ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasRightHomology
inst✝ : S₂.HasRightHomology
⊢ (rightHomologyMap' φ S₁.rightHomologyData S₂.rightHomologyData).op = leftHomologyMap' (𝟙 S₂.op) S₂.rightHomologyData.op S₂.op.leftHomologyData ≫ leftHomologyMap' (opMap φ) S₂.op.leftHomologyData S₁.op.leftHomologyData ≫ leftHomologyMap' (𝟙 S₁.op) S₁.op.leftHomologyData S₁.rightHomologyData.op
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
F : Type u_3
NormedAddCommGroup F
NormedSpace 𝕜 F
f : E → F
f' : E →L[𝕜] F
x : E
s : Set E
h : HasFDerivWithinAt f f' s x
hs : s ∈ 𝓝 x
⊢ HasFDerivAt f f' x
x : ℝ
⊢ ↑x - ⊥ = ⊤
α : Type u_1
Fintype α
DecidableEq α
⊢ Equiv.Perm.cycleType 1 = 0
q a b c : ℚ
⊢ 1 /. 1 = 1
α : Type u_1
l : Filter α
p : α → Prop
⊢ Filter.Tendsto p l (𝓝 True) ↔ ∀ᶠ (x : α) in l, p x
R : Type u_1
A : Type u_2
B : Type u_3
CommSemiring R
StarRing R
TopologicalSpace A
Semiring A
Algebra R A
StarRing A
StarModule R A
TopologicalSemiring A
ContinuousStar A
TopologicalSpace B
Semiring B
Algebra R B
StarRing B
StarModule R B
TopologicalSemiring B
ContinuousStar B
s : StarSubalgebra R A
φ : A →⋆ₐ[R] B
hφ : IsClosedMap ⇑φ
⊢ (StarSubalgebra.map φ s).topologicalClosure ≤ StarSubalgebra.map φ s.topologicalClosure
K : Type u_1
RCLike K
⊢ (starRingEnd K) RCLike.I = -RCLike.I
X : Type u_1
TopologicalSpace X
RegularSpace X
K U : Set X
hK : IsCompact K
hU : U ∈ 𝓝ˢ K
⊢ ∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U
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))
⊢ ↑((((Substructure.inclusion ⋯).comp (g.toEmbedding.comp PM).equivRange.toEmbedding).comp NP) n) = ↑((((Substructure.inclusion ⋯).comp QM.equivRange.toEmbedding).comp NQ) n)
α : Type u_1
β : Type u_2
Preorder α
Preorder β
f : α →o β
⊢ (Setoid.r ⇒ Setoid.r) ⇑f ⇑f
C : Type u
CategoryTheory.Category.{v, u} C
CategoryTheory.Limits.HasFiniteProducts C
CategoryTheory.Limits.HasPullbacks C
X : CategoryTheory.Dial C
⊢ X.leftUnitor.inv.f =
  CategoryTheory.Limits.prod.lift (CategoryTheory.Limits.terminal.from X.src) (CategoryTheory.CategoryStruct.id X.src)
ι : Type u
f : ι → Cardinal.{max u v}
c : Cardinal.{max u v}
hι : Cardinal.lift.{v, u} #ι < c.ord.cof
hf : ∀ (i : ι), f i < c
⊢ iSup f < c
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
CommMonoid G
TopologicalSpace G
ChartedSpace H G
SmoothMul 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
x : M
t : Finset ι
f : ι → M → G
n : ℕ∞
h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x
⊢ ContMDiffAt I' I n (∏ i ∈ t, f i) x
α : Type u
t : TopologicalSpace α
a : α
s : Set α
b : Set (Set α)
hb : TopologicalSpace.IsTopologicalBasis b
hs : s ∈ b
ha : a ∈ s
⊢ s ∈ 𝓝 a
p : ℝ
hp : 1 ≤ p
⊢ ConvexOn ℝ (Set.Ici 0) fun x => x ^ p
ι : Type u_3
⊢ StrictMono ⇑Finsupp.toMultiset
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✝³ : NormedField 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : Nonempty ι
t : TopologicalSpace E
p : SeminormFamily 𝕜 E ι
hp : WithSeminorms p
i : ι
⊢ Continuous ⇑(p i)
m n k : ℕ
⊢ (Nat.fib m).gcd (Nat.fib (n + k * m)) = (Nat.fib m).gcd (Nat.fib n)
R : Type u
S : Type v
T : Type w
A : Type z
A' : Type u_1
B : Type u_2
a b : R
n : ℕ
inst✝⁶ : CommSemiring R
inst✝⁵ : Semiring A
inst✝⁴ : CommSemiring A'
inst✝³ : Semiring B
inst✝² : Algebra R A
inst✝¹ : Algebra R A'
inst✝ : Algebra R B
p✝ q✝ : R[X]
x : A
p q : R[X]
hpq : p.comp q = X
hqp : q.comp p = X
⊢ R[X] ≃ₐ[R] R[X]
C : Type u
CategoryTheory.Category.{v, u} C
CategoryTheory.Limits.HasZeroObject C
CategoryTheory.HasShift C ℤ
CategoryTheory.Preadditive C
∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : CategoryTheory.Pretriangulated C
T T' : CategoryTheory.Pretriangulated.Triangle C
φ : T ⟶ T'
hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
hT' : T' ∈ CategoryTheory.Pretriangulated.distinguishedTriangles
h₁ : CategoryTheory.IsIso φ.hom₁
h₂ : CategoryTheory.IsIso φ.hom₂
⊢ CategoryTheory.IsIso φ.hom₃
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : T1Space X
s t : Set X
⊢ 𝓝ˢ s = 𝓝ˢ t ↔ s = t
M : Type u_4
N : Type u_5
MulOneClass M
MulOneClass N
f : M →* N
a b : M
⊢ f (a * b) = f a * f b
R : Type u_1
M : Type u_8
M₂ : Type u_10
M₃ : Type u_11
CommSemiring R
AddCommMonoid M
AddCommMonoid M₂
AddCommMonoid M₃
Module R M
Module R M₂
Module R M₃
e₁ : M ≃ₗ[R] M₂
e₂ : M₂ ≃ₗ[R] M₃
⊢ e₁.conj ≪≫ₗ e₂.conj = (e₁ ≪≫ₗ e₂).conj
F : Type u_1
inst✝² : Field F
E : Type u_2
inst✝¹ : Field E
inst✝ : Algebra F E
p q : F[X]
hpq : Splits (algebraMap F q.SplittingField) p
hp : IsSolvable p.Gal
hq : IsSolvable (Polynomial.map (algebraMap F p.SplittingField) q).Gal
K : Type u_1 := p.SplittingField
L : Type u_1 := q.SplittingField
this : Fact (Splits (algebraMap F L) p)
ϕ : (L ≃ₐ[K] L) ≃* (Polynomial.map (algebraMap F K) q).Gal := (IsSplittingField.algEquiv L (Polynomial.map (algebraMap F K) q)).autCongr
⊢ IsSolvable q.Gal
𝕜 : Type u_1
𝕜' : Type u_2
D : Type u_3
E : Type u_4
F : Type u_5
G : Type u_6
V : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f✝ : E → F
hf_temperate : Function.HasTemperateGrowth f✝
n : ℕ
k : ℕ → ℕ
C : ℕ → ℝ
f : ∀ (n : ℕ) (x : E), ‖iteratedFDeriv ℝ n f✝ x‖ ≤ C n * (1 + ‖x‖) ^ k n
C' : ℝ := max 0 ((Finset.range (n + 1)).sup' ⋯ C)
hC' : 0 ≤ C'
N : ℕ
hN : N ∈ Finset.range n.succ
x : E
⊢ (1 + ‖x‖) ^ k N ≤ (1 + ‖x‖) ^ (Finset.range (n + 1)).sup k
α : Type u_1
OrderedCommMonoid α
A B : Set α
hA : BddAbove A
hB : BddAbove B
⊢ BddAbove (A * B)
α : Type u_2
β : Type u_3
DecidableEq β
AddGroup α
AddAction α β
a : α
s : Finset β
⊢ (a +ᵥ s).card = s.card
α : Type u_1
Mul α
Preorder α
CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α
h : a * b ≤ c
hle : d ≤ a
⊢ d * b ≤ c
α : Type u_3
Preorder α
LocallyFiniteOrderTop α
a : α
Fintype ↑(Set.Ioi a)
⊢ (Set.Ioi a).toFinset = Finset.Ioi a
R : Type u_1
Monoid R
S : Submonoid R
OreLocalization.OreSet S
X : Type u_2
AddGroup X
DistribMulAction R X
r : X
s : ↥S
⊢ -(r /ₒ s) = -r /ₒ s
X✝ : Type u
Y : Type v
Z : Type u_1
W : Type u_2
ε : Type u_3
ζ : Type u_4
ι : Type u_5
π : ι → Type u_6
κ : Type u_7
inst✝ : TopologicalSpace X✝
T : (i : ι) → TopologicalSpace (π i)
f✝ : X✝ → (i : ι) → π i
X : Type u_8
f : X → (i : ι) → π i
⊢ induced f Pi.topologicalSpace = ⨅ i, induced (fun x => f x i) inferInstance
n : ℕ
a b : Fin n
⊢ (Finset.Ioc a b).card = ↑b - ↑a
𝕜 : Type u_2
E : Type u_3
LinearOrderedField 𝕜
AddCommGroup E
Module 𝕜 E
TopologicalSpace E
TopologicalAddGroup E
ContinuousConstSMul 𝕜 E
s : Set E
hs : Convex 𝕜 s
x y : E
hx : x ∈ closure s
hy : y ∈ interior s
⊢ openSegment 𝕜 x y ⊆ interior s
R✝ : Type u_1
inst✝² : AddMonoidWithOne R✝
p✝ a✝ b✝ : ℕ
R : Type u_1
inst✝¹ : AddMonoidWithOne R
p : ℕ
inst✝ : CharP R p
a b : ℕ
h : a ≡ b [MOD p]
hle : a ≤ b
⊢ ↑a = ↑b
a b c : ℕ
r : ℤ
ha : a % 2 = 0
hb : b % 8 = 5
hc : a / 2 = c
hr : jacobiSymNat c b = r
⊢ jacobiSym (↑a) b = -jacobiSym (↑a / 2) b
a b x : PGame
xl xr : Type u
xL : xl → PGame
xR : xr → PGame
yl yr : Type u
yL : yl → PGame
yR : yr → PGame
i : Ordinal.{u}
hi : i < (mk yl yr yL yR).birthday
j : (mk yl yr yL yR).LeftMoves
hj : i ≤ ((mk yl yr yL yR).moveLeft j).birthday
⊢ (mk xl xr xL xR).birthday ♯ i < max (lsub fun b => (mk xl xr xL xR).birthday ♯ (yL b).birthday) (lsub fun b => (mk xl xr xL xR).birthday ♯ (yR b).birthday)
α : Type u_1
Preorder α
TopologicalSpace α
Topology.IsLower α
⊢ inst✝ = Topology.lower α
R : Type u_1
A : Type u_2
B : Type u_3
CommSemiring R
Semiring A
Algebra R A
Semiring B
Algebra R B
CoalgebraStruct R A
CoalgebraStruct R B
⊢ Function.Injective AlgHomClass.toAlgHom
α : Type u_1
LinearOrder α
PredOrder α
IsPredArchimedean α
r : α → α → Prop
n m : α
hr : Reflexive r
h1 : ∀ i ∈ Set.Ioc m n, r i (Order.pred i)
h2 : ∀ i ∈ Set.Ioc n m, r (Order.pred i) i
⊢ Relation.TransGen r n m
G : Type u_2
DivInvMonoid G
MeasurableSpace G
MeasurableMul G
g : G
⊢ Measurable fun h => h / g
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ Pell.xn a1 (n * k) ≡ Pell.xn a1 n ^ k [MOD Pell.yn a1 n ^ 2] ∧
  Pell.yn a1 (n * k) ≡ k * Pell.xn a1 n ^ (k - 1) * Pell.yn a1 n [MOD Pell.yn a1 n ^ 3]
C : Type u
inst✝ : Category.{v, u} C
X Y Z : LocallyRingedSpace
f : X ⟶ Z
g : Y ⟶ Z
H : IsOpenImmersion f
H' : Set.range ⇑g.val.base ⊆ Set.range ⇑f.val.base
l : Y ⟶ X
hl : l ≫ f = g
⊢ l = lift f g H'
k : Type u₁
G : Type u₂
Semiring k
Mul G
A : Type u₃
NonUnitalNonAssocSemiring A
Module k A
IsScalarTower k A A
SMulCommClass k A A
F : MonoidAlgebra k G →ₙₐ[k] A
⊢ (MonoidAlgebra.liftMagma k).symm F = F.toMulHom.comp (MonoidAlgebra.ofMagma k G)
α : Type u_1
E : Type u_2
F : Type u_3
NormedAddCommGroup E
NormedSpace ℝ E
NormedAddCommGroup F
NormedSpace ℝ F
m : MeasurableSpace α
μ : MeasureTheory.Measure α
