ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<GOAL>IsLocalizedModule S f ↔ IsBaseChange A f
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>h : IsBaseChange A f<GOAL>IsLocalizedModule S f
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>h : IsBaseChange A f<VAR>this : IsBaseChange A (LocalizedModule.mkLinearMap S M)<GOAL>IsLocalizedModule S f
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>h : IsBaseChange A f<VAR>this : IsBaseChange A (LocalizedModule.mkLinearMap S M)<VAR>e : LocalizedModule S M ≃ₗ[R] M' := LinearEquiv.restrictScalars R (this.equiv.symm ≪≫ₗ h.equiv)<GOAL>IsLocalizedModule S f
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>h : IsBaseChange A f<VAR>this : IsBaseChange A (LocalizedModule.mkLinearMap S M)<VAR>e : LocalizedModule S M ≃ₗ[R] M' := LinearEquiv.restrictScalars R (this.equiv.symm ≪≫ₗ h.equiv)<GOAL>f = ↑e ∘ₗ LocalizedModule.mkLinearMap S M
<VAR>R : Type u_1<VAR>inst✝¹¹ : CommSemiring R<VAR>S : Submonoid R<VAR>A : Type u_2<VAR>inst✝¹⁰ : CommRing A<VAR>inst✝⁹ : Algebra R A<VAR>inst✝⁸ : IsLocalization S A<VAR>M : Type u_3<VAR>inst✝⁷ : AddCommMonoid M<VAR>inst✝⁶ : Module R M<VAR>inst✝⁵ : Module A M<VAR>inst✝⁴ : IsScalarTower R A M<VAR>M' : Type u_4<VAR>inst✝³ : AddCommMonoid M'<VAR>inst✝² : Module R M'<VAR>inst✝¹ : Module A M'<VAR>inst✝ : IsScalarTower R A M'<VAR>f : M →ₗ[R] M'<VAR>h : IsBaseChange A f<VAR>this : IsBaseChange A (LocalizedModule.mkLinearMap S M)<VAR>e : LocalizedModule S M ≃ₗ[R] M' := LinearEquiv.restrictScalars R (this.equiv.symm ≪≫ₗ h.equiv)<VAR>x✝ : M<GOAL>f x✝ = (↑e ∘ₗ LocalizedModule.mkLinearMap S M) x✝
<VAR>𝕜 : Type u<VAR>inst✝⁸ : NontriviallyNormedField 𝕜<VAR>E : Type uE<VAR>inst✝⁷ : NormedAddCommGroup E<VAR>inst✝⁶ : NormedSpace 𝕜 E<VAR>F : Type uF<VAR>inst✝⁵ : NormedAddCommGroup F<VAR>inst✝⁴ : NormedSpace 𝕜 F<VAR>G : Type uG<VAR>inst✝³ : NormedAddCommGroup G<VAR>inst✝² : NormedSpace 𝕜 G<VAR>X : Type uX<VAR>inst✝¹ : NormedAddCommGroup X<VAR>inst✝ : NormedSpace 𝕜 X<VAR>s s₁ t u : Set E<VAR>f f₁ : E → F<VAR>g : F → G<VAR>x x₀ : E<VAR>c : F<VAR>m n : ℕ∞<VAR>p : E → FormalMultilinearSeries 𝕜 E F<GOAL>ContDiffAt 𝕜 1 f x ↔ ∃ f', ∃ u ∈ 𝓝 x, ContinuousOn f' u ∧ ∀ x ∈ u, HasFDerivAt f (f' x) x
<VAR>R : Type u_1<VAR>A : Type u_2<VAR>inst✝² : CommSemiring R<VAR>inst✝¹ : Semiring A<VAR>inst✝ : Algebra R A<VAR>s : NonUnitalSubalgebra R A<GOAL>Subalgebra.toSubmodule (adjoin R ↑s) = span R {1} ⊔ s.toSubmodule
<VAR>ι : Type u_1<VAR>ι' : Type u_2<VAR>k : Type u_3<VAR>V : Type u_4<VAR>P : Type u_5<VAR>inst✝⁴ : AddCommGroup V<VAR>inst✝³ : AffineSpace V P<VAR>inst✝² : DivisionRing k<VAR>inst✝¹ : Module k V<VAR>inst✝ : CharZero k<VAR>b : AffineBasis ι k P<VAR>s : Finset ι<VAR>i : ι<VAR>hi : i ∈ s<GOAL>(b.coord i) (Finset.centroid k s ⇑b) = (↑s.card)⁻¹
