ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
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 : Ring R
inst✝ : Ring S
f : R →+* S
X : ModuleCat R
Y : ModuleCat S
g : (restrictScalars f).obj Y ⟶ X
y1 y2 : ↑Y
s : S
⊢ g (s • (y1 + y2)) = ({ toFun := fun s => g (s • y1), map_add' := ⋯, map_smul' := ⋯ } + { toFun := fun s => g (s • y2), map_add' := ⋯, map_smul' := ⋯ }) s
m : Type u_2
n : Type u_3
R : Type u_5
Fintype n
CommRing R
A : Matrix m n R
⊢ A.rank = FiniteDimensional.finrank R ↥(Submodule.span R (Set.range Aᵀ))
A : Type u_7
B : Type u_8
C : Type u_9
CommRing A
CommRing B
CommRing C
Algebra A B
Algebra A C
e : B ≃ₐ[A] C
x : B
⊢ (Algebra.trace A C) (e x) = (Algebra.trace A B) x
C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
inst✝⁷ : MonoidalCategory C
inst✝⁶ : BraidedCategory C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
inst✝⁴ : MonoidalCategory D
inst✝³ : BraidedCategory D
E : Type u₃
inst✝² : Category.{v₃, u₃} E
inst✝¹ : MonoidalCategory E
inst✝ : BraidedCategory E
X₁ X₂ : C
⊢ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ (λ_ X₂).hom = ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫ (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom
n : Type u'
α : Type v
Fintype n
DecidableEq n
CommRing α
A : Matrix n n α
Invertible A
u v : n → α
hM : u = A.mulVec v
⊢ A⁻¹.mulVec u = v
R : Type u_1
CommRing R
a b : R[X]
⊢ -a.wronskian b = b.wronskian a
X : Type u_1
Y : Type u_2
f : X → Y
TopologicalSpace X
TopologicalSpace Y
hf : QuotientMap f
s : Set Y
⊢ IsClosed (f ⁻¹' s) ↔ IsClosed s
x : ℝ
y : ℝ
hy : 0 ≤ y
⊢ ‖x‖₊ * y.toNNReal = ‖x * y‖₊
A : Type u_2
AddMonoid A
H1 : Finset.HasAntidiagonal A
H2 : Finset.HasAntidiagonal A
⊢ Finset.antidiagonal = Finset.antidiagonal
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Field K
T : Type u_4
inst✝¹ : CommRing T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁴ : TopologicalSpace α
ι : Type u_5
π : ι → Type u_6
inst✝³ : (i : ι) → TopologicalSpace (π i)
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace γ
inst✝ : TopologicalSpace δ
f : α → β
a : α
⊢ ContinuousWithinAt f {a} a
α : Type u_1
β : Type u_2
l : Filter α
k : α → β
f : α → β
TopologicalSpace β
LinearOrderedField β
OrderTopology β
hk : Filter.Tendsto k l Filter.atTop
⊢ Asymptotics.SuperpolynomialDecay l k f ↔
  ∀ (z : ℕ), Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l fun a => |k a ^ z * f a|
α : Type u_1
UniformSpace α
⊢ UniformSpace.Completion.map id = id
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
p✝ p₀✝ p₁ p₂ p₀ p : P
⊢ InnerProductGeometry.angle (p₀ -ᵥ p₀) (p -ᵥ p₀) = π / 2
α : Type u
s t : Set α
h : s ⊆ t
⊢ t \ (t \ s) = s
R : Type uR
M : Type uM
ι : Type uι
CommSemiring R
AddCommMonoid M
Module R M
DecidableEq ι
b : Basis ι R M
i j : ι
⊢ (b.toDual (b i)) (b j) = if i = j then 1 else 0
α : Type u_3
β : Type u_4
γ : Type u_5
Lattice α
Lattice β
Lattice γ
f : LatticeHom β γ
g : LatticeHom α β
⊢ { toFun := ⇑(f.comp g), map_sup' := ⋯ } = { toFun := ⇑f, map_sup' := ⋯ }.comp { toFun := ⇑g, map_sup' := ⋯ }
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
inst✝ : Nonempty ι
s : Set β
t : ι → Set β
⊢ ⋂ i, s ∩ (t i)ᶜ = ⋂ i, s \ t i
G : Type u_1
G' : Type u_2
G'' : Type u_3
inst✝⁶ : Group G
inst✝⁵ : Group G'
inst✝⁴ : Group G''
A : Type u_4
inst✝³ : AddGroup A
G₁ : Type u_5
G₂ : Type u_6
G₃ : Type u_7
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G₃
f : G₁ →* G₂
f_inv : G₂ → G₁
hf : Function.RightInverse f_inv ⇑f
g : G₁ →* G₃
hg : f.ker ≤ g.ker
x : G₁
⊢ f_inv (f x) * x⁻¹ ∈ g.ker
C : Type u_1
inst✝² : Category.{?u.26563, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C
Δ Δ' : SimplexCategoryᵒᵖ
θ : Δ ⟶ Δ'
⊢ ((N₁ ⋙ Γ₂).obj X).X.map θ ≫ (fun Δ => (Γ₀.splitting K[X]).desc Δ fun A => PInfty.f (unop A.fst).len ≫ X.map A.e.op) Δ' = (fun Δ => (Γ₀.splitting K[X]).desc Δ fun A => PInfty.f (unop A.fst).len ≫ X.map A.e.op) Δ ≫ ((toKaroubi (SimplicialObject C)).obj X).X.map θ
α : Type u_1
DecidableEq α
s t : Finset α
n : ℕ
⊢ (s ∩ t).sym n = s.sym n ∩ t.sym n
V : Type u_1
P : Type u_2
NormedAddCommGroup V
InnerProductSpace ℝ V
MetricSpace P
NormedAddTorsor V P
p₁ p₂ p₃ : P
h : Wbtw ℝ p₁ p₂ p₃
hp₂p₃ : p₂ ≠ p₃
⊢ EuclideanGeometry.angle p₂ p₃ p₁ = 0
α : Type u_1
a b c d : ℝ≥0∞
r p q : ℝ≥0
⊢ ⋃ n, Ioc a ↑n = Ioi a \ {⊤}
R : Type u_1
inst✝³ : Semiring R
S : Submonoid R
inst✝² : OreSet S
X : Type u_2
inst✝¹ : AddCommMonoid X
inst✝ : Module R X
r₁ : X
s₁ : ↥S
r₂ : R
s₂ : ↥S
r₃ : R
s₃ : ↥S
ra : R
sa : ↥S
ha : ↑sa * ↑s₂ = ra * ↑s₃
rb : R
sb : ↥S
hb : ↑sb * sa • r₂ = rb * ↑s₁
hs₃rasb : ↑sb * ra * ↑s₃ ∈ S
ha' : ↑(sb * sa * s₂) = ↑sb * ra * ↑s₃
rc : R
sc : ↥S
hc : ↑sc * (↑sb * ra) • r₃ = rc * ↑s₁
hc' : ((↑sb * ra) • r₃ /ₒ (sb * sa * s₂)) • (r₁ /ₒ s₁) = rc • r₁ /ₒ (sc * (sb * sa * s₂))
⊢ ((sa • r₂ + ra • r₃) /ₒ (sa * s₂)) • (r₁ /ₒ s₁) = rb • r₁ /ₒ (sb * (sa * s₂)) + rc • r₁ /ₒ (sc * (sb * sa * s₂))
k : Type u
Field k
K : Type v
Field K
IsSepClosed K
f : k →+* K
p : k[X]
h : p.Separable
⊢ Polynomial.Splits f p
α : Type u_1
β : Type u_2
γ : Type u_3
c : γ
⊢ (Equiv.sumAssoc α β γ) (Sum.inr c) = Sum.inr (Sum.inr c)
R : Type uR
A : Type uA
B : Type uB
CommRing R
Ring A
Algebra R A
Ring B
Algebra R B
z : ℤ
⊢ ↑z = 1 ⊗ₜ[R] ↑z
α : Type u_1
inst✝¹ : DecidableEq α
inst✝ : Fintype α
