ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/Computability/Primrec.lean
this : Primrec' fun v => (fun a b => b - a) v.head v.tail.head

⊢ Primrec' fun v => v.head - v.tail.head
-- Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : LocallyIntegrableOn f s μ
hg : ContinuousOn g s
hs : IsOpen s

⊢ LocallyIntegrableOn (fun x => g x * f x) s μ
-- Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
X : Type u_1
Y : Type u_2
E : Type u_3
F : Type u_4
R : Type u_5
inst✝¹⁰ : MeasurableSpace X
inst✝⁹ : TopologicalSpace X
inst✝⁸ : MeasurableSpace Y
inst✝⁷ : TopologicalSpace Y
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
f✝ g✝ : X → E
μ : Measure X
s✝ : Set X
inst✝⁴ : OpensMeasurableSpace X
A K : Set X
inst✝³ : LocallyCompactSpace X
inst✝² : T2Space X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
f g : X → R
s : Set X
hf : ∀ k ⊆ s, IsCompact k → IntegrableOn f k μ
hg : ContinuousOn g s
hs : IsOpen s

⊢ ∀ k ⊆ s, IsCompact k → IntegrableOn (fun x => g x * f x) k μ
-- Mathlib/RingTheory/HahnSeries/PowerSeries.lean
Γ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : StrictOrderedSemiring Γ
σ : Type u_3
inst✝ : Finite σ
f g : HahnSeries (σ →₀ ℕ) R
n : σ →₀ ℕ

⊢ (MvPowerSeries.coeff R n) ({ toFun := fun f => f.coeff, invFun := fun f => { coeff := f, isPWO_support' := ⋯ }, left_inv := ⋯, right_inv := ⋯ }.toFun (f * g)) = (MvPowerSeries.coeff R n) ({ toFun := fun f => f.coeff, invFun := fun f => { coeff := f, isPWO_support' := ⋯ }, left_inv := ⋯, right_inv := ⋯ }.toFun f * { toFun := fun f => f.coeff, invFun := fun f => { coeff := f, isPWO_support' := ⋯ }, left_inv := ⋯, right_inv := ⋯ }.toFun g)
-- Mathlib/RingTheory/HahnSeries/PowerSeries.lean
Γ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : StrictOrderedSemiring Γ
σ : Type u_3
inst✝ : Finite σ
f g : HahnSeries (σ →₀ ℕ) R
n : σ →₀ ℕ

⊢ (MvPowerSeries.coeff R n) (f * g).coeff = ∑ p ∈ antidiagonal n, (MvPowerSeries.coeff R p.1) f.coeff * (MvPowerSeries.coeff R p.2) g.coeff
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n✝ m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p q : MvPolynomial σ R
n : σ
f : MvPolynomial σ R
d : ℕ

⊢ degreeOf n f ≤ d ↔ ∀ m ∈ f.support, m n ≤ d
-- Mathlib/RingTheory/WittVector/Defs.lean
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
x y : 𝕎 R

⊢ (x + y).coeff 0 = x.coeff 0 + y.coeff 0
-- Mathlib/Topology/ContinuousOn.lean
α✝ : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝² : TopologicalSpace α✝
ι✝ : Type u_5
π : ι✝ → Type u_6
inst✝¹ : (i : ι✝) → TopologicalSpace (π i)
ι : Type u_7
α : ι → Type u_8
inst✝ : (i : ι) → TopologicalSpace (α i)
s : (i : ι) → Set (α i)
I : Set ι
hs : ∀ i ∈ I, Dense (s i)

⊢ Dense (I.pi s)
-- Mathlib/Data/Complex/Exponential.lean
x✝ y : ℂ
x : ℝ

⊢ (cos ↑x).im = 0
-- Mathlib/MeasureTheory/Constructions/Projective.lean
ι : Type u_1
α : ι → Type u_2
inst✝ : (i : ι) → MeasurableSpace (α i)
P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)
I J : Finset ι
hP : IsProjectiveMeasureFamily P
S : Set ((i : { x // x ∈ I }) → α ↑i)
T : Set ((i : { x // x ∈ J }) → α ↑i)
hT : MeasurableSet T
h_eq : cylinder I S = cylinder J T
hJI : J ⊆ I

⊢ (P I) S = (P J) T
-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β

⊢ (corec' g x).dest = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x
-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β

⊢ (TypeVec.id ::: corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g)) <$$> Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g (Sum.inr x) = (TypeVec.id ::: Sum.elim _root_.id (corec' g)) <$$> g x
-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β

⊢ corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g) = Sum.elim _root_.id (corec' g)
-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x : β
i : Cofix F α

⊢ corec (MvFunctor.map (TypeVec.id ::: fun x => x) ∘ Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> val.dest) fun val => g val) (Sum.inl i) = i
-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
β : Type u
g : β → F (α ::: (Cofix F α ⊕ β))
x i : β

⊢ corec (MvFunctor.map (TypeVec.id ::: fun x => x) ∘ Sum.rec (fun val => (TypeVec.id ::: Sum.inl) <$$> val.dest) fun val => g val) (Sum.inr i) = corec (Sum.elim (MvFunctor.map (TypeVec.id ::: Sum.inl) ∘ dest) g) (Sum.inr i)
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x y : ℝ
h : x * y < 1

⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x y : ℝ
h : x * y < 1

⊢ arctan (tan (arctan x + arctan y)) = arctan ((x + y) / (1 - x * y))
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x y : ℝ
h : x * y < 1

⊢ -(π / 2) < arctan x + arctan y
-- Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
x y : ℝ
h : x * y < 1

⊢ arctan x + arctan y < π / 2
-- Mathlib/Analysis/NormedSpace/MatrixExponential.lean
𝕂 : Type u_1
m : Type u_2
n : Type u_3
p : Type u_4
n' : m → Type u_5
𝔸 : Type u_6
inst✝¹¹ : Fintype m
inst✝¹⁰ : DecidableEq m
inst✝⁹ : Fintype n
inst✝⁸ : DecidableEq n
inst✝⁷ : (i : m) → Fintype (n' i)
inst✝⁶ : (i : m) → DecidableEq (n' i)
inst✝⁵ : Field 𝕂
inst✝⁴ : Ring 𝔸
inst✝³ : TopologicalSpace 𝔸
inst✝² : TopologicalRing 𝔸
inst✝¹ : Algebra 𝕂 𝔸
inst✝ : T2Space 𝔸
v : (i : m) → Matrix (n' i) (n' i) 𝔸

⊢ exp 𝕂 (blockDiagonal' v) = blockDiagonal' (exp 𝕂 v)
-- Mathlib/Analysis/Calculus/Rademacher.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s : Set E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f

⊢ ∀ᵐ (x : E) ∂μ, DifferentiableAt ℝ f x
-- Mathlib/Analysis/Calculus/Rademacher.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s✝ : Set E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
s_count : s.Countable
s_dense : Dense s

⊢ ∀ᵐ (x : E) ∂μ, DifferentiableAt ℝ f x
-- Mathlib/Analysis/Calculus/Rademacher.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s✝ : Set E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
s_count : s.Countable
s_dense : Dense s
hs : sphere 0 1 ⊆ closure s

⊢ ∀ᵐ (x : E) ∂μ, DifferentiableAt ℝ f x
-- Mathlib/Analysis/Calculus/Rademacher.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s✝ : Set E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
s_count : s.Countable
s_dense : Dense s
hs : sphere 0 1 ⊆ closure s

⊢ ∀ (a : E), (∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) a v) → DifferentiableAt ℝ f a
-- Mathlib/Analysis/Calculus/Rademacher.lean
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f g : E → ℝ
s✝ : Set E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
hf : LipschitzWith C f
s : Set E
s_count : s.Countable
s_dense : Dense s
hs : sphere 0 1 ⊆ closure s
x : E
L : E →L[ℝ] ℝ
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v

⊢ DifferentiableAt ℝ f x
-- Mathlib/Data/Fintype/Lattice.lean
ι : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
s✝ : Finset α
inst✝ : DecidableEq α
s : Finset ι
f : ι → Finset α
a : α

⊢ a ∈ s.inf f ↔ ∀ i ∈ s, a ∈ f i
-- Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
x : ℝ≥0∞
n : ℕ
z : ℝ

⊢ x ^ (↑n * z) = (x ^ n) ^ z
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν'✝ ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
h1 : μ ≪ ν
ν' : Measure α
s : Set α
hs : (ν + ν') s = 0

⊢ μ s = 0
-- Mathlib/MeasureTheory/Measure/MeasureSpace.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ι : Type u_5
R : Type u_6
R' : Type u_7
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν'✝ ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
h1 : μ ≪ ν
ν' : Measure α
s : Set α
hs : ν s = 0 ∧ ν' s = 0

⊢ μ s = 0
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α : Type u
inst✝³ : Group α
inst✝² : LinearOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c : α
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

⊢ a / b ≤ a⁻¹ * b ↔ a ≤ b
-- Mathlib/Algebra/Order/Group/Unbundled/Basic.lean
α : Type u
inst✝³ : Group α
inst✝² : LinearOrder α
inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1
a b c : α
inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

⊢ a * a ≤ b * b ↔ a ≤ b
-- Mathlib/NumberTheory/EulerProduct/Basic.lean
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
hsum : Summable fun x => ‖f x‖

⊢ HasProd (fun p => (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)
-- Mathlib/NumberTheory/EulerProduct/Basic.lean
F : Type u_1
inst✝¹ : NormedField F
inst✝ : CompleteSpace F
f : ℕ →*₀ F
hsum : Summable fun x => ‖f x‖
H : (fun p => (1 - f ↑p)⁻¹) = fun p => ∑' (e : ℕ), f (↑p ^ e)

⊢ HasProd (fun p => (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)
-- Mathlib/Algebra/CharP/ExpChar.lean
R✝ : Type u
R : Type u_1
A : Type u_2
inst✝² : Semiring R
inst✝¹ : Semiring A
f : R →+* A
H : Function.Injective ⇑f
p : ℕ
inst✝ : ExpChar A p

⊢ ExpChar R p
-- Mathlib/MeasureTheory/Measure/MutuallySingular.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ μ : Measure α

⊢ μ ⟂ₘ μ ↔ μ = 0
-- Mathlib/MeasureTheory/Measure/MutuallySingular.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ μ : Measure α

⊢ μ ⟂ₘ μ → μ = 0
-- Mathlib/MeasureTheory/Measure/MutuallySingular.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s = 0
hμs_compl : μ sᶜ = 0

⊢ μ = 0
-- Mathlib/MeasureTheory/Measure/MutuallySingular.lean
α : Type u_1
m0 : MeasurableSpace α
μ✝ μ₁ μ₂ ν ν₁ ν₂ μ : Measure α
s : Set α
hs : MeasurableSet s
hμs : μ s = 0
hμs_compl : μ sᶜ = 0

⊢ μ univ = 0
-- Mathlib/CategoryTheory/Equivalence.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
e : C ≌ D
W X X' Y Y' Z : C
f : W ⟶ X
g : X ⟶ Y
h : Y ⟶ Z
f' : W ⟶ X'
g' : X' ⟶ Y'
h' : Y' ⟶ Z

⊢ f ≫ g ≫ h ≫ e.unit.app Z = f' ≫ g' ≫ h' ≫ e.unit.app Z ↔ f ≫ g ≫ h = f' ≫ g' ≫ h'
-- Mathlib/Algebra/Homology/ShortComplex/Exact.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.45519, u_1} C
inst✝³ : Category.{?u.45523, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
cc : CokernelCofork S.f
hcc : IsColimit cc

⊢ Mono ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))
-- Mathlib/Algebra/Homology/ShortComplex/Exact.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.45519, u_1} C
inst✝³ : Category.{?u.45523, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
cc : CokernelCofork S.f
hcc : IsColimit cc
this : S.HasHomology

