awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
S : Set α
K : ℝ
x : α
hx :
  Tendsto (β := ENNReal)
    (fun (a : Set α) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal) ((μ : (a : Set α) → ENNReal) (S ∩ a) : ENNReal)
        ((μ : (a : Set α) → ENNReal) a : ENNReal))
    (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 1)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
⊢ Tendsto (β := ENNReal)
    (fun (j : ι) ↦
      HDiv.hDiv (α := ENNReal) (β := ENNReal) ((μ : (a : Set α) → ENNReal) (S ∩ closedBall (w j) (δ j)) : ENNReal)
        ((μ : (a : Set α) → ENNReal) (closedBall (w j) (δ j)) : ENNReal))
    l (𝓝 1)

Tactic:
hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Premises used: IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt, Filter.Tendsto.comp
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
S : Set α
K : ℝ
h :
  ∀ (a : α),
    Membership.mem (γ := Set α)
      {x : α |
        Membership.mem (γ := Set α)
            {x : α |
              (fun (x : α) ↦
                  Tendsto (β := ENNReal)
                    (fun (a : Set α) ↦
                      HDiv.hDiv (α := ENNReal) (β := ENNReal) ((μ : (a : Set α) → ENNReal) (S ∩ a) : ENNReal)
                        ((μ : (a : Set α) → ENNReal) a : ENNReal))
                    (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 1))
                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 (β := ENNReal)
                          (fun (j : ι) ↦
                            HDiv.hDiv (α := ENNReal) (β := ENNReal)
                              ((μ : (a : Set α) → ENNReal) (S ∩ closedBall (w j) (δ j)) : ENNReal)
                              ((μ : (a : Set α) → ENNReal) (closedBall (w j) (δ j)) : ENNReal))
                          l (𝓝 1))
                x}
            x}
      a
h₁ h₂ :
  ∀ (x : α)
    (hx :
      Tendsto (β := ENNReal)
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ENNReal) (β := ENNReal) ((μ : (a : Set α) → ENNReal) (S ∩ a) : ENNReal)
            ((μ : (a : Set α) → ENNReal) a : ENNReal))
        (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 1))
    {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
    (xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)),
    Tendsto (β := ENNReal)
      (fun (j : ι) ↦
        HDiv.hDiv (α := ENNReal) (β := ENNReal) ((μ : (a : Set α) → ENNReal) (S ∩ closedBall (w j) (δ j)) : ENNReal)
          ((μ : (a : Set α) → ENNReal) (closedBall (w j) (δ j)) : ENNReal))
      l (𝓝 1)
⊢ ∀ᵐ (x : α) ∂μ.restrict S,
    ∀ {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 (β := ENNReal)
            (fun (j : ι) ↦
              HDiv.hDiv (α := ENNReal) (β := ENNReal)
                ((μ : (a : Set α) → ENNReal) (S ∩ closedBall (w j) (δ j)) : ENNReal)
                ((μ : (a : Set α) → ENNReal) (closedBall (w j) (δ j)) : ENNReal))
            l (𝓝 1)

Tactic:
filter_upwards [(vitaliFamily μ K).ae_tendsto_measure_inter_div S] with x hx ι l w δ δlim
    xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Premises used: IsUnifLocDoublingMeasure.vitaliFamily, VitaliFamily.ae_tendsto_measure_inter_div
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
f : α → E
hf : LocallyIntegrable f μ
K : ℝ
x : α
hx :
  Tendsto (β := ℝ) (fun (a : Set α) ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 0)
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)
⊢ Tendsto (β := ℝ) (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)

Tactic:
hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Premises used: Set.Mem, setOf, MeasureTheory.average, IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt, Filter.Tendsto.comp
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
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 (β := ℝ) (fun (a : Set α) ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ)
                    (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 0))
                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 (β := ℝ) (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0))
                x}
            x}
      a
h₁ h₂ :
  ∀ (x : α)
    (hx :
      Tendsto (β := ℝ) (fun (a : Set α) ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x)
        (𝓝 0))
    {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
    (xmem : ∀ᶠ (j : ι) in l, x ∈ closedBall (w j) (K * δ j)),
    Tendsto (β := ℝ) (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)
⊢ ∀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 (β := ℝ) (fun (j : ι) ↦ ⨍ (y : α) in closedBall (w j) (δ j), ‖f y - f x‖ ∂μ) l (𝓝 0)

Tactic:
filter_upwards [(vitaliFamily μ K).ae_tendsto_average_norm_sub hf] with x hx ι l w δ δlim
    xmem using hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Premises used: IsUnifLocDoublingMeasure.vitaliFamily, VitaliFamily.ae_tendsto_average_norm_sub
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
x : α
hx : Tendsto (fun (a : Set α) ↦ ⨍ (y : α) in a, f y ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 (f x))
ι : Type u_3
l : Filter ι
w : ι → α
δ : ι → ℝ
δlim : Tendsto δ l (𝓝[>] 0)
xmem : ∀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))

Tactic:
hx.comp (tendsto_closedBall_filterAt μ _ _ δlim xmem)

Premises used: Set.Mem, setOf, MeasureTheory.average, IsUnifLocDoublingMeasure.tendsto_closedBall_filterAt, Filter.Tendsto.comp
-- Module: Mathlib.MeasureTheory.Covering.DensityTheorem
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
h₁ h₂ :
  ∀ (x : α)
    (hx : Tendsto (fun (a : Set α) ↦ ⨍ (y : α) in a, f y ∂μ) (VitaliFamily.filterAt (vitaliFamily μ K) x) (𝓝 (f x)))
    {ι : Type u_3} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (δlim : Tendsto δ l (𝓝[>] 0))
    (xmem : ∀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))
⊢ ∀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))

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)

Premises used: IsUnifLocDoublingMeasure.vitaliFamily, VitaliFamily.ae_tendsto_average
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
⊢ 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
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)}
⊢ 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

Premises used: not_lt, Filter.not_eventually, nonpos_iff_eq_zero
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)}
change : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : 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∞)

Tactic:
change μ s = 0

Premises used: DFunLike.coe
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
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
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Tactic:
rw [hs] at hx
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
⊢ ∀ (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

Premises used: Set.mem_setOf_eq, VitaliFamily.frequently_filterAt_iff, gt_iff_lt
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
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
⊢ ∃ t ∈ v.setsAt x ∩ f x, t ⊆ closedBall x ε

Tactic:
rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
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:
⟨a, ⟨a_sets, μa⟩, ax⟩

Premises used: And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
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:
exact ⟨a, ⟨a_sets, μa⟩, ax⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)}
⊢ 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⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)

Tactic:
le_antisymm ?_ bot_le

Premises used: le_antisymm, bot_le
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
refine : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
refine le_antisymm ?_ bot_le

Premises used: le_antisymm, bot_le
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
    (∑' (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)) :
      ℝ≥0∞)

Tactic:
h.measure_le_tsum

Premises used: VitaliFamily.FineSubfamilyOn.measure_le_tsum
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
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
⊢ 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:
congr
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
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∞)
⊢ 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:
ext1 x
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
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:
h.covering_mem x.2

Premises used: Set.Mem, setOf, VitaliFamily.FineSubfamilyOn.covering_mem, Subtype.property
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
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:
exact h.covering_mem x.2

Premises used: VitaliFamily.FineSubfamilyOn.covering_mem, Subtype.property
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)), 0) 0

Tactic:
simp only [tsum_zero]

Premises used: tsum_zero