T T' : Set α → E →L[ℝ] F
C C' : ℝ
hT : MeasureTheory.DominatedFinMeasAdditive μ T C
hT' : MeasureTheory.DominatedFinMeasAdditive μ T' C'
h : T = T'
f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)
⊢ (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT) f = (MeasureTheory.L1.SimpleFunc.setToL1SCLM α E μ hT') f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_5
mγ : MeasurableSpace γ
κ : ProbabilityTheory.Kernel α β
ca : α × γ
g : β → ℝ≥0∞
⊢ ∫⁻ (b : β), g b ∂(ProbabilityTheory.Kernel.prodMkRight γ κ) ca = ∫⁻ (b : β), g b ∂κ ca.1
C : Type u_1
inst✝⁵ : Category.{u_6, u_1} C
D : Type u_2
inst✝⁴ : Category.{u_7, u_2} D
E : Type u_3
inst✝³ : Category.{?u.106010, u_3} E
J : GrothendieckTopology C
K : GrothendieckTopology D
L : GrothendieckTopology E
A : Type u_4
inst✝² : Category.{u_5, u_4} A
G : C ⥤ D
inst✝¹ : G.IsCoverDense K
inst✝ : G.IsLocallyFull K
ℱ : Dᵒᵖ ⥤ A
ℱ' : Sheaf K A
α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val
X : Cᵒᵖ
⊢ yoneda.map ((whiskerLeft G.op (sheafHom α)).app X) = yoneda.map (α.app X)
α : Type u_1
β : Type u_2
s t : Set α
f : α → β
hft : Fintype ↑s
h : ∃ a, s.toFinset = {a}
⊢ ∃ a, s = {a}
n : ℕ+
K : Type u
L : Type v
Field K
CommRing L
IsDomain L
Algebra K L
IsCyclotomicExtension {n} K L
ζ : L
hζ : IsPrimitiveRoot ζ ↑n
C : Type u_1
CommRing C
IsDomain C
Algebra K C
hirr : Irreducible (Polynomial.cyclotomic (↑n) K)
φ' : L →ₐ[K] C
⊢ ↑((hζ.embeddingsEquivPrimitiveRoots C hirr) φ') = φ' ζ
p : ℕ+
⊢ p.Prime → 1 < p
F : Type u_1
inst✝³ : Field F
E : Type u_2
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : FiniteDimensional F E
h : IntermediateField.fixedField ⊤ = ⊥
⊢ IsGalois F E
α : Type u_1
MetricSpace α
m0 : MeasurableSpace α
μ : MeasureTheory.Measure α
v : VitaliFamily μ
x : α
o : Set α
hx : o ∈ 𝓝 x
⊢ ∀ᶠ (a : Set α) in v.filterAt x, a ⊆ o
x y : ℝ
n : ℕ
⊢ log ↑n = n.factorization.sum fun p t => ↑t * log ↑p
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Sort y
s t u : Set α
inst✝⁹ : TopologicalSpace α
inst✝⁸ : MeasurableSpace α
inst✝⁷ : BorelSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : MeasurableSpace β
inst✝⁴ : BorelSpace β
inst✝³ : MeasurableSpace δ
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : SecondCountableTopology α
f : δ → α
hf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)
⊢ inst✝⁸ = MeasurableSpace.generateFrom ?convert_2
R : Type u
A : Type v
CommSemiring R
Semiring A
Algebra R A
S T : Subalgebra R A
h : ∀ (x : A), x ∈ S ↔ x ∈ T
⊢ S = T
K : Type u
L : Type v
E : Type z
Field K
Field L
Field E
Algebra K L
Algebra K E
Module.Finite K L
IsAlgClosed E
pb : PowerBasis K L
Algebra.IsSeparable K L
e : Fin pb.dim ≃ (L →ₐ[K] E)
⊢ (algebraMap K E) (Algebra.discr K ⇑pb.basis) =
  (-1) ^ (FiniteDimensional.finrank K L * (FiniteDimensional.finrank K L - 1) / 2) *
    ∏ i : Fin pb.dim, ∏ j ∈ Finset.Ioi i, ((e j) pb.gen - (e i) pb.gen) * ((e i) pb.gen - (e j) pb.gen)
C : Type u_1
CategoryTheory.Category.{u_2, u_1} C
X X' : CategoryTheory.SimplicialObject C
n : ℕ
Z : C
f : AlgebraicTopology.DoldKan.MorphComponents X n Z
g : X' ⟶ X
⊢ (f.preComp g).a = g.app (Opposite.op [n + 1]) ≫ f.a
α : Type u_1
β : Type u_2
γ : Type u_3
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
f : r ≺i s
g : s ≼i t
⊢ (f.ltLe g).top = g f.top
𝕜 : Type u_1
E : Type u_2
inst✝⁹ : DenselyNormedField 𝕜
inst✝⁸ : NonUnitalNormedRing E
inst✝⁷ : StarRing E
inst✝⁶ : CstarRing E
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : IsScalarTower 𝕜 E E
inst✝³ : SMulCommClass 𝕜 E E
inst✝² : StarRing 𝕜
inst✝¹ : CstarRing 𝕜
inst✝ : StarModule 𝕜 E
x : Unitization 𝕜 E
⊢ sSup ((fun x_1 => ‖((algebraMap 𝕜 (E →L[𝕜] E)) x.fst + (mul 𝕜 E) x.snd) x_1‖) '' Metric.closedBall 0 1) ≤ √‖(algebraMap 𝕜 (E →L[𝕜] E)) (x⋆ * x).fst + (mul 𝕜 E) (x⋆ * x).snd‖
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
F : Type u_11
NormedAddCommGroup F
NormedSpace 𝕜 F
G : Type u_12
TopologicalSpace G
J : ModelWithCorners 𝕜 F G
N : Type u_13
TopologicalSpace N
ChartedSpace G N
p : M × N
⊢ SmoothAt (I.prod J) I Prod.fst p
ι : Type u_1
inst✝¹ : LinearOrder ι
inst✝ : LocallyFiniteOrder ι
i : ι
hi : succFn i ≤ i
j : ι
x✝ : i ≤ j
hij_lt : i < j
h_succFn_eq : succFn i = i
h_glb : IsGLB (↑(Finset.Ioc i j)) i
hi_mem : i ∈ Finset.Ioc i j
⊢ False
α : Type u_1
E : Type u_3
MeasurableSpace α
NormedAddCommGroup E
f : α → E
μ : MeasureTheory.Measure α
l : Filter α
l.IsMeasurablyGenerated
hfm : StronglyMeasurableAtFilter f l μ
hμ : μ.FiniteAtFilter l
b : E
hf : Filter.Tendsto f l (𝓝 b)
⊢ MeasureTheory.IntegrableAtFilter f l μ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : CharP R p
inst✝ : PerfectRing R p
f : 𝕎 R
n : ℕ
⊢ ((map ↑(_root_.frobeniusEquiv R p).symm) (frobenius f)).coeff n = f.coeff n
α : Type u_1
TopologicalSpace α
LinearOrder α
OrderTopology α
DenselyOrdered α
a b : α
H : a < b
⊢ (𝓝[Set.Ioo a b] b).NeBot
C : Type u
inst✝³ : Category.{v, u} C
I : C
F : Cᵒᵖ ⥤ Type w
hF : IsSheafFor F (ofArrows Empty.elim fun a => instIsEmptyEmpty.elim a)
hI : IsInitial I
α : Type
X : α → C
c : Cofan X
hc : IsColimit c
inst✝² : (ofArrows X c.inj).hasPullbacks
inst✝¹ : HasInitial C
inst✝ : ∀ (i : α), Mono (c.inj i)
hd : Pairwise fun i j => IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)
this : HasCoproduct X
h : (Pi.lift fun i => F.map (c.inj i).op) = F.map (Pi.lift fun i => (c.inj i).op) ≫ piComparison F fun i => op (X i)
hh : (Pi.lift fun i => (c.inj i).op) = (productIsProduct fun x => op (X x)).lift c.op
⊢ (piComparison F fun x => op (X x)) = F.map ((opCoproductIsoProduct' hc (productIsProduct fun x => op (X x))).inv ≫ (hc.desc c).op ≫ (opCoproductIsoProduct' hc (productIsProduct fun x => op (X x))).hom) ≫ piComparison F fun i => op (X i)
R : Type u
A : Type v
CommSemiring R
Semiring A
Algebra R A
S : Subalgebra R A
x : A
⊢ x ∈ S.toSubsemiring ↔ x ∈ S
p' : ℕ
h : lucasLehmerResidue (p' + 2) = 0
⊢ orderOf (ωUnit (p' + 2)) = 2 ^ (p' + 2)
α : Type v
Preorder α
b : α
⊢ ⋃ a, Set.Icc a b = Set.Iic b
α : Type u_1
inst✝ : AddGroupWithOne α
p a : PosNum
e : p.pred' = Num.pos a
ep : p = (Num.pos a).succ'
⊢ -↑(Num.casesOn (Num.pos a) 1 bit1) = -↑p + -↑p + 1
n : ℕ
⊢ ∏ p ∈ n.primeFactors, p ∣ n
⊢ (fun τ => jacobiTheta τ - 1) =O[comap im atTop] fun τ => rexp (-π * τ.im)
α : Type u_1
β : Type u_2
ι : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
p : ℝ≥0∞
f : α → β
hf : Memℒp f p μ
hmeas : StronglyMeasurable f
ε : ℝ
hε : 0 < ε
hp_ne_zero : ¬p = 0
hp_ne_top : ¬p = ⊤
M : ℝ
hM' : 0 ≤ M
hM : ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x => ‖f x‖ ^ p.toReal) x‖₊ ∂μ ≤ ENNReal.ofReal (ε ^ p.toReal)
⊢ ∫⁻ (x : α), ↑‖{x | M ^ (1 / p.toReal) ≤ ↑‖f x‖₊}.indicator f x‖₊ ^ p.toReal ∂μ ≤ ENNReal.ofReal (ε ^ p.toReal)
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
F : Type u_3
NormedAddCommGroup F
NormedSpace 𝕜 F
f : E → F
x : E
𝕜' : Type u_6
NontriviallyNormedField 𝕜'
NormedAlgebra 𝕜 𝕜'
NormedSpace 𝕜' F
IsScalarTower 𝕜 𝕜' F
c : E → 𝕜'
hc : DifferentiableAt 𝕜 c x
hf : DifferentiableAt 𝕜 f x
⊢ fderiv 𝕜 (fun y => c y • f y) x = c x • fderiv 𝕜 f x + (fderiv 𝕜 c x).smulRight (f x)
α : Type u_1
Group α
LinearOrder α
CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a : α
h : 1 < a
⊢ 1 < mabs a
α : Type u_2
α' : Type u_3
β : Type u_4
TopologicalSpace α
TopologicalSpace α'
Zero β
f : α → β
T2Space α'
hf : HasCompactSupport f
g : α → α'
cont : Continuous g
inj : Function.Injective g
⊢ tsupport (Function.extend g f 0) = g '' tsupport f
R✝ : Type u_1
R : Type u_2
inst✝ : CommSemiring R
n : ℕ
f g : R⟦X⟧
⊢ trunc n (↑(trunc n f) * ↑(trunc n g)) = trunc n (f * g)
α✝ : Type u_1
β : Type u_2
m✝ : MeasurableSpace α✝
α : Type u_3
m : MeasurableSpace α
μ ν : Measure α
inst✝ : μ.HaveLebesgueDecomposition ν
hμν : μ ≪ ν
f : α → ℝ≥0∞
hf : AEMeasurable f ν
⊢ ∫⁻ (x : α), μ.rnDeriv ν x * f x ∂ν = ∫⁻ (x : α), f x ∂ν.withDensity (μ.rnDeriv ν)
α : Type u_1
β : Type u_2
fa : α → α
fb : β → β
f : α → β
h : Function.Semiconj f fa fb
⊢ Set.MapsTo fb (Set.range f) (Set.range f)
M : Type u_5
κ : Type u_6
AddCommMonoid M
S : Finset M
Finite κ
C : M → κ
⊢ ∃ a > 0, ∃ b c, ∀ s ∈ S, C (a • s + b) = c
P : Game → Prop
α✝¹ β✝¹ : Type u_1
a✝³ : α✝¹ → PGame
a✝² : β✝¹ → PGame
α✝ β✝ : Type u_1
a✝¹ : α✝ → PGame
a✝ : β✝ → PGame
h : (∀ (i : (mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (-mk α✝ β✝ a✝¹ a✝).LeftMoves), P (-⟦(mk α✝¹ β✝¹ a✝³ a✝²).mulOption (-mk α✝ β✝ a✝¹ a✝) i j⟧)) ∧ ∀ (i : (-mk α✝¹ β✝¹ a✝³ a✝²).LeftMoves) (j : (mk α✝ β✝ a✝¹ a✝).LeftMoves), P (-⟦(-mk α✝¹ β✝¹ a✝³ a✝²).mulOption (mk α✝ β✝ a✝¹ a✝) i j⟧)
i : α✝¹
j : β✝
⊢ ⟦mk α✝¹ β✝¹ a✝³ a✝² * a✝ j⟧ = -⟦mk α✝¹ β✝¹ a✝³ a✝² * -a✝ j⟧
M : Type u_1
ι : Type u_2
R : Type u_3
inst✝ : CommSemiring R
