ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b c : α
n : ℤ

⊢ (c ∈ Set.Ico a (a + p) ∧ ∃ z, b - z • p = c) → toIcoMod hp a b = c
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a b : α
n✝ n : ℤ
hc : b - n • p ∈ Set.Ico a (a + p)

⊢ toIcoMod hp a b = b - n • p
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ

⊢ l ≤ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i
-- Mathlib/Analysis/SpecificLimits/Normed.lean
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
inst✝² : OrderedRing E
inst✝¹ : TopologicalSpace E
inst✝ : OrderClosedTopology E
l : E
f : ℕ → E
hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)
hfm : Monotone f
k : ℕ
ha : Antitone fun n => ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i

⊢ l ≤ ∑ i ∈ Finset.range (2 * k), (-1) ^ i * f i
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
z : ℂ
hre : z.re < 0
him : z.im = 0

⊢ Tendsto log (𝓝[{z | z.im < 0}] z) (𝓝 (↑(Real.log (abs z)) - ↑π * I))
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
z : ℂ
hre : z.re < 0
him : z.im = 0

⊢ 𝓝 (↑(Real.log (abs z)) - ↑π * I) = 𝓝 ((ofReal' ∘ fun x => Real.log (abs x)) z + ↑(-π) * I)
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
z : ℂ
hre : z.re < 0
him : z.im = 0

⊢ abs z ≠ 0
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)

⊢ convexBodySumFun x = ∑ w : { w // w.IsReal }, ‖x.1 w‖ + 2 * ∑ w : { w // w.IsComplex }, ‖x.2 w‖
-- Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)

⊢ ∑ x_1 ∈ Finset.subtype IsReal Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x + ∑ x_1 ∈ Finset.subtype (fun x => x.IsComplex) Finset.univ, ↑(↑x_1).mult * (normAtPlace ↑x_1) x = ∑ x_1 ∈ Finset.subtype IsReal Finset.univ, ‖x.1 x_1‖ + ∑ i ∈ Finset.subtype IsComplex Finset.univ, 2 * ‖x.2 i‖
-- Mathlib/Order/Part.lean
α : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
inst✝ : Preorder α
β γ : Type u_4
f : α → Part (β → γ)
g : α → Part β
hf : Monotone f
hg : Monotone g

⊢ Monotone fun x => Seq.seq (f x) fun x_1 => g x
-- Mathlib/Topology/Order/DenselyOrdered.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : LinearOrder α
inst✝¹ : OrderTopology α
inst✝ : DenselyOrdered α
a✝ b : α
s : Set α
a : α
ha : (Iio a).Nonempty

⊢ frontier (Iio a) = {a}
-- Mathlib/Geometry/Manifold/Algebra/Monoid.lean
𝕜 : Type u_1
inst✝¹⁶ : NontriviallyNormedField 𝕜
H : Type u_2
inst✝¹⁵ : TopologicalSpace H
E : Type u_3
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedSpace 𝕜 E
I : ModelWithCorners 𝕜 E H
G✝ : Type u_4
inst✝¹² : Mul G✝
inst✝¹¹ : TopologicalSpace G✝
inst✝¹⁰ : ChartedSpace H G✝
inst✝⁹ : SmoothMul I G✝
E' : Type u_5
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedSpace 𝕜 E'
H' : Type u_6
inst✝⁶ : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E' H'
M : Type u_7
inst✝⁵ : TopologicalSpace M
inst✝⁴ : ChartedSpace H' M
g✝ h✝ : G✝
G : Type u_8
inst✝³ : Semigroup G
inst✝² : TopologicalSpace G
inst✝¹ : ChartedSpace H G
inst✝ : SmoothMul I G
g h x✝ : G

⊢ (𝑳 I (g * h)) x✝ = ((𝑳 I g).comp (𝑳 I h)) x✝
-- Mathlib/Algebra/Periodic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
f g : α → β
c c₁ c₂ x : α
inst✝² : AddCommMonoid α
inst✝¹ : DivisionSemiring γ
inst✝ : Module γ α
h : Periodic f c
a : γ

⊢ Periodic (fun x => f (a⁻¹ • x)) (a • c)
-- 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
o : Ordinal.{u_3}

⊢ 1 ≤ o.card ↔ 1 ≤ o
-- Mathlib/Topology/UniformSpace/Separation.lean
α : Type u
β : Type v
γ : Type w
inst✝³ : UniformSpace α
inst✝² : UniformSpace β
inst✝¹ : UniformSpace γ
ι : Type u_1
inst✝ : T0Space α
V₀ : Set (α × α)
V₀_in : V₀ ∈ 𝓤 α
f : ι → α
hf : Pairwise fun x y => (f x, f y) ∉ V₀
x y : ι
h : f x ≠ f y

⊢ (f x, f y) ∉ V₀
-- Mathlib/Geometry/Manifold/Instances/Sphere.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E

⊢ ContDiff ℝ ⊤ (stereoInvFunAux v)
-- Mathlib/Geometry/Manifold/Instances/Sphere.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
h₀ : ContDiff ℝ ⊤ fun w => ‖w‖ ^ 2

⊢ ContDiff ℝ ⊤ (stereoInvFunAux v)
-- Mathlib/Geometry/Manifold/Instances/Sphere.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
h₀ : ContDiff ℝ ⊤ fun w => ‖w‖ ^ 2
h₁ : ContDiff ℝ ⊤ fun w => (‖w‖ ^ 2 + 4)⁻¹

⊢ ContDiff ℝ ⊤ (stereoInvFunAux v)
-- Mathlib/Geometry/Manifold/Instances/Sphere.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
v : E
h₀ : ContDiff ℝ ⊤ fun w => ‖w‖ ^ 2
h₁ : ContDiff ℝ ⊤ fun w => (‖w‖ ^ 2 + 4)⁻¹
h₂ : ContDiff ℝ ⊤ fun w => 4 • w + (‖w‖ ^ 2 - 4) • v

⊢ ContDiff ℝ ⊤ (stereoInvFunAux v)
-- Mathlib/Order/BooleanAlgebra.lean
α : Type u
β : Type u_1
w x y z : α
inst✝ : BooleanAlgebra α

⊢ x ⊓ y ⊔ x ⊓ yᶜ = x
-- Mathlib/Analysis/Convex/Strong.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
a b m : ℝ
x y : E
f : E → ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1

⊢ a * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) + m / 2 * ‖a • x + b • y‖ ^ 2 = a * f x + b * f y - m / 2 * a * b * ‖x - y‖ ^ 2
-- Mathlib/Analysis/Convex/Strong.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
a b m : ℝ
x y : E
f : E → ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1

⊢ a * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) + m / 2 * (a ^ 2 * ‖x‖ ^ 2 + 2 * (a * (b * ⟪x, y⟫_ℝ)) + b ^ 2 * ‖y‖ ^ 2) = a * f x + b * f y - m / 2 * a * b * (‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2)
-- Mathlib/Analysis/Convex/Strong.lean
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
φ ψ : ℝ → ℝ
s : Set E
b m : ℝ
x y : E
f : E → ℝ
hb : 0 ≤ b
ha : 0 ≤ 1 - b
hab : 1 - b + b = 1

⊢ (1 - b) * (f x - m / 2 * ‖x‖ ^ 2) + b * (f y - m / 2 * ‖y‖ ^ 2) + m / 2 * ((1 - b) ^ 2 * ‖x‖ ^ 2 + 2 * ((1 - b) * (b * ⟪x, y⟫_ℝ)) + b ^ 2 * ‖y‖ ^ 2) = (1 - b) * f x + b * f y - m / 2 * (1 - b) * b * (‖x‖ ^ 2 - 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2)
-- Mathlib/SetTheory/Cardinal/Ordinal.lean
o₁ o₂ : Ordinal.{u_1}

⊢ aleph o₁ + aleph o₂ = aleph (max o₁ o₂)
-- Mathlib/Algebra/GroupWithZero/Units/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n✝ : ℕ
n : ℤ
ha : a ≠ 0

⊢ a ^ (-n) * a ^ n = 1
-- Mathlib/Algebra/GroupWithZero/Units/Basic.lean
α : Type u_1
M₀ : Type u_2
G₀ : Type u_3
M₀' : Type u_4
G₀' : Type u_5
F : Type u_6
F' : Type u_7
inst✝¹ : MonoidWithZero M₀
inst✝ : GroupWithZero G₀
a b c d : G₀
m n✝ : ℕ
n : ℤ
ha : a ≠ 0

⊢ (a ^ n)⁻¹ * a ^ n = 1
-- Mathlib/Topology/Compactness/Compact.lean
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
hs : IsCompact s
l : Filter X
y : X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x = y

⊢ l ≤ 𝓝 y
-- Mathlib/Topology/Compactness/Compact.lean
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
hs : IsCompact s
l : Filter X
y : X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x = y
f : Ultrafilter X
hf : ↑f ≤ l

⊢ ↑f ≤ 𝓝 y
-- Mathlib/Topology/Compactness/Compact.lean
X : Type u
Y : Type v
ι : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s t : Set X
hs : IsCompact s
l : Filter X
y : X
hmem : s ∈ l
h : ∀ x ∈ s, ClusterPt x l → x = y
f : Ultrafilter X
hf : ↑f ≤ l
x : X
hxs : x ∈ s
hx : ↑f ≤ 𝓝 x

⊢ x = y
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : TopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : HasContinuousInv₀ R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝

⊢ IsUnit (cfc f a) ↔ ∀ x ∈ spectrum R a, f x ≠ 0
-- Mathlib/Analysis/CstarAlgebra/ContinuousFunctionalCalculus/Unital.lean
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹⁰ : Semifield R
inst✝⁹ : StarRing R
inst✝⁸ : MetricSpace R
inst✝⁷ : TopologicalSemiring R
inst✝⁶ : ContinuousStar R
inst✝⁵ : HasContinuousInv₀ R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
inst✝ : ContinuousFunctionalCalculus R p
f : R → R
a : A
hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝
ha : autoParam (p a) _auto✝

⊢ 0 ∉ f '' spectrum R a ↔ ∀ x ∈ spectrum R a, f x ≠ 0
-- Mathlib/Data/Option/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
f : α → β
o : Option α
h : (Option.map f o).isSome = true

⊢ o.isSome = true
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s

⊢ f x ∈ adjoin R (⇑f '' (s ∪ {1}))
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s
r : R

⊢ (fun a => f a ∈ adjoin R (⇑f '' (s ∪ {1}))) ((algebraMap R A) r)
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s
r : R
this : f 1 ∈ adjoin R (⇑f '' (s ∪ {1}))

⊢ (fun a => f a ∈ adjoin R (⇑f '' (s ∪ {1}))) ((algebraMap R A) r)
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s
r : R
this : f 1 ∈ adjoin R (⇑f '' (s ∪ {1}))

⊢ f ((algebraMap R A) r) = r • f 1
-- Mathlib/RingTheory/Adjoin/Basic.lean
R : Type uR
S : Type uS
A : Type uA
B : Type uB
inst✝⁸ : CommSemiring R
inst✝⁷ : CommSemiring S
inst✝⁶ : Semiring A
inst✝⁵ : Semiring B
inst✝⁴ : Algebra R S
inst✝³ : Algebra R A
inst✝² : Algebra S A
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R S A
s✝ t s : Set A
x : A
f : A →ₗ[R] B
hf : ∀ (a₁ a₂ : A), f (a₁ * a₂) = f a₁ * f a₂
h : x ∈ adjoin R s
r : R
this : f 1 ∈ adjoin R (⇑f '' (s ∪ {1}))

⊢ f (r • 1) = r • f 1
-- Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
R : Type u_1
inst✝⁴ : CommSemiring R
M : Type u_2
inst✝³ : AddCommMonoid M
inst✝² : Module R M
A : Type u_3
inst✝¹ : Semiring A
inst✝ : Algebra R A
f : M →ₗ[R] A

⊢ ((lift R) f).toLinearMap ∘ₗ ι R = f
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ

⊢ (proj 𝒜 i) a ^ m ∈ 𝒜 (m * i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ

⊢ f ^ i ∈ 𝒜 (m * i)
-- Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
𝒜 : ℕ → Submodule R A
inst✝ : GradedAlgebra 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ

⊢ f ^ i ∈ 𝒜 (i * m)
-- Mathlib/Algebra/CharP/Defs.lean
R : Type u_1
inst✝¹ : NonAssocRing R
p : ℕ
inst✝ : CharP R p
k : ℕ

⊢ ↑k = ↑(k % p + p * (k / p))
-- Mathlib/Algebra/CharP/Defs.lean
R : Type u_1
inst✝¹ : NonAssocRing R
p : ℕ
inst✝ : CharP R p
k : ℕ

⊢ ↑(k % p + p * (k / p)) = ↑(k % p)
-- Mathlib/Topology/Compactification/OnePoint.lean
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X

⊢ ¬Continuous ⇑CofiniteTopology.of.symm
-- Mathlib/Topology/Compactification/OnePoint.lean
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X

⊢ ∃ x, ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 x) (𝓝 (CofiniteTopology.of.symm x))
-- Mathlib/Topology/Compactification/OnePoint.lean
X : Type u_1
inst✝² : TopologicalSpace X
s : Set (OnePoint X)
t : Set X
inst✝¹ : Infinite X
inst✝ : DiscreteTopology X
inhabited_h : Inhabited X

⊢ ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 (CofiniteTopology.of ↑default)) (𝓝 (CofiniteTopology.of.symm (CofiniteTopology.of ↑default)))
-- Mathlib/Control/Traversable/Basic.lean
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η η' : ApplicativeTransformation F G
h : ∀ (α : Type u) (x : F α), (fun {α} => η.app α) x = (fun {α} => η'.app α) x

⊢ η = η'
-- Mathlib/Control/Traversable/Basic.lean
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η η' : ApplicativeTransformation F G
h : ∀ (α : Type u) (x : F α), (fun {α} => η.app α) x = (fun {α} => η'.app α) x

⊢ (fun {α} => η.app α) = fun {α} => η'.app α
-- Mathlib/Control/Traversable/Basic.lean
F : Type u → Type v
inst✝³ : Applicative F
inst✝² : LawfulApplicative F
G : Type u → Type w
inst✝¹ : Applicative G
inst✝ : LawfulApplicative G
η η' : ApplicativeTransformation F G
h : ∀ (α : Type u) (x : F α), (fun {α} => η.app α) x = (fun {α} => η'.app α) x
α : Type u

⊢ η.app α = η'.app α
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p✝ p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
F : Filter α
φ : β → α
p : Filter β
u : α → X
inst✝ : p.NeBot
h : Tendsto φ p F
H : Tendsto (u ∘ φ) p (𝓝 x)

⊢ MapClusterPt x F u
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p✝ p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
F : Filter α
φ : β → α
p : Filter β
u : α → X
inst✝ : p.NeBot
h : Tendsto φ p F
H : Tendsto (u ∘ φ) p (𝓝 x)
this : map (u ∘ φ) p ≤ map u F

⊢ MapClusterPt x F u
-- Mathlib/Topology/Basic.lean
X : Type u
Y : Type v
ι : Sort w
α : Type u_1
β : Type u_2
x : X
s s₁ s₂ t : Set X
p✝ p₁ p₂ : X → Prop
inst✝¹ : TopologicalSpace X
F : Filter α
φ : β → α
p : Filter β
u : α → X
inst✝ : p.NeBot
h : Tendsto φ p F
H : Tendsto (u ∘ φ) p (𝓝 x)
this✝ : map (u ∘ φ) p ≤ map u F
this : map (u ∘ φ) p ≤ 𝓝 x ⊓ map u F

