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

Goal state before:
pp: case zero
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
m m' : ℕ
h : m = m' + 1
⊢ P.extensionProductIter m = P.extensionProduct (P.extensionProductIter m')

Tactic:
h,

Goal state after:
case zero
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
m m' : ℕ
h : m = m' + 1
⊢ P.extensionProductIter (m' + 1) = P.extensionProduct (P.extensionProductIter m')
Module: Mathlib.Algebra.Homology.HomotopyCategory.Acyclic

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ (quasiIso C (ComplexShape.up ℤ)).IsCompatibleWithShift ℤ

Tactic:
rw [quasiIso_eq_trW_subcategoryAcyclic]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
⊢ (subcategoryAcyclic C).trW.IsCompatibleWithShift ℤ
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
n m a a' a'' : ℤ
ha' : n + a = a'
ha'' : m + a' = a''
⊢ ShiftSequence.shiftIso C (m + n) a a'' ⋯ =
    Functor.isoWhiskerRight (shiftFunctorAdd (HomologicalComplex C (up ℤ)) m n) (homologyFunctor C (up ℤ) a) ≪≫
      (CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) m).associator
          (CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) n) (homologyFunctor C (up ℤ) a) ≪≫
        (CategoryTheory.shiftFunctor (HomologicalComplex C (up ℤ)) m).isoWhiskerLeft
            (ShiftSequence.shiftIso C n a a' ha') ≪≫
          ShiftSequence.shiftIso C m a' a'' ha''

Tactic:
ext K
    dsimp [homologyMap]
    simp only [ShiftSequence.shiftIso_hom_app, id_comp,
      ← ShortComplex.homologyMap_comp, shiftFunctorAdd'_eq_shiftFunctorAdd,
      shiftShortComplexFunctorIso_add'_hom_app n m _ rfl a a' a'' ha' ha'' K]

Goal state after:
Module: Mathlib.CategoryTheory.Shift.SingleFunctors

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
E' : Type u_4
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
inst✝⁷ : Category.{v_4, u_4} E'
A : Type u_5
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift D A
inst✝⁴ : HasShift E A
inst✝³ : HasShift E' A
F G✝ H : SingleFunctors C D A
G G' : D ⥤ E
e : G ≅ G'
inst✝² : G.CommShift A
inst✝¹ : G'.CommShift A
inst✝ : NatTrans.CommShift e.hom A
n a a' : A
ha' : n + a = a'
X : C
⊢ (((F.postcomp G).shiftIso n a a' ha').hom ≫ ((F.functor a).isoWhiskerLeft e).hom).app X =
    (whiskerRight ((F.functor a').isoWhiskerLeft e).hom (shiftFunctor E n) ≫
          ((F.postcomp G').shiftIso n a a' ha').hom).app
      X

Tactic:
simp [NatTrans.shift_app e.hom n]

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

Goal state before:
pp: case refine_2
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
X' X Y : C
f₁ f₂ : X ⟶ Y
s : X' ⟶ X
Z : C
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X'
H : Triangle.mk s g h ∈ distinguishedTriangles
mem : P Z
hf₁ : s ≫ f₁ = s ≫ f₂
hf₂ : s ≫ (f₁ - f₂) = 0
q : (Triangle.mk s g h).obj₃ ⟶ Y
hq : f₁ - f₂ = (Triangle.mk s g h).mor₂ ≫ q
Y' : C
r : Y ⟶ Y'
t : Y' ⟶ (shiftFunctor C 1).obj (Triangle.mk s g h).obj₃
mem' : Triangle.mk q r t ∈ distinguishedTriangles
⊢ f₁ ≫ r = f₂ ≫ r

Tactic:
have eq := comp_distTriang_mor_zero₁₂ _ mem'
      dsimp at eq
      rw [← sub_eq_zero, ← sub_comp, hq, assoc, eq, comp_zero]

Goal state after:
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
Y Z✝ Z' : C
f₁ f₂ : Y ⟶ Z✝
s : Z✝ ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z✝
h : Z' ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g s h ∈ distinguishedTriangles
mem : P Z
⊢ (f₁ - f₂) ≫ s = 0

Tactic:
sub_comp,

Goal state after:
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
Y Z✝ Z' : C
f₁ f₂ : Y ⟶ Z✝
s : Z✝ ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z✝
h : Z' ⟶ (shiftFunctor C 1).obj Z
H : Triangle.mk g s h ∈ distinguishedTriangles
mem : P Z
⊢ f₁ ≫ s - f₂ ≫ s = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
X : C
⊢ 𝟙 (Qh.obj (((HomotopyCategory.singleFunctors C).functor n).obj X)) ≫
      𝟙 (HomologicalComplexUpToQuasiIso.Q.obj (((CochainComplex.singleFunctors C).functor n).obj X)) =
    𝟙 (((singleFunctors C).functor n).obj X)

Tactic:
erw [Category.id_comp]

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
n : ℤ
X : C
⊢ 𝟙 (HomologicalComplexUpToQuasiIso.Q.obj (((CochainComplex.singleFunctors C).functor n).obj X)) =
    𝟙 (((singleFunctors C).functor n).obj X)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹⁷ : Category.{v_1, u_1} C
inst✝¹⁶ : HasZeroObject C
inst✝¹⁵ : HasShift C ℤ
inst✝¹⁴ : Preadditive C
inst✝¹³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹² : Pretriangulated C
D : Type u_2
inst✝¹¹ : Category.{v_2, u_2} D
inst✝¹⁰ : Preadditive D
inst✝⁹ : HasZeroObject D
inst✝⁸ : HasShift D ℤ
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁶ : Pretriangulated D
E : Type u_3
inst✝⁵ : Category.{v_3, u_3} E
inst✝⁴ : HasShift E ℤ
P : ObjectProperty C
inst✝³ : P.IsTriangulated
F : C ⥤ D
inst✝² : F.CommShift ℤ
inst✝¹ : F.IsTriangulated
inst✝ : F.Full
n : ℤ
⊢ F.essImage ≤ F.essImage.shift n

Tactic:
by
        rintro Y ⟨X, ⟨e⟩⟩
        exact ⟨X⟦n⟧, ⟨(F.commShiftIso n).app _ ≪≫ (shiftFunctor D n).mapIso e⟩⟩

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
i : ι
i' : ι'
hi : e.r i' = some i
⊢ e.f i = i'

Tactic:
by
  by_cases h : ∃ (k : ι), e.f k = i'
  · obtain ⟨k, rfl⟩ := h
    rw [r_f] at hi
    congr 1
    simpa using hi.symm
  · simp [e.r_eq_none i' (by simpa using h)] at hi

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'
inst✝ : e.IsRelIff
j k : ι
hj : ¬e.BoundaryGE j
hk : c.next j = k
⊢ ¬e.BoundaryGE k

Tactic:
by
  by_cases hjk : c.Rel j k
  · exact e.not_boundaryGE_next hjk
  · subst hk
    simpa only [c.next_eq_self j hjk] using hj

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

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j : ι
hij : c.prev j = j
hij' : ¬c.Rel j j
hj' : c'.Rel (c'.prev (e.f j)) (e.f j)
hj : c'.Rel (c'.prev (e.f j)) (e.f j) → ∃ x, c'.Rel (e.f x) (e.f j)
i : ι
hi : c.Rel i j
⊢ False

Tactic:
c.prev_eq' hi

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j : ι
hij' : ¬c.Rel j j
hj' : c'.Rel (c'.prev (e.f j)) (e.f j)
hj : c'.Rel (c'.prev (e.f j)) (e.f j) → ∃ x, c'.Rel (e.f x) (e.f j)
i : ι
hij : i = j
hi : c.Rel i j
⊢ False
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: case right
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ ∀ (k : ι), ¬c'.Rel (e.f j) (e.f k)

Tactic:
intro k

Goal state after:
case right
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
k : ι
⊢ ¬c'.Rel (e.f j) (e.f k)
Module: Mathlib.Algebra.Homology.Embedding.TruncLEHomology

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 : Abelian C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝ : e.IsTruncLE
i' j' : ι'
hij' : c'.Rel i' j'
hj : ∃ j, e.f j = j'
⊢ ⋯.δ i' j' hij' = 0

Tactic:
obtain ⟨j, rfl⟩ := hj
    rw [← cancel_mono (homologyMap (K.ιTruncLE e) (e.f j)), zero_comp]
    exact (K.shortComplexTruncLE_shortExact e).δ_comp i' _ hij'

Goal state after:
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
⊢ IsIso (Q.map (e.inv ≫ s))

Tactic:
Functor.map_comp

Goal state after:
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
⊢ IsIso (Q.map e.inv ≫ Q.map s)
Module: Mathlib.Algebra.Homology.DerivedCategory.Fractions

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
n : ℤ
inst✝ : X.IsStrictlyLE n
X' : CochainComplex C ℤ
s : X' ⟶ X
hs : IsIso (Q.map s)
g : X' ⟶ Y
⊢ ∃ X'_1,
    ∃ (_ : X'_1.IsStrictlyLE n),
      ∃ s_1, ∃ (x : IsIso (Q.map s_1)), ∃ g_1, inv (Q.map s) ≫ Q.map g = inv (Q.map s_1) ≫ Q.map g_1

Tactic:
have : IsIso (Q.map (CochainComplex.truncLEMap s n)) := by
    rw [isIso_Q_map_iff_quasiIso, CochainComplex.quasiIso_truncLEMap_iff]
    rw [isIso_Q_map_iff_quasiIso] at hs
    infer_instance

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasDerivedCategory C
X Y : CochainComplex C ℤ
n : ℤ
inst✝ : X.IsStrictlyLE n
X' : CochainComplex C ℤ
s : X' ⟶ X
hs : IsIso (Q.map s)
g : X' ⟶ Y
this : IsIso (Q.map (CochainComplex.truncLEMap s n))
⊢ ∃ X'_1,
    ∃ (_ : X'_1.IsStrictlyLE n),
      ∃ s_1, ∃ (x : IsIso (Q.map s_1)), ∃ g_1, inv (Q.map s) ≫ Q.map g = inv (Q.map s_1) ≫ Q.map g_1
Module: Mathlib.CategoryTheory.Localization.HomEquiv

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
f : Y ⟶ X
hf : W f
⊢ (homEquiv W L₁ L₂) ((isoOfHom L₁ W f hf).inv ≫ L₁.map f) = 𝟙 (L₂.obj X)

Tactic:
isoOfHom_inv_hom_id,

Goal state after:
C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
f : Y ⟶ X
hf : W f
⊢ (homEquiv W L₁ L₂) (𝟙 (L₁.obj X)) = 𝟙 (L₂.obj X)
Module: Mathlib.CategoryTheory.Localization.SmallShiftedHom

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
L : C ⥤ D
inst✝² : L.IsLocalization W
inst✝¹ : L.CommShift M
X Y : C
inst✝ : HasSmallLocalizedShiftedHom W M X Y
f : X ⟶ Y
⊢ L.map (ShiftedHom.mk₀ 0 ⋯ f) ≫ (Functor.commShiftIso L 0).hom.app Y = ShiftedHom.mk₀ 0 ⋯ (L.map f)

Tactic:
Functor.map_comp

Goal state after:
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁶ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
L : C ⥤ D
inst✝² : L.IsLocalization W
inst✝¹ : L.CommShift M
X Y : C
inst✝ : HasSmallLocalizedShiftedHom W M X Y
f : X ⟶ Y
⊢ (L.map f ≫ L.map ((shiftFunctorZero' C 0 ⋯).inv.app Y)) ≫ (Functor.commShiftIso L 0).hom.app Y =
    ShiftedHom.mk₀ 0 ⋯ (L.map f)
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁✝ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
f₁ :
  A₁ ⟶
    ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
            (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
        ⋯ 0 ⋯).X₁
hf₁ :
  π₁ ≫ f₂ =
    f₁ ≫
      ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
              (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
          ⋯ 0 ⋯).f
⊢ f₂ = 0

Tactic:
dsimp at hf₁

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ : R₁.map' 0 2 mono_of_epi_of_mono_of_mono'._proof_2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
hR₁' :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₁✝ : Mono (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11)
h₃ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
f₂ : A ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₁ : C
π₁ : A₁ ⟶ A
w✝ : Epi π₁
f₁ :
  A₁ ⟶
    ((mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
            (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).sc
        ⋯ 0 ⋯).X₁
hf₁ : π₁ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ f₂ = 0
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ (π₂ ≫ π₁ ≫ g₁ - f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0

Tactic:
sub_comp,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝¹ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝ : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
⊢ (π₂ ≫ π₁ ≫ g₁) ≫ R₂.map' 1 2 ⋯ ⋯ - (f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.SingleTriangle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ ((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
      triangleOfSESδ ⋯ ≫
        (shiftFunctor (DerivedCategory C) 1).map
          (((CochainComplex.singleFunctors C).functor 0 ⋙ Q).map f.τ₁ ≫
            ((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)

Tactic:
Functor.map_comp

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
S : ShortComplex C
hS : S.ShortExact
S₁ S₂ : ShortComplex C
h₁ : S₁.ShortExact
h₂ : S₂.ShortExact
f : S₁ ⟶ S₂
⊢ ((singleFunctorsPostcompQIso C).hom.hom 0).app S₁.X₃ ≫
      triangleOfSESδ ⋯ ≫
        (shiftFunctor (DerivedCategory C) 1).map (((CochainComplex.singleFunctors C).functor 0 ⋙ Q).map f.τ₁) ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁) =
    (singleFunctor C 0).map f.τ₃ ≫
      ((singleFunctorsPostcompQIso C).hom.hom 0).app S₂.X₃ ≫
        triangleOfSESδ ⋯ ≫
          (shiftFunctor (DerivedCategory C) 1).map (((singleFunctorsPostcompQIso C).inv.hom 0).app S₂.X₁)
Module: Mathlib.CategoryTheory.Shift.Opposite

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b : A
⊢ (shiftFunctorAdd (OppositeShift C A) a b).hom.app X = ((shiftFunctorAdd C a b).inv.app (Opposite.unop X)).op

Tactic:
← cancel_mono ((shiftFunctorAdd (OppositeShift C A) a b).inv.app X),

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
A : Type u_2
inst✝¹ : AddMonoid A
inst✝ : HasShift C A
X : OppositeShift C A
a b : A
⊢ (shiftFunctorAdd (OppositeShift C A) a b).hom.app X ≫ (shiftFunctorAdd (OppositeShift C A) a b).inv.app X =
    ((shiftFunctorAdd C a b).inv.app (Opposite.unop X)).op ≫ (shiftFunctorAdd (OppositeShift C A) a b).inv.app X
Module: Mathlib.CategoryTheory.Shift.Pullback

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ (shiftFunctorZero (PullbackShift C φ) A).inv.app X =
    (shiftFunctorZero' C (φ 0) ⋯).inv.app X ≫ (pullbackShiftIso C φ 0 (φ 0) ⋯).inv.app X

Tactic:
rw [pullbackShiftFunctorZero_inv_app]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
A : Type u_2
B : Type u_3
inst✝² : AddMonoid A
inst✝¹ : AddMonoid B
inst✝ : HasShift C B
φ : A →+ B
X : PullbackShift C φ
⊢ (shiftFunctorZero C B).inv.app X ≫ (pullbackShiftIso C φ 0 0 ⋯).inv.app X =
    (shiftFunctorZero' C (φ 0) ⋯).inv.app X ≫ (pullbackShiftIso C φ 0 (φ 0) ⋯).inv.app X
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
X : D
⊢ a + (b + a) = 0 + a

Tactic:
h,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
X : D
⊢ a + 0 = 0 + a
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
X : D
⊢ b + a = -a + a

Tactic:
h,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
X : D
⊢ 0 = -a + a
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
Y : D
⊢ b + a = -a + a

Tactic:
h,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
a b : A
h : b + a = 0
inst✝ : F.CommShift A
Y : D
⊢ 0 = -a + a