awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
linarith : 20 * R (c a) < 8 * R0 + dist (c a) x → 2 / 2 = 1
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
have linarith₁ : 20 * R (c a) < 8 * R0 + dist (c a) x → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith [Idist_v a hav, (ut' (vu hav)).2]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
linarith : 20 * R (c a) < 8 * R0 + dist (c a) x → 2 / 2 = 1
linarith₁ : 20 * R (c a) < 8 * R0 + dist (c a) x → HMul.hMul (α := ℝ) 1 2 = 2
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
have linarith₂ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
Tactic: linarith [Idist_v a hav, (ut' (vu hav)).2]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
linarith : 20 * R (c a) < 8 * R0 + dist (c a) x → 2 / 2 = 1
linarith₁ : 20 * R (c a) < 8 * R0 + dist (c a) x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
have linarith₃ : 20 * R (c a) < 8 * R0 + dist (c a) x →
  2 * (R x - R0) + (0 - R0) + 19 * (1 * R0 - 2 * r a) + 2 * (20 * R (c a) - (8 * R0 + dist (c a) x)) +
        2 * (dist (c a) x - (r a + R x)) +
      40 * (r a - R (c a)) =
    0
Tactic: linarith [Idist_v a hav, (ut' (vu hav)).2]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
linarith : 20 * R (c a) < 8 * R0 + dist (c a) x → 2 / 2 = 1
linarith₁ : 20 * R (c a) < 8 * R0 + dist (c a) x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
linarith₃ :
  20 * R (c a) < 8 * R0 + dist (c a) x →
    2 * (R x - R0) + (0 - R0) + 19 * (1 * R0 - 2 * r a) + 2 * (20 * R (c a) - (8 * R0 + dist (c a) x)) +
          2 * (dist (c a) x - (r a + R x)) +
        40 * (r a - R (c a)) =
      0
linarith₄ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
have linarith₅ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 19 0
Tactic: linarith [Idist_v a hav, (ut' (vu hav)).2]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
linarith : 20 * R (c a) < 8 * R0 + dist (c a) x → 2 / 2 = 1
linarith₁ : 20 * R (c a) < 8 * R0 + dist (c a) x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
linarith₃ :
  20 * R (c a) < 8 * R0 + dist (c a) x →
    2 * (R x - R0) + (0 - R0) + 19 * (1 * R0 - 2 * r a) + 2 * (20 * R (c a) - (8 * R0 + dist (c a) x)) +
          2 * (dist (c a) x - (r a + R x)) +
        40 * (r a - R (c a)) =
      0
linarith₄ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
linarith₅ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 19 0
linarith₆ linarith₇ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 2 0
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
have linarith₈ : 20 * R (c a) < 8 * R0 + dist (c a) x → GT.gt (α := ℝ) 40 0
Tactic: linarith [Idist_v a hav, (ut' (vu hav)).2]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
⊢ B b ⊆ closedBall x (8 * R0)
have inr : closedBall (c b) (r b) ⊆ closedBall x (8 * R0)
Tactic: refine (hB b (ut bu)).trans ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
⊢ closedBall (c b) (r b) ⊆ closedBall x (8 * R0)
have inr : r b + dist (c b) x ≤ 8 * R0
Tactic: apply closedBall_subset_closedBall'
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (hypothesis)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
⊢ r b + dist (c b) x ≤ 8 * R0
have this : r b ≤ R0
Tactic: have : r b ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨bu, hbx⟩)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
⊢ r b + dist (c b) x ≤ 8 * R0
have linarith : 8 * R0 < r b + dist (c b) x → 2 / 2 = 1
Tactic: linarith [Idist_v b ⟨bu, hbx⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
linarith : 8 * R0 < r b + dist (c b) x → 2 / 2 = 1
⊢ r b + dist (c b) x ≤ 8 * R0
have linarith₁ : 8 * R0 < r b + dist (c b) x → HMul.hMul (α := ℝ) 1 2 = 2
Tactic: linarith [Idist_v b ⟨bu, hbx⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
linarith : 8 * R0 < r b + dist (c b) x → 2 / 2 = 1
linarith₁ : 8 * R0 < r b + dist (c b) x → HMul.hMul (α := ℝ) 1 2 = 2
⊢ r b + dist (c b) x ≤ 8 * R0
have linarith₂ : 8 * R0 < r b + dist (c b) x → GT.gt (α := ℝ) 2 0
Tactic: linarith [Idist_v b ⟨bu, hbx⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
linarith : 8 * R0 < r b + dist (c b) x → 2 / 2 = 1
linarith₁ : 8 * R0 < r b + dist (c b) x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : 8 * R0 < r b + dist (c b) x → GT.gt (α := ℝ) 2 0
⊢ r b + dist (c b) x ≤ 8 * R0
have linarith₃ : 8 * R0 < r b + dist (c b) x →
  R x - R0 + 5 * (0 - R0) + 2 * (r b - R0) + (8 * R0 - (r b + dist (c b) x)) + (dist (c b) x - (r b + R x)) = 0
Tactic: linarith [Idist_v b ⟨bu, hbx⟩]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Vitali (goal)
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
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : v.Nonempty
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : (B b ∩ ball x (R x)).Nonempty
this : r b ≤ R0
linarith : 8 * R0 < r b + dist (c b) x → 2 / 2 = 1
linarith₁ : 8 * R0 < r b + dist (c b) x → HMul.hMul (α := ℝ) 1 2 = 2
linarith₂ : 8 * R0 < r b + dist (c b) x → GT.gt (α := ℝ) 2 0
linarith₃ :
  8 * R0 < r b + dist (c b) x →
    R x - R0 + 5 * (0 - R0) + 2 * (r b - R0) + (8 * R0 - (r b + dist (c b) x)) + (dist (c b) x - (r b + R x)) = 0
⊢ r b + dist (c b) x ≤ 8 * R0
have linarith₄ : 8 * R0 < r b + dist (c b) x → GT.gt (α := ℝ) 5 0
Tactic: linarith [Idist_v b ⟨bu, hbx⟩]