ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K✝
inst✝³¹ : Field L✝
inst✝³⁰ : Algebra A K✝
inst✝²⁹ : IsFractionRing A K✝
inst✝²⁸ : Algebra B L✝
inst✝²⁷ : Algebra K✝ L✝
inst✝²⁶ : Algebra A L✝
inst✝²⁵ : IsScalarTower A B L✝
inst✝²⁴ : IsScalarTower A K✝ L✝
inst✝²³ : IsIntegralClosure B A L✝
inst✝²² : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝¹⁰ : IsIntegralClosure B A L
this✝⁹ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁸ : Algebra Aₘ K := f.toAlgebra
this✝⁷ : IsScalarTower A Aₘ K
this✝⁶ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝⁵ : Algebra Bₘ L := g.toAlgebra
this✝⁴ : IsScalarTower B Bₘ L
this✝³ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this✝² : IsScalarTower Aₘ K L
this✝¹ : IsScalarTower Aₘ Bₘ L
this✝ : IsFractionRing Bₘ L := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (algebraMapSubmonoid B M) Bₘ L
this : FiniteDimensional K L

⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K✝
inst✝³¹ : Field L✝
inst✝³⁰ : Algebra A K✝
inst✝²⁹ : IsFractionRing A K✝
inst✝²⁸ : Algebra B L✝
inst✝²⁷ : Algebra K✝ L✝
inst✝²⁶ : Algebra A L✝
inst✝²⁵ : IsScalarTower A B L✝
inst✝²⁴ : IsScalarTower A K✝ L✝
inst✝²³ : IsIntegralClosure B A L✝
inst✝²² : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝¹¹ : IsIntegralClosure B A L
this✝¹⁰ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁹ : Algebra Aₘ K := f.toAlgebra
this✝⁸ : IsScalarTower A Aₘ K
this✝⁷ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝⁶ : Algebra Bₘ L := g.toAlgebra
this✝⁵ : IsScalarTower B Bₘ L
this✝⁴ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this✝³ : IsScalarTower Aₘ K L
this✝² : IsScalarTower Aₘ Bₘ L
this✝¹ : IsFractionRing Bₘ L := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (algebraMapSubmonoid B M) Bₘ L
this✝ : FiniteDimensional K L
this : IsIntegralClosure Bₘ Aₘ L

⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)
-- Mathlib/RingTheory/IntegralClosure/IntegralRestrict.lean
A : Type u_1
K✝ : Type u_2
L✝ : Type u_3
B : Type u_4
inst✝³⁵ : CommRing A
inst✝³⁴ : CommRing B
inst✝³³ : Algebra A B
inst✝³² : Field K✝
inst✝³¹ : Field L✝
inst✝³⁰ : Algebra A K✝
inst✝²⁹ : IsFractionRing A K✝
inst✝²⁸ : Algebra B L✝
inst✝²⁷ : Algebra K✝ L✝
inst✝²⁶ : Algebra A L✝
inst✝²⁵ : IsScalarTower A B L✝
inst✝²⁴ : IsScalarTower A K✝ L✝
inst✝²³ : IsIntegralClosure B A L✝
inst✝²² : FiniteDimensional K✝ L✝
Aₘ : Type u_5
Bₘ : Type u_6
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : NoZeroSMulDivisors A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : NoZeroSMulDivisors Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_4 := FractionRing B
this✝¹¹ : IsIntegralClosure B A L
this✝¹⁰ : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
this✝⁹ : Algebra Aₘ K := f.toAlgebra
this✝⁸ : IsScalarTower A Aₘ K
this✝⁷ : IsFractionRing Aₘ K := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization M Aₘ K
g : Bₘ →+* L := IsLocalization.map L (RingHom.id B) ⋯
this✝⁶ : Algebra Bₘ L := g.toAlgebra
this✝⁵ : IsScalarTower B Bₘ L
this✝⁴ : Algebra Aₘ L := ((algebraMap K L).comp f).toAlgebra
this✝³ : IsScalarTower Aₘ K L
this✝² : IsScalarTower Aₘ Bₘ L
this✝¹ : IsFractionRing Bₘ L := IsFractionRing.isFractionRing_of_isDomain_of_isLocalization (algebraMapSubmonoid B M) Bₘ L
this✝ : FiniteDimensional K L
this : IsIntegralClosure Bₘ Aₘ L