⊢ Mono ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))
-- Mathlib/Algebra/Homology/ShortComplex/Exact.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.45519, u_1} C
inst✝³ : Category.{?u.45523, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
cc : CokernelCofork S.f
hcc : IsColimit cc
this : S.HasHomology

⊢ Arrow.mk ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯)) ≅ Arrow.mk S.fromOpcycles
-- Mathlib/Algebra/Homology/ShortComplex/Exact.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.45519, u_1} C
inst✝³ : Category.{?u.45523, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
cc : CokernelCofork S.f
hcc : IsColimit cc
this : S.HasHomology

⊢ (hcc.coconePointUniqueUpToIso S.opcyclesIsCokernel).hom ≫ (Arrow.mk S.fromOpcycles).hom = (Arrow.mk ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))).hom ≫ (Iso.refl (Arrow.mk ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))).right).hom
-- Mathlib/Algebra/Homology/ShortComplex/Exact.lean
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{?u.45519, u_1} C
inst✝³ : Category.{?u.45523, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
hS : S.Exact
inst✝ : HasZeroObject C
cc : CokernelCofork S.f
hcc : IsColimit cc
this : S.HasHomology

⊢ Cofork.π cc ≫ (hcc.coconePointUniqueUpToIso S.opcyclesIsCokernel).hom ≫ (Arrow.mk S.fromOpcycles).hom = Cofork.π cc ≫ (Arrow.mk ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))).hom ≫ (Iso.refl (Arrow.mk ((hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯))).right).hom
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p q : MvPolynomial σ R

⊢ totalDegree 0 = 0
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p q : MvPolynomial σ R

⊢ (C 0).totalDegree = 0
-- Mathlib/Algebra/Order/Ring/Abs.lean
α : Type u_1
inst✝ : LinearOrderedRing α
n : ℕ
a b c : α

⊢ |a| < |b| ↔ a * a < b * b
-- Mathlib/Algebra/Order/Ring/Abs.lean
α : Type u_1
inst✝ : LinearOrderedRing α
n : ℕ
a b c : α

⊢ |a| < |b| ↔ |a| * |a| < |b| * |b|
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
h1 : x + 2 * x = 3 * x

⊢ cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
h1 : x + 2 * x = 3 * x

⊢ cosh x * cosh (2 * x) + sinh x * sinh (2 * x) = 4 * cosh x ^ 3 - 3 * cosh x
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
h1 : x + 2 * x = 3 * x

⊢ cosh x * (cosh x ^ 2 + sinh x ^ 2) + sinh x * (2 * sinh x * cosh x) = 4 * cosh x ^ 3 - 3 * cosh x
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2

⊢ cosh x * (cosh x ^ 2 + sinh x ^ 2) + sinh x * (2 * sinh x * cosh x) = 4 * cosh x ^ 3 - 3 * cosh x
-- Mathlib/Data/Complex/Exponential.lean
x y : ℂ
h1 : x + 2 * x = 3 * x
h2 : sinh x * (2 * sinh x * cosh x) = 2 * cosh x * sinh x ^ 2

⊢ cosh x * (cosh x ^ 2 + (cosh x ^ 2 - 1)) + 2 * cosh x * (cosh x ^ 2 - 1) = 4 * cosh x ^ 3 - 3 * cosh x
-- Mathlib/CategoryTheory/Limits/Cones.lean
J : Type u₁
inst✝³ : Category.{v₁, u₁} J
K : Type u₂
inst✝² : Category.{v₂, u₂} K
C : Type u₃
inst✝¹ : Category.{v₃, u₃} C
D : Type u₄
inst✝ : Category.{v₄, u₄} D
F : J ⥤ C
e : K ≌ J
s : Cocone (e.functor ⋙ F)
k : K

