awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h_1 : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = (↑i : ℕ)
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = (↑i : ℕ)
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < jy }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_j : p.color jy = sInf (univ \ A)
h :
  (closedBall (p.c (p.index jx)) (p.r (p.index jx)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty →
    ¬p.color jy = p.color jx
contrapose! :
  ∀
    (h :
      ¬Function.onFun.{u + 1, u_1 + 1, 1} (β := Set α) Disjoint (fun (j : β) ↦ closedBall (q.c j) (q.r j)) (p.index jx)
          (p.index jy)),
    (closedBall (p.c (p.index jx)) (p.r (p.index jx)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty ∧
      p.color jy = p.color jx
⊢ Function.onFun.{u + 1, u_1 + 1, 1} (β := Set α) Disjoint (fun (j : β) ↦ closedBall (q.c j) (q.r j)) (p.index jx)
    (p.index jy)

Tactic:
contrapose! h
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h_1 : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = (↑i : ℕ)
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = (↑i : ℕ)
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ (j : { j : Ordinal.{u} // j < jy }),
    ⋃ (_ :
      (closedBall (p.c (p.index (↑j : Ordinal.{u}))) (p.r (p.index (↑j : Ordinal.{u}))) ∩
          closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color (↑j : Ordinal.{u})}
color_j : p.color jy = sInf (univ \ A)
h :
  ¬Function.onFun.{u + 1, u_1 + 1, 1} (β := Set α) Disjoint (fun (j : β) ↦ closedBall (q.c j) (q.r j)) (p.index jx)
      (p.index jy)
⊢ (closedBall (p.c (p.index jx)) (p.r (p.index jx)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty ∧
    p.color jy = p.color jx

Tactic:
simpa only [jxi, jyi, and_true, eq_self_iff_true, ← not_disjoint_iff_nonempty_inter] using h

Premises used: and_true, Set.not_disjoint_iff_nonempty_inter, eq_self_iff_true
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
⊢ range q.c ⊆ ⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)

Tactic:
range_subset_iff.2 fun b => ?_

Premises used: Iff.mpr, Set.range_subset_iff
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
inr : ∀ (b : β), Membership.mem (γ := Set α) (⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)) (q.c b)
⊢ range q.c ⊆ ⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)

Tactic:
refine range_subset_iff.2 fun b => ?_

Premises used: Iff.mpr, Set.range_subset_iff
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
b : β
⊢ ∃ a < p.lastStep, dist (p.c b) (p.c (p.index a)) < p.r (p.index a)

Tactic:
simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,
        Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b

Premises used: Subtype.coe_mk, Besicovitch.TauPackage.iUnionUpTo, Set.mem_iUnion, Subtype.exists, Besicovitch.TauPackage.mem_iUnionUpTo_lastStep, exists_prop, Metric.mem_ball
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
b : β
inr :
  ∀ (a : Ordinal.{u}) (ha : a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a)),
    Membership.mem (γ := Set α) (⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)) (q.c b)
⊢ Membership.mem (γ := Set α) (⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)) (q.c b)

Tactic:
obtain ⟨a, ha⟩ :
      ∃ a : Ordinal, a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a) := by
      simpa only [iUnionUpTo, exists_prop, mem_iUnion, mem_ball, Subtype.exists,
        Subtype.coe_mk] using p.mem_iUnionUpTo_lastStep b

Premises used: Besicovitch.TauPackage.lastStep, Ordinal
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
b : β
a : Ordinal.{u}
ha : a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a)
inr :
  ∃ (i : Fin N) (i_1 : Ordinal.{u}),
    p.color i_1 = (↑i : ℕ) ∧ i_1 < p.lastStep ∧ dist (q.c b) (q.c (p.index i_1)) < q.r (p.index i_1)
⊢ Membership.mem (γ := Set α) (⋃ (i : Fin N), ⋃ j ∈ s i, ball (q.c j) (q.r j)) (q.c b)

Tactic:
simp only [s, exists_prop, mem_iUnion, mem_ball, mem_singleton_iff, biUnion_and',
      exists_eq_left, iUnion_exists, exists_and_left]

Premises used: Set.mem_iUnion, exists_and_left, exists_eq_left, exists_prop, Metric.mem_ball, Set.mem_singleton_iff, Set.iUnion_exists, Set.biUnion_and'
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
b : β
a : Ordinal.{u}
ha : a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a)
⊢ ∃ (i : Fin N) (i_1 : Ordinal.{u}),
    p.color i_1 = (↑i : ℕ) ∧ i_1 < p.lastStep ∧ dist (q.c b) (q.c (p.index i_1)) < q.r (p.index i_1)

Tactic:
⟨⟨p.color a, p.color_lt ha.1 hN⟩, a, rfl, ha⟩

Premises used: Fin.mk, And.left, Besicovitch.TauPackage.color, And.intro, Exists.intro, Besicovitch.TauPackage.color_lt, rfl
-- Module: Mathlib.MeasureTheory.Covering.Besicovitch
Goal:
α : Type u_1
inst : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
q : BallPackage β α
h : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : (i : Fin N) → Set β :=
  fun (i : Fin N) ↦ ⋃ (k : Ordinal.{u}), ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = (↑i : ℕ)), {p.index k}
b : β
a : Ordinal.{u}
ha : a < p.lastStep ∧ dist (p.c b) (p.c (p.index a)) < p.r (p.index a)
⊢ ∃ (i : Fin N) (i_1 : Ordinal.{u}),
    p.color i_1 = (↑i : ℕ) ∧ i_1 < p.lastStep ∧ dist (q.c b) (q.c (p.index i_1)) < q.r (p.index i_1)

Tactic:
exact ⟨⟨p.color a, p.color_lt ha.1 hN⟩, a, rfl, ha⟩
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
inl :
  ∀ (hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)),
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
inr :
  ∀ (hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)),
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
rcases le_or_gt (μ s) 0 with (hμs | hμs)

Premises used: le_or_gt, OfNat.ofNat, DFunLike.coe
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
have : μ s = 0 := le_bot_iff.1 hμs
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ (↑∅ : Set α) ⊆ s

Tactic:
simp only [Finset.coe_empty, empty_subset]

Premises used: Finset.coe_empty, Set.empty_subset
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
inl :
  (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ (x : α), ⋃ (_ : Membership.mem (γ := Finset α) ∅ x), closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
inl₁ : (↑∅ : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
refine ⟨∅, by simp only [Finset.coe_empty, empty_subset], ?_, ?_⟩
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ (x : α), ⋃ (_ : Membership.mem (γ := Finset α) ∅ x), closedBall x (r x)) ≤
    HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)

Tactic:
simp only [this, Finset.notMem_empty, diff_empty, iUnion_false, iUnion_empty,
        nonpos_iff_eq_zero, mul_zero]

Premises used: Set.diff_empty, MulZeroClass.mul_zero, Finset.notMem_empty, Set.iUnion_false, Set.iUnion_empty, nonpos_iff_eq_zero
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ (↑∅ : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
simp only [Finset.coe_empty, pairwiseDisjoint_empty]

Premises used: Finset.coe_empty, Set.pairwiseDisjoint_empty
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
inr :
  IsEmpty α →
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
inr₁ :
  Nonempty α →
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
cases isEmpty_or_nonempty α

Premises used: isEmpty_or_nonempty
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : IsEmpty α
⊢ LT.lt (α := ℝ≥0∞) 0 0

Tactic:
simp only [eq_empty_of_isEmpty s, measure_empty] at hμs

Premises used: MeasureTheory.measure_empty, Set.eq_empty_of_isEmpty
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (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 : LT.lt (α := ℝ≥0∞) 0 0
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
(lt_irrefl _ hμs).elim

Premises used: lt_irrefl, False.elim
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (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 : LT.lt (α := ℝ≥0∞) 0 0
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
exact (lt_irrefl _ hμs).elim

Premises used: lt_irrefl, False.elim
-- 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
μ : Measure α
inst_4 : 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 : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α 0 τ)
inhabit : ∀ (inhabited_h : Inhabited α), False
⊢ False

Tactic:
inhabit α
-- 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
μ : Measure α
inst_4 : 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 : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α 0 τ)
inhabited_h : Inhabited α
⊢ False

Tactic:
not_isEmpty_of_nonempty _ hN

Premises used: not_isEmpty_of_nonempty
-- 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
μ : Measure α
inst_4 : 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 : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α 0 τ)
inhabited_h : Inhabited α
⊢ False

Tactic:
exact not_isEmpty_of_nonempty _ hN

Premises used: not_isEmpty_of_nonempty
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
⊢ N ≠ 0

Tactic:
have Npos : N ≠ 0 := by
    rintro rfl
    inhabit α
    exact not_isEmpty_of_nonempty _ hN
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
Npos : N ≠ 0
⊢ ∃ (o : Set α),
    s ⊆ o ∧ MeasurableSet o ∧ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)

Tactic:
exists_measurable_superset μ s

Premises used: MeasureTheory.exists_measurable_superset
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
Npos : N ≠ 0
inr :
  ∀ (o : Set α) (so : s ⊆ o) (omeas : MeasurableSet o)
    (μo : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)),
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

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

Premises used: MeasureTheory.exists_measurable_superset, DFunLike.coe, Set
-- 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
μ : Measure α
inst_4 : IsFiniteMeasure μ
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty.{u_1 + 1} (SatelliteConfig α N τ)
s : Set α
r : α → ℝ
rpos : ∀ x ∈ s, 0 < r x
rle : ∀ x ∈ s, r x ≤ 1
hμs : LT.lt (α := ℝ≥0∞) (0 : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
h : Nonempty α
Npos : N ≠ 0
o : Set α
so : s ⊆ o
omeas : MeasurableSet o
μo : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) o : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
a : BallPackage (↑s : Type u_1) α :=
  { c := fun (x : (↑s : Type u_1)) ↦ (↑x : α), r := fun (x : (↑s : Type u_1)) ↦ r (↑x : α), rpos := ⋯, r_bound := 1,
    r_le := ⋯ }
inr :
  ∀ (u : Fin N → Set (↑s : Type u_1))
    (hu : ∀ (i : Fin N), (u i).PairwiseDisjoint (α := Set α) fun (j : (↑s : Type u_1)) ↦ closedBall (a.c j) (a.r j))
    (hu' : range a.c ⊆ ⋃ (i : Fin N), ⋃ j ∈ u i, ball (a.c j) (a.r j)),
    ∃ (t : Finset α),
      (↑t : Set α) ⊆ s ∧
        (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
            HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
              ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
          (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)
⊢ ∃ (t : Finset α),
    (↑t : Set α) ⊆ s ∧
      (μ : (a : Set α) → ℝ≥0∞) (s \ ⋃ x ∈ t, closedBall x (r x)) ≤
          HMul.hMul (β := ℝ≥0∞) ((↑N : ℝ≥0∞) / HAdd.hAdd (α := ℝ≥0∞) (↑N : ℝ≥0∞) 1)
            ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ∧
        (↑t : Set α).PairwiseDisjoint (α := Set α) fun (x : α) ↦ closedBall x (r x)

Tactic:
rcases exist_disjoint_covering_families hτ hN a with ⟨u, hu, hu'⟩

Premises used: Besicovitch.exist_disjoint_covering_families