ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K_hat R K
hx : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, x v ∈ adicCompletionIntegers K v

⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, (-x) v ∈ adicCompletionIntegers K v
-- Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean
R : Type u_1
K : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
x : K_hat R K
hx : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, x v ∈ adicCompletionIntegers K v
h : ∀ (v : HeightOneSpectrum R), -x v ∈ adicCompletionIntegers K v ↔ x v ∈ adicCompletionIntegers K v

⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, (-x) v ∈ adicCompletionIntegers K v
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ Continuous fun x => x * log x
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ ∀ (x : ℝ), ContinuousAt (fun x => x * log x) x
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
x : ℝ

⊢ ContinuousAt (fun x => x * log x) x
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
x : ℝ
hx : x ≠ 0

⊢ ContinuousAt (fun x => x * log x) x
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ ContinuousAt (fun x => x * log x) 0
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ Filter.Tendsto (fun x => x * log x) (𝓝 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ Filter.Tendsto (fun x => log x * x) (𝓝 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean

⊢ Filter.Tendsto (fun x => log x * x) (𝓝[Set.univ] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

⊢ Filter.Tendsto (fun x => log x * x) (𝓝[Set.univ] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

⊢ Filter.Tendsto (fun x => log x * x) (𝓝[<] 0 ⊔ 𝓝[>] 0 ⊔ 𝓝[{0}] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

⊢ (Filter.Tendsto (fun x => log x * x) (𝓝[<] 0) (𝓝 0) ∧ Filter.Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)) ∧ Filter.Tendsto (fun x => log x * x) (pure 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

⊢ Filter.Tendsto (fun x => log x * x) (𝓝[>] 0) (𝓝 0)
-- Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean
this : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}

⊢ Filter.Tendsto (fun x => log x * x) (pure 0) (𝓝 0)
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
C : ℝ
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C

⊢ eLpNorm f p μ ≤ μ Set.univ ^ p.toReal⁻¹ * ENNReal.ofReal C
-- Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ ν : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
f : α → F
C : ℝ
hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C

⊢ eLpNorm f p μ ≤ ENNReal.ofReal C * μ Set.univ ^ p.toReal⁻¹
-- Mathlib/Order/Heyting/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝ : GeneralizedCoheytingAlgebra α
a✝ b✝ c✝ d✝ a b c d : α

⊢ (a \ b) \ c ≤ d ↔ a \ (b ⊔ c) ≤ d
-- Mathlib/Data/Rel.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : Rel α β
s : Set β

⊢ s ∩ r.codom ⊆ r.image (r.preimage s)
-- Mathlib/Data/Rel.lean
α : Type u_1
β : Type u_2
γ : Type u_3
r : Rel α β
s : Set β

⊢ s ∩ r.inv.dom ⊆ r.inv.preimage (r.preimage s)
-- Mathlib/Order/Interval/Set/OrdConnectedComponent.lean
α : Type u_1
inst✝ : LinearOrder α
s t : Set α
x y z : α

⊢ (⇑ofDual ⁻¹' s).ordSeparatingSet (⇑ofDual ⁻¹' t) = ⇑ofDual ⁻¹' s.ordSeparatingSet t
-- Mathlib/Algebra/Ring/InjSurj.lean
α : Type u_1
β : Type u_2
inst✝¹³ : Zero β
inst✝¹² : One β
inst✝¹¹ : Add β
inst✝¹⁰ : Mul β
inst✝⁹ : Neg β
inst✝⁸ : Sub β
inst✝⁷ : SMul ℕ β
inst✝⁶ : SMul ℤ β
inst✝⁵ : Pow β ℕ
inst✝⁴ : NatCast β
inst✝³ : IntCast β
f : α → β
hf : Surjective f
inst✝² : Mul α
inst✝¹ : Add α
inst✝ : RightDistribClass α
add : ∀ (x y : α), f (x + y) = f x + f y
mul : ∀ (x y : α), f (x * y) = f x * f y
x y z : α

⊢ (f x + f y) * f z = f x * f z + f y * f z
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a

⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left red parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a

⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b

⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right red a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b

⊢ RBNode.Ordered cmp (parent.ins (node red a x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a

⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
parent : Path α
x : α
b a : RBNode α
ha : RBNode.Ordered cmp a
hp : Ordered cmp parent
xb : All (fun x_1 => cmpLT cmp x x_1) b
xp : RootOrdered cmp parent x
bp : All (RootOrdered cmp parent) b
hb : RBNode.Ordered cmp b
H : All (RootOrdered cmp (left black parent x b)) a
ax : All (fun a => cmpLT cmp a x) a
ap : All (RootOrdered cmp parent) a

⊢ RBNode.Ordered cmp (parent.ins (a.balance1 x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b

⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
a : RBNode α
x : α
parent : Path α
b : RBNode α
hb : RBNode.Ordered cmp b
hp : Ordered cmp parent
ax : All (fun x_1 => cmpLT cmp x_1 x) a
xp : RootOrdered cmp parent x
ap : All (RootOrdered cmp parent) a
ha : RBNode.Ordered cmp a
H : All (RootOrdered cmp (right black a x parent)) b
xb : All (cmpLT cmp x) b
bp : All (RootOrdered cmp parent) b

⊢ RBNode.Ordered cmp (parent.ins (a.balance2 x b))
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ

⊢ ∫⁻ (x : α), g x ∂μ - ∫⁻ (x : α), f x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ

⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f g : ↥A

⊢ ↑f ⊓ ↑g ∈ A.topologicalClosure
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : CompactSpace X
A : Subalgebra ℝ C(X, ℝ)
f g : ↥A

⊢ 2⁻¹ • (↑f + ↑g - |↑g - ↑f|) ∈ A.topologicalClosure
-- 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✝ : NoMaxOrder α
a b : α
h : a < b

⊢ frontier (Ioc a b) = {a, b}
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝¹ : CommRing R
W : WeierstrassCurve R
A : Type v
inst✝ : CommRing A
φ : R →+* A
C C' : VariableChange R

⊢ { u := (Units.map ↑φ) (C.u * C'.u), r := φ (C.r * ↑C'.u ^ 2 + C'.r), s := φ (↑C'.u * C.s + C'.s), t := φ (C.t * ↑C'.u ^ 3 + C.r * C'.s * ↑C'.u ^ 2 + C'.t) } = { u := (Units.map ↑φ) C.u * (Units.map ↑φ) C'.u, r := φ C.r * ↑((Units.map ↑φ) C'.u) ^ 2 + φ C'.r, s := ↑((Units.map ↑φ) C'.u) * φ C.s + φ C'.s, t := φ C.t * ↑((Units.map ↑φ) C'.u) ^ 3 + φ C.r * φ C'.s * ↑((Units.map ↑φ) C'.u) ^ 2 + φ C'.t }
-- Mathlib/Data/Rat/Cast/Defs.lean
F : Type u_1
ι : Type u_2
α : Type u_3
β : Type u_4
inst✝³ : FunLike F α β
inst✝² : DivisionRing α
inst✝¹ : DivisionRing β
inst✝ : RingHomClass F α β
f : F
q : ℚ

⊢ f ↑q = ↑q
-- Mathlib/Topology/CompactOpen.lean
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
T : Type u_5
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : TopologicalSpace T
K✝ : Set X
U✝ : Set Y
f : C(X, Y)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen U

⊢ IsOpen {x | MapsTo (⇑(x.comp f)) K U}
-- Mathlib/Data/Multiset/Bind.lean
α : Type u_1
β : Type v
γ : Type u_2
δ : Type u_3
a : α
s t : Multiset α
f g : α → Multiset β

⊢ (s.bind fun x => 0) = 0
-- Mathlib/Algebra/Group/Units.lean
α : Type u
M : Type u_1
N : Type u_2
inst✝ : CommMonoid M
x y : M
hu : IsUnit (x * y)

⊢ IsUnit (y * x)
-- Mathlib/AlgebraicTopology/SimplexCategory.lean
n m : SimplexCategory
f : n ⟶ m

⊢ Mono f ↔ Function.Injective ⇑(Hom.toOrderHom f)
-- Mathlib/AlgebraicTopology/SimplexCategory.lean
n m : SimplexCategory
f : n ⟶ m

⊢ Mono (skeletalEquivalence.functor.map f) ↔ Function.Injective ⇑(Hom.toOrderHom f)
-- Mathlib/AlgebraicTopology/SimplexCategory.lean
n m : SimplexCategory
f : n ⟶ m

⊢ Mono (skeletalFunctor.map f) ↔ Function.Injective ⇑(Hom.toOrderHom f)
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex

⊢ Nat.card ↥(_root_.commutator G) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G) + 1)
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) * ((center G).subgroupOf (_root_.commutator G)).index ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G)) * (center G).index
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣ (center G).index ^ ((center G).index * Nat.card ↑(commutatorSet G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)

⊢ Nat.card ↥((center G).subgroupOf (_root_.commutator G)) ∣ (center G).index ^ Group.rank ↥((center G).subgroupOf (_root_.commutator G))
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)

⊢ ∀ (g : ↥((center G).subgroupOf (_root_.commutator G))), g ^ (center G).index = 1
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))

⊢ g ^ (center G).index = 1
-- Mathlib/GroupTheory/Schreier.lean
G : Type u_1
inst✝¹ : Group G
H : Subgroup G
R S : Set G
inst✝ : Finite ↑(commutatorSet G)
hG : ¬(center G).index = 0
this✝¹ : (center G).FiniteIndex
h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index
this✝ : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex
h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)
h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)
g : ↥((center G).subgroupOf (_root_.commutator G))
this : ↑↑g ∈ (MonoidHom.transferCenterPow G).ker

⊢ g ^ (center G).index = 1
-- Mathlib/Algebra/Star/SelfAdjoint.lean
R : Type u_1
A : Type u_2
inst✝¹ : Mul R
inst✝ : StarMul R
x : R

⊢ IsSelfAdjoint (x * star x)
-- Mathlib/LinearAlgebra/Basis/Flag.lean
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
n : ℕ
b : Basis (Fin n) R M

⊢ b.flag (Fin.last n) = ⊤
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : ℝ

⊢ f.measure (Iic x) = ofReal (↑f x - l)
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : ℝ

⊢ Tendsto (fun x_1 => f.measure (Ioc x_1 x)) atBot (𝓝 (ofReal (↑f x - l)))
-- Mathlib/MeasureTheory/Measure/Stieltjes.lean
f : StieltjesFunction
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : ℝ

⊢ Tendsto (fun x_1 => ofReal (↑f x - ↑f x_1)) atBot (𝓝 (ofReal (↑f x - l)))
-- Mathlib/Topology/Sheaves/Skyscraper.lean
X : TopCat
p₀ : ↑X
inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)
C : Type v
inst✝² : Category.{u, v} C
A : C
inst✝¹ : HasTerminal C
inst✝ : HasColimits C
x y : C
f : x ⟶ y
U : (OpenNhds p₀)ᵒᵖ

⊢ colimit.ι (((whiskeringLeft (OpenNhds p₀)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion p₀).op).obj ((skyscraperPresheafFunctor p₀).obj x)) U ≫ (skyscraperPresheafFunctor p₀ ⋙ Presheaf.stalkFunctor C p₀).map f ≫ (fun c => (skyscraperPresheafStalkOfSpecializes p₀ c ⋯).hom) y = colimit.ι (((whiskeringLeft (OpenNhds p₀)ᵒᵖ (Opens ↑X)ᵒᵖ C).obj (OpenNhds.inclusion p₀).op).obj ((skyscraperPresheafFunctor p₀).obj x)) U ≫ (fun c => (skyscraperPresheafStalkOfSpecializes p₀ c ⋯).hom) x ≫ (𝟭 C).map f
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this : lift.{max u v, u} (Module.rank K (ℕ → K)) ≤ lift.{u, max u v} (Module.rank K (ι → K))

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this : lift.{v, u} (Module.rank K (ℕ → K)) ≤ Module.rank K (ι → K)

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this : lift.{v, u} (Module.rank K (ℕ → K)) ≤ Module.rank K (ι → K)
key : lift.{v, u} (max ℵ₀ #K) ≤ Module.rank K (ι → K)

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this : lift.{v, u} (Module.rank K (ℕ → K)) ≤ Module.rank K (ι → K)
key : ℵ₀ ≤ Module.rank K (ι → K) ∧ lift.{v, u} #K ≤ Module.rank K (ι → K)

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this✝ : lift.{v, u} (Module.rank K (ℕ → K)) ≤ Module.rank K (ι → K)
key : ℵ₀ ≤ Module.rank K (ι → K) ∧ lift.{v, u} #K ≤ Module.rank K (ι → K)
this : Infinite ιK

