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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
set :
  let W := blimsup Y₁ atTop p \ Z i;
  Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (blimsup Y₁ atTop p \ Z i) : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
set W := atTop.blimsup Y₁ p \ Z i

Premises used: Filter.blimsup, Filter.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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
by_contra : ∀ (contra : ¬Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)), False
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) W : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
by_contra contra
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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 ∈ W,
    ∀ {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)

Tactic:
Measure.exists_mem_of_measure_ne_zero_of_ae contra
      (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ W 2)

Premises used: IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div, MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae, OfNat.ofNat
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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∞)
obtain :
  ∀ (d : α) (hd : d ∈ W)
    (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)),
    False
⊢ False

Tactic:
obtain ⟨d, hd, hd'⟩ : ∃ d, d ∈ W ∧ ∀ {ι : Type _} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
      Tendsto δ l (𝓝[>] 0) → (∀ᶠ j in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1) :=
    Measure.exists_mem_of_measure_ne_zero_of_ae contra
      (IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div μ W 2)

Premises used: IsUnifLocDoublingMeasure.ae_tendsto_measure_inter_div, MeasureTheory.Measure.exists_mem_of_measure_ne_zero_of_ae, OfNat.ofNat
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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 : d ∈ W
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)
⊢ d ∈ blimsup Y₁ atTop p

Tactic:
replace hd : d ∈ blimsup Y₁ atTop p := ((mem_diff _).mp hd).1
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
obtain : ∀ (f : ℕ → ℕ) (hf : ∀ (i : ℕ), d ∈ Y₁ (f i) ∧ p (f i) ∧ f i ∈ Ici i), False
⊢ False

Tactic:
obtain ⟨f : ℕ → ℕ, hf⟩ := exists_forall_mem_of_hasBasis_mem_blimsup' atTop_basis hd

Premises used: Filter.atTop_basis, Filter.exists_forall_mem_of_hasBasis_mem_blimsup'
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf : ∀ (i : ℕ), d ∈ Y₁ (f i) ∧ p (f i) ∧ f i ∈ Ici i
⊢ (∀ (x : ℕ), d ∈ Y₁ (f x)) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x

Tactic:
simp only [forall_and] at hf

Premises used: forall_and
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf : (∀ (x : ℕ), d ∈ Y₁ (f x)) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x
⊢ ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))

Tactic:
obtain ⟨hf₀ : ∀ j, d ∈ cthickening (r₁ (f j)) (s (f j)), hf₁, hf₂ : ∀ j, j ≤ f j⟩ := hf
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf : (∀ (x : ℕ), d ∈ Y₁ (f x)) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
⊢ ∀ (x : ℕ), p (f x)

Tactic:
obtain ⟨hf₀ : ∀ j, d ∈ cthickening (r₁ (f j)) (s (f j)), hf₁, hf₂ : ∀ j, j ≤ f j⟩ := hf
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf : (∀ (x : ℕ), d ∈ Y₁ (f x)) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
⊢ ∀ (j : ℕ), j ≤ f j

Tactic:
obtain ⟨hf₀ : ∀ j, d ∈ cthickening (r₁ (f j)) (s (f j)), hf₁, hf₂ : ∀ j, j ≤ f j⟩ := hf
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
⊢ Tendsto f atTop atTop

Tactic:
have hf₃ : Tendsto f atTop atTop :=
    tendsto_atTop_atTop.mpr fun j => ⟨f j, fun i hi => (hf₂ j).trans (hi.trans <| hf₂ i)⟩
-- 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
hr : Tendsto r₁ atTop (𝓝[>] 0)
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
⊢ Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)

Tactic:
replace hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0) := hr.comp hf₃
-- 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
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
⊢ ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)

Tactic:
replace hMr : ∀ᶠ j in atTop, M * r₁ (f j) ≤ r₂ (f j) := hf₃.eventually hMr
-- 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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
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)
j : ℕ
⊢ 0 (f j) ≤ r₁ (f j)

Tactic:
specialize hrp (f 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
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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
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)
j : ℕ
hrp : 0 (f j) ≤ r₁ (f j)
⊢ 0 ≤ r₁ (f j)

Tactic:
rw [Pi.zero_apply] at hrp

Premises used: Pi.zero_apply
-- 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
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
hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))
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)
j : ℕ
hrp : 0 ≤ r₁ (f j)
inl : ∀ (hr0 : 0 = r₁ (f j)), ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
inr : ∀ (hrp' : 0 < r₁ (f j)), ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
⊢ ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))

Tactic:
rcases eq_or_lt_of_le hrp with (hr0 | hrp')

Premises used: eq_or_lt_of_le