l3lab/ntp-mathlib huggingface dataset:
/-- The topological Besicovitch covering theorem: there exist finitely many families of disjoint
balls covering all the centers in a package. More specifically, one can use `N` families if there
are no satellite configurations with `N+1` points. -/
theorem exist_disjoint_covering_families {N : ℕ} {τ : ℝ} (hτ : 1 < τ)
    (hN : IsEmpty (SatelliteConfig α N τ)) (q : BallPackage β α) :
    ∃ s : Fin N → Set β,
      (∀ i : Fin N, (s i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)) ∧
        range q.c ⊆ ⋃ i : Fin N, ⋃ j ∈ s i, ball (q.c j) (q.r j) := by
  -- first exclude the trivial case where `β` is empty (we need non-emptiness for the transfinite
  -- induction, to be able to choose garbage when there is no point left).
  cases isEmpty_or_nonempty β
  · refine' ⟨fun _ => ∅, fun _ => pairwiseDisjoint_empty, _⟩
    rw [← image_univ, eq_empty_of_isEmpty (univ : Set β)]
    simp
  -- Now, assume `β` is nonempty.
  let p : TauPackage β α :=
    { q with
      τ
      one_lt_tau := hτ }
  -- we use for `s i` the balls of color `i`.
  let s := fun i : Fin N =>
    ⋃ (k : Ordinal.{u}) (_ : k < p.lastStep) (_ : p.color k = i), ({p.index k} : Set β)
  refine' ⟨s, fun i => _, _⟩
  · -- show that balls of the same color are disjoint
    intro x hx y hy x_ne_y
    obtain ⟨jx, jx_lt, jxi, rfl⟩ :
      ∃ jx : Ordinal, jx < p.lastStep ∧ p.color jx = i ∧ x = p.index jx := by
      simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hx
    obtain ⟨jy, jy_lt, jyi, rfl⟩ :
      ∃ jy : Ordinal, jy < p.lastStep ∧ p.color jy = i ∧ y = p.index jy := by
      simpa only [exists_prop, mem_iUnion, mem_singleton_iff] using hy
    wlog jxy : jx ≤ jy generalizing jx jy
    · exact (this jy jy_lt jyi hy jx jx_lt jxi hx x_ne_y.symm (le_of_not_le jxy)).symm
    replace jxy : jx < jy
    · rcases lt_or_eq_of_le jxy with (H | rfl); · { exact H }; · { exact (x_ne_y rfl).elim }
    

Proof state:
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α :=
  {
    toBallPackage :=
      { c := q.c, r := q.r, rpos := (_ : ∀ (b : β), 0 < BallPackage.r q b), r_bound := q.r_bound,
        r_le := (_ : ∀ (b : β), BallPackage.r q b ≤ q.r_bound) },
    τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i => ⋃ k, ⋃ (_ : k < lastStep p), ⋃ (_ : color p k = ↑i), {index p k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < lastStep p
jxi : color p jx = ↑i
hx : index p jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < lastStep p
jyi : color p jy = ↑i
hy : index p jy ∈ s i
x_ne_y : index p jx ≠ index p jy
jxy : jx < jy
⊢ (Disjoint on fun j => closedBall (BallPackage.c q j) (BallPackage.r q j)) (index p jx) (index p jy)

Next tactic:
let A : Set ℕ :=
      ⋃ (j : { j // j < jy })
        (_ : (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩
          closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
        {p.color j}
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  

Proof state:
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
rcases le_or_lt (μ s) 0 with (hμs | hμs)
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · 

Proof state:
case inl
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : ↑↑μ s ≤ 0
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
have : μ s = 0 := le_bot_iff.1 hμs
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    

Proof state:
case inl
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : ↑↑μ s ≤ 0
this : ↑↑μ s = 0
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by 

Proof state:
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : ↑↑μ s ≤ 0
this : ↑↑μ s = 0
⊢ ↑∅ ⊆ s

Next tactic:
simp only [Finset.coe_empty, empty_subset]
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · 

Proof state:
case inl.refine'_1
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : ↑↑μ s ≤ 0
this : ↑↑μ s = 0
⊢ ↑↑μ (s \ ⋃ x ∈ ∅, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s

Next tactic:
simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · 

Proof state:
case inl.refine'_2
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : ↑↑μ s ≤ 0
this : ↑↑μ s = 0
⊢ PairwiseDisjoint ↑∅ fun x => closedBall x (r x)

Next tactic:
simp only [Finset.coe_empty, pairwiseDisjoint_empty]
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  

Proof state:
case inr
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
cases isEmpty_or_nonempty α
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · 

Proof state:
case inr.inl
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : IsEmpty α
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    

Proof state:
case inr.inl
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
h✝ : IsEmpty α
hμs : 0 < 0
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
exact (lt_irrefl _ hμs).elim
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  

Proof state:
case inr.inr
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
have Npos : N ≠ 0 := by
    rintro rfl
    inhabit α
    exact not_isEmpty_of_nonempty _ hN
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  have Npos : N ≠ 0 := by
    

Proof state:
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
⊢ N ≠ 0

Next tactic:
rintro rfl
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  have Npos : N ≠ 0 := by
    rintro rfl
    

Proof state:
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
τ : ℝ
hτ : 1 < τ
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
hN : IsEmpty (SatelliteConfig α 0 τ)
⊢ False

Next tactic:
inhabit α
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  have Npos : N ≠ 0 := by
    rintro rfl
    inhabit α
    

Proof state:
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
τ : ℝ
hτ : 1 < τ
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
hN : IsEmpty (SatelliteConfig α 0 τ)
inhabited_h : Inhabited α
⊢ False

Next tactic:
exact not_isEmpty_of_nonempty _ hN
/-- Consider, for each `x` in a set `s`, a radius `r x ∈ (0, 1]`. Then one can find finitely
many disjoint balls of the form `closedBall x (r x)` covering a proportion `1/(N+1)` of `s`, if
there are no satellite configurations with `N+1` points.
-/
theorem exist_finset_disjoint_balls_large_measure (μ : Measure α) [IsFiniteMeasure μ] {N : ℕ}
    {τ : ℝ} (hτ : 1 < τ) (hN : IsEmpty (SatelliteConfig α N τ)) (s : Set α) (r : α → ℝ)
    (rpos : ∀ x ∈ s, 0 < r x) (rle : ∀ x ∈ s, r x ≤ 1) :
    ∃ t : Finset α,
      ↑t ⊆ s ∧
        μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ N / (N + 1) * μ s ∧
          (t : Set α).PairwiseDisjoint fun x => closedBall x (r x) := by
  -- exclude the trivial case where `μ s = 0`.
  rcases le_or_lt (μ s) 0 with (hμs | hμs)
  · have : μ s = 0 := le_bot_iff.1 hμs
    refine' ⟨∅, by simp only [Finset.coe_empty, empty_subset], _, _⟩
    · simp only [this, Finset.not_mem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]
    · simp only [Finset.coe_empty, pairwiseDisjoint_empty]
  cases isEmpty_or_nonempty α
  · simp only [eq_empty_of_isEmpty s, measure_empty] at hμs
    exact (lt_irrefl _ hμs).elim
  have Npos : N ≠ 0 := by
    rintro rfl
    inhabit α
    exact not_isEmpty_of_nonempty _ hN
  -- introduce a measurable superset `o` with the same measure, for measure computations
  

Proof state:
case inr.inr
α : Type u_1
inst✝⁴ : MetricSpace α
β : Type u
inst✝³ : SecondCountableTopology α
inst✝² : MeasurableSpace α
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : 0 < ↑↑μ s
h✝ : Nonempty α
Npos : N ≠ 0
⊢ ∃ t,
    ↑t ⊆ s ∧
      ↑↑μ (s \ ⋃ x ∈ t, closedBall x (r x)) ≤ ↑N / (↑N + 1) * ↑↑μ s ∧ PairwiseDisjoint ↑t fun x => closedBall x (r x)

Next tactic:
obtain ⟨o, so, omeas, μo⟩ : ∃ o : Set α, s ⊆ o ∧ MeasurableSet o ∧ μ o = μ s :=
    exists_measurable_superset μ s