⊢ Module.rank K (ι → K) = #(ι → K)
-- Mathlib/LinearAlgebra/Dimension/DivisionRing.lean
K R : Type u
V V₁ V₂ V₃ : Type v
V' V'₁ : Type v'
V'' : Type v''
ι✝ : Type w
ι' : Type w'
η : Type u₁'
φ : η → Type u_1
inst✝ : DivisionRing K
ι : Type v
hι : Infinite ι
ιK : Type (max u v)
bK : Basis ιK K (ι → K)
e : ℕ ↪ ι
this✝ : lift.{v, u} (Module.rank K (ℕ → K)) ≤ Module.rank K (ι → K)
key : ℵ₀ ≤ Module.rank K (ι → K) ∧ lift.{v, u} #K ≤ Module.rank K (ι → K)
this : Infinite ιK

⊢ lift.{v, u} #K ≤ Module.rank K (ι → K)
-- Mathlib/CategoryTheory/Whiskering.lean
C✝ : Type u₁
inst✝³ : Category.{v₁, u₁} C✝
D : Type u₂
inst✝² : Category.{v₂, u₂} D
E : Type u₃
inst✝¹ : Category.{v₃, u₃} E
F : D ⥤ E
hF : F.FullyFaithful
C : Type u_1
inst✝ : Category.{?u.29894, u_1} C
X✝ Y✝ : C ⥤ D
f : ((CategoryTheory.whiskeringRight C D E).obj F).obj X✝ ⟶ ((CategoryTheory.whiskeringRight C D E).obj F).obj Y✝
x✝¹ x✝ : C
g : x✝¹ ⟶ x✝

⊢ X✝.map g ≫ (fun X => hF.preimage (f.app X)) x✝ = (fun X => hF.preimage (f.app X)) x✝¹ ≫ Y✝.map g
-- Mathlib/CategoryTheory/Whiskering.lean
C✝ : Type u₁
inst✝³ : Category.{v₁, u₁} C✝
D : Type u₂
inst✝² : Category.{v₂, u₂} D
E : Type u₃
inst✝¹ : Category.{v₃, u₃} E
F : D ⥤ E
hF : F.FullyFaithful
C : Type u_1
inst✝ : Category.{?u.29894, u_1} C
X✝ Y✝ : C ⥤ D
f : ((CategoryTheory.whiskeringRight C D E).obj F).obj X✝ ⟶ ((CategoryTheory.whiskeringRight C D E).obj F).obj Y✝
x✝¹ x✝ : C
g : x✝¹ ⟶ x✝

⊢ F.map (X✝.map g ≫ (fun X => hF.preimage (f.app X)) x✝) = F.map ((fun X => hF.preimage (f.app X)) x✝¹ ≫ Y✝.map g)
-- Mathlib/CategoryTheory/Whiskering.lean
C✝ : Type u₁
inst✝³ : Category.{v₁, u₁} C✝
D : Type u₂
inst✝² : Category.{v₂, u₂} D
E : Type u₃
inst✝¹ : Category.{v₃, u₃} E
F : D ⥤ E
hF : F.FullyFaithful
C : Type u_1
inst✝ : Category.{?u.29894, u_1} C
X✝ Y✝ : C ⥤ D
f : ((CategoryTheory.whiskeringRight C D E).obj F).obj X✝ ⟶ ((CategoryTheory.whiskeringRight C D E).obj F).obj Y✝
x✝¹ x✝ : C
g : x✝¹ ⟶ x✝

⊢ F.map (X✝.map g ≫ hF.preimage (f.app x✝)) = F.map (hF.preimage (f.app x✝¹) ≫ Y✝.map g)
-- Mathlib/CategoryTheory/Whiskering.lean
C✝ : Type u₁
inst✝³ : Category.{v₁, u₁} C✝
D : Type u₂
inst✝² : Category.{v₂, u₂} D
E : Type u₃
inst✝¹ : Category.{v₃, u₃} E
F : D ⥤ E
hF : F.FullyFaithful
C : Type u_1
inst✝ : Category.{?u.29894, u_1} C
X✝ Y✝ : C ⥤ D
f : ((CategoryTheory.whiskeringRight C D E).obj F).obj X✝ ⟶ ((CategoryTheory.whiskeringRight C D E).obj F).obj Y✝
x✝¹ x✝ : C
g : x✝¹ ⟶ x✝

⊢ F.map (X✝.map g) ≫ f.app x✝ = f.app x✝¹ ≫ F.map (Y✝.map g)