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 ⟂ₘ μ
A :
  ∀ ε > 0,
    ∀ᵐ (x : α) ∂μ,
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
u : ℕ → ℝ≥0
left : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
x : α
hx :
  ∀ (n : ℕ),
    ∀ᶠ (a : Set α) in v.filterAt x,
      (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < (↑w : ℝ≥0∞)
w_lt : (↑w : ℝ≥0∞) < z
⊢ ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) < z
have h : ∀ (n : ℕ) (hn : u n < w),
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) < z
Tactic: obtain ⟨n, hn⟩ : ∃ n, u n < w := ((tendsto_order.1 u_lim).2 w (ENNReal.coe_pos.1 w_pos)).exists
-- 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 ⟂ₘ μ
A :
  ∀ ε > 0,
    ∀ᵐ (x : α) ∂μ,
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
u : ℕ → ℝ≥0
left : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
x : α
hx :
  ∀ (n : ℕ),
    ∀ᶠ (a : Set α) in v.filterAt x,
      (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < (↑w : ℝ≥0∞)
w_lt : (↑w : ℝ≥0∞) < z
n : ℕ
hn : u n < w
⊢ ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) < z
have h : ∀ (a : Set α),
  (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) →
    LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) →
      LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (∞ : 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∞) < z
Tactic: filter_upwards [hx n, h'x, v.eventually_measure_lt_top x]
-- 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 ⟂ₘ μ
A :
  ∀ ε > 0,
    ∀ᵐ (x : α) ∂μ,
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
u : ℕ → ℝ≥0
left : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
x : α
hx :
  ∀ (n : ℕ),
    ∀ᶠ (a : Set α) in v.filterAt x,
      (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < (↑w : ℝ≥0∞)
w_lt : (↑w : ℝ≥0∞) < z
n : ℕ
hn : u n < w
a : Set α
ha : (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
μa_pos : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
μa_lt_top : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (∞ : 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∞) < z
have h : HMul.hMul (β := ℝ≥0∞) (↑w : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
  HMul.hMul (β := ℝ≥0∞) z ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
Tactic: grw [ENNReal.div_lt_iff (.inl μa_pos.ne') (.inl μa_lt_top.ne), ha, hn]
-- 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 ⟂ₘ μ
A :
  ∀ ε > 0,
    ∀ᵐ (x : α) ∂μ,
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑ε : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
u : ℕ → ℝ≥0
left : StrictAnti u
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ (n : ℕ),
      ∀ᶠ (a : Set α) in v.filterAt x,
        (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
x : α
hx :
  ∀ (n : ℕ),
    ∀ᶠ (a : Set α) in v.filterAt x,
      (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
z : ℝ≥0∞
hz : z > 0
w : ℝ≥0
w_pos : 0 < (↑w : ℝ≥0∞)
w_lt : (↑w : ℝ≥0∞) < z
n : ℕ
hn : u n < w
a : Set α
ha : (ρ : (a : Set α) → ℝ≥0∞) a < HMul.hMul (β := ℝ≥0∞) (↑(u n) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
μa_pos : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
μa_lt_top : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑w : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) <
    HMul.hMul (β := ℝ≥0∞) z ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
have h : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: gcongr
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
have apply : ∀ x ∈ s, ∃ u ∈ 𝓝[s] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: apply measure_null_of_locally_null s fun x _ => ?_
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h : x ∈ s
⊢ ∃ u ∈ 𝓝[s] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
have obtain : ∀ (o : Set α) (xo : x ∈ o) (o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o)
  (μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)),
  ∃ u ∈ 𝓝[s] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: obtain ⟨o, xo, o_open, μo⟩ : ∃ o : Set α, x ∈ o ∧ IsOpen o ∧ μ o < ∞ :=
    Measure.exists_isOpen_measure_lt_top μ x
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∃ u ∈ 𝓝[s] x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) u : ℝ≥0∞) (0 : ℝ≥0∞)
have refine : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: refine ⟨s ∩ o, inter_mem_nhdsWithin _ (o_open.mem_nhds xo), ?_⟩
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s' : Set α := s ∩ o
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)
have by_contra : ∀ (h : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)), False
Tactic: by_contra h
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h_1 : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s' : Set α := s ∩ o
h : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ False
have apply : LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
Tactic: apply lt_irrefl (ρ s')
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h_1 : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s' : Set α := s ∩ o
h : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) <
    HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
have hinf : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: gcongr
-- 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
c d : ℝ≥0
hcd : c < d
s : Set α
hc :
  ∀ x ∈ s,
    ∃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∞)
hd :
  ∀ x ∈ s,
    ∃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
x : α
h_1 : x ∈ s
o : Set α
xo : x ∈ o
o_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] o
μo : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) (∞ : ℝ≥0∞)
s' : Set α := s ∩ o
h : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (s ∩ o) : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ LT.lt (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
have ρ₁ : HMul.hMul (β := ℝ≥0∞) (↑c : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞) <
  HMul.hMul (β := ℝ≥0∞) (↑d : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s' : ℝ≥0∞)
Tactic: calc
    ρ s' ≤ c * μ s' := v.measure_le_of_frequently_le (c • μ) hρ s' fun x hx => hc x hx.1
    _ < d * μ s' := by gcongr; exact measure_ne_top_of_subset inter_subset_right μo.ne
    _ ≤ ρ s' := v.measure_le_of_frequently_le ρ smul_absolutelyContinuous s' fun x hx ↦ hd x hx.1
-- 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,
    ∃ (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 obtain : ∀ (w : Set ℝ≥0∞) (w_count : w.Countable) (w_dense : Dense w),
  0 ∉ w →
    ∀ (w_top : ∞ ∉ w),
      ∀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)
Tactic: obtain ⟨w, w_count, w_dense, _, w_top⟩ :
    ∃ w : Set ℝ≥0∞, w.Countable ∧ Dense w ∧ 0 ∉ w ∧ ∞ ∉ w :=
    ENNReal.exists_countable_dense_no_zero_top
-- 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
⊢ ∀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 I : ∀ x ∈ w, x ≠ ∞
Tactic: have I : ∀ x ∈ w, x ≠ ∞ := fun x xs hx => w_top (hx ▸ xs)
-- 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 ∈ w
d : ℝ≥0∞
hd : d ∈ w
hcd : 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∞))
have lift : ∀ (c : ℝ≥0) (hc : (↑c : ℝ≥0∞) ∈ w) (hcd : (↑c : ℝ≥0∞) < 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 : ℝ≥0∞)) ∧
        ∃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: lift c to ℝ≥0 using I c hc
Cleared hypotheses: c, hc, hcd
-- 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 ≠ ∞
d : ℝ≥0∞
hd : d ∈ w
c : ℝ≥0
hc : (↑c : ℝ≥0∞) ∈ w
hcd : (↑c : ℝ≥0∞) < 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 : ℝ≥0∞)) ∧
        ∃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∞))
have lift : ∀ (d : ℝ≥0) (hd : (↑d : ℝ≥0∞) ∈ w) (hcd : (↑c : ℝ≥0∞) < (↑d : ℝ≥0∞)),
  ∀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 : 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∞))
Tactic: lift d to ℝ≥0 using I d hd
Cleared hypotheses: d, hd, hcd
-- 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∞)
⊢ ∀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 : 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∞))
have hc₁ : ∀ (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∞)
Tactic: apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd)
-- 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∞)
hc₁ :
  ∀ (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∞)
⊢ ∀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 : 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∞))
have hd₁ : ∀ (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
Tactic: apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd)