-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : OpensMeasurableSpace α
inst_4 : HasBesicovitchCovering α
μ : Measure α
inst_5 : IsFiniteMeasure μ
f : α → Set ℝ
s : Set α
hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
P : Finset (α × ℝ) → Prop :=
  fun (t : Finset (α × ℝ)) ↦
    ((↑t : Set (α × ℝ)).PairwiseDisjoint (α := Set α) fun (p : α × ℝ) ↦ closedBall p.1 p.2) ∧
      (∀ p ∈ t, p.1 ∈ s) ∧ ∀ p ∈ t, p.2 ∈ f p.1
⊢ ∀ (t : Finset (α × ℝ)),
    P t →
      ∃ (u : Finset (α × ℝ)),
        t ⊆ u ∧
          P u ∧
            (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ u, closedBall p.1 p.2) ≤
              HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
                ((μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ p ∈ t, closedBall p.1 p.2) : ℝ≥0∞)

Tactic:
have :
      ∀ t : Finset (α × ℝ), P t → ∃ u : Finset (α × ℝ), t ⊆ u ∧ P u ∧
        μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ u), closedBall p.1 p.2) ≤
          N / (N + 1) * μ (s \ ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2) := by
    intro t ht
    set B := ⋃ (p : α × ℝ) (_ : p ∈ t), closedBall p.1 p.2 with hB
    have B_closed : IsClosed B := isClosed_biUnion_finset fun i _ => isClosed_closedBall
    set s' := s \ B
    have : ∀ x ∈ s', ∃ r ∈ f x ∩ Ioo 0 1, Disjoint B (closedBall x r) := by
      intro x hx
      have xs : x ∈ s := ((mem_diff x).1 hx).1
      rcases eq_empty_or_nonempty B with (hB | hB)
      · rcases hf x xs 1 zero_lt_one with ⟨r, hr, h'r⟩
        exact ⟨r, ⟨hr, h'r⟩, by simp only [hB, empty_disjoint]⟩
      · let r := infDist x B
        have : 0 < min r 1 :=
          lt_min ((B_closed.notMem_iff_infDist_pos hB).1 ((mem_diff x).1 hx).2) zero_lt_one
        rcases hf x xs _ this with ⟨r, hr, h'r⟩
        refine ⟨r, ⟨hr, ⟨h'r.1, h'r.2.trans_le (min_le_right _ _)⟩⟩, ?_⟩
        rw [disjoint_comm]
        exact disjoint_closedBall_of_lt_infDist (h'r.2.trans_le (min_le_left _ _))
    choose! r hr using this
    obtain ⟨v, vs', hμv, hv⟩ :
      ∃ v : Finset α,
        ↑v ⊆ s' ∧
          μ (s' \ ⋃ x ∈ v, closedBall x (r x)) ≤ N / (N + 1) * μ s' ∧
            (v : Set α).PairwiseDisjoint fun x : α => closedBall x (r x) :=
      haveI rI : ∀ x ∈ s', r x ∈ Ioo (0 : ℝ) 1 := fun x hx => (hr x hx).1.2
      exist_finset_disjoint_balls_large_measure μ hτ hN s' r (fun x hx => (rI x hx).1) fun x hx =>
        (rI x hx).2.le
    refine ⟨t ∪ Finset.image (fun x => (x, r x)) v, Finset.subset_union_left, ⟨?_, ?_, ?_⟩, ?_⟩
    · simp only [Finset.coe_union, pairwiseDisjoint_union, ht.1, true_and, Finset.coe_image]
      constructor
      · intro p hp q hq hpq
        rcases (mem_image _ _ _).1 hp with ⟨p', p'v, rfl⟩
        rcases (mem_image _ _ _).1 hq with ⟨q', q'v, rfl⟩
        refine hv p'v q'v fun hp'q' => ?_
        rw [hp'q'] at hpq
        exact hpq rfl
      · intro p hp q hq hpq
        rcases (mem_image _ _ _).1 hq with ⟨q', q'v, rfl⟩
        apply disjoint_of_subset_left _ (hr q' (vs' q'v)).2
        rw [hB, ← Finset.set_biUnion_coe]
        exact subset_biUnion_of_mem (u := fun x : α × ℝ => closedBall x.1 x.2) hp
    · intro p hp
      rcases Finset.mem_union.1 hp with (h'p | h'p)
      · exact ht.2.1 p h'p
      · rcases Finset.mem_image.1 h'p with ⟨p', p'v, rfl⟩
        exact ((mem_diff _).1 (vs' (Finset.mem_coe.2 p'v))).1
    · intro p hp
      rcases Finset.mem_union.1 hp with (h'p | h'p)
      · exact ht.2.2 p h'p
      · rcases Finset.mem_image.1 h'p with ⟨p', p'v, rfl⟩
        exact (hr p' (vs' p'v)).1.1
    · convert! hμv using 2
      rw [Finset.set_biUnion_union, ← diff_diff, Finset.set_biUnion_finset_image]