mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: case neg
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
X : C
h : ¬a ≤ b
⊢ IsIso ((t.truncGELTToLTGE a b).app X)

Tactic:
·

Goal state after:
case neg
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
X : C
h : ¬a ≤ b
⊢ IsIso ((t.truncGELTToLTGE a b).app X)
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
inst✝ : IsTriangulated C
a b : ℤ
X : C
h : b < a
⊢ b - 1 < a

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.ModelCategory.Injective

Goal state before:
pp: case inr
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Abelian C
A : CochainComplex C ℤ
hA : CochainComplex.plus C A
B : CochainComplex C ℤ
hB : CochainComplex.plus C B
X : CochainComplex C ℤ
hX : CochainComplex.plus C X
Y : CochainComplex C ℤ
hY : CochainComplex.plus C Y
i : A ⟶ B
inst✝¹ : Mono (ObjectProperty.homMk i)
hi : Mono i
p : X ⟶ Y
inst✝ : Fibration (ObjectProperty.homMk p)
hp : degreewiseEpiWithInjectiveKernel p
hip : WeakEquivalence (ObjectProperty.homMk i) ∨ WeakEquivalence (ObjectProperty.homMk p)
t : A ⟶ X
b : B ⟶ Y
sq✝ : CommSq (ObjectProperty.homMk t) (ObjectProperty.homMk i) (ObjectProperty.homMk p) (ObjectProperty.homMk b)
sq : CommSq t i p b
sq' : ∀ (n : ℤ), CommSq (t.f n) (i.f n) (p.f n) (b.f n)
this✝¹ : ∀ (n : ℤ), ⋯.HasLift
β : Cocycle (cokernel i) (kernel p) 1 :=
  Lifting.cocycle₁ sq (fun n ↦ { l := ⋯.lift, fac_left := ⋯, fac_right := ⋯ }) (cokernelIsCokernel i) (kernelIsKernel p)
this✝ : ∀ (n : ℤ), Injective ((kernel p).X n)
this : (kernel p).IsKInjective
h✝ : QuasiIso p
⊢ ∃ α, δ 0 1 α = ↑β

Tactic:
·

Goal state after:
case inr
C : Type u_1
inst✝³ : Category.{u_2, u_1} C
inst✝² : Abelian C
A : CochainComplex C ℤ
hA : CochainComplex.plus C A
B : CochainComplex C ℤ
hB : CochainComplex.plus C B
X : CochainComplex C ℤ
hX : CochainComplex.plus C X
Y : CochainComplex C ℤ
hY : CochainComplex.plus C Y
i : A ⟶ B
inst✝¹ : Mono (ObjectProperty.homMk i)
hi : Mono i
p : X ⟶ Y
inst✝ : Fibration (ObjectProperty.homMk p)
hp : degreewiseEpiWithInjectiveKernel p
hip : WeakEquivalence (ObjectProperty.homMk i) ∨ WeakEquivalence (ObjectProperty.homMk p)
t : A ⟶ X
b : B ⟶ Y
sq✝ : CommSq (ObjectProperty.homMk t) (ObjectProperty.homMk i) (ObjectProperty.homMk p) (ObjectProperty.homMk b)
sq : CommSq t i p b
sq' : ∀ (n : ℤ), CommSq (t.f n) (i.f n) (p.f n) (b.f n)
this✝¹ : ∀ (n : ℤ), ⋯.HasLift
β : Cocycle (cokernel i) (kernel p) 1 :=
  Lifting.cocycle₁ sq (fun n ↦ { l := ⋯.lift, fac_left := ⋯, fac_right := ⋯ }) (cokernelIsCokernel i) (kernelIsKernel p)
this✝ : ∀ (n : ℤ), Injective ((kernel p).X n)
this : (kernel p).IsKInjective
h✝ : QuasiIso p
⊢ ∃ α, δ 0 1 α = ↑β
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsCofibrant X
h✝ : LeftHomotopyRel f g
Q : PathObject Y
inst✝ : Q.IsGood
P : Cylinder X := ⋯.choose
h : ⋯.choose.LeftHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π ≫ f) h.h)
⊢ (P.i₁ ≫ sq.lift) ≫ Q.p₁ = g

Tactic:
by
      have := sq.fac_right =≫ prod.snd
      rw [Category.assoc, Q.p_snd, prod.lift_snd] at this
      simp [this]

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.Quotient

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
W : MorphismProperty C
homRel : HomRel C
inst✝² : HomRel.IsStableUnderPrecomp homRel
inst✝¹ : HomRel.IsStableUnderPostcomp homRel
inst✝ : W.HasQuotient homRel
⊢ W = (W.quotient homRel).inverseImage (Quotient.functor homRel)

Tactic:
ext
  exact (quotient_iff _ _ _).symm

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case left
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.fst.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.fst.f i

pp: case right
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.snd.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.snd.f i

Tactic:
·

Goal state after:
case left
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.fst.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.fst.f i
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Injective fun f ↦ f.postcomp g

pp: case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Surjective fun f ↦ f.postcomp g

Tactic:
·

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Injective fun f ↦ f.postcomp g
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case right
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.snd.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.snd.f i

Tactic:
·

Goal state after:
case right
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ ((π K L n₁).f i ≫ biprod.inr.f i) ≫ biprod.snd.f i = 𝟙 ((mid K L n₁).X i) ≫ biprod.snd.f i
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
i : ℤ
hi✝ : i ≤ n₀
hi : i < n₀
⊢ QuasiIsoAt (π K L n₁) i

pp: case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
⊢ QuasiIsoAt (π K L n₁) i

Tactic:
·

Goal state after:
case inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
i : ℤ
hi✝ : i ≤ n₀
hi : i < n₀
⊢ QuasiIsoAt (π K L n₁) i
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Surjective fun f ↦ f.postcomp g

Tactic:
·

Goal state after:
case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsCofibrant X
g : Y ⟶ Z
inst✝¹ : Fibration g
inst✝ : WeakEquivalence g
⊢ Function.Surjective fun f ↦ f.postcomp g
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
⊢ QuasiIsoAt (π K L n₁) i

Tactic:
·

Goal state after:
case inr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
⊢ QuasiIsoAt (π K L n₁) i
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
⊢ i ≠ n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr.refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (π K L n₁) i ≫ homologyMap (σ K L n₁) i = 𝟙 (homology (mid K L n₁) i)

pp: case inr.refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (σ K L n₁) i ≫ homologyMap (π K L n₁) i = 𝟙 (homology L i)

Tactic:
·

Goal state after:
case inr.refine_1
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (π K L n₁) i ≫ homologyMap (σ K L n₁) i = 𝟙 (homology (mid K L n₁) i)
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: case inr.refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (σ K L n₁) i ≫ homologyMap (π K L n₁) i = 𝟙 (homology L i)

Tactic:
·

Goal state after:
case inr.refine_2
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
inst✝ : EnoughInjectives C
n₁ i : ℤ
hn₁ : i + 1 = n₁
hi : i ≤ i
this : homologyMap biprod.inl i = 0
⊢ homologyMap (σ K L n₁) i ≫ homologyMap (π K L n₁) i = 𝟙 (homology L i)
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Injective fun g ↦ g.precomp f

pp: case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Surjective fun g ↦ g.precomp f

Tactic:
·

Goal state after:
case left
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Injective fun g ↦ g.precomp f
Module: Mathlib.AlgebraicTopology.ModelCategory.Homotopy

Goal state before:
pp: case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Surjective fun g ↦ g.precomp f

Tactic:
·

Goal state after:
case right
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : ModelCategory C
X Y Z : C
inst✝² : IsFibrant Z
f : X ⟶ Y
inst✝¹ : Cofibration f
inst✝ : WeakEquivalence f
⊢ Function.Surjective fun g ↦ g.precomp f
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
A : C
x₁ : A ⟶ K.X n₁
x✝ : x₁ ≫ K.d n₁ (n₁ + 1) = 0
y₀ : A ⟶ (mid K L n₁).X n₀
hy₀ : x₁ ≫ (ι f n₁).f n₁ = y₀ ≫ (mid K L n₁).d n₀ n₁
⊢ n₀ ≠ n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝¹ : EnoughInjectives C
inst✝ : Mono f
n₀ n₁ : ℤ
hf : ∀ i ≤ n₀, QuasiIsoAt f i
hn₁ : n₀ + 1 = n₁
i : ℤ
hi : i ≤ n₀
⊢ i ≠ n₁

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n : ℤ
⊢ Mono ((p f n).f n)

Tactic:
by
  simp only [p, mkHomToSingle_f, mono_comp_iff_of_mono]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
⊢ i - 1 + 1 = i

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
⊢ i + -1 = i - 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
⊢ i + 0 = i

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
hi : i ≤ n
⊢ i + -1 = i - 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
hi : i ≤ n
⊢ i - 1 ≠ n

Tactic:
by lia

Goal state after:
Module: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X : C
h : (trivialCofibrations C).MapFactorizationData (fibrations C) (terminal.from X)
⊢ ∃ X', ∃ (_ : IsFibrant X'), ∃ i, Cofibration i ∧ WeakEquivalence i

Tactic:
refine ⟨h.Z, ?_, h.i, inferInstance, inferInstance⟩

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X : C
h : (trivialCofibrations C).MapFactorizationData (fibrations C) (terminal.from X)
⊢ IsFibrant h.Z
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n i : ℤ
hi : i ≤ n
⊢ i - 1 + 1 = i

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n q : ℤ
hq : q ≤ n
⊢ q - 1 + 1 = q

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n q : ℤ
hq : q ≤ n
⊢ q - 1 ≠ n

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n q : ℤ
hq : q < n
⊢ q ≠ n

Tactic:
by lia

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n q : ℤ
hq : q < n
⊢ q ≤ n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ IsIso α → ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))

pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α

Tactic:
·

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ IsIso α → ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝ : EnoughInjectives C
n : ℤ
this✝ : IsIso ((S f n).homologyπ n)
this : Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)
⊢ n - 1 < n

Tactic:
by lia

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α

Tactic:
·

Goal state after:
case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasRightResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ IsIso α → ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))

pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α

Tactic:
·

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ IsIso α → ∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))
Module: Mathlib.CategoryTheory.Localization.Resolution

Goal state before:
pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α

Tactic:
·

Goal state after:
case mpr
C₁ : Type u_1
C₂ : Type u_2
D₂ : Type u_4
H : Type u_5
inst✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_4, u_4} D₂
inst✝² : Category.{v_5, u_5} H
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
inst✝ : Φ.HasLeftResolutions
F G : D₂ ⥤ H
α : F ⟶ G
⊢ (∀ (X₁ : C₁), IsIso (α.app (L₂.obj (Φ.functor.obj X₁)))) → IsIso α
Module: Mathlib.Algebra.Homology.Factorizations.CM5a

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝¹ : EnoughInjectives C
n : ℤ
inst✝ : Mono f
T : ShortComplex C :=
  { X₁ := homology K n, X₂ := homology L n, X₃ := homology (cokernel f) n, f := homologyMap f n,
    g := homologyMap (cokernel.π f) n, zero := ⋯ }
φ : T ⟶ homologyShortComplex f n :=
  { τ₁ := 𝟙 T.X₁, τ₂ := 𝟙 T.X₂, τ₃ := homologyMap ((cokernel f).πTruncGE n ≫ p f n) n, comm₁₂ := ⋯, comm₂₃ := ⋯ }
⊢ Mono φ.τ₃ ∧ IsIso φ.τ₂ ∧ Epi φ.τ₁

Tactic:
dsimp [φ]
    rw [homologyMap_comp]
    exact ⟨inferInstance, inferInstance, inferInstance⟩

Goal state after:
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: case mp
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
⊢ (w.whiskerVertical α.hom β.inv).GuitartExact → w.GuitartExact

pp: case mpr
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
⊢ w.GuitartExact → (w.whiskerVertical α.hom β.inv).GuitartExact

Tactic:
·

Goal state after:
case mp
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
⊢ (w.whiskerVertical α.hom β.inv).GuitartExact → w.GuitartExact
Module: Mathlib.CategoryTheory.GuitartExact.VerticalComposition

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
h : (w.whiskerVertical α.hom β.inv).GuitartExact
⊢ w = (w.whiskerVertical α.hom β.inv).whiskerVertical α.inv β.hom

Tactic:
ext X₁

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_4, u_4} D₁
inst✝ : Category.{v_5, u_5} D₂
T : C₁ ⥤ D₁
L : C₁ ⥤ C₂
R : D₁ ⥤ D₂
B : C₂ ⥤ D₂
w : TwoSquare T L R B
L' : C₁ ⥤ C₂
R' : D₁ ⥤ D₂
α : L ≅ L'
β : R ≅ R'
h : (w.whiskerVertical α.hom β.inv).GuitartExact
X₁ : C₁
⊢ w.natTrans.app X₁ = ((w.whiskerVertical α.hom β.inv).whiskerVertical α.inv β.hom).natTrans.app X₁