⊢ (((whiskering e.inverse ⋙ precompose (F.leftUnitor.inv ≫ whiskerRight e.counitIso.inv F ≫ (e.inverse.associator e.functor F).inv)) ⋙ whiskering e.functor).obj s).ι.app k ≫ (Iso.refl (((whiskering e.inverse ⋙ precompose (F.leftUnitor.inv ≫ whiskerRight e.counitIso.inv F ≫ (e.inverse.associator e.functor F).inv)) ⋙ whiskering e.functor).obj s).pt).hom = ((𝟭 (Cocone (e.functor ⋙ F))).obj s).ι.app k
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0

⊢ edist z x = edist z y
-- Mathlib/Topology/EMetricSpace/Basic.lean
α : Type u
β : Type v
X : Type u_1
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0

⊢ edist x z = edist y z
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop

⊢ Cardinal.univ.{u, v} ≤ univ.{u, v}.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{max (u + 1) v}
h : c < Cardinal.univ.{u, v}

⊢ c < univ.{u, v}.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{u}
h : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}

⊢ Cardinal.lift.{max (u + 1) v, u} c < univ.{u, v}.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{u}
h : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof

⊢ Cardinal.lift.{max (u + 1) v, u} c < univ.{u, v}.cof
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{u}
h : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof

⊢ Cardinal.lift.{u + 1, u} c < #↑S
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{u}
h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h : Cardinal.lift.{u + 1, u} c ≥ #↑S

⊢ False
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α : Type u_1
r : α → α → Prop
c : Cardinal.{u}
h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e : Cardinal.lift.{u + 1, u} a = #↑S

⊢ False
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
c : Cardinal.{u}
h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e✝ : Cardinal.lift.{u + 1, u} a = #↑S
α : Type u
e : Cardinal.lift.{u + 1, u} ⟦α⟧ = #↑S

⊢ False
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
c : Cardinal.{u}
h✝ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e✝ : Cardinal.lift.{u + 1, u} a = #↑S
α : Type u
e : Cardinal.lift.{u + 1, u} ⟦α⟧ = #↑S
f : ULift.{u + 1, u} α ≃ ↑S

⊢ False
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
c : Cardinal.{u}
h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e✝ : Cardinal.lift.{u + 1, u} a = #↑S
α : Type u
e : Cardinal.lift.{u + 1, u} ⟦α⟧ = #↑S
f✝ : ULift.{u + 1, u} α ≃ ↑S
f : α ≃ ↑S
g : α → Ordinal.{u} := fun a => ↑(f a)
o : Ordinal.{u} := succ (sup g)
b : Ordinal.{u}
h : b ∈ S
l : ¬(fun x x_1 => x < x_1) b o

⊢ False
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
c : Cardinal.{u}
h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e✝ : Cardinal.lift.{u + 1, u} a = #↑S
α : Type u
e : Cardinal.lift.{u + 1, u} ⟦α⟧ = #↑S
f✝ : ULift.{u + 1, u} α ≃ ↑S
f : α ≃ ↑S
g : α → Ordinal.{u} := fun a => ↑(f a)
o : Ordinal.{u} := succ (sup g)
b : Ordinal.{u}
h : b ∈ S
l : ¬(fun x x_1 => x < x_1) b o

⊢ b ≤ sup g
-- Mathlib/SetTheory/Cardinal/Cofinality.lean
α✝ : Type u_1
r : α✝ → α✝ → Prop
c : Cardinal.{u}
h✝¹ : Cardinal.lift.{max (u + 1) v, u} c < Cardinal.univ.{u, v}
S : Set Ordinal.{u}
H : Unbounded (fun x x_1 => x < x_1) S
Se : #↑S = (type fun x x_1 => x < x_1).cof
h✝ : Cardinal.lift.{u + 1, u} c ≥ #↑S
a : Cardinal.{u}
e✝ : Cardinal.lift.{u + 1, u} a = #↑S
α : Type u
e : Cardinal.lift.{u + 1, u} ⟦α⟧ = #↑S
f✝ : ULift.{u + 1, u} α ≃ ↑S
f : α ≃ ↑S
g : α → Ordinal.{u} := fun a => ↑(f a)
o : Ordinal.{u} := succ (sup g)
b : Ordinal.{u}
h : b ∈ S
l : ¬(fun x x_1 => x < x_1) b o

⊢ g (f.symm ⟨b, h⟩) ≤ sup g
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
P X Y Z : C
f : P ⟶ X
g : P ⟶ Y
inst✝² : HasBinaryProduct X Y
inst✝¹ : HasBinaryProduct (G.obj X) (G.obj Y)
inst✝ : PreservesLimit (pair X Y) G

⊢ (iso G X Y).inv ≫ G.map prod.snd = prod.snd
-- Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
P X Y Z : C
f : P ⟶ X
g : P ⟶ Y
inst✝² : HasBinaryProduct X Y
inst✝¹ : HasBinaryProduct (G.obj X) (G.obj Y)
inst✝ : PreservesLimit (pair X Y) G

⊢ 𝟙 (G.obj (X ⨯ Y)) ≫ G.map prod.snd = (iso G X Y).hom ≫ prod.snd
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p✝ q p : MvPolynomial σ R
i : σ
c : R

⊢ Multiset.count i (C c * p).degrees ≤ Multiset.count i p.degrees
-- Mathlib/Algebra/MvPolynomial/Degrees.lean
R : Type u
S : Type v
σ : Type u_1
τ : Type u_2
r : R
e : ℕ
n m : σ
s : σ →₀ ℕ
inst✝ : CommSemiring R
p✝ q p : MvPolynomial σ R
i : σ
c : R

⊢ p.degrees = (C c).degrees + p.degrees
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝² : CommRing R
E : EllipticCurve R
j : R
inst✝¹ : Invertible j
inst✝ : Invertible (j - 1728)

⊢ (ofJ' j).j = j
-- Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean
R : Type u
inst✝² : CommRing R
E : EllipticCurve R
j : R
inst✝¹ : Invertible j
inst✝ : Invertible (j - 1728)

⊢ j * (j ^ 2 * (j - 1728) ^ 9) = (j * (j - 1728) ^ 3) ^ 3
-- .lake/packages/batteries/Batteries/Data/String/Lemmas.lean
α : Type u_1
f : α → Char → α
init : α
x✝ : Substring
h✝ : x✝.Valid
w✝² w✝¹ w✝ : List Char
h : ValidFor w✝² w✝¹ w✝ x✝

⊢ Substring.foldl f init x✝ = List.foldl f init x✝.toString.data
-- Mathlib/NumberTheory/Modular.lean
g : SL(2, ℤ)
z : ℍ
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)

⊢ ¬cd 0 ^ 2 + (-cd 1) ^ 2 = 0
-- Mathlib/NumberTheory/Modular.lean
g : SL(2, ℤ)
z : ℍ
cd : Fin 2 → ℤ
hcd : IsCoprime (cd 0) (cd 1)

⊢ ¬cd 0 ^ 2 + cd 1 ^ 2 = 0
-- Mathlib/NumberTheory/Divisors.lean
n : ℕ
a✝ : ℕ × ℕ

⊢ a✝ ∈ divisorsAntidiagonal 1 ↔ a✝ ∈ {(1, 1)}
-- Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
n : ℕ

⊢ eulerMascheroniSeq n < eulerMascheroniConstant
-- Mathlib/NumberTheory/Harmonic/EulerMascheroni.lean
n : ℕ

⊢ eulerMascheroniSeq n.succ ≤ eulerMascheroniConstant
-- Mathlib/Topology/MetricSpace/HausdorffDistance.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s t u : Set α
x y : α
Φ : α → β
h : y ∈ s

⊢ infDist x (s ∩ closedBall x (dist y x)) = infDist x s
-- Mathlib/Topology/MetricSpace/HausdorffDistance.lean
ι : Sort u_1
α : Type u
β : Type v
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
s t u : Set α
x y : α
Φ : α → β
h : y ∈ s ∩ closedBall x (dist y x)

⊢ infDist x (s ∩ closedBall x (dist y x)) = infDist x s