ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
𝕜₁ : Type u_1
inst✝²⁰ : NontriviallyNormedField 𝕜₁
𝕜₂ : Type u_2
inst✝¹⁹ : NontriviallyNormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
iσ : RingHomIsometric σ
B : Type u_3
F₁ : Type u_4
inst✝¹⁸ : NormedAddCommGroup F₁
inst✝¹⁷ : NormedSpace 𝕜₁ F₁
E₁ : B → Type u_5
inst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)
inst✝¹⁵ : (x : B) → Module 𝕜₁ (E₁ x)
inst✝¹⁴ : TopologicalSpace (TotalSpace F₁ E₁)
F₂ : Type u_6
inst✝¹³ : NormedAddCommGroup F₂
inst✝¹² : NormedSpace 𝕜₂ F₂
E₂ : B → Type u_7
inst✝¹¹ : (x : B) → AddCommGroup (E₂ x)
inst✝¹⁰ : (x : B) → Module 𝕜₂ (E₂ x)
inst✝⁹ : TopologicalSpace (TotalSpace F₂ E₂)
inst✝⁸ : TopologicalSpace B
e₁ e₁' : Trivialization F₁ TotalSpace.proj
e₂ e₂' : Trivialization F₂ TotalSpace.proj
inst✝⁷ : (x : B) → TopologicalSpace (E₁ x)
inst✝⁶ : FiberBundle F₁ E₁
inst✝⁵ : (x : B) → TopologicalSpace (E₂ x)
ita : ∀ (x : B), TopologicalAddGroup (E₂ x)
inst✝⁴ : FiberBundle F₂ E₂
inst✝³ : Trivialization.IsLinear 𝕜₁ e₁
inst✝² : Trivialization.IsLinear 𝕜₁ e₁'
inst✝¹ : Trivialization.IsLinear 𝕜₂ e₂
inst✝ : Trivialization.IsLinear 𝕜₂ e₂'
b : B
hb : b ∈ e₁.baseSet ∩ e₂.baseSet
L : F₁ →SL[σ] F₂
⊢ cast ⋯ (↑(continuousLinearMap σ e₁ e₂).symm (b, L)).snd = (Trivialization.symmL 𝕜₂ e₂ b).comp (L.comp (Trivialization.continuousLinearMapAt 𝕜₁ e₁ b))
R✝ : Type u₁
S✝ : Type u₂
inst✝³ : CommRing R✝
inst✝² : CommRing S✝
f✝ : R✝ →+* S✝
R : Type u₁
S : Type u₂
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
Y : ModuleCat S
m1 : Module R S := Module.compHom S f
m2 : Module R ↑Y := Module.compHom (↑Y) f
s : S
r : R
y : ↑Y
⊢ s • f r • y = f r • s • y
α : Type u_1
inst✝ : NormedLatticeAddCommGroup α
a b c d : α
⊢ |a ⊓ b - c ⊓ d| ≤ ||a - c| + |b - d||
f g : Ordinal.{u} → Ordinal.{u}
hf : Ordinal.IsNormal f
hg : Ordinal.IsNormal g
⊢ f = g ↔ f 0 = g 0 ∧ ∀ (a : Ordinal.{u}), f a = g a → f (Order.succ a) = g (Order.succ a)
s : String
p : String.Pos
h : ¬s.atEnd p = true
⊢ s.get' p h = s.get p
⊢ Subgroup.normalClosure {⟨Equiv.swap 0 4 * Equiv.swap 1 3, ⋯⟩} = ⊤
R : Type r
CommRing R
W : WeierstrassCurve R
⊢ W.φ 2 = Polynomial.C Polynomial.X * W.ψ₂ ^ 2 - Polynomial.C W.Ψ₃
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasPullbacks C
P : MorphismProperty C
hP : P.StableUnderBaseChange
inst✝ : P.RespectsIso
X Y S : C
f : X ⟶ S
g : Y ⟶ S
h : P.diagonal g
⊢ P.diagonal (pullback.fst f g)
C : Type u_1
CategoryTheory.Category.{u_2, u_1} C
B X : C
f : X ⟶ B
⊢ CategoryTheory.EffectiveEpi f ↔
  CategoryTheory.EffectiveEpiFamily (fun x => X) fun x =>
    match x with
    | PUnit.unit => f
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n => {x | Integrable (f x) ν}.indicator fun x => SimpleFunc.integral ν (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
h2f' : Tendsto f' atTop (𝓝 fun x => ∫ (y : β), f x y ∂ν)
⊢ StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν
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
hp : Fact (1 ≤ p)
H : ∀ (f : ℕ → α → E), (∀ (n : ℕ), Memℒp (f n) p μ) → ∀ (B : ℕ → ℝ≥0∞), ∑' (i : ℕ), B i < ⊤ → (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) → ∃ f_lim, Memℒp f_lim p μ ∧ Tendsto (fun n => eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n => (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
hf : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) < B N
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n => ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
hB1 : ∑' (i : ℕ), B1 i < ⊤
f1 : ℕ → α → E := fun n => ↑↑(f n)
⊢ ∃ f_lim, Memℒp f_lim p μ ∧ Tendsto (fun n => eLpNorm (↑↑(f n) - f_lim) p μ) atTop (𝓝 0)
X : Type u_1
Y : Type u_2
TopologicalSpace X
TopologicalSpace Y
f : X → Y
T2Space Y
compactlyGenerated : ∀ (s : Set Y), IsClosed s ↔ ∀ ⦃K : Set Y⦄, IsCompact K → IsClosed (s ∩ K)
⊢ IsProperMap f ↔ Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)
α : Type u_1
β : Type u_2
G : SimpleGraph α
f : α ↪ β
hG : G.EdgeDisjointTriangles
⊢ (SimpleGraph.map f G).EdgeDisjointTriangles
w : ℂ
h : 0 = 0 → w.re = 0 → w = 0
⊢ abs (0 ^ w) = 0 ^ w.re / rexp (arg 0 * w.im)
𝕜 : Type u_1
B : Type u_2
B' : Type u_3
F : Type u_4
M : Type u_5
E : B → Type u_6
inst✝¹⁶ : NontriviallyNormedField 𝕜
EB : Type u_7
inst✝¹⁵ : NormedAddCommGroup EB
inst✝¹⁴ : NormedSpace 𝕜 EB
HB : Type u_8
inst✝¹³ : TopologicalSpace HB
IB : ModelWithCorners 𝕜 EB HB
inst✝¹² : TopologicalSpace B
inst✝¹¹ : ChartedSpace HB B
inst✝¹⁰ : SmoothManifoldWithCorners IB B
EM : Type u_9
inst✝⁹ : NormedAddCommGroup EM
inst✝⁸ : NormedSpace 𝕜 EM
HM : Type u_10
inst✝⁷ : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
inst✝⁶ : TopologicalSpace M
inst✝⁵ : ChartedSpace HM M
Is : SmoothManifoldWithCorners IM M
n : ℕ∞
inst✝⁴ : (x : B) → AddCommMonoid (E x)
inst✝³ : (x : B) → Module 𝕜 (E x)
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : (x : B) → TopologicalSpace (E x)
a : VectorPrebundle 𝕜 F E
ha : IsSmooth IB a
e✝ e'✝ : Pretrivialization F TotalSpace.proj
this✝¹ : TopologicalSpace (TotalSpace F E) := a.totalSpaceTopology
this✝ : FiberBundle F E := a.toFiberBundle
this : VectorBundle 𝕜 F E := toVectorBundle a
e : Pretrivialization F TotalSpace.proj
he : e ∈ a.toFiberPrebundle.pretrivializationAtlas
e' : Pretrivialization F TotalSpace.proj
he' : e' ∈ a.toFiberPrebundle.pretrivializationAtlas
b : B
hb : b ∈ e.baseSet ∩ e'.baseSet
v : F
⊢ (↑(a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he') { proj := b, snd := (a.toFiberPrebundle.trivializationOfMemPretrivializationAtlas he).symm b v }).2 = (↑e' { proj := b, snd := e.symm b v }).2
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
x : α
ε : ℝ≥0
⊢ EMetric.ball x ↑ε = EMetric.ball x (ENNReal.ofReal ↑ε)
J : Type w
C : Type u
CategoryTheory.Category.{v, u} C
X Y : C
f : J → (X ⟶ Y)
CategoryTheory.Limits.HasWideCoequalizer f
j₁ j₂ : J
⊢ CategoryTheory.CategoryStruct.comp (f j₁) (CategoryTheory.Limits.wideCoequalizer.π f) =
  CategoryTheory.CategoryStruct.comp (f j₂) (CategoryTheory.Limits.wideCoequalizer.π f)
σ : Type u_1
R : Type u_2
Semiring R
φ : MvPowerSeries σ R
h : IsUnit φ
⊢ IsUnit ((MvPowerSeries.constantCoeff σ R) φ)
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
c : ℝ≥0∞
hc : c ≠ 0
hc_ne_top : c ≠ ⊤
h : μ.rnDeriv (c • ν) =ᶠ[ae ν] c⁻¹ • μ.rnDeriv ν
x : α
hx_eq : μ.rnDeriv (c • ν) x = (c⁻¹ • μ.rnDeriv ν) x
hx_pos : 0 < μ.rnDeriv ν x
hx_ne_top : μ.rnDeriv ν x < ⊤
⊢ -log c.toReal + log (μ.rnDeriv ν x).toReal = log (μ.rnDeriv ν x).toReal - log c.toReal
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
F : Type u_3
NormedAddCommGroup F
NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
c : F
⊢ DifferentiableWithinAt 𝕜 (fun y => c + f y) s x ↔ DifferentiableWithinAt 𝕜 f s x
α : Type u_1
E : Type u_2
MeasurableSpace α
μ : MeasureTheory.Measure α
NormedAddCommGroup E
NormedSpace ℝ E
s : ℕ → Set α
f : α → E
hsm : ∀ (i : ℕ), MeasurableSet (s i)
h_anti : Antitone s
hfi : MeasureTheory.IntegrableOn f (s 0) μ
⊢ Filter.Tendsto (fun i => ∫ (a : α) in s i, f a ∂μ) Filter.atTop (nhds (∫ (a : α) in ⋂ n, s n, f a ∂μ))
α : Type u
β : α → Type v
inst✝ : DecidableEq α
a : α
xs : List (Sigma β)
tail_ih✝ : a ∈ xs.toFinmap ↔ ∃ b, ⟨a, b⟩ ∈ xs
fst_i : α
snd_i : β fst_i
⊢ a = fst_i → ∃ x, HEq x snd_i
U : Type u_1
inst✝ : Quiver U
u v v' w : U
p : Path u v
p' : Path u v'
e : v ⟶ w
e' : v' ⟶ w
h : p.cons e = p'.cons e'
⊢ HEq e e'
α : Type u_1
inst✝ : SubtractionMonoid α
a b c : α
h : a + -b = c
⊢ a - b = c
k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
Ring k
AddCommGroup V₁
Module k V₁
AddTorsor V₁ P₁
AddCommGroup V₂
Module k V₂
AddTorsor V₂ P₂
TopologicalSpace P₁
TopologicalSpace P₂
e : P₁ ≃ᵃL[k] P₂
x : P₂
y : P₁
⊢ e.symm x = y ↔ x = e y
C : Type u
CategoryTheory.Category.{v, u} C
f g : CategoryTheory.Arrow C
CategoryTheory.Limits.HasImage f.hom
CategoryTheory.Limits.HasImage g.hom
sq : f ⟶ g
CategoryTheory.Limits.HasImageMap sq
⊢ CategoryTheory.Limits.factorThruImage f.hom ≫ CategoryTheory.Limits.image.map sq =
  sq.left ≫ CategoryTheory.Limits.factorThruImage g.hom
α : Type u_2
LinearOrderedSemiring α
FloorSemiring α
⊢ Nat.floor ⁻¹' {0} = Set.Iio 1
α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α
⊢ a * a⁻¹ ≤ a * b ↔ 1 ≤ a * b
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ ((PreservesCoequalizer.iso (tensorLeft P.X) (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).inv ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫ coequalizer.π (TensorBimod.actRight P Q ▷ L.X) ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ TensorBimod.X P Q ◁ L.actLeft)) ⋯) ≫ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ⋯ = coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X) ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft) ≫ 𝟙 (TensorBimod.X P (Q.tensorBimod L))
V : Type u_1
V₁ : Type u_2
V₂ : Type u_3
V₃ : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : SeminormedAddCommGroup V₁
inst✝¹ : SeminormedAddCommGroup V₂
inst✝ : SeminormedAddCommGroup V₃
f g : NormedAddGroupHom V₁ V₂
x : V₁
C : ℝ
_Cpos : 0 < C
hC : ‖f x‖ ≤ C * ‖x‖
h : ¬‖x‖ = 0
⊢ ‖f x‖ ≤ ‖f‖ * ‖x‖
F : Type u_1
K : Type u_2
inst✝¹¹ : Field F
inst✝¹⁰ : Field K
inst✝⁹ : Algebra F K
E : Type u_3
inst✝⁸ : Field E
inst✝⁷ : Algebra F E
inst✝⁶ : Algebra K E
inst✝⁵ : IsScalarTower F K E
E' : Type u_4
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
inst✝¹ : Field E'
inst✝ : Algebra F E'
h : ∀ (x : E), IsIntegral F x ∧ Splits (algebraMap F E) (minpoly F x)
f : E ≃ₐ[F] E'
⊢ ∀ (x : E'), IsIntegral F x ∧ Splits (algebraMap F E') (minpoly F x)
α✝ : Type u_1
β✝ : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁷ : CommMonoid α✝
inst✝⁶ : TopologicalSpace α✝
f✝ g : β✝ → α✝
a✝ b✝ : α✝
s : Finset β✝
inst✝⁵ : ContinuousMul α✝
α : Type u_5
β : Type u_6
inst✝⁴ : TopologicalSpace α
inst✝³ : CommMonoid α
inst✝² : T2Space α
inst✝¹ : ContinuousMul α
inst✝ : DecidableEq β
f : β → α
a : α
hf : HasProd f a
b : β
a' : α
hf' : HasProd (fun n => if n = b then 1 else f n) a'
⊢ HasProd (update f b 1) a'
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I J : FractionalIdeal R⁰ K
hI : I ≠ 0
h : I ≤ J
hJ : ¬J = 0
this : J⁻¹ * I ≤ J⁻¹ * J
⊢ count K v J ≤ count K v I
M : Type u_1
N : Type u_2
Add M
Add N
ι : Type u_5
f : AddHom M N
hf : Function.Injective ⇑f
S : ι → AddSubsemigroup M
⊢ AddSubsemigroup.comap f (⨆ i, AddSubsemigroup.map f (S i)) = iSup S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : MonoidalCategory C
inst✝ : BraidedCategory C
X Y Z : C
⊢ (β_ X (Y ⊗ Z)).hom ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom ≫ (α_ Z Y X).inv = X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom ≫ (α_ Z Y X).inv
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
NormedAddCommGroup E
NormedAddCommGroup E'
NormedAddCommGroup F
f : G → E
g g' : G → E'
NontriviallyNormedField 𝕜
NormedSpace 𝕜 E
NormedSpace 𝕜 E'
NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
MeasurableSpace G
μ : MeasureTheory.Measure G
NormedSpace ℝ F
AddGroup G
hfg : MeasureTheory.ConvolutionExists f g L μ
hfg' : MeasureTheory.ConvolutionExists f g' L μ
⊢ MeasureTheory.convolution f (g + g') L μ = MeasureTheory.convolution f g L μ + MeasureTheory.convolution f g' L μ
n m : ℕ
a b : Fin (n + 1)
i : Fin n
hl : ↑i < ↑a
hu : ↑b < ↑i + 1
⊢ ↑b < ↑a
C : Type u₁
CategoryTheory.Category.{v₁, u₁} C
CategoryTheory.MonoidalCategory C
D : Type u₂
CategoryTheory.Category.{v₂, u₂} D
e : C ≌ D
X Y : C
⊢ (CategoryTheory.Monoidal.toTransported e).μ X Y =
  e.symm.unit.app (e.functor.obj (e.inverse.obj (e.functor.obj X) ⊗ e.inverse.obj (e.functor.obj Y))) ≫
    CategoryTheory.inv
        (e.functor.map (e.unitIso.hom.app (e.inverse.obj (e.functor.obj X) ⊗ e.inverse.obj (e.functor.obj Y)))) ≫
      e.functor.map (e.symm.counit.app X ⊗ e.symm.counit.app Y)
α : Type u_1
MulZeroClass α
Preorder α
PosMulMono α
x y : α
hx : 0 ≤ x
hy : 0 ≤ y
⊢ ⟨x, hx⟩ * ⟨y, hy⟩ = ⟨x * y, ⋯⟩
𝕜 : Type u_1
E : Type u_2
F : Type u_3
𝓕 : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : ProperSpace E
inst✝² : ProperSpace F
f : 𝓕
inst✝¹ : FunLike 𝓕 E F
inst✝ : CocompactMapClass 𝓕 E F
ε : ℝ
⊢ ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : T5Space Y
e : X → Y
he : _root_.Embedding e
this : CompletelyNormalSpace X
⊢ ∀ ⦃s t : Set X⦄, Disjoint (closure s) t → Disjoint s (closure t) → Disjoint (𝓝ˢ s) (𝓝ˢ t)
M : Type u_1
S : Set M
Monoid M
⊢ ↑(Submonoid.centralizer S) = S.centralizer
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
a : Unit
s : ℕ → ℚ
x✝ : Monotone s
hs_tendsto : Tendsto s atTop atTop
⊢ Tendsto (fun m => ∫ (c : α), (preCDF ρ (s m) (a, c).2).toReal ∂(Kernel.const Unit ρ.fst) a) atTop (𝓝 (((Kernel.const Unit ρ.fst) a) univ).toReal)
p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
hx : x ≠ 0
repr : ↑x = ↑↑(unitCoeff hx) * ↑p ^ x.valuation
⊢ ↑p ^ x.valuation.natAbs = ↑p ^ x.valuation
α : Type u_1
β : Type u_2
Ω : Type u_3
F : Type u_4
MeasurableSpace Ω
StandardBorelSpace Ω
Nonempty Ω
NormedAddCommGroup F
mα : MeasurableSpace α
μ : MeasureTheory.Measure α
MeasureTheory.IsFiniteMeasure μ
X : α → β
Y : α → Ω
mβ : MeasurableSpace β
f : β × Ω → F
hX : AEMeasurable X μ
hY : AEMeasurable Y μ
hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ)
⊢ ∀ᵐ (a : α) ∂μ, MeasureTheory.Integrable (fun ω => f (X a, ω)) ((ProbabilityTheory.condDistrib Y X μ) (X a))
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousSMul 𝕜 E
U : Set E
inst✝ : RegularSpace E
s : Set E
hs : s ∈ 𝓝 0 ∧ IsClosed s
⊢ ∃ i', (i' ∈ 𝓝 0 ∧ IsClosed i' ∧ Balanced 𝕜 i') ∧ id i' ⊆ id s
ι : Type u_2
α : Type u_4
G : SimpleGraph α
DecidableRel G.Adj
DecidableEq α
s : Finset ι
t : Finset α
f : ι → Finset α
⊢ G.interedges (s.biUnion f) t = s.biUnion fun a => G.interedges (f a) t
α' : Type u_3
β : Type u_4
TopologicalSpace α'
One β
T2Space α'
TopologicalSpace β
U : Set α'
hU : IsOpen U
f : ↑U → β
cont : Continuous f
supp : HasCompactMulSupport f
⊢ Continuous (Function.extend Subtype.val f 1)
α : Type u
f g : Filter α
⊢ f = g ↔ ∀ (s : Set α), s ∈ f ↔ s ∈ g
C : Type u₁
CategoryTheory.Category.{v₁, u₁} C
D : Type u₂
CategoryTheory.Category.{v₂, u₂} D
X : C × D
⊢ (CategoryTheory.Prod.symmetry C D).hom.app X = CategoryTheory.CategoryStruct.id X
R : Type w
inst✝² : Ring R
J : Type u
inst✝¹ : Category.{v, u} J
F : J ⥤ ModuleCat R
inst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrp)
i j : J
f : i ⟶ j
⊢ F.map f ≫ (fun j => homMk (colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j) ⋯) j = (fun j => homMk (colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j) ⋯) i ≫ ((Functor.const J).obj (mkOfSMul (coconePointSMul F))).map f
σ : Type u_1
R : Type u
CommSemiring R
s : Set σ
⊢ MvPolynomial.supported R s = Algebra.adjoin R (MvPolynomial.X '' s)
J : Type u₁
CategoryTheory.Category.{v₁, u₁} J
C : Type u₃
CategoryTheory.Category.{v₃, u₃} C
F : CategoryTheory.Functor J C
c : CategoryTheory.Limits.Cone F
⊢ c.mapConeToUnder.inv.hom = CategoryTheory.CategoryStruct.id c.pt
E : Type u_1
inst✝ : SeminormedCommGroup E
ε δ : ℝ
s t : Set E
x y : E
⊢ ball 1 δ * s = thickening δ s
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : DecidableEq α
inst✝ : Fintype α
f✝ f g : Perm α
⊢ f = g → (g.IsCycle ↔ f.IsCycle)
E : Type u_1
NormedAddCommGroup E
NormedSpace ℂ E
R : ℝ
z w : ℂ
f : ℂ → E
⊢ ∫ (θ : ℝ) in 0 ..2 * π, Complex.circleTransform R z w f θ =
  (2 * ↑π * Complex.I)⁻¹ • ∮ (z : ℂ) in C(z, R), (z - w)⁻¹ • f z
X : Scheme
inst✝ : ∀ (x : ↑↑X.toPresheafedSpace), _root_.IsReduced ↑(X.presheaf.stalk x)
U : X.Opens
s : ↑Γ(X, U)
hs : IsNilpotent s
x : ↥U
⊢ (X.sheaf.presheaf.germ x) s = (X.sheaf.presheaf.germ x) 0
Γ : Type u_1
R : Type u_2
PartialOrder Γ
Zero R
a : Γ
r : R
h : r ≠ 0
⊢ ((HahnSeries.single a) r).support = {a}
R : Type v
NonAssocSemiring R
Star R
self : StarSubsemiring R
a : R
⊢ a ∈ self.carrier → star a ∈ self.carrier
ι : Type u_1
R : Type u_2
CommMonoidWithZero R
PartialOrder R
ZeroLEOneClass R
PosMulStrictMono R
Nontrivial R
f g : ι → R
s : Finset ι
hf : ∀ i ∈ s, 0 < f i
hfg : ∀ i ∈ s, f i < g i
h_ne : s.Nonempty
⊢ ∏ i ∈ s, f i < ∏ i ∈ s, g i
α : Type u_2
β : Type u_3
o : Option α
f : α → Set β
⊢ ⋂ x ∈ o.toFinset, f x = ⋂ x ∈ o, f x
ι : Type u_1
Ω : Type u_2
inst✝¹ : MeasurableSpace Ω
inst✝ : PseudoEMetricSpace Ω
μ ν : Measure Ω
⊢ levyProkhorovDist μ ν = levyProkhorovDist ν μ
a r : ℝ
ha : 0 < a
hr : 0 < r
leftSide : ∫⁻ (x : ℝ) in Iio 0, gammaPDF a r x = 0
rightSide : ∫⁻ (x : ℝ) in Ici 0, gammaPDF a r x = ∫⁻ (x : ℝ) in Ici 0, ENNReal.ofReal (r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)))
⊢ 0 ≤ᶠ[ae (ℙ.restrict (Ici 0))] fun x => r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x))
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type w
inst✝¹ : Semiring R
inst✝ : Linear R C
X Y Z : C
f : X ⟶ Y
r : Rˣ
g : Y ⟶ Z
⊢ f ≫ (r • g) = r • f ≫ g
Γ : Type u_1
R : Type u_2
LinearOrderedCancelAddCommMonoid Γ
Ring R
IsDomain R
x : HahnSeries Γ R
hx : x ≠ 0
⊢ (HahnSeries.addVal Γ R) x = ↑x.order
g : Equiv.Perm (Fin 5)
h : 3 ∈ g.cycleType
⊢ (g * g).IsThreeCycle
m✝ n a b c d m : ℕ
h : a ≡ b [MOD n]
⊢ a ^ m ≡ b ^ m [MOD n]
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f ⊤ μ
hmeas : StronglyMeasurable f
hbdd : eLpNormEssSup f μ < ⊤
⊢ eLpNormEssSup ({x | (eLpNorm f ⊤ μ + 1).toReal ≤ ↑‖f x‖₊}.indicator f) μ = 0
M : Type u_1
AddCommMonoid M
S : AddSubmonoid M
N : Type u_2
AddCommMonoid N
f : S.LocalizationMap N
z : N
⊢ ∃ x, z + f.toMap ↑x.2 = f.toMap x.1
α : Type u_1
MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
x : α
⊢ ∀ᶠ (a : Set α) in v.filterAt x, MeasurableSet a
m : Type u_2
R : Type u_7
α : Type v
Fintype m
DecidableEq m
CommSemiring R
Semiring α
Algebra R α
⊢ (AlgHom.id R α).mapMatrix = AlgHom.id R (Matrix m m α)
α : Type u_1
DecidableEq α
Fintype α
G : SimpleGraph α
DecidableRel G.Adj
ε : ℝ
hG : ↑(G.cliqueFinset 3).card < SimpleGraph.triangleRemovalBound ε * ↑(Fintype.card α) ^ 3
⊢ ∃ G' ≤ G, ∃ x, ↑G.edgeFinset.card - ↑G'.edgeFinset.card < ε * ↑(Fintype.card α ^ 2) ∧ G'.CliqueFree 3
α : Type u_1
l : List α
c : Composition l.length
⊢ List.map List.length (l.splitWrtComposition c) = c.blocks
ι : Type u_1
α : Type u_2
s : Finset α
f : α → ℚ
hf : ∀ a ∈ s, 0 ≤ f a
⊢ ∏ i ∈ s, f i = ∏ a ∈ s, ↑(f a).toNNRat
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
ι' : Sort u_5
ι₂ : Sort u_6
κ : ι → Sort u_7
κ₁ : ι → Sort u_8
κ₂ : ι → Sort u_9
κ' : ι' → Sort u_10
s : ι → Set (Set α)
⊢ ⋃₀ ⋃ i, s i = ⋃ i, ⋃₀ s i
I : Type u
LinearOrder I
IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
l : Profinite.NobelingProof.Products I
o₁ o₂ : Ordinal.{u}
h : o₁ ≤ o₂
hlt : ∀ i ∈ ↑l, Profinite.NobelingProof.ord I i < o₁
⊢ (Profinite.NobelingProof.πs' C h)
    (Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o₁)
      l) =
  Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o₂) l
α : Type u
β : Type v
γ : Type w
s✝ : WSeq α
s1 s2 : Computation ℕ
l : List α
s : WSeq α
h : s1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ s2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))
⊢ Computation.BisimO (fun c1 c2 => ∃ l s, c1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ c2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))) s1.destruct s2.destruct
