ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/NumberTheory/LSeries/MellinEqDirichlet.lean
ι : Type u_1
inst✝ : Countable ι
a : ι → ℂ
p : ι → ℝ
F : ℝ → ℂ
s : ℂ
hp : ∀ (i : ι), a i = 0 ∨ 0 < p i
hs : 0 < s.re
hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(rexp (-p i * t))) (F t)
h_sum : Summable fun i => ‖a i‖ / p i ^ s.re

⊢ Summable fun i => ∫ (a_1 : ℝ) in Ioi 0, ‖(fun i t => ↑t ^ (s - 1) * (a i * ↑(rexp (-p i * t)))) i a_1‖ ∂volume
-- Mathlib/Order/WithBot.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : LT α
a✝ b : α
a : WithTop α

⊢ ¬⊤ < a
-- Mathlib/Order/WithBot.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : LT α
a✝ b : α
a : WithTop α

⊢ ¬WithTop.toDual a < WithTop.toDual ⊤
-- Mathlib/NumberTheory/ModularForms/SlashActions.lean
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ

⊢ ((f * g) ∣[k1 + k2] A) x = ((↑↑A).det • f ∣[k1] A * g ∣[k2] A) x
-- Mathlib/NumberTheory/ModularForms/SlashActions.lean
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ

⊢ f (A • x) * g (A • x) * ↑(↑↑A).det ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) = ↑(↑↑A).det * (f (A • x) * ↑(↑↑A).det ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * ↑(↑↑A).det ^ (k2 - 1) * denom A x ^ (-k2))
-- Mathlib/NumberTheory/ModularForms/SlashActions.lean
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(↑↑A).det

⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) = d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2))
-- Mathlib/NumberTheory/ModularForms/SlashActions.lean
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(↑↑A).det
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)

⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) = d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2))
-- Mathlib/NumberTheory/ModularForms/SlashActions.lean
Γ : Subgroup SL(2, ℤ)
k : ℤ
f✝ : ℍ → ℂ
k1 k2 : ℤ
A : ↥GL(2, ℝ)⁺
f g : ℍ → ℂ
x : ℍ
d : ℂ := ↑(↑↑A).det
h1 : d ^ (k1 + k2 - 1) = d * d ^ (k1 - 1) * d ^ (k2 - 1)
h22 : denom A x ^ (-(k1 + k2)) = denom A x ^ (-k1) * denom A x ^ (-k2)

⊢ f (A • x) * g (A • x) * d ^ (k1 + k2 - 1) * denom A x ^ (-(k1 + k2)) = d * (f (A • x) * d ^ (k1 - 1) * denom A x ^ (-k1)) * (g (A • x) * d ^ (k2 - 1) * denom A x ^ (-k2))
-- Mathlib/AlgebraicGeometry/OpenImmersion.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f

⊢ f ''ᵁ ⊤ = f.opensRange
-- Mathlib/AlgebraicGeometry/OpenImmersion.lean
C : Type u
inst✝ : Category.{v, u} C
X Y : Scheme
f : X.Hom Y
H : IsOpenImmersion f

⊢ ↑(f ''ᵁ ⊤) = ↑f.opensRange
-- Mathlib/GroupTheory/Coxeter/Basic.lean
B : Type u_1
B' : Type u_2
e : B ≃ B'
W : Type u_3
H : Type u_4
inst✝¹ : Group W
inst✝ : Group H
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
this : FreeGroup.of i * FreeGroup.of i ∈ M.relationsSet

⊢ cs.simple i * cs.simple i = 1
-- Mathlib/GroupTheory/Coxeter/Basic.lean
B : Type u_1
B' : Type u_2
e : B ≃ B'
W : Type u_3
H : Type u_4
inst✝¹ : Group W
inst✝ : Group H
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
this✝ : FreeGroup.of i * FreeGroup.of i ∈ M.relationsSet
this : ↑(FreeGroup.of i * FreeGroup.of i) = 1

⊢ cs.simple i * cs.simple i = 1
-- Mathlib/GroupTheory/Coxeter/Basic.lean
B : Type u_1
B' : Type u_2
e : B ≃ B'
W : Type u_3
H : Type u_4
inst✝¹ : Group W
inst✝ : Group H
M : CoxeterMatrix B
cs : CoxeterSystem M W
i : B
this✝ : FreeGroup.of i * FreeGroup.of i ∈ M.relationsSet
this : ↑(FreeGroup.of i * FreeGroup.of i) = 1

⊢ cs.mulEquiv.symm (PresentedGroup.of i) * cs.mulEquiv.symm (PresentedGroup.of i) = 1
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Semiring R
p q r : R[X]
h : a * p.leadingCoeff ≠ 0

⊢ (C a * p).natDegree = p.natDegree
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Semiring R
p q r : R[X]
h : a * p.leadingCoeff ≠ 0

⊢ p.natDegree ∈ (C a * p).support
-- Mathlib/Algebra/Polynomial/Degree/Lemmas.lean
R : Type u
S : Type v
ι : Type w
a b : R
m n : ℕ
inst✝ : Semiring R
p q r : R[X]
h : a * p.leadingCoeff ≠ 0

⊢ (C a * p).coeff p.natDegree ≠ 0
-- Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : CommMonoid α
inst✝⁴ : TopologicalSpace α
f✝ g✝ : β → α
a✝ b : α
s : Finset β
inst✝³ : CommMonoid γ
inst✝² : TopologicalSpace γ
G : Type u_5
inst✝¹ : FunLike G α γ
inst✝ : MonoidHomClass G α γ
g : G
hg : Inducing ⇑g
f : β → α
a : α

⊢ HasProd (⇑g ∘ f) (g a) ↔ HasProd f a
-- Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝⁵ : CommMonoid α
inst✝⁴ : TopologicalSpace α
f✝ g✝ : β → α
a✝ b : α
s : Finset β
inst✝³ : CommMonoid γ
inst✝² : TopologicalSpace γ
G : Type u_5
inst✝¹ : FunLike G α γ
inst✝ : MonoidHomClass G α γ
g : G
hg : Inducing ⇑g
f : β → α
a : α

⊢ Tendsto (fun s => g (∏ x ∈ s, f x)) atTop (𝓝 (g a)) ↔ Tendsto (fun s => ∏ x ∈ s, f x) atTop (𝓝 a)
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ

⊢ (W.ΨSq n).natDegree ≤ n.natAbs ^ 2 - 1
-- Mathlib/RingTheory/FiniteType.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Semiring A
inst✝⁸ : Semiring B
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_1
inst✝ : Finite ι

⊢ FiniteType R (FreeAlgebra R ι)
-- Mathlib/RingTheory/FiniteType.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
M : Type uM
N : Type uN
inst✝¹¹ : CommSemiring R
inst✝¹⁰ : CommSemiring S
inst✝⁹ : Semiring A
inst✝⁸ : Semiring B
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : AddCommMonoid N
inst✝¹ : Module R N
ι : Type u_1
inst✝ : Finite ι
val✝ : Fintype ι

⊢ FiniteType R (FreeAlgebra R ι)
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f✝ g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
f : X → ℝ
hf : 0 ≤ᶠ[ae (μ.restrict s)] f
hfi : IntegrableOn f s μ

⊢ 0 < ∫ (x : X) in s, f x ∂μ ↔ 0 < μ (support f ∩ s)
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f✝ g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
f : X → ℝ
hf : 0 ≤ᶠ[ae (μ.restrict s)] f
hfi : IntegrableOn f s μ

⊢ NullMeasurableSet (support f) (μ.restrict s)
-- Mathlib/MeasureTheory/Integral/SetIntegral.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
inst✝² : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f✝ g : X → E
s t : Set X
μ ν : Measure X
l l' : Filter X
f : X → ℝ
hf : 0 ≤ᶠ[ae (μ.restrict s)] f
hfi : IntegrableOn f s μ

⊢ NullMeasurableSet (f ⁻¹' {0}ᶜ) (μ.restrict s)
-- Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Category.{?u.64705, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
j : ℤ
A : C
f : (mappingCone φ).X j ⟶ A
i : ℤ
hij : j + 1 = i

⊢ f = (↑(fst φ)).v j i hij ≫ (inl φ).v i j ⋯ ≫ f + (snd φ).v j j ⋯ ≫ (inr φ).f j ≫ f
-- Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
a : UnitAddCircle
s : ℂ
hs' : s ≠ 1 ∨ a ≠ 0