m : M
a : R[M]
r : R
⊢ a = Finsupp.single m r ↔ (Finsupp.lsingle m) r = a
α : Type u_1
F : Type u_4
𝕜 : Type u_14
Norm F
NormedField 𝕜
g : α → F
l : Filter α
c : 𝕜
f : α → 𝕜
hc : c ≠ 0
⊢ (fun x => c * f x) =Θ[l] g ↔ f =Θ[l] g
R : Type u_1
S : Type u_2
x y : R
M : Type v
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
p : Submodule R M
hp : p ≤ Submodule.comap f p
k : ℕ
hk : f ^ k = 0
⊢ p.mapQ p f hp ^ k = 0
m : ℤ
⊢ ‖↑↑m‖ = ‖↑m‖
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝³ : EMetricSpace X
inst✝² : EMetricSpace Y
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
d₁ d₂ : ℝ
h : d₁ < d₂
s : Set X
H : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ⊤
c : ℝ≥0
hc : c ≠ 0
this : 0 < ↑c ^ (d₂ - d₁)⁻¹
⊢ (fun r => r ^ d₂) ≤ᶠ[𝓝[≥] 0] ↑c • fun r => r ^ d₁
T : Type u₁
CategoryTheory.Category.{v₁, u₁} T
X : T
f : CategoryTheory.Over X
⊢ f.iteratedSliceEquiv.unitIso =
  CategoryTheory.NatIso.ofComponents
    (fun g =>
      CategoryTheory.Over.isoMk
        (CategoryTheory.Over.isoMk
          (CategoryTheory.Iso.refl ((CategoryTheory.Functor.id (CategoryTheory.Over f)).obj g).left.left) ⋯)
        ⋯)
    ⋯
α : Type u_1
⊢ Function.Injective Finset.Colex.ofColex
α β : Type u
c : Cardinal.{?u.169944}
ι : Type u_1
f g : ι → Cardinal.{u_2}
H : ∀ (i : ι), f i < g i
x✝ : sum f ≥ prod g
F : ((i : ι) → Quotient.out (g i)) ↪ (i : ι) × Quotient.out (f i)
⊢ False
α : Type u_1
β : Type u_2
Preorder α
Preorder β
f g : α → β
hf : Monotone f
hg : Antitone g
⊢ Antitone fun x => Set.Ico (f x) (g x)
α : Type u_1
β : Type u_2
ι : Type u_3
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
h : CauchySeq fun n => ∑ k ∈ Finset.range n, f k
⊢ ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
S : ShortComplex C
cg : KernelFork S.g
hg : IsLimit cg
cf : CokernelCofork S.f
hf : IsColimit cf
⊢ S.Exact ↔ Fork.ι cg ≫ Cofork.π cf = 0
α : Type u_1
Fintype α
Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
⊢ ∀ᶠ (n : ℕ) in Filter.atTop,
  ∀ (i : α),
    b i ^ AkraBazziRecurrence.p a b * ↑n ^ AkraBazziRecurrence.p a b * (1 + AkraBazziRecurrence.smoothingFn ↑n) ≤
      ↑(r i n) ^ AkraBazziRecurrence.p a b * (1 + AkraBazziRecurrence.smoothingFn ↑(r i n))
α : Type u_1
m n : PosNum
h : m = n
⊢ ↑m < ↑n ↔ Ordering.eq = Ordering.lt
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
hUl : ∀ x ∈ s, (U x)ᶜ ∈ l
hxU : ∀ x ∈ s, x ∈ U x
hUo : ∀ x ∈ s, IsOpen (U x)
t : Finset X
hts : ∀ x ∈ t, x ∈ s
hst : s ⊆ ⋃ x ∈ t, U x
⊢ ∀ i ∈ t, (U i)ᶜ ∈ l
ι : Type u_1
R : Type u_4
CommSemiring R
s : ι → Type u_7
(i : ι) → AddCommMonoid (s i)
(i : ι) → Module R (s i)
⊢ PiTensorProduct.reindex R s (Equiv.refl ι) = LinearEquiv.refl R (PiTensorProduct R fun i => s i)
α : Type u_2
β : Type u_3
Mul α
Add α
LE α
Mul β
Add β
LE β
e : α ≃+*o β
⊢ e.symm.symm = e
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
hT : T.IsSymmetric
