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 : 0 < C
suffices :
  ∃ η < 1,
    ∀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∞) ≤
        (↑η : ℝ≥0∞)
⊢ False

Tactic:
suffices ∃ η < (1 : ℝ≥0),
      ∀ᶠ j in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ η by
    obtain ⟨η, hη, hη'⟩ := this
    replace hη' : 1 ≤ η := by
      simpa only [ENNReal.one_le_coe_iff] using
        le_of_tendsto (hd' w (fun j => r₁ (f j)) hr <| Eventually.of_forall hw') hη'
    exact (lt_self_iff_false _).mp (lt_of_lt_of_le hη hη')
-- 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 : 0 < C
⊢ ∃ η < 1,
    ∀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∞) ≤
        (↑η : ℝ≥0∞)

Tactic:
⟨1 - C⁻¹, tsub_lt_self zero_lt_one (inv_pos.mpr hC), ?_⟩

Premises used: Iff.mpr, HSub.hSub, inv_pos, zero_lt_one, Inv.inv, And.intro, Exists.intro, OfNat.ofNat, tsub_lt_self
-- 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 : 0 < C
refine :
  ∀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∞)
⊢ ∃ η < 1,
    ∀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∞) ≤
        (↑η : ℝ≥0∞)

Tactic:
refine ⟨1 - C⁻¹, tsub_lt_self zero_lt_one (inv_pos.mpr hC), ?_⟩
-- 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 : 0 < C
⊢ C ≠ 0

Tactic:
replace hC : C ≠ 0 := ne_of_gt hC
-- 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))
⊢ ∀ (j : ℕ), b j ⊆ B j

Tactic:
have h₁ : ∀ j, b j ⊆ B j := fun j =>
    closedBall_subset_closedBall (mul_le_of_le_one_left (hrp (f j)) hM'.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
⊢ ∀ (j : ℕ), W ∩ B j ⊆ B j

Tactic:
have h₂ : ∀ j, W ∩ B j ⊆ B j := fun j => inter_subset_right
-- 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
apply : ∀ᶠ (x : ℕ) in atTop, M * r₁ (f x) ≤ r₂ (f x) → Disjoint (b x) (W ∩ B x)
⊢ ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)

Tactic:
apply hMr.mp

Premises used: Filter.Eventually.mp
-- 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
rw : ∃ (a : ℕ), ∀ b_1 ≥ a, M * r₁ (f b_1) ≤ r₂ (f b_1) → Disjoint (b b_1) (W ∩ B b_1)
⊢ ∀ᶠ (x : ℕ) in atTop, M * r₁ (f x) ≤ r₂ (f x) → Disjoint (b x) (W ∩ B x)

Tactic:
rw [eventually_atTop]

Premises used: Filter.eventually_atTop
-- 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
⊢ ∃ (a : ℕ), ∀ b_1 ≥ a, M * r₁ (f b_1) ≤ r₂ (f b_1) → Disjoint (b b_1) (W ∩ B b_1)

Tactic:
⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) ?_⟩

Premises used: Filter.blimsup, Disjoint.inf_right', Filter.atTop, Exists.intro, Disjoint.inf_right
-- 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
fun : ∀ (j : ℕ) (hj : j ≥ i) (hj' : M * r₁ (f j) ≤ r₂ (f j)), Disjoint (b j) fun (a : α) ↦ a ∈ Z i → False
⊢ ∃ (a : ℕ), ∀ b_1 ≥ a, M * r₁ (f b_1) ≤ r₂ (f b_1) → Disjoint (b b_1) (W ∩ B b_1)

Tactic:
refine
      ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) ?_⟩
-- 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)
rw : b j ⊆ Z i
⊢ Disjoint (b j) (Z i)ᶜ

Tactic:
rw [disjoint_compl_right_iff_subset]

Premises used: Set.disjoint_compl_right_iff_subset
-- 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)
⊢ b j ⊆ Z i

Tactic:
(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