ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/LinearAlgebra/Matrix/PosDef.lean
m : Type u_1
n : Type u_2
R : Type u_3
𝕜 : Type u_4
inst✝⁶ : Fintype m
inst✝⁵ : Fintype n
inst✝⁴ : CommRing R
inst✝³ : PartialOrder R
inst✝² : StarRing R
inst✝¹ : StarOrderedRing R
inst✝ : RCLike 𝕜
M : Matrix n n R
e : m ≃ n
h : (M.submatrix ⇑e ⇑e).PosSemidef

⊢ M.PosSemidef
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → β
h : Integrable f (μ + ν)

⊢ Integrable f μ
-- Mathlib/MeasureTheory/Function/L1Space.lean
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → β
h : Memℒp f 1 (μ + ν)

⊢ Memℒp f 1 μ
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ

⊢ (logEmbedding K) x = 0 ↔ x ∈ torsion K
-- Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (𝓞 K)ˣ

⊢ (logEmbedding K) x = 0 ↔ ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1
-- Mathlib/Analysis/SpecialFunctions/Log/Base.lean
b✝ x y a b c : ℝ
h₁ : b ≠ 0
h₂ : b ≠ 1
h₃ : b ≠ -1

⊢ log b / log a * (log c / log b) = log c / log a
-- Mathlib/Topology/MetricSpace/HausdorffDimension.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
C K r : ℝ≥0
f✝ : X → Y
s✝ t : Set X
inst✝ : SecondCountableTopology X
f : X → Y
s : Set X
hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t

⊢ dimH (f '' s) ≤ dimH s
-- Mathlib/Topology/MetricSpace/HausdorffDimension.lean
ι : Type u_1
X : Type u_2
Y : Type u_3
inst✝² : EMetricSpace X
inst✝¹ : EMetricSpace Y
C K r : ℝ≥0
f✝ : X → Y
s✝ t : Set X
inst✝ : SecondCountableTopology X
f : X → Y
s : Set X
hf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t
this : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t

⊢ dimH (f '' s) ≤ dimH s
-- Mathlib/Data/Nat/Dist.lean
n m : ℕ

⊢ n ≤ m + n.dist m
-- Mathlib/Data/Nat/Dist.lean
n m : ℕ

⊢ n ≤ m + m.dist n
-- Mathlib/Analysis/Normed/Group/Uniform.lean
𝓕 : Type u_1
α : Type u_2
E : Type u_3
F : Type u_4
inst✝¹ : SeminormedCommGroup E
inst✝ : SeminormedCommGroup F
a a₁✝ a₂✝ b b₁✝ b₂✝ : E
r r₁ r₂ : ℝ
a₁ a₂ b₁ b₂ : E

⊢ |dist a₁ b₁ - dist a₂ b₂| ≤ dist (a₁ * a₂) (b₁ * b₂)
-- Mathlib/Topology/UniformSpace/UniformConvergence.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f s x
hg : Tendsto g p (𝓝[s] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u

⊢ Tendsto (fun n => F n (g n)) p (𝓝 (f x))
-- Mathlib/Topology/UniformSpace/UniformConvergence.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f s x
hg : Tendsto g p (𝓝[s] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β

⊢ u₀ ∈ map (fun x_1 => (f x, F x_1 (g x_1))) p
-- Mathlib/Topology/UniformSpace/UniformConvergence.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f s x
hg : Tendsto g p (𝓝[s] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β
u₁ : Set (β × β)
h₁ : u₁ ∈ 𝓤 β
u₁₀ : u₁ ○ u₁ ⊆ u₀

⊢ u₀ ∈ map (fun x_1 => (f x, F x_1 (g x_1))) p
-- Mathlib/Topology/UniformSpace/UniformConvergence.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s✝ s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f s✝ x
hg : Tendsto g p (𝓝[s✝] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s✝] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β
u₁ : Set (β × β)
h₁ : u₁ ∈ 𝓤 β
u₁₀ : u₁ ○ u₁ ⊆ u₀
s : Set α
sx : s ∈ 𝓝[s✝] x
hs : ∀ᶠ (n : ι) in p, ∀ y ∈ s, (f y, F n y) ∈ u₁
A : ∀ᶠ (n : ι) in p, g n ∈ s

⊢ u₀ ∈ map (fun x_1 => (f x, F x_1 (g x_1))) p
-- Mathlib/Topology/UniformSpace/UniformConvergence.lean
α : Type u
β : Type v
γ : Type w
ι : Type x
inst✝¹ : UniformSpace β
F : ι → α → β
f : α → β
s✝ s' : Set α
x : α
p : Filter ι
p' : Filter α
g : ι → α
inst✝ : TopologicalSpace α
h : ContinuousWithinAt f s✝ x
hg : Tendsto g p (𝓝[s✝] x)
hunif : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s✝] x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, F n y) ∈ u
u₀ : Set (β × β)
hu₀ : u₀ ∈ 𝓤 β
u₁ : Set (β × β)
h₁ : u₁ ∈ 𝓤 β
u₁₀ : u₁ ○ u₁ ⊆ u₀
s : Set α
sx : s ∈ 𝓝[s✝] x
hs : ∀ᶠ (n : ι) in p, ∀ y ∈ s, (f y, F n y) ∈ u₁
A : ∀ᶠ (n : ι) in p, g n ∈ s
B : ∀ᶠ (n : ι) in p, (f x, f (g n)) ∈ u₁

⊢ u₀ ∈ map (fun x_1 => (f x, F x_1 (g x_1))) p
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : MeasurableSet s

⊢ s ∈ ae (μ.restrict s)
-- Mathlib/MeasureTheory/Measure/Restrict.lean
R : Type u_1
α : Type u_2
β : Type u_3
δ : Type u_4
γ : Type u_5
ι : Type u_6
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t s : Set α
hs : MeasurableSet s

⊢ ∃ t₁ ∈ ae μ, ∃ t₂, s ⊆ t₂ ∧ s = t₁ ∩ t₂
-- Mathlib/Data/Stream/Init.lean
α : Type u
β : Type v
δ : Type w
a : α
s : Stream' α
x✝ : a ∈ s.even
n : ℕ
h : (fun b => a = b) (s.even.get n)

⊢ (fun b => a = b) (s.get (2 * n))
-- 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]
a : L
h : map i f = C a
this✝ : DecidableEq L := Classical.decEq L
ha : ¬a = 0
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 = 0
-- 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]
a : L
h : map i f = C a
this✝ : DecidableEq L := Classical.decEq L
ha : ¬a = 0
g✝ : L[X]
hg : Irreducible g✝
x✝ : g✝ ∣ map i f
p : L[X]
hp : map i f = g✝ * p
this : g✝.degree = 0 ∧ p.degree = 0

⊢ g✝.degree = 0
-- Mathlib/Algebra/Homology/QuasiIso.lean
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
c : ComplexShape ι
K✝ L✝ : HomologicalComplex C c
e✝ : HomotopyEquiv K✝ L✝
inst✝² : ∀ (i : ι), K✝.HasHomology i
inst✝¹ : ∀ (i : ι), L✝.HasHomology i
inst✝ : CategoryWithHomology C
K L : HomologicalComplex C c
e : HomotopyEquiv K L

⊢ quasiIso C c e.hom
-- Mathlib/Algebra/Homology/QuasiIso.lean
ι : Type u_1
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
c : ComplexShape ι
K✝ L✝ : HomologicalComplex C c
e✝ : HomotopyEquiv K✝ L✝
inst✝² : ∀ (i : ι), K✝.HasHomology i
inst✝¹ : ∀ (i : ι), L✝.HasHomology i
inst✝ : CategoryWithHomology C
K L : HomologicalComplex C c
e : HomotopyEquiv K L

⊢ QuasiIso e.hom
-- Mathlib/Algebra/Order/Floor.lean
F : Type u_1
α : Type u_2
β : Type u_3
inst✝¹ : LinearOrderedRing α
inst✝ : FloorRing α
z : ℤ
a✝ : α
m : ℤ
a : α

⊢ fract (↑m + a) = fract a
-- Mathlib/LinearAlgebra/Quotient.lean
R : Type u_1
M : Type u_2
r : R
x y : M
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
p p' : Submodule R M
R₂ : Type u_3
M₂ : Type u_4
inst✝² : Ring R₂
inst✝¹ : AddCommGroup M₂
inst✝ : Module R₂ M₂
τ₁₂ : R →+* R₂

⊢ comap p.mkQ (map p.mkQ p') = p ⊔ p'
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝ : DecidableEq α
s✝ s t : Multiset α
_l₁ _l₂ : List α

⊢ _l₁ ≈ _l₂ ↔ ∀ (a : α), count a ⟦_l₁⟧ = count a ⟦_l₂⟧
-- Mathlib/Data/Multiset/Basic.lean
α : Type u_1
β : Type v
γ : Type u_2
inst✝ : DecidableEq α
s✝ s t : Multiset α
_l₁ _l₂ : List α

⊢ _l₁ ≈ _l₂ ↔ ∀ (a : α), List.count a _l₁ = List.count a _l₂
-- Mathlib/Algebra/Order/ToIntervalMod.lean
α : Type u_1
inst✝ : LinearOrderedAddCommGroup α
hα : Archimedean α
p : α
hp : 0 < p
a✝ b✝ c✝ : α
n : ℤ
a b c : α

⊢ toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
hn : n ≠ 0

⊢ 2 * ↑(WeierstrassCurve.expDegree n) = ↑n ^ 2 - if Even n then 4 else 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
hn : 2 * n ≠ 0

⊢ 2 * ↑(WeierstrassCurve.expDegree (2 * n)) = ↑(2 * n) ^ 2 - if Even (2 * n) then 4 else 1
-- Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℕ
hn : 2 * n + 1 ≠ 0

⊢ 2 * ↑(WeierstrassCurve.expDegree (2 * n + 1)) = ↑(2 * n + 1) ^ 2 - if Even (2 * n + 1) then 4 else 1
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ (pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ pullback.snd (f ≫ g) (Scheme.Opens.ι U) = ((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv ≫ pullback.snd g (Scheme.Opens.ι U)
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ (pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U)) ≫ pullback.snd g (Scheme.Opens.ι U) = ((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv ≫ pullback.snd g (Scheme.Opens.ι U)
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ (((pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv) ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U))) ≫ pullback.snd g (Scheme.Opens.ι U) = (((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv) ≫ pullback.snd g (Scheme.Opens.ι U)
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ ((pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv) ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U)) = ((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ (((pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv) ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U))) ≫ pullback.fst g (Scheme.Opens.ι U) = (((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv) ≫ pullback.fst g (Scheme.Opens.ι U)
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ (pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U)) ≫ pullback.fst g (Scheme.Opens.ι U) = (pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι ≫ (pullbackRestrictIsoRestrict g U).inv ≫ pullback.fst g (Scheme.Opens.ι U)
-- Mathlib/AlgebraicGeometry/Restrict.lean
C : Type u₁
inst✝ : Category.{v, u₁} C
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
U : Opens ↑↑Z.toPresheafedSpace

⊢ ((f ≫ g) ⁻¹ᵁ U).ι ≫ f = (f ⁻¹ᵁ g ⁻¹ᵁ U).ι ≫ f
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionMonoid α
a b c d : α
n : ℕ

⊢ 1 ^ ↑n = 1
-- Mathlib/Algebra/Group/Basic.lean
α : Type u_1
β : Type u_2
G : Type u_3
M : Type u_4
inst✝ : DivisionMonoid α
a b c d : α
n : ℕ

⊢ 1 ^ Int.negSucc n = 1
-- Mathlib/Logic/Equiv/Fintype.lean
α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x

⊢ q (e.extendSubtype x)
-- Mathlib/Logic/Equiv/Fintype.lean
α : Type u_1
β : Type u_2
inst✝² : Finite α
p q : α → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
e : { x // p x } ≃ { x // q x }
x : α
hx : p x

⊢ e.extendSubtype x = ↑(e ⟨x, hx⟩)
-- Mathlib/CategoryTheory/Adjunction/Basic.lean
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
X' X : C
Y Y' : D
f : F.obj X ⟶ Y
g : Y ⟶ Y'

⊢ (adj.homEquiv X Y') (f ≫ g) = (adj.homEquiv X Y) f ≫ G.map g
-- Mathlib/Topology/MetricSpace/GromovHausdorff.lean
p q : GHSpace

⊢ dist p q = ghDist p.Rep q.Rep
-- Mathlib/Data/DFinsupp/Order.lean
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → CanonicallyLinearOrderedAddCommMonoid (α i)
inst✝ : DecidableEq ι
f g : Π₀ (i : ι), α i
a✝ : ι

⊢ a✝ ∈ (f ⊔ g).support ↔ a✝ ∈ f.support ∪ g.support
-- Mathlib/Data/DFinsupp/Order.lean
ι : Type u_1
α : ι → Type u_2
inst✝¹ : (i : ι) → CanonicallyLinearOrderedAddCommMonoid (α i)
inst✝ : DecidableEq ι
f g : Π₀ (i : ι), α i
a✝ : ι

⊢ ¬f a✝ ⊔ g a✝ = ⊥ ↔ ¬f a✝ = ⊥ ∨ ¬g a✝ = ⊥
-- Mathlib/Algebra/Polynomial/Smeval.lean
R : Type u_1
inst✝⁶ : Semiring R
p✝ : R[X]
r : R
p q✝ : R[X]
S : Type u_2
inst✝⁵ : NonAssocSemiring S
inst✝⁴ : Module R S
inst✝³ : IsScalarTower R S S
inst✝² : SMulCommClass R S S
inst✝¹ : Pow S ℕ
inst✝ : NatPowAssoc S
x : S
q : ℕ[X]

⊢ ∀ (n : ℕ), q.smeval ↑n = ↑(q.smeval n)
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n✝ p✝ n p : ℕ
hn : n ≠ 0
hp : Prime p
h : p ^ (n.factorization p + 1) ∣ n

⊢ False
-- Mathlib/Data/Nat/Factorization/Defs.lean
a b m n✝ p✝ n p : ℕ
hn : n ≠ 0
hp : Prime p
h : (p ^ (n.factorization p + 1)).factorization ≤ n.factorization

⊢ False
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a

⊢ c ∈ (P.avoid b).parts ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \ b = c
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a

⊢ (∃ x ∈ P.parts, ¬c = ⊥ ∧ x \ b = c) ↔ ∃ d ∈ P.parts, ¬d ≤ b ∧ d \ b = c
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b c : α
P : Finpartition a
d : α

⊢ d \ b = c → (¬c = ⊥ ↔ ¬d ≤ b)
-- Mathlib/Order/Partition/Finpartition.lean
α : Type u_1
inst✝¹ : GeneralizedBooleanAlgebra α
inst✝ : DecidableEq α
a b : α
P : Finpartition a
d : α

⊢ ¬d \ b = ⊥ ↔ ¬d ≤ b
-- Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean
C : Type u_1
inst✝¹ : Category.{u_2, u_1} C
inst✝ : Abelian C
S S₁ S₂ S₃ : SnakeInput C
f : S₁ ⟶ S₂

⊢ (pullback.fst S₁.L₁.g S₁.v₀₁.τ₃ ≫ S₁.v₁₂.τ₂) ≫ f.f₂.τ₂ = pullback.map S₁.L₁.g S₁.v₀₁.τ₃ S₂.L₁.g S₂.v₀₁.τ₃ f.f₁.τ₂ f.f₀.τ₃ f.f₁.τ₃ ⋯ ⋯ ≫ pullback.fst S₂.L₁.g S₂.v₀₁.τ₃ ≫ S₂.v₁₂.τ₂
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ M.actRight ▷ N₁.X ≫ M.X ◁ f.hom = (M.X ⊗ Y.X) ◁ f.hom ≫ M.actRight ▷ N₂.X
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ M.X ◁ f.hom = (M.X ⊗ Y.X) ◁ f.hom ≫ (α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ M.X ◁ N₁.actLeft ≫ M.X ◁ f.hom = M.X ◁ Y.X ◁ f.hom ≫ M.X ◁ N₂.actLeft
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ M.X ◁ (Y.X ◁ f.hom ≫ N₂.actLeft) = M.X ◁ Y.X ◁ f.hom ≫ M.X ◁ N₂.actLeft
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (M.tensorBimod N₁).actLeft ≫ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) = X.X ◁ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ≫ (M.tensorBimod N₂).actLeft
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (tensorLeft X.X).map (coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft)) ≫ (M.tensorBimod N₁).actLeft ≫ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) = (tensorLeft X.X).map (coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft)) ≫ X.X ◁ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ≫ (M.tensorBimod N₂).actLeft
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ TensorBimod.actLeft M N₁ ≫ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) = X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ X.X ◁ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ≫ TensorBimod.actLeft M N₂
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ ((α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft)) ≫ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) = X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ X.X ◁ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ≫ TensorBimod.actLeft M N₂
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one = X.X ◁ coequalizer.π (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ X.X ◁ colimMap (parallelPairHom (M.actRight ▷ N₁.X) ((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft) ((M.X ⊗ Y.X) ◁ f.hom) (M.X ◁ f.hom) ⋯ ⋯) ≫ TensorBimod.actLeft M N₂
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one = (X.X ◁ M.X ◁ f.hom ≫ X.X ◁ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one) ≫ TensorBimod.actLeft M N₂
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one = X.X ◁ M.X ◁ f.hom ≫ (α_ X.X M.X N₂.X).inv ≫ M.actLeft ▷ N₂.X ≫ coequalizer.π (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)
-- Mathlib/CategoryTheory/Monoidal/Bimod.lean
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon_ C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon_ C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂

⊢ (α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one = (((α_ X.X M.X N₁.X).inv ≫ (X.X ⊗ M.X) ◁ f.hom) ≫ M.actLeft ▷ N₂.X) ≫ coequalizer.π (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)