⊢ DifferentiableAt ℂ (cosZeta a) s
-- Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
a : UnitAddCircle
s : ℂ
hs'✝ : s ≠ 1 ∨ a ≠ 0
hs' : s ≠ 1

⊢ DifferentiableAt ℂ (cosZeta a) s
-- Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean
a : UnitAddCircle
hs' : 1 ≠ 1 ∨ a ≠ 0

⊢ DifferentiableAt ℂ (cosZeta a) 1
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h✝ : a ∈ s
n : ℕ
h : some a ∈ s.get? n

⊢ ∃ s', some (a, s') ∈ (s.drop n).destruct
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h✝ : a ∈ s
n : ℕ
h : some a ∈ s.get? n
o : Option (α × WSeq α)
om : o ∈ (s.drop n).destruct

⊢ ∃ s', some (a, s') ∈ (s.drop n).destruct
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s : WSeq α
a : α
h✝ : a ∈ s
n : ℕ
h : some a ∈ s.get? n
o : Option (α × WSeq α)
om : o ∈ (s.drop n).destruct
this : Prod.fst <$> o = some a

⊢ ∃ s', some (a, s') ∈ (s.drop n).destruct
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : TopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f g : R → R
a✝ : A
ha✝ : autoParam (p a✝) _auto✝
hf : autoParam (ContinuousOn f (spectrum R a✝)) _auto✝
hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝
r : R
a : A
ha : autoParam (p a) _auto✝

⊢ cfc (fun a => eval a (C r)) a = (algebraMap R A) r
-- Mathlib/Algebra/Group/Nat.lean
m n : ℕ

⊢ Even (n + 1) ↔ ¬Even n
-- Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
R✝ : Type u_1
inst✝³ : CommRing R✝
inst✝² : IsDomain R✝
R : Type u_2
inst✝¹ : CommRing R
inst✝ : IsDomain R

⊢ cyclotomic' 0 R = 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

⊢ 𝓤 α = comap (fun x => x.1 / x.2) (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

⊢ comap ((fun x => x.2 / x.1) ∘ Prod.swap) (𝓝 1) = comap (fun x => x.1 / x.2) (𝓝 1)
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s : WSeq α

⊢ s.length = Computation.map List.length s.toList
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s : WSeq α

⊢ Computation.IsBisimulation fun c1 c2 => ∃ l s, c1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ c2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s✝ : WSeq α
s1 s2 : Computation ℕ
l : List α
s : WSeq α
h : s1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ s2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))

⊢ Computation.BisimO (fun c1 c2 => ∃ l s, c1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ c2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))) s1.destruct s2.destruct
-- Mathlib/Data/Seq/WSeq.lean
α : Type u
β : Type v
γ : Type w
s✝ : WSeq α
s1 s2 : Computation ℕ
l : List α
s : WSeq α
h : s1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ s2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))

⊢ Computation.BisimO (fun c1 c2 => ∃ l s, c1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ c2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))) (Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s)).destruct (Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))).destruct
-- 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
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f

⊢ ∫⁻ (a : α), f a ∂count = ∑' (a : α), f a
-- 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
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f

⊢ ∑' (i : α), ∫⁻ (a : α), f a ∂dirac i = ∑' (a : α), f a
-- 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
inst✝ : MeasurableSpace α
f : α → ℝ≥0∞
hf : Measurable f

⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a
-- Mathlib/CategoryTheory/Sites/Preserves.lean
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)
hF' : Nonempty (PreservesLimit (Discrete.functor fun x => op (X x)) F)
x✝ : PreservesLimit (Discrete.functor fun x => op (X x)) F := hF'.some

⊢ IsSheafFor F (ofArrows X c.inj)
-- 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)

⊢ o.oangle (x + y) y = ↑(Real.arctan (‖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)

⊢ (-o).oangle y (x + y) = ↑(Real.arctan (‖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)

⊢ (-o).oangle y (y + x) = ↑(Real.arctan (‖x‖ / ‖y‖))
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : Num

⊢ m.cmp n = Ordering.eq ↔ m = n
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : Num
this : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)

⊢ m.cmp n = Ordering.eq ↔ m = n
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : Num

⊢ Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m) → (m.cmp n = Ordering.eq ↔ m = n)
-- Mathlib/Data/Fin/Basic.lean
n m : ℕ
j : Fin n
i : Fin (n + 1)
hi : i ≠ last n

⊢ j < i.castPred hi ↔ j.castSucc < i
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
R : Type u_1
A : Type u_2
inst✝³ : CommSemiring R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
n : ℕ
hn : 0 < n

⊢ ↑(basicOpen 𝒜 (f ^ n)) = ↑(basicOpen 𝒜 f)
-- Mathlib/CategoryTheory/Limits/Cones.lean
J : Type u₁
inst✝³ : Category.{v₁, u₁} J
K : Type u₂
inst✝² : Category.{v₂, u₂} K
C : Type u₃
inst✝¹ : Category.{v₃, u₃} C
D : Type u₄
inst✝ : Category.{v₄, u₄} D
F : J ⥤ C
e : K ≌ J
s : Cone (e.functor ⋙ F)
k : K

⊢ (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app k = (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫ ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app k
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing T
inst✝¹² : Algebra R S
inst✝¹¹ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
ι κ : Type w
inst✝⁷ : Fintype ι
F : Type u_6
inst✝⁶ : Field F
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra L F
inst✝³ : Algebra R F
inst✝² : IsScalarTower R L F
pb✝ : PowerBasis K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
e : trace K L = 0
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L

⊢ False
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing T
inst✝¹² : Algebra R S
inst✝¹¹ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
ι κ : Type w
inst✝⁷ : Fintype ι
F : Type u_6
inst✝⁶ : Field F
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra L F
inst✝³ : Algebra R F
inst✝² : IsScalarTower R L F
pb✝ : PowerBasis K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
e : trace K L = 0
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L

⊢ (traceMatrix K ⇑pb.basis).det = 0
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing T
inst✝¹² : Algebra R S
inst✝¹¹ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
ι κ : Type w
inst✝⁷ : Fintype ι
F : Type u_6
inst✝⁶ : Field F
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra L F
inst✝³ : Algebra R F
inst✝² : IsScalarTower R L F
pb✝ : PowerBasis K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
e : trace K L = 0
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L

⊢ Nonempty (Fin pb.dim)
-- Mathlib/RingTheory/Trace/Basic.lean
R : Type u_1
S : Type u_2
T : Type u_3
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : CommRing T
inst✝¹² : Algebra R S
inst✝¹¹ : Algebra R T
K : Type u_4
L : Type u_5
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
ι κ : Type w
inst✝⁷ : Fintype ι
F : Type u_6
inst✝⁶ : Field F
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra L F
inst✝³ : Algebra R F
inst✝² : IsScalarTower R L F
pb✝ : PowerBasis K L
inst✝¹ : FiniteDimensional K L
inst✝ : Algebra.IsSeparable K L
e : trace K L = 0
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L

⊢ 0 < finrank K L
-- Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean

⊢ Integrable (fun x => (1 + x ^ 2)⁻¹) volume
-- Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean

⊢ Integrable (fun x => (1 + ‖x‖ ^ 2) ^ (-2 / 2)) volume
-- Mathlib/Data/Matroid/Map.lean
α : Type u_1
β : Type u_2
f : α → β
E I s : Set α
M : Matroid α
N : Matroid β
hf : InjOn f M.E
I₀ : Set α
hI₀ : M.Indep I₀

⊢ ∃ I₀_1, M.Indep I₀_1 ∧ BijOn f I₀_1 (f '' I₀)
-- Mathlib/Data/Finset/Card.lean
α : Type u_1
β : Type u_2
R : Type u_3
s t✝ u : Finset α
f✝ : α → β
n : ℕ
t : Finset β
f : (a : α) → a ∈ s → β
hf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t
hinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂
hst : t.card ≤ s.card

⊢ ∀ b ∈ t, ∃ a, ∃ (ha : a ∈ s), b = f a ha
-- Mathlib/Analysis/Convex/Deriv.lean
S : Set ℝ
f : ℝ → ℝ
x y f' : ℝ
hfc : StrictConcaveOn ℝ S f
hx : x ∈ S
hy : y ∈ S
hxy : x < y
hfd : HasDerivWithinAt f f' S x

⊢ slope f x y < f'
-- Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n : ℤ
γ : Cocycle K L n
a n' : ℤ
hn' : n' + a = n

