awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
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))
⊢ ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
have inr : ∀ (hrp' : 0 < 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')
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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)
hr0 : 0 = r₁ (f j)
⊢ ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
have hf₄ : d ∈ cthickening (r₁ (f j)) (s (f j))
Tactic: specialize hf₀ j
Cleared hypotheses: hf₀
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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
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)
hr0 : 0 = r₁ (f j)
hf₀ : d ∈ cthickening (r₁ (f j)) (s (f j))
⊢ ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
have hf₄ : d ∈ s (f j)
Tactic: rw [← hr0, cthickening_zero, (hs (f j)).closure_eq] at hf₀
Cleared hypotheses: hf₀
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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)
⊢ False
have hf₄ : ∀ (j : ℕ), ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
Tactic: replace hf₀ : ∀ j, ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j)) := by
    intro j
    specialize hrp (f j)
    rw [Pi.zero_apply] at hrp
    rcases eq_or_lt_of_le hrp with (hr0 | hrp')
    · specialize hf₀ j
      rw [← hr0, cthickening_zero, (hs (f j)).closure_eq] at hf₀
      exact ⟨d, hf₀, by simp [← hr0]⟩
    · simpa using mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))
        (by positivity) (lt_two_mul_self hrp') (hf₀ j))
Cleared hypotheses: hf₀
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
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)
hf₀ : ∀ (j : ℕ), ∃ w ∈ s (f j), d ∈ closedBall w (2 * r₁ (f j))
⊢ False
have choose : ∀ (w : (j : ℕ) → α) (hw : ∀ (j : ℕ), w j ∈ s (f j)) (hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))), False
Tactic: choose w hw hw' using hf₀
Cleared hypotheses: hf₀
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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
⊢ False
have hC : 0 < C
Tactic: have hC : 0 < C :=
    lt_of_lt_of_le zero_lt_one (IsUnifLocDoublingMeasure.one_le_scalingConstantOf μ M⁻¹)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
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
this :
  ∃ 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
have obtain : ∀ (η : ℝ≥0) (hη : η < 1)
  (hη' :
    ∀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: obtain ⟨η, hη, hη'⟩ := this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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 : ℝ≥0
hη : η < 1
hη' :
  ∀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
have hη'₁ : 1 ≤ η
Tactic: 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'
Cleared hypotheses: hη'
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
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
⊢ False
have 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∞)
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')
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (goal)
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∞)
have 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∞)
Tactic: refine ⟨1 - C⁻¹, tsub_lt_self zero_lt_one (inv_pos.mpr hC), ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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
⊢ ∀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∞)
have hC₁ : C ≠ 0
Tactic: replace hC : C ≠ 0 := ne_of_gt hC
Cleared hypotheses: hC
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.LiminfLimsup (hypothesis)
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))
⊢ ∀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∞)
have h₁ : ∀ (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)