ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Analysis/Asymptotics/Asymptotics.lean
α : Type u_1
β : Type u_2
E : Type u_3
F : Type u_4
G : Type u_5
E' : Type u_6
F' : Type u_7
G' : Type u_8
E'' : Type u_9
F'' : Type u_10
G'' : Type u_11
E''' : Type u_12
R : Type u_13
R' : Type u_14
𝕜 : Type u_15
𝕜' : Type u_16
inst✝¹³ : Norm E
inst✝¹² : Norm F
inst✝¹¹ : Norm G
inst✝¹⁰ : SeminormedAddCommGroup E'
inst✝⁹ : SeminormedAddCommGroup F'
inst✝⁸ : SeminormedAddCommGroup G'
inst✝⁷ : NormedAddCommGroup E''
inst✝⁶ : NormedAddCommGroup F''
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : SeminormedRing R
inst✝³ : SeminormedAddGroup E'''
inst✝² : SeminormedRing R'
inst✝¹ : NormedDivisionRing 𝕜
inst✝ : NormedDivisionRing 𝕜'
c c' c₁ c₂ : ℝ
f✝ : α → E
g : α → F
k : α → G
f' : α → E'
g' : α → F'
k' : α → G'
f'' : α → E''
g'' : α → F''
k'' : α → G''
l l' : Filter α
f : E'' → F''
s : Set E''
x₀ : E''
n : ℕ
h : f =O[𝓝[s] x₀] fun x => ‖x - x₀‖ ^ n
hx₀ : x₀ ∈ s
hn : n ≠ 0

⊢ ‖x₀ - x₀‖ ^ n = 0
-- Mathlib/CategoryTheory/Monoidal/Center.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : MonoidalCategory C
X₁ Y₁ X₂ Y₂ : Center C
f : X₁ ⟶ Y₁
g : X₂ ⟶ Y₂
U : C

⊢ (f.f ⊗ g.f) ▷ U ≫ ((Y₁.tensorObj Y₂).snd.β U).hom = ((X₁.tensorObj X₂).snd.β U).hom ≫ U ◁ (f.f ⊗ g.f)
-- Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean
R : Type u_1
inst✝¹⁰ : CommRing R
M : Type u_2
inst✝⁹ : AddCommGroup M
inst✝⁸ : Module R M
Q : QuadraticForm R M
n : ℕ
A : Type u_3
inst✝⁷ : Semiring A
inst✝⁶ : Algebra R A
M₁ : Type u_4
M₂ : Type u_5
M₃ : Type u_6
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup M₃
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R M₃
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Q₃ : QuadraticForm R M₃
m : M₁

⊢ ((map (QuadraticMap.Isometry.id Q₁)).toLinearMap ∘ₗ ι Q₁) m = ((AlgHom.id R (CliffordAlgebra Q₁)).toLinearMap ∘ₗ ι Q₁) m
-- Mathlib/LinearAlgebra/Dimension/Free.lean
R : Type u
M M₁ : Type v
M' : Type v'
inst✝¹⁰ : Ring R
inst✝⁹ : StrongRankCondition R
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : Free R M
inst✝⁵ : AddCommGroup M'
inst✝⁴ : Module R M'
inst✝³ : Free R M'
inst✝² : AddCommGroup M₁
inst✝¹ : Module R M₁
inst✝ : Free R M₁
n : ℕ
hn : finrank R M = n.succ

⊢ 0 < n.succ
-- Mathlib/Algebra/Algebra/Spectrum.lean
R : Type u
A : Type v
inst✝³ : CommSemiring R
inst✝² : Ring A
inst✝¹ : Algebra R A
inst✝ : Subsingleton A
a : A

⊢ σ a = ∅
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ

⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x

⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)

⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y

⊢ multiplicity 2 (x ^ 2 ^ i + y ^ 2 ^ i) = ↑1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y

⊢ 2 ^ 1 ∣ x ^ 2 ^ i + y ^ 2 ^ i
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
i : ℕ
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y

⊢ ¬2 ^ (1 + 1) ∣ x ^ 2 ^ i + y ^ 2 ^ i
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ

⊢ ¬2 ^ (1 + 1) ∣ x ^ 2 ^ (i + 1) + y ^ 2 ^ (i + 1)
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x y : ℤ
hx : ¬2 ∣ x
hxy : 4 ∣ x - y
hx_odd : Odd x
hxy_even : Even (x - y)
hy_odd : Odd y
i : ℕ

⊢ ∀ (x : ℤ), Odd x → x ^ 2 ^ (i + 1) % 4 = 1
-- Mathlib/NumberTheory/Multiplicity.lean
R : Type u_1
n : ℕ
x✝ y : ℤ
hx✝ : ¬2 ∣ x✝
hxy : 4 ∣ x✝ - y
hx_odd : Odd x✝
hxy_even : Even (x✝ - y)
hy_odd : Odd y
i : ℕ
x : ℤ
hx : Odd x

⊢ x ^ 2 ^ (i + 1) % 4 = 1
-- Mathlib/MeasureTheory/Measure/WithDensity.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ : Measure α
m : MeasurableSpace α
μ ν : Measure α
f : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s

⊢ ((μ + ν).withDensity f) s = (μ.withDensity f + ν.withDensity f) s
-- Mathlib/Analysis/Complex/LocallyUniformLimit.lean
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)

⊢ cderiv r (f - g) z = cderiv r f z - cderiv r g z
-- Mathlib/Analysis/Complex/LocallyUniformLimit.lean
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
h1 : ContinuousOn (fun w => ((w - z) ^ 2)⁻¹) (sphere z r)

⊢ cderiv r (f - g) z = cderiv r f z - cderiv r g z
-- Mathlib/Analysis/Complex/LocallyUniformLimit.lean
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
h1 : ContinuousOn (fun w => ((w - z) ^ 2)⁻¹) (sphere z r)

⊢ ((2 * ↑π * I)⁻¹ • ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w) = (2 * ↑π * I)⁻¹ • ((∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • f w) - ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • g w)
-- Mathlib/Analysis/Complex/LocallyUniformLimit.lean
E : Type u_1
ι : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
U K : Set ℂ
z : ℂ
M r δ : ℝ
φ : Filter ι
F : ι → ℂ → E
f g : ℂ → E
hr : 0 < r
hf : ContinuousOn f (sphere z r)
hg : ContinuousOn g (sphere z r)
h1 : ContinuousOn (fun w => ((w - z) ^ 2)⁻¹) (sphere z r)

⊢ (∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w) = (∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • f w) - ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • g w
-- Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
a x✝ z x : ℂ

⊢ abs x * Real.cos x.arg = x.re
-- Mathlib/Order/UpperLower/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s s₁ s₂ : LowerSet α
t t₁ t₂ : LowerSet β
x : α × β

⊢ s ×ˢ t = ⊥ ↔ s = ⊥ ∨ t = ⊥
-- Mathlib/Order/UpperLower/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
κ : ι → Sort u_5
inst✝¹ : Preorder α
inst✝ : Preorder β
s s₁ s₂ : LowerSet α
t t₁ t₂ : LowerSet β
x : α × β

⊢ ↑(s ×ˢ t) = ↑⊥ ↔ ↑s = ↑⊥ ∨ ↑t = ↑⊥
-- Mathlib/NumberTheory/Ostrowski.lean
f g : MulRingNorm ℚ
x✝ : ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n
c : ℝ
hc : 0 < c
h : ∀ (n : ℕ), f ↑n ^ c = g ↑n
x : ℚ

⊢ f x ^ c = g x
-- Mathlib/Data/Nat/Periodic.lean
a : ℕ

⊢ Periodic (fun n => n % a) a
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)

⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)
x✝ : K

⊢ ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)| = (B / ↑(finrank ℚ K)) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)
x✝ : K

⊢ ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial * √|↑(discr K)| / ↑(finrank ℚ K) ^ finrank ℚ K = (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal / ↑(finrank ℚ K) ^ finrank ℚ K
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)
x✝ : K

⊢ ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial * √|↑(discr K)| = (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal
-- Mathlib/NumberTheory/NumberField/Discriminant.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)
x✝ : K

⊢ (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal = ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial * √|↑(discr K)|
-- Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
z : ℍ
n : ℤ

⊢ ModularGroup.T ^ n • z = ↑n +ᵥ z
-- Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean
z : ℍ
n : ℤ

⊢ (↑((algebraMap ℤ ℝ) (↑↑(ModularGroup.T ^ n) 0 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑↑(ModularGroup.T ^ n) 0 1))) / (↑((algebraMap ℤ ℝ) (↑↑(ModularGroup.T ^ n) 1 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑↑(ModularGroup.T ^ n) 1 1))) = ↑z + ↑↑n
-- Mathlib/Data/List/Sublists.lean
α : Type u
β : Type v
γ : Type w
n : ℕ
a : α
l : List α

⊢ (sublistsLen (n + 1) (a :: l)).length = (a :: l).length.choose (n + 1)
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C

⊢ (Compatibility.υ toKaroubiCompN₂IsoN₁).hom = Γ₂N₁.hom
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C

⊢ Γ₂N₁.inv ≫ (Compatibility.υ toKaroubiCompN₂IsoN₁).hom = 𝟙 (N₁ ⋙ Γ₂)
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ (Γ₂N₁.inv ≫ (Compatibility.υ toKaroubiCompN₂IsoN₁).hom).app X = (𝟙 (N₁ ⋙ Γ₂)).app X
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ Γ₂N₁.inv.app X ≫ (Compatibility.υ toKaroubiCompN₂IsoN₁).hom.app X = (𝟙 (N₁ ⋙ Γ₂)).app X
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ ((Γ₂N₂ToKaroubiIso.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X)) ≫ (Compatibility.υ toKaroubiCompN₂IsoN₁).hom.app X = (𝟙 (N₁ ⋙ Γ₂)).app X
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ Γ₂N₂ToKaroubiIso.inv.app X ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X) ≫ Γ₂N₂.hom.app ((toKaroubiEquivalence (SimplicialObject C)).functor.obj X) ≫ Preadditive.DoldKan.equivalence.inverse.map (toKaroubiCompN₂IsoN₁.hom.app X) = (𝟙 (N₁ ⋙ Γ₂)).app X
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ Γ₂N₂ToKaroubiIso.inv.app X ≫ (𝟙 (N₂ ⋙ Γ₂)).app ((toKaroubi (SimplicialObject C)).obj X) ≫ Preadditive.DoldKan.equivalence.inverse.map (toKaroubiCompN₂IsoN₁.hom.app X) = (𝟙 (N₁ ⋙ Γ₂)).app X
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ Γ₂.map (toKaroubiCompN₂IsoN₁.inv.app X) ≫ Preadditive.DoldKan.equivalence.inverse.map (toKaroubiCompN₂IsoN₁.hom.app X) = 𝟙 ((N₁ ⋙ Γ₂).obj X)
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ Γ₂.map (toKaroubiCompN₂IsoN₁.inv.app X) ≫ Γ₂.map (toKaroubiCompN₂IsoN₁.hom.app X) = 𝟙 ((N₁ ⋙ Γ₂).obj X)
-- Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Preadditive C
inst✝¹ : IsIdempotentComplete C
inst✝ : HasFiniteCoproducts C
X : SimplicialObject C

⊢ 𝟙 (Γ₂.obj (N₁.obj X)) = 𝟙 ((N₁ ⋙ Γ₂).obj X)
-- Mathlib/NumberTheory/FermatPsp.lean
a b : ℕ
ha : 2 ≤ a
hb : 2 < b

⊢ 2 ≤ (a ^ b + 1) / (a + 1)
-- Mathlib/NumberTheory/FermatPsp.lean
a b : ℕ
ha : 2 ≤ a
hb : 2 < b

⊢ 2 * a.succ ≤ a ^ b + 1
-- Mathlib/NumberTheory/FermatPsp.lean
a b : ℕ
ha : 2 ≤ a
hb : 2 < b

⊢ Nat.mul 2 a + 1 ≤ a ^ b
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁴ : Category.{v₃, u₃} E
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
inst✝¹ : HasZeroMorphisms E
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X : C
hX : IsZero X

⊢ IsZero (F.obj X)
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/Zero.lean
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁵ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁴ : Category.{v₃, u₃} E
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
inst✝¹ : HasZeroMorphisms E
F : C ⥤ D
inst✝ : F.PreservesZeroMorphisms
X : C
hX : 𝟙 X = 0

⊢ 𝟙 (F.obj X) = 0
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b

⊢ liminf (fun i => (xs i).toReal) F = (liminf xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
liminf_le : liminf xs F ≤ b

⊢ liminf (fun i => (xs i).toReal) F = (liminf xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
liminf_le : liminf xs F ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)

⊢ liminf (fun i => (xs i).toReal) F = (liminf xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
liminf_le : liminf xs F ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (liminf xs F).toReal = b.truncateToReal (liminf xs F)

⊢ liminf (fun i => (xs i).toReal) F = (liminf xs F).toReal
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
liminf_le : liminf xs F ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (liminf xs F).toReal = b.truncateToReal (liminf xs F)

⊢ liminf (fun a => b.truncateToReal (xs a)) F = b.truncateToReal (liminf xs F)
-- Mathlib/Topology/Instances/ENNReal.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
ι : Type u_5
F : Filter ι
inst✝ : F.NeBot
b : ℝ≥0∞
b_ne_top : b ≠ ⊤
xs : ι → ℝ≥0∞
le_b : ∀ᶠ (i : ι) in F, xs i ≤ b
liminf_le : liminf xs F ≤ b
aux : ∀ᶠ (i : ι) in F, (xs i).toReal = b.truncateToReal (xs i)
aux' : (liminf xs F).toReal = b.truncateToReal (liminf xs F)
key : b.truncateToReal (liminf xs F) = liminf (b.truncateToReal ∘ xs) F

⊢ liminf (fun a => b.truncateToReal (xs a)) F = b.truncateToReal (liminf xs 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)
s : Finset I
a : I
z : I → Bool
hz : z ∈ C
hx : ↑⟨Proj (fun x => x ∈ s) z, ⋯⟩ a = true

⊢ e (π C fun x => x ∈ s) a ∈ factors C s ⟨Proj (fun x => x ∈ s) z, ⋯⟩
-- 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)
s : Finset I
a : I
z : I → Bool
hz : z ∈ C
hx : ↑⟨Proj (fun x => x ∈ s) z, ⋯⟩ a = true

⊢ ∃ a_1 ∈ s, (if Proj (fun x => x ∈ s) z a_1 = true then e (π C fun x => x ∈ s) a_1 else 1 - e (π C fun x => x ∈ s) a_1) = e (π C fun x => x ∈ s) a
-- 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)
s : Finset I
a : I
z : I → Bool
hz : z ∈ C
hx : ↑⟨Proj (fun x => x ∈ s) z, ⋯⟩ a = true

⊢ a ∈ s
-- Mathlib/RingTheory/MvPowerSeries/Basic.lean
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
m n : σ →₀ ℕ
h : m ≠ n
a : R

⊢ (coeff R m) ((monomial R n) a) = 0
-- Mathlib/Order/Interval/Set/Basic.lean
α : Type u_1
β : Type u_2
inst✝ : PartialOrder α
a b c : α
h : a < b

⊢ insert a (Ioo a b) = Ico a b
-- Mathlib/Data/Nat/Choose/Multinomial.lean
α : Type u_1
s : Finset α
f : α → ℕ
a✝ b✝ : α
n a b c : ℕ

⊢ multinomial univ ![a, b, c] = (a + b + c)! / (a ! * b ! * c !)
-- Mathlib/Data/Nat/Choose/Multinomial.lean
α : Type u_1
s : Finset α
f : α → ℕ
a✝ b✝ : α
n a b c : ℕ

⊢ (![a, b, c] 0 + ![a, b, c] 1 + ![a, b, c] 2)! / ((![a, b, c] 0)! * (![a, b, c] 1)! * (![a, b, c] 2)!) = (a + b + c)! / (a ! * b ! * c !)
-- Mathlib/Algebra/Lie/OfAssociative.lean
R : Type u
L : Type v
M : Type w
inst✝⁸ : CommRing R
inst✝⁷ : LieRing L
inst✝⁶ : LieAlgebra R L
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
inst✝³ : LieRingModule L M
inst✝² : LieModule R L M
A : Type v
inst✝¹ : Ring A
inst✝ : Algebra R A
a b : A

⊢ ((ad R A) a) b = ((LinearMap.mulLeft R - LinearMap.mulRight R) a) b
-- Mathlib/NumberTheory/LSeries/Dirichlet.lean
s : ℂ
hs : 1 < s.re

⊢ L (fun n => ↑(ζ n)) s * L (fun n => ↑(μ n)) s = 1
-- Mathlib/NumberTheory/LSeries/Dirichlet.lean
s : ℂ
hs : 1 < s.re

⊢ L ((fun x => ↑(ζ x)) ⍟ fun n => ↑(μ n)) s = 1
-- Mathlib/Data/Num/Lemmas.lean
α : Type u_1
m n : PosNum
h : ↑m = ↑n

⊢ pos m = pos n
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
ι : Type u_2
π : ι → Set (Set α)
S : Set ι
i : ι
his : i ∈ S

⊢ π i ⊆ piiUnionInter π S
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
ι : Type u_2
π : ι → Set (Set α)
S : Set ι
i : ι
his : i ∈ S
h_ss : {i} ⊆ S

⊢ π i ⊆ piiUnionInter π S
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
ι : Type u_2
π : ι → Set (Set α)
S : Set ι
i : ι
his : i ∈ S
h_ss : {i} ⊆ S

⊢ π i ⊆ piiUnionInter π {i}
-- Mathlib/MeasureTheory/PiSystem.lean
α : Type u_1
ι : Type u_2
π : ι → Set (Set α)
S : Set ι
i : ι
his : i ∈ S
h_ss : {i} ⊆ S

⊢ π i ⊆ π i ∪ {univ}
-- Mathlib/Topology/Algebra/Group/Basic.lean
G : Type w
H : Type x
α : Type u
β : Type v
inst✝² : TopologicalSpace G
inst✝¹ : Group G
inst✝ : ContinuousMul G
h : IsClosed {1}
x : G

⊢ IsClosed {x}
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝ : CompleteLattice α
h : IsCoatomistic αᵒᵈ
b : α

⊢ ∃ s, b = sSup s ∧ ∀ a ∈ s, IsAtom a
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝ : CompleteLattice α
h : IsAtomistic α
b : αᵒᵈ

⊢ ∃ s, b = sInf s ∧ ∀ a ∈ s, IsCoatom a
-- Mathlib/CategoryTheory/Subobject/Lattice.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasPullbacks C
A : C
f₁ f₂ : Subobject A
inst✝ : HasBinaryProduct f₁ f₂

⊢ (f₁ ⨯ f₂) = f₁ ⊓ f₂
-- Mathlib/CategoryTheory/Subobject/Lattice.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasPullbacks C
A : C
f₁ f₂ : Subobject A
inst✝ : HasBinaryProduct f₁ f₂

⊢ (f₁ ⨯ f₂) ≤ f₁ ⊓ f₂
-- Mathlib/CategoryTheory/Subobject/Lattice.lean
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasPullbacks C
A : C
f₁ f₂ : Subobject A
inst✝ : HasBinaryProduct f₁ f₂

⊢ f₁ ⊓ f₂ ≤ (f₁ ⨯ f₂)
-- Mathlib/MeasureTheory/Measure/VectorMeasure.lean
α : Type u_1
β : Type u_2
m : MeasurableSpace α
M : Type u_3
inst✝² : AddCommMonoid M
inst✝¹ : TopologicalSpace M
inst✝ : T2Space M
v : VectorMeasure α M
f : ℕ → Set α
A B : Set α
hA : MeasurableSet A
hB : MeasurableSet B
h : A ⊆ B

⊢ ↑v A + ↑v (B \ A) = ↑v B
-- Mathlib/Combinatorics/Quiver/Covering.lean
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type ?u.7311
inst✝ : Quiver W
ψ : V ⥤q W
u : U

⊢ φ.symmetrify.costar u = ⇑(symmetrifyCostar (φ.obj u)).symm ∘ Sum.map (φ.costar u) (φ.star u) ∘ ⇑(symmetrifyCostar u)
-- Mathlib/Combinatorics/Quiver/Covering.lean
U : Type u_1
inst✝² : Quiver U
V : Type u_2
inst✝¹ : Quiver V
φ : U ⥤q V
W : Type ?u.7311
inst✝ : Quiver W
ψ : V ⥤q W
u : U

⊢ ⇑(symmetrifyCostar (φ.obj u)) ∘ φ.symmetrify.costar u = Sum.map (φ.costar u) (φ.star u) ∘ ⇑(symmetrifyCostar u)
-- Mathlib/Data/Real/ConjExponents.lean
a b p q : ℝ
h : p.IsConjExponent q
ha : 0 < a
hb : 0 < b
hab : a + b = 1

⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1
-- Mathlib/Logic/Equiv/Option.lean
α : Type u_1
β : Type u_2
γ : Type u_3
e : Option α ≃ Option β
x : α
h : e (some x) = none

⊢ some (e.removeNone_aux x) = e none
-- Mathlib/SetTheory/Cardinal/Basic.lean
α✝ β✝ : Type u
a b : Cardinal.{v}
α β : Type v

⊢ lift.{u, v} #α ≤ lift.{u, v} #β ↔ #α ≤ #β
-- Mathlib/SetTheory/Cardinal/Basic.lean
α✝ β✝ : Type u
a b : Cardinal.{v}
α β : Type v

⊢ lift.{max v u, v} #α ≤ lift.{max v u, v} #β ↔ #α ≤ #β
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y

⊢ (mkMetric fun x => ⊤) = ⊤
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y

⊢ ⊤ ≤ ⨆ r, ⨆ (_ : r > 0), ⊤
-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝¹ : EMetricSpace X
inst✝ : EMetricSpace Y

⊢ ∀ (b : OuterMeasure X), (∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b) → ⊤ ≤ b
-- Mathlib/MeasureTheory/Function/Jacobian.lean
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
s : Set E
f✝ : E → E
f' : E → E →L[ℝ] E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hs : MeasurableSet s
f : E ≃ᵐ E
g : E → ℝ
hg : ∀ᵐ (x : E) ∂μ, x ∈ ⇑f '' s → 0 ≤ g x
hg_int : IntegrableOn g (⇑f '' s) μ
hf' : ∀ x ∈ s, HasFDerivWithinAt (⇑f) (f' x) s x

⊢ (Measure.map (⇑f.symm) (μ.withDensity fun x => ENNReal.ofReal (g x))) s = ENNReal.ofReal (∫ (x : E) in s, |(f' x).det| * g (f x) ∂μ)
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
f : α → β
h : (f '' s).ncard = s.ncard
hs : autoParam s.Finite _auto✝

⊢ InjOn f s
-- Mathlib/Data/Set/Card.lean
α : Type u_1
β : Type u_2
s t : Set α
f : α → β
h : (f '' s).encard = s.encard
hs : autoParam s.Finite _auto✝

⊢ InjOn f s
-- Mathlib/Order/Basic.lean
ι : Type u_1
α : Type u_2
β : Type u_3
π : ι → Type u_4
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → Preorder (π i)
x y : (i : ι) → π i
i : ι
a b : π i

⊢ update x i a ≤ update x i b ↔ a ≤ b
-- Mathlib/Probability/Independence/Kernel.lean
α : Type u_1
Ω : Type u_2
ι : Type u_3
_mα : MeasurableSpace α
f : ι → Set Ω
_mΩ : MeasurableSpace Ω
κ : Kernel α Ω
μ : Measure α

⊢ iIndep (fun i => MeasurableSpace.comap (fun x => x ∈ f i) ⊤) κ μ ↔ iIndepSet f κ μ
-- Mathlib/Tactic/NormNum/Pow.lean
a b✝ c : ℕ
α : Type u_1
inst✝ : Ring α
an : ℤ
ad b : ℕ
inv✝ : Invertible ↑ad
this : Invertible (↑ad ^ b)

⊢ IsRat ((↑an * ⅟↑ad) ^ b) (an ^ b) (ad ^ b)