awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.DensityTheorem (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : MeasurableSpace α
μ : Measure α
inst_2 : IsUnifLocDoublingMeasure μ
inst_3 : SecondCountableTopology α
inst_4 : BorelSpace α
inst_5 : IsLocallyFiniteMeasure μ
E : Type u_2
inst_6 : NormedAddCommGroup E
inst_7 : NormedSpace ℝ E
inst_8 : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) →
        (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
          Tendsto (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x))
have h : ∀ (a : α),
  Membership.mem (γ := Set α)
    {x : α |
      Membership.mem (γ := Set α)
          {x : α |
            (fun (x : α) ↦
                Tendsto (fun (a : Set α) ↦ ⨍ (y : α) in a, f y ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x)
                  (𝓝 (f x)))
              x}
          x →
        Membership.mem (γ := Set α)
          {x : α |
            (fun (x : α) ↦
                ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
                  Tendsto δ l (𝓝[>] 0) →
                    (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
                      Tendsto (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x)))
              x}
          x}
    a
Tactic: filter_upwards [(vitaliFamily μ K).ae_tendsto_average hf] with x hx ι l w δ δlim xmem using
    hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.DensityTheorem (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : MeasurableSpace α
μ : Measure α
inst_2 : IsUnifLocDoublingMeasure μ
inst_3 : SecondCountableTopology α
inst_4 : BorelSpace α
inst_5 : IsLocallyFiniteMeasure μ
E : Type u_2
inst_6 : NormedAddCommGroup E
inst_7 : NormedSpace ℝ E
inst_8 : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
h :
  ∀ (a : α),
    Membership.mem (γ := Set α)
      {x : α |
        Membership.mem (γ := Set α)
            {x : α |
              (fun (x : α) ↦
                  Tendsto (fun (a : Set α) ↦ ⨍ (y : α) in a, f y ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x)
                    (𝓝 (f x)))
                x}
            x →
          Membership.mem (γ := Set α)
            {x : α |
              (fun (x : α) ↦
                  ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
                    Tendsto δ l (𝓝[>] 0) →
                      (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
                        Tendsto (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x)))
                x}
            x}
      a
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) →
        (∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)) →
          Tendsto (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f x))
have h₁ : ∀ (a : α),
  Tendsto (fun (a : Set α) ↦ ⨍ (y : α) in a, f y ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) a) (𝓝 (f a)) →
    ∀ {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) →
        (∀ᶠ (j : ι) in l, a ∈ closedBall (w j) (K * δ j)) →
          Tendsto (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), f y ∂μ) l (𝓝 (f a))
Tactic: filter_upwards [(vitaliFamily μ K).ae_tendsto_average hf] with x hx ι l w δ δlim xmem using
    hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)
-- 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
⊢ ∀ᵐ (x : α) ∂μ, ∀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∞)
have hs : let s : Set α :=
  {x : α | ¬∀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∞)};
s = {x : α | ¬∀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∞)}
Tactic: set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ¬∀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∞)}
⊢ ∀ᵐ (x : α) ∂μ, ∀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∞)
have hs₁ : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
Tactic: simp -zeta only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
Cleared hypotheses: hs
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
⊢ ∀ᵐ (x : α) ∂μ, ∀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∞)
have change : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: change μ s = 0
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
x : α
hx : x ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε
have rw : ∀
  (hx :
    Membership.mem (γ := Set α)
      {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)} x)
  (ε : ℝ) (εpos : ε > 0), ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε
Tactic: rw [hs] at hx
Cleared hypotheses: hx, ε, εpos
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
x : α
hx :
  Membership.mem (γ := Set α)
    {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)} x
ε : ℝ
εpos : ε > 0
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε
have hx₁ : ∀ (ε : ℝ), 0 < ε → ∃ t ∈ v.setsAt x, t ⊆ closedBall x ε ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: simp only [frequently_filterAt_iff, gt_iff_lt, mem_setOf_eq] at hx
Cleared hypotheses: hx
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
x : α
ε : ℝ
εpos : ε > 0
hx :
  ∀ (ε : ℝ),
    0 < ε → ∃ t ∈ v.setsAt x, t ⊆ closedBall x ε ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε
have rcases : ∀ (a : Set α) (a_sets : a ∈ v.setsAt x) (ax : a ⊆ closedBall x ε)
  (μa : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)), ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε
Tactic: rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
have h : v.FineSubfamilyOn f s
Tactic: have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
h : v.FineSubfamilyOn f s
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
have refine : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: refine le_antisymm ?_ bot_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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
h : v.FineSubfamilyOn f s
⊢ Eq (α := ℝ≥0∞)
    (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)),
