l3lab/ntp-mathlib huggingface dataset:
instance instDistribLattice : DistribLattice (Booleanisation α) where
  inf_le_left _ _ := inf_le_left
  inf_le_right _ _ := inf_le_right
  le_inf _ _ _ := le_inf
  le_sup_inf x y z := match x, y, z with
    | lift a, lift b, lift c => LE.lift le_sup_inf
    | lift a, lift b, comp c => LE.lift $ by simp [sup_left_comm, sup_comm]
    | lift a, comp b, lift c => LE.lift $ by simp [sup_left_comm, sup_comm (a := b \ a)]
    | lift a, comp b, comp c => LE.comp $ by rw [sup_sdiff]
    | comp a, lift b, lift c => LE.comp $ by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y z : Booleanisation α
a b c : α
⊢ a \ (b ⊓ c) ≤ a \ b ⊔ a \ c

Next tactic:
rw [sdiff_inf]
instance instDistribLattice : DistribLattice (Booleanisation α) where
  inf_le_left _ _ := inf_le_left
  inf_le_right _ _ := inf_le_right
  le_inf _ _ _ := le_inf
  le_sup_inf x y z := match x, y, z with
    | lift a, lift b, lift c => LE.lift le_sup_inf
    | lift a, lift b, comp c => LE.lift $ by simp [sup_left_comm, sup_comm]
    | lift a, comp b, lift c => LE.lift $ by simp [sup_left_comm, sup_comm (a := b \ a)]
    | lift a, comp b, comp c => LE.comp $ by rw [sup_sdiff]
    | comp a, lift b, lift c => LE.comp $ by rw [sdiff_inf]
    | comp a, lift b, comp c => LE.comp $ by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y z : Booleanisation α
a b c : α
⊢ a \ (b \ c) ≤ a \ b ⊔ a ⊓ c

Next tactic:
rw [sdiff_sdiff_right']
instance instDistribLattice : DistribLattice (Booleanisation α) where
  inf_le_left _ _ := inf_le_left
  inf_le_right _ _ := inf_le_right
  le_inf _ _ _ := le_inf
  le_sup_inf x y z := match x, y, z with
    | lift a, lift b, lift c => LE.lift le_sup_inf
    | lift a, lift b, comp c => LE.lift $ by simp [sup_left_comm, sup_comm]
    | lift a, comp b, lift c => LE.lift $ by simp [sup_left_comm, sup_comm (a := b \ a)]
    | lift a, comp b, comp c => LE.comp $ by rw [sup_sdiff]
    | comp a, lift b, lift c => LE.comp $ by rw [sdiff_inf]
    | comp a, lift b, comp c => LE.comp $ by rw [sdiff_sdiff_right']
    | comp a, comp b, lift c => LE.comp $ by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y z : Booleanisation α
a b c : α
⊢ a \ (c \ b) ≤ a ⊓ b ⊔ a \ c

Next tactic:
rw [sdiff_sdiff_right', sup_comm]
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y : Booleanisation α
a✝ b : α
x : Booleanisation α
a : α
⊢ lift a ⊓ (lift a)ᶜ ≤ ⊥

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y : Booleanisation α
a✝ b : α
x : Booleanisation α
a : α
⊢ comp a ⊓ (comp a)ᶜ ≤ ⊥

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y : Booleanisation α
a✝ b : α
x : Booleanisation α
a : α
⊢ ⊤ ≤ lift a ⊔ (lift a)ᶜ

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by simp
    | comp a => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y : Booleanisation α
a✝ b : α
x : Booleanisation α
a : α
⊢ ⊤ ≤ comp a ⊔ (comp a)ᶜ

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by simp
    | comp a => by simp
  sdiff_eq x y := match x, y with
    | lift a, lift b => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y : Booleanisation α
a b : α
⊢ lift a \ lift b = lift a ⊓ (lift b)ᶜ

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by simp
    | comp a => by simp
  sdiff_eq x y := match x, y with
    | lift a, lift b => by simp
    | lift a, comp b => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y : Booleanisation α
a b : α
⊢ lift a \ comp b = lift a ⊓ (comp b)ᶜ

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by simp
    | comp a => by simp
  sdiff_eq x y := match x, y with
    | lift a, lift b => by simp
    | lift a, comp b => by simp
    | comp a, lift b => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y : Booleanisation α
a b : α
⊢ comp a \ lift b = comp a ⊓ (lift b)ᶜ

Next tactic:
simp
instance instBooleanAlgebra : BooleanAlgebra (Booleanisation α) where
  le_top _ := le_top
  bot_le _ := bot_le
  inf_compl_le_bot x := match x with
    | lift a => by simp
    | comp a => by simp
  top_le_sup_compl x := match x with
    | lift a => by simp
    | comp a => by simp
  sdiff_eq x y := match x, y with
    | lift a, lift b => by simp
    | lift a, comp b => by simp
    | comp a, lift b => by simp
    | comp a, comp b => by 

