ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{u_3, u_1} C
inst✝⁷ : Category.{u_4, u_2} D
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroMorphisms D
S : ShortComplex C
h₁✝ : S.LeftHomologyData
h₂✝ : S.RightHomologyData
F✝ : C ⥤ D
inst✝⁴ : F✝.PreservesZeroMorphisms
hl : S.LeftHomologyData
hr : S.RightHomologyData
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
γ : HomologyMapData (𝟙 (S.map F)) (S.map F).homologyData (S.homologyData.map F)
eq : leftHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.left (S.homologyData.left.map F) ≫ F.map S.homologyData.iso.hom = (S.map F).homologyData.iso.hom ≫ rightHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.right (S.homologyData.right.map F)
⊢ (S.map F).homologyData.iso.hom ≫ rightHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.right (S.homologyData.right.map F) ≫ F.map S.homologyData.iso.inv = leftHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.left (S.homologyData.left.map F)
R : Type u_1
R' : Type u_2
𝕜 : Type u_3
𝕜₂ : Type u_4
𝕜₃ : Type u_5
𝕝 : Type u_6
E : Type u_7
E₂ : Type u_8
E₃ : Type u_9
F : Type u_10
G : Type u_11
ι : Type u_12
inst✝⁵ : NormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : NormedSpace ℝ 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : Module ℝ E
inst✝ : IsScalarTower ℝ 𝕜 E
p : Seminorm 𝕜 E
x : E
r : ℝ
y : E
⊢ p (y + -x) < r ↔ y ∈ {x_1 | (⇑p ∘ fun z => z + -x) x_1 < r}
α : Type u_1
β : Type u_2
e : PartialEquiv α β
s : Set α
⊢ (e.restr s).target = e.target ∩ ↑e.symm ⁻¹' s
α : Type u_1
CancelCommMonoidWithZero α
NormalizationMonoid α
UniqueFactorizationMonoid α
x y : α
hx : x ≠ 0
hy : y ≠ 0
⊢ DvdNotUnit x y ↔ UniqueFactorizationMonoid.normalizedFactors x < UniqueFactorizationMonoid.normalizedFactors y
α : Type u_2
β : Type u_3
VAdd α β
f : Filter α
g h : Filter β
⊢ h ≤ f +ᵥ g ↔ ∀ ⦃s : Set α⦄, s ∈ f → ∀ ⦃t : Set β⦄, t ∈ g → s +ᵥ t ∈ h
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
H : Type u_3
TopologicalSpace H
I : ModelWithCorners 𝕜 E H
M : Type u_4
TopologicalSpace M
ChartedSpace H M
E' : Type u_5
NormedAddCommGroup E'
NormedSpace 𝕜 E'
H' : Type u_6
TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M' : Type u_7
TopologicalSpace M'
ChartedSpace H' M'
f f₁ : M → M'
s : Set M
x : M
n : ℕ∞
h₁ : ∀ y ∈ s, f₁ y = f y
hx : f₁ x = f x
⊢ ContMDiffWithinAt I I' n f₁ s x ↔ ContMDiffWithinAt I I' n f s x
J : Type u_1
C : Type u_2
CategoryTheory.Category.{u_3, u_1} J
CategoryTheory.Category.{u_4, u_2} C
CategoryTheory.Limits.HasZeroMorphisms C
F : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)
⊢ ((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).X₁ = F.comp CategoryTheory.ShortComplex.π₁
α : Type u
β : Type u_1
γ : Type u_2
δ : Type u_3
ε : Type u_4
l₁ : List α
h : l₁.length ≤ [].length
⊢ (l₁.zip []).unzip.1 = l₁
ι : Type u_1
π : ι → Type u_4
(i : ι) → HImp (π i)
a b : (i : ι) → π i
i : ι
⊢ (a ⇨ b) i = a i ⇨ b i
X : Type u_1
inst✝³ : TopologicalSpace X
inst✝² : MeasurableSpace X
inst✝¹ : OpensMeasurableSpace X
μ : Measure X
inst✝ : IsFiniteMeasure μ
f : X →ᵇ ℝ
c : ℝ
⊢ ∫ (x : X), (f + const X c) x ∂μ = ∫ (x : X), f x ∂μ + (μ Set.univ).toReal • c
R : Type u_1
M : Type u_2
CommSemiring R
AddCommMonoid M
Module R M
B : LinearMap.BilinForm R M
H : B.IsRefl
x y : M
⊢ (B x) y = 0 → (B y) x = 0
ι : Type u_1
α : Type u_2
inst✝¹ : PartialOrder α
inst✝ : LocallyFiniteOrder α
a b c : α
h : a < b
⊢ Ico a b = cons a (Ioo a b) ⋯
H : Type u_1
M : Type u_2
H' : Type u_3
TopologicalSpace H
TopologicalSpace M
ChartedSpace H M
TopologicalSpace H'
G : StructureGroupoid H
G' : StructureGroupoid H'
e e' : PartialHomeomorph M H
P : (H → H') → Set H → H → Prop
s : Set M
x : M
hG : G.LocalInvariantProp G' P
g : M → H'
he : e ∈ StructureGroupoid.maximalAtlas M G
xe : x ∈ e.source
he' : e' ∈ StructureGroupoid.maximalAtlas M G
xe' : x ∈ e'.source
⊢ P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P (g ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)
k : Type u_1
V : Type u_2
P : Type u_3
Ring k
AddCommGroup V
Module k V
AffineSpace V P
n : ℕ
s : Affine.Simplex k P n
⊢ s.reindex (Equiv.refl (Fin (n + 1))) = s
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
inst✝ : MeasureSpace α
s✝ t s : Set α
hs : MeasurableSet s
⊢ Measure.map Subtype.val volume = volume.restrict s
n : ℕ
F : TypeVec.{u} n → Type u_1
q : MvQPF F
h : IsUniform
α : TypeVec.{u} n
a : (P F).A
f : (P F).B a ⟹ α
i✝ : Fin2 n
u : α i✝
⊢ (∀ (a_1 : (P F).A) (f_1 : (P F).B a_1 ⟹ α), abs ⟨a_1, f_1⟩ = abs ⟨a, f⟩ → u ∈ f_1 i✝ '' univ) ↔ u ∈ f i✝ '' univ
τ : Type u_1
α : Type u_2
β : Type u_3
ι : Type u_4
inst✝ : TopologicalSpace β
f : Filter τ
ϕ : τ → α → β
s s₁ s₂ : Set α
c : Set β
hc₁ : IsCompact c
n : Set β
hn₁ : IsOpen n
hn₂ : ω f ϕ s ⊆ n
v : Set τ
hv₁ : v ∈ f
hv₂ : closure (image2 ϕ v s) ⊆ c
k : Set β := closure (image2 ϕ v s)
hk : IsCompact (k \ n)
j : Set τ → Set β := fun u => (closure (image2 ϕ (u ∩ v) s))ᶜ
⊢ ∃ u ∈ f, closure (image2 ϕ u s) ⊆ n
α : Type u_1
BooleanAlgebra α
a b : α
⊢ toBoolRing (a ∆ b) = toBoolRing a + toBoolRing b
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : CanonicallyLinearOrderedAddCommMonoid M
w : σ → M
φ : MvPolynomial σ R
hw : NonTorsionWeight w
m : σ →₀ ℕ
x : σ
⊢ m x ≠ 0 → m x • w x = 0 ↔ m x = 0
R : Type u_1
Semiring R
d df dg : ℕ
a b : R
f g : R[X]
h_mul_left : f.natDegree ≤ df
h_mul_right : g.natDegree ≤ dg
⊢ f.coeff df = a → g.coeff dg = b → df + dg ≤ d → (f * g).coeff d = if d = df + dg then a * b else 0
α : Type u_1
E : Type u_2
F : Type u_3
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : MeasureTheory.Measure α
NormedAddCommGroup E
NormedAddCommGroup F
g : E → F
c : ℝ≥0
Fact (1 ≤ p)
hg : LipschitzWith c g
g0 : g 0 = 0
⊢ Continuous (hg.compLp g0)
i : ℤ
j : ℤ
hj : j ≠ 0
⊢ 0 < i.gcd j
α : Type u_2
β : Type u_3
LE α
LE β
e : α ≃o β
⊢ e.symm.trans e = OrderIso.refl β
n m l : ℕ
hln : l ≤ n
⊢ filter (fun x => decide (l ≤ x)) (Ico n m) = Ico (max n l) m
α : Type u_1
CompleteLattice α
a : α
s : Set α
⊢ sInf (insert a s) = a ⊓ sInf s
R : Type u_1
m₁ : Type u_3
m₂ : Type u_4
n₁ : Type u_6
n₂ : Type u_7
B₁₁ : Matrix m₁ n₁ R
B₁₂ : Matrix m₁ n₂ R
B₂₁ : Matrix m₂ n₁ R
B₂₂ : Matrix m₂ n₂ R
⊢ (B₁₁.fromColumns B₁₂).fromRows (B₂₁.fromColumns B₂₂) = Matrix.fromBlocks B₁₁ B₁₂ B₂₁ B₂₂
H : Type u_1
TopologicalSpace H
G : StructureGroupoid H
ClosedUnderRestriction G
⊢ G.LocalInvariantProp G G.IsLocalStructomorphWithinAt
α : Type ua
β : Type ub
γ : Type uc
δ : Type ud
ι : Sort u_1
inst✝ : UniformSpace α
t : Set (α × α)
h : t ∈ 𝓤 α
w : Set (α × α)
w_in : w ∈ 𝓤 α
w_symm : SymmetricRel w
r : w ○ w ○ w ⊆ t
⊢ Set (α × α)
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
s t : Set E
c : ℝ
hc : 0 ≤ c
x : E
⊢ (c * gauge s x / (c * gauge t x) * c) • x = (c * (gauge s x / gauge t x)) • x
ι : Type u_1
α : Type u
β : Type v
γ : Type w
l₁ l₂ : List α
p : α → Bool
a b : α
as : List α
h✝ : filter p (a :: b :: as) ≠ []
h' : p ((b :: as).getLast ⋯) = true
h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []
⊢ (if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯
C : Type u
CategoryTheory.Category.{v, u} C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
g' : Z ⟶ W
CategoryTheory.Limits.HasPushout f g
CategoryTheory.Limits.HasPushout (CategoryTheory.Limits.pushout.inr f g) g'
CategoryTheory.Limits.HasPushout f (g ≫ g')
⊢ CategoryTheory.Limits.pushout.inl f g ≫
    CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.pushout.inr f g) g' ≫
      (CategoryTheory.Limits.pushoutLeftPushoutInrIso f g g').hom =
  CategoryTheory.Limits.pushout.inl f (g ≫ g')
x y : Cardinal.{u_1}
⊢ Cardinal.toNat (x * y) = Cardinal.toNat x * Cardinal.toNat y
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : MeasureTheory.Measure α
NormedAddCommGroup E
β : Type u_5
MeasurableSpace β
μb : MeasureTheory.Measure β
f : α → β
g : ↥(MeasureTheory.Lp E p μb)
hf : MeasureTheory.MeasurePreserving f μ μb
⊢ ↑((MeasureTheory.Lp.compMeasurePreserving f hf) g) = (↑g).compMeasurePreserving f hf
α : Type u_1
M : Matroid α
B B₁ B₂ : Set α
inst✝ : M.RkPos
hB : M.Base B
⊢ B.Nonempty
𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁴ : NontriviallyNormedField 𝕜
inst✝¹³ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
σ' : 𝕜' →+* 𝕜
inst✝¹² : RingHomInvPair σ σ'
inst✝¹¹ : RingHomInvPair σ' σ
inst✝¹⁰ : RingHomIsometric σ
inst✝⁹ : RingHomIsometric σ'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F✝ : Type u_4
inst✝⁶ : NormedAddCommGroup F✝
inst✝⁵ : NormedSpace 𝕜' F✝
f : E →SL[σ] F✝
inst✝⁴ : CompleteSpace F✝
inst✝³ : CompleteSpace E
F : Type u_5
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace F
g : E →ₗ[𝕜] F
hg : IsClosed ↑g.graph
⊢ Continuous ⇑g
ι : Type u_1
𝕜 : Type u_2
NontriviallyNormedField 𝕜
H : Type u_3
TopologicalSpace H
E : Type u_4
NormedAddCommGroup E
NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G : Type u_5
AddCommMonoid G
TopologicalSpace G
ChartedSpace H G
SmoothAdd I G
E' : Type u_6
NormedAddCommGroup E'
NormedSpace 𝕜 E'
H' : Type u_7
TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_8
TopologicalSpace M
ChartedSpace H' M
s : Set M
x : M
t : Finset ι
f : ι → M → G
h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x
⊢ SmoothWithinAt I' I (fun x => ∑ i ∈ t, f i x) s x
ι : Type u_1
I✝ J✝ : Box ι
p : Box ι → Prop
I : Box ι
H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
hpI : ¬p I
s : Box ι → Set ι
hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))
J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I
⊢ False
R : Type u₁
L : Type u₂
CommRing R
LieRing L
LieAlgebra R L
A : Type u₃
Ring A
Algebra R A
f : L →ₗ⁅R⁆ A
x : L
⊢ ((UniversalEnvelopingAlgebra.lift R) f) ((UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ x)) = f x
θ ψ : Real.Angle
h : 2 • θ = 2 • ψ
⊢ |θ.sin| = |ψ.sin|
k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝⁴ : DivisionRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
inst✝ : Fintype ι
p : ι → P
hp : AffineIndependent k p
h✝ : IsEmpty ι
⊢ Fintype.card ι ≤ finrank k ↥(vectorSpan k (Set.range p)) + 1
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
inst✝ : F.IsEquivalence
X Y : C
f : X ⟶ Y
⊢ F.inv.map (F.map f) = (F.asEquivalence.functor ⋙ F.asEquivalence.inverse).map f
ι : Type u
s : Finset ι
w z : ι → ℝ
x : ℝ
hw' : ∑ i ∈ s, w i = 1
hx : ∀ i ∈ s, w i ≠ 0 → z i = x
⊢ ∑ i ∈ s, w i * x = x
G : Type u_1
α : Type u_2
AddGroup G
AddAction G α
g : G
a : α
⊢ AddAction.stabilizer G (g +ᵥ a) = AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (AddAction.stabilizer G a)
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
D' : Type u₃
inst✝⁴ : Category.{v₃, u₃} D'
L : C ⥤ D
inst✝³ : L.IsLocalization W
X Y Z : C
inst✝² : HasSmallLocalizedHom W X Y
inst✝¹ : HasSmallLocalizedHom W Y Z
inst✝ : HasSmallLocalizedHom W X Z
this✝ : Small.{w, max u₁ v₁} (W.Q.obj X ⟶ W.Q.obj Y) := small_of_hasSmallLocalizedHom W W.Q X Y
this : Small.{w, max u₁ v₁} (W.Q.obj Y ⟶ W.Q.obj Z) := small_of_hasSmallLocalizedHom W W.Q Y Z
α : W.Q.obj X ⟶ W.Q.obj Y
β : W.Q.obj Y ⟶ W.Q.obj Z
⊢ (homEquiv W W.Q L) ((homEquiv W W.Q W.Q) ((homEquiv W W.Q W.Q) α ≫ (homEquiv W W.Q W.Q) β)) = (homEquiv W W.Q L) α ≫ (homEquiv W W.Q L) β
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Semiring A
inst✝⁹ : Semiring B
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : Algebra R B
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : Algebra S A
inst✝ : IsScalarTower R S A
hA : FiniteType R A
⊢ FiniteType S A
α : Type u
PseudoEMetricSpace α
K : Set α
hK : IsCompact K
⊢ (𝓝ˢ K).HasBasis (fun δ => 0 < δ) fun δ => Metric.thickening δ K
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✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SFinite ν
s : Set (α × β)
h : (μ.prod ν) s = 0
⊢ (fun x => ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0
E : Type u_3
NormedAddCommGroup E
NormedSpace ℝ E
a b : ℝ
f g : ℝ → E
μ : MeasureTheory.Measure ℝ
h : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → f x = g x
⊢ ∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ
α : Type u_1
DecidableEq α
s t : Finset α
⊢ Finset.Ico s t = Finset.filter (fun x => s ⊆ x) t.ssubsets
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
r : R
n : ℕ
⊢ 0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0
α : Type u_1
β : Type u_2
X✝ Y✝ : PartialFun
f : X✝ ⟶ Y✝
a : (𝟭 PartialFun).obj X✝
b : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝
⊢ (b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔ b ∈ (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val) ⟨some a, ⋯⟩
α : Type u_1
Preorder α
a x : α
h : x ∈ Set.Iio a
⊢ x ∈ Set.Iic a
S : Type u_1
R : Type u_3
M : Type u_4
N : Type u_5
CommRing R
AddCommGroup M
AddCommGroup N
Module R M
Module R N
Q : QuadraticMap R M N
CommSemiring S
Algebra S R
Module S M
IsScalarTower S R M
Module S N
IsScalarTower S R N
a : S
x y : M
⊢ QuadraticMap.polar (⇑Q) (a • x) y = a • QuadraticMap.polar (⇑Q) x y
C : Type u_1
D : Type u_2
inst✝¹ : Category.{?u.25278, u_1} C
inst✝ : Category.{?u.25282, u_2} D
F G : C ⥤ Cat
α : F ⟶ G
X : Grothendieck F
⊢ (eqToHom ⋯).app X.fiber ≫ eqToHom ⋯ = eqToHom ⋯
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : MeasureTheory.Measure α
NormedAddCommGroup E
TopologicalSpace α
BorelSpace α
SecondCountableTopologyEither α E
MeasureTheory.IsFiniteMeasure μ
f : α →ᵇ E
⊢ ‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖₊ ≤ MeasureTheory.measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖₊
α : Type u_1
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
⊢ ⊤ + μ = ⊤
R : Type u
ι : Type v
M₂ : Type w₂
M₃ : Type w₃
Semiring R
AddCommMonoid M₂
AddCommMonoid M₃
Module R M₂
Module R M₃
TopologicalSpace M₂
TopologicalSpace M₃
Subsingleton ι
i : ι
f : M₂ →L[R] M₃
⊢ ((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f).toMultilinearMap = (MultilinearMap.ofSubsingleton R M₂ M₃ i) ↑f
σ : Type u_1
R : Type u_2
Semiring R
φ : MvPowerSeries σ R
s : σ
⊢ Commute φ (MvPowerSeries.X s)
μ : YoungDiagram
x : ℕ
hx : ∃ a ∈ List.range (μ.colLen 0), μ.rowLen a = x
⊢ 0 < x
α : Type u_1
x y : List α
h : x <+: y
hx : x ≠ []
⊢ y ≠ []
R : Type u_1
M : Type u_2
CommSemiring R
σ : Type u_3
AddCommMonoid M
w : σ → M
m : M
⊢ MvPolynomial.weightedHomogeneousSubmodule R w m = Finsupp.supported R R {d | (Finsupp.weight w) d = m}
𝕜 : Type u_1
E : Type u_2
F : Type u_3
RCLike 𝕜
NormedAddCommGroup E
NormedAddCommGroup F
InnerProductSpace 𝕜 E
InnerProductSpace 𝕜 F
CompleteSpace E
CompleteSpace F
A : E →L[𝕜] F
x : E
⊢ ‖A x‖ = √(RCLike.re ⟪((ContinuousLinearMap.adjoint A).comp A) x, x⟫_𝕜)
G : Type u_1
P : Type u_2
AddGroup G
T : AddTorsor G P
p q : P
⊢ p -ᵥ q ≠ 0 ↔ p ≠ q
α : Type u_1
G : Type u_2
Group G
MulAction G α
g h : G
comm : Commute g h
⊢ (MulAction.fixedBy α g)ᶜ ∈ MulAction.fixedBy (Set α) h
C : Type u
CategoryTheory.Category.{v, u} C
CategoryTheory.Limits.HasTerminal C
CategoryTheory.Limits.HasBinaryProducts C
⊢ comonEquiv.unitIso = CategoryTheory.NatIso.ofComponents (fun A => iso_cartesianComon_ A) ⋯
K : Type u_1
RCLike K
z : K
⊢ RCLike.normSq z = 0 ↔ z = 0
α : Type u_1
β : Type u_2
t₁ t₂ : TopologicalSpace α
g : β → α
⊢ TopologicalSpace.induced g (t₁ ⊓ t₂) = TopologicalSpace.induced g t₁ ⊓ TopologicalSpace.induced g t₂
R : Type u
L : Type v
CommRing R
LieRing L
LieAlgebra R L
L' : LieSubalgebra R L
⊢ 0 ∈ L'
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → ℝ≥0∞
hfm : AEMeasurable f μ
hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤
⊢ Memℒp (fun x => (f x).toReal) 1 μ
α : Type u_1
β : Type u_2
f : α → β → β
inst✝¹ : BEq α
inst✝ : Hashable α
m : Imp α β
k : α
h : m.size = m.buckets.size
⊢ m.size = ((m.buckets.update (mkIdx ⋯ (hash k).toUSize).val AssocList.nil ⋯).update (mkIdx ⋯ (hash k).toUSize).val (AssocList.modify k f m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat]) ⋯).size
ι : Sort u_1
α : Type u_2
β : Type u_3
MeasurableSpace α
MeasurableSpace β
f : ι → α → β
μ : MeasureTheory.Measure α
p : α → (ι → β) → Prop
Countable ι
hf : ∀ (i : ι), AEMeasurable (f i) μ
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x
⊢ μ (aeSeqSet hf p)ᶜ = 0
M : Type u_5
N : Type u_6
AddZeroClass M
AddZeroClass N
⊢ ⇑AddEquiv.prodComm = Prod.swap
α : Type u_1
inst✝ : DecidableEq α
l✝ : List α
x✝ : α
l l' : List α
h : l ~r l'
hn : l.Nodup
x : α
hx : x ∈ l
⊢ l.next x hx = l'.next x ⋯
α β : Type u
f : α → β
x : FreeAddGroup α
⊢ f <$> (-x) = -f <$> x
α : Type u_1
M : Type u_5
N : Type u_7
One M
One N
f : α → M × N
⊢ Function.mulSupport f = (Function.mulSupport fun x => (f x).1) ∪ Function.mulSupport fun x => (f x).2
ι : Type u_1
G : ι → Type u_2
H : Type u_3
(i : ι) → Group (G i)
Group H
φ : (i : ι) → H →* G i
d : Monoid.PushoutI.NormalWord.Transversal φ
DecidableEq ι
(i : ι) → DecidableEq (G i)
w : Monoid.CoprodI.Word G
i : ι
h : H
hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → g ∈ d.set i
hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (Monoid.CoprodI.of ((φ i) h) • w).toList → g ∈ d.set j
⊢ h = 1
G : Type u_1
Group G
S : Set G
hS : Subgroup.closure S = ⊤
⊢ Subgroup.center G = ⨅ g, Subgroup.centralizer ↑(Subgroup.zpowers ↑g)
𝕜 : Type u_1
inst✝¹⁰ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁹ : NormedAddCommGroup D
inst✝⁸ : NormedSpace 𝕜 D
E : Type uE
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type uF
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type uG
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
s✝ s₁ t u : Set E
f✝ f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
f : ι → E → F
s : Finset ι
h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x
⊢ ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 ≤ -1
hs' : -1 ≠ 0
⊢ (1 + -1) ^ p < 1 + p * -1
α : Type u_1
DecidableEq α
s : Finset α
a : α
n : ℕ
h : a ∉ s
m : { x // x ∈ (insert a s).sym n }
⊢ ↑((Finset.symInsertEquiv h) m).snd = (Sym.filterNe a ↑m).snd
α : Type u
as : List α
a : α
⊢ as.concat a = as ++ a :: List.nil
A : Type u_4
CommRing A
IsDomain A
UniqueFactorizationMonoid A
K : Type u_5
Field K
Algebra A K
IsFractionRing A K
x : K
⊢ IsRelPrime (IsFractionRing.num A x) ↑(IsFractionRing.den A x)
α : Type u_1
inst✝⁸ : TopologicalSpace α
inst✝⁷ : NormalSpace α
inst✝⁶ : R1Space α
inst✝⁵ : MeasurableSpace α
inst✝⁴ : BorelSpace α
E : Type u_2
inst✝³ : NormedAddCommGroup E
μ : Measure α
p : ℝ≥0∞
inst✝² : NormedSpace ℝ E
inst✝¹ : WeaklyLocallyCompactSpace α
inst✝ : μ.Regular
f : α → E
hf : Integrable f μ
ε : ℝ
hε : 0 < ε
⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ Integrable g μ
α : Type u_1
β : Type u_2
γ : Type u_3
n✝ m✝ l n m : ℕ
⊢ (fun x => x = m) = Eq m
α : Type u
inst✝ : DecidableEq α
β : Type v
n : ℕ
f g : Perm (Fin n)
⊢ (∏ x ∈ finPairsLT n, if (f * g) x.fst ≤ (f * g) x.snd then -1 else 1) = (∏ x ∈ finPairsLT n, if f x.fst ≤ f x.snd then -1 else 1) * ∏ x ∈ finPairsLT n, if g⁻¹ x.fst ≤ g⁻¹ x.snd then -1 else 1
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
ha : Polynomial.eval a F = 0
z' : ℤ_[p]
hz' : Polynomial.eval z' F = 0
hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖
h : ℤ_[p] := z' - a
q : ℤ_[p]
hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2
this✝ : (Polynomial.eval a (Polynomial.derivative F) + q * h) * h = 0
hne : ¬h = 0
this : Polynomial.eval a (Polynomial.derivative F) + q * h = 0
⊢ Polynomial.eval a (Polynomial.derivative F) = -q * h
C : Type u
CategoryTheory.Category.{v, u} C
⊢ CategoryTheory.Square.arrowArrowEquivalence'.inverse = CategoryTheory.Square.fromArrowArrowFunctor'
a b d : Int
hda : d ∣ a
hdb : d ∣ b
⊢ a.div d = b.div d ↔ a = b
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace
⊢ ((pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv) ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U)) = ((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv
R : Type u
Ring R
p : R[X]
⊢ (-p).degree = p.degree
α : Type u_1
f : α → ℚ → ℝ
a : α
self : ProbabilityTheory.IsRatStieltjesPoint f a
⊢ Monotone (f a)
G : Type u_1
AddGroup G
TopologicalSpace G
NonarchimedeanAddGroup G
K : Type u_3
AddGroup K
TopologicalSpace K
NonarchimedeanAddGroup K
U : Set (G × K)
hU : U ∈ 𝓝 0
⊢ ∃ V W, ↑V ×ˢ ↑W ⊆ U
α : Type u_1
E : Type u_4
𝕜 : Type u_6
MeasurableSpace α
NormedAddCommGroup E
p : ℝ≥0∞
μ : MeasureTheory.Measure α
NormedRing 𝕜
Module 𝕜 E
BoundedSMul 𝕜 E
Fact (1 ≤ p)
⊢ BoundedSMul 𝕜 ↥(MeasureTheory.Lp.simpleFunc E p μ)
E : Type uE
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
H : Type uH
inst✝⁵ : TopologicalSpace H
I : ModelWithCorners ℝ E H
M : Type uM
inst✝⁴ : TopologicalSpace M
inst✝³ : ChartedSpace H M
inst✝² : SmoothManifoldWithCorners I M
c : M
f : SmoothBumpFunction I c
x✝ : M
inst✝¹ : T2Space M
inst✝ : SmoothManifoldWithCorners I M
x : M
hx : x ∈ tsupport ↑f
⊢ ContMDiffAt I 𝓘(ℝ, ℝ) ⊤ (↑f) x
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
x : E
⊢ DifferentiableAt 𝕜 (fun x => x) x
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ x + 1 < rexp x
X : Scheme
inst✝ : IsIntegral X
U : X.Opens
x : ↥U
y : ↑(X.presheaf.obj (op U))
hy : (X.presheaf.germ x) y = 0
⊢ y = 0
α : Type u_1
MeasurableSpace α
s : MeasureTheory.SignedMeasure α
i : Set α
n : ℕ
hn :
  ¬MeasureTheory.VectorMeasure.restrict s
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤
      MeasureTheory.VectorMeasure.restrict 0
        (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
⊢ 1 /
    (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s
          (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +
      1) <
  ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n.succ)
α : Type u_1
𝕜 : Type u_2
inst✝¹ : LinearOrderedField 𝕜
G : SimpleGraph α
inst✝ : DecidableRel G.Adj
ε : 𝕜
s t : Finset α
a b : α
h : ¬G.IsUniform ε s t
⊢ (⋯.choose, ⋯.choose).2 ⊆ t
R : Type u_1
L : Type u_3
M : Type u_4
CommRing R
LieRing L
LieAlgebra R L
AddCommGroup M
Module R M
LieRingModule L M
LieModule R L M
Module.Free R M
Module.Finite R M
LieAlgebra.IsNilpotent R L
IsDomain R
IsPrincipalIdealRing R
h : LinearMap.ker (LieModule.traceForm R L M) = ⊥
⊢ IsLieAbelian L
p : ℕ
q : ℚ
hq : ¬q = 0
⊢ 0 ≤ ↑p
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : Ω → ℝ
hint : Integrable X ℙ
hnonneg : 0 ≤ X
K : ℕ
A : Tendsto (fun N => ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))
N : ℕ
hN : N ∈ Set.Ici K
⊢ ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)
α : Type u_1
Mul α
LE α
CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a : α
b c : α
⊢ a * b ≤ a * c ↔ b ≤ c
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : LT α
a✝ b : α
a : WithTop α
⊢ ¬⊤ < a
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), P.IsPrime → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
⊢ False
α : Type u_1
β : Type u_3
γ : Type u_5
MeasurableSpace α
MeasurableSpace β
MeasurableSpace γ
X : α → β
Y : α → γ
μ : MeasureTheory.Measure α
hY : AEMeasurable Y μ
⊢ (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).fst = MeasureTheory.Measure.map X μ