P : Finpartition univ
u : Finset α
ε✝ : ℝ
l✝ : ℕ
ε : ℝ
hε : 0 < ε
l : ℕ
⊢ 0 < log 4
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
hj₃ : ↑C⁻¹ * μ (B j) + μ (W ∩ B j) ≤ μ (B j)
⊢ μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)
μ : ℝ
⊢ ProbabilityTheory.gaussianReal μ 0 = MeasureTheory.Measure.dirac μ
α : Type u_1
Fintype α
DecidableEq α
σ τ : Equiv.Perm α
⊢ (σ * τ * σ⁻¹).support.card = τ.support.card
R : Type u_1
S : Type u_4
Semiring R
Monoid S
M : Type u_6
TopologicalSpace M
AddCommMonoid M
Module R M
N₂ : Type u_9
TopologicalSpace N₂
AddCommMonoid N₂
Module R N₂
N₃ : Type u_10
TopologicalSpace N₃
AddCommMonoid N₃
Module R N₃
DistribMulAction S N₃
SMulCommClass R S N₃
ContinuousConstSMul S N₃
DistribMulAction S N₂
ContinuousConstSMul S N₂
SMulCommClass R S N₂
LinearMap.CompatibleSMul N₂ N₃ S R
hₗ : N₂ →L[R] N₃
c : S
fₗ : M →L[R] N₂
⊢ hₗ.comp (c • fₗ) = c • hₗ.comp fₗ
R : Type u_1
inst✝³ : CommRing R
M : Submonoid R
S✝ : Type u_2
inst✝² : CommRing S✝
inst✝¹ : Algebra R S✝
inst✝ : IsLocalization M S✝
S : Finset R
hS : Ideal.span ↑S = ⊤
I : Ideal R
⊢ I ≤ ⨅ f ∈ S, Ideal.comap (algebraMap R (Localization.Away f)) (Ideal.map (algebraMap R (Localization.Away f)) I)
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₂ : R₂.Exact
hR₁' : Epi (R₁.map' 1 2 ⋯ ⋯)
h₀ : Epi (app' φ 0 ⋯)
h₁ : Epi (app' φ 2 ⋯)
ψ : mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0 ⟶ mk₃ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) 0 := homMk₃ (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) (𝟙 ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).obj' 3 ⋯)) ⋯ ⋯ ⋯
⊢ (ShortComplex.mk ((mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯) ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map' 0 1 ⋯ ⋯) ((mk₂ ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 1 2 ⋯ ⋯) ((mk₃ (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) 0).map' 2 3 ⋯ ⋯)).map' 1 2 ⋯ ⋯) ⋯).Exact
z : ℍ
hz : z ∈ ModularGroup.fdo
n : ℤ
⊢ 1 < Complex.normSq ↑(ModularGroup.T ^ n • z)
C : Type u₁
CategoryTheory.Category.{v₁, u₁} C
X Y : Cᴹᵒᵖ
f : X ⟶ Y
⊢ f.unmop.mop = f
α : Type u_2
β : Type u_3
TopologicalSpace α
Preorder α
TopologicalSpace β
Preorder β
f : EsakiaHom α β
⊢ f.toFun = ⇑f
k : Type u_1
DivisionRing k
ι : Type u_4
s : Finset ι
Fintype ι
⊢ ∑ i : ι, Finset.centroidWeightsIndicator k s i = ∑ i ∈ s, Finset.centroidWeights k s i
α : Type u_1
β : Type u_2
l : List α
d : α
⊢ (fun o => o.getD d) '' range l.get? = insert d {x | x ∈ l}
α : Type u_1
β : Type u_2
F : Type u_4
PseudoEMetricSpace α
PseudoEMetricSpace β
FunLike F α β
DilationClass F α β
f : F
γ : Type u_6
TopologicalSpace γ
g : γ → α
s : Set γ
⊢ ContinuousOn (⇑f ∘ g) s ↔ ContinuousOn g s
𝕜 : Type u_1
E : Type u_2
RCLike 𝕜
NormedAddCommGroup E
InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
⊢ U ⟂ ⊥
P : Type u_1
Preorder P
F : Order.PFilter P
⊢ (↑F).Nonempty
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
x : 𝕜
m : ℕ
hf : AnalyticAt 𝕜 (fun z => (z - x) ^ m • f z) x
n : ℕ
hg : AnalyticAt 𝕜 (fun z => (z - x) ^ n • g z) x
this : (fun z => (z - x) ^ max m n • (f + g) z) = fun z => (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z
⊢ AnalyticAt 𝕜 (fun z => (z - x) ^ (max m n - m) • (z - x) ^ m • f z + (z - x) ^ (max m n - n) • (z - x) ^ n • g z) x
⊢ ∀ {X Y : Pointed} (f : X ⟶ Y) (a : Option X.X), (pointedToTwoPFst.map f).toFun a = Option.map f.toFun a
ι : Type u_1
α : Type u_2
β : Type u_3
f g : Perm α
s✝ t : Set α
a b x y : α
s : Finset α
hf : f.IsCycleOn ↑s
ha : a ∈ s
m n : ℤ
⊢ (f ^ m) a = (f ^ n) a ↔ (f ^ (n - m)) a = a
s : Finset ℕ
m n✝ p n d : ℕ
h : n.minSqFac = some d
this : n.MinSqFacProp n.minSqFac
⊢ Prime d
α : Type u_1
DecidableEq α
s t u : Multiset α
⊢ s ∪ t ≤ u ↔ s ≤ u ∧ t ≤ u
C : Type u
CategoryTheory.Category.{v, u} C
⊢ ∀ {X Y Z : CategoryTheory.Square C} (f : X.Hom Y) (g : Y.Hom Z),
  (CategoryTheory.CategoryStruct.comp f g).τ₃ = CategoryTheory.CategoryStruct.comp f.τ₃ g.τ₃
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕝 : Type u_3
𝕝₂ : Type u_4
E : Type u_5
F : Type u_6
G : Type u_7
ι : Type u_8
ι' : Type u_9
inst✝¹¹ : Nonempty ι
inst✝¹⁰ : Nonempty ι'
inst✝⁹ : NormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
inst✝² : Preorder ι
inst✝¹ : LocallyFiniteOrderBot ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
⊢ Seminorm.IsBounded (fun i => (Finset.Iic i).sup p) p LinearMap.id
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
l : Filter X
hs : IsCompact s
H : ∀ x ∈ s, Disjoint (𝓝 x) l
U : X → Set X
hxU : ∀ x ∈ s, x ∈ U x ∧ IsOpen (U x)
hUl : ∀ x ∈ s, (U x)ᶜ ∈ l
⊢ Disjoint (𝓝ˢ s) l
α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
RCLike 𝕜
NormedAddCommGroup E
NormedSpace 𝕜 E
NormedAddCommGroup F
CompleteSpace F
u : α → ℝ
NormedSpace 𝕜 F
f : α → E → F
x₀ : E
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
hf0 : Summable fun n => f n x₀
⊢ (fderiv 𝕜 fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), fderiv 𝕜 (f n) x
α : Type u
DecidableEq α
M : Type u_1
AddCommMonoid M
n : α →₀ ℕ
f : (α →₀ ℕ) → (α →₀ ℕ) → M
⊢ ∑ p ∈ Finset.antidiagonal n, f p.1 p.2 = ∑ p ∈ Finset.antidiagonal n, f p.2 p.1
α : Type u_1
Div α
a b : α
⊢ OrderDual.toDual (a / b) = OrderDual.toDual a / OrderDual.toDual b
M : Type u_1
Monoid M
N : Type u_2
Monoid N
φ : M →* N
A : Type u_4
AddMonoid A
DistribMulAction M A
B : Type u_5
AddMonoid B
DistribMulAction N B
f g : A →ₑ+[φ] B
h : ↑f = ↑g
⊢ f = g
ι : Type u_1
α : Type u_2
LinearOrderedRing α
s : Finset ι
σ : Equiv.Perm ι
f g : ι → α
hfg : MonovaryOn f g ↑s
hσ : {x | σ x ≠ x} ⊆ ↑s
⊢ ∑ i ∈ s, f i • g (σ i) < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn f (g ∘ ⇑σ) ↑s
α : Type u_1
F : Type u_5
MeasurableSpace α
NormedAddCommGroup F
p : ℝ
f : MeasureTheory.SimpleFunc α F
μ : MeasureTheory.Measure α
⊢ MeasureTheory.eLpNorm' (↑f) p μ = (∑ y ∈ f.range, ↑‖y‖₊ ^ p * μ (↑f ⁻¹' {y})) ^ (1 / p)
C : Type u
CategoryTheory.Category.{v, u} C
A : Type u'
CategoryTheory.Category.{v', u'} A
CategoryTheory.ConcreteCategory A
F G : Cᵒᵖ ⥤ A
f : F ⟶ G
U : Cᵒᵖ
s : (CategoryTheory.forget A).obj (G.obj U)
V : C
g : V ⟶ Opposite.unop U
hg : (CategoryTheory.Presheaf.imageSieve f s).arrows g
⊢ (f.app (Opposite.op V)) (CategoryTheory.Presheaf.localPreimage f s g hg) = (G.map g.op) s
α : Type u_1
r : α → α → Prop
o : Ordinal.{u}
f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}
⊢ (o.blsub f).cof ≤ Cardinal.lift.{v, u} #(Quotient.out o).α
x : ℝ
⊢ Real.toEReal ⁻¹' Set.Ico ↑x ⊤ = Set.Ici x
a : ℕ
a1 : 1 < a
i n : ℕ
npos : 0 < n
j : ℕ
ij : i < j + 1
j2n : j + 1 ≤ 2 * n
jnn : j + 1 ≠ n
ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)
k : ℕ
kn : k > n
k2n : k ≤ 2 * n
k2nl : 2 * n - k < n := lt_of_add_lt_add_right (let_fun this := Eq.mpr (id (congrArg (fun _a => _a < n + k) (tsub_add_cancel_of_le k2n))) (Eq.mpr (id (congrArg (fun _a => _a < n + k) (two_mul n))) (add_lt_add_left kn n)); this)
xle : xn a1 (2 * n - k) ≤ xn a1 n
⊢ xn a1 k + xn a1 (2 * n - k) ≡ xn a1 n - xn a1 (2 * n - k) + xn a1 (2 * n - k) [MOD xn a1 n]
ι : Type u_1
ι' : Type u_2
α : Type u_3
β : Type u_4
γ : Type u_5
inst✝ : OrderedRing α
l : Filter β
f g : β → α
hf : Tendsto f l atTop
hg : Tendsto g l atBot
this : Tendsto (fun x => f x * (Neg.neg ∘ g) x) l atTop
⊢ Tendsto (fun x => f x * g x) l atBot
K : Type u_1
Field K
NumberField K
IsCyclotomicExtension {3} ℚ K
ζ : K
hζ : IsPrimitiveRoot ζ ↑3
S' : FermatLastTheoremForThreeGen.Solution' hζ
DecidableRel fun a b => a ∣ b
⊢ ∃ S₁, S₁.multiplicity = S'.multiplicity
K : Type u_1
Field K
NumberField K
α : K
⊢ NumberField.house α = ↑(Finset.univ.sup' ⋯ fun φ => ‖φ α‖₊)
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
x : Set ℝ
⊢ x ∈ ⋃ a, {Ioi ↑a} ↔ x ∈ Ioi '' (Rat.cast '' univ)
R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝¹ p✝ q✝ : R[X]
ι : Type u_1
p q : R[X]
⊢ p.degree + q.degree ≤ ↑p.natDegree + ↑q.natDegree
C : Type u_1
CategoryTheory.Category.{u_2, u_1} C
CategoryTheory.Preadditive C
CategoryTheory.Limits.HasKernels C
X Y : C
CategoryTheory.Simple X
CategoryTheory.Simple Y
f : X ⟶ Y
w : f ≠ 0
⊢ CategoryTheory.IsIso f
α : Type u
s : Set α
a : α
⊢ Cardinal.mk ↑(insert a s) ≤ Cardinal.mk ↑s + 1
G : Type u_1
Group G
H : Subgroup G
⊢ ⊤.IsComplement' H ↔ H = ⊥
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s t u : Set α
x y : α
Φ : α → β
r : ℝ
hr : 0 ≤ r
H1 : ∀ x ∈ s, infDist x t ≤ r
H2 : ∀ x ∈ t, infDist x s ≤ r
h1 : ¬hausdorffEdist s t = ⊤
hs : s.Nonempty
ht : t.Nonempty
this : hausdorffEdist s t ≤ ENNReal.ofReal r
⊢ hausdorffDist s t ≤ r
R : Type u_3
A : Type u_4
AddZeroClass R
Add A
m₁ m₂ : A
⊢ ↑(m₁ + m₂) = ↑m₁ + ↑m₂
α : Type u_9
self : CompleteSemilatticeSup α
s : Set α
a : α
⊢ a ∈ s → a ≤ sSup s
α : Type u
β : Type v
TopologicalSpace α
PseudoMetricSpace β
AddMonoid β
BoundedAdd β
ContinuousAdd β
⊢ ∀ (a : BoundedContinuousFunction α β) (a_1 : α), BoundedContinuousFunction.coeFnAddHom a a_1 = a a_1
α : Type u
x y : α
BooleanAlgebra α
⊢ Disjoint x yᶜ ↔ x ≤ y
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C : Set (I → Bool)
s : Finset I
x y : ↑(π C fun x => x ∈ s)
h : y ≠ x
⊢ (List.map (⇑(LocallyConstant.evalMonoidHom y)) (factors C s x)).prod = 0
R : Type u
CommRing R
A : Type u_2
B : Type u_3
CommRing A
CommRing B
Algebra R A
Algebra R B
Algebra A B
IsScalarTower R A B
h : Function.Surjective ⇑(algebraMap A B)
⊢ Function.Surjective ⇑(KaehlerDifferential.mapBaseChange R A B)
α : Type u_1
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
f : α → ℝ≥0∞
f_meas : Measurable f
g : α → ℝ≥0∞
⊢ ∫⁻ (a : α), g a ∂μ.withDensity f ≤ ∫⁻ (a : α), (f * g) a ∂μ
K : Type u_1
R : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Fintype K
p✝ : ℕ
inst✝³ : Fact (Nat.Prime p✝)
inst✝² : Algebra (ZMod p✝) K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
n : ℕ
hcard : q = p ^ n
x : K
⊢ (frobenius K p ^ n) x = x ^ p ^ n
κ : Type w
A : Type u
B : Type v
CommRing A
CommRing B
Algebra A B
Fintype κ
DecidableEq κ
b : Basis κ A B
⊢ Algebra.traceMatrix A ⇑b = (BilinForm.toMatrix b) (Algebra.traceForm A B)
R : Type u
S : Type v
T : Type w
a✝ b : R
n : ℕ
inst✝ : CommSemiring R
a p P : R[X]
h : ∀ (r : R), r • P = 0 → r = 0
Q : R[X]
hQ : P * Q = 0
⊢ ∀ (i : ℕ), P.coeff i • Q = 0
C : Type u
CategoryTheory.Category.{v, u} C
D : Type u'
CategoryTheory.Category.{v', u'} D
W : CategoryTheory.MorphismProperty C
L : CategoryTheory.Functor Cᵒᵖ Dᵒᵖ
h : W.op.IsInvertedBy L
⊢ W.IsInvertedBy L.unop
ι : Type u_1
M : Type u_3
Semiring M
⊢ AddMonoidAlgebra.toDirectSum 0 = 0
x y : ℂ
s1 : cos ((x + y) / 2 + (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)
s2 : cos ((x + y) / 2 - (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)
⊢ cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2)
C : Type u
inst✝² : Category.{v, u} C
α : Type w
inst✝¹ : HasFiniteCoproducts C
inst✝ : HasFilteredColimitsOfSize.{w, w, v, u} C
F : Discrete α ⥤ C
s : Cocone F
m : { pt := colimit (liftToFinset F), ι := Discrete.natTrans fun j => Sigma.ι (fun x => F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinset F) {j} }.pt ⟶ s.pt
h : ∀ (j : Discrete α), { pt := colimit (liftToFinset F), ι := Discrete.natTrans fun j => Sigma.ι (fun x => F.obj ↑x) ⟨j, ⋯⟩ ≫ colimit.ι (liftToFinset F) {j} }.ι.app j ≫ m = s.ι.app j
t : Finset (Discrete α)
⊢ colimit.ι { obj := fun s => ∐ fun x => F.obj ↑x, map := fun {x Y} h => Sigma.desc fun y => Sigma.ι (fun x => F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ } t ≫ m = colimit.ι { obj := fun s => ∐ fun x => F.obj ↑x, map := fun {x Y} h => Sigma.desc fun y => Sigma.ι (fun x => F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ } t ≫ colimit.desc { obj := fun s => ∐ fun x => F.obj ↑x, map := fun {x Y} h => Sigma.desc fun y => Sigma.ι (fun x => F.obj ↑x) ⟨↑y, ⋯⟩, map_id := ⋯, map_comp := ⋯ } { pt := s.pt, ι := { app := fun t => Sigma.desc fun x => s.ι.app ↑x, naturality := ⋯ } }
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
p : FormalMultilinearSeries 𝕜 E F
r : ℝ≥0
this✝ : (∃ a ∈ Ioo 0 1, (fun n => ‖p n‖ * ↑r ^ n) =o[atTop] fun x => a ^ x) ↔ ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n
t : ℝ≥0
C : ℝ
hC : ∀ (n : ℕ), ‖p n‖ * ↑t ^ n ≤ C
rt : ↑r / ↑t < 1
this : 0 < ↑t
⊢ ∃ a < 1, ∃ C, (0 < C ∨ 0 < 1) ∧ ∀ (n : ℕ), |‖p n‖ * ↑r ^ n| ≤ C * a ^ n
α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : OrderedAddCommMonoid M
s : SignedMeasure α
i j : Set α
hi : ↑s i < 0
hi₁ : MeasurableSet i
h : ¬restrict s i ≤ restrict 0 i
hn : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
A : Set α := i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
hA : A = i \ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l
bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)
hn' : ∀ (n : ℕ), ¬restrict s (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)
h₂ : ↑s A ≤ ↑s i
h₃' : Summable fun n => 1 / (↑(bdd n) + 1)
h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop
h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop
A_meas : MeasurableSet A
⊢ restrict s A ≤ restrict 0 A
α : Type u_3
m : MeasurableSpace α
G₀ : Type u_4
GroupWithZero G₀
MeasurableSpace G₀
MeasurableInv G₀
f : α → G₀
⊢ (Measurable fun x => (f x)⁻¹) ↔ Measurable f
𝕜 : Type u_1
RCLike 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ Metric.closedBall 0 r, ‖f z‖ ≤ c
z : E
⊢ ‖f z‖ ≤ c / r * ‖z‖
R : Type u_1
S : Type u_2
F : Type u_3
K : Type u_4
P Q : Cubic R
a b c d a' b' c' d' : R
inst✝ : Semiring R
n : ℕ
hn : 3 < n
⊢ ¬n = 3
α : Type u
β : Type v
LinearOrder α
LinearOrder β
f : α → β
a b : α
hf : Monotone f
⊢ f (max a b) = max (f a) (f b)
R : Type u_3
CommRing R
x : R
⊢ ↑(-x) = -↑x
R : Type u
Ring R
⊢ ⊤.jacobson = ⊤
α : Type u_1
β : Type u_2
xs : List α
a b : α
h : s(a, b) ∈ xs.sym2
⊢ b ∈ xs
a b : ℝ
hab : a < b
l : Filter ℝ
f f' g g' : ℝ → ℝ
hdf : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ f x
hg' : ∀ᶠ (x : ℝ) in atTop, deriv g x ≠ 0
hftop : Tendsto f atTop (𝓝 0)
hgtop : Tendsto g atTop (𝓝 0)
hdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l
hdg : ∀ᶠ (x : ℝ) in atTop, DifferentiableAt ℝ g x
hdf' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (deriv f x) x
⊢ Tendsto (fun x => f x / g x) atTop l
M : Type u_5
N : Type u_6
P : Type u_7
MulOneClass M
MulOneClass N
CommMonoid P
f : M × N →* P
⊢ (f.comp (MonoidHom.inl M N)).coprod (f.comp (MonoidHom.inr M N)) = f
k : Type u₁
G : Type u₂
CommSemiring k
AddMonoid G
A : Type u₃
Semiring A
Algebra k A
F : AddMonoidAlgebra k G →ₐ[k] A
⊢ F = (AddMonoidAlgebra.lift k G A) ((↑F).comp (AddMonoidAlgebra.of k G))
R : Type u
S : Type v
T : Type w
a b : R
n : ℕ
inst✝¹ : CommRing R
inst✝ : IsDomain R
p q : R[X]
hn : 0 < n
⊢ 1 ∈ nthRootsFinset n R
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p₁ p₂ : P
v : V
⊢ v ∈ vectorSpan k {p₁, p₂} ↔ ∃ r, r • (p₁ -ᵥ p₂) = v
α : Sort u
β : Sort v
e : α ≃ β
f : α → α
⊢ ∀ (a : β), e.conj f a = e (f (e.symm a))
α : Type u_1
β : Type u_2
l₁ : List α
l₂ : List β
⊢ l₁.length ≤ l₂.length → List.map Prod.fst (l₁.zip l₂) = l₁
α : Type u_2
r : α → α → Prop
s t : Set α
⊢ (s ∪ t).PartiallyWellOrderedOn r ↔ s.PartiallyWellOrderedOn r ∧ t.PartiallyWellOrderedOn r
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝ : UniformSpace β
u₁ u₂ : UniformSpace γ
⊢ ⨅ i, uniformSpace α γ = ⨅ b, bif b then uniformSpace α γ else uniformSpace α γ
α : Type u_1
inst✝⁵ : MetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
s : Set α
t : Set α := toMeasurable μ s
A : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 (t.indicator 1 x))
B : ∀ᵐ (x : α) ∂μ.restrict s, t.indicator 1 x = 1
x : α
hx : Tendsto (fun a => μ (t ∩ a) / μ a) (v.filterAt x) (𝓝 1)
h'x : t.indicator 1 x = 1
a✝ : Set α
ha : MeasurableSet a✝
⊢ μ (t ∩ a✝) = μ (s ∩ a✝)
R : Type u_1
M : Type u_2
CommRing R
AddCommGroup M
Module R M
p : PrimeSpectrum R
⊢ p ∈ Module.support R M ↔ ∃ m, (Submodule.span R {m}).annihilator ≤ p.asIdeal