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
    

Proof state:
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
⊢ ∃ 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 g : α → Set ℝ := fun x => {r | 0 < r ∧ closedBall x r ∈ f 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}
    

Proof state:
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}
⊢ ∃ 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:
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⟩⟩
/-- 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
      

Proof state:
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}
⊢ ∀ x ∈ s, ∀ δ > 0, Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
intro x xs δ δpos
/-- 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
      

Proof state:
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
obtain ⟨t, tf, ht⟩ : ∃ (t : Set α), t ∈ f x ∧ t ⊆ closedBall x (δ / 2) :=
        ffine x xs (δ / 2) (half_pos δpos)
/-- 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)
      

Proof state:
case 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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
t : Set α
tf : t ∈ f x
ht : t ⊆ closedBall x (δ / 2)
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
obtain ⟨r, rpos, rfl⟩ : ∃ r : ℝ, 0 < r ∧ closedBall x r = t := by simpa using fsubset x xs tf
/-- 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 

Proof state:
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
t : Set α
tf : t ∈ f x
ht : t ⊆ closedBall x (δ / 2)
⊢ ∃ r, 0 < r ∧ closedBall x r = t

Next tactic:
simpa using fsubset x xs tf
/-- 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
      

Proof state:
case 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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
rpos : 0 < r
tf : closedBall x r ∈ f x
ht : closedBall x r ⊆ closedBall x (δ / 2)
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
rcases le_total r (δ / 2) with (H | H)
/-- 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)
      · 

Proof state:
case intro.intro.intro.intro.inl
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
rpos : 0 < r
tf : closedBall x r ∈ f x
ht : closedBall x r ⊆ closedBall x (δ / 2)
H : r ≤ δ / 2
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
exact ⟨r, ⟨rpos, tf⟩, ⟨rpos, H.trans_lt (half_lt_self δpos)⟩⟩
/-- 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)⟩⟩
      · 

Proof state:
case intro.intro.intro.intro.inr
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
rpos : 0 < r
tf : closedBall x r ∈ f x
ht : closedBall x r ⊆ closedBall x (δ / 2)
H : δ / 2 ≤ r
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
have : closedBall x r = closedBall x (δ / 2) :=
          Subset.antisymm ht (closedBall_subset_closedBall H)
/-- 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)
        

Proof state:
case intro.intro.intro.intro.inr
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
rpos : 0 < r
tf : closedBall x r ∈ f x
ht : closedBall x r ⊆ closedBall x (δ / 2)
H : δ / 2 ≤ r
this : closedBall x r = closedBall x (δ / 2)
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
rw [this] at tf
/-- 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
        

Proof state:
case intro.intro.intro.intro.inr
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}
x : α
xs : x ∈ s
δ : ℝ
δpos : δ > 0
r : ℝ
rpos : 0 < r
tf : closedBall x (δ / 2) ∈ f x
ht : closedBall x r ⊆ closedBall x (δ / 2)
H : δ / 2 ≤ r
this : closedBall x r = closedBall x (δ / 2)
⊢ Set.Nonempty (g x ∩ Ioo 0 δ)

Next tactic:
refine' ⟨δ / 2, ⟨half_pos δpos, tf⟩, ⟨half_pos δpos, half_lt_self δpos⟩⟩
/-- 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⟩⟩
    

Proof state:
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, (∀ 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:
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