ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Mathlib/MeasureTheory/Integral/Lebesgue.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✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α → ℝ≥0∞
a_mble : Measurable a
c : ℝ≥0∞
tsum_le_c : ∫⁻ (a_1 : α), a a_1 ∂count ≤ c
ε : ℝ≥0∞
ε_ne_zero : ε ≠ 0
ε_ne_top : ε ≠ ⊤
\u22a2 count {i | ε ≤ a i} ≤ c / ε
-- Mathlib/MeasureTheory/Integral/Lebesgue.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✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α → ℝ≥0∞
a_mble : Measurable a
c : ℝ≥0∞
tsum_le_c : ∫⁻ (a_1 : α), a a_1 ∂count ≤ c
ε : ℝ≥0∞
ε_ne_zero : ε ≠ 0
ε_ne_top : ε ≠ ⊤
\u22a2 (∫⁻ (a_1 : α), a a_1 ∂count) / ε ≤ c / ε
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
x✝ : ℕ
\u22a2 Tendsto (fun x => x.eraseIdx x✝) (𝓝 []) (𝓝 ([].eraseIdx x✝))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
x✝ : ℕ
\u22a2 Tendsto (fun x => x.eraseIdx x✝) (pure []) (𝓝 ([].eraseIdx x✝))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
a : α
l : List α
\u22a2 Tendsto (fun x => x.eraseIdx 0) (𝓝 (a :: l)) (𝓝 ((a :: l).eraseIdx 0))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
a : α
l : List α
\u22a2 Tendsto (fun p => (p.1 :: p.2).eraseIdx 0) (𝓝 a ×ˢ 𝓝 l) (𝓝 ((a :: l).eraseIdx 0))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
n : ℕ
a : α
l : List α
\u22a2 Tendsto (fun x => x.eraseIdx (n + 1)) (𝓝 (a :: l)) (𝓝 ((a :: l).eraseIdx (n + 1)))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
n : ℕ
a : α
l : List α
\u22a2 Tendsto (fun p => (p.1 :: p.2).eraseIdx (n + 1)) (𝓝 a ×ˢ 𝓝 l) (𝓝 ((a :: l).eraseIdx (n + 1)))
-- Mathlib/Topology/List.lean
α : Type u_1
β : Type u_2
inst✝¹ : TopologicalSpace α
inst✝ : TopologicalSpace β
n : ℕ
a : α
l : List α
\u22a2 Tendsto (fun p => p.1 :: p.2.eraseIdx n) (𝓝 a ×ˢ 𝓝 l) (𝓝 (a :: l.eraseIdx n))
-- Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean
s : ℝ
\u22a2 Complex.Gamma ↑s = ↑(Gamma s)
-- .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean
m n : Nat
H : 0 < m.gcd n
\u22a2 (m / m.gcd n).Coprime (n / m.gcd n)
-- Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean
V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
x : V
hx : x ≠ 0
θ : Real.Angle
\u22a2 x = (o.rotation θ) x ↔ θ = 0
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R✝ : Type u
S : Type v
T : Type w
a b : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p✝ q✝ : R✝[X]
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
\u22a2 (p /ₘ q).leadingCoeff = p.leadingCoeff
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R✝ : Type u
S : Type v
T : Type w
a b : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p✝ q✝ : R✝[X]
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
\u22a2 (p /ₘ q).leadingCoeff = p.leadingCoeff
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R✝ : Type u
S : Type v
T : Type w
a b : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p✝ q✝ : R✝[X]
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
\u22a2 (p /ₘ q).leadingCoeff = (p %ₘ q + q * (p /ₘ q)).leadingCoeff
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R✝ : Type u
S : Type v
T : Type w
a b : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p✝ q✝ : R✝[X]
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
\u22a2 (p %ₘ q).degree < (q * (p /ₘ q)).degree
-- Mathlib/Algebra/Polynomial/RingDivision.lean
R✝ : Type u
S : Type v
T : Type w
a b : R✝
n : ℕ
inst✝² : CommRing R✝
inst✝¹ : IsDomain R✝
p✝ q✝ : R✝[X]
R : Type u
inst✝ : CommRing R
p q : R[X]
hmonic : q.Monic
hdegree : q.degree ≤ p.degree
a✝ : Nontrivial R
h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0
\u22a2 (p %ₘ q).degree < p.degree
