ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Fintype σ
inst✝² : Fintype τ
n : ℕ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (t.biUnion fun i => (Finsupp.single i 1).support) = (∑ i ∈ s, Finsupp.single i 1).support

⊢ ∀ (i₁ i₂ : σ), i₁ ≠ i₂ → Disjoint (Finsupp.single i₁ 1).support (Finsupp.single i₂ 1).support
-- Mathlib/RingTheory/MvPolynomial/Symmetric.lean
σ : Type u_1
R : Type u_2
τ : Type u_3
S : Type u_4
inst✝⁵ : CommSemiring R
inst✝⁴ : CommSemiring S
inst✝³ : Fintype σ
inst✝² : Fintype τ
n : ℕ
inst✝¹ : DecidableEq σ
inst✝ : Nontrivial R
s t : Finset σ
hst : s ≠ t
h : ∑ i ∈ t, Finsupp.single i 1 = ∑ i ∈ s, Finsupp.single i 1
this : (∑ i ∈ t, Finsupp.single i 1).support = (∑ i ∈ s, Finsupp.single i 1).support

⊢ ∀ (i₁ i₂ : σ), i₁ ≠ i₂ → Disjoint (Finsupp.single i₁ 1).support (Finsupp.single i₂ 1).support
-- Mathlib/Probability/Moments.lean
Ω : Type u_1
ι : Type u_2
m : MeasurableSpace Ω
X : Ω → ℝ
p : ℕ
μ : Measure Ω
t : ℝ

⊢ mgf X μ 0 = (μ Set.univ).toReal
-- Mathlib/CategoryTheory/Limits/IsLimit.lean
J : Type u₁
inst✝² : Category.{v₁, u₁} J
K : Type u₂
inst✝¹ : Category.{v₂, u₂} K
C : Type u₃
inst✝ : Category.{v₃, u₃} C
F : J ⥤ C
t : Cocone F
X : C
h : coyoneda.obj (op X) ⋙ uliftFunctor.{u₁, v₃} ≅ F.cocones
s : Cocone F
s_pt : C
s_ι : F ⟶ (const J).obj s_pt

⊢ h.hom.app s_pt { down := (h.inv.app s_pt s_ι).down } = s_ι
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
ι : Type u_2
F : ι → β → α
f : β → α
p : Filter ι

⊢ TendstoUniformly F f p ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ (x : β), edist (f x) (F n x) < ε
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : { x // x ∈ s } → R

⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i
-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Field K
inst✝ : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : { x // x ∈ s } → R
y : R
hy : ∀ (i : ι) (hi : i ∈ s), (Ideal.Quotient.mk (P i ^ e i)) y = (Ideal.Quotient.mk (P ↑⟨i, hi⟩ ^ e ↑⟨i, hi⟩)) (x ⟨i, hi⟩)

⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u}
b : Ordinal.{max u v}
h : b ≤ lift.{v, u} a

⊢ b.card ≤ Cardinal.lift.{v, u} a.card
-- Mathlib/SetTheory/Ordinal/Basic.lean
α : Type u
β : Type u_1
γ : Type u_2
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
a : Ordinal.{u}
b : Ordinal.{max u v}
h : b ≤ lift.{v, u} a

⊢ b.card ≤ (lift.{v, u} a).card
-- Mathlib/Data/Fintype/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
s t : Set α
inst✝¹ : Fintype ↑s
inst✝ : Fintype ↑t

⊢ s.toFinset ⊆ t.toFinset ↔ s ⊆ t
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ

⊢ f ∈ eval C '' {m | m < l} ↔ f ∈ ⇑(πs C o) '' (eval (π C fun x => ord I x < o) '' {m | m < l})
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ

⊢ (∃ x < l, eval C x = f) ↔ ∃ a < l, (πs C o) (eval (π C fun x => ord I x < o) a) = f
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ

⊢ ∀ (a : Products I), a < l ∧ eval C a = f ↔ a < l ∧ (πs C o) (eval (π C fun x => ord I x < o) a) = f
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I

⊢ m < l ∧ eval C m = f ↔ m < l ∧ (πs C o) (eval (π C fun x => ord I x < o) m) = f
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I

⊢ m < l → (eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f)
-- 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)
l : Products I
o : Ordinal.{u}
hl : ∀ i ∈ ↑l, ord I i < o
f : LocallyConstant ↑C ℤ
m : Products I
hm : m < l

⊢ eval C m = f ↔ (πs C o) (eval (π C fun x => ord I x < o) m) = f
-- Mathlib/Order/Interval/Finset/Fin.lean
n : ℕ
a b : Fin n

⊢ (Ioo a b).card = ↑b - ↑a - 1
-- Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f

⊢ Tendsto (seminormFromConst_seq c f c) atTop (𝓝 (f c))
-- Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hseq : seminormFromConst_seq c f c = fun _n => f c

⊢ Tendsto (seminormFromConst_seq c f c) atTop (𝓝 (f c))
-- Mathlib/Analysis/Normed/Ring/SeminormFromConst.lean
R : Type u_1
inst✝ : CommRing R
c : R
f : RingSeminorm R
hf1 : f 1 ≤ 1
hc : f c ≠ 0
hpm : IsPowMul ⇑f
hseq : seminormFromConst_seq c f c = fun _n => f c

⊢ Tendsto (fun _n => f c) atTop (𝓝 (f c))
-- Mathlib/Data/Matroid/Restrict.lean
α : Type u_1
M : Matroid α
R I J✝ X Y B J : Set α
e : α
hIX : M.Basis I X
hJX : M.Basis J X
he : e ∈ I \ J

⊢ ∃ f ∈ J \ I, M.Basis (insert f (I \ {e})) X
-- Mathlib/Data/Matroid/Restrict.lean
α : Type u_1
M : Matroid α
R I J✝ X Y B J : Set α
e : α
hIX : M.Basis I X
hJX : M.Basis J X
he : e ∈ I \ J
y : α
hy : y ∈ J \ I
h : (M ↾ X).Base (insert y (I \ {e}))

⊢ ∃ f ∈ J \ I, M.Basis (insert f (I \ {e})) X
-- Mathlib/Analysis/Complex/ReImTopology.lean
a : ℝ

⊢ closure {z | a < z.re} = {z | a ≤ z.re}
-- Mathlib/Probability/StrongLaw.lean
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
A : ℝ
x : α

⊢ |truncation f A x| ≤ |f x|
-- Mathlib/Probability/StrongLaw.lean
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f✝ f : α → ℝ
A : ℝ
x : α

⊢ |if f x ∈ Set.Ioc (-A) A then f x else 0| ≤ |f x|
-- Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasStrictTerminalObjects C
I : C
hI : IsTerminal I
A : C
f g : I ⟶ A

⊢ f = g
-- Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasStrictTerminalObjects C
I : C
hI : IsTerminal I
A : C
f g : I ⟶ A
this : IsIso f

⊢ f = g
-- Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasStrictTerminalObjects C
I : C
hI : IsTerminal I
A : C
f g : I ⟶ A
this✝ : IsIso f
this : IsIso g

⊢ f = g
-- Mathlib/CategoryTheory/NatIso.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F G : C ⥤ D
X✝ Y : C
α : F ⟶ G
x✝ : IsIso α
X : C

⊢ (inv α).app X = inv (α.app X)
-- Mathlib/CategoryTheory/NatIso.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F G : C ⥤ D
X✝ Y : C
α : F ⟶ G
x✝ : IsIso α
X : C

⊢ α.app X ≫ (inv α).app X = 𝟙 (F.obj X)
-- Mathlib/CategoryTheory/NatIso.lean
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
E : Type u₃
inst✝ : Category.{v₃, u₃} E
F G : C ⥤ D
X✝ Y : C
α : F ⟶ G
x✝ : IsIso α
X : C

⊢ (α ≫ inv α).app X = 𝟙 (F.obj X)
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝³ : TopologicalSpace X
𝕜 : Type u_2
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Zero ↑s
h0 : ↑0 = 0
inst✝ : CompactSpace ↑s

⊢ Dense ↑(adjoin 𝕜 {ContinuousMapZero.id h0})
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝³ : TopologicalSpace X
𝕜 : Type u_2
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Zero ↑s
h0 : ↑0 = 0
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s

⊢ Dense ↑(adjoin 𝕜 {ContinuousMapZero.id h0})
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝³ : TopologicalSpace X
𝕜 : Type u_2
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Zero ↑s
h0 : ↑0 = 0
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s

⊢ ⇑toContinuousMapHom ⁻¹' ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0'⟩)) = Set.univ
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝³ : TopologicalSpace X
𝕜 : Type u_2
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Zero ↑s
h0 : ↑0 = 0
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
f : C(↑s, 𝕜)₀

⊢ f ∈ ⇑toContinuousMapHom ⁻¹' ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0'⟩))
-- Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean
X : Type u_1
inst✝³ : TopologicalSpace X
𝕜 : Type u_2
inst✝² : RCLike 𝕜
s : Set 𝕜
inst✝¹ : Zero ↑s
h0 : ↑0 = 0
inst✝ : CompactSpace ↑s
h0' : 0 ∈ s
f : C(↑s, 𝕜)₀

⊢ f ⟨0, h0'⟩ = 0
-- Mathlib/NumberTheory/LegendreSymbol/Basic.lean
p : ℕ
inst✝ : Fact (Nat.Prime p)
x y : ZMod p
hy : y ≠ 0
hxy : x ^ 2 / y ^ 2 = -y ^ 2 / y ^ 2

⊢ (x / y) ^ 2 = -1
-- Mathlib/Algebra/BigOperators/Group/List.lean
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✝ : CommGroup α

⊢ [].reverse.alternatingProd = [].alternatingProd ^ (-1) ^ ([].length + 1)
-- Mathlib/Algebra/BigOperators/Group/List.lean
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✝ : CommGroup α
a : α
l : List α

⊢ (a :: l).reverse.alternatingProd = (a :: l).alternatingProd ^ (-1) ^ ((a :: l).length + 1)
-- Mathlib/Algebra/BigOperators/Group/List.lean
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✝ : CommGroup α
a : α
l : List α

⊢ (l.alternatingProd ^ (-1) ^ l.length)⁻¹ * a ^ (-1) ^ l.length = (a / l.alternatingProd) ^ (-1) ^ l.length
-- Mathlib/Data/Nat/Cast/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : NonAssocSemiring α
m n : ℕ

⊢ ↑(m * n) = ↑m * ↑n
-- Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean

⊢ MeasurePreserving (⇑measurableEquivPi) volume volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean

⊢ volume = Measure.map (⇑measurableEquivPi.symm) volume
-- Mathlib/MeasureTheory/Measure/Lebesgue/Complex.lean

⊢ ((Pi.basisFun ℝ (Fin 2)).map basisOneI.equivFun.toContinuousLinearEquiv.symm.toLinearEquiv).addHaar = volume
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ

⊢ Tendsto (fun n => ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
A : Tendsto (fun n => ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))) atTop (𝓝 (Complex.sin (↑π * z)))

⊢ Tendsto (fun n => ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
A : Tendsto (fun n => ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))) atTop (𝓝 (Complex.sin (↑π * z)))
this : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)

⊢ Tendsto (fun n => ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))

⊢ Tendsto (fun n => ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))

⊢ Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))) atTop (𝓝 1)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))

⊢ Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) atTop (𝓝 1)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))

⊢ Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) atTop (𝓝 1)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))

⊢ (fun n => (∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) = fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * Complex.cos (2 * z * ↑x)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean
z : ℂ
this✝ : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)
A : Tendsto (fun n => (↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ((∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n)))) atTop (𝓝 (Complex.sin (↑π * z) * 1))
this : ContinuousOn (fun x => Complex.cos (2 * z * ↑x)) (Icc 0 (π / 2))

⊢ 𝓝 1 = 𝓝 (Complex.cos (2 * z * ↑0))
-- Mathlib/Data/Set/Pointwise/SMul.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Group α
inst✝ : MulAction α β
s✝ t✝ A B : Set β
a : α
x✝ : β
s t : Set α
x : α

⊢ x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x
-- Mathlib/Data/Set/Pointwise/SMul.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝¹ : Group α
inst✝ : MulAction α β
s✝ t✝ A B : Set β
a : α
x✝ : β
s t : Set α
x : α

⊢ (x • s ∩ t).Nonempty ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : OrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y z : P
f : P ≃ᵃ[R] P'

⊢ Sbtw R (f x) (f y) (f z) ↔ Sbtw R x y z
-- Mathlib/Analysis/Convex/Between.lean
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁶ : OrderedRing R
inst✝⁵ : AddCommGroup V
inst✝⁴ : Module R V
inst✝³ : AddTorsor V P
inst✝² : AddCommGroup V'
inst✝¹ : Module R V'
inst✝ : AddTorsor V' P'
x y z : P
f : P ≃ᵃ[R] P'

⊢ Function.Injective ⇑↑f
-- Mathlib/Data/List/Zip.lean
α : Type u
β : Type u_1
γ : Type u_2
δ : Type u_3
ε : Type u_4
l : List (α × β)

⊢ l.unzip.2 = map Prod.snd l
-- Mathlib/Data/Matroid/Dual.lean
α : Type u_1
M : Matroid α
I X : Set α
h : M✶.RkPos
hB : M.Base M.E

⊢ False
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
a : α

⊢ f a = ⨆ n, ↑(eapprox f n) 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
f : α → ℝ≥0∞
hf : Measurable f

⊢ ∫⁻ (a : α), ⨆ n, ↑(eapprox f n) a ∂μ = ⨆ n, ∫⁻ (a : α), ↑(eapprox f n) a ∂μ
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f

⊢ ∀ (n : ℕ), Measurable ↑(eapprox f n)
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f

⊢ Monotone fun n => ↑(eapprox f n)
-- Mathlib/MeasureTheory/Integral/Lebesgue.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
hf : Measurable f
n : ℕ

⊢ ∫⁻ (a : α), ↑(eapprox f n) a ∂μ = (eapprox f n).lintegral μ
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f g : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = h ≫ g
inst✝² : HasEqualizer f g
inst✝¹ : HasEqualizer (G.map f) (G.map g)
inst✝ : PreservesLimit (parallelPair f g) G

⊢ (iso G f g).inv ≫ G.map (equalizer.ι f g) = equalizer.ι (G.map f) (G.map g)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Equalizers.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y Z : C
f g : X ⟶ Y
h : Z ⟶ X
w : h ≫ f = h ≫ g
inst✝² : HasEqualizer f g
inst✝¹ : HasEqualizer (G.map f) (G.map g)
inst✝ : PreservesLimit (parallelPair f g) G

⊢ 𝟙 (G.obj (equalizer f g)) ≫ G.map (equalizer.ι f g) = (iso G f g).hom ≫ equalizer.ι (G.map f) (G.map g)
-- 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
hp : p ≠ 0
h : p.degree < q.degree

⊢ p.natDegree < q.natDegree
-- 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
hp : p ≠ 0
h : p.degree < q.degree
hq : q ≠ 0

⊢ p.natDegree < q.natDegree
-- Mathlib/Algebra/Lie/Submodule.lean
R : Type u
L : Type v
M : Type w
inst✝⁶ : CommRing R
inst✝⁵ : LieRing L
inst✝⁴ : LieAlgebra R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
N N' : LieSubmodule R L M
I J : LieIdeal R L

⊢ IsCompl N N' ↔ IsCompl ↑N ↑N'
-- Mathlib/Data/Analysis/Topology.lean
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
T : TopologicalSpace α
F : Realizer α
s : Set α
a : α

⊢ s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s
-- Mathlib/Data/Analysis/Topology.lean
α : Type u_1
β : Type u_2
σ : Type u_3
τ : Type u_4
T : TopologicalSpace α
F : Realizer α
s : Set α
a : α
this : s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s

⊢ s ∈ 𝓝 a ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s
-- Mathlib/Data/Finset/Image.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq β
f✝ g : α → β
s : Finset α
t : Finset β
a : α
b c : β
f : α → β
x : β

⊢ (x ∈ s.biUnion fun a => {f a}) ↔ x ∈ image f s
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
p k : ℕ
hp : Nat.Prime p
hk : k ≠ 0
h : ¬k = 1

⊢ μ (p ^ k) = 0
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
p k : ℕ
hp : Nat.Prime p
hk : k ≠ 0
h : ¬k = 1

⊢ ¬Squarefree (p ^ k)
-- Mathlib/NumberTheory/ArithmeticFunction.lean
R : Type u_1
p k : ℕ
hp : Nat.Prime p
hk : k ≠ 0
h : ¬k = 1

⊢ ¬Squarefree p ∨ ¬k = 1
-- Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean
β : Type u_1
G : Type u_2
M : Type u_3
inst✝² : DivInvMonoid G
inst✝¹ : Preorder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : G
H : 1 ≤ x
n : ℕ

⊢ 1 ≤ x ^ ↑n
-- Mathlib/Algebra/Order/Monoid/Unbundled/Pow.lean
β : Type u_1
G : Type u_2
M : Type u_3
inst✝² : DivInvMonoid G
inst✝¹ : Preorder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
x : G
H : 1 ≤ x
n : ℕ

⊢ 1 ≤ x ^ n
-- Mathlib/Data/Set/Basic.lean
α : Type u
β : Type v
γ : Type w
ι : Sort x
a b : α
s s₁ s₂ t t₁ t₂ u : Set α

⊢ s ∩ {a} = ∅ ↔ a ∉ s
-- Mathlib/Analysis/Calculus/FDeriv/Add.lean
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
G : Type u_4
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
G' : Type u_5
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f f₀ f₁ g : E → F
f' f₀' f₁' g' e : E →L[𝕜] F
x : E
s t : Set E
L L₁ L₂ : Filter E
hg : DifferentiableAt 𝕜 g x

⊢ DifferentiableAt 𝕜 (fun y => f y - g y) x ↔ DifferentiableAt 𝕜 f x
-- Mathlib/NumberTheory/LegendreSymbol/AddCharacter.lean
R : Type u
inst✝² : CommRing R
R' : Type v
inst✝¹ : CommMonoid R'
R'' : Type u_1
inst✝ : CommMonoid R''
φ : AddChar R R'
f : R' →* R''
hφ : φ.IsPrimitive
hf : Function.Injective ⇑f
a : R
ha : a ≠ 0

⊢ (f.compAddChar φ).mulShift a ≠ 1
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle

⊢ f.cycleOf x = 1 ↔ f x = x
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle

⊢ (∀ (x_1 : α), (if f.SameCycle x x_1 then f x_1 else x_1) = x_1) ↔ f x = x
-- Mathlib/GroupTheory/Perm/Cycle/Factors.lean
ι : Type u_1
α : Type u_2
β : Type u_3
f✝ g : Perm α
x y✝ : α
inst✝² : DecidableRel f✝.SameCycle
inst✝¹ : DecidableRel g.SameCycle
f : Perm α
inst✝ : DecidableRel f.SameCycle
h : f x = x
y : α

⊢ (if f.SameCycle x y then f y else y) = y
-- Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A : C
k : S.X₂ ⟶ A
hk : S.f ≫ k = 0
inst✝ : S.HasRightHomology

⊢ S.g = S.g ≫ 𝟙 S.X₃
-- Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
α : Type u_1
inst✝² : TopologicalSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → ℂ
x : ℂ
h₁ : DifferentiableAt ℂ f x
h₂ : f x ∈ slitPlane

⊢ deriv (log ∘ f) x = logDeriv f x
-- Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean
α : Type u_1
inst✝² : TopologicalSpace α
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → ℂ
x : ℂ
h₁ : DifferentiableAt ℂ f x
h₂ : f x ∈ slitPlane
A : deriv (fun t => log (f t)) x = deriv f x / f x

⊢ deriv (log ∘ f) x = logDeriv f x
-- Mathlib/Analysis/Normed/Module/Dual.lean
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r

⊢ polar 𝕜 (closedBall 0 r) = closedBall 0 r⁻¹
-- Mathlib/Analysis/Normed/Module/Dual.lean
𝕜✝ : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜✝
E✝ : Type u_2
inst✝⁴ : SeminormedAddCommGroup E✝
inst✝³ : NormedSpace 𝕜✝ E✝
𝕜 : Type u_3
E : Type u_4
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
hr : 0 < r

⊢ polar 𝕜 (closedBall 0 r) ⊆ closedBall 0 r⁻¹