ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
t : R[X]
ht : t ∈ Submonoid.closure {X}

⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨t, ht⟩)
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
n : ℕ
ht : X ^ n ∈ Submonoid.closure {X}

⊢ IsUnit ((algebraMap R[X] R[T;T⁻¹]) ↑⟨X ^ n, ht⟩)
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f : R[T;T⁻¹]

⊢ ∃ x, f * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
n : ℕ

⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
n : ℕ
this : X ^ n ∈ Submonoid.closure {X}

⊢ ∃ x, toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑x.2 = (algebraMap R[X] R[T;T⁻¹]) x.1
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f : R[X]
n : ℕ
this : X ^ n ∈ Submonoid.closure {X}

⊢ toLaurent f * T (-↑n) * (algebraMap R[X] R[T;T⁻¹]) ↑(f, ⟨X ^ n, this⟩).2 = (algebraMap R[X] R[T;T⁻¹]) (f, ⟨X ^ n, this⟩).1
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f g : R[X]

⊢ (algebraMap R[X] R[T;T⁻¹]) f = (algebraMap R[X] R[T;T⁻¹]) g → ∃ c, ↑c * f = ↑c * g
-- Mathlib/Algebra/Polynomial/Laurent.lean
R : Type u_1
inst✝ : CommSemiring R
f g : R[X]

⊢ f = g → ∃ c, ↑c * f = ↑c * g
-- Mathlib/CategoryTheory/Limits/Types.lean
J : Type v
inst✝¹ : Category.{w, v} J
F : J ⥤ Type u
inst✝ : HasColimit F
j j' : J
x : F.obj j
x' : F.obj j'
f : j ⟶ j'
w : F.map f x = x'

⊢ colimit.ι F j x = colimit.ι F j' x'
-- Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁶ : NormedAddCommGroup E✝
inst✝⁵ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
y : (i : ι) → E i
x✝ : y ∈ EMetric.ball 0 ⊤

⊢ (∑ i ∈ Finset.range (Fintype.card ι + 1), (f.toFormalMultilinearSeries i) fun x => y) = f (0 + y)
-- Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁶ : NormedAddCommGroup E✝
inst✝⁵ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
y : (i : ι) → E i
x✝ : y ∈ EMetric.ball 0 ⊤

⊢ ((f.toFormalMultilinearSeries (Fintype.card ι)) fun x => y) = f y
-- Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
𝕜 : Type u_1
inst✝⁷ : NontriviallyNormedField 𝕜
E✝ : Type u
inst✝⁶ : NormedAddCommGroup E✝
inst✝⁵ : NormedSpace 𝕜 E✝
F : Type v
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
ι : Type u_2
E : ι → Type u_3
inst✝² : (i : ι) → NormedAddCommGroup (E i)
inst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E F
y : (i : ι) → E i
x✝ : y ∈ EMetric.ball 0 ⊤

⊢ ∀ b ∈ Finset.range (Fintype.card ι + 1), b ≠ Fintype.card ι → ((f.toFormalMultilinearSeries b) fun x => y) = 0
-- Mathlib/Order/Filter/Ultrafilter.lean
α : Type u
β : Type v
γ : Type u_1
f✝ g : Ultrafilter α
s t : Set α
p q : α → Prop
f : Ultrafilter α
h₀ : optParam (Injective id) ⋯

⊢ range id ∈ f
-- Mathlib/Order/Filter/Ultrafilter.lean
α : Type u
β : Type v
γ : Type u_1
f✝ g : Ultrafilter α
s t : Set α
p q : α → Prop
f : Ultrafilter α
h₀ : optParam (Injective id) ⋯

⊢ univ ∈ f
-- Mathlib/Algebra/Field/Subfield.lean
K : Type u
L : Type v
M : Type w
inst✝² : DivisionRing K
inst✝¹ : DivisionRing L
inst✝ : DivisionRing M
f : K →+* L
s : Subfield L
h : s ≤ f.fieldRange

⊢ map f (comap f s) = s
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → E
hf : Memℒp f p μ

⊢ edist (Memℒp.toLp f hf) 0 = eLpNorm f p μ
-- Mathlib/MeasureTheory/Function/LpSpace.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → E
hf : Memℒp f p μ

⊢ f = f - 0
-- Mathlib/Data/Int/Cast/Basic.lean
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ

⊢ ↑(↑m + ↑n) = ↑↑m + ↑↑n
-- Mathlib/Data/Int/Cast/Basic.lean
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ

⊢ ↑(↑m + -[n+1]) = ↑↑m + ↑-[n+1]
-- Mathlib/Data/Int/Cast/Basic.lean
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ

⊢ ↑(-[m+1] + ↑n) = ↑-[m+1] + ↑↑n
-- Mathlib/Data/Int/Cast/Basic.lean
R : Type u
inst✝ : AddGroupWithOne R
m n : ℕ

