l3lab/ntp-mathlib huggingface dataset:
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      

Proof state:
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∀ a ∈ t, r a < 0
a : ι
ha : a ∈ t
b : ι
x✝¹ : b ∈ t
x✝ : a ≠ b
⊢ (Disjoint on fun a => closedBall (x a) (r a)) a b

Next tactic:
simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint]
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by 

Proof state:
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∀ a ∈ t, r a < 0
a : ι
ha : a ∈ t
⊢ closedBall (x a) (r a) ⊆ closedBall (x a) (5 * r a)

Next tactic:
simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  

Proof state:
case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ¬∀ a ∈ t, r a < 0
⊢ ∃ u,
    ∃ (_ : u ⊆ t),
      (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
push_neg at ht
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  

Proof state:
case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
⊢ ∃ u,
    ∃ (_ : u ⊆ t),
      (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
let t' := { a ∈ t | 0 ≤ r a }
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  

Proof state:
case neg
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
⊢ ∃ u,
    ∃ (_ : u ⊆ t),
      (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  

Proof state:
case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
⊢ ∃ u,
    ∃ (_ : u ⊆ t),
      (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine' ⟨b, bu, _⟩
    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
    apply closedBall_subset_closedBall'
    linarith
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    

Proof state:
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
⊢ ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
intro a ha
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    

Proof state:
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
a : ι
ha : a ∈ t'
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
rcases hu a ha with ⟨b, bu, hb, rb⟩
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    

Proof state:
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))
rb : r a ≤ 2 * r b
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
refine' ⟨b, bu, _⟩
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine' ⟨b, bu, _⟩
    

Proof state:
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))
rb : r a ≤ 2 * r b
⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine' ⟨b, bu, _⟩
    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
    

Proof state:
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))
rb : r a ≤ 2 * r b
this : dist (x a) (x b) ≤ r a + r b
⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
apply closedBall_subset_closedBall'
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine' ⟨b, bu, _⟩
    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
    apply closedBall_subset_closedBall'
    

Proof state:
case intro.intro.intro.h
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))
rb : r a ≤ 2 * r b
this : dist (x a) (x b) ≤ r a + r b
⊢ r a + dist (x a) (x b) ≤ 5 * r b

Next tactic:
linarith
/-- Vitali covering theorem, closed balls version: given a family `t` of closed balls, one can
extract a disjoint subfamily `u ⊆ t` so that all balls in `t` are covered by the 5-times
dilations of balls in `u`. -/
theorem exists_disjoint_subfamily_covering_enlargment_closedBall [MetricSpace α] (t : Set ι)
    (x : ι → α) (r : ι → ℝ) (R : ℝ) (hr : ∀ a ∈ t, r a ≤ R) :
    ∃ (u : _) (_ : u ⊆ t),
      (u.PairwiseDisjoint fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
  rcases eq_empty_or_nonempty t with (rfl | _)
  · exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
  by_cases ht : ∀ a ∈ t, r a < 0
  · exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],
      fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩
  push_neg at ht
  let t' := { a ∈ t | 0 ≤ r a }
  rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2
      one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_closedBall_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩
  have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by
    intro a ha
    rcases hu a ha with ⟨b, bu, hb, rb⟩
    refine' ⟨b, bu, _⟩
    have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb
    apply closedBall_subset_closedBall'
    linarith
  

Proof state:
case neg.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝ : MetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
h✝ : Set.Nonempty t
ht : ∃ a ∈ t, 0 ≤ r a
t' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)
⊢ ∃ u,
    ∃ (_ : u ⊆ t),
      (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)

Next tactic:
refine' ⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => _⟩