<VAR>ι : Type uι<VAR>inst✝⁵ : Fintype ι<VAR>𝕜 : Type u𝕜<VAR>inst✝⁴ : NontriviallyNormedField 𝕜<VAR>E : ι → Type uE<VAR>inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)<VAR>inst✝² : (i : ι) → NormedSpace 𝕜 (E i)<VAR>F : Type uF<VAR>inst✝¹ : SeminormedAddCommGroup F<VAR>inst✝ : NormedSpace 𝕜 F<VAR>f : ContinuousMultilinearMap 𝕜 E F<VAR>x : ⨂[𝕜] (i : ι), E i<GOAL>((liftIsometry 𝕜 E F) f) x = (lift f.toMultilinearMap) x
<VAR>ι : Sort u_1<VAR>α : Type u<VAR>β : Type v<VAR>inst✝ : PseudoEMetricSpace α<VAR>δ ε : ℝ<VAR>s t : Set α<VAR>x : α<VAR>E : Set α<GOAL>closure E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E
<VAR>ι : Sort u_1<VAR>α : Type u<VAR>β : Type v<VAR>inst✝ : PseudoEMetricSpace α<VAR>δ ε : ℝ<VAR>s t : Set α<VAR>x : α<VAR>E : Set α<GOAL>cthickening 0 E = ⋂ δ, ⋂ (_ : 0 < δ), cthickening δ E
<VAR>X : Type u<VAR>inst✝¹ : Lattice X<VAR>inst✝ : JordanHolderLattice X<VAR>s₁ s₂ : CompositionSeries X<VAR>h : ∀ (x : X), x ∈ s₁ ↔ x ∈ s₂<GOAL>(toList s₁).Perm (toList s₂)
<VAR>a b c : ℕ<VAR>r : ℤ<VAR>ha : a % 2 = 0<VAR>hb : b % 8 = 3<VAR>hc : a / 2 = c<VAR>hr : jacobiSymNat c b = r<GOAL>jacobiSymNat a b = -r
<VAR>a b c : ℕ<VAR>r : ℤ<VAR>ha : a % 2 = 0<VAR>hb : b % 8 = 3<VAR>hc : a / 2 = c<VAR>hr : jacobiSymNat c b = r<GOAL>jacobiSym (↑a) b = -jacobiSym (↑a / 2) b
<VAR>a b c : ℕ<VAR>r : ℤ<VAR>ha : a % 2 = 0<VAR>hb : b % 8 = 3<VAR>hc : a / 2 = c<VAR>hr : jacobiSymNat c b = r<GOAL>b % 2 = 1
<VAR>a b c : ℕ<VAR>r : ℤ<VAR>ha : a % 2 = 0<VAR>hb : b % 8 = 3<VAR>hc : a / 2 = c<VAR>hr : jacobiSymNat c b = r<GOAL>2 ∣ 8
<VAR>x y : ℂ<GOAL>cexp (-x) = (cexp x)⁻¹
<VAR>x y : ℂ<GOAL>cexp (x + -x) = cexp x * (cexp x)⁻¹
<VAR>α✝ : Sort u_1<VAR>β : Sort u_2<VAR>ι : Type u_3<VAR>α : ι → Sort u_4<VAR>s : (i : ι) → Setoid (α i)<VAR>p : ((i : ι) → Quotient (s i)) → Prop<VAR>f : (i : ι) → Quotient (s i)<VAR>h : ∀ (a : (i : ι) → α i), p fun i => ⟦a i⟧<GOAL>p f
<VAR>α✝ : Sort u_1<VAR>β : Sort u_2<VAR>ι : Type u_3<VAR>α : ι → Sort u_4<VAR>s : (i : ι) → Setoid (α i)<VAR>p : ((i : ι) → Quotient (s i)) → Prop<VAR>f : (i : ι) → Quotient (s i)<VAR>h : ∀ (a : (i : ι) → α i), p fun i => ⟦a i⟧<GOAL>p fun i => ⟦(f i).out⟧
<VAR>a b c : Ordinal.{u}<GOAL>a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c'
<VAR>a b c : Ordinal.{u}<GOAL>a < max (b.blsub fun a' x => a' ♯ c) (c.blsub fun b' x => b ♯ b') ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c'
<VAR>x : ℝ≥0∞<VAR>n : ℤ<VAR>z : ℝ<GOAL>x ^ (↑n * z) = (x ^ n) ^ z
<VAR>R : Type u_1<VAR>F : Type u<VAR>K : Type v<VAR>L : Type w<VAR>inst✝² : CommRing K<VAR>inst✝¹ : Field L<VAR>inst✝ : Field F<VAR>i : K →+* L<VAR>f g : K[X]<VAR>hf : Splits i f<VAR>hg : Splits i g<VAR>this : DecidableEq L := Classical.decEq L<VAR>h : ¬map i (f * g) = 0<VAR>p : L[X]<VAR>hp : Irreducible p<VAR>hpf : p ∣ map i (f * g)<GOAL>map i f * map i g = map i (f * g)
