ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g₁ g₂ : Fix F α → β
h : ∀ (x : F (α ::: Fix F α)), (TypeVec.id ::: g₁) <$$> x = (TypeVec.id ::: g₂) <$$> x → g₁ (mk x) = g₂ (mk x)
x : (P F).W α
⊢ ∀ (a : (P F).A) (f' : (P F).drop.B a ⟹ α) (f : (P F).last.B a → (P F).W α), (∀ (i : (P F).last.B a), g₁ (Quot.mk Setoid.r (f i)) = g₂ (Quot.mk Setoid.r (f i))) → g₁ (Quot.mk Setoid.r ((P F).wMk a f' f)) = g₂ (Quot.mk Setoid.r ((P F).wMk a f' f))
G : Type u_1
inst✝ : Mul G
A B : Finset G
h : 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2
hA : 0 < A.card
hB : 0 < B.card
hc : 1 < A.card ∨ 1 < B.card
p : G × G
hp : p.1 ∈ A ∧ p.2 ∈ B
w✝ : G × G
left✝¹ : w✝ ∈ A ×ˢ B
left✝ : p ≠ w✝
hu : UniqueMul A B p.1 p.2
right✝ : UniqueMul A B w✝.1 w✝.2
⊢ ∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b
n : ℕ∞
⊢ ContDiff ℝ n Real.cos
R : Type u_1
CommSemiring R
p : ℕ
Fact (Nat.Prime p)
CharP R p
x y : R
⊢ (x + y) ^ p = x ^ p + y ^ p
ι : Type u_1
V : Type u
CategoryTheory.Category.{v, u} V
CategoryTheory.Limits.HasZeroMorphisms V
c : ComplexShape ι
A B : HomologicalComplex V c
f : A.Hom B
i j : ι
⊢ f.f i ≫ B.d i j = A.d i j ≫ f.f j
ι : Type u_1
α : ι → Type u_2
inst✝² : (i : ι) → Zero (α i)
inst✝¹ : (i : ι) → Preorder (α i)
r : ι → ι → Prop
inst✝ : IsStrictOrder ι r
x y : Π₀ (i : ι), α i
hlt✝ : x < y
hle : ⇑x ≤ ⇑y
j : ι
hlt : x j < y j
⊢ ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
α : Type u_1
β : Type u_2
CommMonoid α
TopologicalSpace α
b : β
DecidablePred fun x => x = b
a : α
⊢ (∏' (b' : β), if b' = b then a else 1) = a
Γ : Type u_1
Inhabited Γ
l₁ l₂ : List Γ
⊢ Turing.BlankRel l₁ l₂ → Turing.BlankRel l₂ l₁
α : Type u_1
I : Type u_2
inst✝¹ : CommMonoid α
inst✝ : DecompositionMonoid α
x y z : α
s : I → α
t : Finset I
⊢ IsRelPrime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsRelPrime x (s i)
α : Type u_1
d : MeasurableSpace.DynkinSystem α
h_inter : ∀ (s₁ s₂ : Set α), d.Has s₁ → d.Has s₂ → d.Has (s₁ ∩ s₂)
⊢ MeasurableSpace.DynkinSystem.ofMeasurableSpace (d.toMeasurableSpace h_inter) = d
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
⊢ Option.map (fun x => match x with | { q := q, Tape := T } => { q := g₁ q, Tape := Tape.map f₁ T }) (match { q := q, Tape := T } with | { q := q, Tape := T } => Option.map (fun x => match x with | (q', a) => { q := q', Tape := match a with | Stmt.move d => Tape.move d T | Stmt.write a => Tape.write a T }) (M q T.head)) = match match { q := q, Tape := T } with | { q := q, Tape := T } => { q := g₁ q, Tape := Tape.map f₁ T } with | { q := q, Tape := T } => Option.map (fun x => match x with | (q', a) => { q := q', Tape := match a with | Stmt.move d => Tape.move d T | Stmt.write a => Tape.write a T }) (match q, T.head with | q, l => Option.map (Prod.map g₁ (Stmt.map f₁)) (M (g₂ q) (f₂.f l)))
G : Type u_1
AddGroup G
N : Type u_5
AddGroup N
f : G →+ N
⊢ Function.Surjective ⇑f.rangeRestrict
M : Type u_1
S T : Set M
inst✝ : Mul M
z₁ z₂ : M
