ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf this : FiniteDimensional k ↥s.direction

⊢ finrank k ↥(vectorSpan k (insert p ↑s)) ≤ finrank k ↥s.direction + 1
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf this : FiniteDimensional k ↥s.direction

⊢ finrank k ↥(affineSpan k (insert p ↑(affineSpan k ↑s))).direction ≤ finrank k ↥s.direction + 1
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf this : FiniteDimensional k ↥s.direction
hs : ↑s = ∅

⊢ finrank k ↥(affineSpan k (insert p ↑(affineSpan k ↑s))).direction ≤ finrank k ↥s.direction + 1
-- Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : DivisionRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
s : AffineSubspace k P
p : P
hf this : FiniteDimensional k ↥s.direction
p₀ : P
hp₀ : p₀ ∈ ↑s

⊢ finrank k ↥(affineSpan k (insert p ↑(affineSpan k ↑s))).direction ≤ finrank k ↥s.direction + 1
-- Mathlib/Data/Set/Subset.lean
ι : Sort u_1
α : Type u_2
A B C : Set α
D E : Set ↑A
S : Set (Set α)
T : Set (Set ↑A)
s : ι → Set α
t : ι → Set ↑A

⊢ A ↓∩ ⋂₀ S = ⋂₀ ((fun x => A ↓∩ x) '' S)
-- Mathlib/RingTheory/Ideal/Norm.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Module.Free R S
inst✝ : Module.Finite R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S

⊢ spanNorm R (I * J) = spanNorm R I * spanNorm R J
-- Mathlib/RingTheory/Ideal/Norm.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Module.Free R S
inst✝ : Module.Finite R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S

⊢ spanNorm R (I * J) ≤ spanNorm R I * spanNorm R J
-- Mathlib/RingTheory/Ideal/Norm.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Module.Free R S
inst✝ : Module.Finite R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : spanNorm R I = ⊤

⊢ spanNorm R (I * J) ≤ spanNorm R I * spanNorm R J
-- Mathlib/RingTheory/Ideal/Norm.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Module.Free R S
inst✝ : Module.Finite R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : spanNorm R I = ⊤

⊢ spanNorm R (I * J) ≤ spanNorm R J
-- Mathlib/RingTheory/Ideal/Norm.lean
R : Type u_1
inst✝⁶ : CommRing R
S : Type u_2
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Module.Free R S
inst✝ : Module.Finite R S
eq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤
I J : Ideal S
hI : spanNorm R I = ⊤
hJ : spanNorm R J = ⊤

⊢ spanNorm R (I * J) ≤ ⊤
-- Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Preadditive C
X : SimplicialObject C
n : ℕ

⊢ (ChainComplex.of (fun n => X _[n]) (AlternatingFaceMapComplex.objD X) ⋯).d (n + 1) n = AlternatingFaceMapComplex.objD X n
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α

⊢ ⨆ n, ↑(eapprox f n) a = f a
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α

⊢ ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k = f a
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α

⊢ ¬f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α
h : f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k

⊢ False
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α
h : f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k
q : ℚ
left✝ : 0 ≤ q
lt_q : ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k < ↑(↑q).toNNReal
q_lt : ↑(↑q).toNNReal < f a

⊢ False
-- Mathlib/MeasureTheory/Function/SimpleFunc.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝ : MeasurableSpace α
K : Type u_5
f : α → ℝ≥0∞
hf : Measurable f
a : α
h : f a > ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k
q : ℚ
left✝ : 0 ≤ q
lt_q : ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k < ↑(↑q).toNNReal
q_lt : ↑(↑q).toNNReal < f a
this : ↑(↑q).toNNReal ≤ ⨆ k, ⨆ (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k

⊢ False
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean

⊢ (↑(-π / 2)).sign = -1
-- Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
x y a b : ℝ
n : ℕ
ha : a ≠ 0

⊢ (fun x => x ^ n / (a * rexp x + b)) ∘ log =ᶠ[atTop] fun x => log x ^ n / (a * x + b)
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S

⊢ TopologicalSpace.IsTopologicalBasis (SetLike.coe '' Set.range basicOpen)
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S

⊢ SetLike.coe '' Set.range basicOpen = Set.range fun r => ↑(basicOpen r)
-- Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
R : Type u
S : Type v
inst✝¹ : CommSemiring R
inst✝ : CommSemiring S

⊢ Set.range (SetLike.coe ∘ basicOpen) = Set.range fun r => ↑(basicOpen r)
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C✝ : Set (I → Bool)
x✝¹ : 0 ≤ Ordinal.type fun x x_1 => x < x_1
C : Set (I → Bool)
x✝ : IsClosed C
hsC : contained C 0

⊢ LinearIndependent ℤ (eval C)
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C✝ : Set (I → Bool)
x✝¹ : 0 ≤ Ordinal.type fun x x_1 => x < x_1
C : Set (I → Bool)
x✝ : IsClosed C
hsC : contained C 0
this : C ⊆ {fun x => false}

⊢ LinearIndependent ℤ (eval C)
-- Mathlib/Topology/Category/Profinite/Nobeling.lean
I : Type u
inst✝¹ : LinearOrder I
inst✝ : IsWellOrder I fun x x_1 => x < x_1
C✝ : Set (I → Bool)
x✝¹ : 0 ≤ Ordinal.type fun x x_1 => x < x_1
C : Set (I → Bool)
x✝ : IsClosed C
hsC : contained C 0
this : C = ∅ ∨ C = {fun x => false}

⊢ LinearIndependent ℤ (eval C)
-- Mathlib/NumberTheory/LSeries/Linearity.lean
f g : ℕ → ℂ
s : ℂ

⊢ term (f - g) s = term f s - term g s
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
S : Set Ordinal.{u}
hS : Unbounded (fun x x_1 => x < x_1) S
o : Ordinal.{u}

⊢ enumOrd S o ∈ S ∩ Ici (o.blsub fun c x => enumOrd S c)
-- Mathlib/SetTheory/Ordinal/Arithmetic.lean
S : Set Ordinal.{u}
hS : Unbounded (fun x x_1 => x < x_1) S
o : Ordinal.{u}

⊢ sInf (S ∩ Ici (o.blsub fun a x => enumOrd S a)) ∈ S ∩ Ici (o.blsub fun c x => enumOrd S c)
-- Mathlib/Order/Atoms/Finite.lean
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
a : α
inst✝ : IsStronglyCoatomic α
ha : (Set.Iic a).Infinite
hfin : {x | x ⋖ a}.Finite

⊢ ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite
-- Mathlib/Order/Atoms/Finite.lean
α : Type u_1
β : Type u_2
inst✝¹ : PartialOrder α
a : α
inst✝ : IsStronglyCoatomic α
ha : (Set.Iic a).Infinite
hfin : {x | OrderDual.toDual a ⋖ OrderDual.toDual x}.Finite

⊢ ∃ b, OrderDual.toDual a ⋖ OrderDual.toDual b ∧ (Set.Iic b).Infinite
-- Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n

⊢ expZeta a (1 - s) = (2 * ↑π) ^ (-s) * Complex.Gamma s * (cexp (↑π * I * s / 2) * hurwitzZeta a s + cexp (-↑π * I * s / 2) * hurwitzZeta (-a) s)
-- Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n

⊢ 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (↑π * s / 2 * I) + cexp (-(↑π * s / 2) * I)) / 2) * hurwitzZetaEven a s + I * (2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * ((cexp (-(↑π * s / 2) * I) - cexp (↑π * s / 2 * I)) * I / 2) * hurwitzZetaOdd a s) = (2 * ↑π) ^ (-s) * Complex.Gamma s * (cexp (↑π * I * s / 2) * (hurwitzZetaEven a s + hurwitzZetaOdd a s) + cexp (-↑π * I * s / 2) * (hurwitzZetaEven a s + -hurwitzZetaOdd a s))
-- Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n
x y z : ℂ

⊢ x * Complex.Gamma s * y * hurwitzZetaEven a s - x * Complex.Gamma s * y * I ^ 2 * hurwitzZetaOdd a s + x * Complex.Gamma s * z * hurwitzZetaEven a s + x * Complex.Gamma s * z * I ^ 2 * hurwitzZetaOdd a s = x * Complex.Gamma s * y * hurwitzZetaEven a s + x * Complex.Gamma s * y * hurwitzZetaOdd a s + (x * Complex.Gamma s * z * hurwitzZetaEven a s - x * Complex.Gamma s * z * hurwitzZetaOdd a s)
-- Mathlib/NumberTheory/LSeries/HurwitzZeta.lean
a : UnitAddCircle
s : ℂ
hs : ∀ (n : ℕ), s ≠ 1 - ↑n
hs' : ∀ (n : ℕ), s ≠ -↑n
x y z : ℂ