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>this : ∀ (x : α), ∃ R, 0 < R ∧ R ≤ 1 ∧ μ (closedBall x (20 * R)) < ⊤<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<GOAL>∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<GOAL>μ (s \ ⋃ a ∈ u, B a) = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<GOAL>∃ u_1 ∈ 𝓝[s \ ⋃ a ∈ u, B a] x, μ u_1 = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<GOAL>∃ u_1 ∈ 𝓝[s \ ⋃ a ∈ u, B a] x, μ u_1 = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<GOAL>∃ u_1 ∈ 𝓝[s \ ⋃ a ∈ u, B a] x, μ u_1 = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<GOAL>∃ u_1 ∈ 𝓝[s \ ⋃ a ∈ u, B a] x, μ u_1 = 0
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<VAR>ε : ℝ≥0∞<VAR>εpos : 0 < ε<GOAL>μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<VAR>ε : ℝ≥0∞<VAR>εpos : 0 < ε<VAR>I : ∑' (a : ↑v), μ (B ↑a) < ⊤<GOAL>μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<VAR>ε : ℝ≥0∞<VAR>εpos : 0 < ε<VAR>I : ∑' (a : ↑v), μ (B ↑a) < ⊤<VAR>w : Finset ↑v<VAR>hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C<GOAL>μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<VAR>ε : ℝ≥0∞<VAR>εpos : 0 < ε<VAR>I : ∑' (a : ↑v), μ (B ↑a) < ⊤<VAR>w : Finset ↑v<VAR>hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C<VAR>M : (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)<GOAL>μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
<VAR>α : Type u_1<VAR>ι : Type u_2<VAR>inst✝⁴ : MetricSpace α<VAR>inst✝³ : MeasurableSpace α<VAR>inst✝² : OpensMeasurableSpace α<VAR>inst✝¹ : SecondCountableTopology α<VAR>μ : Measure α<VAR>inst✝ : IsLocallyFiniteMeasure μ<VAR>s : Set α<VAR>t : Set ι<VAR>C : ℝ≥0<VAR>r : ι → ℝ<VAR>c : ι → α<VAR>B : ι → Set α<VAR>hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)<VAR>μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)<VAR>ht : ∀ a ∈ t, (interior (B a)).Nonempty<VAR>h't : ∀ a ∈ t, IsClosed (B a)<VAR>hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x<VAR>R : α → ℝ<VAR>hR0 : ∀ (x : α), 0 < R x<VAR>hR1 : ∀ (x : α), R x ≤ 1<VAR>hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤<VAR>t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}<VAR>u : Set ι<VAR>ut' : u ⊆ t'<VAR>u_disj : u.PairwiseDisjoint B<VAR>hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b<VAR>ut : u ⊆ t<VAR>u_count : u.Countable<VAR>x : α<VAR>x✝ : x ∈ s \ ⋃ a ∈ u, B a<VAR>v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}<VAR>vu : v ⊆ u<VAR>K : ℝ<VAR>μK : μ (closedBall x K) < ⊤<VAR>hK : ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K<VAR>ε : ℝ≥0∞<VAR>εpos : 0 < ε<VAR>I : ∑' (a : ↑v), μ (B ↑a) < ⊤<VAR>w : Finset ↑v<VAR>hw : ∑' (a : { a // a ∉ w }), μ (B ↑↑a) < ε / ↑C<VAR>M : (s \ ⋃ a ∈ u, B a) ∩ ball x (R x) ⊆ ⋃ a, closedBall (c ↑↑a) (3 * r ↑↑a)<VAR>this : Countable ↑v<GOAL>μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ ε
<VAR>U : Type u_1<VAR>inst✝ : Quiver U<VAR>u v v' w : U<VAR>p : Path u v<VAR>p' : Path u v'<VAR>e : v ⟶ w<VAR>e' : v' ⟶ w<VAR>h : p.cons e = p'.cons e'<GOAL>Path.cast ⋯ ⋯ p = p'