ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
U : Set G
hU : U ∈ {s | IsOpen s}
U' : Opens G
this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }
⊢ ⨆ b, ↑(μ.toFun ↑b) + μ.outerMeasure (↑U' \ U) ≤ μ.outerMeasure ↑U'
α : Type u_1
a b c d : α
inst✝³ : MulZeroClass α
inst✝² : LinearOrder α
inst✝¹ : PosMulMono α
inst✝ : MulPosMono α
hab : 0 < a * b
ha : 0 < a
⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0
α : Type u_1
AddCommGroup α
s t : Set α
hs : s.Finite
ht' : t.Finite
ht : t.Nonempty
⊢ ∃ u, Nat.card ↑u * Nat.card ↑t ≤ Nat.card ↑(s + t) ∧ s ⊆ u + t - t ∧ u.Finite
n p : ℕ
pp : Nat.Prime p
hn : n ≠ 0
⊢ {i | i ≠ 0 ∧ p ^ i ∣ n} = Set.Icc 1 (n.factorization p)
α : Type u_1
inst✝² : SemilatticeInf α
inst✝¹ : OrderTop α
inst✝ : DecidableEq α
a : α
s : Multiset α
⊢ (ndinsert a s).inf = a ⊓ s.inf
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ✝ : Kernel α β
f✝ : α → β → ℝ≥0∞
inst✝¹ : Countable ι
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
f : ι → α → β → ℝ≥0∞
hf : ∀ (i : ι), Measurable (Function.uncurry (f i))
h_sum_a : ∀ (a : α), Summable fun n => f n a
h_sum : Summable fun n => f n
a : α
s : Set β
hs : MeasurableSet s
this : ∫⁻ (b : β) in s, tsum (fun n => f n) a b ∂κ a = ∫⁻ (b : β) in s, ∑' (n : ι), (fun b => f n a b) b ∂κ a
⊢ ∑' (i : ι), ∫⁻ (a_1 : β) in s, f i a a_1 ∂κ a = ∑' (n : ι), ((κ.withDensity (f n)) a) s
n d : PosNum
q r : Num
h₁ : ↑r + ↑d * (↑q + ↑q) = ↑n
h₂ : ↑r < 2 * ↑d
this : ∀ {r₂ : Num}, Num.ofZNum' (r.sub' (Num.pos d)) = some r₂ ↔ ↑r = ↑r₂ + ↑d
⊢ ↑(match Num.ofZNum' (r.sub' (Num.pos d)) with | some r' => (q.bit1, r') | none => (q.bit0, r)).2 + ↑d * ↑(match Num.ofZNum' (r.sub' (Num.pos d)) with | some r' => (q.bit1, r') | none => (q.bit0, r)).1 = ↑n ∧ ↑(match Num.ofZNum' (r.sub' (Num.pos d)) with | some r' => (q.bit1, r') | none => (q.bit0, r)).2 < ↑d
α : Type u_1
β : Type u_2
inst✝ : MeasurableSpace α
j₁ j₂ : JordanDecomposition α
hj : j₁.toSignedMeasure = j₂.toSignedMeasure
S : Set α
hS₁ : MeasurableSet S
hS₂ : VectorMeasure.restrict j₁.toSignedMeasure S ≤ VectorMeasure.restrict 0 S
hS₃ : VectorMeasure.restrict 0 Sᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Sᶜ
hS₄ : j₁.posPart S = 0
hS₅ : j₁.negPart Sᶜ = 0
T : Set α
hT₁ : MeasurableSet T
hT₂ : VectorMeasure.restrict j₁.toSignedMeasure T ≤ VectorMeasure.restrict 0 T
hT₃ : VectorMeasure.restrict 0 Tᶜ ≤ VectorMeasure.restrict j₁.toSignedMeasure Tᶜ
hT₄ : j₂.posPart T = 0
hT₅ : j₂.negPart Tᶜ = 0
hST₁ : ↑j₁.toSignedMeasure (symmDiff Sᶜ Tᶜ) = 0
⊢ j₁.posPart = j₂.posPart
α : Type u_2
SemilatticeInf α
⊢ Monotone ⇑infClosure
ι : Type u_1
R : Type u_2
A : Type u_3
AddCommMonoid ι
DecidableEq ι
CommRing R
CommRing A
Algebra R A
𝒜 : ι → Submodule R A
GradedAlgebra 𝒜
x : Submonoid A
f : HomogeneousLocalization.NumDenSameDeg 𝒜 x
h : f.den = 0
⊢ HomogeneousLocalization.mk f = 0
R : Type u_1
ι : Type u_5
κ : Type u_6
CommSemiring R
i : ι × κ
r : R
⊢ (finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) = Finsupp.single i.1 1 ⊗ₜ[R] Finsupp.single i.2 r
a : ℕ
a1 : 1 < a
n : ℕ
⊢ (Pell.pellZd a1 n).im = ↑(Pell.yn a1 n)
F : Type u_1
Field F
S : Set F
self : IsSubfield S
x : F
⊢ x ∈ S → x⁻¹ ∈ S
R : Type u
K : Type u'
M : Type v
V : Type v'
M₂ : Type w
V₂ : Type w'
M₃ : Type y
V₃ : Type y'
M₄ : Type z
ι : Type x
M₅ : Type u_1
M₆ : Type u_2
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M
inst✝ : Module R M₂
p : Submodule R M
q : Submodule R M₂
⊢ range (snd R M M₂) = ⊤
f : StieltjesFunction
a b : ℝ
⊢ { toOuterMeasure := f.outer, m_iUnion := ⋯, trim_le := ⋯ } (Ioc a b) = ofReal (↑f b - ↑f a)
Ω : Type u_1
ι : Type u_2
_mΩ : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
β : Type u_10
m : MeasurableSpace β
Sub β
MeasurableSub₂ β
f : ι → Ω → β
hf_indep : ProbabilityTheory.iIndepFun (fun x => m) f μ
hf_meas : ∀ (i : ι), Measurable (f i)
i j k : ι
hik : i ≠ k
hjk : j ≠ k
⊢ ProbabilityTheory.IndepFun (f i - f j) (f k) μ
a x z✝ z : ℂ
⊢ -(π / 2) < z.arg ↔ 0 < z.re ∨ 0 ≤ z.im
α : Type u_1
β : Type u_2
SemilatticeSup α
SemilatticeSup β
a₁ a₂ : α
⊢ Sum.inlₗ (a₁ ⊔ a₂) = Sum.inlₗ a₁ ⊔ Sum.inlₗ a₂
α : Type u_1
β : Type u_2
p : TwoPointing α
q : TwoPointing β
⊢ (p.prod q).fst = (p.fst, q.fst)
X : Type u_1
TopologicalSpace X
s : Set X
⊢ IsSeqClosed s ↔ seqClosure s = s
α : Type u_1
m : MeasureTheory.OuterMeasure α
⊢ m.measureOf = ⇑m
P : MorphismProperty Scheme
Q : AffineTargetMorphismProperty
inst✝¹ : HasAffineProperty P Q
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
hP' : Q.StableUnderBaseChange
X Y S : Scheme
f : X ⟶ S
g : Y ⟶ S
inst✝ : IsAffine S
H : Q g
this✝ : Q.IsLocal := isLocal_affineProperty P
i : (X.affineCover.pullbackCover (pullback.fst f g)).J
e : pullback (pullback.fst f g) (X.affineCover.map i) ≅ pullback (X.affineCover.map i ≫ f) g := pullbackSymmetry (pullback.fst f g) (X.affineCover.map i) ≪≫ pullbackRightPullbackFstIso f g (X.affineCover.map i)
this : e.hom ≫ pullback.fst (X.affineCover.map i ≫ f) g = X.affineCover.pullbackHom (pullback.fst f g) i
⊢ Q g
α : Type u_1
LinearOrder α
x y : α
β : Type u_3
LinearOrder β
x' y' : β
h : cmp x y = cmp x' y'
⊢ x < y ↔ x' < y'
α : Type u
o : Option α
⊢ o.isNone = true → o = none
R : Type u_1
CommRing R
f : R → ℝ
c : ℝ
f_nonneg : 0 ≤ f
f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y
x : R
hx : f x = 0
y : R
⊢ f (x * y) = 0
ι : Type u
X : Type v
inst✝⁴ : TopologicalSpace X
E : Type u_1
inst✝³ : AddCommMonoid E
inst✝² : SMulWithZero ℝ E
inst✝¹ : TopologicalSpace E
inst✝ : ContinuousSMul ℝ E
s✝ : Set X
f : PartitionOfUnity ι X s✝
s : Set X
ρ : PartitionOfUnity ι X s
x₀ : X
I : Finset ι
hI : ∀ᶠ (x : X) in 𝓝 x₀, (support fun i => (ρ i) x) ⊆ ↑I
⊢ ∃ I, (∀ᶠ (x : X) in 𝓝[s] x₀, ∑ i ∈ I, (ρ i) x = 1) ∧ ∀ᶠ (x : X) in 𝓝 x₀, (support fun x_1 => (ρ x_1) x) ⊆ ↑I
X : Type u_1
Y : Type u_2
TopologicalSpace X
TopologicalSpace Y
f : X → Y
s : Set X
h : Continuous f
⊢ closure (f '' closure s) = closure (f '' s)
K : Type u_1
RCLike K
x : ℝ
⊢ ↑x < 0 ↔ x < 0
α : Type u
PartialOrder α
OrderTop α
a : α
⊢ a = ⊤ ∨ a < ⊤
G : Type u_3
Group G
h : Group.FG G
⊢ ∃ S, S.card = Group.rank G ∧ Subgroup.closure ↑S = ⊤
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
inst✝ : MeasurableSpace δ
μa : Measure α
μb : Measure β
μc : Measure γ
μd : Measure δ
μ : Measure α
f : α → α
s : Set α
hf : MeasurePreserving f μ μ
hs : NullMeasurableSet s μ
n : ℕ
hvol : μ univ < ↑n * μ s
A : ∀ (m : ℕ), NullMeasurableSet (f^[m] ⁻¹' s) μ
B : ∀ (m : ℕ), μ (f^[m] ⁻¹' s) = μ s
this : μ univ < ∑ m ∈ Finset.range n, μ (f^[m] ⁻¹' s)
i : ℕ
hi : i < n
j : ℕ
hj : j < n
hij : i ≠ j
x : α
hxi : f^[i] x ∈ s
hxj : f^[j] x ∈ s
⊢ ∃ x ∈ s, ∃ m ∈ Ioo 0 n, f^[m] x ∈ s
F : PFunctor.{u}
Inhabited F.M
x y : F.M
H : ∀ (ps : PFunctor.Approx.Path F), PFunctor.M.iselect ps x = PFunctor.M.iselect ps y
⊢ x = y
R : Type u
CommSemiring R
𝓟 : Ideal R
f : R[X]
hf : f.Monic
h : 𝓟 ≠ ⊤
hmem : ∀ {n : ℕ}, n < f.natDegree → f.coeff n ∈ 𝓟
hnot_mem : f.coeff 0 ∉ 𝓟 ^ 2
⊢ f.IsEisensteinAt 𝓟
α : Type u_1
β : Type u_2
f : α → β
c : α
DivisionSemiring α
h : Function.Periodic f c
a : α
⊢ Function.Periodic (fun x => f (x * a)) (c * a⁻¹)
x : ℝ
⊢ 0 ≤ Real.exp x
b : ℂ
V : Type u_1
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : FiniteDimensional ℝ V
inst✝² : MeasurableSpace V
inst✝¹ : BorelSpace V
ι : Type u_2
inst✝ : Fintype ι
hb : 0 < b.re
c : ℂ
w : EuclideanSpace ℝ ι
this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume
⊢ ∫ (v : EuclideanSpace ℝ ι), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) = (↑π / b) ^ (↑(Fintype.card ι) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))
R : Type u_1
M : Type u_2
CommSemiring R
AddCommMonoid M
Module R M
i : ℕ
h : i = i
⊢ TensorPower.cast R M h = LinearEquiv.refl R (⨂[R]^i M)
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
t : Set β
⊢ e.IsImage s t ↔ e.source ∩ ↑e ⁻¹' t = e.source ∩ s
α : Type u_1
β : Type u_2
inst✝⁶ : SemilatticeInf α
inst✝⁵ : BoundedOrder α
inst✝⁴ : DecidableRel fun x x_1 => x ≤ x_1
inst✝³ : SemilatticeInf β
inst✝² : BoundedOrder β
inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1
s t : Finset α
a : α
inst✝ : DecidableEq α
hs : a ∉ upperClosure ↑s
ht : a ∉ upperClosure ↑t
⊢ a ∉ upperClosure ↑(s ∪ t)
𝕜 : Type u
NontriviallyNormedField 𝕜
F : Type v
NormedAddCommGroup F
NormedSpace 𝕜 F
f : 𝕜 → F
x : 𝕜
s : Set 𝕜
h : ¬DifferentiableWithinAt 𝕜 f s x
⊢ derivWithin f s x = 0
α : Type u_1
β : Type u_2
R : α → β → Prop
l : List α
l₁ l₂ : List β
h : List.Forall₂ R l (l₁ ++ l₂)
⊢ List.Forall₂ R (List.take l₁.length l) l₁
α : Type u_2
β : Type u_3
CommMonoid α
CommMonoid β
s : Set α
t : Set β
f : α → β
hf : IsMulFreimanIso 2 s t f
⊢ ThreeGPFree s ↔ ThreeGPFree t
R : Type u
CommRing R
E : EllipticCurve R
C : WeierstrassCurve.VariableChange R
⊢ (E.variableChange C).a₆ =
  ↑C.u⁻¹ ^ 6 * (E.a₆ + C.r * E.a₄ + C.r ^ 2 * E.a₂ + C.r ^ 3 - C.t * E.a₃ - C.t ^ 2 - C.r * C.t * E.a₁)
F : Type u_1
α : Type u_2
β : Type u_3
R : Type u_4
inst✝² : Semiring α
inst✝¹ : Semiring β
a b : α
m : ℕ
inst✝ : IsCancelAdd α
hab : a + b = 0
k : ℕ
ih : a ^ (2 * k + 1) + b ^ (2 * k + 1) = 0
⊢ a ^ (2 * (k + 1) + 1) + b ^ (2 * (k + 1) + 1) = 0
C : Type u
CategoryTheory.Category.{v, u} C
X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
w : CategoryTheory.Limits.WalkingCospan
⊢ (CategoryTheory.Limits.cospan f g).map (CategoryTheory.Limits.WalkingCospan.Hom.id w) =
  𝟙 ((CategoryTheory.Limits.cospan f g).obj w)
α : Type u
β : Type v
f : α → β
s : Set α
h : Set.InjOn f s
⊢ PartENat.card ↑(f '' s) = PartENat.card ↑s
x y : ℝ
h : 1 < x * y
hx : 0 < x
hy : 0 < y
k : arctan (-((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))) + π = arctan x + arctan y
⊢ (x + y) / (1 - x * y) = -((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))
C : Type u₁
CategoryTheory.Category.{v₁, u₁} C
CategoryTheory.MonoidalCategory C
D : Type u₂
CategoryTheory.Category.{v₂, u₂} D
CategoryTheory.MonoidalCategory D
F : C ⥤ D
ε : 𝟙_ D ⟶ F.obj (𝟙_ C)
μ :
  (X Y : C) →
    CategoryTheory.MonoidalCategory.tensorObj (F.obj X) (F.obj Y) ⟶
      F.obj (CategoryTheory.MonoidalCategory.tensorObj X Y)
μ_natural :
  ∀ {X Y X' Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y'),
    CategoryTheory.MonoidalCategory.tensorHom (F.map f) (F.map g) ≫ μ Y Y' =
      μ X X' ≫ F.map (CategoryTheory.MonoidalCategory.tensorHom f g) := by
  aesop_cat
associativity :
  ∀ (X Y Z : C),
    CategoryTheory.MonoidalCategory.tensorHom (μ X Y) (𝟙 (F.obj Z)) ≫
        μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z ≫
          F.map (CategoryTheory.MonoidalCategory.associator X Y Z).hom =
      (CategoryTheory.MonoidalCategory.associator (F.obj X) (F.obj Y) (F.obj Z)).hom ≫
        CategoryTheory.MonoidalCategory.tensorHom (𝟙 (F.obj X)) (μ Y Z) ≫
          μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z) := by
  aesop_cat
left_unitality :
  ∀ (X : C),
    (CategoryTheory.MonoidalCategory.leftUnitor (F.obj X)).hom =
      CategoryTheory.MonoidalCategory.tensorHom ε (𝟙 (F.obj X)) ≫
        μ (𝟙_ C) X ≫ F.map (CategoryTheory.MonoidalCategory.leftUnitor X).hom := by
  aesop_cat
right_unitality :
  ∀ (X : C),
    (CategoryTheory.MonoidalCategory.rightUnitor (F.obj X)).hom =
      CategoryTheory.MonoidalCategory.tensorHom (𝟙 (F.obj X)) ε ≫
        μ X (𝟙_ C) ≫ F.map (CategoryTheory.MonoidalCategory.rightUnitor X).hom := by
  aesop_cat
⊢ ∀ (a : C),
  (CategoryTheory.LaxMonoidalFunctor.ofTensorHom F ε μ μ_natural associativity left_unitality right_unitality).obj a =
    F.obj a
o : Ordinal.{u_1}
IH : ∀ k < o, -nim k = nim k
⊢ (-mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) = mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)
R : Type u
Semiring R
p q : R[X]
h : p.leadingCoeff + q.leadingCoeff ≠ 0
⊢ (p + q).degree = max p.degree q.degree
n p : ℕ
⊢ p ∈ n.primeFactors ↔ p ∈ n.primeFactorsList
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
x : E
s : Set E
ι : Type u_6
Fintype ι
F' : ι → Type u_7
(i : ι) → NormedAddCommGroup (F' i)
(i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
h : ∀ (i : ι), DifferentiableWithinAt 𝕜 (φ i) s x
hs : UniqueDiffWithinAt 𝕜 s x
⊢ fderivWithin 𝕜 (fun x i => φ i x) s x = ContinuousLinearMap.pi fun i => fderivWithin 𝕜 (φ i) s x
F : Type u
Field F
W : WeierstrassCurve.Affine F
x₁ x₂ y₁ y₂ : F
h₁ : W.Nonsingular x₁ y₁
h₂ : W.Nonsingular x₂ y₂
hy : y₁ ≠ W.negY x₂ y₂
⊢ WeierstrassCurve.Affine.Point.some h₁ + WeierstrassCurve.Affine.Point.some h₂ = -WeierstrassCurve.Affine.Point.some ⋯
α : Type u_1
β : Type u_2
p : α → Prop
f : α → β
x : Option α
H : ∀ (a : α), a ∈ x → p a
⊢ Option.pmap (fun a x => f a) x H = Option.map f x
T : ℝ
hT : Fact (0 < T)
x : AddCircle T
ε : ℝ
hε : ε ≤ 0
⊢ closedBall x ε =ᶠ[ae volume] ball x ε
𝕜 : Type u_1
inst✝¹⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
H : Type u_3
inst✝¹² : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
inst✝¹¹ : TopologicalSpace M
inst✝¹⁰ : ChartedSpace H M
E' : Type u_5
inst✝⁹ : NormedAddCommGroup E'
inst✝⁸ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁷ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
inst✝⁶ : TopologicalSpace M'
inst✝⁵ : ChartedSpace H' M'
E'' : Type u_8
inst✝⁴ : NormedAddCommGroup E''
inst✝³ : NormedSpace 𝕜 E''
H'' : Type u_9
inst✝² : TopologicalSpace H''
I'' : ModelWithCorners 𝕜 E'' H''
M'' : Type u_10
inst✝¹ : TopologicalSpace M''
inst✝ : ChartedSpace H'' M''
f f₀ f₁ : M → M'
x : M
s t : Set M
g : M' → M''
u : Set M'
Is : SmoothManifoldWithCorners I M
I's : SmoothManifoldWithCorners I' M'
I''s : SmoothManifoldWithCorners I'' M''
f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)
g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))
h : MDifferentiableWithinAt I I' f s x
hs : s ∈ 𝓝 x
⊢ MDifferentiableAt I I' f x
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ✝ : Type u_5
γ' : Type u_6
δ : Type u_7
δ' : Type u_8
ε : Type u_9
ε' : Type u_10
ζ : Type u_11
ζ' : Type u_12
ν : Type u_13
inst✝⁷ : DecidableEq α'
inst✝⁶ : DecidableEq β'
inst✝⁵ : DecidableEq γ✝
inst✝⁴ : DecidableEq γ'
inst✝³ : DecidableEq δ
inst✝² : DecidableEq δ'
inst✝¹ : DecidableEq ε
inst✝ : DecidableEq ε'
f✝ f'✝ : α → β → γ✝
g✝ g'✝ : α → β → γ✝ → δ
s s' : Finset α
t t' : Finset β
u✝ u' : Finset γ✝
a a' : α
b b' : β
c : γ✝
γ : Type u_14
u : Finset γ
f : δ → γ → ε
g : α → β → δ
f' : α → ε' → ε
g' : β → γ → ε'
h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c)
⊢ image2 f (image2 g ↑s ↑t) ↑u = image2 f' (↑s) (image2 g' ↑t ↑u)
b : ℂ
hb : b.re < 0
c d : ℂ
hb' : b ≠ 0
h : ∀ (x : ℝ), cexp (b * ↑x ^ 2 + c * ↑x + d) = cexp (- -b * (↑x + c / (2 * b)) ^ 2) * cexp (d - c ^ 2 / (4 * b))
⊢ (∫ (a : ℝ), cexp (- -b * (↑a + (↑(c / (2 * b)).re + ↑(c / (2 * b)).im * I)) ^ 2)) * cexp (d - c ^ 2 / (4 * b)) = (↑π / -b) ^ (1 / 2) * cexp (d - c ^ 2 / (4 * b))
R : Type u
CommRing R
S : Type u_1
Ring S
Algebra R S
h : PowerBasis R S
⊢ ((Algebra.leftMulMatrix h.basis) h.gen).charpoly = minpoly R h.gen
C : Type u₁
CategoryTheory.Category.{v₁, u₁} C
CategoryTheory.MonoidalCategory C
D : Type u₂
CategoryTheory.Category.{v₂, u₂} D
CategoryTheory.MonoidalCategory D
F : CategoryTheory.MonoidalFunctor C D
X : C
⊢ ∀ (X_1 : C), (F.commTensorRight X).hom.app X_1 = F.μ X_1 X
X : Scheme
R : CommRingCat
inst✝ : IsDomain ↑R
⊢ closure {⊥} = ⊤
Γ : Type u_1
Inhabited Γ
L R : Turing.ListBlank Γ
n : ℕ
⊢ (Turing.Tape.mk' L R).nth ↑n = R.nth n
L : Language
K : Set (Bundled L.Structure)
M : Type w
inst✝¹ : L.Structure M
N✝ : Type w
inst✝ : L.Structure N✝
h : L.IsUltrahomogeneous M
N P Q : Bundled L.Structure
NP : ↑N ↪[L] ↑P
NQ : ↑N ↪[L] ↑Q
Nfg : Structure.FG L ↑N
Pfg : Structure.FG L ↑P
PM : ↑P ↪[L] M
Qfg : Structure.FG L ↑Q
QM : ↑Q ↪[L] M
g : M ≃[L] M
hg : (QM.comp NQ).comp (PM.comp NP).equivRange.symm.toEmbedding = g.toEmbedding.comp (PM.comp NP).toHom.range.subtype
s : L.Substructure M := (g.toHom.comp PM.toHom).range ⊔ QM.toHom.range
n : ↑N
hgn : QM (NQ n) = g (PM (NP n))
⊢ ↑(Set.inclusion ⋯ ((g.toEmbedding.comp PM).equivRange.toEmbedding (NP n))) = ↑(Set.inclusion ⋯ (QM.equivRange (NQ n)))
G : Type u
α : Type v
β : Type w
inst✝ : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this✝ : Fact (Nat.Prime p)
this : Finite G
m : ℕ
hm : Nat.card G = p ^ m
⊢ p ^ n ∣ Nat.card G
x y : ℝ
hx : x < -1
⊢ 0 < log x
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (WittVector.polyOfInterest p n).vars =
  (↑p ^ (n + 1) *
      (WittVector.wittMul p (n + 1) + ↑p ^ (n + 1) * MvPolynomial.X (0, n + 1) * MvPolynomial.X (1, n + 1) -
          MvPolynomial.X (0, n + 1) * (MvPolynomial.rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -
        MvPolynomial.X (1, n + 1) * (MvPolynomial.rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)))).vars
F : Type u_1
inst✝ : Field F
S : Set F
p : F
hp : p ∈ Ring.closure S
q : F
hq : q ∈ Ring.closure S
⊢ (p / q)⁻¹ ∈ closure S
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
o : Ordinal.{u}
hC : IsClosed C
hsC : contained C (Order.succ o)
ho : o < Ordinal.type fun x x_1 => x < x_1
f : LocallyConstant ↑C ℤ
hf : ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuous_toFun := ⋯ }) f) = ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₀ C ho, continuous_toFun := ⋯ }) f)
C₀C : ↑(C0 C ho) → ↑C := fun x => ⟨↑x, ⋯⟩
h₀ : Continuous C₀C
C₁C : ↑(π (C1 C ho) fun x => ord I x < o) → ↑C := fun x => ⟨SwapTrue o ↑x, ⋯⟩
h₁ : Continuous C₁C
⊢ (π C fun x => ord I x < o) ⊆ C0 C ho ∪ π (C1 C ho) fun x => ord I x < o
R : Type u_1
K : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
inst✝¹ : Module.Free R M
inst✝ : Module.Finite R M
x y z : L
⊢ (trace R M) ((φ x * φ y - φ y * φ x) * φ z) = (trace R M) (φ x * (φ y * φ z)) - (trace R M) (φ y * (φ x * φ z))
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommSemiring R
inst✝ : NoZeroDivisors R
p q : R[X]
hm : p.Monic
hnd : p.natDegree = 2
h✝ : Nontrivial R
⊢ (¬∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → g.natDegree ∉ Ioc 0 (2 / 2)) ↔ ∃ c₁ c₂, p.coeff 0 = c₁ * c₂ ∧ p.coeff 1 = c₁ + c₂
G : Type u_1
H : Type u_2
AddGroup G
N : AddSubgroup G
AddGroup H
f : G →+ H
hN : N = f.ker
tH : AddMonoid.IsTorsion H
tN : AddMonoid.IsTorsion ↥N
⊢ AddMonoid.IsTorsion G
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : ProbabilityTheory.Kernel α (β × ℝ)
ν : ProbabilityTheory.Kernel α β
f : α × β → StieltjesFunction
ProbabilityTheory.IsFiniteKernel κ
hf : ProbabilityTheory.IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) Set.univ ∂ν a = (κ a) (s ×ˢ Set.univ)
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P Q U : Cᵒᵖ ⥤ Type w
X Y : C
S : Sieve X
R✝ R : Presieve X
⊢ (IsSeparatedFor P R ∧ ∀ (x : FamilyOfElements P R), x.Compatible → ∃ t, x.IsAmalgamation t) ↔ IsSheafFor P (generate R).arrows
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
⊢ (∏ x ∈ finPairsLT n, if 1 x.fst ≤ 1 x.snd then -1 else 1) = ∏ _x ∈ finPairsLT n, 1
a b : ℝ
n : ℕ
f : ℝ → ℝ
μ ν : Measure ℝ
inst✝ : IsLocallyFiniteMeasure μ
c d : ℝ
r : ℂ
h : 0 ≤ r.re
h2 : ¬0 ∉ [[a, b]]
h' : 0 = r.re
⊢ IntervalIntegrable (fun x => ↑x ^ r) μ a b
R : Type u
L : Type v
CommRing R
LieRing L
LieAlgebra R L
x : L
⊢ x ∈ ⊤
α : Type u_1
C : Set (Set α)
s t : Set α
I : Finset (Set α)
hC : IsSetSemiring C
hs : s ∈ C
hI : ↑I ⊆ C
⊢ Disjoint I (hC.diffFinset₀ hs hI)
ι : Sort u_1
c : ι → Set (↑unitInterval × ↑unitInterval)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : Set.univ ⊆ ⋃ i, c i
⊢ ∃ t,
  t 0 = 0 ∧
    Monotone t ∧
      (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ×ˢ Set.Icc (t m) (t (m + 1)) ⊆ c i
C : Type u
inst✝³ : Category.{v, u} C
D : Type u_1
inst✝² : Category.{u_2, u_1} D
P : MorphismProperty C
inst✝¹ : P.RespectsIso
F : C ⥤ D
inst✝ : F.IsEquivalence
⊢ (P.map F).inverseImage F = P
R : Type u
inst✝¹ : CommRing R
W' : Jacobian R
F : Type v
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 ≠ Q x * P z ^ 2
⊢ W.addXYZ P Q = addZ P Q • ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), 1]
C : Type u
CategoryTheory.Category.{v, u} C
P : C
⊢ Reflexive (CategoryTheory.Abelian.PseudoEqual P)
C : Type u_1
CategoryTheory.Category.{u_2, u_1} C
CategoryTheory.Limits.HasZeroMorphisms C
S : CategoryTheory.ShortComplex C
hf : S.f = 0
hg : S.g = 0
⊢ (CategoryTheory.ShortComplex.RightHomologyData.ofZeros S hf hg).H = S.X₂
α : Type u_1
β : Type u_2
UniformSpace β
𝔖 : Set (Set α)
h : ⋃₀ 𝔖 = Set.univ
⊢ UniformContinuous ⇑(UniformOnFun.toFun 𝔖)
𝕜 : Type u_1
E : Type u_2
F : Type u_3
NormedCommRing 𝕜
AddCommMonoid E
AddCommMonoid F
Module 𝕜 E
Module 𝕜 F
B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜
⊢ GaloisConnection (⇑OrderDual.toDual ∘ B.polar) (B.flip.polar ∘ ⇑OrderDual.ofDual)
R : Type u_1
inst✝⁶ : Ring R
E : Type u_2
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module R E
F : Type u_3
inst✝³ : AddCommGroup F
inst✝² : Module R F
G : Type u_4
