ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>M : Type u_2<VAR>M' : Type u_3<VAR>M'' : Type u_4<VAR>inst✝¹⁰ : AddCommMonoid M<VAR>inst✝⁹ : AddCommMonoid M'<VAR>inst✝⁸ : AddCommMonoid M''<VAR>A : Type u_5<VAR>inst✝⁷ : CommSemiring A<VAR>inst✝⁶ : Algebra R A<VAR>inst✝⁵ : Module A M'<VAR>inst✝⁴ : IsLocalization S A<VAR>inst✝³ : Module R M<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module R M''<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>g : M →ₗ[R] M''<VAR>e : M' ≃ₗ[R] M''<VAR>hf : IsLocalizedModule S f<VAR>x : M''<VAR>p : M × ↥S<VAR>h : p.2 • e.symm x = f p.1<GOAL>∃ x_1, x_1.2 • x = (↑e ∘ₗ f) x_1.1
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>M : Type u_2<VAR>M' : Type u_3<VAR>M'' : Type u_4<VAR>inst✝¹⁰ : AddCommMonoid M<VAR>inst✝⁹ : AddCommMonoid M'<VAR>inst✝⁸ : AddCommMonoid M''<VAR>A : Type u_5<VAR>inst✝⁷ : CommSemiring A<VAR>inst✝⁶ : Algebra R A<VAR>inst✝⁵ : Module A M'<VAR>inst✝⁴ : IsLocalization S A<VAR>inst✝³ : Module R M<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module R M''<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>g : M →ₗ[R] M''<VAR>e : M' ≃ₗ[R] M''<VAR>hf : IsLocalizedModule S f<VAR>x₁✝ x₂✝ : M<VAR>h : (↑e ∘ₗ f) x₁✝ = (↑e ∘ₗ f) x₂✝<GOAL>∃ c, c • x₁✝ = c • x₂✝
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>M : Type u_2<VAR>M' : Type u_3<VAR>M'' : Type u_4<VAR>inst✝¹⁰ : AddCommMonoid M<VAR>inst✝⁹ : AddCommMonoid M'<VAR>inst✝⁸ : AddCommMonoid M''<VAR>A : Type u_5<VAR>inst✝⁷ : CommSemiring A<VAR>inst✝⁶ : Algebra R A<VAR>inst✝⁵ : Module A M'<VAR>inst✝⁴ : IsLocalization S A<VAR>inst✝³ : Module R M<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module R M''<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>g : M →ₗ[R] M''<VAR>e : M' ≃ₗ[R] M''<VAR>hf : IsLocalizedModule S f<VAR>x₁✝ x₂✝ : M<VAR>h : f x₁✝ = f x₂✝<GOAL>∃ c, c • x₁✝ = c • x₂✝
<VAR>α : Type u<VAR>inst✝ : Monoid α<VAR>a b c u u₁ u₂ : αˣ<VAR>h : ↑u₁ = ↑u₂<GOAL>↑u₁ * ↑u₂⁻¹ = 1
<VAR>ι : Type u_1<VAR>n : ℕ<VAR>s✝ : Finset ι<VAR>s : Multiset ℤ<VAR>hs : 2 * n - 1 ≤ Multiset.card s<GOAL>∃ t ≤ s, Multiset.card t = n ∧ ↑n ∣ t.sum
<VAR>ι : Type u_1<VAR>n : ℕ<VAR>s✝ : Finset ι<VAR>s : Multiset ℤ<VAR>hs : 2 * n - 1 ≤ Multiset.card s<VAR>t : Finset (ℤ × ℕ)<VAR>hts : t ⊆ s.toEnumFinset<VAR>ht : t.card = n ∧ ↑n ∣ ∑ i ∈ t, i.1<GOAL>∃ t ≤ s, Multiset.card t = n ∧ ↑n ∣ t.sum
<VAR>R : Type u<VAR>S : Type v<VAR>σ : Type u_1<VAR>τ : Type u_2<VAR>r : R<VAR>e : ℕ<VAR>n m : σ<VAR>s : σ →₀ ℕ<VAR>inst✝¹ : CommSemiring R<VAR>p q : MvPolynomial σ R<VAR>inst✝ : DecidableEq σ<VAR>φ ψ : MvPolynomial σ R<GOAL>(φ * ψ).vars ⊆ φ.vars ∪ ψ.vars
<VAR>R : Type u<VAR>S : Type v<VAR>σ : Type u_1<VAR>τ : Type u_2<VAR>r : R<VAR>e : ℕ<VAR>n m : σ<VAR>s : σ →₀ ℕ<VAR>inst✝¹ : CommSemiring R<VAR>p q : MvPolynomial σ R<VAR>inst✝ : DecidableEq σ<VAR>φ ψ : MvPolynomial σ R<GOAL>(φ * ψ).degrees ⊆ φ.degrees + ψ.degrees