⊢ (algebraMap Aₘ K) ((algebraMap A Aₘ) ((intTrace A B) x)) = (algebraMap Aₘ K) ((intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x))
-- Mathlib/Order/UpperLower/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝ : LE α
s t : UpperSet α
a : α
f : (i : ι) → κ i → UpperSet α

⊢ (⨅ i, ⨅ j, f i j).compl = ⨅ i, ⨅ j, (f i j).compl
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M

⊢ ∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioc 0 M, (μ {a | t ≤ f a}).toReal
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M

⊢ ∫ (t : ℝ) in Ioi 0, (μ {a | t ≤ f a}).toReal = ∫ (t : ℝ) in Ioc 0 M, (μ {a | t ≤ f a}).toReal
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M

⊢ ∀ᵐ (x : ℝ), x ∈ Ioi 0 \ Ioc 0 M → (μ {a | x ≤ f a}).toReal = 0
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M

⊢ (μ {a | t ≤ f a}).toReal = 0
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t

⊢ (μ {a | t ≤ f a}).toReal = 0
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
obs : μ {a | M < f a} = 0

⊢ (μ {a | t ≤ f a}).toReal = 0
-- Mathlib/MeasureTheory/Integral/Layercake.lean
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᶠ[ae μ] f
f_bdd : f ≤ᶠ[ae μ] fun x => M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
obs : μ {a | M < f a} = 0

⊢ μ {a | t ≤ f a} = 0 ∨ μ {a | t ≤ f a} = ⊤
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ

⊢ m.toNat ≤ n ↔ m ≤ ↑n
-- Mathlib/Data/Int/Defs.lean
a b c d m n✝ : ℤ
n : ℕ

⊢ m ≤ ↑n ∧ 0 ≤ ↑n ↔ m ≤ ↑n
-- Mathlib/CategoryTheory/Bicategory/Strict.lean
B : Type u
inst✝ : Bicategory B
a b c : B
f : a ⟶ b
h : b ⟶ c

⊢ eqToHom ⋯ ▷ h = eqToHom ⋯
-- Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean
z τ : ℂ

⊢ jacobiTheta₂'' (-z) τ = -jacobiTheta₂'' z τ
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R

⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R

⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)

⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c)

⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- Mathlib/Probability/Martingale/BorelCantelli.lean
Ω : Type u_1
m0 : MeasurableSpace Ω
μ : Measure Ω
ℱ : Filtration ℕ m0
f : ℕ → Ω → ℝ
ω✝ : Ω
r : ℝ
R : ℝ≥0
inst✝ : IsFiniteMeasure μ
hf : Martingale f ℱ μ
hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R
hbdd' : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(-f) (i + 1) ω - (-f) i ω| ≤ ↑R
hup : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
hdown : ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c)
ω : Ω
hω₁ : BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)
hω₂ : BddAbove (Set.range fun n => (-f) n ω) ↔ ∃ c, Tendsto (fun n => (-f) n ω) atTop (𝓝 c)

⊢ BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- Mathlib/CategoryTheory/EssentiallySmall.lean
C✝ : Type u
inst✝² : Category.{v, u} C✝
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Quiver.IsThin C

⊢ EssentiallySmall.{w, v, u} C ↔ Small.{w, u} (Skeleton C)
-- Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
f : E → F
R : ℝ
hR : 0 ≤ R

⊢ ∫ (x : E), f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ (x : E), f x ∂μ
-- Mathlib/Algebra/GroupWithZero/Units/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : MonoidWithZero M₀
x y z : M₀
h : IsUnit z
h1 : x = y * inverse z

⊢ x * z = y
-- Mathlib/Algebra/GroupWithZero/Units/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝ : MonoidWithZero M₀
x y z : M₀
h : IsUnit z
h1 : x * z = y

⊢ x = y * inverse z
-- Mathlib/Algebra/Order/Interval/Set/Monoid.lean
M : Type u_1
inst✝¹ : OrderedCancelAddCommMonoid M
inst✝ : ExistsAddOfLE M
a b c d : M

⊢ (fun x => a + x) '' Ico b c = Ico (a + b) (a + c)
-- Mathlib/Topology/Algebra/InfiniteSum/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : CommMonoid α
inst✝ : TopologicalSpace α
f g : β → α
a b : α
s : Finset β
h : ¬Multipliable f

⊢ ∏' (b : β), f b = 1
-- Mathlib/Topology/Bornology/Absorbs.lean
G₀ : Type u_1
α : Type u_2
inst✝² : GroupWithZero G₀
inst✝¹ : Bornology G₀
inst✝ : MulAction G₀ α
s t u : Set α
S : Set (Set α)
hS : S.Finite

⊢ Absorbs G₀ (⋂₀ S) t ↔ ∀ s ∈ S, Absorbs G₀ s t
-- Mathlib/Order/Filter/Archimedean.lean
α : Type u_1
R : Type u_2
l : Filter α
f : α → R
r : R
inst✝¹ : LinearOrderedRing R
inst✝ : Archimedean R
hr : r < 0
hf : Tendsto f l atTop

⊢ Tendsto (fun x => f x * r) l atBot
-- Mathlib/Topology/Irreducible.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t u v : Set X
hu : IsOpen u
hv : IsOpen v

⊢ (closure s ∩ u).Nonempty → (closure s ∩ v).Nonempty → (closure s ∩ (u ∩ v)).Nonempty ↔ (s ∩ u).Nonempty → (s ∩ v).Nonempty → (s ∩ (u ∩ v)).Nonempty
-- Mathlib/Topology/Irreducible.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t u v : Set X
hu : IsOpen u
hv : IsOpen v

⊢ IsOpen (u ∩ v)
-- Mathlib/Topology/Irreducible.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t u v : Set X
hu : IsOpen u
hv : IsOpen v

⊢ IsOpen v
-- Mathlib/Topology/Irreducible.lean
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t u v : Set X
hu : IsOpen u
hv : IsOpen v

⊢ IsOpen u
-- Mathlib/Algebra/Order/Ring/Basic.lean
α : Type u_1
M : Type u_2
R : Type u_3
inst✝ : LinearOrderedSemiring R
a b : R
m n : ℕ
ha : 0 ≤ a
hn : n ≠ 0

⊢ 1 ≤ a ^ n ↔ 1 ≤ a
-- Mathlib/Order/InitialSeg.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder β s
f : r ↪r s
a : α

⊢ ↑(⋯.fix (fun a IH => let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b}; let_fun this := ⋯; ⟨⋯.min S ⋯, ⋯⟩) a) ∈ {b | ∀ (a' : α), r a' a → s (↑(⋯.fix (fun a IH => let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b}; let_fun this := ⋯; ⟨⋯.min S ⋯, ⋯⟩) a')) b}
-- Mathlib/Order/InitialSeg.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder β s
f : r ↪r s
a : α

⊢ ↑(let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑((fun y x => ⋯.fix (fun a IH => let S := {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b}; let_fun this := ⋯; ⟨⋯.min S ⋯, ⋯⟩) y) a_1 h)) b}; let_fun this := ⋯; ⟨⋯.min S ⋯, ⋯⟩) ∈ {b | ∀ (a' : α), r a' a → s (↑(⋯.fix (fun a IH => let S := {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b}; let_fun this := ⋯; ⟨⋯.min S ⋯, ⋯⟩) a')) b}
-- Mathlib/Order/InitialSeg.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
inst✝ : IsWellOrder β s
f : r ↪r s
a : α

