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

Goal state before:
pp: C : Type u_1
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : HasZeroObject C
inst✝¹³ : HasShift C ℤ
inst✝¹² : Preadditive C
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹⁰ : Pretriangulated C
D : Type u_2
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Preadditive D
inst✝⁷ : HasZeroObject D
inst✝⁶ : HasShift D ℤ
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated D
E : Type u_3
inst✝³ : Category.{v_3, u_3} E
inst✝² : HasShift E ℤ
P : ObjectProperty C
inst✝¹ : IsTriangulated C
inst✝ : P.IsTriangulated
T₁ T₃ : Triangle C
mem₁ : T₁ ∈ distinguishedTriangles
mem₃ : T₃ ∈ distinguishedTriangles
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₅ : Triangle.mk a g₅ h₅ ∈ distinguishedTriangles
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ (shiftFunctor C 1).obj T₁.obj₂
mem₄ : Triangle.mk b g₄ h₄ ∈ distinguishedTriangles
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ (shiftFunctor C 1).obj T₁.obj₁
mem₂ : Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂ ∈ distinguishedTriangles
H : Octahedron ⋯ mem₁ mem₄ mem₂
H' : Octahedron ⋯ mem₅ mem₃ mem₂
φ : T₁ ⟶ T₃ := H.triangleMorphism₁ ≫ H'.triangleMorphism₂
⊢ ∃ c, ∃ (_ : P.trW c), T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor C 1).map a = c ≫ T₃.mor₃

Tactic:
exact ⟨φ.hom₃, P.trW.comp_mem _ _ (trW.mk P H.mem mem₄') (trW.mk' P H'.mem mem₅'),
    by simpa [φ] using φ.comm₂, by simpa [φ] using φ.comm₃⟩

Goal state after:
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
inst✝ : Abelian V
X Y Z : Vᵒᵖ
f : X ⟶ Y
g : Y ⟶ Z
w : f ≫ g = 0
⊢ imageToKernel g.unop f.unop ⋯ =
    (imageSubobjectIso g.unop ≪≫ (imageUnopUnop g).symm).hom ≫
      (cokernel.desc f (factorThruImage g) ⋯).unop ≫ (kernelSubobjectIso f.unop ≪≫ kernelUnopUnop f).inv

Tactic:
ext
  dsimp only [imageUnopUnop]
  simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelUnopUnop_inv, Category.assoc,
    imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, cokernel.π_desc, Iso.unop_inv,
    ← unop_comp, factorThruImage_comp_imageUnopOp_inv, Quiver.Hom.unop_op, imageSubobject_arrow]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i' : ι'
j : ι
hj : c'.Rel i' (e.f j)
hi' : ∀ (i : ι), e.f i ≠ i'
⊢ e.BoundaryGE j

Tactic:
by
  constructor
  · simpa only [c'.prev_eq' hj] using hj
  · intro i hi
    apply hi' i
    rw [← c'.prev_eq' hj, c'.prev_eq' hi]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' j' : ι'
hij' : c'.Rel i' j'
i : ι
hi : e.f i = i'
hj' : ¬∃ j, e.f j = j'
⊢ ∀ (i : ι), e.f i ≠ j'

Tactic:
simpa using hj'

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

Goal state before:
pp: case neg
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 : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
inst✝ : DecidableEq ι
e : c.Embedding c'
X : C
i : ι
k' : ι'
j : ι
hk : ¬∃ k, e.f k = k'
⊢ (((single C c i).obj X).extend e).d (e.f j) k' = 0

Tactic:
· exact IsZero.eq_of_tgt (isZero_extend_X _ _ _ (by tauto)) _ _

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' j' : ι'
hij' : c'.Rel i' j'
hi' : ¬∃ i, e.f i = i'
this : (L.extend e).d i' j' = 0
hj' : ¬∃ j, e.f j = j'
⊢ ∀ (i : ι), e.f i ≠ j'

Tactic:
simpa using hj'

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i : ι
i' : ι'
h : e.f i = i'
⊢ f ψ i = (K.restrictionXIso e h).hom ≫ ψ.f i' ≫ (L.extendXIso e h).hom

