι → ℝ) → E
μ : Measure (ι → ℝ)
inst_4 : IsLocallyFiniteMeasure μ
hc :
  ∀ᵐ (x : ι → ℝ) ∂μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I),
    ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x
ε : ℝ
ε0 : ε > 0
ε₁ : ℝ
ε₁0 : 0 < ε₁
hε₁ : HMul.hMul (α := ℝ) (β := ℝ) (2 : ℝ) ((μ.toBoxAdditive : (a : Box ι) → ℝ) I : ℝ) * ε₁ < ε
C : ℝ
hC :
  ∀ (x : ι → ℝ), Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x → ‖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 : ι → ℝ |
    Membership.mem (γ := Set (ι → ℝ)) ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I : Set (ι → ℝ)) x ∧
      ¬ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) x}
μ' : Measure (ι → ℝ) := μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I)
⊢ Eq (α := ℝ≥0∞) ((μ' : (a : Set (ι → ℝ)) → ℝ≥0∞) D : ℝ≥0∞) (0 : ℝ≥0∞)
have rcases : ∀ (V : Set (ι → ℝ)) (ae : V ∈ ae (μ.restrict ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I)))
  (hV : ∀ y ∈ V, ContinuousWithinAt f ((Box.Icc : (a : Box ι) → Set (ι → ℝ)) I) y),
  Eq (α := ℝ≥0∞) ((μ' : (a : Set (ι → ℝ)) → ℝ≥0∞) D : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: rcases eventually_iff_exists_mem.1 hc with ⟨V, ae, hV⟩