awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
t' : Set ι := {a : ι | a ∈ t ∧ 0 ≤ r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ closedBall (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (closedBall (x a) (r a) ∩ closedBall (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a < 0
b : ι
rb : b ∈ t ∧ 0 ≤ r b
c : ι
cu : c ∈ u
right : closedBall (x b) (r b) ⊆ closedBall (x c) (τ * r c)
⊢ ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (τ * r b)

Tactic:
exact ⟨c, cu, by simp only [closedBall_eq_empty.2 h'a, empty_subset]⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
inl :
  ∀ (hr : ∀ (a : ι), Membership.mem (γ := Set ι) ∅ a → r a ≤ R),
    ∃ u ⊆ ∅,
      (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
        ∀ (a : ι), Membership.mem (γ := Set ι) ∅ a → ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
inr :
  t.Nonempty →
    ∃ u ⊆ t,
      (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases eq_empty_or_nonempty t with (rfl | _)

Premises used: Set.eq_empty_or_nonempty, rfl
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
x : ι → α
r : ι → ℝ
R τ : ℝ
hτ : 3 < τ
hr : ∀ (a : ι), Membership.mem (γ := Set ι) ∅ a → r a ≤ R
⊢ ∀ (a : ι),
    Membership.mem (γ := Set ι) ∅ a →
      ∃ (b : ι), Membership.mem (γ := Set ι) ∅ b ∧ ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
simp
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
x : ι → α
r : ι → ℝ
R τ : ℝ
hτ : 3 < τ
hr : ∀ (a : ι), Membership.mem (γ := Set ι) ∅ a → r a ≤ R
⊢ ∃ u ⊆ ∅,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ (a : ι), Membership.mem (γ := Set ι) ∅ a → ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
pos :
  ∀ (ht : ∀ a ∈ t, r a ≤ 0),
    ∃ u ⊆ t,
      (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
neg :
  ∀ (ht : ∃ a ∈ t, 0 < r a),
    ∃ u ⊆ t,
      (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
by_cases! ht : ∀ a ∈ t, r a ≤ 0
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∀ a ∈ t, r a ≤ 0
a : ι
ha : a ∈ t
b : ι
h_1 : b ∈ t
h_2 : a ≠ b
⊢ Function.onFun.{u_2 + 1, u_1 + 1, 1} (β := Set α) Disjoint (fun (a : ι) ↦ ball (x a) (r a)) a b

Tactic:
simp only [ball_eq_empty.2 (ht a ha), empty_disjoint, Function.onFun]

Premises used: Iff.mpr, Function.onFun, Metric.ball_eq_empty, Set.empty_disjoint
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∀ a ∈ t, r a ≤ 0
a : ι
ha : a ∈ t
⊢ ball (x a) (r a) ⊆ ball (x a) (τ * r a)

Tactic:
simp only [ball_eq_empty.2 (ht a ha), empty_subset]

Premises used: Iff.mpr, Set.empty_subset, Metric.ball_eq_empty
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∀ a ∈ t, r a ≤ 0
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
exact ⟨t, Subset.rfl, fun a ha b _ _ => by
      simp only [ball_eq_empty.2 (ht a ha), empty_disjoint, Function.onFun],
      fun a ha => ⟨a, ha, by simp only [ball_eq_empty.2 (ht a ha), empty_subset]⟩⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
linarith : (τ - 1) / 2 ≤ 1 → 2 / 2 = 1
linarith₁ : (τ - 1) / 2 ≤ 1 → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : (τ - 1) / 2 ≤ 1 → GT.gt (α := ℝ) 2 0
linarith₃ : (τ - 1) / 2 ≤ 1 → 3 - τ + (1 * τ - HMul.hMul (α := ℝ) 1 1 - HMul.hMul (α := ℝ) 2 1) = 0
⊢ 1 < (τ - 1) / 2

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
neg :
  ∀ (u : Set ι) (ut' : u ⊆ t') (u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a))
    (hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b),
    ∃ u ⊆ t,
      (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
        ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases exists_disjoint_subfamily_covering_enlargement (fun a => ball (x a) (r a)) t' r
      ((τ - 1) / 2) (by linarith) (fun a ha => ha.2.le) R (fun a ha => hr a ha.1) fun a ha =>
      ⟨x a, mem_ball_self ha.2⟩ with
    ⟨u, ut', u_disj, hu⟩

Premises used: HSub.hSub, Metric.mem_ball_self, HDiv.hDiv, LT.lt.le, Metric.ball, Vitali.exists_disjoint_subfamily_covering_enlargement, And.right, And.left, Exists.intro, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
rcases :
  ∀ (b : ι) (bu : b ∈ u) (hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty) (rb : r a ≤ (τ - 1) / 2 * r b),
    ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases hu a ha with ⟨b, bu, hb, rb⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
⟨b, bu, ?_⟩

Premises used: And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
refine : ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
refine ⟨b, bu, ?_⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
⊢ dist (x a) (x b) < r a + r b

Tactic:
have : dist (x a) (x b) < r a + r b := dist_lt_add_of_nonempty_ball_inter_ball hb
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h_1 : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
this : dist (x a) (x b) < r a + r b
h : r a + dist (x a) (x b) ≤ τ * r b
⊢ ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
apply ball_subset_ball'

Premises used: Metric.ball_subset_ball'
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
a : ι
ha : a ∈ t'
b : ι
bu : b ∈ u
hb : (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty
rb : r a ≤ (τ - 1) / 2 * r b
this : dist (x a) (x b) < r a + r b
linarith : τ * r b < r a + dist (x a) (x b) → 2 / 2 = 1
linarith₁ : τ * r b < r a + dist (x a) (x b) → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : τ * r b < r a + dist (x a) (x b) → HMul.hMul (α := ℝ) 2 1 = 2
linarith₃ : τ * r b < r a + dist (x a) (x b) → GT.gt (α := ℝ) 2 0
linarith₄ :
  τ * r b < r a + dist (x a) (x b) →
    2 * r a - (1 * τ - HMul.hMul (α := ℝ) 1 1) * (1 * r b) + (dist (x a) (x b) - (r a + r b)) +
        (τ * r b - (r a + dist (x a) (x b))) =
      0
⊢ r a + dist (x a) (x b) ≤ τ * r b

Tactic:
linarith
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
⊢ ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
have A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * 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_lt_add_of_nonempty_ball_inter_ball hb
    apply ball_subset_ball'
    linarith
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => ?_⟩

Premises used: HasSubset.Subset.trans, And.left, And.intro, Exists.intro
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
neg : ∀ (a : ι) (ha : a ∈ t), ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ u ⊆ t,
    (PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)) ∧
      ∀ a ∈ t, ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
refine ⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => ?_⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
neg : ∀ (h'a : 0 < r a), ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
neg₁ : ∀ (h'a : r a ≤ 0), ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases lt_or_ge 0 (r a) with (h'a | h'a)

Premises used: lt_or_ge, OfNat.ofNat
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : 0 < r a
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
A a ⟨ha, h'a⟩

Premises used: And.intro
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : 0 < r a
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
exact A a ⟨ha, h'a⟩

Premises used: And.intro
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
ht : ∃ a ∈ t, 0 < r a
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
neg : ∀ (b : ι) (rb : b ∈ t ∧ 0 < r b), ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases ht with ⟨b, rb⟩
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
b : ι
rb : b ∈ t ∧ 0 < r b
neg : ∀ (c : ι) (cu : c ∈ u), ball (x b) (r b) ⊆ ball (x c) (τ * r c) → ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩

Premises used: And.right, And.left, And.intro
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
b : ι
rb : b ∈ t ∧ 0 < r b
c : ι
cu : c ∈ u
right : ball (x b) (r b) ⊆ ball (x c) (τ * r c)
⊢ ball (x a) (r a) ⊆ ball (x c) (τ * r c)

Tactic:
simp only [ball_eq_empty.2 h'a, empty_subset]

Premises used: Iff.mpr, Set.empty_subset, Metric.ball_eq_empty
-- Module: Mathlib.MeasureTheory.Covering.Vitali
Goal:
α : Type u_1
ι : Type u_2
inst : PseudoMetricSpace α
t : Set ι
x : ι → α
r : ι → ℝ
R : ℝ
hr : ∀ a ∈ t, r a ≤ R
τ : ℝ
hτ : 3 < τ
h : t.Nonempty
t' : Set ι := {a : ι | a ∈ t ∧ 0 < r a}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint.{u_1, u_2} (α := Set α) u fun (a : ι) ↦ ball (x a) (r a)
hu : ∀ a ∈ t', ∃ b ∈ u, (ball (x a) (r a) ∩ ball (x b) (r b)).Nonempty ∧ r a ≤ (τ - 1) / 2 * r b
A : ∀ a ∈ t', ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)
a : ι
ha : a ∈ t
h'a : r a ≤ 0
b : ι
rb : b ∈ t ∧ 0 < r b
c : ι
cu : c ∈ u
right : ball (x b) (r b) ⊆ ball (x c) (τ * r c)
⊢ ∃ b ∈ u, ball (x a) (r a) ⊆ ball (x b) (τ * r b)

Tactic:
exact ⟨c, cu, by simp only [ball_eq_empty.2 h'a, empty_subset]⟩
-- 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
x : α
⊢ ∃ (R : ℝ), 0 < R ∧ R ≤ 1 ∧ LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R)) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
((eventually_le_nhds one_pos).and ?_).exists_gt

Premises used: Filter.Eventually.and, eventually_le_nhds, one_pos, Filter.Eventually.exists_gt
-- 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
x : α
refine : ∀ᶠ (R : ℝ) in 𝓝 0, LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R)) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∃ (R : ℝ), 0 < R ∧ R ≤ 1 ∧ LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R)) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
refine ((eventually_le_nhds one_pos).and ?_).exists_gt

Premises used: Filter.Eventually.and, eventually_le_nhds, one_pos, Filter.Eventually.exists_gt
-- 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
x : α
⊢ ∀ᶠ (R : ℝ) in 𝓝 0, LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R)) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
(tendsto_closedBall_smallSets x).comp ?_ (μ.finiteAt_nhds x).eventually

Premises used: MeasureTheory.Measure.finiteAt_nhds, Set.Mem, Filter.Eventually, setOf, Set.preimage, tendsto_closedBall_smallSets, MeasureTheory.Measure.FiniteAtFilter.eventually, Filter.Tendsto.comp, Metric.closedBall
-- 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
x : α
refine : Tendsto (α := ℝ) (HMul.hMul (α := ℝ) 20) (𝓝 0) (𝓝 0)
⊢ ∀ᶠ (R : ℝ) in 𝓝 0, LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x (20 * R)) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
refine (tendsto_closedBall_smallSets x).comp ?_ (μ.finiteAt_nhds x).eventually

Premises used: MeasureTheory.Measure.finiteAt_nhds, tendsto_closedBall_smallSets, MeasureTheory.Measure.FiniteAtFilter.eventually, Filter.Tendsto.comp
-- 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
x : α
⊢ Continuous (X := ℝ) (HMul.hMul (α := ℝ) 20)

Tactic:
fun_prop