Tactic:
subst h
  simp [f, restrictionXIso]

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✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : HasZeroMorphisms C
e : c.Embedding c'
K : HomologicalComplex C c
inst✝ : ∀ (i : ι), Projective (K.X i)
i' : ι'
i : ι
hi : e.f i = i'
⊢ Projective ((K.extend e).X i')

Tactic:
exact Projective.of_iso (K.extendXIso e hi).symm inferInstance

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L : HomologicalComplex C c
f : K ⟶ L
⊢ { f := fun i ↦ (K.extendXIso e ⋯).inv ≫ ((e.extendFunctor C).map f).f (e.f i) ≫ (L.extendXIso e ⋯).hom, comm' := ⋯ } =
    f

Tactic:
ext i

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
⊢ { f := fun i ↦ (K.extendXIso e ⋯).inv ≫ ((e.extendFunctor C).map f).f (e.f i) ≫ (L.extendXIso e ⋯).hom, comm' := ⋯ }.f
      i =
    f.f i
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
⊢ e.homRestrict (e.liftExtend φ hφ) = φ

Tactic:
ext i

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i : ι
⊢ (e.homRestrict (e.liftExtend φ hφ)).f i = φ.f i
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K K' : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
α : K' ⟶ K
ψ : K ⟶ L.extend e
⊢ e.homRestrict (α ≫ ψ) = restrictionMap α e ≫ e.homRestrict ψ

Tactic:
ext i

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K K' : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
α : K' ⟶ K
ψ : K ⟶ L.extend e
i : ι
⊢ (e.homRestrict (α ≫ ψ)).f i = (restrictionMap α e ≫ e.homRestrict ψ).f i
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L L' : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
β : L ⟶ L'
⊢ e.homRestrict (ψ ≫ extendMap β e) = e.homRestrict ψ ≫ β

Tactic:
ext i

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L L' : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
β : L ⟶ L'
i : ι
⊢ (e.homRestrict (ψ ≫ extendMap β e)).f i = (e.homRestrict ψ ≫ β).f i
Module: Mathlib.Algebra.Homology.Embedding.RestrictionHomology

Goal state before:
pp: 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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst✝¹ : K.HasHomology j'
inst✝ : (K.restriction e).HasHomology j
⊢ K.iCycles j' ≫ K.d j' k' ≫ (K.restrictionXIso e hk').inv = 0

Tactic:
iCycles_d_assoc,

Goal state after:
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 : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsRelIff
i j k : ι
hi : c.prev j = i
hk : c.next j = k
i' j' k' : ι'
hi' : e.f i = i'
hj' : e.f j = j'
hk' : e.f k = k'
hi'' : c'.prev j' = i'
hk'' : c'.next j' = k'
inst✝¹ : K.HasHomology j'
inst✝ : (K.restriction e).HasHomology j
⊢ 0 ≫ (K.restrictionXIso e hk').inv = 0
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

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✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : c.Rel j k
hk'' : e.f k = k'
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0

Tactic:
have : f' ≫ (K.extendXIso e hk'').inv = f'' := by
      apply Cofork.IsColimit.hom_ext hcocone
      rw [reassoc_of% hf', ← cancel_epi (extendXIso K e hj').hom, hf'',
        K.extend_d_eq e hj' hk'']

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : c.Rel j k
hk'' : e.f k = k'
this : f' ≫ (K.extendXIso e hk'').inv = f''
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0
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'
⊢ truncGE'Map (𝟙 K) e = 𝟙 (K.truncGE' e)

Tactic:
ext i

Goal state after:
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 : ι
⊢ (truncGE'Map (𝟙 K) e).f i = (𝟙 (K.truncGE' e)).f i
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✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
⊢ truncGE'Map (φ ≫ φ') e = truncGE'Map φ e ≫ truncGE'Map φ' e

Tactic:
ext i

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i : ι
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i
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✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
⊢ K.restrictionToTruncGE' e ≫ truncGE'Map φ e = restrictionMap φ e ≫ L.restrictionToTruncGE' e

