ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : I.toRightDerivedZero' F = (injectiveResolution X).toRightDerivedZero' F ≫ HomologicalComplex.cyclesMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
\u22a2 (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) = I.toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : I.toRightDerivedZero' F = (injectiveResolution X).toRightDerivedZero' F ≫ HomologicalComplex.cyclesMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ : (I.isoRightDerivedToHomotopyCategoryObj F).hom = (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))
\u22a2 (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) = I.toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex).homologyπ 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map (I.isoRightDerivedToHomotopyCategoryObj F).inv
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : I.toRightDerivedZero' F = (injectiveResolution X).toRightDerivedZero' F ≫ HomologicalComplex.cyclesMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ : (I.isoRightDerivedToHomotopyCategoryObj F).hom = (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))
\u22a2 (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex) ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) 0).map ((F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) = (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
-- Mathlib/CategoryTheory/Abelian/RightDerived.lean
C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u_1
inst✝⁴ : Category.{u_2, u_1} D
inst✝³ : Abelian C
inst✝² : HasInjectiveResolutions C
inst✝¹ : Abelian D
X : C
I : InjectiveResolution X
F : C ⥤ D
inst✝ : F.Additive
h₁ : I.toRightDerivedZero' F = (injectiveResolution X).toRightDerivedZero' F ≫ HomologicalComplex.cyclesMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0
h₂ : (I.isoRightDerivedToHomotopyCategoryObj F).hom = (F.mapHomologicalComplex (ComplexShape.up ℕ) ⋙ HomotopyCategory.quotient D (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))
\u22a2 (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ (HomologicalComplex.homologyFunctor D (ComplexShape.up ℕ) 0).map ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex) = (injectiveResolution X).toRightDerivedZero' F ≫ ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj (injectiveResolution X).cocomplex).homologyπ 0 ≫ HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up ℕ)).map (desc (𝟙 X) I (injectiveResolution X))) 0 ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) 0).inv.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj I.cocomplex)
-- Mathlib/Order/Filter/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort u_5
s : Set α
t : Set β
f : Filter α
g : Filter β
f₁ f₂ : Filter α
g₁ g₂ : Filter β
\u22a2 f₁ ×ˢ g₁ ⊓ f₂ ×ˢ g₂ = (f₁ ⊓ f₂) ×ˢ (g₁ ⊓ g₂)
-- Mathlib/Analysis/CstarAlgebra/Multiplier.lean
𝕜 : Type u_1
A : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NonUnitalNormedRing A
inst✝² : NormedSpace 𝕜 A
inst✝¹ : SMulCommClass 𝕜 A A
inst✝ : IsScalarTower 𝕜 A A
a : 𝓜(𝕜, A)
\u22a2 a.toProd ∈ {lr | ∀ (x y : A), lr.2 x * y = x * lr.1 y}
-- Mathlib/RingTheory/Polynomial/Chebyshev.lean
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℤ
\u22a2 T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
-- Mathlib/RingTheory/Polynomial/Chebyshev.lean
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℤ
h₁ : U R (n + 1) = X * U R n + T R (n + 1)
\u22a2 T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
-- Mathlib/RingTheory/Polynomial/Chebyshev.lean
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℤ
h₁ : U R (n + 1) = X * U R n + T R (n + 1)
h₂ : U R (n + 1 + 1) = X * U R (n + 1) + T R (n + 1 + 1)
\u22a2 T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
-- Mathlib/RingTheory/Polynomial/Chebyshev.lean
R : Type u_1
S : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing S
n : ℤ
h₁ : U R (n + 1) = X * U R n + T R (n + 1)
h₂ : U R (n + 1 + 1) = X * U R (n + 1) + T R (n + 1 + 1)
h₃ : U R (n + 2) = 2 * X * U R (n + 1) - U R n
\u22a2 T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n
-- Mathlib/Topology/Algebra/UniformGroup.lean
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
\u22a2 𝓤 α = comap (fun x => x.2 / x.1) (𝓝 1)
-- Mathlib/Topology/Algebra/UniformGroup.lean
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
\u22a2 𝓤 α = comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α)
-- Mathlib/Topology/Algebra/UniformGroup.lean
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
\u22a2 map (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α) ≤ 𝓤 α
-- Mathlib/Topology/Algebra/UniformGroup.lean
α : Type u_1
β : Type u_2
inst✝² : UniformSpace α
inst✝¹ : Group α
inst✝ : UniformGroup α
\u22a2 comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α) ≤ 𝓤 α
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
hr : r < 0
\u22a2 o.oangle (r • x) y = o.oangle (-x) y
-- Mathlib/Algebra/Homology/HomotopyCofiber.lean
C : Type u_1
inst✝⁶ : Category.{u_5, u_1} C
inst✝⁵ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝⁴ : HasHomotopyCofiber φ
inst✝³ : DecidableRel c.Rel
inst✝² : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝¹ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
D : Type u_3
inst✝ : Category.{u_4, u_3} D
H : HomologicalComplex C c ⥤ D
hH : (homotopyEquivalences C c).IsInvertedBy H
this : IsIso (H.map (π K))
\u22a2 H.map (ι₀ K) = H.map (ι₁ K)
-- Mathlib/LinearAlgebra/Dual.lean
K : Type uK
inst✝⁴ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
W W' : Subspace K V₁
h : IsCompl W W'
\u22a2 IsCompl (dualAnnihilator W) (dualAnnihilator W')
-- Mathlib/LinearAlgebra/Dual.lean
K : Type uK
inst✝⁴ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
W W' : Subspace K V₁
h : W ⊓ W' = ⊥ ∧ W ⊔ W' = ⊤
\u22a2 dualAnnihilator W ⊓ dualAnnihilator W' = ⊥ ∧ dualAnnihilator W ⊔ dualAnnihilator W' = ⊤
-- Mathlib/LinearAlgebra/Dual.lean
K : Type uK
inst✝⁴ : Field K
V₁ : Type uV₁
V₂ : Type uV₂
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
W W' : Subspace K V₁
h : W ⊓ W' = ⊥ ∧ W ⊔ W' = ⊤
\u22a2 ⊥ = ⊥ ∧ ⊤ = ⊤
-- Mathlib/NumberTheory/Modular.lean
g : SL(2, ℤ)
z : ℍ
\u22a2 (T⁻¹ • z).re = z.re - 1
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
x y z : S
\u22a2 (z𝖣x + y) = (z𝖣x) + z𝖣y
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
x y z : S
\u22a2 single x z + single y z - single (x + y) z ∈ kerTotal R S
-- Mathlib/RingTheory/Kaehler/Basic.lean
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
M : Type u_1
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : Module S M