      (μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)))
    (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)), 0)
have e_f : Eq.{u_1 + 1} (α := (↑(h.index (f := f) (s := s)) : Type u_1) → ℝ≥0∞)
  (fun (x : (↑(h.index (f := f) (s := s)) : Type u_1)) ↦
    (μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)))
  fun (x : (↑(h.index (f := f) (s := s)) : Type u_1)) ↦ 0
Tactic: congr
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
h : v.FineSubfamilyOn f s
⊢ Eq.{u_1 + 1} (α := (↑(h.index (f := f) (s := s)) : Type u_1) → ℝ≥0∞)
    (fun (x : (↑(h.index (f := f) (s := s)) : Type u_1)) ↦
      (μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)))
    fun (x : (↑(h.index (f := f) (s := s)) : Type u_1)) ↦ 0
have e_f : ∀ (x : (↑(h.index (f := f) (s := s)) : Type u_1)),
  Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)) : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: ext1 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
h : v.FineSubfamilyOn f s
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
have μ₁ : Eq (α := ℝ≥0∞)
  (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)),
    (μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)))
  (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)), 0)
Tactic: calc
    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
    _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2
    _ = 0 := by simp only [tsum_zero]
-- 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
s : Set α := {x : α | ¬∀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∞)}
hs : s = {x : α | ∃ᶠ (x : Set α) in v.filterAt x, Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) x : ℝ≥0∞) (0 : ℝ≥0∞)}
f : α → Set (Set α) := fun (x : α) ↦ {a : Set α | Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞)}
h : v.FineSubfamilyOn f s
μ₁ :
  Eq (α := ℝ≥0∞)
    (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)),
      (μ : (a : Set α) → ℝ≥0∞) (h.covering (f := f) (s := s) (↑x : α × Set α)))
    (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)), 0)
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
have μ₂ : Eq (α := ℝ≥0∞) (∑' (x : (↑(h.index (f := f) (s := s)) : Type u_1)), 0) 0
Tactic: calc
    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
    _ = ∑' x : h.index, 0 := by congr; ext1 x; exact h.covering_mem x.2
    _ = 0 := by simp only [tsum_zero]
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
have apply : ∀ (ε : ℝ≥0) (εpos : 0 < ε),
  LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞) →
    (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: apply ENNReal.le_of_forall_pos_le_add fun ε εpos _ => ?_
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
have obtain : ∀ (U : Set α) (sU : s ⊆ U) (U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U)
  (νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)),
  (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: obtain ⟨U, sU, U_open, νU⟩ : ∃ (U : Set α), s ⊆ U ∧ IsOpen U ∧ ν U ≤ ν s + ε :=
    exists_isOpen_le_add s ν (ENNReal.coe_pos.2 εpos).ne'
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
f : α → Set (Set α) :=
  fun (x : α) ↦
    {a : Set α | LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ∧ a ⊆ U}
⊢ v.FineSubfamilyOn f s
have apply : ∀ (x : α) (hx : x ∈ s), ∃ᶠ (t : Set α) in v.filterAt x, t ∈ f x
Tactic: apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
f : α → Set (Set α) :=
  fun (x : α) ↦
    {a : Set α | LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ∧ a ⊆ U}
x : α
hx : x ∈ s
⊢ ∃ᶠ (t : Set α) in v.filterAt x, t ∈ f x
have this : ∃ᶠ (x_1 : Set α) in v.filterAt x,
  LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ∧
    x_1 ∈ v.setsAt x ∧ x_1 ⊆ U
Tactic: have :=
      (hs x hx).and_eventually
        ((v.eventually_filterAt_mem_setsAt x).and
          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
f : α → Set (Set α) :=
  fun (x : α) ↦
    {a : Set α | LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ∧ a ⊆ U}
x : α
hx : x ∈ s
this :
  ∃ᶠ (x_1 : Set α) in v.filterAt x,
    LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ∧
      x_1 ∈ v.setsAt x ∧ x_1 ⊆ U
⊢ ∃ᶠ (t : Set α) in v.filterAt x, t ∈ f x
have apply : ∀ (x_1 : Set α),
  LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) x_1 : ℝ≥0∞) ∧
      x_1 ∈ v.setsAt x ∧ x_1 ⊆ U →
    x_1 ∈ f x
Tactic: apply Frequently.mono this
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
f : α → Set (Set α) :=
  fun (x : α) ↦
    {a : Set α | LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ∧ a ⊆ U}
⊢ (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
have h : v.FineSubfamilyOn f s
Tactic: have h : v.FineSubfamilyOn f s := by
    apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_
    have :=
      (hs x hx).and_eventually
        ((v.eventually_filterAt_mem_setsAt x).and
          (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))
    apply Frequently.mono this
    rintro a ⟨ρa, _, aU⟩
    exact ⟨ρa, aU⟩
-- 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
ρ ν : Measure α
inst_3 : IsLocallyFiniteMeasure ν
hρ : ρ ≪ μ
s : Set α
hs :
  ∀ x ∈ s,
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
ε : ℝ≥0
εpos : 0 < ε
h_1 : LT.lt (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
sU : s ⊆ U
U_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
νU : (ν : (a : Set α) → ℝ≥0∞) U ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
f : α → Set (Set α) :=
  fun (x : α) ↦
    {a : Set α | LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ∧ a ⊆ U}
h : v.FineSubfamilyOn f s
⊢ (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
have haveI : ∀ (this : Encodable (↑(h.index (f := f) (s := s)) : Type u_1)),
  (ρ : (a : Set α) → ℝ≥0∞) s ≤ HAdd.hAdd (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (↑ε : ℝ≥0∞)
Tactic: haveI : Encodable h.index := h.index_countable.toEncodable