Tactic:
ext i

Goal state after:
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 : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
⊢ (K.restrictionToTruncGE' e ≫ truncGE'Map φ e).f i = (restrictionMap φ e ≫ L.restrictionToTruncGE' e).f i
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

Goal state before:
pp: 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 : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L : HomologicalComplex C c
φ : K ⟶ L
e : c.Embedding c'
inst✝¹ : ∀ (j : ι), K.HasHomology j
inst✝ : ∀ (j : ι), L.HasHomology j
h : ∀ (i : ι), QuasiIsoAt (extendMap φ e) (e.f i)
j' : ι'
hj' : ¬∃ j, e.f j = j'
⊢ ∀ (j : ι), e.f j ≠ j'

Tactic:
simpa using hj'

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

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K L : HomologicalComplex C c
φ : K ⟶ L
hK : K.IsStrictlySupportedOutside e₂
hL : L.IsStrictlySupportedOutside e₁
⊢ φ = 0

Tactic:
ext i

Goal state after:
ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K L : HomologicalComplex C c
φ : K ⟶ L
hK : K.IsStrictlySupportedOutside e₂
hL : L.IsStrictlySupportedOutside e₁
i : ι
⊢ φ.f i = HomologicalComplex.Hom.f 0 i
Module: Mathlib.Algebra.Homology.DerivedCategory.Fractions

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : CochainComplex C ℤ
f : Q.obj X ⟶ Q.obj Y
⊢ ∃ X' s, ∃ (x : IsIso (Q.map s)), ∃ g, f = inv (Q.map s) ≫ Q.map g

Tactic:
have ⟨φ, hφ⟩ := Localization.exists_rightFraction Qh (HomotopyCategory.quasiIso C _) f

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
X Y : CochainComplex C ℤ
f : Q.obj X ⟶ Q.obj Y
φ : (HomotopyCategory.quasiIso C (ComplexShape.up ℤ)).RightFraction { as := X } { as := Y }
hφ : f = φ.map Qh ⋯
⊢ ∃ X' s, ∃ (x : IsIso (Q.map s)), ∃ g, f = inv (Q.map s) ≫ Q.map g
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K : CochainComplex C ℤ
inst✝² : HasZeroObject C
n : ℤ
inst✝¹ : K.IsStrictlyGE n
inst✝ : K.IsStrictlyLE n
⊢ mkHomToSingle (𝟙 (K.X n)) ⋯ ≫ mkHomFromSingle (𝟙 (K.X n)) ⋯ = 𝟙 K

Tactic:
ext i

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
K : CochainComplex C ℤ
inst✝² : HasZeroObject C
n : ℤ
inst✝¹ : K.IsStrictlyGE n
inst✝ : K.IsStrictlyLE n
i : ℤ
⊢ (mkHomToSingle (𝟙 (K.X n)) ⋯ ≫ mkHomFromSingle (𝟙 (K.X n)) ⋯).f i = (𝟙 K).f i
Module: Mathlib.Algebra.Homology.DerivedCategory.FullyFaithful

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
A B : C
X : CochainComplex C ℤ
w✝² : X.IsStrictlyGE n
w✝¹ : X.IsStrictlyLE n
s : X ⟶ (CochainComplex.singleFunctor C n).obj A
w✝ : IsIso (Q.map s)
g : X ⟶ (CochainComplex.singleFunctor C n).obj B
A₀ : C
e : X ≅ (HomologicalComplex.single C (ComplexShape.up ℤ) n).obj A₀
φ : A₀ ⟶ A
hφ : (CochainComplex.singleFunctor C n).map φ = e.inv ≫ s
this : IsIso ((singleFunctor C n).map φ)
⊢ IsIso ((singleFunctor C n ⋙ homologyFunctor C n).map φ)

Tactic:
dsimp; infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.SingleHomology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
ι : Type u_1
inst✝ : DecidableEq ι
c : ComplexShape ι
j : ι
A B : C
f : A ⟶ B
⊢ cyclesMap ((single C c j).map f) j ≫ (singleObjCyclesSelfIso c j B).hom = (singleObjCyclesSelfIso c j A).hom ≫ f

Tactic:
rw [← cancel_mono (singleObjCyclesSelfIso c j B).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
    ← cancel_mono (iCycles _ _)]
  simp only [cyclesMap_i, singleObjCyclesSelfIso, Iso.trans_hom, iCyclesIso_hom, Iso.trans_inv,
    assoc, iCyclesIso_inv_hom_id, comp_id, single_map_f_self]

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Fractions

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
h : a < b
inst✝¹ : X.IsStrictlyLE a
inst✝ : Y.IsStrictlyGE b
X' : CochainComplex C ℤ
w✝¹ : X'.IsStrictlyLE a
s : X' ⟶ X
w✝ : IsIso (Q.map s)
g : X' ⟶ Y
⊢ g = 0

Tactic:
ext i

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
h : a < b
inst✝¹ : X.IsStrictlyLE a
inst✝ : Y.IsStrictlyGE b
X' : CochainComplex C ℤ
w✝¹ : X'.IsStrictlyLE a
s : X' ⟶ X
w✝ : IsIso (Q.map s)
g : X' ⟶ Y
i : ℤ
⊢ g.f i = HomologicalComplex.Hom.f 0 i
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₁ i)
h₂ : Mono (homologyMap φ.τ₂ i)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₁ j)
⊢ Mono (homologyMap φ.τ₃ i)

Tactic:
by_cases hi : ∃ j, c.Rel i j
  · obtain ⟨j, hij⟩ := hi
    apply mono_of_epi_of_mono_of_mono
      ((δlastFunctor ⋙ δlastFunctor).map (mapComposableArrows₅ φ hS₁ hS₂ i j hij))
    · exact (composableArrows₅_exact hS₁ i j hij).δlast.δlast
    · exact (composableArrows₅_exact hS₂ i j hij).δlast.δlast
    · exact h₁
    · exact h₂
    · exact h₃ _ hij
  · refine mono_of_epi_of_epi_of_mono (mapComposableArrows₂ φ i)
      (composableArrows₂_exact hS₁ i) (composableArrows₂_exact hS₂ i) ?_ h₁ h₂
    have := hS₁.epi_g
    apply epi_homologyMap_of_epi_of_not_rel
    simpa using hi

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

Goal state before:
pp: C₁ : Type u₁
inst✝¹⁷ : Category.{v₁, u₁} C₁
C₂ : Type u₂
inst✝¹⁶ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
M : Type w'
inst✝¹⁵ : AddMonoid M
inst✝¹⁴ : HasShift C₁ M
inst✝¹³ : HasShift C₂ M
inst✝¹² : Φ.functor.CommShift M
X₁ Y₁ Z₁ : C₁
X₂ Y₂ Z₂ : C₂
inst✝¹¹ : HasSmallLocalizedShiftedHom W₁ M X₁ Y₁
inst✝¹⁰ : HasSmallLocalizedShiftedHom W₂ M X₂ X₂
inst✝⁹ : HasSmallLocalizedShiftedHom W₂ M X₂ Y₂
inst✝⁸ : HasSmallLocalizedShiftedHom W₂ M Y₂ Y₂
eX : Φ.functor.obj X₁ ≅ X₂
eY : Φ.functor.obj Y₁ ≅ Y₂
eZ : Φ.functor.obj Z₁ ≅ Z₂
inst✝⁷ : W₁.IsCompatibleWithShift M
inst✝⁶ : W₂.IsCompatibleWithShift M
inst✝⁵ : HasSmallLocalizedShiftedHom W₁ M Y₁ Z₁
inst✝⁴ : HasSmallLocalizedShiftedHom W₂ M Z₂ Z₂
inst✝³ : HasSmallLocalizedShiftedHom W₂ M Y₂ Z₂
inst✝² : HasSmallLocalizedShiftedHom W₁ M X₁ Z₁
inst✝¹ : HasSmallLocalizedShiftedHom W₁ M Z₁ Z₁
inst✝ : HasSmallLocalizedShiftedHom W₂ M X₂ Z₂
a b c : M
f : SmallShiftedHom W₁ X₁ Y₁ a
g : SmallShiftedHom W₁ Y₁ Z₁ b
h : b + a = c
⊢ Φ.smallShiftedHomMap eX eZ (f.comp g h) = (Φ.smallShiftedHomMap eX eY f).comp (Φ.smallShiftedHomMap eY eZ g) h

