-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
inst_1 : MeasurableSpace α
inst_2 : OpensMeasurableSpace α
inst_3 : SecondCountableTopology α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB :
  ∀ a ∈ t,
    (μ : (a : Set α) → ℝ≥0∞) (closedBall (c a) (3 * r a)) ≤
      HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (B a) : ℝ≥0∞)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : (x : α) → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R x)) : ℝ≥0∞) (∞ : ℝ≥0∞)
t' : Set ι := {a : ι | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
h : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a : ι | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
⊢ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆
    ⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι))

Tactic:
have M : (s \ ⋃ a ∈ u, B a) ∩
      ball x (R x) ⊆ ⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a) := by
    intro z hz
    set k := ⋃ (a : v) (_ : a ∈ w), B a
    have k_closed : IsClosed k := isClosed_biUnion_finset fun i _ => h't _ (ut (vu i.2))
    have z_notmem_k : z ∉ k := by
      simp only [k, not_exists, exists_prop, mem_iUnion, forall_exists_index,
        SetCoe.exists, not_and, exists_and_right]
      intro b hbv _ h'z
      have : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ⋃ a ∈ u, B a :=
        mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)
      simpa only [diff_inter_self]
    -- since the elements of `w` are closed and finitely many, one can find a small ball around `z`
    -- not intersecting them
    have : ball x (R x) \ k ∈ 𝓝 z := by
      apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _
      exact (mem_diff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩
    obtain ⟨d, dpos, hd⟩ : ∃ d, 0 < d ∧ closedBall z d ⊆ ball x (R x) \ k :=
      nhds_basis_closedBall.mem_iff.1 this
    -- choose an element `a` of the family `t` contained in this small ball
    obtain ⟨a, hat, ad, rfl⟩ : ∃ a ∈ t, r a ≤ min d (R z) ∧ c a = z :=
      hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))
    have ax : B a ⊆ ball x (R x) := by
      refine (hB a hat).trans ?_
      refine Subset.trans ?_ (hd.trans Set.diff_subset)
      gcongr
      exact ad.trans (min_le_left _ _)
    -- it intersects an element `b` of `u` with comparable diameter, by definition of `u`
    obtain ⟨b, bu, ab, bdiam⟩ : ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b :=
      hu a ⟨hat, ad.trans (min_le_right _ _)⟩
    have bv : b ∈ v := by
      refine ⟨bu, ab.mono ?_⟩
      rw [inter_comm]
      exact inter_subset_inter_right _ ax
    let b' : v := ⟨b, bv⟩
    -- `b` cannot belong to `w`, as the elements of `w` do not intersect `closedBall z d`,
    -- contrary to `b`
    have b'_notmem_w : b' ∉ w := by
      intro b'w
      have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w
      have : (ball x (R x) \ k ∩ k).Nonempty := by
        apply ab.mono (inter_subset_inter _ b'k)
        refine ((hB _ hat).trans ?_).trans hd
        gcongr
        exact ad.trans (min_le_left _ _)
      simpa only [diff_inter_self, Set.not_nonempty_empty]
    let b'' : { a // a ∉ w } := ⟨b', b'_notmem_w⟩
    -- since `a` and `b` have comparable diameters, it follows that `z` belongs to the
    -- enlargement of `b`
    have zb : c a ∈ closedBall (c b) (3 * r b) := by
      rcases ab with ⟨e, ⟨ea, eb⟩⟩
      have A : dist (c a) e ≤ r a := mem_closedBall'.1 (hB a hat ea)
      have B : dist e (c b) ≤ r b := mem_closedBall.1 (hB b (ut bu) eb)
      simp only [mem_closedBall]
      linarith only [dist_triangle (c a) e (c b), A, B, bdiam]
    suffices H : closedBall (c b'') (3 * r b'')
        ⊆ ⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a) from H zb
    exact subset_iUnion (fun a : { a // a ∉ w } => closedBall (c a) (3 * r a)) b''