hz₁ : z₁ ∈ center M
hz₂ : z₂ ∈ center M
a c : M
⊢ a * z₁ * (z₂ * c) = a * (z₁ * (z₂ * c))
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Ring R
p✝ q✝ p q : R[X]
hq : q.Monic
this : DecidableEq R := Classical.decEq R
h : q.degree ≤ p.degree ∧ p ≠ 0
_wf : (p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).degree < p.degree
ih : ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux hq).2 = p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree)) - q * ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))) /ₘ q)
⊢ p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree)) - q * ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))) /ₘ q) = p - q * if h : q.Monic then (if q.degree ≤ p.degree ∧ ¬p = 0 then (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree) + ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux ⋯).1, ((p - q * (C p.leadingCoeff * X ^ (p.natDegree - q.natDegree))).divModByMonicAux ⋯).2) else (0, p)).1 else 0
G✝ : Type w
H : Type x
α : Type u
β : Type v
inst✝⁵ : TopologicalSpace G✝
inst✝⁴ : Group G✝
inst✝³ : TopologicalGroup G✝
inst✝² : TopologicalSpace α
f : α → G✝
s : Set α
x : α
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
hinv : Tendsto (fun x => x⁻¹) (𝓝 1) (𝓝 1)
hleft : ∀ (x₀ : G), 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)
hconj : ∀ (x₀ : G), Tendsto (fun x => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)
x₀ : G
⊢ ContinuousAt (fun a => a⁻¹) x₀
α : Type u_1
inst✝ : LinearOrderedCommGroupWithZero α
a b c d : α
m n : ℕ
hab : a ≤ b
hb : b ≠ 0
hcd : c < d
hd : d ≠ 0
ha : a = 0
⊢ a * c < b * d
𝕜 : Type u
NontriviallyNormedField 𝕜
CompleteSpace 𝕜
EM : Type u_1
NormedAddCommGroup EM
NormedSpace 𝕜 EM
HM : Type u_2
TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
TopologicalSpace M
ChartedSpace HM M
x : M
⊢ nonunits ↑((smoothSheafCommRing IM 𝓘(𝕜, 𝕜) M 𝕜).presheaf.stalk x) =
  ↑(RingHom.ker (smoothSheafCommRing.eval IM 𝓘(𝕜, 𝕜) M 𝕜 x))
G : Type u_1
H : Type u_2
α : Type u_3
β : Type u_4
E : Type u_5
inst✝⁴ : Group G
inst✝³ : MulAction G α
s : Set α
x : α
inst✝² : Group H
inst✝¹ : MulAction H α
inst✝ : SMulCommClass H G α
g : H
⊢ fundamentalInterior G (g • s) = g • fundamentalInterior G s
p : ℝ × ℝ
hp_fst : 0 < p.1
⊢ (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2)
n : ℕ
hn : Odd n
⊢ commProb (DihedralGroup n) = (↑n + 3) / (4 * ↑n)
A : Type u_1
A' : Type u_2
B' : Type u_4
CategoryTheory.Category.{u_5, u_1} A
CategoryTheory.Category.{u_6, u_2} A'
CategoryTheory.Category.{u_7, u_4} B'
eA : A ≌ A'
e' : A' ≌ B'
X : A
⊢ (AlgebraicTopology.DoldKan.Compatibility.equivalence₀ eA e').unitIso.hom.app X =
  eA.unitIso.hom.app X ≫ eA.inverse.map (e'.unitIso.hom.app (eA.functor.obj X))
α : Type u_1
AddCommSemigroup α
PartialOrder α
ExistsAddOfLE α
CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
Sub α
OrderedSub α
a b c : α
hab : AddLECancellable (a - b)
h₁ : b ≤ a
h : c < b
⊢ a - b < a - c
s : ℝ
hs : 0 < s