⊢ MapClusterPt x F u
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
θ : ℂ

⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1

⊢ cos θ = 0 ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1

⊢ (∃ n, 2 * I * θ = ↑π * I + ↑n * (2 * ↑π * I)) ↔ ∃ k, θ = (2 * ↑k + 1) * ↑π / 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
x : ℤ

⊢ 2 * I * θ = ↑π * I + ↑x * (2 * ↑π * I) ↔ θ = (2 * ↑x + 1) * ↑π / 2
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean
θ : ℂ
h : (cexp (θ * I) + cexp (-θ * I)) / 2 = 0 ↔ cexp (2 * θ * I) = -1
x : ℤ

⊢ ↑π * I + ↑x * (2 * ↑π * I) = 2 * I * ((2 * ↑x + 1) * ↑π / 2)
-- Mathlib/RingTheory/DedekindDomain/Dvr.lean
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : IsDomain A
inst✝⁵ : Field K
inst✝⁴ : IsDedekindDomain A
P : Ideal A
hP : P ≠ ⊥
pP : P.IsPrime
Aₘ : Type u_4
inst✝³ : CommRing Aₘ
inst✝² : IsDomain Aₘ
inst✝¹ : Algebra A Aₘ
inst✝ : IsLocalization.AtPrime Aₘ P

⊢ DiscreteValuationRing Aₘ
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
K : Set F
inst✝ : CompleteSpace F

⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ici x) x}
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
K : Set F
inst✝ : CompleteSpace F
this : IsComplete univ

⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ici x) x}
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
F : Type u_1
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
f : ℝ → F
K : Set F
inst✝ : CompleteSpace F
this : IsComplete univ
x✝ : ℝ

⊢ DifferentiableWithinAt ℝ f (Ici x✝) x✝ ↔ DifferentiableWithinAt ℝ f (Ici x✝) x✝ ∧ derivWithin f (Ici x✝) x✝ ∈ univ
-- Mathlib/Data/List/Cycle.lean
α : Type u_1
inst✝ : DecidableEq α
l : List α
x : α
h : l.Nodup

⊢ pmap l.prev l ⋯ = l.rotate (l.length - 1)
-- Mathlib/Data/List/Cycle.lean
α : Type u_1
inst✝ : DecidableEq α
l : List α
x : α
h : l.Nodup

⊢ (pmap l.prev l ⋯).length = (l.rotate (l.length - 1)).length
-- Mathlib/Data/List/Cycle.lean
α : Type u_1
inst✝ : DecidableEq α
l : List α
x : α
h : l.Nodup

⊢ ∀ (n : ℕ) (h₁ : n < (pmap l.prev l ⋯).length) (h₂ : n < (l.rotate (l.length - 1)).length), (pmap l.prev l ⋯).nthLe n h₁ = (l.rotate (l.length - 1)).nthLe n h₂
-- Mathlib/NumberTheory/Modular.lean
g✝ : SL(2, ℤ)
z : ℍ
R : Type u_1
inst✝ : CommRing R
g : SL(2, R)

⊢ IsCoprime (↑g 1 0) (↑g 1 1)
-- Mathlib/NumberTheory/Modular.lean
g✝ : SL(2, ℤ)
z : ℍ
R : Type u_1
inst✝ : CommRing R
g : SL(2, R)

⊢ -↑g 0 1 * ↑g 1 0 + ↑g 0 0 * ↑g 1 1 = 1
-- Mathlib/NumberTheory/Modular.lean
g✝ : SL(2, ℤ)
z : ℍ
R : Type u_1
inst✝ : CommRing R
g : SL(2, R)

⊢ (↑g).det = 1
-- Mathlib/CategoryTheory/Category/Cat.lean
X : Type u

⊢ { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (𝟙 X) = 𝟙 ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X)
-- Mathlib/CategoryTheory/Category/Cat.lean
X : Type u

⊢ autoParam (∀ (X_1 Y : ↑({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X)) (f : X_1 ⟶ Y), ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (𝟙 X)).map f = eqToHom ⋯ ≫ (𝟙 ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X)).map f ≫ eqToHom ⋯) _auto✝
-- Mathlib/CategoryTheory/Category/Cat.lean
X : Type u

⊢ ∀ (X_1 : ↑({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X)), ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (𝟙 X)).obj X_1 = (𝟙 ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X)).obj X_1
-- Mathlib/CategoryTheory/Category/Cat.lean
X✝ Y✝ Z✝ : Type u
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝

⊢ { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (f ≫ g) = { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map f ≫ { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map g
-- Mathlib/CategoryTheory/Category/Cat.lean
X✝ Y✝ Z✝ : Type u
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝

⊢ autoParam (∀ (X Y : ↑({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X✝)) (f_1 : X ⟶ Y), ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (f ≫ g)).map f_1 = eqToHom ⋯ ≫ ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map f ≫ { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map g).map f_1 ≫ eqToHom ⋯) _auto✝
-- Mathlib/CategoryTheory/Category/Cat.lean
X✝ Y✝ Z✝ : Type u
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝

⊢ ∀ (X : ↑({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.obj X✝)), ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map (f ≫ g)).obj X = ({ obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map f ≫ { obj := fun X => Cat.of (Discrete X), map := fun {X Y} f => id (Discrete.functor (Discrete.mk ∘ f)) }.map g).obj X
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
π : δ → Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : (a : δ) → MeasurableSpace (π a)
inst✝ : MeasurableSpace γ
f : δ' ≃ δ

⊢ Measurable ⇑(piCongrLeft π f)
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
π : δ → Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : (a : δ) → MeasurableSpace (π a)
inst✝ : MeasurableSpace γ
f : δ' ≃ δ

⊢ ∀ (a : δ), Measurable fun x => (piCongrLeft π f) x a
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
π : δ → Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : (a : δ) → MeasurableSpace (π a)
inst✝ : MeasurableSpace γ
f : δ' ≃ δ
i : δ

⊢ Measurable fun x => (piCongrLeft π f) x i
-- Mathlib/MeasureTheory/MeasurableSpace/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
δ' : Type u_5
ι : Sort uι
s t u : Set α
π : δ → Type u_6
inst✝² : MeasurableSpace α
inst✝¹ : (a : δ) → MeasurableSpace (π a)
inst✝ : MeasurableSpace γ
f : δ' ≃ δ
i : δ

⊢ Measurable fun x => cast ⋯ (x (f.symm i))
-- Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
x : ℂ
hx₁ : -π < x.im
hx₂ : x.im ≤ π

⊢ log (cexp x) = x
-- Mathlib/Topology/DiscreteSubset.lean
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
f : X → Y
inst✝ : DiscreteTopology X
hf : Tendsto f (cocompact X) (cocompact Y)

⊢ cofinite = cocompact X
-- Mathlib/RingTheory/Multiplicity.lean
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hdvd : a ∣ b
heq : 0 = multiplicity a b

⊢ multiplicity a b < 1
-- Mathlib/RingTheory/Multiplicity.lean
α : Type u_1
β : Type u_2
inst✝³ : Monoid α
inst✝² : Monoid β
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : α
hdvd : a ∣ b
heq : 0 = multiplicity a b

⊢ a ^ 1 ∣ b
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ

⊢ ‖jacobiTheta₂_term n z τ‖ ≤ rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ

⊢ ↑n ^ 2 * T + -(2 * S * ↑|n|) ≤ ↑n ^ 2 * τ.im + 2 * ↑n * z.im
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ

⊢ -(2 * S * ↑|n|) ≤ 2 * ↑n * z.im
-- Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean
S T : ℝ
hT : 0 < T
z τ : ℂ
hz : |z.im| ≤ S
hτ : T ≤ τ.im
n : ℤ

⊢ ↑|n| * -S ≤ ↑n * z.im