Proof state:
α : Type u_1
inst✝ : GeneralizedBooleanAlgebra α
x✝ y✝ : Booleanisation α
a✝ b✝ : α
x y : Booleanisation α
a b : α
⊢ comp a \ comp b = comp a ⊓ (comp b)ᶜ

Next tactic:
simp
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a

Next tactic:
set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a

Next tactic:
simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
⊢ ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a

Next tactic:
change μ s = 0
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
⊢ ↑↑μ s = 0

Next tactic:
let f : α → Set (Set α) := fun _ => {a | μ a = 0}
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
⊢ ↑↑μ s = 0

Next tactic:
have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
⊢ FineSubfamilyOn v f s

Next tactic:
intro x hx ε εpos
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
x : α
hx : x ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ a ∈ setsAt v x ∩ f x, a ⊆ closedBall x ε

Next tactic:
rw [hs] at hx
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
x : α
hx : x ∈ {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
ε : ℝ
εpos : ε > 0
⊢ ∃ a ∈ setsAt v x ∩ f x, a ⊆ closedBall x ε

Next tactic:
simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
x : α
ε : ℝ
εpos : ε > 0
hx : ∀ (ε : ℝ), 0 < ε → ∃ a ∈ setsAt v x, a ⊆ closedBall x ε ∧ ↑↑μ a = 0
⊢ ∃ a ∈ setsAt v x ∩ f x, a ⊆ closedBall x ε

Next tactic:
rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    

Proof state:
case intro.intro.intro
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
x : α
ε : ℝ
εpos : ε > 0
hx : ∀ (ε : ℝ), 0 < ε → ∃ a ∈ setsAt v x, a ⊆ closedBall x ε ∧ ↑↑μ a = 0
a : Set α
a_sets : a ∈ setsAt v x
ax : a ⊆ closedBall x ε
μa : ↑↑μ a = 0
⊢ ∃ a ∈ setsAt v x ∩ f x, a ⊆ closedBall x ε

Next tactic:
exact ⟨a, ⟨a_sets, μa⟩, ax⟩
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
h : FineSubfamilyOn v f s
⊢ ↑↑μ s = 0

Next tactic:
refine' le_antisymm _ bot_le
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
  refine' le_antisymm _ bot_le
  

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
h : FineSubfamilyOn v f s
⊢ ↑↑μ s ≤ 0

Next 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, add_zero]
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
  refine' le_antisymm _ bot_le
  calc
    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
    _ = ∑' x : h.index, 0 := by 

Proof state:
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
h : FineSubfamilyOn v f s
⊢ ∑' (x : ↑(FineSubfamilyOn.index h)), ↑↑μ (FineSubfamilyOn.covering h ↑x) = ∑' (x : ↑(FineSubfamilyOn.index h)), 0

Next tactic:
congr
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
  refine' le_antisymm _ bot_le
  calc
    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
    _ = ∑' x : h.index, 0 := by congr; 

Proof state:
case e_f
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
h : FineSubfamilyOn v f s
⊢ (fun x => ↑↑μ (FineSubfamilyOn.covering h ↑x)) = fun x => 0

Next tactic:
ext1 x
/-- For almost every point `x`, sufficiently small sets in a Vitali family around `x` have positive
measure. (This is a nontrivial result, following from the covering property of Vitali families). -/
theorem ae_eventually_measure_pos [SecondCountableTopology α] :
    ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, 0 < μ a := by
  set s := {x | ¬∀ᶠ a in v.filterAt x, 0 < μ a} with hs
  simp (config := { zeta := false }) only [not_lt, not_eventually, nonpos_iff_eq_zero] at hs
  change μ s = 0
  let f : α → Set (Set α) := fun _ => {a | μ a = 0}
  have h : v.FineSubfamilyOn f s := by
    intro x hx ε εpos
    rw [hs] at hx
    simp only [frequently_filterAt_iff, exists_prop, gt_iff_lt, mem_setOf_eq] at hx
    rcases hx ε εpos with ⟨a, a_sets, ax, μa⟩
    exact ⟨a, ⟨a_sets, μa⟩, ax⟩
  refine' le_antisymm _ bot_le
  calc
    μ s ≤ ∑' x : h.index, μ (h.covering x) := h.measure_le_tsum
    _ = ∑' x : h.index, 0 := by congr; ext1 x; 

Proof state:
case e_f.h
α : Type u_1
inst✝² : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : SecondCountableTopology α
s : Set α := {x | ¬∀ᶠ (a : Set α) in filterAt v x, 0 < ↑↑μ a}
hs : s = {x | ∃ᶠ (x : Set α) in filterAt v x, ↑↑μ x = 0}
f : α → Set (Set α) := fun x => {a | ↑↑μ a = 0}
h : FineSubfamilyOn v f s
x : ↑(FineSubfamilyOn.index h)
⊢ ↑↑μ (FineSubfamilyOn.covering h ↑x) = 0

Next tactic:
exact h.covering_mem x.2