⊢ s.doublingGamma = Real.Gamma s
α : Type u_1
E : α → Type u_2
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
f : (i : α) → E i
hfq : Memℓp f 0
hpq : 0 ≤ ⊤
⊢ Memℓp f ⊤
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
G : Type u_6
k : Type u_7
R : Type u_8
inst✝¹ : Fintype ι
inst✝ : OrderedCommMonoid M
f : ι → M
hf : f ≤ 1
⊢ (∀ i ∈ Finset.univ, f i = 1) ↔ f = 1
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a : α
snd : β
⊢ (G.boxProdRight H a) snd = (a, snd)
α : Type u_1
f : Perm α
n : ℕ
x : α
⊢ f ((f ^ n) x) = (f ^ n) x ↔ f x = x
α : Type u_1
inst✝¹ : Lattice α
inst✝ : IsWeakUpperModularLattice α
a b : α
⊢ b ⊓ a ⋖ a → b ⊓ a ⋖ b → b ⋖ b ⊔ a
R M : Type u
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
I : Ideal R
F F' : I.Filtration M
inst✝² : IsNoetherianRing R
inst✝¹ : LocalRing R
inst✝ : Module.Finite R M
h : I ≠ ⊤
x : M
hx : x ∈ ⨅ i, I ^ i • ⊤
⊢ x ∈ ⊥
R : Type u_1
inst✝ : Semiring R
k✝ : ℕ
f✝ : R[X]
k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (k + 1)! • (↑((n + (k + 1)).choose (k + 1)) * f.coeff (n + (k + 1))) = k ! • (↑((n + 1 + k).choose (n + 1)) * f.coeff (n + 1 + k) * (↑n + 1))
α : Type v
G : Type u
AddGroup G
AddAction G α
j : ℤ
g : G
a : α
⊢ j • g +ᵥ a = a ↔ ↑(AddAction.period g a) ∣ j
x : ℝ
⊢ Ioo ⊥ ↑↑x = Ioo ⊥ ↑x
α : Type u_1
β : Type u_2
f : α → β
⊢ Function.Bijective f ↔ Set.BijOn f Set.univ Set.univ
α : Type u_1
BEq α
LawfulBEq α
x : α
l₁ l₂ : List α
⊢ x ∈ l₁ ∪ l₂ ↔ x ∈ l₁ ∨ x ∈ l₂
ι : Type u_1
κ : Type u_2
α : Type u_3
inst✝³ : OrderedCommGroup α
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalGroup α
inst✝ : OrderClosedTopology α
f g : ι → α
a₁ a₂ : α
i : ι
h : f ≤ g
hi : f i < g i
hf : HasProd f a₁
hg : HasProd g a₂
this : update f i 1 ≤ update g i 1
⊢ a₁ < a₂
C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₁, u₂} D
G : D ⥤ C
F : C ⥤ D
adj : F ⊣ G
A : adj.toMonad.Algebra
B : D
inst✝ : HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))
f : F.obj A.A ⟶ B
⊢ F.map A.a ≫ f = adj.counit.app (F.obj A.A) ≫ f ↔ G.map (F.map ((adj.homEquiv A.A B) f)) ≫ G.map (adj.counit.app B) = A.a ≫ (adj.homEquiv A.A B) f
ι : Type u_1
ι' : Type u_2
R : Type u_3
M : Type u_6
M' : Type u_7
Semiring R
AddCommMonoid M
Module R M
AddCommMonoid M'
Module R M'
b : Basis ι R M
b' : Basis ι' R M'
x : M × M'
i : ι'
⊢ ((b.prod b').repr x) (Sum.inr i) = (b'.repr x.2) i
α : Type u_1
β : Type u_2
inst✝ : DecidableEq α
l✝ : List α
x✝ : α
l : List α
x : α
h : l.formPerm x ≠ x
⊢ x ∈ l
G : Type u_1
Group G
H : Subgroup G
⊢ H.relindex ⊤ = H.index
α : Type u
β : Type v
Preorder α
Preorder β
f : α → β
a : α
hf : StrictMono f
ha : IsMax (f a)
⊢ IsMax a
a : ℝ
ha : a < -1
c : ℝ
hc : 0 < c
hd : ∀ x ∈ Ici c, HasDerivAt (fun t => t ^ (a + 1) / (a + 1)) (x ^ a) x
ht : Tendsto (fun t => t ^ (a + 1) / (a + 1)) atTop (𝓝 (0 / (a + 1)))
⊢ -c ^ (a + 1) / (a + 1) = 0 / (a + 1) - c ^ (a + 1) / (a + 1)
R : Type u
inst✝ : CommRing R
f a b : R
hb : b ∈ Submonoid.powers f
c d : R
hd : d ∈ Submonoid.powers f
h_eq : const R a b (PrimeSpectrum.basicOpen f) ⋯ = const R c d (PrimeSpectrum.basicOpen f) ⋯
I : Ideal R := { carrier := {r | r * (d * a) = r * (b * c)}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
⊢ ∀ x ∈ PrimeSpectrum.zeroLocus ↑I, f ∈ x.asIdeal
M : Type u_2
β : Type u_3
α : Type u_4
MeasurableSpace M
MeasurableSpace β
SMul M β
m : MeasurableSpace α
g : α → β
MeasurableSMul M β
μ : MeasureTheory.Measure α
hf : AEMeasurable g μ
c : M
⊢ AEMeasurable (c • g) μ
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
i j : n
⊢ ∑ σ : Perm n, ↑↑(sign σ) * ∏ i_1 : n, A.updateRow i (Pi.single j 1) (σ i_1) i_1 = ∑ σ : Perm n, ↑↑(sign σ) * ∏ i_1 : n, A.updateColumn j (Pi.single i 1) (σ i_1) i_1
G : Type u_1
TopologicalSpace G
AddGroup G
TopologicalAddGroup G
MeasurableSpace G
BorelSpace G
μ' μ : MeasureTheory.Measure G
μ.IsAddHaarMeasure
MeasureTheory.IsFiniteMeasureOnCompacts μ'
μ'.IsAddLeftInvariant
f : G → ℝ
hf : Continuous f
h'f : HasCompactSupport f
int_nonzero : ∫ (x : G), f x ∂μ ≠ 0
⊢ ↑(μ'.addHaarScalarFactor μ) = (∫ (x : G), f x ∂μ') / ∫ (x : G), f x ∂μ
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
U : Set E
f : ℂ → ℂ
g : E → ℂ
z₀ w : ℂ
ε r m : ℝ
h : DiffContOnCl ℂ f (ball z₀ r)
hr : 0 < r
hf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖
hz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀
v : ℂ
hv : v ∈ ball (f z₀) (ε / 2)
h1 : DiffContOnCl ℂ (fun z => f z - v) (ball z₀ r)
h2 : ContinuousOn (fun z => ‖f z - v‖) (closedBall z₀ r)
h3 : AnalyticOn ℂ f (ball z₀ r)
h4 : ∀ z ∈ sphere z₀ r, ε / 2 ≤ ‖f z - v‖
h5 : ‖f z₀ - v‖ < ε / 2
z : ℂ
hz1 : z ∈ ball z₀ r
hz2 : IsLocalMin (fun z => ‖f z - v‖) z
h6 : ∀ᶠ (y : ℂ) in 𝓝 z, DifferentiableAt ℂ (fun z => f z - v) y
key : ∀ᶠ (z_1 : ℂ) in 𝓝 z, f z_1 - v = f z - v
h7 : ∃ᶠ (w : ℂ) in 𝓝[≠] z, f w = f z
⊢ False
G₀ : Type u_3
GroupWithZero G₀
TopologicalSpace G₀
HasContinuousInv₀ G₀
ContinuousMul G₀
⊢ ContinuousOn (fun p => p.1 / p.2) {p | p.2 ≠ 0}
α : Type u_1
m : MeasurableSpace α
μ : MeasureTheory.Measure α
E : Type u_4
NormedAddCommGroup E
NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g : α → E
⊢ MeasureTheory.AEStronglyMeasurable g (μ.withDensity fun x => ↑(f x)) ↔
  MeasureTheory.AEStronglyMeasurable (fun x => ↑(f x) • g x) μ
α : Type u
β : Type v
γ : Type u_1
f g : Ultrafilter α
s t : Set α
p q : α → Prop
⊢ sᶜ ∈ f ↔ s ∉ f
V : Type u_1
G₁ G₂ : SimpleGraph V
Fintype ↑G₁.edgeSet
Fintype ↑G₂.edgeSet
⊢ G₁.edgeFinset ⊂ G₂.edgeFinset ↔ G₁ < G₂
R : Type u_1
A : Type u_2
B : Type u_3
C : Type u_4
CommSemiring R
Semiring A
Algebra R A
Semiring B
Algebra R B
Semiring C
Algebra R C
f : A →ₐ[R] B
g : A →ₐ[R] C
⊢ (AlgHom.snd R B C).comp (f.prod g) = g
R : ℝ
hR : 0 < R
z x : ℂ
f : ℂ → ℂ
hx : x ∈ Metric.ball z R
hf : ContinuousOn f (Metric.sphere z R)
⊢ ∃ B ε,
  0 < ε ∧
    Metric.ball x ε ⊆ Metric.ball z R ∧ ∀ (t : ℝ), ∀ y ∈ Metric.ball x ε, ‖Complex.circleTransformDeriv R z y f t‖ ≤ B
V : Type u_1