⊢ x * Complex.Gamma s * y * hurwitzZetaEven a s - x * Complex.Gamma s * y * -1 * hurwitzZetaOdd a s + x * Complex.Gamma s * z * hurwitzZetaEven a s + x * Complex.Gamma s * z * -1 * hurwitzZetaOdd a s = x * Complex.Gamma s * y * hurwitzZetaEven a s + x * Complex.Gamma s * y * hurwitzZetaOdd a s + (x * Complex.Gamma s * z * hurwitzZetaEven a s - x * Complex.Gamma s * z * hurwitzZetaOdd a s)
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ s.singularPart μ + μ.withDensityᵥ (s.rnDeriv μ) = s.toJordanDecomposition.posPart.toSignedMeasure - s.toJordanDecomposition.negPart.toSignedMeasure
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ (s.toJordanDecomposition.posPart.singularPart μ + μ.withDensity (s.toJordanDecomposition.posPart.rnDeriv μ)).toSignedMeasure - (μ.withDensity (s.toJordanDecomposition.negPart.rnDeriv μ) + s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure = s.toJordanDecomposition.posPart.toSignedMeasure - s.toJordanDecomposition.negPart.toSignedMeasure
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ AEMeasurable (s.toJordanDecomposition.negPart.rnDeriv μ) μ
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ ∫⁻ (x : α), s.toJordanDecomposition.negPart.rnDeriv μ x ∂μ ≠ ⊤
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ AEMeasurable (s.toJordanDecomposition.posPart.rnDeriv μ) μ
-- Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ ν : Measure α
s t : SignedMeasure α
inst✝ : s.HaveLebesgueDecomposition μ

⊢ ∫⁻ (x : α), s.toJordanDecomposition.posPart.rnDeriv μ x ∂μ ≠ ⊤
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
h : p.leadingCoeff ^ 0 ≠ 0

⊢ (p ^ 0).degree = 0 • p.degree
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
h : 1 ≠ 0

⊢ (C 1).degree = 0 • p.degree
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
n : ℕ
h : p.leadingCoeff ^ (n + 1) ≠ 0

⊢ (p ^ (n + 1)).degree = (n + 1) • p.degree
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
n : ℕ
h : p.leadingCoeff ^ (n + 1) ≠ 0
h₁ : p.leadingCoeff ^ n ≠ 0

⊢ (p ^ (n + 1)).degree = (n + 1) • p.degree
-- Mathlib/Algebra/Polynomial/Degree/Definitions.lean
R : Type u
S : Type v
a b c d : R
n✝ m : ℕ
inst✝ : Semiring R
p✝ p q : R[X]
ι : Type u_1
n : ℕ
h : p.leadingCoeff ^ (n + 1) ≠ 0
h₁ : p.leadingCoeff ^ n ≠ 0
h₂ : (p ^ n).leadingCoeff * p.leadingCoeff ≠ 0

⊢ (p ^ (n + 1)).degree = (n + 1) • p.degree
-- Mathlib/Tactic/NormNum/Eq.lean
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
n✝¹ n✝ : ℤ
h : decide (n✝¹ = n✝) = false

⊢ ¬↑n✝¹ = ↑n✝
-- Mathlib/CategoryTheory/Limits/Shapes/Pullback/CommSq.lean
C : Type u₁
inst✝ : Category.{v₁, u₁} C
Z X Y P : C
f : Z ⟶ X
g : Z ⟶ Y
inl : X ⟶ P
inr : Y ⟶ P
h : IsPushout f g inl inr
Z' X' Y' P' : C
f' : Z' ⟶ X'
g' : Z' ⟶ Y'
inl' : X' ⟶ P'
inr' : Y' ⟶ P'
e₁ : Z ≅ Z'
e₂ : X ≅ X'
e₃ : Y ≅ Y'
e₄ : P ≅ P'
commf : f ≫ e₂.hom = e₁.hom ≫ f'
commg : g ≫ e₃.hom = e₁.hom ≫ g'
comminl : inl ≫ e₄.hom = e₂.hom ≫ inl'
comminr : inr ≫ e₄.hom = e₃.hom ≫ inr'