<VAR>α✝ : Type u_1<VAR>β : Type u_2<VAR>m : Type u_3<VAR>n : Type u_4<VAR>R : Type u_5<VAR>α : Type u_6<VAR>inst✝¹ : Zero α<VAR>inst✝ : One α<GOAL>(circulant fun i => if ↑i = 0 then 1 else 0) = 1
<VAR>α✝ : Type u_1<VAR>β : Type u_2<VAR>m : Type u_3<VAR>n✝ : Type u_4<VAR>R : Type u_5<VAR>α : Type u_6<VAR>inst✝¹ : Zero α<VAR>inst✝ : One α<VAR>n : ℕ<GOAL>(circulant fun i => if ↑i = 0 then 1 else 0) = 1
<VAR>α✝ : Type u_1<VAR>β : Type u_2<VAR>m : Type u_3<VAR>n✝ : Type u_4<VAR>R : Type u_5<VAR>α : Type u_6<VAR>inst✝¹ : Zero α<VAR>inst✝ : One α<VAR>n : ℕ<GOAL>(circulant fun i => if ↑i = 0 then 1 else 0) = circulant (Pi.single 0 1)
<VAR>α✝ : Type u_1<VAR>β : Type u_2<VAR>m : Type u_3<VAR>n✝ : Type u_4<VAR>R : Type u_5<VAR>α : Type u_6<VAR>inst✝¹ : Zero α<VAR>inst✝ : One α<VAR>n : ℕ<VAR>j j✝ : Fin (n + 1)<GOAL>circulant (fun i => if ↑i = 0 then 1 else 0) j j✝ = circulant (Pi.single 0 1) j j✝
<VAR>C : Type u<VAR>inst✝³ : Category.{v, u} C<VAR>J : GrothendieckTopology C<VAR>A : Type u_1<VAR>B : Type u_2<VAR>inst✝² : Category.{u_3, u_1} A<VAR>inst✝¹ : Category.{u_4, u_2} B<VAR>F : A ⥤ B<VAR>G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A<VAR>adj₁ : G₁ ⊣ sheafToPresheaf J A<VAR>G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B<VAR>adj₂ : G₂ ⊣ sheafToPresheaf J B<VAR>inst✝ : J.HasSheafCompose F<GOAL>J.PreservesSheafification F ↔ IsIso (sheafComposeNatTrans J F adj₁ adj₂)
<VAR>C : Type u<VAR>inst✝³ : Category.{v, u} C<VAR>J : GrothendieckTopology C<VAR>A : Type u_1<VAR>B : Type u_2<VAR>inst✝² : Category.{u_3, u_1} A<VAR>inst✝¹ : Category.{u_4, u_2} B<VAR>F : A ⥤ B<VAR>G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A<VAR>adj₁ : G₁ ⊣ sheafToPresheaf J A<VAR>G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B<VAR>adj₂ : G₂ ⊣ sheafToPresheaf J B<VAR>inst✝ : J.HasSheafCompose F<GOAL>(∀ (P : Cᵒᵖ ⥤ A), IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F))) ↔ ∀ (X : Cᵒᵖ ⥤ A), IsIso ((sheafComposeNatTrans J F adj₁ adj₂).app X)
<VAR>C : Type u<VAR>inst✝³ : Category.{v, u} C<VAR>J : GrothendieckTopology C<VAR>A : Type u_1<VAR>B : Type u_2<VAR>inst✝² : Category.{u_3, u_1} A<VAR>inst✝¹ : Category.{u_4, u_2} B<VAR>F : A ⥤ B<VAR>G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A<VAR>adj₁ : G₁ ⊣ sheafToPresheaf J A<VAR>G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B<VAR>adj₂ : G₂ ⊣ sheafToPresheaf J B<VAR>inst✝ : J.HasSheafCompose F<GOAL>∀ (a : Cᵒᵖ ⥤ A), IsIso (G₂.map (whiskerRight (adj₁.unit.app a) F)) ↔ IsIso ((sheafComposeNatTrans J F adj₁ adj₂).app a)
<VAR>C : Type u<VAR>inst✝³ : Category.{v, u} C<VAR>J : GrothendieckTopology C<VAR>A : Type u_1<VAR>B : Type u_2<VAR>inst✝² : Category.{u_3, u_1} A<VAR>inst✝¹ : Category.{u_4, u_2} B<VAR>F : A ⥤ B<VAR>G₁ : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A<VAR>adj₁ : G₁ ⊣ sheafToPresheaf J A<VAR>G₂ : (Cᵒᵖ ⥤ B) ⥤ Sheaf J B<VAR>adj₂ : G₂ ⊣ sheafToPresheaf J B<VAR>inst✝ : J.HasSheafCompose F<VAR>P : Cᵒᵖ ⥤ A<GOAL>IsIso (G₂.map (whiskerRight (adj₁.unit.app P) F)) ↔ IsIso ((sheafComposeNatTrans J F adj₁ adj₂).app P)