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
u : V
p : G.Walk u u
ht : p.IsEulerian
⊢ Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ = ∅ ∨ (Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ).card = 2
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : Fintype α
s t : Finset α
⊢ univ = ∅ ↔ IsEmpty α
a b c d m n k : ℕ
p q : ℕ → Prop
hb : 0 < b
⊢ a < b * c + b ↔ a ≤ b * c + (b - 1)
q r : ℚ
s : ℚ := (q.num * ↑r.den + r.num * ↑q.den) /. (↑q.den * ↑r.den)
hs : ↑q.den * ↑r.den ≠ 0
c : ℤ
c_mul_num : q.num * ↑r.den + r.num * ↑q.den = c * (↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).num
c_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).den
a✝ : ¬c = 0
h : q + r = s
⊢ ↑(↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).den * s.num = (↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).num * ↑s.den
V : Type u
G : SimpleGraph V
v w : V
hvw : G.Adj v w
⊢ (G.subgraphOfAdj hvw).edgeSet = {s(v, w)}
G : Type u_7
H : Type u_8
F : Type u_9
FunLike F G H
AddGroup G
SubtractionMonoid H
AddMonoidHomClass F G H
f : F
a : G
⊢ f (-a) = -f a
R : Type u_4
S : Type u_5
CommRing R
LocalRing R
Field S
f : R →+* S
IsLocalRingHom f
⊢ (LocalRing.ResidueField.lift f).comp (LocalRing.residue R) = f
R : Type u_1
S : Type u_2
L : Type u_4
EuclideanDomain R
CommRing S
IsDomain S
Field L
algRL : Algebra R L
Algebra R S
Algebra S L
ist : IsScalarTower R S L
iic : IsIntegralClosure S R L
abv : AbsoluteValue R ℤ
ι : Type u_5
DecidableEq ι
Fintype ι
bS : Basis ι R S
adm : abv.IsAdmissible
Infinite R
DecidableEq R
IsDedekindDomain S
Algebra.IsAlgebraic R L
I : ↥(Ideal S)⁰
⊢ ∃ J, ClassGroup.mk0 I = ClassGroup.mk0 J ∧ (algebraMap R S) (∏ m ∈ ClassGroup.finsetApprox bS adm, m) ∈ ↑J
α : Type u_1
M✝ : Matroid α
I✝ B✝ X✝ : Set α
M : Matroid α
X I' : Set α
hI'E : I' ⊆ M.E
B : Set α
hB : M.Base B
hI'B : Disjoint I' B
hI'X : I' ⊆ X
I : Set α
hI : M.Basis I (M.E \ X)
B' : Set α
hB' : M.Base B'
hIB' : I ⊆ B'
hB'IB : B' ⊆ I ∪ B
J : Set α
hJE : J ⊆ M.E
B'' : Set α
hB'' : M.Base B''
hdj : Disjoint J B''
a✝ : I' ⊆ J
hJX : J ⊆ X
hssJ : X ∩ M.E ⊆ B' ∪ J
hI' : B'' ∩ X ∪ B' \ X ⊆ B'
B₁ : Set α
hB₁ : M.Base B₁
hI'B₁ : B'' ∩ X ∪ B' \ X ⊆ B₁
hB₁I : B₁ ⊆ B'' ∪ B' \ X
⊢ J ⊆ X \ B'
ι : Type u_1
α : Type u_2
inst✝³ : MeasurableSpace α
f✝ : α → α
s : Set α
μ✝ : Measure α
inst✝² : TopologicalSpace α
inst✝¹ : SecondCountableTopology α
inst✝ : OpensMeasurableSpace α
f : α → α
μ : Measure α
h : Conservative f μ
this : ∀ s ∈ countableBasis α, ∀ᵐ (x : α) ∂μ, x ∈ s → ∃ᶠ (n : ℕ) in atTop, f^[n] x ∈ s
⊢ ∀ᵐ (x : α) ∂μ, ∀ s ∈ 𝓝 x, ∃ᶠ (n : ℕ) in atTop, f^[n] x ∈ s
a✝ b✝ : Prop
α : Sort u_1
a b : α
p : Prop
⊢ p ∧ a = b ↔ p ∧ b = a
R : Type u_1
CommSemiring R
A : Type u_2
Semiring A
TopologicalSpace A
B : Type u_3
Semiring B
TopologicalSpace B
Algebra R A
Algebra R B
f : A →A[R] B
f' : A → B
h : f' = ⇑f
⊢ f.copy f' h = f
C : Type u_1
inst✝⁴ : Category.{u_2, u_1} C
inst✝³ : Preadditive C
K K' : ChainComplex C ℕ