⊢ f' ≫ inl' = g' ≫ inr'
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x

⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)

⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))

⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x

⊢ x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x

⊢ f x ≤ 0
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f

⊢ f x ≤ 0
-- Mathlib/NumberTheory/Bertrand.lean
x : ℝ
x_large : 512 ≤ x
f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x
hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)
hf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))
h5 : 0 < x
h : ConcaveOn ℝ (Set.Ioi 0.5) f

⊢ ∃ x1 x2, 0.5 < x1 ∧ x1 < x2 ∧ x2 ≤ x ∧ 0 ≤ f x1 ∧ f x2 ≤ 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0

⊢ HasStrictFDerivAt (fun x => x.1 ^ x.2) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ) p
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)

⊢ HasStrictFDerivAt (fun x => x.1 ^ x.2) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ) p
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)

⊢ HasStrictFDerivAt (fun x => rexp (log x.1 * x.2) * cos (x.2 * π)) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ) p
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)
e_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup
he✝ : NormedField.toNormedSpace = NormedAlgebra.toNormedSpace'

⊢ (p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ = rexp (log p.1 * p.2) • -sin (p.2 * π) • π • ContinuousLinearMap.snd ℝ ℝ ℝ + cos (p.2 * π) • rexp (log p.1 * p.2) • (log p.1 • ContinuousLinearMap.snd ℝ ℝ ℝ + p.2 • p.1⁻¹ • ContinuousLinearMap.fst ℝ ℝ ℝ)
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)
e_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup
he✝ : NormedField.toNormedSpace = NormedAlgebra.toNormedSpace'

⊢ (p.2 * (p.1 ^ p.2 / p.1)) • ContinuousLinearMap.fst ℝ ℝ ℝ + (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ = (rexp (log p.1 * p.2) * -sin (p.2 * π) * π + p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ + (p.1 ^ p.2 * p.2 * p.1⁻¹) • ContinuousLinearMap.fst ℝ ℝ ℝ
-- Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
x y z : ℝ
p : ℝ × ℝ
hp : p.1 < 0
this : (fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => rexp (log x.1 * x.2) * cos (x.2 * π)
e_7✝ : normedAddCommGroup = NonUnitalNormedRing.toNormedAddCommGroup
he✝ : NormedField.toNormedSpace = NormedAlgebra.toNormedSpace'

⊢ (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ + (p.2 * (p.1 ^ p.2 * p.1⁻¹)) • ContinuousLinearMap.fst ℝ ℝ ℝ = (rexp (log p.1 * p.2) * -sin (p.2 * π) * π + p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℝ ℝ ℝ + (p.1 ^ p.2 * p.2 * p.1⁻¹) • ContinuousLinearMap.fst ℝ ℝ ℝ
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E

⊢ Complex.abs ((o.kahler x) y) = ‖x‖ * ‖y‖
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E

⊢ Complex.normSq ((o.kahler x) y) = (‖x‖ * ‖y‖) ^ 2
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E

⊢ 0 ≤ Complex.abs ((o.kahler x) y)
-- Mathlib/Analysis/InnerProductSpace/TwoDim.lean
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x y : E

⊢ 0 ≤ ‖x‖ * ‖y‖
-- Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
α : Type u
a : α
inst✝¹ : Group α
G : Type u_1
inst✝ : Group G
x✝ : Fintype G
p : ℕ
hp : Fact (Nat.Prime p)
h : Fintype.card G = p
g g' : G
hg : g ≠ 1

⊢ g' ∈ Submonoid.powers g
-- Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean
𝕜 : Type u_1
α : Type u_2
β : Type u_3
inst✝² : TopologicalSpace β
inst✝¹ : AddGroup β
inst✝ : ContinuousNeg β
l : Filter α
f : α → β
hf : l.ZeroAtFilter f

⊢ l.ZeroAtFilter (-f)
-- Mathlib/Data/List/Rotate.lean
α : Type u
l : List α
n : ℕ

⊢ l.reverse ~r (l.rotate n).reverse
-- Mathlib/Algebra/Ring/NegOnePow.lean
n : ℤ

⊢ |↑n.negOnePow| = 1
-- Mathlib/LinearAlgebra/UnitaryGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A : Matrix n n α

⊢ ↑(toGL 1) = ↑1
-- Mathlib/LinearAlgebra/UnitaryGroup.lean
n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
A : Matrix n n α

⊢ LinearMap.id = ↑1
-- Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean
α : Type u_1
a b c d : α
inst✝³ : MulOneClass α
inst✝² : Zero α
inst✝¹ : Preorder α
inst✝ : PosMulStrictMono α
ha : 0 < a
h : 1 < b

⊢ a < a * b
-- Mathlib/MeasureTheory/Measure/Regular.lean
α✝ : Type u_1
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : μ.InnerRegularWRT pa qa
f : α → β
hf : AEMeasurable f μ
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (f '' K)
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < (map f μ) U

⊢ ∃ K ⊆ U, pb K ∧ r < (map f μ) K
-- Mathlib/MeasureTheory/Measure/Regular.lean
α✝ : Type u_1
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : μ.InnerRegularWRT pa qa
f : α → β
hf : AEMeasurable f μ
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (f '' K)
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < μ (f ⁻¹' U)

⊢ ∃ K ⊆ U, pb K ∧ r < (map f μ) K
-- Mathlib/MeasureTheory/Measure/Regular.lean
α✝ : Type u_1
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : μ.InnerRegularWRT pa qa
f : α → β
hf : AEMeasurable f μ
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (f '' K)
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < μ (f ⁻¹' U)
K : Set α
hKU : K ⊆ f ⁻¹' U
hKc : pa K
hK : r < μ K

⊢ ∃ K ⊆ U, pb K ∧ r < (map f μ) K
-- Mathlib/MeasureTheory/Measure/Regular.lean
α✝ : Type u_1
m : MeasurableSpace α✝
μ✝ : Measure α✝
p q : Set α✝ → Prop
U✝ : Set α✝
ε : ℝ≥0∞
α : Type u_2
β : Type u_3
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
pa qa : Set α → Prop
H : μ.InnerRegularWRT pa qa
f : α → β
hf : AEMeasurable f μ
pb qb : Set β → Prop
hAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)
hAB' : ∀ (K : Set α), pa K → pb (f '' K)
hB₂ : ∀ (U : Set β), qb U → MeasurableSet U
U : Set β
hU : qb U
r : ℝ≥0∞
hr : r < μ (f ⁻¹' U)
K : Set α
hKU : K ⊆ f ⁻¹' U
hKc : pa K
hK : r < μ K

⊢ r < (map f μ) (f '' K)
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
parent✝ : Path α
v✝ : α
r✝ x✝ : RBNode α

⊢ RBNode.Ordered cmp ((left c✝ parent✝ v✝ r✝).fill x✝) ↔ Ordered cmp (left c✝ parent✝ v✝ r✝) ∧ RBNode.Ordered cmp x✝ ∧ All (RootOrdered cmp (left c✝ parent✝ v✝ r✝)) x✝
-- .lake/packages/batteries/Batteries/Data/RBMap/Alter.lean
α : Type u_1
cmp : α → α → Ordering
c✝ : RBColor
parent✝ : Path α
v✝ : α
r✝ x✝ : RBNode α

⊢ Ordered cmp parent✝ ∧ (All (fun x => cmpLT cmp x v✝) x✝ ∧ All (fun x => cmpLT cmp v✝ x) r✝ ∧ RBNode.Ordered cmp x✝ ∧ RBNode.Ordered cmp r✝) ∧ RootOrdered cmp parent✝ v✝ ∧ All (RootOrdered cmp parent✝) x✝ ∧ All (RootOrdered cmp parent✝) r✝ ↔ (Ordered cmp parent✝ ∧ All (fun x => cmpLT cmp v✝ x) r✝ ∧ RootOrdered cmp parent✝ v✝ ∧ All (RootOrdered cmp parent✝) r✝ ∧ RBNode.Ordered cmp r✝) ∧ RBNode.Ordered cmp x✝ ∧ All (fun x => cmpLT cmp x v✝) x✝ ∧ All (RootOrdered cmp parent✝) x✝