ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/GroupTheory/SpecificGroups/Alternating.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩

⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
-- Mathlib/GroupTheory/SpecificGroups/Alternating.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * ↑⟨σ, hσ✝⟩ * π⁻¹ = ↑⟨τ, hτ⟩

⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
-- Mathlib/GroupTheory/SpecificGroups/Alternating.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ

⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
-- Mathlib/GroupTheory/SpecificGroups/Alternating.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = 1

⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
-- Mathlib/GroupTheory/SpecificGroups/Alternating.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
σ : Perm α
hσ✝ : σ ∈ alternatingGroup α
hσ : (↑⟨σ, hσ✝⟩).support.card + 2 ≤ card α
τ : Perm α
hτ : τ ∈ alternatingGroup α
hc : IsConj ↑⟨σ, hσ✝⟩ ↑⟨τ, hτ⟩
π : Perm α
hπ : π * σ * π⁻¹ = τ
h : sign π = -1

⊢ IsConj ⟨σ, hσ✝⟩ ⟨τ, hτ⟩
-- Mathlib/Data/Set/Countable.lean
α : Type u
β : Type v
γ : Type w
ι : Sort x
s : Set α
hc : s.Countable
hs : s.Nonempty
f : ℕ → ↑s
hf : Surjective f

⊢ s = range (Subtype.val ∘ f)
-- Mathlib/Analysis/ODE/PicardLindelof.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : PicardLindelof E
f : v.FunSpace

⊢ Continuous f.vComp
-- Mathlib/Analysis/ODE/PicardLindelof.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : PicardLindelof E
f : v.FunSpace
this : Continuous ((fun x => (↑x, f.toFun x)) ∘ v.proj)

⊢ Continuous f.vComp
-- Mathlib/Analysis/ODE/PicardLindelof.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
v : PicardLindelof E
f : v.FunSpace
this : Continuous ((fun x => (↑x, f.toFun x)) ∘ v.proj)
x : ℝ

⊢ ((fun x => (↑x, f.toFun x)) ∘ v.proj) x ∈ Icc v.tMin v.tMax ×ˢ closedBall v.x₀ ↑v.R
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : TopologicalSpace α
h : RestrictGenTopology 𝔖

⊢ IsClosed {f | Continuous ((toFun 𝔖) f)}
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s✝ s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : TopologicalSpace α
h : RestrictGenTopology 𝔖
f : α →ᵤ[𝔖] β
u : Filter (α →ᵤ[𝔖] β)
x✝ : u.NeBot
hu : u ≤ 𝓟 {f | Continuous ((toFun 𝔖) f)}
huf : u ≤ 𝓝 f
s : Set α
hs : s ∈ 𝔖

⊢ ContinuousOn ((toFun 𝔖) f) s
-- Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
s✝ s' : Set α
x : α
p : Filter ι
g : ι → α
inst✝¹ : UniformSpace β
𝔖 : Set (Set α)
inst✝ : TopologicalSpace α
h : RestrictGenTopology 𝔖
f : α →ᵤ[𝔖] β
u : Filter (α →ᵤ[𝔖] β)
x✝ : u.NeBot
hu : u ≤ 𝓟 {f | Continuous ((toFun 𝔖) f)}
huf : ∀ s ∈ 𝔖, TendstoUniformlyOn (⇑(toFun 𝔖) ∘ id) ((toFun 𝔖) f) u s
s : Set α
hs : s ∈ 𝔖

⊢ ContinuousOn ((toFun 𝔖) f) s
-- Mathlib/Analysis/Convex/Strict.lean
𝕜 : Type u_1
𝕝 : Type u_2
E : Type u_3
F : Type u_4
β : Type u_5
inst✝⁶ : OrderedRing 𝕜
inst✝⁵ : TopologicalSpace E
inst✝⁴ : TopologicalSpace F
inst✝³ : AddCommGroup E
inst✝² : AddCommGroup F
inst✝¹ : Module 𝕜 E
inst✝ : Module 𝕜 F
s t : Set E
x✝ y✝ : E
hs : StrictConvex 𝕜 s
f : E →ᵃ[𝕜] F
hf : IsOpenMap ⇑f
x : E
hx : x ∈ s
y : E
hy : y ∈ s
hxy : f x ≠ f y
a b : 𝕜
ha : 0 < a
hb : 0 < b
hab : a + b = 1

⊢ a • f x + b • f y ∈ interior (⇑f '' s)
-- Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M