-- Mathlib/Algebra/Polynomial/Inductions.lean
R : Type u
S : Type v
T : Type w
A : Type z
a b : R
n : ℕ
inst✝ : Semiring R
p q : R[X]
\u22a2 (C a * X ^ n).divX = if n = 0 then 0 else C a * X ^ (n - 1)
-- Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean
a : ℤ
\u22a2 J(a | 1) = 1
-- Mathlib/Logic/Equiv/Fin.lean
m n : ℕ
\u22a2 (finRotate n.succ) 0 = 1
-- Mathlib/Algebra/Lie/Weights/Basic.lean
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : LieAlgebra.IsNilpotent R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
χ : L → R
m : M
\u22a2 m ∈ ↑(weightSpace M (χ ∘ ⇑⊤.incl)) ↔ m ∈ ↑(weightSpace M χ)
-- Mathlib/Algebra/Lie/Weights/Basic.lean
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : LieAlgebra.IsNilpotent R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
χ : L → R
m : M
\u22a2 (∀ (a : L) (b : a ∈ ⊤), ∃ k, (((toEnd R (↥⊤) M) ⟨a, b⟩ - (χ ∘ ⇑⊤.incl) ⟨a, b⟩ • 1) ^ k) m = 0) ↔ ∀ (x : L), ∃ k, (((toEnd R L M) x - χ x • 1) ^ k) m = 0
-- Mathlib/Algebra/Lie/Weights/Basic.lean
K : Type u_1
R : Type u_2
L : Type u_3
M : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
inst✝⁴ : LieAlgebra.IsNilpotent R L
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : LieRingModule L M
inst✝ : LieModule R L M
χ : L → R
m : M
\u22a2 ∀ (a : L), (∀ (b : a ∈ ⊤), ∃ k, (((toEnd R (↥⊤) M) ⟨a, b⟩ - (χ ∘ ⇑⊤.incl) ⟨a, b⟩ • 1) ^ k) m = 0) ↔ ∃ k, (((toEnd R L M) a - χ a • 1) ^ k) m = 0
-- Mathlib/Data/PNat/Factors.lean
u v : PrimeMultiset
n : ℕ+ := u.prod
m : ℕ+ := v.prod
\u22a2 (u ⊓ v).prod = n.gcd m
-- Mathlib/Data/PNat/Factors.lean
u v : PrimeMultiset
n : ℕ+ := u.prod
m : ℕ+ := v.prod
this : u = n.factorMultiset
\u22a2 (u ⊓ v).prod = n.gcd m
-- Mathlib/Data/PNat/Factors.lean
u v : PrimeMultiset
n : ℕ+ := u.prod
m : ℕ+ := v.prod
this : u = n.factorMultiset
\u22a2 (n.factorMultiset ⊓ v).prod = n.gcd m
-- Mathlib/Data/PNat/Factors.lean
u v : PrimeMultiset
n : ℕ+ := u.prod
m : ℕ+ := v.prod
this✝ : u = n.factorMultiset
this : v = m.factorMultiset
\u22a2 (n.factorMultiset ⊓ v).prod = n.gcd m
-- Mathlib/Data/PNat/Factors.lean
u v : PrimeMultiset
n : ℕ+ := u.prod
m : ℕ+ := v.prod
this✝ : u = n.factorMultiset
this : v = m.factorMultiset
\u22a2 (n.factorMultiset ⊓ m.factorMultiset).prod = n.gcd m
-- Mathlib/Data/Nat/Choose/Lucas.lean
n k p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
ha₁ : n < p ^ a
ha₂ : k < p ^ a
\u22a2 n.choose k ≡ ∏ i ∈ range a, (n / p ^ i % p).choose (k / p ^ i % p) [MOD p]
-- Mathlib/Data/Nat/Choose/Lucas.lean
n k p : ℕ
inst✝ : Fact (Nat.Prime p)
a : ℕ
ha₁ : n < p ^ a
ha₂ : k < p ^ a
\u22a2 ↑(n.choose k) ≡ ↑(∏ i ∈ range a, (n / p ^ i % p).choose (k / p ^ i % p)) [ZMOD ↑p]
-- Mathlib/Data/ENNReal/Real.lean
a b c d : ℝ≥0∞
r p q : ℝ≥0
hle : a ≤ b + c
hb : b = ⊤ → a = ⊤
hc : c = ⊤ → a = ⊤
\u22a2 a.toReal ≤ b.toReal + c.toReal
-- Mathlib/Data/ENNReal/Real.lean
a b c d : ℝ≥0∞
r p q : ℝ≥0
hle : a ≤ b + c
hb : b = ⊤ → a = ⊤
hc : c = ⊤ → a = ⊤
\u22a2 b + c = ⊤ → a = ⊤
-- Mathlib/Topology/DiscreteQuotient.lean
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
S A B : DiscreteQuotient X
h : A ≤ B
a : Quotient A.toSetoid
\u22a2 A.proj ⁻¹' {a} ⊆ B.proj ⁻¹' {ofLE h a}
-- Mathlib/Topology/DiscreteQuotient.lean
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
S A B : DiscreteQuotient X
h : A ≤ B
a : X
\u22a2 A.proj ⁻¹' {A.proj a} ⊆ B.proj ⁻¹' {ofLE h (A.proj a)}
-- Mathlib/Topology/DiscreteQuotient.lean
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
S A B : DiscreteQuotient X
h : A ≤ B
a : X
\u22a2 setOf (A.Rel a) ⊆ setOf (B.Rel a)
-- Mathlib/Data/Set/Function.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
fa : α → α
fb : β → β
f : α → β
g : β → γ
s t : Set α
h : Semiconj f fa fb
ha : Bijective fa
hf : Injective f
\u22a2 BijOn fb (range f) (range f)
-- Mathlib/Data/Set/Function.lean
α : Type u_1
β : Type u_2
γ : Type u_3
ι : Sort u_4
π : α → Type u_5
fa : α → α
fb : β → β
f : α → β
g : β → γ
s t : Set α
h : Semiconj f fa fb
ha : Bijective fa
hf : Injective f
\u22a2 BijOn fb (f '' univ) (f '' univ)
-- Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
n : ℕ
x : Rˣ
\u22a2 (v.valuationOfNeZeroMod n) ↑((Units.map ↑(algebraMap R K)) x) = 1
-- Mathlib/Data/Finset/Prod.lean
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq α
s t : Finset α
x : α × α
a : α
\u22a2 {a}.offDiag = ∅
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝ : Fintype α
\u22a2 Nonempty (Field α) ↔ IsPrimePow ‖α‖
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝ : Fintype α
\u22a2 IsPrimePow ‖α‖ → Nonempty (Field α)
-- Mathlib/FieldTheory/Cardinality.lean
α : Type u_1
inst✝ : Fintype α
p n : ℕ
hp : Prime p
hn : 0 < n
hα : p ^ n = ‖α‖
this : Fact (Nat.Prime p)
\u22a2 Nonempty (Field α)
-- Mathlib/RingTheory/UniqueFactorizationDomain.lean
α : Type u_1
inst✝³ : CancelCommMonoidWithZero α
inst✝² : UniqueFactorizationMonoid α
inst✝¹ : DecidableEq (Associates α)
inst✝ : (p : Associates α) → Decidable (Irreducible p)
m p : Associates α
h₁ : m ≠ 0
h₂ : Irreducible p
k : ℕ
\u22a2 p ^ k ≤ m ↔ k ≤ p.count m.factors
-- Mathlib/RingTheory/Ideal/Operations.lean
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I J K L : Ideal R
s : Multiset (Ideal R)
P : Ideal R
hp : P.IsPrime
\u22a2 Multiset.prod 0 ≤ P ↔ ∃ I ∈ 0, I ≤ P
-- Mathlib/RingTheory/Ideal/Operations.lean
R : Type u
ι : Type u_1
inst✝ : CommSemiring R
I✝ J K L : Ideal R
s✝ : Multiset (Ideal R)
P : Ideal R
hp : P.IsPrime
I : Ideal R
s : Multiset (Ideal R)
ih : s.prod ≤ P ↔ ∃ I ∈ s, I ≤ P
\u22a2 (I ::ₘ s).prod ≤ P ↔ ∃ I_1 ∈ I ::ₘ s, I_1 ≤ P
-- Mathlib/Topology/FiberBundle/Constructions.lean
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
\u22a2 Pullback.TotalSpace.topologicalSpace F E f = induced (pullbackTotalSpaceEmbedding f) instTopologicalSpaceProd
-- Mathlib/Topology/FiberBundle/Constructions.lean
B : Type u
F : Type v
E : B → Type w₁
B' : Type w₂
f✝ : B' → B
inst✝¹ : TopologicalSpace B'
inst✝ : TopologicalSpace (TotalSpace F E)
f : B' → B
\u22a2 induced TotalSpace.proj inst✝¹ ⊓ induced (Pullback.lift f) inst✝ = induced (Prod.fst ∘ pullbackTotalSpaceEmbedding f) inst✝¹ ⊓ induced (Prod.snd ∘ pullbackTotalSpaceEmbedding f) inst✝
-- 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
N : Bimod Y Z
\u22a2 ((isoOfIso { hom := AssociatorBimod.hom M (regular Y) N, inv := AssociatorBimod.inv M (regular Y) N, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom ≫ M.whiskerLeft (isoOfIso { hom := LeftUnitorBimod.hom N, inv := LeftUnitorBimod.inv N, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom).hom = (whiskerRight (isoOfIso { hom := RightUnitorBimod.hom M, inv := RightUnitorBimod.inv M, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom N).hom
-- 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
N : Bimod Y Z
\u22a2 coequalizer.π ((M.tensorBimod (regular Y)).actRight ▷ N.X) ((α_ (M.tensorBimod (regular Y)).X Y.X N.X).hom ≫ (M.tensorBimod (regular Y)).X ◁ N.actLeft) ≫ ((isoOfIso { hom := AssociatorBimod.hom M (regular Y) N, inv := AssociatorBimod.inv M (regular Y) N, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom ≫ M.whiskerLeft (isoOfIso { hom := LeftUnitorBimod.hom N, inv := LeftUnitorBimod.inv N, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom).hom = coequalizer.π ((M.tensorBimod (regular Y)).actRight ▷ N.X) ((α_ (M.tensorBimod (regular Y)).X Y.X N.X).hom ≫ (M.tensorBimod (regular Y)).X ◁ N.actLeft) ≫ (whiskerRight (isoOfIso { hom := RightUnitorBimod.hom M, inv := RightUnitorBimod.inv M, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom N).hom
-- 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
N : Bimod Y Z
\u22a2 coequalizer.π (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) ≫ coequalizer.desc (AssociatorBimod.homAux M (regular Y) N) ⋯ ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.π (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) ≫ colimMap (parallelPairHom (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) (RightUnitorBimod.hom M ▷ Y.X ▷ N.X) (RightUnitorBimod.hom M ▷ N.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
N : Bimod Y Z
\u22a2 AssociatorBimod.homAux M (regular Y) N ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.π (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) ≫ colimMap (parallelPairHom (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) (RightUnitorBimod.hom M ▷ Y.X ▷ N.X) (RightUnitorBimod.hom M ▷ N.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
N : Bimod Y Z
\u22a2 ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul)).inv ≫ coequalizer.desc ((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.π (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) ≫ colimMap (parallelPairHom (TensorBimod.actRight M (regular Y) ▷ N.X) ((α_ (TensorBimod.X M (regular Y)) Y.X N.X).hom ≫ TensorBimod.X M (regular Y) ◁ N.actLeft) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) (RightUnitorBimod.hom M ▷ Y.X ▷ N.X) (RightUnitorBimod.hom M ▷ N.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
N : Bimod Y Z
\u22a2 ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul)).inv ≫ coequalizer.desc ((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = RightUnitorBimod.hom M ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one
-- 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
N : Bimod Y Z
\u22a2 ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul)).inv ≫ coequalizer.desc ((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.desc M.actRight ⋯ ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one
-- 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
N : Bimod Y Z
\u22a2 (tensorRight N.X).map (coequalizer.π (M.actRight ▷ (regular Y).X) ((α_ M.X Y.X (regular Y).X).hom ≫ M.X ◁ (regular Y).actLeft)) ≫ ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul)).inv ≫ coequalizer.desc ((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (regular Y) N) ((α_ M.X Y.X (TensorBimod.X (regular Y) N)).hom ≫ M.X ◁ TensorBimod.actLeft (regular Y) N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = (tensorRight N.X).map (coequalizer.π (M.actRight ▷ (regular Y).X) ((α_ M.X Y.X (regular Y).X).hom ≫ M.X ◁ (regular Y).actLeft)) ≫ coequalizer.desc M.actRight ⋯ ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one
-- 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
N : Bimod Y Z
\u22a2 coequalizer.π (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul) ▷ N.X ≫ ((PreservesCoequalizer.iso (tensorRight N.X) (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul)).inv ≫ coequalizer.desc ((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) ((α_ M.X Y.X (TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)).hom ≫ M.X ◁ TensorBimod.actLeft { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) ((α_ M.X Y.X (TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)).hom ≫ M.X ◁ TensorBimod.actLeft { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.π (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul) ▷ N.X ≫ coequalizer.desc M.actRight ⋯ ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one