l3lab/ntp-mathlib huggingface dataset:
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
⊢ ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (PairwiseDisjoint t fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ ↑↑μ (s \ ⋃ p ∈ t, p.2) = 0

Next tactic:
let F : α → α × Set α := fun x => (x, closedBall x (r x))
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    

Proof state:
case intro.intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
⊢ ∃ t, (∀ p ∈ t, p.1 ∈ s) ∧ (PairwiseDisjoint t fun p => p.2) ∧ (∀ p ∈ t, p.2 ∈ f p.1) ∧ ↑↑μ (s \ ⋃ p ∈ t, p.2) = 0

Next tactic:
refine' ⟨F '' t, _, _, _, _⟩
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
⊢ ∀ p ∈ F '' t, p.1 ∈ s

Next tactic:
rintro - ⟨x, hx, rfl⟩
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · rintro - ⟨x, hx, rfl⟩; 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_1.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
x : α
hx : x ∈ t
⊢ (F x).1 ∈ s

Next tactic:
exact ts hx
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · rintro - ⟨x, hx, rfl⟩; exact ts hx
    · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_2
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
⊢ PairwiseDisjoint (F '' t) fun p => p.2

Next tactic:
rintro p ⟨x, hx, rfl⟩ q ⟨y, hy, rfl⟩ hxy
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · rintro - ⟨x, hx, rfl⟩; exact ts hx
    · rintro p ⟨x, hx, rfl⟩ q ⟨y, hy, rfl⟩ hxy
      

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_2.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
x : α
hx : x ∈ t
y : α
hy : y ∈ t
hxy : F x ≠ F y
⊢ (Disjoint on fun p => p.2) (F x) (F y)

Next tactic:
exact tdisj hx hy (ne_of_apply_ne F hxy)
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · rintro - ⟨x, hx, rfl⟩; exact ts hx
    · rintro p ⟨x, hx, rfl⟩ q ⟨y, hy, rfl⟩ hxy
      exact tdisj hx hy (ne_of_apply_ne F hxy)
    · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_3
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
⊢ ∀ p ∈ F '' t, p.2 ∈ f p.1

Next tactic:
rintro - ⟨x, hx, rfl⟩
/-- In a space with the Besicovitch covering property, the set of closed balls with positive radius
forms a Vitali family. This is essentially a restatement of the measurable Besicovitch theorem. -/
protected def vitaliFamily (μ : Measure α) [SigmaFinite μ] : VitaliFamily μ where
  setsAt x := (fun r : ℝ => closedBall x r) '' Ioi (0 : ℝ)
  MeasurableSet' := by
    intro x y hy
    obtain ⟨r, _, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    exact isClosed_ball.measurableSet
  nonempty_interior := by
    intro x y hy
    obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = y := by
      simpa only [mem_image, mem_Ioi] using hy
    simp only [Nonempty.mono ball_subset_interior_closedBall, rpos, nonempty_ball]
  Nontrivial x ε εpos := ⟨closedBall x ε, mem_image_of_mem _ εpos, Subset.refl _⟩
  covering := by
    intro s f fsubset ffine
    let g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
    have A : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty := by
      intro x xs δ δpos
      obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
      obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
      rcases le_total r (δ / 2) with (H | H)
      · exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
      · have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
        rw [this] at tf
        refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
    obtain ⟨t, r, _, ts, tg, μt, tdisj⟩ :
      ∃ (t : Set α) (r : α → ℝ),
        t.Countable ∧
          t ⊆ s ∧
            (∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1) ∧
              μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧
                t.PairwiseDisjoint fun x => closedBall x (r x) :=
      exists_disjoint_closedBall_covering_ae μ g s A (fun _ => 1) fun _ _ => zero_lt_one
    let F : α → α × Set α := fun x => (x, closedBall x (r x))
    refine' ⟨F '' t, _, _, _, _⟩
    · rintro - ⟨x, hx, rfl⟩; exact ts hx
    · rintro p ⟨x, hx, rfl⟩ q ⟨y, hy, rfl⟩ hxy
      exact tdisj hx hy (ne_of_apply_ne F hxy)
    · rintro - ⟨x, hx, rfl⟩; 

Proof state:
case intro.intro.intro.intro.intro.intro.refine'_3.intro.intro
α : Type u_1
inst✝⁵ : MetricSpace α
β : Type u
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : HasBesicovitchCovering α
μ : Measure α
inst✝ : SigmaFinite μ
s : Set α
f : α → Set (Set α)
fsubset : ∀ x ∈ s, f x ⊆ (fun x => (fun r => closedBall x r) '' Ioi 0) x
ffine : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ f x, a ⊆ closedBall x ε
g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f x}
A : ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)
t : Set α
r : α → ℝ
left✝ : Set.Countable t
ts : t ⊆ s
tg : ∀ x ∈ t, r x ∈ g x ∩ Ioo 0 1
μt : ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) = 0
tdisj : PairwiseDisjoint t fun x => closedBall x (r x)
F : α → α × Set α := fun x => (x, closedBall x (r x))
x : α
hx : x ∈ t
⊢ (F x).2 ∈ f (F x).1

Next tactic:
exact (tg x hx).1.2