inst✝¹ : AddCommGroup G
inst✝ : Module R G
f : E →ₗ.[R] F
⊢ Submodule.map (LinearMap.fst R F E) (Submodule.map (LinearEquiv.prodComm R E F) f.graph) = Submodule.map (LinearMap.fst R F E) (Submodule.map (↑(LinearEquiv.prodComm R E F)) f.graph)
α : Type u
β : Type v
X : Type u_1
ι : Type u_2
inst✝ : PseudoMetricSpace α
⊢ Function.swap dist = dist
K : Type u_1
L : Type u_2
Field K
Field L
Algebra K L
S : IntermediateField K L
⊢ 0 ∈ S
G : Type u_1
α : Type u_3
Group G
MulAction G α
MeasurableSpace α
s : Set α
μ : MeasureTheory.Measure α
MeasurableSpace G
MeasurableSMul G α
MeasureTheory.SMulInvariantMeasure G α μ
Countable G
h : MeasureTheory.IsFundamentalDomain G s μ
f : α → ℝ≥0∞
t : Set α
⊢ ∫⁻ (x : α) in t, f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in g • t ∩ s, f (g⁻¹ • x) ∂μ
R : Type u_1
S : Type u_2
CommRing R
CommRing S
Algebra R S
x : S
hx : ¬IsIntegral R x
⊢ minpolyDiv R x = 0
α : Type u
β : Type v
Preorder β
f : α → β
s : Set α
a : α
t : Set α
hf : IsMaxOn f t a
h : s ⊆ t
⊢ IsMaxOn f s a
R : CommRingCat
⊢ (AlgebraicGeometry.StructureSheaf.globalSectionsIso ↑R).hom = AlgebraicGeometry.StructureSheaf.toOpen ↑R ⊤
α : Type u_1
LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
⊢ toIocMod hp (-a) b = p - toIcoMod hp a (-b)
r : ℝ
hr : 0 < r
x : ℝ
hx : x ∈ Set.Ioi 0
⊢ log x / x ^ (-r) = (fun x => log x * x ^ r) x
n k s : ℕ
hkn : k ≤ n
hsk : s ≤ k
h : 0 < (n - k)! * (k - s)! * s !
⊢ n ! = n.choose s * s ! * ((n - s).choose (k - s) * (k - s)! * (n - s - (k - s))!)
G : Type u
Group G
N : Subgroup G
nN : N.Normal
h : N ≤ Subgroup.comap (MonoidHom.id G) N := ⋯
⊢ QuotientGroup.map N N (MonoidHom.id G) h = MonoidHom.id (G ⧸ N)
α : Type u
β : Type u_1
w x y z : α
inst✝ : GeneralizedBooleanAlgebra α
⊢ x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = ⊥
X : Type u_1
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : OpensMeasurableSpace X
inst✝ : CompletelyRegularSpace X
x : X
L : Filter X
h : ¬Tendsto id L (𝓝 x)
U : Set X
U_nhd : U ∈ 𝓝 x
hU : ∃ᶠ (x : X) in L, x ∉ U
⊢ ¬Tendsto diracProba L (𝓝 (diracProba x))
α : Type u_1
M : Type u_2
R : Type u_5
Semiring R
AddCommMonoid M
Module R M
a : α
b : M
⊢ (Finsupp.lsingle a) b = Finsupp.single a b
M : Type u_1
S T : Set M
inst✝ : Mul M
z₁ z₂ : M
hz₁ : z₁ ∈ center M
hz₂ : z₂ ∈ center M
a c : M
⊢ a * z₁ * z₂ * c = a * z₁ * (z₂ * c)
α β : Type u
n : ℕ
⊢ Function.FromTypes (fun x => α) β = Function.OfArity α β n
n : Type u_3
α : Type u_5
DecidableEq n
Semiring α
i j : n
c : α
Fintype n
b : n
M : Matrix n n α
⊢ (Matrix.stdBasisMatrix i j c * M) i b = c * M j b
y : ℝ
h : 0 < y
⊢ some 0 ^ y = some 0
ι : Type u_1
𝕜 : Type u_2
inst✝⁴ : RCLike 𝕜
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
cplt : CompleteSpace E
G : ι → Type u_4
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
F : ι → Submodule 𝕜 E
hV : IsHilbertSum 𝕜 G V
W₀ : Π₀ (i : ι), G i
⊢ ↑(hV.linearIsometryEquiv (hV.linearIsometryEquiv.symm (W₀.sum (lp.single 2)))) = ⇑W₀