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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
z : α
hz : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
z_notmem_k : z ∉ k
this : ball x (R x) \ k ∈ 𝓝 z
d : ℝ
dpos : 0 < d
hd : closedBall z d ⊆ ball x (R x) \ k
⊢ Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    z
have obtain : ∀ (a : ι) (hat : a ∈ t) (hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) (z_notmem_k : c a ∉ k)
  (this : ball x (R x) \ k ∈ 𝓝 (c a)) (hd : closedBall (c a) d ⊆ ball x (R x) \ k) (ad : r a ≤ min d (R (c a))),
  Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    (c a)
Tactic: obtain ⟨a, hat, ad, rfl⟩ : ∃ a ∈ t, r a ≤ min d (R z) ∧ c a = z :=
      hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))
Cleared hypotheses: z, hz, z_notmem_k, this, hd
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ B a ⊆ ball x (R x)
have refine : closedBall (c a) (r a) ⊆ ball x (R x)
Tactic: refine (hB a hat).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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ ball x (R x)
have refine : closedBall (c a) (r a) ⊆ closedBall (c a) d
Tactic: refine Subset.trans ?_ (hd.trans Set.diff_subset)
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ closedBall (c a) d
have h : r a ≤ d
Tactic: gcongr
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    (c a)
have ax : B a ⊆ ball x (R x)
Tactic: have ax : B a ⊆ ball x (R x) := by
      refine (hB a hat).trans ?_
      refine Subset.trans ?_ (hd.trans Set.diff_subset)
      gcongr
      exact ad.trans (min_le_left _ _)
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
⊢ Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    (c a)
have obtain : ∀ (b : ι) (bu : b ∈ u) (ab : (B a ∩ B b).Nonempty) (bdiam : r a ≤ 2 * r b),
  Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    (c a)
Tactic: obtain ⟨b, bu, ab, bdiam⟩ : ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b :=
      hu a ⟨hat, ad.trans (min_le_right _ _)⟩
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
⊢ b ∈ v
have refine : B a ∩ B b ⊆ B b ∩ ball x (R x)
Tactic: refine ⟨bu, ab.mono ?_⟩
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
⊢ B a ∩ B b ⊆ B b ∩ ball x (R x)
have rw : B b ∩ B a ⊆ B b ∩ ball x (R x)
Tactic: rw [inter_comm]
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
⊢ Membership.mem (γ := Set α)
    (⋃ (a : { a : (↑v : Type u_2) // a ∉ w }),
      closedBall (c (↑(↑a : (↑v : Type u_2)) : ι)) (3 * r (↑(↑a : (↑v : Type u_2)) : ι)))
    (c a)
have bv : b ∈ v
Tactic: have bv : b ∈ v := by
      refine ⟨bu, ab.mono ?_⟩
      rw [inter_comm]
      exact inter_subset_inter_right _ ax
-- 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
K : ℝ
μK : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closedBall x K) : ℝ≥0∞) (∞ : ℝ≥0∞)
hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : LT.lt (α := ℝ≥0∞) (∑' (a : (↑v : Type u_2)), (μ : (a : Set α) → ℝ≥0∞) (B (↑a : ι))) ∞
w : Finset (↑v : Type u_2)
hw :
  ∑' (a : { a : (↑v : Type u_2) // a ∉ w }), (μ : (a : Set α) → ℝ≥0∞) (B (↑(↑a : (↑v : Type u_2)) : ι)) <
    ε / (↑C : ℝ≥0∞)
k : Set α := ⋃ a ∈ w, B (↑a : ι)
k_closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ a ∈ u, B a) ∩ ball x (R x)
z_notmem_k : c a ∉ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : (B a ∩ B b).Nonempty
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : (↑v : Type u_2) := Subtype.mk (p := fun (x : ι) ↦ x ∈ v) b bv
b'w : b' ∈ w
⊢ False
have b'k : B (↑b' : ι) ⊆ k
Tactic: have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w