f : K ⟶ K'
Δ✝ Δ'✝ Δ''✝ : SimplexCategory
inst✝² : HasFiniteCoproducts C
Δ Δ' : SimplexCategoryᵒᵖ
A : Splitting.IndexSet Δ
θ : Δ ⟶ Δ'
Δ'' : SimplexCategory
e : unop Δ' ⟶ Δ''
i : Δ'' ⟶ unop A.fst
inst✝¹ : Epi e
inst✝ : Mono i
fac : e ≫ i = θ.unop ≫ A.e
⊢ (splitting K).ι (unop A.fst).len ≫ (obj K).map (A.e.op ≫ θ) = Termwise.mapMono K i ≫ (splitting K).ι Δ''.len ≫ map K (Splitting.IndexSet.mk e).e.op
K : Type u_1
inst✝³ : Field K
inst✝² : NumberField K
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
p : IntermediateField ℚ A → Prop
S : Set { F // p F }
T : Set A
hT : Finite ↑T
h : ∀ F ∈ S, ∃ x ∈ T, ↑F = ℚ⟮x⟯
x✝¹ x✝ : ↑S
h_eq : (fun x => match x with | ⟨F, hF⟩ => ⟨⋯.choose, ⋯⟩) x✝¹ = (fun x => match x with | ⟨F, hF⟩ => ⟨⋯.choose, ⋯⟩) x✝
⊢ ↑↑x✝¹ = ℚ⟮⋯.choose⟯
α : Type u
a b : α
s : Set α
ha : a ∉ s
⊢ insert a s = insert b s ↔ a = b
C : Type u
inst✝⁴ : Category.{v, u} C
A : C ⥤ Type w
I : Type u₁
inst✝³ : Category.{v₁, u₁} I
inst✝² : Small.{w, u₁} I
F : I ⥤ A.Elements
inst✝¹ : HasLimitsOfShape I C
inst✝ : PreservesLimitsOfShape I A
i i' : I
f : i ⟶ i'
⊢ ↑(((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElement F⟩).map f ≫ (fun i => ⟨limit.π (F ⋙ π A) i, ⋯⟩) i') = ↑((fun i => ⟨limit.π (F ⋙ π A) i, ⋯⟩) i ≫ F.map f)
𝕜 : Type u_1
NontriviallyNormedField 𝕜
E : Type u_2
NormedAddCommGroup E
NormedSpace 𝕜 E
x : E
𝔸 : Type u_6
NormedRing 𝔸
NormedAlgebra 𝕜 𝔸
a : E → 𝔸
ha : DifferentiableAt 𝕜 a x
b : 𝔸
⊢ DifferentiableAt 𝕜 (fun y => a y * b) x
𝕜 : Type u_1
E : Type u_2
NontriviallyNormedField 𝕜
SeminormedAddCommGroup E
NormedSpace 𝕜 E
𝕜' : Type u_3
NormedField 𝕜'
NormedAlgebra 𝕜 𝕜'
NormedSpace 𝕜' E
IsScalarTower 𝕜 𝕜' E
⊢ ‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ ≤ 1
R : Type u
⊢ Function.Injective (@NonAssocRing.toNonAssocSemiring R)
A : Type u₁
B : Type u₂
C : Type u₃
D : Type u₄
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} B
inst✝¹ : Category.{v₃, u₃} C
inst✝ : Category.{v₄, u₄} D
G : A ⥤ C
H : B ⥤ D
L₁ : A ⥤ B
R₁ : B ⥤ A
L₂ : C ⥤ D
R₂ : D ⥤ C
L₃ : C ⥤ D
R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : G ⋙ L₂ ⟶ L₁ ⋙ H
β : L₃ ⟶ L₂
b : B
vcomp : (mateEquiv adj₁ adj₃) (whiskerLeft G (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv) ≫ whiskerRight α (𝟭 D)) = whiskerRight ((mateEquiv adj₁ adj₂) α) (𝟭 C) ≫ whiskerLeft H ((mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv))
vcompb : adj₃.unit.app (G.obj (R₁.obj b)) ≫ R₃.map (β.app (G.obj (R₁.obj b))) ≫ R₃.map (α.app (R₁.obj b)) ≫ R₃.map (H.map (adj₁.counit.app b)) = adj₂.unit.app (G.obj (R₁.obj b)) ≫ R₂.map (α.app (R₁.obj b)) ≫ R₂.map (H.map (adj₁.counit.app b)) ≫ adj₃.unit.app (R₂.obj (H.obj b)) ≫ R₃.map (β.app (R₂.obj (H.obj b))) ≫ R₃.map (adj₂.counit.app (H.obj b))