inst✝ : IsScalarTower R S M
x y z : S
\u22a2 z • (single x 1 + single y 1 - single (x + y) 1) ∈ kerTotal R S
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l : Filter α
f : α → β
g : β → γ
a : Filter α
b : Filter β
c : Filter γ
hfg : Tendsto (g ∘ f) a c
hg : comap g c ≤ b
\u22a2 Tendsto f a b
-- Mathlib/Order/Filter/Basic.lean
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
l : Filter α
f : α → β
g : β → γ
a : Filter α
b : Filter β
c : Filter γ
hfg : a ≤ comap (g ∘ f) c
hg : comap g c ≤ b
\u22a2 a ≤ comap f b
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
\u22a2 ↑(LinearEquiv.det (o.rotation θ).toLinearEquiv) = ↑1
-- Mathlib/Data/Set/Lattice.lean
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
f : α → β
s : (i : ι) → κ i → Set β
\u22a2 f ⁻¹' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f ⁻¹' s i j
-- Mathlib/Order/Radical.lean
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
\u22a2 f (⨅ a ∈ {H | IsCoatom H}, a) = ⨅ a ∈ {H | IsCoatom H}, a
-- Mathlib/Order/Radical.lean
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
\u22a2 ⨅ i ∈ {H | IsCoatom H}, f i = ⨅ a ∈ {H | IsCoatom H}, a
-- Mathlib/Order/Radical.lean
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
\u22a2 α ≃ β
-- Mathlib/Order/Radical.lean
α : Type u_1
inst✝¹ : CompleteLattice α
β : Type u_2
inst✝ : CompleteLattice β
f : α ≃o β
\u22a2 ∀ (x : α), ⨅ (_ : ?e x ∈ {H | IsCoatom H}), ?e x = ⨅ (_ : x ∈ {H | IsCoatom H}), f x
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SFinite ν
inst✝² : SFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
\u22a2 QuasiMeasurePreserving (fun p => p.1 / p.2) (μ.prod ν) μ
-- Mathlib/MeasureTheory/Group/Prod.lean
G : Type u_1
inst✝⁶ : MeasurableSpace G
inst✝⁵ : Group G
inst✝⁴ : MeasurableMul₂ G
μ ν : Measure G
inst✝³ : SFinite ν
inst✝² : SFinite μ
s : Set G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsMulRightInvariant
y : G
\u22a2 QuasiMeasurePreserving (fun x => (x, y).1 / (x, y).2) μ μ
-- Mathlib/LinearAlgebra/Dual.lean
R : Type uR
M : Type uM
K : Type uK
V : Type uV
ι : Type uι
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : DecidableEq ι
b : Basis ι R M
inst✝ : _root_.Finite ι
l : Dual R M
i : ι
\u22a2 (b.dualBasis.repr l) i = l (b i)
-- Mathlib/SetTheory/Game/Impartial.lean
G : PGame
inst✝ : G.Impartial
H : PGame
\u22a2 G ≈ H ↔ G + H ≈ 0
-- Mathlib/SetTheory/Game/Impartial.lean
G : PGame
inst✝ : G.Impartial
H : PGame
\u22a2 0 = ⟦G + H⟧ ↔ ⟦G + H⟧ = ⟦0⟧
-- Mathlib/Data/Matrix/Basic.lean
l : Type u_1
m : Type u_2
n : Type u_3
o : Type u_4
m' : o → Type u_5
n' : o → Type u_6
R : Type u_7
S : Type u_8
α : Type v
β : Type w
γ : Type u_9
inst✝² : Fintype m
inst✝¹ : Fintype n
inst✝ : NonUnitalSemiring α
u : m → α
w : n → α
v : Matrix m n α
\u22a2 (fun j => u ⬝ᵥ fun i => v i j) ⬝ᵥ w = u ⬝ᵥ fun i => v i ⬝ᵥ w
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : LinearOrder α
inst✝² : OrderTopology α
inst✝¹ : DenselyOrdered α
a✝ b✝ : α
s : Set α
inst✝ : NoMinOrder α
a b x : α
\u22a2 Ico a b ∈ 𝓝 x ↔ x ∈ Ioo a b
-- Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
R : Type u_1
k : Type u_2
V1 : Type u_3
P1 : Type u_4
V2 : Type u_5
P2 : Type u_6
V3 : Type u_7
P3 : Type u_8
inst✝⁵ : CommRing k
inst✝⁴ : AddCommGroup V1
inst✝³ : AffineSpace V1 P1
inst✝² : AddCommGroup V2
inst✝¹ : Module k V1
inst✝ : Module k V2
c : P1
r₁ r₂ : k
p : P1
\u22a2 (homothety c (r₁ * r₂)) p = (homothety c r₁) ((homothety c r₂) p)
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : o.oangle x y = ↑(π / 2)
\u22a2 ‖y‖ / (o.oangle (x + y) y).cos = ‖x + y‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
\u22a2 ‖y‖ / ((-o).oangle y (x + y)).cos = ‖x + y‖
-- Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
h : (-o).oangle y x = ↑(π / 2)
\u22a2 ‖y‖ / ((-o).oangle y (y + x)).cos = ‖y + x‖
-- Mathlib/Analysis/BoundedVariation.lean
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : b ≤ a
\u22a2 variationOnFromTo f s a b ≤ 0
-- Mathlib/Analysis/BoundedVariation.lean
α : Type u_1
inst✝¹ : LinearOrder α
E : Type u_2
inst✝ : PseudoEMetricSpace E
f : α → E
s : Set α
a b : α
h : b ≤ a
\u22a2 -variationOnFromTo f s b a ≤ 0
-- Mathlib/RingTheory/MvPowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n✝ : σ →₀ ℕ
φ ψ : MvPowerSeries σ R
n : σ →₀ ℕ
\u22a2 (coeff R n) (0 * φ) = (coeff R n) 0
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
\u22a2 I = ⊥ ∨ I = ⊤
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
\u22a2 ¬I = ⊤ → I = ⊥
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
\u22a2 I ≠ ⊤ → I = ⊥
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
\u22a2 1 ∉ I → I = ⊥
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
\u22a2 I = ⊥
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
\u22a2 I ≤ ⊥
-- Mathlib/RingTheory/Ideal/Basic.lean
α : Type u
β : Type v
K : Type u
inst✝ : DivisionSemiring K
I : Ideal K
h1 : 1 ∉ I
r : K
hr : r ∈ I
H : ¬r = 0
\u22a2 r ∈ ⊥
-- Mathlib/Algebra/Quaternion.lean
S : Type u_1
T : Type u_2
R : Type u_3
inst✝ : CommRing R
c₁ c₂ r x y z : R
a b c : ℍ[R,c₁,c₂]
\u22a2 star a + a = 2 * ↑a.re
-- Mathlib/CategoryTheory/Shift/SingleFunctors.lean
C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝⁷ : Category.{u_7, u_1} C
inst✝⁶ : Category.{u_6, u_2} D
inst✝⁵ : Category.{?u.58445, u_3} E
inst✝⁴ : Category.{?u.58449, u_4} E'
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift D A
inst✝¹ : HasShift E A
inst✝ : HasShift E' A
F G H : SingleFunctors C D A
e : F ≅ G
n : A
X : C
\u22a2 (e.inv.hom n).app X ≫ (e.hom.hom n).app X = 𝟙 ((G.functor n).obj X)
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
\u22a2 Nat.card ↥↑P = p ^ (Nat.card G).factorization p
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Nat.card ↥↑P = p ^ n
\u22a2 Nat.card ↥↑P = p ^ (Nat.card G).factorization p
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Nat.card ↥↑P = p ^ n
\u22a2 Nat.card ↥↑P ∣ p ^ (Nat.card G).factorization p
-- Mathlib/GroupTheory/Sylow.lean
G : Type u
α : Type v
β : Type w
inst✝¹ : Group G
inst✝ : Finite G
p : ℕ
hp : Fact (Nat.Prime p)
P : Sylow p G
n : ℕ
heq : Nat.card ↥↑P = p ^ n
\u22a2 Nat.card ↥↑P ∣ Nat.card G
-- Mathlib/Analysis/LocallyConvex/Bounded.lean
𝕜✝ : Type u_1
𝕜' : Type u_2
E✝ : Type u_3
E' : Type u_4
F : Type u_5
ι✝ : Type u_6
𝕜 : Type u_7
ι : Type u_8
E : ι → Type u_9
inst✝³ : NormedDivisionRing 𝕜
inst✝² : (i : ι) → AddCommGroup (E i)
inst✝¹ : (i : ι) → Module 𝕜 (E i)
inst✝ : (i : ι) → TopologicalSpace (E i)
S : Set ((i : ι) → E i)
\u22a2 IsVonNBounded 𝕜 S ↔ ∀ (i : ι), IsVonNBounded 𝕜 (eval i '' S)
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
\u22a2 -llr μ ν =ᶠ[ae μ] llr ν μ
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
x : α
hx : (μ.rnDeriv ν)⁻¹ x = ν.rnDeriv μ x
\u22a2 (-llr μ ν) x = llr ν μ x
-- Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
f : α → ℝ
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hμν : μ ≪ ν
x : α
hx : (μ.rnDeriv ν)⁻¹ x = ν.rnDeriv μ x
\u22a2 log (μ.rnDeriv ν x)⁻¹.toReal = log (ν.rnDeriv μ x).toReal
-- Mathlib/Analysis/Calculus/LogDeriv.lean
𝕜 : Type u_1
𝕜' : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
x : 𝕜
n : ℤ
\u22a2 logDeriv (fun x => x ^ n) x = ↑n / x
-- Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
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 : SFinite ν
inst✝ : SFinite μ
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
\u22a2 Integrable f (μ.prod ν) ↔ (∀ᵐ (y : β) ∂ν, Integrable (fun x => f (x, y)) μ) ∧ Integrable (fun y => ∫ (x : α), ‖f (x, y)‖ ∂μ) ν
-- Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
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 : SFinite ν
inst✝ : SFinite μ
f : α × β → E
h1f : AEStronglyMeasurable f (μ.prod ν)
\u22a2 Integrable f (μ.prod ν) ↔ Integrable (fun z => f z.swap) (ν.prod μ)
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
\u22a2 (Measure.pi μ) (eval i ⁻¹' s) = 0
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
t : Set (α i)
hst : s ⊆ t
left✝ : MeasurableSet t
hμt : (μ i) t = 0
\u22a2 (Measure.pi μ) (eval i ⁻¹' s) = 0
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
t : Set (α i)
hst : s ⊆ t
left✝ : MeasurableSet t
hμt : (μ i) t = 0
\u22a2 (Measure.pi μ) (eval i ⁻¹' t) = 0
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
t : Set (α i)
hst : s ⊆ t
left✝ : MeasurableSet t
hμt : (μ i) t = 0
\u22a2 ∏ i_1 : ι, (μ i_1) (update (fun j => univ) i t i_1) = 0
-- Mathlib/MeasureTheory/Constructions/Pi.lean
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝² : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝¹ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
i : ι
s : Set (α i)
hs : (μ i) s = 0
t : Set (α i)
hst : s ⊆ t
left✝ : MeasurableSet t
hμt : (μ i) t = 0
\u22a2 (μ i) (update (fun j => univ) i t i) = 0
-- Mathlib/Geometry/Euclidean/Sphere/Basic.lean
V : Type u_1
P : Type u_2
inst✝ : MetricSpace P
p : P
\u22a2 Cospherical {p}
-- Mathlib/Geometry/Euclidean/Sphere/Basic.lean
V : Type u_1
P : Type u_2
inst✝ : MetricSpace P
p : P
\u22a2 ∃ radius, ∀ p_1 ∈ {p}, dist p_1 p = radius
-- Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean
ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : ContinuousSMul 𝕜 F
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
\u22a2 IsVonNBounded 𝕜 (⇑f '' s)
-- Mathlib/ModelTheory/FinitelyGenerated.lean
L : Language
M : Type u_1
inst✝¹ : L.Structure M
N : Type u_2
inst✝ : L.Structure N
f : M ↪[L] N
s : L.Substructure M
t : Set N
h1 : t.Countable
h2 : (closure L).toFun t = Substructure.map f.toHom s
\u22a2 s.CG
-- Mathlib/ModelTheory/FinitelyGenerated.lean
L : Language
M : Type u_1
inst✝¹ : L.Structure M
N : Type u_2
inst✝ : L.Structure N
f : M ↪[L] N
s : L.Substructure M
t : Set N
h1 : t.Countable
h2 : (closure L).toFun t = Substructure.map f.toHom s
\u22a2 ∃ S, S.Countable ∧ (closure L).toFun S = s