⊢ ↑-[m + n + 1+1] = ↑-[m+1] + ↑-[n+1]
-- Mathlib/SetTheory/Ordinal/NaturalOps.lean
a b : Ordinal.{u_1}

⊢ a ≤ a ♯ b
-- Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
D₁ : Type u₄
D₂ : Type u₅
D₃ : Type u₆
inst✝⁸ : Category.{v₁, u₁} C₁
inst✝⁷ : Category.{v₂, u₂} C₂
inst✝⁶ : Category.{v₃, u₃} C₃
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
inst✝³ : Category.{v₆, u₆} D₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝ : Φ.functor.IsEquivalence
h : W₂ ≤ W₁.map Φ.functor

⊢ Φ.IsLocalizedEquivalence
-- Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean
C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
D₁ : Type u₄
D₂ : Type u₅
D₃ : Type u₆
inst✝⁸ : Category.{v₁, u₁} C₁
inst✝⁷ : Category.{v₂, u₂} C₂
inst✝⁶ : Category.{v₃, u₃} C₃
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
inst✝³ : Category.{v₆, u₆} D₃
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝ : Φ.functor.IsEquivalence
h : W₂ ≤ W₁.map Φ.functor
this : (Φ.functor ⋙ W₂.Q).IsLocalization W₁

⊢ Φ.IsLocalizedEquivalence
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)

⊢ coeSubmodule (FractionRing B) (differentIdeal A B) = 1 / Submodule.traceDual A (FractionRing A) 1
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this : IsIntegralClosure B A (FractionRing B)

⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝ : IsIntegralClosure B A (FractionRing B)
this : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1

⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : (fun a => ↑a) (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ ↑1

⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1

⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / Submodule.traceDual A (FractionRing A) ↑1 ≤ ↑1

⊢ 1 / Submodule.traceDual A (FractionRing A) 1 ≤ LinearMap.range (Algebra.linearMap B (FractionRing B))
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑1 / ↑(FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1

⊢ 1 ≠ 0
-- Mathlib/RingTheory/DedekindDomain/Different.lean
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝²² : CommRing A
inst✝²¹ : Field K
inst✝²⁰ : CommRing B
inst✝¹⁹ : Field L
inst✝¹⁸ : Algebra A K
inst✝¹⁷ : Algebra B L
inst✝¹⁶ : Algebra A B
inst✝¹⁵ : Algebra K L
inst✝¹⁴ : Algebra A L
inst✝¹³ : IsScalarTower A K L
inst✝¹² : IsScalarTower A B L
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsFractionRing A K
inst✝⁹ : IsIntegralClosure B A L
inst✝⁸ : IsFractionRing B L
inst✝⁷ : FiniteDimensional K L
inst✝⁶ : Algebra.IsSeparable K L
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : IsDedekindDomain B
inst✝³ : NoZeroSMulDivisors A B
inst✝² : Algebra.IsIntegral A B
inst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
this✝¹ : IsIntegralClosure B A (FractionRing B)
this✝ : (FractionalIdeal.dual A (FractionRing A) 1)⁻¹ ≤ 1
this : ↑(1 / FractionalIdeal.dual A (FractionRing A) 1) ≤ ↑1

⊢ 1 ≠ 0
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
y : ℝ
h : 0 < y

⊢ ⊤ ^ y = ⊤
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)

⊢ Measurable fun a => (f a).measure
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s

⊢ Measurable fun b => (f b).measure s
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure

⊢ Measurable fun b => (f b).measure s
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure

⊢ (fun s => Measurable fun b => (f b).measure s) ∅
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure

⊢ ∀ t ∈ range Iic, (fun s => Measurable fun b => (f b).measure s) t
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure

⊢ ∀ (t : Set ℝ), MeasurableSet t → (fun s => Measurable fun b => (f b).measure s) t → (fun s => Measurable fun b => (f b).measure s) tᶜ
-- Mathlib/Probability/Kernel/Disintegration/CdfToKernel.lean
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α (β × ℝ)
ν✝ : Kernel α β
x✝ : MeasurableSpace β
f✝ : α × β → StieltjesFunction
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α → StieltjesFunction
hf : ∀ (q : ℝ), Measurable fun a => ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
s : Set ℝ
hs : MeasurableSet s
this : ∀ (a : α), IsProbabilityMeasure (f a).measure

⊢ ∀ (f_1 : ℕ → Set ℝ), Pairwise (Disjoint on f_1) → (∀ (i : ℕ), MeasurableSet (f_1 i)) → (∀ (i : ℕ), (fun s => Measurable fun b => (f b).measure s) (f_1 i)) → (fun s => Measurable fun b => (f b).measure s) (⋃ i, f_1 i)
-- Mathlib/Data/Int/Cast/Basic.lean
R : Type u
inst✝¹ : AddGroupWithOne R
n : ℕ
inst✝ : n.AtLeastTwo

