awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
d : ℝ≥0
hd : (↑d : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)
⊢ ∀ (x : α),
    Membership.mem (γ := Set α)
        {x : α |
            (fun (x : α) ↦
                ¬((∃ᶠ (a : Set α) in v.filterAt x,
                      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                          ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                        (↑c : ℝ≥0∞)) ∧
                    ∃ᶠ (a : Set α) in v.filterAt x,
                      (↑d : ℝ≥0∞) <
                        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                          ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)))
              x}ᶜ
        x →
      ∃ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
have hc₁ : ∀ (x : α),
  (∃ᶠ (a : Set α) in v.filterAt x,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
        (↑c : ℝ≥0∞)) →
    (∃ᶠ (x : Set α) in v.filterAt x,
        (↑d : ℝ≥0∞) <
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)) →
      ∃ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
Tactic: simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
        mem_setOf_eq, mem_compl_iff, not_forall]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
d : ℝ≥0
hd : (↑d : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)
x : α
h1x :
  ∃ᶠ (a : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
      (↑c : ℝ≥0∞)
h :
  ∃ᶠ (x : Set α) in v.filterAt x,
    (↑d : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)
⊢ ∃ᶠ (a : Set α) in v.filterAt x,
    (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
have apply : ∀ (a : Set α)
  (ha :
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
      (↑c : ℝ≥0∞)),
  (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
Tactic: apply h1x.mono fun a ha => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
d : ℝ≥0
hd : (↑d : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)
x : α
h1x :
  ∃ᶠ (a : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
      (↑c : ℝ≥0∞)
h :
  ∃ᶠ (x : Set α) in v.filterAt x,
    (↑d : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)
a : Set α
ha :
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
    (↑c : ℝ≥0∞)
⊢ (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
have refine : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞) ∨ (↑c : ℝ≥0∞) ≠ ∞
Tactic: refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
d : ℝ≥0
hd : (↑d : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)
⊢ ∀ (x : α),
    Membership.mem (γ := Set α)
        {x : α |
            (fun (x : α) ↦
                ¬((∃ᶠ (a : Set α) in v.filterAt x,
                      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                          ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                        (↑c : ℝ≥0∞)) ∧
                    ∃ᶠ (a : Set α) in v.filterAt x,
                      (↑d : ℝ≥0∞) <
                        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                          ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)))
              x}ᶜ
        x →
      ∃ᶠ (a : Set α) in v.filterAt x,
        HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a
have hd₁ : ∀ (x : α),
  (∃ᶠ (a : Set α) in v.filterAt x,
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
        (↑c : ℝ≥0∞)) →
    (∃ᶠ (x : Set α) in v.filterAt x,
        (↑d : ℝ≥0∞) <
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)) →
      ∃ᶠ (a : Set α) in v.filterAt x,
        HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a
Tactic: simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
        mem_setOf_eq, mem_compl_iff, not_forall]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
d : ℝ≥0
hd : (↑d : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)
x : α
h :
  ∃ᶠ (a : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
      (↑c : ℝ≥0∞)
h2x :
  ∃ᶠ (x : Set α) in v.filterAt x,
    (↑d : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞)
⊢ ∃ᶠ (a : Set α) in v.filterAt x,
    HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a
have apply : ∀ (a : Set α)
  (ha :
    (↑d : ℝ≥0∞) <
      HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)),
  HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ≤ (ρ : (a : Set α) → ℝ≥0∞) a
Tactic: apply h2x.mono fun a ha => ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
⊢ ∀ᵐ (x : α) ∂μ,
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)
have A : ∀ c ∈ w,
  ∀ d ∈ w,
    c < d →
      ∀ᵐ (x : α) ∂μ,
        ¬((∃ᶠ (a : Set α) in v.filterAt x,
              HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                  ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                c) ∧
            ∃ᶠ (a : Set α) in v.filterAt x,
              d <
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                  ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
Tactic: have A : ∀ c ∈ w, ∀ d ∈ w, c < d → ∀ᵐ x ∂μ,
      ¬((∃ᶠ a in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ a in v.filterAt x, d < ρ a / μ a) := by
    intro c hc d hd hcd
    lift c to ℝ≥0 using I c hc
    lift d to ℝ≥0 using I d hd
    apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd)
    · simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
        mem_setOf_eq, mem_compl_iff, not_forall]
      intro x h1x _
      apply h1x.mono fun a ha => ?_
      refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
      simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
    · simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
        mem_setOf_eq, mem_compl_iff, not_forall]
      intro x _ h2x
      apply h2x.mono fun a ha => ?_
      exact ENNReal.mul_le_of_le_div ha.le
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x,
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                  c) ∧
              ∃ᶠ (a : Set α) in v.filterAt x,
                d <
                  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
⊢ ∀ᵐ (x : α) ∂μ,
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)
have B : ∀ᵐ (x : α) ∂μ,
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ¬((∃ᶠ (a : Set α) in v.filterAt x,
              HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                  ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                c) ∧
            ∃ᶠ (a : Set α) in v.filterAt x,
              d <
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                  ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
Tactic: have B : ∀ᵐ x ∂μ, ∀ c ∈ w, ∀ d ∈ w, c < d →
      ¬((∃ᶠ a in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ a in v.filterAt x, d < ρ a / μ a) := by
    simpa only [ae_ball_iff w_count, ae_all_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left : 0 ∉ w
w_top : ∞ ∉ w
I : ∀ x ∈ w, x ≠ ∞
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x,
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                  c) ∧
              ∃ᶠ (a : Set α) in v.filterAt x,
                d <
                  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ c ∈ w,
      ∀ d ∈ w,
        c < d →
          ¬((∃ᶠ (a : Set α) in v.filterAt x,
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                  c) ∧
              ∃ᶠ (a : Set α) in v.filterAt x,
                d <
                  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
⊢ ∀ᵐ (x : α) ∂μ,
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)
have h : ∀ (a : α),
  (∀ c ∈ w,
      ∀ d ∈ w,
        c < d →
          ¬((∃ᶠ (a : Set α) in v.filterAt a,
                HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
                  c) ∧
              ∃ᶠ (a : Set α) in v.filterAt a,
                d <
                  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
                    ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))) →
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt a) (𝓝 c)
Tactic: filter_upwards [B]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
⊢ ∀ᵐ (x : α) ∂μ,
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt x) (𝓝 (v.limRatio ρ x))
have h : ∀ (a : α),
  (∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt a) (𝓝 c)) →
    Tendsto
      (fun (a : Set α) ↦
        HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
      (v.filterAt a) (𝓝 (v.limRatio ρ a))
Tactic: filter_upwards [v.ae_tendsto_div hρ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
⊢ ∃ (a : Set α) (b : Set α),
    MeasurableSet a ∧
      MeasurableSet b ∧
        {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ a ∧
          {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧
            Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (a ∩ b) : ℝ≥0∞) (0 : ℝ≥0∞)
have refine_1 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n))
Tactic: refine
    ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
      toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
refine_1 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n))
⊢ ∃ (a : Set α) (b : Set α),
    MeasurableSet a ∧
      MeasurableSet b ∧
        {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ a ∧
          {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧
            Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (a ∩ b) : ℝ≥0∞) (0 : ℝ≥0∞)
have refine_2 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))
Tactic: refine
    ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
      toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
refine_1 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n))
refine_2 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))
⊢ ∃ (a : Set α) (b : Set α),
    MeasurableSet a ∧
      MeasurableSet b ∧
        {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ a ∧
          {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧
            Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (a ∩ b) : ℝ≥0∞) (0 : ℝ≥0∞)
have refine_3 : {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
Tactic: refine
    ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
      toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
refine_1 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n))
refine_2 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))
refine_3 : {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
⊢ ∃ (a : Set α) (b : Set α),
    MeasurableSet a ∧
      MeasurableSet b ∧
        {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ a ∧
          {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧
            Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (a ∩ b) : ℝ≥0∞) (0 : ℝ≥0∞)
have refine_4 : {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
Tactic: refine
    ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
      toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
refine_1 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n))
refine_2 : MeasurableSet (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))
refine_3 : {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
refine_4 : {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
⊢ ∃ (a : Set α) (b : Set α),
    MeasurableSet a ∧
      MeasurableSet b ∧
        {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ⊆ a ∧
          {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ⊆ b ∧
            Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (a ∩ b) : ℝ≥0∞) (0 : ℝ≥0∞)
have refine_5 : Eq (α := ℝ≥0∞)
  ((μ : (a : Set α) → ℝ≥0∞)
      ((toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
        (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))) :
    ℝ≥0∞)
  (0 : ℝ≥0∞)
Tactic: refine
    ⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
      toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} x
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
have pos : ∀ (h : x ∈ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
Tactic: by_cases h : x ∈ s