⊢ ⋯.min {b | ∀ (a_1 : α), r a_1 a → s (↑(⋯.fix (fun a IH => ⟨⋯.min {b | ∀ (a_2 : α) (h : r a_2 a), s (↑(IH a_2 h)) b} ⋯, ⋯⟩) a_1)) b} ⋯ ∈ {b | ∀ (a' : α), r a' a → s (↑(⋯.fix (fun a IH => ⟨⋯.min {b | ∀ (a_1 : α) (h : r a_1 a), s (↑(IH a_1 h)) b} ⋯, ⋯⟩) a')) b}
-- Mathlib/FieldTheory/Separable.lean
F : Type u
inst✝³ : Field F
K : Type v
inst✝² : Field K
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]

⊢ (map f p).Separable ↔ p.Separable
-- Mathlib/FieldTheory/Separable.lean
F : Type u
inst✝³ : Field F
K : Type v
inst✝² : Field K
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]
H : (map f p).Separable

⊢ p.Separable
-- Mathlib/FieldTheory/Separable.lean
F : Type u
inst✝³ : Field F
K : Type v
inst✝² : Field K
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]
H : (map f p).Separable
m : Ideal S
hm : m.IsMaximal

⊢ p.Separable
-- Mathlib/FieldTheory/Separable.lean
F : Type u
inst✝³ : Field F
K : Type v
inst✝² : Field K
S : Type u_1
inst✝¹ : CommRing S
inst✝ : Nontrivial S
f : F →+* S
p : F[X]
H : (map f p).Separable
m : Ideal S
hm : m.IsMaximal
this : (map (Ideal.Quotient.mk m) (map f p)).Separable

⊢ p.Separable
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m n₁ n₂ n₁₂ : ℤ
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q

⊢ (comp 0 z₂ h).v p q hpq = v 0 p q hpq
-- Mathlib/CategoryTheory/Category/Basic.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g : X ⟶ X

⊢ g ≫ f = f ↔ g = 𝟙 X
-- Mathlib/CategoryTheory/Category/Basic.lean
C : Type u
inst✝¹ : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
inst✝ : Mono f
g : X ⟶ X

⊢ f = 𝟙 X ≫ f
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω

⊢ Continuous toLevyProkhorov
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω

⊢ ∀ (x : ProbabilityMeasure Ω), ContinuousAt toLevyProkhorov x
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω

⊢ ContinuousAt toLevyProkhorov P
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω

⊢ ∀ ε > 0, ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open : ∀ (J : Set ℕ), IsOpen (thickening (ε / 3) (⋃ j ∈ J, Es j))

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε
-- Mathlib/MeasureTheory/Measure/LevyProkhorovMetric.lean
ι : Type u_1
Ω : Type u_2
inst✝³ : PseudoMetricSpace Ω
inst✝² : SeparableSpace Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : OpensMeasurableSpace Ω
P : ProbabilityMeasure Ω
ε : ℝ
ε_pos : ε > 0
third_ε_pos : 0 < ε / 3
third_ε_pos' : 0 < ENNReal.ofReal (ε / 3)
Es : ℕ → Set Ω
Es_mble : ∀ (n : ℕ), MeasurableSet (Es n)
Es_bdd : ∀ (n : ℕ), Bornology.IsBounded (Es n)
Es_diam : ∀ (n : ℕ), diam (Es n) ≤ ε / 3
Es_cover : ⋃ n, Es n = univ
Es_disjoint : Pairwise fun n m => Disjoint (Es n) (Es m)
N : ℕ
hN : ↑P (⋃ j ∈ Iio N, Es j)ᶜ < ENNReal.ofReal (ε / 3)
Js_finite : {J | J ⊆ Iio N}.Finite
Gs : Set (Set Ω) := (fun J => thickening (ε / 3) (⋃ j ∈ J, Es j)) '' {J | J ⊆ Iio N}
Gs_open : ∀ (J : Set ℕ), IsOpen (thickening (ε / 3) (⋃ j ∈ J, Es j))
mem_nhds_P : ∀ (G : Set Ω), IsOpen G → {Q | ↑P G < ↑Q G + ENNReal.ofReal (ε / 3)} ∈ 𝓝 P

⊢ ∀ᶠ (x : ProbabilityMeasure Ω) in 𝓝 P, dist x.toLevyProkhorov P.toLevyProkhorov < ε