⊢ ↑(OfNat.ofNat n) = OfNat.ofNat n
-- .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
α : Type u_1
le : α → α → Bool
r : Nat
a : α
c : HeapNode α
s : Heap α

⊢ (cons r a c s).realSize = (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize + 1
-- .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
α : Type u_1
le : α → α → Bool
r : Nat
a : α
c : HeapNode α
s : Heap α
this : Batteries.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1)

⊢ (cons r a c s).realSize = (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize + 1
-- .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean
α : Type u_1
le : α → α → Bool
r : Nat
a : α
c : HeapNode α
s : Heap α

⊢ Batteries.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1) → (cons r a c s).realSize = (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize + 1
-- Mathlib/Data/Matrix/Rank.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
R : Type u_5
inst✝⁴ : Fintype n
inst✝³ : Fintype o
inst✝² : CommRing R
inst✝¹ : StrongRankCondition R
inst✝ : Fintype m
f : n → m
e : n ≃ m
A : Matrix m m R

⊢ (A.submatrix f ⇑e).rank ≤ A.rank
-- Mathlib/Data/Matrix/Rank.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
R : Type u_5
inst✝⁴ : Fintype n
inst✝³ : Fintype o
inst✝² : CommRing R
inst✝¹ : StrongRankCondition R
inst✝ : Fintype m
f : n → m
e : n ≃ m
A : Matrix m m R

⊢ finrank R ↥(Submodule.map (LinearMap.funLeft R R f) (LinearMap.range A.mulVecLin)) ≤ finrank R ↥(LinearMap.range A.mulVecLin)
-- Mathlib/CategoryTheory/Limits/HasLimits.lean
J : Type u₁
inst✝⁴ : Category.{v₁, u₁} J
K : Type u₂
inst✝³ : Category.{v₂, u₂} K
C : Type u
inst✝² : Category.{v, u} C
F✝ : J ⥤ C
J' : Type u₂
inst✝¹ : Category.{v₂, u₂} J'
e : J ≌ J'
inst✝ : HasLimitsOfShape J C
F : J' ⥤ C

⊢ HasLimit F
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : BoundedOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a : α
hs : a ∈ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t

⊢ (s ⊼ t).truncatedSup a = s.truncatedSup a ⊓ t.truncatedSup a
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : BoundedOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a : α
hs : a ∈ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t

⊢ ((filter (fun x => a ≤ x) s ⊼ filter (fun x => a ≤ x) t).sup' ⋯ fun x => id x) = (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : BoundedOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a : α
hs : a ∈ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t

⊢ ((image (Function.uncurry fun x x_1 => x ⊓ x_1) (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t)).sup' ⋯ fun x => id x) = (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2
-- Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean
α : Type u_1
β : Type u_2
inst✝³ : DistribLattice α
inst✝² : BoundedOrder α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a : α
hs : a ∈ lowerClosure ↑s
ht : a ∈ lowerClosure ↑t

⊢ (filter (fun x => a ≤ x) s ×ˢ filter (fun x => a ≤ x) t).sup' ⋯ ((fun x => id x) ∘ Function.uncurry fun x x_1 => x ⊓ x_1) = (filter (fun b => a ≤ b) s ×ˢ filter (fun b => a ≤ b) t).sup' ⋯ fun i => id i.1 ⊓ id i.2
-- Mathlib/RingTheory/FiniteType.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N

⊢ adjoin R ↑{X} = ⊤
-- Mathlib/RingTheory/FiniteType.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹⁰ : CommSemiring R
inst✝⁹ : CommSemiring S
inst✝⁸ : Semiring A
inst✝⁷ : Semiring B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N

⊢ adjoin R {X} = ⊤
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
s : S
n : ℕ
h : s ∈ Set.range ⇑f

⊢ (monomial n) s ∈ lifts f
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
r : R
h : f r ∈ Set.range ⇑f

⊢ (monomial n) (f r) ∈ lifts f
-- Mathlib/Algebra/Polynomial/Lifts.lean
R : Type u
inst✝¹ : Semiring R
S : Type v
inst✝ : Semiring S
f : R →+* S
n : ℕ
r : R
h : f r ∈ Set.range ⇑f

⊢ (mapRingHom f) ((monomial n) r) = (monomial n) (f r)
-- Mathlib/Order/Interval/Set/UnorderedInterval.lean
α : Type u_1
β : Type u_2
inst✝ : DistribLattice α
a a₁ a₂ b b₁ b₂ c x : α

⊢ b ∈ [[a, c]] → c ∈ [[a, b]] → b = c