ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<GOAL>μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<GOAL>μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<VAR>hB : μ (B j) = ⊤<GOAL>μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<VAR>hB : μ (B j) ≠ ⊤<GOAL>μ (W ∩ B j) / μ (B j) ≤ ↑(1 - C⁻¹)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<VAR>hB : μ (B j) ≠ ⊤<GOAL>μ (W ∩ B j) ≤ ↑(1 - C⁻¹) * μ (B j)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<VAR>hB : μ (B j) ≠ ⊤<GOAL>μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)<VAR>hB : ↑C⁻¹ * μ (B j) ≠ ⊤<GOAL>μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hB : ↑C⁻¹ * μ (B j) ≠ ⊤<VAR>hj₁ : Disjoint (b j) (W ∩ B j)<VAR>hj₂ : μ (B j) ≤ ↑C * μ (b j)<GOAL>μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
<VAR>α : Type u_1<VAR>inst✝⁵ : MetricSpace α<VAR>inst✝⁴ : SecondCountableTopology α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : BorelSpace α<VAR>μ : Measure α<VAR>inst✝¹ : IsLocallyFiniteMeasure μ<VAR>inst✝ : IsUnifLocDoublingMeasure μ<VAR>p : ℕ → Prop<VAR>s : ℕ → Set α<VAR>hs : ∀ (i : ℕ), IsClosed (s i)<VAR>r₁ r₂ : ℕ → ℝ<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)<VAR>Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)<VAR>Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j<VAR>i : ℕ<VAR>W : Set α := blimsup Y₁ atTop p \ Z i<VAR>contra : ¬μ W = 0<VAR>d : α<VAR>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 (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)<VAR>hd : d ∈ blimsup Y₁ atTop p<VAR>f : ℕ → ℕ<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<VAR>hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)<VAR>hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)<VAR>w : ℕ → α<VAR>hw : ∀ (j : ℕ), w j ∈ s (f j)<VAR>hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))<VAR>C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹<VAR>hC : C ≠ 0<VAR>b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))<VAR>B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))<VAR>h₁ : ∀ (j : ℕ), b j ⊆ B j<VAR>h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j<VAR>h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)<VAR>h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)<VAR>j : ℕ<VAR>hB : ↑C⁻¹ * μ (B j) ≠ ⊤<VAR>hj₁ : Disjoint (b j) (W ∩ B j)<VAR>hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)<GOAL>μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)