⊢ 1 ≤ evenOdd Q 0
-- Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M

⊢ 1 ≤ LinearMap.range (ι Q) ^ ↑⟨0, ⋯⟩
-- Mathlib/Analysis/InnerProductSpace/PiL2.lean
ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁹ : _root_.RCLike 𝕜
E : Type u_4
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : InnerProductSpace 𝕜 E
E' : Type u_5
inst✝⁶ : NormedAddCommGroup E'
inst✝⁵ : InnerProductSpace 𝕜 E'
F : Type u_6
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
F' : Type u_7
inst✝² : NormedAddCommGroup F'
inst✝¹ : InnerProductSpace ℝ F'
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
v : EuclideanSpace 𝕜 ι

⊢ ∑ i : ι, v i • b i = b.repr.symm v
-- Mathlib/Algebra/Ring/Subring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : Ring R
s : Set R

⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)
-- Mathlib/Algebra/Ring/Subring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : Ring R
s : Set R

⊢ sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}
-- Mathlib/Algebra/Ring/Subring/MulOpposite.lean
ι : Sort u_1
R : Type u_2
inst✝ : Ring R
s : Set R
a : Subring Rᵐᵒᵖ

⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0

⊢ Tendsto (fun n => ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2 * T_gen p F a ^ 2 ^ n) atTop (𝓝 0)
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0

⊢ Tendsto (fun n => ‖Polynomial.eval a (Polynomial.derivative F)‖ * ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n) atTop (𝓝 (‖Polynomial.eval a (Polynomial.derivative F)‖ * 0))
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0

⊢ Tendsto (fun n => ‖Polynomial.eval a (Polynomial.derivative F)‖ * (‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n)) atTop (𝓝 (‖Polynomial.eval a (Polynomial.derivative F)‖ * 0))
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0

⊢ Tendsto (fun x => ‖Polynomial.eval a (Polynomial.derivative F)‖) atTop (𝓝 ‖Polynomial.eval a (Polynomial.derivative F)‖)
-- Mathlib/NumberTheory/Padics/Hensel.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
F : Polynomial ℤ_[p]
a : ℤ_[p]
hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2
hnsol : Polynomial.eval a F ≠ 0

⊢ Tendsto (fun x => ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ x) atTop (𝓝 0)
-- Mathlib/Algebra/Polynomial/Smeval.lean
R✝ : Type u_1
inst✝⁴ : Semiring R✝
p q : R✝[X]
S : Type u_2
inst✝³ : AddCommMonoid S
inst✝² : Pow S ℕ
inst✝¹ : Module R✝ S
x : S
R : Type u_3
inst✝ : Semiring R
r : R

⊢ leval r = smeval.linearMap R r
-- Mathlib/Algebra/Polynomial/Smeval.lean
R✝ : Type u_1
inst✝⁴ : Semiring R✝
p q : R✝[X]
S : Type u_2
inst✝³ : AddCommMonoid S
inst✝² : Pow S ℕ
inst✝¹ : Module R✝ S
x : S
R : Type u_3
inst✝ : Semiring R
r : R

⊢ ∀ (x : R[X]), (leval r) x = (smeval.linearMap R r) x
-- Mathlib/Algebra/Polynomial/Smeval.lean
R✝ : Type u_1
inst✝⁴ : Semiring R✝
p q : R✝[X]
S : Type u_2
inst✝³ : AddCommMonoid S
inst✝² : Pow S ℕ
inst✝¹ : Module R✝ S
x : S
R : Type u_3
inst✝ : Semiring R
r : R
x✝ : R[X]

⊢ (leval r) x✝ = (smeval.linearMap R r) x✝
-- Mathlib/Data/Set/Pointwise/Interval.lean
α : Type u_1
inst✝ : LinearOrderedField α
a✝ a c : α
h : c < 0

⊢ (fun x => c * x) ⁻¹' Ioi a = Iio (a / c)
-- Mathlib/Topology/Algebra/Module/CharacterSpace.lean
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A

⊢ φ x = ψ x
-- Mathlib/Topology/Algebra/Module/CharacterSpace.lean
𝕜 : Type u_1
A : Type u_2
inst✝⁷ : CommRing 𝕜
inst✝⁶ : NoZeroDivisors 𝕜
inst✝⁵ : TopologicalSpace 𝕜
inst✝⁴ : ContinuousAdd 𝕜
inst✝³ : ContinuousConstSMul 𝕜 𝕜
inst✝² : TopologicalSpace A
inst✝¹ : Ring A
inst✝ : Algebra 𝕜 A
φ ψ : ↑(characterSpace 𝕜 A)
h : RingHom.ker φ = RingHom.ker ψ
x : A
this : x - (algebraMap 𝕜 A) (ψ x) ∈ RingHom.ker φ

⊢ φ x = ψ x
-- Mathlib/CategoryTheory/Monoidal/Tor.lean
C : Type u_1
inst✝⁵ : Category.{u_2, u_1} C
inst✝⁴ : MonoidalCategory C
inst✝³ : Abelian C
inst✝² : MonoidalPreadditive C
inst✝¹ : HasProjectiveResolutions C
X Y : C
inst✝ : Projective X
n : ℕ

⊢ IsZero (((Tor' C (n + 1)).obj X).obj Y)
-- Mathlib/Data/Nat/Defs.lean
a b c d m k✝ : ℕ
p q : ℕ → Prop
k : ℕ
h1 : 0 < m * k
h2 : m * k < m

⊢ False
-- Mathlib/Data/Nat/Defs.lean
a b c d m k : ℕ
p q : ℕ → Prop
h1 : 0 < m * 0
h2 : m * 0 < m

⊢ False
-- Mathlib/Data/Nat/Defs.lean
a b c d m k✝ : ℕ
p q : ℕ → Prop
k : ℕ
h1 : 0 < m * k
h2 : m * k < m
hk : k > 0

⊢ False
-- Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean
F : Type u_1
inst✝² : Field F
inst✝¹ : Finite F
inst✝ : DecidableEq F

⊢ ringChar ℂ ≠ ringChar F
-- Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j

⊢ (summand a b σ) v + (summand a b (swap i j • σ)) v = 0
-- Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)

⊢ (summand a b (Quotient.mk'' σ)) v + (summand a b (swap i j • Quotient.mk'' σ)) v = 0
-- Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)

⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v + (Perm.sign (swap i j • σ) • MultilinearMap.domDomCongr (swap i j • σ) ((↑a).domCoprod ↑b)) v = 0
-- Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)

⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v + ((-1 * Perm.sign σ) • MultilinearMap.domDomCongr (swap i j * σ) ((↑a).domCoprod ↑b)) v = 0
-- Mathlib/LinearAlgebra/Alternating/DomCoprod.lean
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
σ✝ : Perm.ModSumCongr ιa ιb
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)

⊢ (Perm.sign σ • (↑a fun i => v (σ (Sum.inl i))) ⊗ₜ[R'] ↑b fun i => v (σ (Sum.inr i))) + -(Perm.sign σ • (↑a fun i_1 => v ((swap i j) (σ (Sum.inl i_1)))) ⊗ₜ[R'] ↑b fun i_1 => v ((swap i j) (σ (Sum.inr i_1)))) = 0
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y z✝ : ℝ
n : ℕ
x z : ℝ
hx1 : 0 ≤ x
hx2 : x ≤ 1
hz : 0 ≤ z

⊢ x ^ z ≤ 1
-- Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
x✝ y z✝ : ℝ
n : ℕ
x z : ℝ
hx1 : 0 ≤ x
hx2 : x ≤ 1
hz : 0 ≤ z

⊢ x ^ z ≤ 1 ^ z
-- Mathlib/Algebra/Lie/Weights/Basic.lean
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : NoZeroSMulDivisors R M
x : L

⊢ CompleteLattice.Independent fun χ => weightSpaceOf M χ x
-- Mathlib/Algebra/Lie/Weights/Basic.lean
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : LieAlgebra.IsNilpotent R L
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : LieRingModule L M
inst✝¹ : LieModule R L M
inst✝ : NoZeroSMulDivisors R M
x : L

⊢ CompleteLattice.Independent fun i => ↑(weightSpaceOf M i x)
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
tU tV : (subsheafToTypes P).presheaf.stalk x
h : stalkToFiber P x tU = stalkToFiber P x tV

⊢ tU = tV
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
tU tV : (subsheafToTypes P).presheaf.stalk x
h : stalkToFiber P x tU = stalkToFiber P x tV
Q : ∃ W s, ∃ (hW : P.pred s), tU = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ tV = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩ := ?refine_1

⊢ tU = tV
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
tU tV : (subsheafToTypes P).presheaf.stalk x
h : stalkToFiber P x tU = stalkToFiber P x tV

⊢ ∃ W s, ∃ (hW : P.pred s), tU = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ tV = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
tV : (subsheafToTypes P).presheaf.stalk x
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) U ⟨fU, hU⟩) = stalkToFiber P x tV

⊢ ∃ W s, ∃ (hW : P.pred s), colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) U ⟨fU, hU⟩ = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ tV = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) U ⟨fU, hU⟩) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) V ⟨fV, hV⟩)

⊢ ∃ W s, ∃ (hW : P.pred s), colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) U ⟨fU, hU⟩ = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) V ⟨fV, hV⟩ = (subsheafToTypes P).presheaf.germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) U ⟨fU, hU⟩) = stalkToFiber P x (colimit.ι (((whiskeringLeft (OpenNhds x)ᵒᵖ (Opens ↑X)ᵒᵖ (Type v)).obj (OpenNhds.inclusion x).op).obj (subsheafToTypes P).presheaf) V ⟨fV, hV⟩)

⊢ ∃ W s, ∃ (hW : P.pred s), colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U ⟨fU, hU⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V ⟨fV, hV⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
w : ∀ (U V : OpenNhds x) (fU : (y : ↥U.obj) → T ↑y), P.pred fU → ∀ (fV : (y : ↥V.obj) → T ↑y), P.pred fV → fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩ → ∃ W iU iV, ∀ (w : ↥W.obj), fU ((fun x_4 => ⟨↑x_4, ⋯⟩) w) = fV ((fun x_4 => ⟨↑x_4, ⋯⟩) w)
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩

⊢ ∃ W s, ∃ (hW : P.pred s), colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U ⟨fU, hU⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V ⟨fV, hV⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
W : OpenNhds x
iU : W ⟶ unop U
iV : W ⟶ unop V
w : ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

⊢ ∃ W s, ∃ (hW : P.pred s), colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U ⟨fU, hU⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩ ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V ⟨fV, hV⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨s, hW⟩
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
W : OpenNhds x
iU : W ⟶ unop U
iV : W ⟶ unop V
w : ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

⊢ (unop (op W)).obj ⟶ (unop U).obj
-- Mathlib/Topology/Sheaves/LocalPredicate.lean
X : TopCat
T : ↑X → Type v
P : LocalPredicate T
x : ↑X
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T ↑x_1
hU : P.pred fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T ↑x_1
hV : P.pred fV
h : fU ⟨x, ⋯⟩ = fV ⟨x, ⋯⟩
W : OpenNhds x
iU : W ⟶ unop U
iV : W ⟶ unop V
w : ∀ (w : ↥W.obj), fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w) = fV ((fun x_1 => ⟨↑x_1, ⋯⟩) w)

⊢ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) U ⟨fU, hU⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨fun w => fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w), ⋯⟩ ∧ colimit.ι ((OpenNhds.inclusion x).op ⋙ subpresheafToTypes P.toPrelocalPredicate) V ⟨fV, hV⟩ = (subpresheafToTypes P.toPrelocalPredicate).germ ⟨x, ⋯⟩ ⟨fun w => fU ((fun x_1 => ⟨↑x_1, ⋯⟩) w), ⋯⟩
-- Mathlib/Probability/Kernel/Disintegration/Integral.lean
α : Type u_1
Ω : Type u_2
E : Type u_3
F : Type u_4
mα : MeasurableSpace α
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf_int : Integrable f ρ

⊢ ∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω => f (a, ω)) (ρ.condKernel a)
-- Mathlib/Probability/Kernel/Disintegration/Integral.lean
α : Type u_1
Ω : Type u_2
E : Type u_3
F : Type u_4
mα : MeasurableSpace α
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf_int : Integrable f ρ
hf_ae : AEStronglyMeasurable f ρ

⊢ ∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω => f (a, ω)) (ρ.condKernel a)
-- Mathlib/Probability/Kernel/Disintegration/Integral.lean
α : Type u_1
Ω : Type u_2
E : Type u_3
F : Type u_4
mα : MeasurableSpace α
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
ρ : Measure (α × Ω)
inst✝ : IsFiniteMeasure ρ
f : α × Ω → F
hf_int : (∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω => f (a, ω)) (ρ.condKernel a)) ∧ Integrable (fun a => ∫ (ω : Ω), ‖f (a, ω)‖ ∂ρ.condKernel a) ρ.fst
hf_ae : AEStronglyMeasurable f ρ

⊢ ∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω => f (a, ω)) (ρ.condKernel a)
-- Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
u : ℂ
hu : 0 < u.re

⊢ u.betaIntegral 1 = 1 / u
-- Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
u : ℂ
hu : 0 < u.re

⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) = 1 / u
-- Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
u : ℂ
hu : 0 < u.re

⊢ (↑1 ^ (u - 1 + 1) - ↑0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u
-- Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean
u : ℂ
hu : 0 < u.re

⊢ -1 < (u - 1).re
-- Mathlib/Algebra/MonoidAlgebra/Basic.lean
k : Type u₁
G : Type u₂
H : Type u_1
R : Type u_2
inst✝¹ : Semiring k
inst✝ : AddZeroClass G
f : k[G]
r : k
x a : G

⊢ a + 0 = x ↔ a = x