⊢ δ n' (n' + 1) ((↑γ).rightShift a n' hn') = 0
-- Mathlib/Analysis/Calculus/Deriv/Mul.lean
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
E : Type w
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
G : Type u_1
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
𝕜' : Type u_2
𝔸 : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : NormedAlgebra 𝕜 𝔸
c d : 𝕜 → 𝔸
c' d' : 𝔸
u v : 𝕜 → 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x

⊢ HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x
-- Mathlib/Analysis/Calculus/Deriv/Mul.lean
𝕜 : Type u
inst✝¹⁰ : NontriviallyNormedField 𝕜
F : Type v
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
E : Type w
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
G : Type u_1
inst✝⁵ : NormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
f f₀ f₁ g : 𝕜 → F
f' f₀' f₁' g' : F
x : 𝕜
s t : Set 𝕜
L L₁ L₂ : Filter 𝕜
𝕜' : Type u_2
𝔸 : Type u_3
inst✝³ : NormedField 𝕜'
inst✝² : NormedRing 𝔸
inst✝¹ : NormedAlgebra 𝕜 𝕜'
inst✝ : NormedAlgebra 𝕜 𝔸
c d : 𝕜 → 𝔸
c' d' : 𝔸
u v : 𝕜 → 𝕜'
hc : HasStrictDerivAt c c' x
hd : HasStrictDerivAt d d' x
this : HasStrictDerivAt (fun y => c y * d y) ((c x • smulRight 1 d' + (smulRight 1 c').smulRight (d x)) 1) x

⊢ HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
B I J X Y : Set α
e : α

⊢ M.Basis B M.E ↔ M.Base B
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
B I J X Y : Set α
e : α

⊢ M.Indep B → ((B ⊆ M.E ∧ ∀ (J : Set α), M.Indep J → B ⊆ J → J ⊆ M.E → B = J) ∧ M.E ⊆ M.E ↔ ∀ (I : Set α), M.Indep I → B ⊆ I → B = I)
-- Mathlib/Data/Matroid/Basic.lean
α : Type u_1
M : Matroid α
B I J X Y : Set α
e : α

⊢ M.Indep B → ((B ⊆ M.E ∧ ∀ (J : Set α), M.Indep J → B ⊆ J → J ⊆ M.E → B = J) ↔ ∀ (I : Set α), M.Indep I → B ⊆ I → B = I)
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedSemiring α
inst✝ : FloorSemiring α
a✝ : α
n : ℕ
a b : α
ha : 0 ≤ a
x✝ : ℕ

⊢ x✝ ∈ Nat.cast ⁻¹' Ioo a b ↔ x✝ ∈ Ioo ⌊a⌋₊ ⌈b⌉₊
-- Mathlib/Data/Nat/Pairing.lean
m n : ℕ

⊢ pair m n < (max m n + 1) ^ 2
-- Mathlib/Data/Nat/Pairing.lean
m n : ℕ

⊢ (if m < n then n * n + m else m * m + (m + n)) < max m n * max m n + (max m n + (max m n + 1))
-- Mathlib/CategoryTheory/Sites/CoverLifting.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
G : C ⥤ D
A : Type w
inst✝⁴ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝³ : G.IsCocontinuous J K
inst✝² : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F
inst✝¹ : G.IsCocontinuous J K
inst✝ : G.IsContinuous J K
F : Sheaf K A

⊢ ((G.sheafAdjunctionCocontinuous A J K).unit.app F).val = (G.op.ranAdjunction A).unit.app F.val
-- Mathlib/CategoryTheory/Sites/CoverLifting.lean
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{u_4, u_1} C
inst✝⁵ : Category.{u_3, u_2} D
G : C ⥤ D
A : Type w
inst✝⁴ : Category.{w', w} A
J : GrothendieckTopology C
K : GrothendieckTopology D
inst✝³ : G.IsCocontinuous J K
inst✝² : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F
inst✝¹ : G.IsCocontinuous J K
inst✝ : G.IsContinuous J K
F : Sheaf K A

⊢ (G.op.ranAdjunction A).unit.app ((sheafToPresheaf K A).obj F) ≫ G.op.ran.map ((G.sheafPushforwardContinuousCompSheafToPresheafIso A J K).symm.hom.app F) ≫ (G.sheafPushforwardCocontinuousCompSheafToPresheafIso A J K).symm.hom.app ((G.sheafPushforwardContinuous A J K).obj F) = (G.op.ranAdjunction A).unit.app F.val