R : Type u
S : Type v
a b✝ : R
m n : ℕ
ι : Type y
inst✝ : Semiring R
p q r : R[X]
b : R
hp : p.leadingCoeff * b = 1
a✝ : Nontrivial R
⊢ (p * C b).leadingCoeff = 1
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
F : Type u_3
NormedAddCommGroup F
NormedSpace 𝕜 F
x : E
iso : E ≃L[𝕜] F
⊢ HasStrictFDerivAt (⇑iso) (↑iso) x
α : Type u_1
β : Type u_2
G : Type u_3
𝓕 : Filter α
inst✝¹ : NormedAddCommGroup G
inst✝ : CompleteSpace G
f : α → β → G
g : β → G
bound : β → ℝ
h_sum : Summable bound
hab : ∀ (k : β), Tendsto (fun x => f x k) 𝓕 (𝓝 (g k))
h_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k
h✝ : IsEmpty β
⊢ Tendsto (fun x => ∑' (k : β), f x k) 𝓕 (𝓝 (∑' (k : β), g k))
α : Type u_2
β : Type u_3
MulZeroOneClass α
MulZeroOneClass β
f g : α →*₀ β
h : ∀ (x : α), f x = g x
⊢ f = g
α : Type u_1
o : Part α
Decidable o.Dom
⊢ o.toOption = none ↔ ¬o.Dom
F : Type u_1
Field F
E : Type u_2
Field E
Algebra F E
S T U : IntermediateField F E
hST : S = T
hTU : T = U
⊢ (IntermediateField.equivOfEq hST).trans (IntermediateField.equivOfEq hTU) = IntermediateField.equivOfEq ⋯
R : Type u_3
Γ₀ : Type u_4
Ring R
LinearOrderedAddCommMonoidWithTop Γ₀
v : AddValuation R Γ₀
x y : R
⊢ v (x * y) = v x + v y
J : Type v
inst✝¹ : SmallCategory J
F : J ⥤ MonCatMax
inst✝ : IsFiltered J
j₁ : J
y : ↑(F.obj j₁)
j₂ : J
x : ↑(F.obj j₂)
j₃ : J
y' : ↑(F.obj j₃)
l : J
f : ⟨j₁, y⟩.fst ⟶ l
g : ⟨j₃, y'⟩.fst ⟶ l
hfg : (F.map f) y = (F.map g) y'
⊢ colimitMulAux F ⟨j₂, x⟩ ⟨j₁, y⟩ = colimitMulAux F ⟨j₂, x⟩ ⟨j₃, y'⟩
R : Type u_2
M : Type u_5
Semiring R
AddCommMonoid M
Module R M
n : ℕ
b : M
⊢ n • b = ↑n • b
α : Type u_1
a : α
l : List α
⊢ a ∈ l ↔ ∃ n, l[n]? = some a
α : Type u_2
SemilatticeInf α
s t : Set α
⊢ (s ⊼ t).Nonempty → t.Nonempty
x : ZFSet
⊢ ↑x.powerset = (↑x).powerset
ι : Type u_1
α : ι → Type u_2
inst✝² : DecidableEq ι
inst✝¹ : (i : ι) → Zero (α i)
s : Finset ι
f : Π₀ (i : ι), α i
t : (i : ι) → Finset (α i)
inst✝ : (i : ι) → DecidableEq (α i)
⊢ (∃ a ∈ s.pi t, { toFun := fun f => DFinsupp.mk s fun i => f ↑i ⋯, inj' := ⋯ } a = f) → f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i
α✝ : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g : α✝ → ℝ≥0∞
α : Type u_4
f : α → ℝ≥0∞
hf : ∑' (x : α), f x ≠ ⊤
f_ne_top : ∀ (n : α), f n ≠ ⊤
⊢ Tendsto f cofinite (𝓝 0)
n : ℕ
α : Type u_1
β : Type u_2
v : Vector α (n + 1)
f : α → β
⊢ (map f v).head = f v.head
α : Type u_2
β : Type u_3
r : α → β → Prop
c d : Concept α β r
⊢ (c ⊓ d).toProd.2 = intentClosure r (c.toProd.1 ∩ d.toProd.1)
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
s : Set α
a b c d : α
hf : MonotoneOn f s
ha : a ∈ s
hb : b ∈ s
⊢ f (max a b) = max (f a) (f b)