ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/RingTheory/AlgebraicIndependent.lean
ι : Type u_1
ι' : Type u_2
R : Type u_3
K✝ : Type u_4
A : Type u_5
A' : Type u_6
A'' : Type u_7
V : Type u
V' : Type u_8
x : ι → A
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing A
inst✝⁸ : CommRing A'
inst✝⁷ : CommRing A''
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R A'
inst✝⁴ : Algebra R A''
a b : R
K : Type u_9
inst✝³ : CommRing K
inst✝² : Algebra R K
inst✝¹ : Algebra K A
inst✝ : IsScalarTower R K A
hinj : Injective ⇑(algebraMap R K)
ai : AlgebraicIndependent K x
this : (aeval x).comp (MvPolynomial.map (algebraMap R K)) = (aeval x).toRingHom

⊢ Injective (⇑(aeval x).toRingHom ∘ ⇑(MvPolynomial.map (algebraMap R K)))
-- Mathlib/Algebra/Ring/Subsemiring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : NonAssocSemiring R
M : Submonoid R
inst✝¹ : NonAssocSemiring S
inst✝ : NonAssocSemiring T
f : R →+* S
t : Subsemiring S
h : t ≤ f.rangeS

⊢ map f (comap f t) = t
-- 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
zero_mem : 0 ∈ s
hx : x ∈ s
hx₀ : x ≠ 0
t : 𝕜
ht₀ : 0 < t
ht₁ : t < 1

⊢ t • x ∈ interior s
-- Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
x : ℝ
hx : Liouville x
p : ℝ

⊢ LiouvilleWith p x
-- Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
x : ℝ
hx : Liouville x
p : ℝ

⊢ LiouvilleWith (↑⌈p⌉₊) x
-- Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
x : ℝ
hx : Liouville x
p : ℝ

⊢ ∀ (x_1 : ℕ), (1 < x_1 ∧ ∃ a, x ≠ ↑a / ↑x_1 ∧ |x - ↑a / ↑x_1| < 1 / ↑x_1 ^ ⌈p⌉₊) → ∃ m, x ≠ ↑m / ↑x_1 ∧ |x - ↑m / ↑x_1| < 1 / ↑x_1 ^ ↑⌈p⌉₊
-- Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
x : ℝ
hx : Liouville x
p : ℝ
b : ℕ
_hb : 1 < b
a : ℤ
hne : x ≠ ↑a / ↑b
hlt : |x - ↑a / ↑b| < 1 / ↑b ^ ⌈p⌉₊

⊢ |x - ↑a / ↑b| < 1 / ↑b ^ ↑⌈p⌉₊
-- Mathlib/Topology/Algebra/Valued/ValuationTopology.lean
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀

⊢ (𝓝 0).HasBasis (fun x => True) fun γ => {x | v x < ↑γ}
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
p : α → Bool
a : α
pa : p a = true
x : α
xs l₂ : List α
h : a ∈ x :: xs
h' : ¬p x = true

⊢ eraseP p (xs ++ l₂) = eraseP p xs ++ l₂
-- .lake/packages/batteries/Batteries/Data/List/Lemmas.lean
α : Type u_1
p : α → Bool
a : α
pa : p a = true
x : α
xs l₂ : List α
h : a ∈ x :: xs
h' : ¬p x = true

⊢ a = x → p x = true
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
a b c d : ℝ≥0∞
r p q : ℝ≥0
x y z ε ε₁ ε₂ : ℝ≥0∞
s : Set ℝ≥0∞
f✝ g✝ : α → ℝ≥0∞
f : α → β
hf : Surjective f
g : β → ℝ≥0∞

⊢ ∑' (y : β), g y = ∑' (y : β), g (f (surjInv hf y))
-- Mathlib/RingTheory/Coprime/Lemmas.lean
R✝ : Type u
I : Type v
inst✝¹ : CommSemiring R✝
x y z : R✝
s : I → R✝
t : Finset I
R : Type u_1
inst✝ : CommRing R
a b : ℕ
h : a.Coprime b

⊢ IsCoprime ↑a ↑b
-- Mathlib/RingTheory/Coprime/Lemmas.lean
R✝ : Type u
I : Type v
inst✝¹ : CommSemiring R✝
x y z : R✝
s : I → R✝
t : Finset I
R : Type u_1
inst✝ : CommRing R
a b : ℕ
h : IsCoprime ↑a ↑b

⊢ IsCoprime ↑a ↑b
-- Mathlib/RingTheory/Coprime/Lemmas.lean
R✝ : Type u
I : Type v
inst✝¹ : CommSemiring R✝
x y z : R✝
s : I → R✝
t : Finset I
R : Type u_1
inst✝ : CommRing R
a b : ℕ
h : IsCoprime ↑a ↑b

⊢ IsCoprime ↑↑a ↑↑b
-- Mathlib/Topology/Order.lean
α : Type u
β : Type v
a : α
f : Filter α
t : TopologicalSpace α

⊢ nhdsAdjoint a f ≤ t ↔ f ≤ (fun t => 𝓝 a) t
-- Mathlib/Topology/Order.lean
α : Type u
β : Type v
a : α
f : Filter α
t : TopologicalSpace α

⊢ nhdsAdjoint a f ≤ t ↔ ∀ (s : Set α), a ∈ s → IsOpen s → s ∈ f
-- Mathlib/Algebra/MvPolynomial/Division.lean
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ

⊢ X i ∣ X j ↔ i = j
-- Mathlib/Algebra/MvPolynomial/Division.lean
σ : Type u_1
R : Type u_2
inst✝¹ : CommSemiring R
inst✝ : Nontrivial R
i j : σ

⊢ Finsupp.single i 1 ≤ Finsupp.single j 1 ↔ i = j
-- Mathlib/Computability/AkraBazzi/AkraBazzi.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r

⊢ Continuous fun p => ∑ i : α, a i * b i ^ p
-- Mathlib/Computability/AkraBazzi/AkraBazzi.lean
α : Type u_1
inst✝¹ : Fintype α
inst✝ : Nonempty α
T : ℕ → ℝ
g : ℝ → ℝ
a b : α → ℝ
r : α → ℕ → ℕ
R : AkraBazziRecurrence T g a b r
i : α
x✝ : i ∈ univ

⊢ Continuous (HPow.hPow (b i))
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Semiring α
x : α
n : ℕ

⊢ (∑ i ∈ range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Semiring α
x : α
n : ℕ
this : (∑ i ∈ range n, (x + 1) ^ i * 1 ^ (n - 1 - i)) * x + 1 ^ n = (x + 1) ^ n

⊢ (∑ i ∈ range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n
-- Mathlib/Algebra/GeomSum.lean
α : Type u
inst✝ : Semiring α
x : α
n : ℕ
this : (∑ i ∈ range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n

⊢ (∑ i ∈ range n, (x + 1) ^ i) * x + 1 = (x + 1) ^ n
-- Mathlib/Topology/Algebra/UniformConvergence.lean
α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖✝ : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
𝔖 : Set (Set α)
h𝔖₁ : 𝔖.Nonempty
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b

⊢ (𝓝 1).HasBasis (fun Si => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, (toFun 𝔖) f x ∈ b Si.2}
-- Mathlib/Topology/Algebra/UniformConvergence.lean
α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖✝ : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
𝔖 : Set (Set α)
h𝔖₁ : 𝔖.Nonempty
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b
this : (𝓤 G).HasBasis p fun i => {x | x.1 / x.2 ∈ b i}

⊢ (𝓝 1).HasBasis (fun Si => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si => {f | ∀ x ∈ Si.1, (toFun 𝔖) f x ∈ b Si.2}
-- Mathlib/Topology/Algebra/UniformConvergence.lean
α : Type u_1
G : Type u_2
ι : Type u_3
inst✝² : Group G
𝔖✝ : Set (Set α)
inst✝¹ : UniformSpace G
inst✝ : UniformGroup G
𝔖 : Set (Set α)
h𝔖₁ : 𝔖.Nonempty
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
p : ι → Prop
b : ι → Set G
h : (𝓝 1).HasBasis p b
this : (𝓤 G).HasBasis p fun i => {x | x.1 / x.2 ∈ b i}
x✝¹ : Set α × ι
x✝ : α →ᵤ[𝔖] G

⊢ (∀ x ∈ x✝¹.1, (toFun 𝔖) x✝ x ∈ b x✝¹.2) ↔ (x✝, 1) ∈ UniformOnFun.gen 𝔖 x✝¹.1 {x | x.1 / x.2 ∈ b x✝¹.2}
-- Mathlib/CategoryTheory/DifferentialObject.lean
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C

⊢ ((CategoryTheory.shiftFunctor C n).map X.d ≫ (shiftComm X.obj 1 n).hom) ≫ (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map X.d ≫ (shiftComm X.obj 1 n).hom) = 0
-- Mathlib/CategoryTheory/DifferentialObject.lean
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝

⊢ ((CategoryTheory.shiftFunctor C n).map X✝.d ≫ (shiftFunctorComm C 1 n).hom.app X✝.obj) ≫ (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map f.f) = (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map Y✝.d ≫ (shiftFunctorComm C 1 n).hom.app Y✝.obj
-- Mathlib/CategoryTheory/DifferentialObject.lean
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ : DifferentialObject S C
f : X✝ ⟶ Y✝

⊢ (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map Y✝.d ≫ (shiftComm Y✝.obj 1 n).hom = (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map Y✝.d ≫ (shiftFunctorComm C 1 n).hom.app Y✝.obj
-- Mathlib/CategoryTheory/DifferentialObject.lean
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X : DifferentialObject S C

⊢ autoParam ((CategoryTheory.shiftFunctor C n).map (𝟙 X.obj) = 𝟙 ((CategoryTheory.shiftFunctor C n).obj X.obj)) _auto✝
-- Mathlib/CategoryTheory/DifferentialObject.lean
S : Type u_1
inst✝³ : AddCommGroupWithOne S
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasShift C S
n : S
X✝ Y✝ Z✝ : DifferentialObject S C
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝

⊢ autoParam ((CategoryTheory.shiftFunctor C n).map (f.f ≫ g.f) = (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map g.f) _auto✝
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ

⊢ IsOpen (A f L r ε)
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ

⊢ ∀ x ∈ A f L r ε, ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r

⊢ ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'

⊢ ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r

⊢ ∃ ε_1 > 0, ball x ε_1 ⊆ A f L r ε
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)

⊢ ∀ y ∈ ball x' s, ∀ z ∈ ball x' s, ‖f z - f y - L (z - y)‖ < ε * r
-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
x' : E
hx' : x' ∈ ball x (r' - s)
B : ball x' s ⊆ ball x r'

⊢ ∀ y ∈ ball x' s, ∀ z ∈ ball x' s, ‖f z - f y - L (z - y)‖ < ε * r
-- Mathlib/MeasureTheory/Measure/Typeclasses.lean
α : Type u_1
β : Type u_2
δ : Type u_3
ι : Type u_4
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ ν ν₁ ν₂ : Measure α
s t : Set α
inst✝ : IsProbabilityMeasure μ
p : α → Prop
f : β → α
hs : NullMeasurableSet s μ

⊢ μ sᶜ = 1 ↔ μ s = 0
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0

⊢ (normalizedFactors a).prod ~ᵤ a
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0

⊢ (Multiset.map (⇑normalize) (Classical.choose ⋯)).prod ~ᵤ a
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0

⊢ (Multiset.map (⇑normalize) (Classical.choose ⋯)).prod ~ᵤ (Classical.choose ⋯).prod
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0

⊢ (Multiset.map (Associates.mk ∘ ⇑normalize) (Classical.choose ⋯)).prod = (Multiset.map Associates.mk (Classical.choose ⋯)).prod
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝² : CancelCommMonoidWithZero α
inst✝¹ : NormalizationMonoid α
inst✝ : UniqueFactorizationMonoid α
a : α
ane0 : a ≠ 0
x✝ : α

⊢ (Associates.mk ∘ ⇑normalize) x✝ = Associates.mk x✝
-- Mathlib/Data/Int/Log.lean
R : Type u_1
inst✝¹ : LinearOrderedSemifield R
inst✝ : FloorSemiring R
b : ℕ

⊢ clog b 1 = 0
-- Mathlib/Combinatorics/Enumerative/Composition.lean
n✝ n : ℕ
this : Fintype.card (Finset (Fin (n - 1))) = 2 ^ (n - 1)

⊢ Fintype.card (CompositionAsSet n) = Fintype.card (Finset (Fin (n - 1)))
-- Mathlib/NumberTheory/Zsqrtd/Basic.lean
d : ℕ
a b : ℤ√↑d

⊢ 0 ≤ a → 0 ≤ b → 0 ≤ a * b
-- Mathlib/NumberTheory/Zsqrtd/Basic.lean
d : ℕ
a b : ℤ√↑d

⊢ a.Nonneg → b.Nonneg → (a * b).Nonneg
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
q : R
hq : q ∈ ↑(Submonoid.closure s)

⊢ 0 * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
q : R
hq : q ∈ ↑(Submonoid.closure s)

⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
q : R
hq : q ∈ ↑(Submonoid.closure s)
p₁ p₂ : R
ihp₁ : p₁ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)
ihp₂ : p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)

⊢ p₁ * q + p₂ * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝¹ : R
h : x✝¹ ∈ closure s
x✝ y : R
hx✝ : x✝ ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
q : R
hq : q ∈ ↑(Submonoid.closure s)
x : R
hx : x * q ∈ AddSubgroup.closure ↑(Submonoid.closure s)
f : -x * q = -(x * q)

⊢ -(x * q) ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)

⊢ x * 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)

⊢ 0 ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
q₁ q₂ : R
ihq₁ : x * q₁ ∈ AddSubgroup.closure ↑(Submonoid.closure s)
ihq₂ : x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)

⊢ x * q₁ + x * q₂ ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Ring/Subring/Basic.lean
R : Type u
S : Type v
T : Type w
inst✝² : Ring R
inst✝¹ : Ring S
inst✝ : Ring T
s : Set R
x✝ : R
h : x✝ ∈ closure s
x y : R
hx : x ∈ AddSubgroup.closure ↑(Submonoid.closure s)
hy : y ∈ AddSubgroup.closure ↑(Submonoid.closure s)
z : R
hz : x * z ∈ AddSubgroup.closure ↑(Submonoid.closure s)
f : x * -z = -(x * z)

⊢ -(x * z) ∈ AddSubgroup.closure ↑(Submonoid.closure s)
-- Mathlib/Algebra/Order/Sub/Canonical.lean
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hbc : AddLECancellable (b - c)
h₁ : b ≤ a
h₂ : c ≤ b

⊢ a = a - b + c + (b - c)
-- Mathlib/MeasureTheory/Integral/Periodic.lean
T : ℝ
hT : Fact (0 < T)

⊢ ENNReal.ofReal T * (addHaarMeasure ⊤) univ = ENNReal.ofReal T
-- Mathlib/MeasureTheory/Integral/Periodic.lean
T : ℝ
hT : Fact (0 < T)

⊢ ENNReal.ofReal T * (addHaarMeasure ⊤) ↑⊤ = ENNReal.ofReal T
-- Mathlib/Algebra/Group/UniqueProds.lean
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
aA : a0 ∈ A
bB : b0 ∈ B
h : ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0
x1 x2 : G
a✝ : (x1, x2) ∈ A ×ˢ B ∧ (x1, x2).1 * (x1, x2).2 = a0 * b0
J : ∀ (y : G × G), y ∈ A ×ˢ B ∧ y.1 * y.2 = a0 * b0 → y = (x1, x2)
x y : G
hx : x ∈ A
hy : y ∈ B
l : x * y = a0 * b0

⊢ x = a0 ∧ y = b0
-- Mathlib/Algebra/Group/UniqueProds.lean
G : Type u_1
H : Type u_2
inst✝¹ : Mul G
inst✝ : Mul H
A B : Finset G
a0 b0 : G
aA : a0 ∈ A
bB : b0 ∈ B
h : ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = a0 * b0
x y : G
hx : x ∈ A
hy : y ∈ B
l : x * y = a0 * b0
a✝ : (a0, b0) ∈ A ×ˢ B ∧ (a0, b0).1 * (a0, b0).2 = a0 * b0
J : ∀ (y : G × G), y ∈ A ×ˢ B ∧ y.1 * y.2 = a0 * b0 → y = (a0, b0)

⊢ x = a0 ∧ y = b0
-- Mathlib/Topology/Order/LeftRight.lean
α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : LinearOrder α
inst✝ : TopologicalSpace β
a : α
f : α → β

⊢ ContinuousAt f a ↔ ContinuousWithinAt f (Iic a) a ∧ ContinuousWithinAt f (Ici a) a
-- Mathlib/Topology/Sheaves/Stalks.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasColimits C
X Y Z : TopCat
f : X ⟶ Y
F : Presheaf C Y
x : ↑X
V : Opens ↑X
hV : x ∈ V

⊢ ((pullback C f).obj F).germ ⟨x, hV⟩ ≫ stalkPullbackInv C f F x = germToPullbackStalk C f F V ⟨x, hV⟩
-- Mathlib/Analysis/Normed/Group/Basic.lean
𝓕 : Type u_1
𝕜 : Type u_2
α : Type u_3
ι : Type u_4
κ : Type u_5
E : Type u_6
F : Type u_7
G : Type u_8
inst✝² : SeminormedGroup E
inst✝¹ : SeminormedGroup F
inst✝ : SeminormedGroup G
s : Set E
a a₁ a₂ b b₁ b₂ : E
r r₁ r₂ : ℝ
u v w : E

⊢ ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖
-- Mathlib/Analysis/InnerProductSpace/Basic.lean
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : _root_.RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x : F
r : ℝ

⊢ ⟪x, r • x⟫_ℝ = r * (‖x‖ * ‖x‖)
-- Mathlib/Data/Prod/Lex.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : (a₁, b₁) < (a₂, b₂)

⊢ toLex (a₁, b₁) < toLex (a₂, b₂)
-- Mathlib/Data/Prod/Lex.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ b₂ : β
h : (a₁, b₁) < (a₁, b₂)

⊢ toLex (a₁, b₁) < toLex (a₁, b₂)
-- Mathlib/Data/Prod/Lex.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝¹ : PartialOrder α
inst✝ : Preorder β
a₁ : α
b₁ : β
a₂ : α
b₂ : β
h : (a₁, b₁) < (a₂, b₂)
ha : (a₁, b₁).1 < (a₂, b₂).1

⊢ toLex (a₁, b₁) < toLex (a₂, b₂)
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α : Type u
inst✝² : Group α
inst✝¹ : LE α
inst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c d : α

⊢ b⁻¹ * a ≤ c ↔ a ≤ b * c
-- 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

⊢ ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i ≤ l
-- 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
hm : Monotone fun n => ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i

⊢ ∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i ≤ l
-- Mathlib/Data/Fin/VecNotation.lean
α : Type u
m n o : ℕ
m' : Type u_1
n' : Type u_2
o' : Type u_3
v : Fin (n + 1) → α
i : Fin (n + 1)

⊢ vecAlt1 ⋯ (vecAppend ⋯ v v) i = (v ∘ fun n_1 => n_1 + n_1 + 1) i
-- Mathlib/Data/Fin/VecNotation.lean
α : Type u
m n o : ℕ
m' : Type u_1
n' : Type u_2
o' : Type u_3
v : Fin (n + 1) → α
i : Fin (n + 1)

⊢ (if h : ↑i + ↑i + 1 < n + 1 then v ⟨↑i + ↑i + 1, ⋯⟩ else v ⟨↑i + ↑i + 1 - (n + 1), ⋯⟩) = v (i + i + 1)
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ Reaches₁ (TM2.step tr) { l := some (head main q), var := s, stk := elim (trList L) [] c d } { l := some q, var := none, stk := elim (trList [L.headI]) [] c d }
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ Option.elim' (elim (trList L) [] c d main = trNat L.headI ∧ trList L.tail = []) (fun a => natEnd a = true ∧ elim (trList L) [] c d main = trNat L.headI ++ a :: trList L.tail) o
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ TransGen (fun a b => b ∈ TM2.step tr a) (TM2.stepAux (tr ((fun x => Λ'.read fun s => (if s = some Γ'.consₗ then id else Λ'.clear (fun x => decide (x = Γ'.consₗ)) main) (unrev q)) o)) o (update (update (update (elim (trList L) [] c d) main (trList L.tail)) rev ((trNat L.headI).reverseAux (elim (trList L) [] c d rev))) rev ((fun s => ((fun x => some Γ'.cons) s).iget) o :: update (update (elim (trList L) [] c d) main (trList L.tail)) rev ((trNat L.headI).reverseAux (elim (trList L) [] c d rev)) rev))) { l := some q, var := none, stk := elim (trList [L.headI]) [] c d }
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ TransGen (fun a b => (match a with | { l := none, var := var, stk := stk } => none | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some ((if o = some Γ'.consₗ then id else Λ'.clear (fun x => decide (x = Γ'.consₗ)) main) (unrev q)), var := o, stk := elim (trList L.tail) ((some Γ'.cons).iget :: (trNat L.headI).reverseAux []) c d } { l := some q, var := none, stk := elim (trNat L.headI ++ [Γ'.cons]) [] c d }
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ TransGen (fun a b => (match a with | { l := none, var := var, stk := stk } => none | { l := some l, var := v, stk := S } => some (TM2.stepAux (tr l) v S)) = some b) { l := some (Λ'.clear (fun x => decide (x = Γ'.consₗ)) main (unrev q)), var := o, stk := elim (trList L.tail) ((some Γ'.cons).iget :: (trNat L.headI).reverseAux []) c d } { l := some q, var := none, stk := elim (trNat L.headI ++ [Γ'.cons]) [] c d }
-- Mathlib/Computability/TMToPartrec.lean
q : Λ'
s : Option Γ'
L : List ℕ
c d : List Γ'
o : Option Γ' := List.casesOn L none fun x x => some Γ'.cons

⊢ ∀ x ∈ elim (trList L.tail) ((some Γ'.cons).iget :: (trNat L.headI).reverseAux []) c d main, decide (x = Γ'.consₗ) = false