⊢ ({ toFun := fun α => whiskerLeft (R₁ ⋙ G) adj₃.unit ≫ whiskerRight (whiskerLeft R₁ α) R₃ ≫ whiskerRight adj₁.counit (H ⋙ R₃), invFun := fun β => whiskerRight adj₁.unit (G ⋙ L₃) ≫ whiskerRight (whiskerLeft L₁ β) L₃ ≫ whiskerLeft (L₁ ⋙ H) adj₃.counit, left_inv := ⋯, right_inv := ⋯ } (whiskerLeft G β ≫ α)).app b = ({ toFun := fun α => whiskerLeft (R₁ ⋙ G) adj₂.unit ≫ whiskerRight (whiskerLeft R₁ α) R₂ ≫ whiskerRight adj₁.counit (H ⋙ R₂), invFun := fun β => whiskerRight adj₁.unit (G ⋙ L₂) ≫ whiskerRight (whiskerLeft L₁ β) L₂ ≫ whiskerLeft (L₁ ⋙ H) adj₂.counit, left_inv := ⋯, right_inv := ⋯ } α ≫ whiskerLeft H ((Trans.trans (Trans.trans (L₃.leftUnitor.homCongr L₂.rightUnitor).symm { toFun := fun α => whiskerLeft (R₂ ⋙ 𝟭 C) adj₃.unit ≫ whiskerRight (whiskerLeft R₂ α) R₃ ≫ whiskerRight adj₂.counit (𝟭 D ⋙ R₃), invFun := fun β => whiskerRight adj₂.unit (𝟭 C ⋙ L₃) ≫ whiskerRight (whiskerLeft L₂ β) L₃ ≫ whiskerLeft (L₂ ⋙ 𝟭 D) adj₃.counit, left_inv := ⋯, right_inv := ⋯ }) (R₂.rightUnitor.homCongr R₃.leftUnitor)) β)).app b
R : Type u
M : Type v
Ring R
AddCommGroup M
Module R M
StrongRankCondition R
ι : Type u_1
v : ι → M
i : LinearIndependent R v
w : Set M
Fintype ↑w
s : Submodule.span R w = ⊤
⊢ #ι ≤ ↑(Fintype.card ↑w)
ι : Sort u_1
V : Type u
G✝ : SimpleGraph V
a b c u v w : V
e✝ : Sym2 V
G₁ G₂ G : SimpleGraph V
s : Set (Sym2 V)
e : Sym2 V
⊢ e ∈ G.edgeSet \ (s \ {e | e.IsDiag}) ↔ e ∈ G.edgeSet \ s
ι : Type u
β : ι → Type v
(i : ι) → Zero (β i)
Fintype ι
⊢ ∀ (a : Π₀ (i : ι), β i) (a_1 : ι), DFinsupp.equivFunOnFintype a a_1 = a a_1
α : Type u
s : Set α
hs : s.Subsingleton
⊢ s.Countable
m : ℕ
inst✝¹ : NeZero (m + 1)
inst✝ : Fintype (ZMod (m + 1))ˣ
⊢ Fintype.card { x // x.val.Coprime (m + 1) } = φ (m + 1)
x y : ℤ[i]
⊢ (toComplex { re := round ↑(↑(x * star y).re / ↑(norm y)), im := round (↑(x * star y).im / ↑(norm y)) }).re = ↑(round (toComplex x / toComplex y).re)
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
h_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J
a₁ : R := choose ⋯
J₁ : Ideal R := choose ⋯
h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁
h_a₁_ne_zero : a₁ ≠ 0
⊢ count K v I = ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)
k : Type u_1
V : Type u_2
P : Type u_3
DivisionRing k
AddCommGroup V
Module k V
AffineSpace V P
s : Set P
FiniteDimensional k ↥(vectorSpan k s)
⊢ Coplanar k s ↔ FiniteDimensional.finrank k ↥(vectorSpan k s) ≤ 2
ι : Type u_1
α : Type u_2
β : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
G : Type u_7
inst✝² : Monoid M
inst✝¹ : Monoid N
inst✝ : Monoid P
l l₁ l₂ : List M
a✝ x : M
xs : List M
i : ℕ
a : M
⊢ ((x :: xs).set (i + 1) a).prod = ((take (i + 1) (x :: xs)).prod * if i + 1 < (x :: xs).length then a else 1) * (drop (i + 1 + 1) (x :: xs)).prod
𝕜 : Type u_1
E : Type u_2
M : Type u_3
H : Type u_4
NontriviallyNormedField 𝕜
NormedAddCommGroup E
NormedSpace 𝕜 E
TopologicalSpace H
TopologicalSpace M
I : ModelWithCorners 𝕜 E H
s : Set M
ChartedSpace H M
x : M
⊢ (extChartAt I x).source ∈ 𝓝[s] x
q : ℍ
⊢ exp ℝ q.re ^ 2 * (Real.cos ‖q.im‖ ^ 2 + Real.sin ‖q.im‖ ^ 2) = exp ℝ q.re ^ 2
p : ℕ → Prop
hf : (setOf p).Infinite
n : ℕ
⊢ IsLeast {i | p i ∧ ∀ k < n, Nat.nth p k < i} (Nat.nth p n)