<VAR>β : Type u_1<VAR>G : Type u_2<VAR>M : Type u_3<VAR>inst✝³ : Monoid M<VAR>inst✝² : LinearOrder M<VAR>inst✝¹ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1<VAR>inst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1<VAR>a b c : M<VAR>h : a * b < c ^ 2<GOAL>min a b < c
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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<GOAL>blimsup Y₁ atTop p ≤ᶠ[ae μ] blimsup Y₂ atTop p
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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<GOAL>∀ (i : ℕ), μ (blimsup Y₁ atTop p \ Z i) = 0
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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<GOAL>μ W = 0
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 : d ∈ W<VAR>hd' : ∀ {ι : Type ?u.7325} {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)<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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>hf : ∀ (i : ℕ), d ∈ Y₁ (f i) ∧ p (f i) ∧ f i ∈ Ici i<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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>hf : (∀ (x : ℕ), d ∈ Y₁ (f x)) ∧ (∀ (x : ℕ), p (f x)) ∧ ∀ (x : ℕ), f x ∈ Ici x<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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>hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<GOAL>False
<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>hr : Tendsto r₁ atTop (𝓝[>] 0)<VAR>hrp : 0 ≤ r₁<VAR>M : ℝ<VAR>hM : 0 < M<VAR>hM' : M < 1<VAR>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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>hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))<VAR>hf₁ : ∀ (x : ℕ), p (f x)<VAR>hf₂ : ∀ (j : ℕ), j ≤ f j<VAR>hf₃ : Tendsto f atTop atTop<GOAL>False
<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>hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i<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 ?u.7325} {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>hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))<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)<GOAL>False
<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 ?u.7325} {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>hf₀ : ∀ (j : ℕ), d ∈ cthickening (r₁ (f j)) (s (f j))<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)<GOAL>False
<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 ?u.7325} {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<GOAL>False
<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 ?u.7325} {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 : 0 < C<GOAL>False
<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 : 0 < C<GOAL>∃ η < 1, ∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑η
<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 : 0 < C<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<GOAL>∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)