awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
refine : ∀ (a : α) (ha : a ∈ cthickening (r₂ (f j)) (s (f j))), a ∈ Z i
⊢ b j ⊆ Z i

Tactic:
refine (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans
      ((cthickening_mono hj' _).trans fun a ha => ?_)

Premises used: Metric.cthickening_mono, HMul.hMul, HasSubset.Subset.trans, Metric.closedBall_subset_cthickening
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
a : α
ha : a ∈ cthickening (r₂ (f j)) (s (f j))
simp : ∃ (i_1 : ℕ), (p i_1 ∧ i ≤ i_1) ∧ a ∈ Y₂ i_1
⊢ a ∈ Z i

Tactic:
simp only [Z, mem_iUnion, exists_prop]

Premises used: Set.mem_iUnion, exists_prop
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
a : α
ha : a ∈ cthickening (r₂ (f j)) (s (f j))
⊢ ∃ (i_1 : ℕ), (p i_1 ∧ i ≤ i_1) ∧ a ∈ Y₂ i_1

Tactic:
⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩

Premises used: And.intro, Exists.intro, LE.le.trans, GE.ge.le
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
j : ℕ
hj : j ≥ i
hj' : M * r₁ (f j) ≤ r₂ (f j)
a : α
ha : a ∈ cthickening (r₂ (f j)) (s (f j))
⊢ ∃ (i_1 : ℕ), (p i_1 ∧ i ≤ i_1) ∧ a ∈ Y₂ i_1

Tactic:
exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)

Tactic:
have h₃ : ∀ᶠ j in atTop, Disjoint (b j) (W ∩ B j) := by
    apply hMr.mp
    rw [eventually_atTop]
    refine
      ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) ?_⟩
    change Disjoint (b j) (Z i)ᶜ
    rw [disjoint_compl_right_iff_subset]
    refine (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans
      ((cthickening_mono hj' _).trans fun a ha => ?_)
    simp only [Z, mem_iUnion, exists_prop]
    exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
⊢ ∀ᶠ (j : ℕ) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)

Tactic:
have h₄ : ∀ᶠ j in atTop, μ (B j) ≤ C * μ (b j) :=
    (hr.eventually (IsUnifLocDoublingMeasure.eventually_measure_le_scaling_constant_mul'
      μ M hM)).mono fun j hj => hj (w j)
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ :
  ∀ᶠ (j : ℕ) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
⊢ ∀ᶠ (j : ℕ) in atTop,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (r₁ (f j))) : ℝ≥0∞)
        ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (r₁ (f j))) : ℝ≥0∞) ≤
      (↑(1 - C⁻¹) : ℝ≥0∞)

Tactic:
(h₃.and h₄).mono fun j hj₀ => ?_

Premises used: Filter.Eventually.and, Filter.Eventually.mono
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ :
  ∀ᶠ (j : ℕ) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
refine :
  ∀ (j : ℕ)
    (hj₀ :
      Disjoint (b j) (W ∩ B j) ∧
        (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)),
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (r₁ (f j))) : ℝ≥0∞)
        ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (r₁ (f j))) : ℝ≥0∞) ≤
      (↑(1 - C⁻¹) : ℝ≥0∞)
⊢ ∀ᶠ (j : ℕ) in atTop,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (r₁ (f j))) : ℝ≥0∞)
        ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (r₁ (f j))) : ℝ≥0∞) ≤
      (↑(1 - C⁻¹) : ℝ≥0∞)

Tactic:
refine (h₃.and h₄).mono fun j hj₀ => ?_

Premises used: Filter.Eventually.and, Filter.Eventually.mono
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ :
  ∀ᶠ (j : ℕ) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
j : ℕ
hj₀ :
  Disjoint (b j) (W ∩ B j) ∧
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
change :
  HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ B j) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) ≤
    (↑(1 - C⁻¹) : ℝ≥0∞)
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (r₁ (f j))) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (r₁ (f j))) : ℝ≥0∞) ≤
    (↑(1 - C⁻¹) : ℝ≥0∞)

Tactic:
change μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)

Premises used: HSub.hSub, Inter.inter, Inv.inv, OfNat.ofNat, DFunLike.coe
-- Module: Mathlib.MeasureTheory.Covering.LiminfLimsup
Goal:
α : Type u_1
inst : PseudoMetricSpace α
inst_1 : SecondCountableTopology α
inst_2 : MeasurableSpace α
inst_3 : BorelSpace α
μ : Measure α
inst_4 : IsLocallyFiniteMeasure μ
inst_5 : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun (i : ℕ) ↦ cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun (i : ℕ) ↦ ⋃ (j : ℕ), ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (β := ℝ≥0∞)
          (fun (j : ι) ↦
            HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ closedBall (w j) (δ j)) : ℝ≥0∞)
              ((μ : (a : Set α) → ℝ≥0∞) (closedBall (w j) (δ j)) : ℝ≥0∞))
          l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : (j : ℕ) → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun (j : ℕ) ↦ closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ :
  ∀ᶠ (j : ℕ) in atTop,
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
j : ℕ
hj₀ :
  Disjoint (b j) (W ∩ B j) ∧
    (μ : (a : Set α) → ℝ≥0∞) (B j) ≤ HMul.hMul (β := ℝ≥0∞) (↑C : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (b j) : ℝ≥0∞)
inl :
  ∀ (hB : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) (∞ : ℝ≥0∞)),
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ B j) : ℝ≥0∞)
        ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) ≤
      (↑(1 - C⁻¹) : ℝ≥0∞)
inr :
  ∀ (hB : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) (∞ : ℝ≥0∞)),
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ B j) : ℝ≥0∞)
        ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) ≤
      (↑(1 - C⁻¹) : ℝ≥0∞)
⊢ HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (W ∩ B j) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (B j) : ℝ≥0∞) ≤
    (↑(1 - C⁻¹) : ℝ≥0∞)

Tactic:
rcases eq_or_ne (μ (B j)) ∞ with (hB | hB)

Premises used: eq_or_ne, Top.top, DFunLike.coe