C : Type u
A : Type u_1
inst✝² : Category.{v, u} C
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
a : A
X : C
⊢ (shiftFunctorAdd C 0 a).inv.app X = (shiftFunctor C a).map ((shiftFunctorZero C A).hom.app X) ≫ eqToHom ⋯
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
⊢ ∀ (i : Set X), IsClosed i ∧ IsCompact i → ∀ (j : Set X), IsClosed j ∧ IsCompact j → ∃ k, (IsClosed k ∧ IsCompact k) ∧ kᶜ ⊆ iᶜ ∧ kᶜ ⊆ jᶜ
⊢ Ordinal.Principal (fun x x_1 => x * x_1) 1
α : Type u_1
cmp : α → α → Ordering
v : α
path : Batteries.RBNode.Path α
hp : Batteries.RBNode.Path.Ordered cmp path
vp : Batteries.RBNode.Path.RootOrdered cmp path v
⊢ Batteries.RBNode.Ordered cmp (path.insertNew v)
M : Type u_3
TopologicalSpace M
Monoid M
ContinuousMul M
a b : M
ha : a * b = 1
⊢ Filter.Tendsto (fun x => x * a) (Filter.cocompact M) (Filter.cocompact M)
J : Type w
C : Type u
CategoryTheory.Category.{v, u} C
X Y : C
f : J → (X ⟶ Y)
Nonempty J
P : C
ι : P ⟶ X
w : ∀ (j₁ j₂ : J), CategoryTheory.CategoryStruct.comp ι (f j₁) = CategoryTheory.CategoryStruct.comp ι (f j₂)
⊢ ∀ (X_1 : CategoryTheory.Limits.WalkingParallelFamily J),
  (CategoryTheory.Limits.Trident.ofι ι w).π.app X_1 =
    CategoryTheory.Limits.WalkingParallelFamily.casesOn X_1 ι
      (CategoryTheory.CategoryStruct.comp ι (f (Classical.arbitrary J)))
ι : Type u_1
α : ι → Type u_2
s : Finset ι
S T : Set ((i : { x // x ∈ s }) → α ↑i)
⊢ MeasureTheory.cylinder s S \ MeasureTheory.cylinder s T = MeasureTheory.cylinder s (S \ T)
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
r✝ : α✝ → α✝ → Prop
s✝ : β → β → Prop
t : γ → γ → Prop
u : δ → δ → Prop
α : Type u
r : α → α → Prop
a b : α
s : α → α → Prop
h₂ : HEq r s
⊢ s ((Equiv.cast ⋯) a) ((Equiv.cast ⋯) b) ↔ r a b
R : Type u
CommSemiring R
s : Set R
⊢ IsClosed (PrimeSpectrum.zeroLocus s)
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
hn : 2 ≤ n + 1
p : FormalMultilinearSeries 𝕜 E F
i : E ≃L[𝕜] F
r a C : ℝ
hr : 0 ≤ r
ha : 0 ≤ a
hC : 0 ≤ C
hp : ∀ (n : ℕ), ‖p n‖ ≤ C * r ^ n
I : ℝ := ‖↑i.symm‖
j : ℕ
a✝¹ : j ∈ Ico 2 (n + 1)
i✝ : Composition j
a✝ : i✝ ∈ {c | 1 < c.length}.toFinset
⊢ ‖p i✝.length‖ * ∏ i_1 : Fin i✝.length, ‖p.rightInv i (i✝.blocksFun i_1)‖ ≤ C * r ^ i✝.length * ∏ j_1 : Fin i✝.length, ‖p.rightInv i (i✝.blocksFun j_1)‖
R : Type u
CommSemiring R
I : Ideal R
⊢ I * Submodule.annihilator I = ⊥
F : Type u → Type v
Applicative F
G : Type u → Type w
Applicative G
η η' : ApplicativeTransformation F G
h : (fun {α} => η.app α) = fun {α} => η'.app α
⊢ η = η'
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : Semigroup α
x y z : α
⊢ ((fun x_1 => x_1 * x) ∘ fun x => x * y) z = z * (y * x)
α β : Type u_3
r : α → α → Prop
s : β → β → Prop
IsWellOrder α r
IsWellOrder β s
⊢ Ordinal.type r = Ordinal.type s ↔ Nonempty (r ≃r s)
ι : Sort u_1
α : Type u
β : Type v
inst✝ : PseudoEMetricSpace α
δ✝ ε : ℝ
s✝ t✝ : Set α
x : α
δ : ℝ
s t : Set α
⊢ cthickening δ (s ∪ t) = cthickening δ s ∪ cthickening δ t