inst✝ : FiniteDimensional 𝕜 E
v : E
μ : Eigenvalues T
w : PiLp 2 fun μ => ↥(eigenspace T (↑T μ))
⊢ T (hT.diagonalization.symm w) = hT.diagonalization.symm fun μ => ↑T μ • w μ
R : Type u_1
M : Type u_2
CommRing R
AddCommGroup M
Module R M
p : Submodule R M
x : M
hx : x ∉ p
hp' : Module.Free R (M ⧸ p)
⊢ ∃ f, f x ≠ 0 ∧ Submodule.map f p = ⊥
R : Type u_1
NonUnitalCommSemiring R
⊢ NonUnitalSubsemiring.center R = ⊤
α : Type u_1
β : Type u_2
g : Filter β
⊢ ⊥ ×ˢ g = ⊥
n : Type u_1
p : Type u_2
R : Type u₂
𝕜 : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : DecidableEq n
inst✝³ : DecidableEq p
inst✝² : CommRing R
r : ℕ
M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜
inst✝¹ : Fintype n
inst✝ : Fintype p
M : Matrix p p 𝕜
e : p ≃ n
L₀ L₀' : List (TransvectionStruct n 𝕜)
D₀ : n → 𝕜
h₀ : (List.map toMatrix L₀).prod * (reindexAlgEquiv 𝕜 𝕜 e) M * (List.map toMatrix L₀').prod = diagonal D₀
this : M = (reindexAlgEquiv 𝕜 𝕜 e.symm) ((reindexAlgEquiv 𝕜 𝕜 e) M)
⊢ (List.map toMatrix (List.map (reindexEquiv e.symm) L₀)).prod * (reindexAlgEquiv 𝕜 𝕜 e.symm) ((reindexAlgEquiv 𝕜 𝕜 e) M) * (List.map toMatrix (List.map (reindexEquiv e.symm) L₀')).prod = diagonal (D₀ ∘ ⇑e)
R : Type u
σ : Type u_1
CommSemiring R
p : MvPolynomial σ R
⊢ (MvPolynomial.aeval MvPolynomial.X) p = p
C : Type u_1
inst✝³ : Category.{u_3, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j k : ι
hij : c.Rel i j
hjk : c.Rel j k
⊢ inlX φ j i hij ≫ d φ i j = -F.d j k ≫ inlX φ k j hjk + φ.f j ≫ inrX φ j
n : ℕ
NeZero n
i : ZMod (2 * n)
⊢ orderOf (QuaternionGroup.xa i) = 4
R : Type u_1
CommRing R
p a b c d x y : R
h1 : p ∣ a * x + b * y
h2 : p ∣ c * x + d * y
⊢ p ∣ (a * d - b * c) * y
R : Type u_1
Semiring R
n : ℤ
⊢ (LaurentPolynomial.T n).degree ≤ ↑n
S : Type u_1
T : Type u_2
inst✝¹ : CommRing S
inst✝ : CommRing T
g : S →+* T
u : Set S
x : S
hx : ∀ (S_1 : Subring S), u ⊆ ↑S_1 → x ∈ S_1
T₁ : Subring T
h₁ : ⇑g '' u ⊆ ↑T₁
⊢ ⇑g '' u ⊆ ↑T₁
b : ℂ
hb : 0 < b.re
full_integral : ∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2) ∂volume + ∫ (x : ℝ) in Iic 0, cexp (-b * ↑x ^ 2) ∂volume = (↑π / b) ^ (1 / 2)
this✝ : MeasurableSet (Ioi 0)
this : ∀ (c : ℝ), ∫ (x : ℝ) in 0 ..c, cexp (-b * ↑x ^ 2) = ∫ (x : ℝ) in -c..0, cexp (-b * ↑x ^ 2)
t1 : Tendsto (fun i => ∫ (x : ℝ) in 0 ..id i, cexp (-b * ↑x ^ 2)) atTop (𝓝 (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2) ∂volume))
t2 : Tendsto (fun c => ∫ (x : ℝ) in 0 ..c, cexp (-b * ↑x ^ 2)) atTop (𝓝 (∫ (x : ℝ) in Iic 0, cexp (-b * ↑x ^ 2)))
⊢ ∫ (x : ℝ) in Iic 0, cexp (-b * ↑x ^ 2) = ∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)
α : Type u
PseudoEMetricSpace α
e : α ≃ᵢ α
x : α
⊢ e (e⁻¹ x) = x
α : Type u_1
β : Type u_2
p : α × β
⊢ { fst := p.fst, snd := p.snd } = p
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0
⊢ Tendsto (fun n => ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n) atTop (𝓝 0)