mathlib-initiative/mathlib-tactics huggingface dataset:
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 : IsFibrant Y
h✝ : RightHomotopyRel f g
Q : Cylinder X
inst✝ : Q.IsGood
P : PathObject Y := ⋯.choose
h : ⋯.choose.RightHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (coprod.desc (f ≫ P.ι) h.h) Q.i P.p₀ (Q.π ≫ f)
this : coprod.inl ≫ Q.i ≫ sq.lift = coprod.inl ≫ coprod.desc (f ≫ P.ι) h.h
⊢ Q.i₀ ≫ sq.lift ≫ P.p₁ = f

Tactic:
Q.inl_i_assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : ModelCategory C
X Y Z : C
f g : X ⟶ Y
inst✝¹ : IsFibrant Y
h✝ : RightHomotopyRel f g
Q : Cylinder X
inst✝ : Q.IsGood
P : PathObject Y := ⋯.choose
h : ⋯.choose.RightHomotopy f g
h' : ⋯.choose.IsGood
sq : CommSq (coprod.desc (f ≫ P.ι) h.h) Q.i P.p₀ (Q.π ≫ f)
this : Q.i₀ ≫ sq.lift = coprod.inl ≫ coprod.desc (f ≫ P.ι) h.h
⊢ Q.i₀ ≫ sq.lift ≫ P.p₁ = 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
n₁ : ℤ
⊢ ι f n₁ ≫ π K L n₁ = f

Tactic:
simp

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
inst✝ : EnoughInjectives C
n₁ q : ℤ
⊢ biprod.inl.f q ≫ (π K L n₁).f q = 0

Tactic:
simp

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
inst✝ : EnoughInjectives C
n₁ i : ℤ
hi : i ≠ n₁
⊢ biprod.inr.f i ≫ (π K L n₁).f i = 𝟙 (L.X i)

Tactic:
simp

Goal state after:
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:
simp

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
inst✝ : EnoughInjectives C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
i : ℤ
hi✝ : i ≤ n₀
hi : i < n₀
⊢ (ComplexShape.up ℤ).prev i = i - 1

Tactic:
simp

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
inst✝ : EnoughInjectives C
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
i : ℤ
hi✝ : i ≤ n₀
hi : i < n₀
⊢ (ComplexShape.up ℤ).next i = i + 1

Tactic:
simp

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

Goal state before:
pp: 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
φ : X ⟶ Z
sq : CommSq (initial.to Y) (initial.to X) g φ
⊢ (fun f ↦ f.postcomp g) (mk sq.lift) = mk φ

Tactic:
simp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsCofibrant X
g : Y ⟶ Z
inst✝² : IsFibrant Y
inst✝¹ : IsFibrant Z
inst✝ : WeakEquivalence g
h : FibrantBrownFactorization g := Classical.arbitrary (FibrantBrownFactorization g)
φ : X ⟶ Y
⊢ ((fun f ↦ f.postcomp h.r) ∘ fun f ↦ f.postcomp h.i) (mk φ) = id (mk φ)

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{u_2, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K : CochainComplex C ℤ
n : ℤ
hn : K.IsStrictlyGE n
L : CochainComplex C ℤ
m : ℤ
hm : L.IsStrictlyGE m
f : { obj := K, property := ⋯ }.obj ⟶ { obj := L, property := ⋯ }.obj
⊢ min (n - 1) m ≤ m

Tactic:
simp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsCofibrant X
g : Y ⟶ Z
inst✝² : IsFibrant Y
inst✝¹ : IsFibrant Z
inst✝ : WeakEquivalence g
h : FibrantBrownFactorization g := Classical.arbitrary (FibrantBrownFactorization g)
hi : Function.Bijective fun f ↦ f.postcomp h.i
φ : X ⟶ Y
⊢ (mk φ).postcomp g = ((fun f ↦ f.postcomp h.p) ∘ fun f ↦ f.postcomp h.i) (mk φ)

Tactic:
simp

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
n₀✝ n₁ : ℤ
hn₁ : n₀✝ + 1 = n₁
hf : ∀ i ≤ n₀✝, QuasiIsoAt f i
n₀ : ℤ := n₁ - 1
⊢ (ComplexShape.up ℤ).prev n₁ = n₀

Tactic:
simp

Goal state after:
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
⊢ n₁ - 1 = n₀
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
⊢ (ComplexShape.up ℤ).next n₁ = n₁ + 1

Tactic:
simp

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

Goal state before:
pp: 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
φ : X ⟶ Z
sq : CommSq φ f (terminal.from Z) (terminal.from Y)
⊢ (fun g ↦ g.precomp f) (mk sq.lift) = mk φ

Tactic:
simp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsFibrant Z
f : X ⟶ Y
inst✝² : IsCofibrant X
inst✝¹ : IsCofibrant Y
inst✝ : WeakEquivalence f
h : CofibrantBrownFactorization f := Classical.arbitrary (CofibrantBrownFactorization f)
φ : Y ⟶ Z
⊢ ((fun g ↦ g.precomp h.s) ∘ fun g ↦ g.precomp h.p) (mk φ) = id (mk φ)

Tactic:
simp

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : ModelCategory C
X Y Z : C
inst✝³ : IsFibrant Z
f : X ⟶ Y
inst✝² : IsCofibrant X
inst✝¹ : IsCofibrant Y
inst✝ : WeakEquivalence f
h : CofibrantBrownFactorization f := Classical.arbitrary (CofibrantBrownFactorization f)
hj : Function.Bijective fun g ↦ g.precomp h.p
φ : Y ⟶ Z
⊢ (mk φ).precomp f = ((fun g ↦ g.precomp h.i) ∘ fun g ↦ g.precomp h.p) (mk φ)

Tactic:
simp

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
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 ⟶ L.X n₀
h₂ : x₁ ≫ f.f n₁ = y₀ ≫ L.d n₀ n₁
h₁ : x₁ ≫ pOpcycles K n₁ = 0
⊢ (ComplexShape.up ℤ).prev n₁ = n₀

Tactic:
simp

Goal state after:
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 ⟶ L.X n₀
h₂ : x₁ ≫ f.f n₁ = y₀ ≫ L.d n₀ n₁
h₁ : x₁ ≫ pOpcycles K n₁ = 0
⊢ n₁ - 1 = n₀
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
⊢ δ (-1) 0 0 + Cochain.ofHom (f ≫ α f n) = 0

Tactic:
simp

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
⊢ ι f n ≫ π f n = f

Tactic:
simp

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
⊢ toHoCat.Full

Tactic:
by dsimp [toHoCat]; 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
⊢ (↑(mappingCocone.inr (α f n))).v (i - 1) i ⋯ ≫ (π f n).f i = 0

Tactic:
simp

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
⊢ (mappingCocone.inl (α f n)).v i i ⋯ ≫ (π f n).f i = 𝟙 (L.X i)

Tactic:
simp

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 Y : C
f : X ⟶ Y
⊢ (f ≫ iResolutionObj Y) ≫ terminal.from (resolutionObj Y) = iResolutionObj X ≫ terminal.from (resolutionObj X)

Tactic:
simp

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✝² x✝¹ : C
x✝ : x✝² ⟶ x✝¹
h : weakEquivalences C x✝
⊢ (weakEquivalences (HoCat C)).inverseImage resolution x✝

Tactic:
by
    simpa only [MorphismProperty.inverseImage_iff, ← weakEquivalence_iff, HoCat.resolution,
      weakEquivalence_toHoCat_map_iff, weakEquivalence_resolutionMap_iff,
      weakEquivalence_homMk_iff] using h

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
⊢ (ComplexShape.up ℤ).prev n = n - 1

Tactic:
simp

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
⊢ (S f n).d (n - 1) n = 0

Tactic:
simp

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
this✝ : IsIso ((S f n).homologyπ n)
this : Mono (homologyπ ((cokernel f).truncGE n) n ≫ homologyMap (p f n) n)
⊢ (ComplexShape.up ℤ).prev n = n - 1

Tactic:
simp

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
n : ℤ
hf : ∀ i < n, QuasiIsoAt f i
inst✝¹ : Mono f
inst✝ : Mono (homologyMap f n)
i : ℤ
hi : i < n
⊢ (ComplexShape.up ℤ).Rel i (i + 1)

Tactic:
simp

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} C₃
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
inst✝³ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
L₂ : C₂ ⥤ C₃
R₂ : D₂ ⥤ D₃
H₃ : C₃ ⥤ D₃
w' : TwoSquare H₂ L₂ R₂ H₃
inst✝² : R₁.EssSurj
inst✝¹ : w.GuitartExact
inst✝ : (w ≫ᵥ w').GuitartExact
Y₂ : D₂
⊢ (w'.structuredArrowDownwards Y₂).Initial

Tactic:
structuredArrowDownwards_initial_iff_of_iso _ (R₁.objObjPreimageIso Y₂).symm

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D₁ : Type u_4
D₂ : Type u_5
D₃ : Type u_6
inst✝⁸ : Category.{v_1, u_1} C₁
inst✝⁷ : Category.{v_2, u_2} C₂
inst✝⁶ : Category.{v_3, u_3} C₃
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
inst✝³ : Category.{v_6, u_6} D₃
H₁ : C₁ ⥤ D₁
L₁ : C₁ ⥤ C₂
R₁ : D₁ ⥤ D₂
H₂ : C₂ ⥤ D₂
w : TwoSquare H₁ L₁ R₁ H₂
L₂ : C₂ ⥤ C₃
R₂ : D₂ ⥤ D₃
H₃ : C₃ ⥤ D₃
w' : TwoSquare H₂ L₂ R₂ H₃
inst✝² : R₁.EssSurj
inst✝¹ : w.GuitartExact
inst✝ : (w ≫ᵥ w').GuitartExact
Y₂ : D₂
⊢ (w'.structuredArrowDownwards (R₁.obj (R₁.objPreimage Y₂))).Initial
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) :=
  TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv
this✝ : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor.obj c
x : L.obj (Φ.functor.obj c) ⟶ X₃
fac✝ : L.map g ≫ 𝟙 (L.obj (Φ.functor.obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow.RightResolution (Arrow.mk g) := Classical.arbitrary (Φ.arrow.RightResolution (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := c, w := 𝟙 (Φ.functor.obj c), hw := ⋯ })
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := ρ.X₁.right, w := Arrow.Hom.right ρ.w, hw := ⋯ })
⊢ TwoSquare.CostructuredArrowDownwards.mk w y c g x fac✝ =
    (fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := c, w := 𝟙 (Φ.functor.obj c), hw := ⋯ }

Tactic:
simp

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
C₄ : Type u₄
inst✝³ : Category.{v₁, u₁} C₁
inst✝² : Category.{v₂, u₂} C₂
inst✝¹ : Category.{v₃, u₃} C₃
inst✝ : Category.{v₄, u₄} C₄
T : C₁ ⥤ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ⥤ C₄
w : TwoSquare T L R B
X₃ X₃' : C₃
e : X₃ ≅ X₃'
⊢ (w.costructuredArrowRightwards X₃).Final ↔ (w.costructuredArrowRightwards X₃').Final

Tactic:
rw [Functor.final_iff_comp_equivalence _ (CostructuredArrow.mapIso (B.mapIso e)).functor,
    Functor.final_iff_equivalence_comp (CostructuredArrow.mapIso e).functor]
  exact Functor.final_natIso_iff
    (NatIso.ofComponents (fun _ ↦ CostructuredArrow.isoMk (Iso.refl _)))

Goal state after:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)
inst✝³ : Φ.arrow.HasRightResolutions
inst✝² : W₂.ContainsIdentities
D : Type u_3
inst✝¹ : Category.{v_3, u_3} D
L : C₂ ⥤ D
inst✝ : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) :=
  TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv
this✝ : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor.obj c
x : L.obj (Φ.functor.obj c) ⟶ X₃
fac✝ : L.map g ≫ 𝟙 (L.obj (Φ.functor.obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow.RightResolution (Arrow.mk g) := Classical.arbitrary (Φ.arrow.RightResolution (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := c, w := 𝟙 (Φ.functor.obj c), hw := ⋯ })
    ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      { X₁ := ρ.X₁.right, w := Arrow.Hom.right ρ.w, hw := ⋯ })
⊢ (CostructuredArrow.left ((fromRightResolution Φ L y).obj { X₁ := ρ.X₁.left, w := Arrow.Hom.left ρ.w, hw := ⋯ })).hom ≫
      Φ.functor.map ρ.X₁.hom =
    (CostructuredArrow.left
        ((fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
          { X₁ := ρ.X₁.right, w := Arrow.Hom.right ρ.w, hw := ⋯ })).hom

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : ModelCategory C
X : C
R₀ : (localizerMorphism C).RightResolution X :=
  { X₁ := mk (HoCat.resolutionObj X), w := HoCat.iResolutionObj X, hw := ⋯ }
R : (localizerMorphism C).RightResolution X
⊢ R.w ≫ terminal.from ((localizerMorphism C).functor.obj R.X₁) =
    R₀.w ≫ terminal.from ((localizerMorphism C).functor.obj R₀.X₁)

Tactic:
simp

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

Goal state before:
pp: C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
X : C
g : R.obj Y₀ ⟶ B.obj X
X₀ : C₀ := T.objPreimage Y₀
e₀ : T.obj X₀ ≅ Y₀ := T.objObjPreimageIso Y₀
S : Type (max 0 v_2) := { f // B.map f = e.inv.app X₀ ≫ R.map e₀.hom ≫ g }
Z : S → CostructuredArrowDownwards e.hom g := fun s ↦ CostructuredArrowDownwards.mk e.hom g X₀ e₀.inv ↑s ⋯
this : Nonempty (CostructuredArrowDownwards e.hom g)
⊢ IsConnected (CostructuredArrowDownwards e.hom g)

Tactic:
refine zigzag_isConnected (fun A₀ A₁ ↦ ?_)

Goal state after:
C₀ : Type u_1
C : Type u_2
H₀ : Type u_3
H : Type u_4
inst✝⁶ : Category.{v_1, u_1} C₀
inst✝⁵ : Category.{v_2, u_2} C
inst✝⁴ : Category.{v_3, u_3} H₀
inst✝³ : Category.{v_4, u_4} H
T : C₀ ⥤ H₀
L : C₀ ⥤ C
R : H₀ ⥤ H
B : C ⥤ H
inst✝² : T.EssSurj
inst✝¹ : T.Full
inst✝ : B.Full
e : T ⋙ R ≅ L ⋙ B
he :
  ∀ ⦃X₀ : C₀⦄ ⦃Y : C⦄ (f₀ f₁ : L.obj X₀ ⟶ Y),
    B.map f₀ = B.map f₁ → ∃ P, T.map P.i₀ = T.map P.i₁ ∧ Nonempty ((P.map L).LeftHomotopy f₀ f₁)
Y₀ : H₀
X : C
g : R.obj Y₀ ⟶ B.obj X
X₀ : C₀ := T.objPreimage Y₀
e₀ : T.obj X₀ ≅ Y₀ := T.objObjPreimageIso Y₀
S : Type (max 0 v_2) := { f // B.map f = e.inv.app X₀ ≫ R.map e₀.hom ≫ g }
Z : S → CostructuredArrowDownwards e.hom g := fun s ↦ CostructuredArrowDownwards.mk e.hom g X₀ e₀.inv ↑s ⋯
this : Nonempty (CostructuredArrowDownwards e.hom g)
A₀ A₁ : CostructuredArrowDownwards e.hom g
⊢ Zigzag A₀ A₁
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.Basic

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝⁶ : Category.{v₁, u₁} C₁
inst✝⁵ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
D₁ : Type u_1
D₂ : Type u_2
inst✝⁴ : Category.{v_1, u_1} D₁
inst✝³ : Category.{v_2, u_2} D₂
L₁ : C₁ ⥤ D₁
L₂ : C₂ ⥤ D₂
inst✝² : L₁.IsLocalization W₁
inst✝¹ : L₂.IsLocalization W₂
F : D₁ ⥤ D₂
inst✝ : Φ.HasRightResolutions
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ F
this :
  Φ.IsRightDerivabilityStructure ↔
    TwoSquare.GuitartExact (CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)).hom
e' : Φ.functor ⋙ W₂.Q ≅ W₁.Q ⋙ Φ.localizedFunctor W₁.Q W₂.Q :=
  CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
E₁ : W₁.Localization ≌ D₁ := uniq W₁.Q L₁ W₁
E₂ : W₂.Localization ≌ D₂ := uniq W₂.Q L₂ W₂
e₁ : W₁.Q ⋙ E₁.functor ≅ L₁ := compUniqFunctor W₁.Q L₁ W₁
e₂ : W₂.Q ⋙ E₂.functor ≅ L₂ := compUniqFunctor W₂.Q L₂ W₂
e'' : (Φ.functor ⋙ W₂.Q) ⋙ E₂.functor ≅ (W₁.Q ⋙ E₁.functor) ⋙ F :=
  Φ.functor.associator W₂.Q E₂.functor ≪≫ Φ.functor.isoWhiskerLeft e₂ ≪≫ e ≪≫ isoWhiskerRight e₁.symm F
e''' : Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor ≅ E₁.functor ⋙ F :=
  liftNatIso W₁.Q W₁ ((Φ.functor ⋙ W₂.Q) ⋙ E₂.functor) ((W₁.Q ⋙ E₁.functor) ⋙ F)
    (Φ.localizedFunctor W₁.Q W₂.Q ⋙ E₂.functor) (E₁.functor ⋙ F) e''
X₁ : C₁
⊢ e₂.inv.app (Φ.functor.obj X₁) ≫
      E₂.functor.map (𝟙 ((Φ.functor ⋙ W₂.Q).obj X₁)) ≫ e₂.hom.app (Φ.functor.obj X₁) ≫ e.hom.app X₁ =
    (TwoSquare.natTrans e.hom).app X₁

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
q : ℕ
i : ℤ
h : i + 1 ≤ n₀ + ↑q
⊢ (ComplexShape.up ℤ).prev i = i - 1

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
inst✝² : Mono f
n₀ : ℤ
inst✝¹ : K.IsStrictlyGE (n₀ + 1)
inst✝ : L.IsStrictlyGE (n₀ + 1)
q : ℕ
i : ℤ
h : i + 1 ≤ n₀ + ↑q
⊢ (ComplexShape.up ℤ).next i = i + 1

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
K L : CochainComplex C ℤ
f : K ⟶ L
inst✝³ : EnoughInjectives C
n : ℤ
inst✝² : K.IsStrictlyGE n
inst✝¹ : L.IsStrictlyGE n
inst✝ : Mono f
⊢ n = n - 1 + 1

Tactic:
simp

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

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
K L : CochainComplex C ℤ
inst✝² : EnoughInjectives C
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
this : K.IsStrictlyGE n
L' : CochainComplex C ℤ
w✝³ : L'.IsStrictlyGE n
p : L' ⟶ L
hp : degreewiseEpiWithInjectiveKernel p
w✝² : QuasiIso p
K' : CochainComplex C ℤ
w✝¹ : K'.IsStrictlyGE n
ι : K ⟶ K'
π : K' ⟶ L'
left✝¹ : Mono ι
left✝ : QuasiIso ι
hπ : degreewiseEpiWithInjectiveKernel π
w✝ : Mono (ι ≫ π)
⊢ ι ≫ π ≫ p = (ι ≫ π) ≫ p

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} D
inst✝² : Category.{v_5, u_2} H
RF RF' : D ⥤ H
F F' : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F' ⟶ L ⋙ RF'
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : RF.IsRightDerivedFunctor α W
τ : F ⟶ F'
⊢ α ≫ L.whiskerLeft (RF.rightDerivedDesc α W RF' (τ ≫ α')) = τ ≫ α'

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} D
inst✝² : Category.{v_5, u_2} H
RF RF' : D ⥤ H
F F' : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F' ⟶ L ⋙ RF'
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : RF.IsRightDerivedFunctor α W
τ : F ⟶ F'
X : C
⊢ α.app X ≫ (RF.rightDerivedDesc α W RF' (τ ≫ α')).app (L.obj X) = τ.app X ≫ α'.app X

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_3
D : Type u_1
H : Type u_2
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : Category.{v_3, u_1} D
inst✝² : Category.{v_5, u_2} H
RF : D ⥤ H
F : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : RF.IsRightDerivedFunctor α W
⊢ α ≫ L.whiskerLeft (RF.rightDerivedNatTrans RF α α W (𝟙 F)) = α ≫ L.whiskerLeft (𝟙 RF)

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Functor.Derived.RightDerived

Goal state before:
pp: C : Type u_1
D : Type u_3
H : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_3, u_3} D
inst✝³ : Category.{v_5, u_2} H
RF RF' RF'' : D ⥤ H
F F' F'' : C ⥤ H
L : C ⥤ D
α : F ⟶ L ⋙ RF
α' : F' ⟶ L ⋙ RF'
α'' : F'' ⟶ L ⋙ RF''
W : MorphismProperty C
inst✝² : L.IsLocalization W
inst✝¹ : RF.IsRightDerivedFunctor α W
inst✝ : RF'.IsRightDerivedFunctor α' W
τ : F ⟶ F'
τ' : F' ⟶ F''
⊢ α ≫ L.whiskerLeft (RF.rightDerivedNatTrans RF' α α' W τ ≫ RF'.rightDerivedNatTrans RF'' α' α'' W τ') =
    α ≫ L.whiskerLeft (RF.rightDerivedNatTrans RF'' α α'' W (τ ≫ τ'))

Tactic:
simp

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 : CochainComplex C ℤ
inst✝¹ : EnoughInjectives C
n : ℤ
inst✝ : K.IsStrictlyGE n
this : HasDerivedCategory C
⊢ n - 1 + 1 = n

Tactic:
simp

Goal state after: