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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₃ X₂ : C
inl : X₁ ⟶ X₂
snd : X₂ ⟶ X₃
mem : Triangle.mk inl snd 0 ∈ distinguishedTriangles
⊢ HasBinaryBiproduct X₁ X₃

Tactic:
obtain ⟨inr : X₃ ⟶ X₂, inr_snd : 𝟙 _ = inr ≫ snd⟩ :=
    Triangle.coyoneda_exact₃ _ mem (𝟙 X₃) (by simp)

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
X₁ X₃ X₂ : C
inl : X₁ ⟶ X₂
snd : X₂ ⟶ X₃
mem : Triangle.mk inl snd 0 ∈ distinguishedTriangles
inr : X₃ ⟶ X₂
inr_snd : 𝟙 X₃ = inr ≫ snd
⊢ HasBinaryBiproduct X₁ X₃
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
⊢ T.rotate ∈ distinguishedTriangles C

Tactic:
obtain ⟨K, L, φ, ⟨e⟩⟩ := hT

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
K L : CochainComplex C ℤ
φ : K ⟶ L
e : T ≅ CochainComplex.mappingCone.triangleh φ
⊢ T.rotate ∈ distinguishedTriangles C
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
K L : CochainComplex C ℤ
φ : K ⟶ L
e : T ≅ CochainComplex.mappingCone.triangleh φ
⊢ T.rotate ∈ distinguishedTriangles C

Tactic:
exact ⟨_, _, _, ⟨(rotate _).mapIso e ≪≫ CochainComplex.mappingCone.rotateTrianglehIso φ⟩⟩

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
n : ℤ
⊢ (Triangle.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).obj T ∈ distinguishedTriangles C

Tactic:
obtain ⟨K, L, φ, ⟨e⟩⟩ := hT

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
n : ℤ
K L : CochainComplex C ℤ
φ : K ⟶ L
e : T ≅ CochainComplex.mappingCone.triangleh φ
⊢ (Triangle.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).obj T ∈ distinguishedTriangles C
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
p : ℤ
s_g : (σ (p + 1)).s ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1))).g = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1))).X₃
⊢ ((↑(mappingCone.fst (homOfDegreewiseSplit S σ))).v p (p + 1) ⋯ ≫ (σ (p + 1)).s -
        (mappingCone.snd (homOfDegreewiseSplit S σ)).v p p ⋯ ≫ (Cochain.ofHom S.f).v (p + 1) (p + 1) ⋯) ≫
      (S.g.f (p + 1) ≫ (mappingCone.inl (homOfDegreewiseSplit S σ)).v (p + 1) p ⋯ -
        (σ (p + 1)).r ≫ (XIsoOfEq S.X₁ ⋯).hom ≫ (mappingCone.inr (homOfDegreewiseSplit S σ)).f p) =
    𝟙 ((mappingCone (homOfDegreewiseSplit S σ)).X p)

Tactic:
have f_r := (σ (p + 1)).f_r

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
p : ℤ
s_g : (σ (p + 1)).s ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1))).g = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1))).X₃
f_r : (S.map (eval C (ComplexShape.up ℤ) (p + 1))).f ≫ (σ (p + 1)).r = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1))).X₁
⊢ ((↑(mappingCone.fst (homOfDegreewiseSplit S σ))).v p (p + 1) ⋯ ≫ (σ (p + 1)).s -
        (mappingCone.snd (homOfDegreewiseSplit S σ)).v p p ⋯ ≫ (Cochain.ofHom S.f).v (p + 1) (p + 1) ⋯) ≫
      (S.g.f (p + 1) ≫ (mappingCone.inl (homOfDegreewiseSplit S σ)).v (p + 1) p ⋯ -
        (σ (p + 1)).r ≫ (XIsoOfEq S.X₁ ⋯).hom ≫ (mappingCone.inr (homOfDegreewiseSplit S σ)).f p) =
    𝟙 ((mappingCone (homOfDegreewiseSplit S σ)).X p)
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ : D
⊢ ∀ {X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : D} {u₁₂ : Y₁ ⟶ X₂} {u₂₃ : X₂ ⟶ X₃} {u₁₃ : Y₁ ⟶ X₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : X₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : X₃ ⟶ Z₂₃}
    {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj X₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : X₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles),
    Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
Y₂

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ : D
⊢ ∀ {X₃ Z₁₂ Z₂₃ Z₁₃ : D} {u₁₂ : Y₁ ⟶ Y₂} {u₂₃ : Y₂ ⟶ X₃} {u₁₃ : Y₁ ⟶ X₃} (comm : u₁₂ ≫ u₂₃ = u₁₃) {v₁₂ : Y₂ ⟶ Z₁₂}
    {w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles) {v₂₃ : X₃ ⟶ Z₂₃}
    {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : X₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles),
    Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X =
    (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X) ≫ (F.shiftIso mn a a'' ⋯).hom.app X

Tactic:
F.shiftIso_add'_hom_app n m mn hnm a a' a'' ha' ha'',

Goal state after:
C : Type u_1
A : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_3, u_3} A
F : C ⥤ A
M : Type u_4
inst✝² : AddMonoid M
inst✝¹ : HasShift C M
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X =
    (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).inv.app X) ≫
      (F.shift a).map ((shiftFunctorAdd' C m n mn hnm).hom.app X) ≫
        (F.shiftIso n a a' ha').hom.app ((shiftFunctor C m).obj X) ≫ (F.shiftIso m a' a'' ha'').hom.app X
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
⊢ Quotient.functor (relations W) ⋙ G₁ = Quotient.functor (relations W) ⋙ G₂

Tactic:
refine Paths.ext_functor ?_ ?_

Goal state after:
case refine_1
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
⊢ (Quotient.functor (relations W) ⋙ G₁).obj = (Quotient.functor (relations W) ⋙ G₂).obj

case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G₁ G₂ : W.Localization ⥤ D
h : W.Q ⋙ G₁ = W.Q ⋙ G₂
⊢ ∀ (a b : LocQuiver W) (e : a ⟶ b),
    (Quotient.functor (relations W) ⋙ G₁).map e.toPath =
      eqToHom ⋯ ≫ (Quotient.functor (relations W) ⋙ G₂).map e.toPath ≫ eqToHom ⋯
Module: Mathlib.CategoryTheory.Localization.Equivalence

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ (Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).hom.app X ≫ β.hom.app X ≫ (Lifting.iso L₂ W₂ L₂ (𝟭 D₂)).inv.app X =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X

Tactic:
dsimp [Lifting.iso]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X ≫ 𝟙 (L₂.obj X) =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X
Module: Mathlib.CategoryTheory.Localization.LocalizerMorphism

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
D₁ : Type u₄
D₂ : Type u₅
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₄, u₄} D₁
inst✝⁴ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝² : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝¹ : CatCommSq Φ.functor L₁ L₂ G
inst✝ : G.IsEquivalence
⊢ (Φ.localizedFunctor W₁.Q W₂.Q).IsEquivalence

Tactic:
rw [Φ.isEquivalence_iff W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q) L₁ L₂ G]
    exact inferInstance

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction₂ X Y
hφ₁ : φ₀.fst.map L ⋯ = φ.fst.map L ⋯
hφ₂ : φ₀.snd.map L ⋯ = φ.snd.map L ⋯
hφ₀ : add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ₀.add.map L ⋯
Z : C
t₁ : φ₀.Y' ⟶ Z
t₂ : φ.Y' ⟶ Z
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
hft' : φ₀.f' ≫ t₁ = φ.f' ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ L.map (φ.f ≫ t₂ + φ₀.f' ≫ t₁) = L.map (φ.f ≫ t₂ + φ.f' ≫ t₂)

Tactic:
hft'

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction₂ X Y
hφ₁ : φ₀.fst.map L ⋯ = φ.fst.map L ⋯
hφ₂ : φ₀.snd.map L ⋯ = φ.snd.map L ⋯
hφ₀ : add' W (φ₀.fst.map L ⋯) (φ₀.snd.map L ⋯) = φ₀.add.map L ⋯
Z : C
t₁ : φ₀.Y' ⟶ Z
t₂ : φ.Y' ⟶ Z
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
hft' : φ₀.f' ≫ t₁ = φ.f' ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ L.map (φ.f ≫ t₂ + φ.f' ≫ t₂) = L.map (φ.f ≫ t₂ + φ.f' ≫ t₂)
Module: Mathlib.CategoryTheory.Localization.LocalizerMorphism

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
C₃ : Type u₃
D₁ : Type u₄
D₂ : Type u₅
inst✝⁷ : Category.{v₁, u₁} C₁
inst✝⁶ : Category.{v₂, u₂} C₂
inst✝⁵ : Category.{v₃, u₃} C₃
inst✝⁴ : Category.{v₄, u₄} D₁
inst✝³ : Category.{v₅, u₅} D₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
W₃ : MorphismProperty C₃
Φ : LocalizerMorphism W₁ W₂
L₁ : C₁ ⥤ D₁
inst✝² : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝¹ : L₂.IsLocalization W₂
G : D₁ ⥤ D₂
inst✝ : Φ.IsInduced
⊢ W₂.op.inverseImage Φ.op.functor = W₁.op

Tactic:
by
    simp [← Φ.inverseImage_eq]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: case hφ₁
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
β : W.LeftFraction Y Z
hβ : g = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.f ≫ γ.s = α.s ≫ γ.f
⊢ α.fst.map L ⋯ ≫ β.map L ⋯ = { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, s := β.s ≫ γ.s, hs := ⋯ }.fst.map L ⋯

Tactic:
exact LeftFraction.map_comp_map_eq_map _ _ _ hγ _

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: case hφ₂
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
β : W.LeftFraction Y Z
hβ : g = β.map L ⋯
γ : W.LeftFraction α.Y' β.Y'
hγ : β.f ≫ γ.s = α.s ≫ γ.f
⊢ α.snd.map L ⋯ ≫ β.map L ⋯ = { Y' := γ.Y', f := α.f ≫ γ.f, f' := α.f' ≫ γ.f, s := β.s ≫ γ.s, hs := ⋯ }.snd.map L ⋯

Tactic:
exact LeftFraction.map_comp_map_eq_map _ _ _ hγ _

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
⊢ P.colimitsClosure J = (P.op.limitsClosure fun a ↦ (J a)ᵒᵖ).unop

Tactic:
refine le_antisymm ?_ ?_
  · apply colimitsClosure_le
    rw [← op_monotone_iff, op_unop]
    apply le_limitsClosure
  · rw [← op_monotone_iff, op_unop]
    apply limitsClosure_le
    rw [op_monotone_iff]
    apply le_colimitsClosure

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ a₁.map F hF ≫ F.map a₁.s ≫ F.map t₁ = a₂.map F hF ≫ F.map a₁.s ≫ F.map t₁

Tactic:
map_comp_map_s_assoc,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
E : Type u_3
inst✝ : Category.{v_3, u_3} E
X Y : C
f : Hom W X Y
F : C ⥤ E
hF : W.IsInvertedBy F
a₁ a₂ : W.LeftFraction X Y
Z : C
t₁ : a₁.Y' ⟶ Z
t₂ : a₂.Y' ⟶ Z
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
h : W (a₁.s ≫ t₁)
this : IsIso (F.map (a₁.s ≫ t₁))
⊢ F.map a₁.f ≫ F.map t₁ = a₂.map F hF ≫ F.map a₁.s ≫ F.map t₁
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
this : IsIso (L.map (z₂.s ≫ z₃.s))
⊢ z₁.map L ⋯ ≫ z₂.map L ⋯ ≫ L.map z₂.s ≫ L.map z₃.s =
    L.map { Y' := z₃.Y', f := z₁.f ≫ z₃.f, s := z₂.s ≫ z₃.s, hs := ⋯ }.f

Tactic:
map_comp_map_s_assoc,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
this : IsIso (L.map (z₂.s ≫ z₃.s))
⊢ z₁.map L ⋯ ≫ L.map z₂.f ≫ L.map z₃.s = L.map { Y' := z₃.Y', f := z₁.f ≫ z₃.f, s := z₂.s ≫ z₃.s, hs := ⋯ }.f
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
this : IsIso (L.map (z₂.s ≫ z₃.s))
⊢ z₁.map L ⋯ ≫ L.map z₁.s ≫ L.map z₃.f = L.map { Y' := z₃.Y', f := z₁.f ≫ z₃.f, s := z₂.s ≫ z₃.s, hs := ⋯ }.f

Tactic:
map_comp_map_s_assoc,

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝¹ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁ : W.LeftFraction X Y
z₂ : W.LeftFraction Y Z
z₃ : W.LeftFraction z₁.Y' z₂.Y'
h₃ : z₂.f ≫ z₃.s = z₁.s ≫ z₃.f
L : C ⥤ D
inst✝ : L.IsLocalization W
this : IsIso (L.map (z₂.s ≫ z₃.s))
⊢ L.map z₁.f ≫ L.map z₃.f = L.map { Y' := z₃.Y', f := z₁.f ≫ z₃.f, s := z₂.s ≫ z₃.s, hs := ⋯ }.f
Module: Mathlib.CategoryTheory.MorphismProperty.Retract

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P Q : MorphismProperty C
inst✝ : Q.IsStableUnderRetracts
⊢ P.retracts ≤ Q ↔ P ≤ Q

Tactic:
by
  constructor
  · exact le_trans P.le_retracts
  · intro h
    exact le_trans (retracts_monotone h) Q.retracts_le

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P₁ P₂ Q : ObjectProperty C
inst✝ : Q.IsTriangulatedClosed₂
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i✝ : C
⊢ P₁.extensionProduct P₂ i✝ ≤ Q.isoClosure i✝

Tactic:
⟨_, _, _, _, _, hT, hY, hZ⟩

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P₁ P₂ Q : ObjectProperty C
inst✝ : Q.IsTriangulatedClosed₂
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i✝ w✝⁴ w✝³ : C
w✝² : w✝⁴ ⟶ i✝
w✝¹ : i✝ ⟶ w✝³
w✝ : w✝³ ⟶ (shiftFunctor C 1).obj w✝⁴
hT : Triangle.mk w✝² w✝¹ w✝ ∈ distinguishedTriangles
hY : P₁ w✝⁴
hZ : P₂ w✝³
⊢ Q.isoClosure i✝
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
P₁ P₂ Q : ObjectProperty C
inst✝¹ : Q.IsTriangulatedClosed₂
inst✝ : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i✝ : C
⊢ P₁.extensionProduct P₂ i✝ ≤ Q i✝

Tactic:
⟨_, _, _, _, _, hT, hY, hZ⟩

Goal state after:
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : HasZeroObject C
inst✝⁵ : HasShift C ℤ
inst✝⁴ : Preadditive C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
P₁ P₂ Q : ObjectProperty C
inst✝¹ : Q.IsTriangulatedClosed₂
inst✝ : Q.IsClosedUnderIsomorphisms
h₁ : P₁ ≤ Q
h₂ : P₂ ≤ Q
i✝ w✝⁴ w✝³ : C
w✝² : w✝⁴ ⟶ i✝
w✝¹ : i✝ ⟶ w✝³
w✝ : w✝³ ⟶ (shiftFunctor C 1).obj w✝⁴
hT : Triangle.mk w✝² w✝¹ w✝ ∈ distinguishedTriangles
hY : P₁ w✝⁴
hZ : P₂ w✝³
⊢ Q i✝
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : IsTriangulated C
n m m' : ℕ
h : m = m' + 1
⊢ P.extensionProductIter (n + m) = (P.extensionProductIter n).extensionProduct (P.extensionProductIter m')

Tactic:
induction n with
  | zero => rw [zero_add, extensionProductIter_zero, h, extensionProductIter_succ]
  | succ n hn => rw [add_assoc, add_comm 1 m, ← add_assoc, extensionProductIter_succ,
    extensionProductIter_succ, hn, extensionProduct_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : P.trW f
n : ℤ
⊢ P.trW ((shiftFunctor C n).map f)

Tactic:
rw [← smul_mem_trW_iff _ _ (n.negOnePow)]

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroObject C
inst✝⁴ : HasShift C ℤ
inst✝³ : Preadditive C
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
P : ObjectProperty C
inst✝ : P.IsStableUnderShift ℤ
X₁ X₂ : C
f : X₁ ⟶ X₂
hf : P.trW f
n : ℤ
⊢ P.trW (n.negOnePow • (shiftFunctor C n).map f)
Module: Mathlib.Algebra.Homology.Embedding.Basic

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
⊢ embeddingDownNat.IsRelIff

Tactic:
dsimp [embeddingDownNat]; infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Extend

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroObject C
inst✝² : HasZeroMorphisms C
inst✝¹ : DecidableEq ι
e : c.Embedding c'
X : C
inst✝ : DecidableEq ι'
i : ι
i' : ι'
h : e.f i = i'
j' : ι'
hj : ∃ j, e.f j = j'
⊢ (mkHomToSingle ((((single C c i).obj X).extendXIso e h).hom ≫ (singleObjXSelf c i X).hom) ⋯ ≫
          mkHomFromSingle ((singleObjXSelf c i X).inv ≫ (((single C c i).obj X).extendXIso e h).inv) ⋯).f
      j' =
    (𝟙 (((single C c i).obj X).extend e)).f j'

Tactic:
obtain ⟨j, hj⟩ := hj
      by_cases hij : j = i
      · obtain rfl : i' = j' := by rw [← hj, hij, h]
        simp
      · exact ((isZero_single_obj_X _ _ _ _ hij).of_iso
          (((single C c i).obj X).extendXIso e hj)).eq_of_src _ _

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i j : ι
hij : c.Rel i j
i' j' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hi : e.BoundaryGE i
⊢ (K.truncGE' e).d i j = (K.truncGE'XIsoOpcycles e hi' hi).hom ≫ K.fromOpcycles i' j' ≫ (K.truncGE'XIso e hj' ⋯).inv

Tactic:
dsimp [truncGE', truncGE'.d]
  rw [dif_pos hij, dif_pos hi]
  subst hi' hj'
  simp [truncGE'XIso, truncGE'XIsoOpcycles]

Goal state after: