ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_5, u_2} D
inst✝⁷ : Category.{u_6, u_3} E
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : Preadditive E
F : C ⥤ D
inst✝³ : F.Additive
inst✝² : F.Full
inst✝¹ : F.EssSurj
G : D ⥤ E
inst✝ : (F ⋙ G).Additive
X Y : D
f g : X ⟶ Y
f' : F.objPreimage X ⟶ F.objPreimage Y
hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv
g' : F.objPreimage X ⟶ F.objPreimage Y
hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv

⊢ G.map (F.map f' + F.map g') = (F ⋙ G).map (f' + g')
-- Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁹ : Category.{u_4, u_1} C
inst✝⁸ : Category.{u_5, u_2} D
inst✝⁷ : Category.{u_6, u_3} E
inst✝⁶ : Preadditive C
inst✝⁵ : Preadditive D
inst✝⁴ : Preadditive E
F : C ⥤ D
inst✝³ : F.Additive
inst✝² : F.Full
inst✝¹ : F.EssSurj
G : D ⥤ E
inst✝ : (F ⋙ G).Additive
X Y : D
f g : X ⟶ Y
f' : F.objPreimage X ⟶ F.objPreimage Y
hf' : F.map f' = (F.objObjPreimageIso X).hom ≫ f ≫ (F.objObjPreimageIso Y).inv
g' : F.objPreimage X ⟶ F.objPreimage Y
hg' : F.map g' = (F.objObjPreimageIso X).hom ≫ g ≫ (F.objObjPreimageIso Y).inv

⊢ G.map (F.map f' + F.map g') = G.map (F.map (f' + g'))
-- Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
H : T ∈ distinguishedTriangles

⊢ T.mor₁ ≫ T.mor₂ = 0
-- Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
H : T ∈ distinguishedTriangles
c : (contractibleTriangle T.obj₁).obj₃ ⟶ T.obj₃
hc : (contractibleTriangle T.obj₁).mor₂ ≫ c = T.mor₁ ≫ T.mor₂ ∧ (contractibleTriangle T.obj₁).mor₃ ≫ (shiftFunctor C 1).map (𝟙 T.obj₁) = c ≫ T.mor₃

⊢ T.mor₁ ≫ T.mor₂ = 0
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝¹ : Inhabited Γ
Λ : Type u_2
inst✝ : Inhabited Λ
M : TM0.Machine Γ Λ
x✝ : Cfg₀
q : Λ
T : Tape Γ
q' : Λ
s : TM0.Stmt Γ
e : M q T.head = some (q', s)

⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝¹ : Inhabited Γ
Λ : Type u_2
inst✝ : Inhabited Λ
M : TM0.Machine Γ Λ
x✝ : Cfg₀
q : Λ
T : Tape Γ
q' : Λ
s : TM0.Stmt Γ
e : M q T.head = some (q', s)

⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T } { l := match match M q' (match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T).head with | some val => true | none => false with | true => some (Λ'.normal q') | false => none, var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝¹ : Inhabited Γ
Λ : Type u_2
inst✝ : Inhabited Λ
M : TM0.Machine Γ Λ
x✝ : Cfg₀
q : Λ
T : Tape Γ
q' : Λ
s : TM0.Stmt Γ
this : TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } = some { l := some (Λ'.normal q'), var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
e : M q T.head = some (q', s)

⊢ Reaches₁ (TM1.step (tr M)) { l := some (Λ'.normal q), var := (), Tape := T } { l := match match M q' (match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T).head with | some val => true | none => false with | true => some (Λ'.normal q') | false => none, var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝¹ : Inhabited Γ
Λ : Type u_2
inst✝ : Inhabited Λ
M : TM0.Machine Γ Λ
x✝ : Cfg₀
q : Λ
T : Tape Γ
q' : Λ
s : TM0.Stmt Γ
this : TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } = some { l := some (Λ'.normal q'), var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
e : M q T.head = some (q', s)

⊢ { l := some (Λ'.act s q'), var := (), Tape := T } ∈ TM1.step (tr M) { l := some (Λ'.normal q), var := (), Tape := T }
-- Mathlib/Computability/TuringMachine.lean
Γ : Type u_1
inst✝¹ : Inhabited Γ
Λ : Type u_2
inst✝ : Inhabited Λ
M : TM0.Machine Γ Λ
x✝ : Cfg₀
q : Λ
T : Tape Γ
q' : Λ
s : TM0.Stmt Γ
this : TM1.step (tr M) { l := some (Λ'.act s q'), var := (), Tape := T } = some { l := some (Λ'.normal q'), var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
e : M q T.head = some (q', s)

⊢ ReflTransGen (fun a b => b ∈ TM1.step (tr M) a) { l := some (Λ'.normal q'), var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T } { l := match match M q' (match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T).head with | some val => true | none => false with | true => some (Λ'.normal q') | false => none, var := (), Tape := match s with | TM0.Stmt.move d => Tape.move d T | TM0.Stmt.write a => Tape.write a T }
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : Group G
a✝ b c d : G
n✝ : ℤ
a : G
m n : ℕ
h : n ≤ m

⊢ a ^ (m - n) * a ^ n = a ^ m
-- Mathlib/Topology/ContinuousFunction/Compact.lean
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : MetricSpace β
inst✝ : NormedAddCommGroup E
f g : C(α, β)
C : ℝ
x : α

⊢ dist (f x) (g x) ≤ dist f g
-- Mathlib/LinearAlgebra/Basis.lean
ι : Type u_1
ι' : Type u_2
R : Type u_3
R₂ : Type u_4
K : Type u_5
M : Type u_6
M' : Type u_7
M'' : Type u_8
V : Type u
V' : Type u_9
inst✝⁴ : Semiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid M'
inst✝ : Module R M'
b b₁ : Basis ι R M
i : ι
c : R
x m : M

⊢ b.sumCoords m = ∑ᶠ (i : ι), (b.coord i) m
-- Mathlib/SetTheory/Game/Nim.lean
G : PGame
inst✝ : G.Impartial

⊢ (-G).grundyValue = G.grundyValue
-- Mathlib/CategoryTheory/Shift/CommShift.lean
C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{u_7, u_1} C
inst✝⁹ : Category.{u_6, u_2} D
inst✝⁸ : Category.{?u.40009, u_3} E
F : C ⥤ D
G : D ⥤ E
A : Type u_4
B : Type u_5
inst✝⁷ : AddMonoid A
inst✝⁶ : AddCommMonoid B
inst✝⁵ : HasShift C A
inst✝⁴ : HasShift D A
inst✝³ : HasShift E A
inst✝² : HasShift C B
inst✝¹ : HasShift D B
inst✝ : F.CommShift A
a b : A

⊢ F.commShiftIso (a + b) = CommShift.isoAdd' ⋯ (F.commShiftIso a) (F.commShiftIso b)
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M✝ : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M✝
M : Type u_13
inst✝ : AddZeroClass M
d : M

⊢ AddEquiv.finsuppUnique.symm d = single () d
-- Mathlib/Data/Finsupp/Defs.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
M✝ : Type u_5
M' : Type u_6
N : Type u_7
P : Type u_8
G : Type u_9
H : Type u_10
R : Type u_11
S : Type u_12
inst✝¹ : AddZeroClass M✝
M : Type u_13
inst✝ : AddZeroClass M
d : M

⊢ (AddEquiv.finsuppUnique.symm d) default = d
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0

⊢ QuasiMeasurePreserving (⇑f) μ μ
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0

⊢ map (⇑f) μ ≪ μ
-- Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
E : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : MeasurableSpace E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
μ : Measure E
inst✝³ : μ.IsAddHaarMeasure
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0

⊢ ENNReal.ofReal |(LinearMap.det f)⁻¹| • μ ≪ μ
-- Mathlib/RingTheory/Polynomial/RationalRoot.lean
A : Type u_1
K : Type u_2
R : Type u_3
S : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : Field K
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid A
inst✝⁵ : Algebra A S
inst✝⁴ : IsLocalization M S
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0

⊢ (p.scaleRoots ↑(den A x)).IsRoot (num A x)
-- Mathlib/RingTheory/Polynomial/RationalRoot.lean
A : Type u_1
K : Type u_2
R : Type u_3
S : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : Field K
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid A
inst✝⁵ : Algebra A S
inst✝⁴ : IsLocalization M S
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0

⊢ eval₂ (algebraMap A K) ((algebraMap A K) (num A x)) (p.scaleRoots ↑(den A x)) = 0
-- Mathlib/RingTheory/Polynomial/RationalRoot.lean
A : Type u_1
K : Type u_2
R : Type u_3
S : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : Field K
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid A
inst✝⁵ : Algebra A S
inst✝⁴ : IsLocalization M S
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0

⊢ (aeval (mk' K (num A x) (den A x))) p = 0
-- Mathlib/RingTheory/Polynomial/RationalRoot.lean
A : Type u_1
K : Type u_2
R : Type u_3
S : Type u_4
inst✝⁹ : CommRing A
inst✝⁸ : Field K
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
M : Submonoid A
inst✝⁵ : Algebra A S
inst✝⁴ : IsLocalization M S
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0

⊢ (aeval x) p = 0
-- Mathlib/Data/Set/Prod.lean
ι : Type u_1
α : ι → Type u_2
β : ι → Type u_3
s s₁ s₂ : Set ι
t t₁ t₂ : (i : ι) → Set (α i)
i : ι

⊢ (s.pi t).Nonempty ↔ ∀ (i : ι), ∃ x, i ∈ s → x ∈ t i
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0

⊢ ∃ n, Associated (p ^ n) x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x

⊢ ∃ n, Associated (p ^ n) x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x

⊢ Associated (p ^ Multiset.card fx) x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associated fx.prod x

⊢ Associated (p ^ Multiset.card fx) x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x

⊢ Associates.mk (p ^ Multiset.card fx) = Associates.mk x
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x

⊢ (Multiset.replicate (Multiset.card fx) (Associates.mk p)).prod = (Multiset.map Associates.mk fx).prod
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x

⊢ Multiset.map Associates.mk fx = Multiset.replicate (Multiset.card fx) (Associates.mk p)
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x

⊢ Multiset.card (Multiset.map Associates.mk fx) = Multiset.card fx ∧ ∀ b ∈ Multiset.map Associates.mk fx, b = Associates.mk p
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x

⊢ ∀ (b : Associates R), ∀ x ∈ fx, Associates.mk x = b → b = Associates.mk p
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
q : R
hq : q ∈ fx

⊢ Associates.mk q = Associates.mk p
-- Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
R : Type u_1
inst✝² : CommRing R
hR : HasUnitMulPowIrreducibleFactorization R
inst✝¹ : IsDomain R
inst✝ : UniqueFactorizationMonoid R
h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q
p : R
hp : Irreducible p
x : R
hx : x ≠ 0
fx : Multiset R
hfx : (∀ b ∈ fx, Irreducible b) ∧ Associated fx.prod x
H : Associates.mk fx.prod = Associates.mk x
q : R
hq : q ∈ fx

⊢ Associated q p
-- Mathlib/Analysis/Normed/Field/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝ : SeminormedRing α
a : α
x✝ : 0 < 1

⊢ ‖a ^ 1‖₊ ≤ ‖a‖₊ ^ 1
-- Mathlib/Analysis/Normed/Field/Basic.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Type u_4
inst✝ : SeminormedRing α
a : α
n : ℕ
x✝ : 0 < n + 2

⊢ ‖a ^ (n + 2)‖₊ ≤ ‖a‖₊ ^ (n + 2)
-- Mathlib/Algebra/Polynomial/Splits.lean
R : Type u_1
F : Type u
K : Type v
L : Type w
inst✝² : CommRing K
inst✝¹ : Field L
inst✝ : Field F
i : K →+* L
f : K[X]
hf : (map i f).degree = 1
g✝ : L[X]
hg : Irreducible g✝
x✝ : g✝ ∣ map i f
p : L[X]
hp : map i f = g✝ * p
this : (map i f).degree = (g✝ * p).degree

⊢ g✝.degree = 1
-- Mathlib/Algebra/Polynomial/Splits.lean
R : Type u_1
F : Type u
K : Type v
L : Type w
inst✝² : CommRing K
inst✝¹ : Field L
inst✝ : Field F
i : K →+* L
f : K[X]
hf : (map i f).degree = 1
g✝ : L[X]
hg : Irreducible g✝
x✝ : g✝ ∣ map i f
p : L[X]
hp : map i f = g✝ * p
this : g✝.degree = 1 ∧ p.degree = 0

⊢ g✝.degree = 1
-- Mathlib/Algebra/Group/Indicator.lean
α : Type u_1
β : Type u_2
ι : Type u_3
M : Type u_4
N : Type u_5
inst✝¹ : One M
inst✝ : One N
s t : Set α
f g : α → M
a : α
h1 : s.mulIndicator f = f
h2 : s ⊆ t

⊢ t.mulIndicator f = f
-- Mathlib/Algebra/Group/Indicator.lean
α : Type u_1
β : Type u_2
ι : Type u_3
M : Type u_4
N : Type u_5
inst✝¹ : One M
inst✝ : One N
s t : Set α
f g : α → M
a : α
h1 : mulSupport f ⊆ s
h2 : s ⊆ t

⊢ mulSupport f ⊆ t
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
x₀ : G

⊢ (𝓝 x₀).HasBasis (fun V => V ∈ B) fun V => (fun y => x₀ * y) '' V
-- Mathlib/Topology/Algebra/FilterBasis.lean
G : Type u
inst✝ : Group G
B✝ B : GroupFilterBasis G
x₀ : G

⊢ (B.N x₀).HasBasis (fun V => V ∈ B) fun V => (fun y => x₀ * y) '' V
-- Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasProducts C
X : TopCat
F : Presheaf C X
ι : Type v'
U : ι → Opens ↑X
c d : Cone (SheafConditionEqualizerProducts.diagram F U)
f : c ⟶ d

⊢ f.hom ≫ 𝟙 d.pt = 𝟙 c.pt ≫ f.hom
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝² : CompleteLattice α
inst✝¹ : IsLowerModularLattice α
inst✝ : IsCoatomistic α

⊢ IsStronglyCoatomic α
-- Mathlib/Order/Atoms.lean
ι : Sort u_1
α : Type u_2
β : Type u_3
inst✝² : CompleteLattice α
inst✝¹ : IsLowerModularLattice α
inst✝ : IsCoatomistic α

⊢ IsStronglyAtomic αᵒᵈ
-- Mathlib/LinearAlgebra/Finsupp.lean
α : Type u_1
M : Type u_2
N : Type u_3
P : Type u_4
R : Type u_5
S : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : Semiring S
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid N
inst✝² : Module R N
inst✝¹ : AddCommMonoid P
inst✝ : Module R P
l : α →₀ M
h : l ∈ supported M R ∅
a✝ : α

⊢ l a✝ = 0 a✝
-- Mathlib/Data/Set/Pointwise/Basic.lean
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : InvolutiveInv α
s✝ t : Set α
a✝ a : α
s : Set α

⊢ (insert a s)⁻¹ = insert a⁻¹ s⁻¹
-- Mathlib/CategoryTheory/Sites/OneHypercover.lean
C : Type u
inst✝¹ : Category.{v, u} C
A : Type u_1
inst✝ : Category.{?u.32278, u_1} A
J : GrothendieckTopology C
X : C
S : J.Cover X
f₁ f₂ : S.preOneHypercover.I₀
x✝ : C
p₁ : x✝ ⟶ S.preOneHypercover.X f₁
p₂ : x✝ ⟶ S.preOneHypercover.X f₂
w : p₁ ≫ S.preOneHypercover.f f₁ = p₂ ≫ S.preOneHypercover.f f₂

⊢ S.preOneHypercover.sieve₁ p₁ p₂ ∈ J.sieves x✝
-- Mathlib/Order/Filter/CardinalInter.lean
ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
p : α → (i : ι) → i ∈ S → Prop

⊢ (∀ᶠ (x : α) in l, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᶠ (x : α) in l, p x i hi
-- Mathlib/Order/Filter/CardinalInter.lean
ι α β : Type u
c : Cardinal.{u}
l : Filter α
inst✝ : CardinalInterFilter l c
S : Set ι
hS : #↑S < c
p : α → (i : ι) → i ∈ S → Prop

⊢ ⋂ i, ⋂ (i_1 : i ∈ S), {x | p x i i_1} ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), {x | p x i hi} ∈ l
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C C₁ C₂ C₃ : HomologicalComplex V c
f : (i : ι) → C₁.X i ≅ C₂.X i
hf : autoParam (∀ (i j : ι), c.Rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom) _auto✝
i : ι

⊢ ({ f := fun i => (f i).hom, comm' := hf } ≫ { f := fun i => (f i).inv, comm' := ⋯ }).f i = (𝟙 C₁).f i
-- Mathlib/Algebra/Homology/HomologicalComplex.lean
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
c : ComplexShape ι
C C₁ C₂ C₃ : HomologicalComplex V c
f : (i : ι) → C₁.X i ≅ C₂.X i
hf : autoParam (∀ (i j : ι), c.Rel i j → (f i).hom ≫ C₂.d i j = C₁.d i j ≫ (f j).hom) _auto✝
i : ι

⊢ ({ f := fun i => (f i).inv, comm' := ⋯ } ≫ { f := fun i => (f i).hom, comm' := hf }).f i = (𝟙 C₂).f i
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
t : Stream' α := s₁.tail ⋈ s₂.tail

⊢ corec (fun x => match x with | (s₁, snd) => s₁.head) (fun x => match x with | (s₁, s₂) => (s₂, s₁.tail)) (s₁, s₂) = s₁.head :: s₂.head :: t
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
t : Stream' α := s₁.tail ⋈ s₂.tail

⊢ (match (s₁, s₂) with | (s₁, snd) => s₁.head) :: corec (fun x => match x with | (s₁, snd) => s₁.head) (fun x => match x with | (s₁, s₂) => (s₂, s₁.tail)) (match (s₁, s₂) with | (s₁, s₂) => (s₂, s₁.tail)) = s₁.head :: s₂.head :: t
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
t : Stream' α := s₁.tail ⋈ s₂.tail

⊢ s₁.head :: corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) (s₂, s₁.tail) = s₁.head :: s₂.head :: t
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
t : Stream' α := s₁.tail ⋈ s₂.tail

⊢ s₁.head :: (s₂, s₁.tail).1.head :: corec (fun x => x.1.head) (fun x => (x.2, x.1.tail)) ((s₂, s₁.tail).2, (s₂, s₁.tail).1.tail) = s₁.head :: s₂.head :: t
-- Mathlib/FieldTheory/PerfectClosure.lean
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x : ℕ × K
n : ℕ

⊢ mk K p x ^ n = mk K p (x.1, x.2 ^ n)
-- Mathlib/Order/Interval/Finset/Basic.lean
ι : Type u_1
α : Type u_2
inst✝¹ : Preorder α
inst✝ : LocallyFiniteOrder α
a a₁ a₂ b b₁ b₂ c x : α
h₁ : a₁ ≤ b₁

⊢ Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂
-- Mathlib/Algebra/Homology/ShortComplex/Basic.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{?u.58352, u_1} C
inst✝² : Category.{?u.58356, u_2} D
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
inst✝ : HasZeroMorphisms D

⊢ S.g.op ≫ S.f.op = 0
-- Mathlib/Algebra/Homology/ShortComplex/Basic.lean
C : Type u_1
D : Type u_2
inst✝³ : Category.{?u.58352, u_1} C
inst✝² : Category.{?u.58356, u_2} D
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
inst✝ : HasZeroMorphisms D

⊢ Quiver.Hom.op 0 = 0
-- Mathlib/Data/List/Range.lean
α : Type u
n : ℕ

⊢ Pairwise (fun x x_1 => x > x_1) (iota n)
-- Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
k : Type u_1
V₁ : Type u_2
P₁ : Type u_3
V₂ : Type u_4
P₂ : Type u_5
V₃ : Type u_6
P₃ : Type u_7
inst✝⁹ : Ring k
inst✝⁸ : AddCommGroup V₁
inst✝⁷ : Module k V₁
inst✝⁶ : AffineSpace V₁ P₁
inst✝⁵ : AddCommGroup V₂
inst✝⁴ : Module k V₂
inst✝³ : AffineSpace V₂ P₂
inst✝² : AddCommGroup V₃
inst✝¹ : Module k V₃
inst✝ : AffineSpace V₃ P₃
f : P₁ →ᵃ[k] P₂
s : AffineSubspace k P₁

⊢ (map f s).direction = Submodule.map f.linear s.direction
-- 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
G : D ⥤ C
iso : e.inverse ≅ G
X : C

⊢ e.functor.map (e.unitIso.hom.app X ≫ iso.hom.app (e.functor.obj X)) ≫ e.functor.map (iso.inv.app (e.functor.obj X)) ≫ e.counitIso.hom.app (e.functor.obj X) = 𝟙 (e.functor.obj X)
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ

⊢ ∑ i : PointsWithCircumcenterIndex (n + 2), mongePointWeightsWithCircumcenter n i = 1
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ

⊢ ↑(n + 2 + 1) * (↑(n + 1))⁻¹ + -2 / ↑(n + 1) = 1
-- Mathlib/Geometry/Euclidean/MongePoint.lean
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
n : ℕ

⊢ ↑n + 2 + 1 + -2 = ↑n + 1
-- Mathlib/SetTheory/Ordinal/Notation.lean
o o' : ONote
m : ℕ
inst✝ : o.NF
h : o.split = (o', m)
a : ONote
n : ℕ
e : o.split' = (a, n)

⊢ ω ∣ o'.repr
-- Mathlib/SetTheory/Ordinal/Notation.lean
o o' : ONote
m : ℕ
inst✝ : o.NF
a : ONote
n : ℕ
h : (scale 1 a, n) = (o', m)
e : o.split' = (a, n)

⊢ ω ∣ o'.repr
-- Mathlib/SetTheory/Ordinal/Notation.lean
o : ONote
m : ℕ
inst✝ : o.NF
a : ONote
n : ℕ
e : o.split' = (a, n)
snd_eq✝ : n = m

⊢ ω ∣ (scale 1 a).repr