Tactic:
by
  apply (SmallShiftedHom.equiv W₂ W₂.Q).injective
  let e := CatCommSq.iso Φ.functor W₁.Q W₂.Q (Φ.localizedFunctor W₁.Q W₂.Q)
  simp only [Φ.equiv_smallShiftedHomMap W₁.Q W₂.Q _ _ (Φ.localizedFunctor W₁.Q W₂.Q) e,
    SmallShiftedHom.equiv_comp, ShiftedHom.map_comp]
  rw [ShiftedHom.comp_assoc _ _ _ _ (zero_add b) (by simpa),
    ShiftedHom.comp_assoc _ _ _ _ h (by simpa),
    ShiftedHom.comp_assoc _ _ _ _ (add_zero b) (by simpa),
    ← ShiftedHom.comp_assoc _ _ _ (add_zero 0) (add_zero b) (by simp),
    ShiftedHom.mk₀_comp_mk₀]
  simp

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
f : S₁ ⟶ S₂
⊢ map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂ = descShortComplex S₁ ≫ f.τ₃

Tactic:
ext i

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S₁ S₂ : ShortComplex (CochainComplex C ℤ)
f : S₁ ⟶ S₂
i : ℤ
⊢ (map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂).f i = (descShortComplex S₁ ≫ f.τ₃).f i
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
J : Type u_1
inst✝ : Fintype J
X : J → C
c : Bicone X
hc : c.IsBilimit
Y : C
n : ℕ
x✝ : (i : J) → Ext (X i) Y n
⊢ (fun e i ↦ (mk₀ (c.ι i)).comp e ⋯) ((fun e ↦ ∑ i, (mk₀ (c.π i)).comp (e i) ⋯) x✝) = x✝

Tactic:
ext i

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
J : Type u_1
inst✝ : Fintype J
X : J → C
c : Bicone X
hc : c.IsBilimit
Y : C
n : ℕ
x✝ : (i : J) → Ext (X i) Y n
i : J
⊢ (fun e i ↦ (mk₀ (c.ι i)).comp e ⋯) ((fun e ↦ ∑ i, (mk₀ (c.π i)).comp (e i) ⋯) x✝) i = x✝ i
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X : C
J : Type u_1
inst✝ : Fintype J
Y : J → C
c : Bicone Y
hc : c.IsBilimit
n : ℕ
x✝ : (i : J) → Ext X (Y i) n
⊢ (fun e i ↦ e.comp (mk₀ (c.π i)) ⋯) ((fun e ↦ ∑ i, (e i).comp (mk₀ (c.ι i)) ⋯) x✝) = x✝

Tactic:
ext i

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X : C
J : Type u_1
inst✝ : Fintype J
Y : J → C
c : Bicone Y
hc : c.IsBilimit
n : ℕ
x✝ : (i : J) → Ext X (Y i) n
i : J
⊢ (fun e i ↦ e.comp (mk₀ (c.π i)) ⋯) ((fun e ↦ ∑ i, (e i).comp (mk₀ (c.ι i)) ⋯) x✝) i = x✝ i
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
X : Cᵒᵖ
n m : ℤ
hnm : n + m = 0
⊢ (shiftFunctorCompIsoId Cᵒᵖ n m hnm).inv.app X =
    ((shiftFunctorCompIsoId C m n ⋯).hom.app (Opposite.unop X)).op ≫
      (shiftFunctorOpIso C m n ⋯).inv.app (Opposite.op ((shiftFunctor C m).obj (Opposite.unop X))) ≫
        (shiftFunctor Cᵒᵖ m).map ((shiftFunctorOpIso C n m hnm).inv.app X)

Tactic:
simp [shiftFunctorCompIsoId, shiftFunctorZero_op_inv_app X,
    shiftFunctorAdd'_op_hom_app X n m 0 hnm m n 0 hnm (by lia) (add_zero 0)]

Goal state after: