mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : W.LeftFraction X Y
⊢ homMk f = f.map (Q W) ⋯

Tactic:
← Q_map_comp_Qinv f.f f.s f.hs,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : W.LeftFraction X Y
⊢ (Q W).map f.f ≫ Qinv f.s ⋯ = f.map (Q W) ⋯
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
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction X Y
⊢ φ.map L ⋯ = ψ.map L ⋯ ↔ LeftFractionRel φ ψ

Tactic:
constructor
  · intro h
    rw [← MorphismProperty.LeftFraction.Localization.map_eq_iff]
    apply map_eq_of_map_eq _ _ _ _ h
  · intro h
    simp only [← Localization.Hom.map_mk _ L (Localization.inverts _ _)]
    congr 1
    exact Quot.sound h

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

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
  ∀ (n : ℤ),
    IsZero
        { X₁ := (F.shift n).obj (Triangle.mk f g h).obj₂, X₂ := (F.shift n).obj (Triangle.mk f g h).obj₃,
            X₃ := (F.shift (n + 1)).obj (Triangle.mk f g h).obj₁, f := (F.shift n).map (Triangle.mk f g h).mor₂,
            g := F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯, zero := ⋯ }.X₂ ↔
      { X₁ := (F.shift n).obj (Triangle.mk f g h).obj₂, X₂ := (F.shift n).obj (Triangle.mk f g h).obj₃,
              X₃ := (F.shift (n + 1)).obj (Triangle.mk f g h).obj₁, f := (F.shift n).map (Triangle.mk f g h).mor₂,
              g := F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯, zero := ⋯ }.f =
          0 ∧
        { X₁ := (F.shift n).obj (Triangle.mk f g h).obj₂, X₂ := (F.shift n).obj (Triangle.mk f g h).obj₃,
              X₃ := (F.shift (n + 1)).obj (Triangle.mk f g h).obj₁, f := (F.shift n).map (Triangle.mk f g h).mor₂,
              g := F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯, zero := ⋯ }.g =
          0
h₂ :
  ∀ (n : ℤ),
    Mono ((F.shift (n + 1)).map (Triangle.mk f g h).mor₁) ↔ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₃ : ∀ (n : ℤ), Epi ((F.shift n).map (Triangle.mk f g h).mor₁) ↔ (F.shift n).map (Triangle.mk f g h).mor₂ = 0
⊢ F.homologicalKernel (Triangle.mk f g h).obj₃ ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)

Tactic:
dsimp at h₁ h₂ h₃ ⊢

Goal state after:
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
h₁ :
  ∀ (n : ℤ),
    IsZero ((F.shift n).obj Z) ↔ (F.shift n).map g = 0 ∧ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₂ : ∀ (n : ℤ), Mono ((F.shift (n + 1)).map f) ↔ F.homologySequenceδ (Triangle.mk f g h) n (n + 1) ⋯ = 0
h₃ : ∀ (n : ℤ), Epi ((F.shift n).map f) ↔ (F.shift n).map g = 0
⊢ F.homologicalKernel Z ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)
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 Q R : ObjectProperty C
inst✝¹ : P.Nonempty
inst✝ : Q.Nonempty
Y : C
f : P.arbitrary ⟶ Y
g : Y ⟶ Q.arbitrary
hT : Triangle.mk f g 0 ∈ distinguishedTriangles
⊢ (P.extensionProduct Q).Nonempty

Tactic:
exact ⟨_, _, _, _, _, _, hT, P.prop_arbitrary, Q.prop_arbitrary⟩

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

Goal state before:
pp: C : Type u_1
inst✝⁸ : Category.{v_1, u_1} C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
M : Type u_4
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
X Y : C
inst✝³ : Preadditive C
inst✝² : Preadditive D
a : M
F : C ⥤ D
inst✝¹ : F.CommShift M
inst✝ : F.Additive
⊢ map 0 F = 0

Tactic:
by
  simp [ShiftedHom.map]

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

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
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (F.shiftIso 0 a a ⋯).hom.app X = (shiftFunctorZero D A).hom.app ((F.functor a).obj X)

Tactic:
rw [shiftIso_zero]
  rfl

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

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
A : Type u_5
inst✝¹ : AddMonoid A
inst✝ : HasShift D A
F : SingleFunctors C D A
a : A
X : C
⊢ (F.shiftIso 0 a a ⋯).inv.app X = (shiftFunctorZero D A).inv.app ((F.functor a).1 X)

Tactic:
rw [shiftIso_zero]
  rfl

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

Goal state before:
pp: case mp
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝¹ : Category.{v_1, u_3} C
inst✝ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
⊢ IsZero K → K.IsStrictlySupported e ∧ K.IsStrictlySupportedOutside e

Tactic:
intro hK
    constructor
    all_goals
      constructor
      intros
      exact (eval _ _ _).map_isZero hK

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'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
K L M : HomologicalComplex C c
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
⊢ extendMap (φ ≫ φ') e = extendMap φ e ≫ extendMap φ' e

Tactic:
ext i'
  by_cases hi' : ∃ i, e.f i = i'
  · obtain ⟨i, hi⟩ := hi'
    simp [extendMap_f _ e hi]
  · simp [extendMap_f_eq_zero _ e i' (fun i hi => hi' ⟨i, hi⟩)]

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

Goal state before:
pp: p : ℤ
n : ℕ
⊢ (embeddingUpIntGE p).BoundaryGE n ↔ n = 0

Tactic:
by
  constructor
  · intro h
    obtain _ | n := n
    · rfl
    · have := h.2 n
      dsimp at this
      lia
  · rintro rfl
    constructor
    · simp
    · intro i hi
      dsimp at hi
      lia

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

Goal state before:
pp: p : ℤ
n : ℕ
⊢ (embeddingUpIntLE p).BoundaryLE n ↔ n = 0

Tactic:
by
  constructor
  · intro h
    obtain _ | n := n
    · rfl
    · have := h.2 n
      dsimp at this
      lia
  · rintro rfl
    constructor
    · simp
    · intro i hi
      dsimp at hi
      lia

Goal state after:
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.restriction e).d i j ≫ (K.restrictionXIso e hj').hom ≫ K.pOpcycles j' = 0

Tactic:
rw [restriction_d_eq _ _ hi' hj', assoc, assoc, Iso.inv_hom_id_assoc,
        d_pOpcycles, comp_zero]

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✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncGE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : K.IsStrictlySupported e
i : ι
hi₁ : c'.Rel (c'.prev (e.f i)) (e.f i)
hi₂ : ∀ (i_1 : ι), ¬c'.Rel (e.f i_1) (e.f i)
j : ι
hj : e.f j = c'.prev (e.f i)
⊢ c'.Rel (e.f j) (e.f i)

Tactic:
simpa only [hj] using hi₁

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

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.IsTruncLE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : ∀ (i' : ι'), L.HasHomology i'
i : ι
hi : ¬e.BoundaryLE i
i' : ι'
h : e.f i = i'
⊢ ((truncLE'Map φ e).f i).op = ((K.truncLE'XIso e h hi).hom ≫ φ.f i' ≫ (L.truncLE'XIso e h hi).inv).op

Tactic:
by simpa using! truncGE'Map_f_eq ((opFunctor C c').map φ.op) e.op (by simpa) h

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

Goal state before:
pp: case mpr
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.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
⊢ K.IsSupported e → QuasiIso (K.πTruncGE e)

Tactic:
· intro
    rw [quasiIso_iff]
    intro i'
    by_cases hi' : ∃ i, e.f i = i'
    · obtain ⟨i, rfl⟩ := hi'
      infer_instance
    · rw [quasiIsoAt_iff_exactAt (K.πTruncGE e) i']
      all_goals exact exactAt_of_isSupported _ e i' (by simpa using hi')

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

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 : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
⊢ QuasiIso (truncLEMap φ e) ↔ ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'

Tactic:
← quasiIso_opFunctor_map_iff

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 : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncLE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
⊢ QuasiIso ((opFunctor C c').map (truncLEMap φ e).op) ↔ ∀ (i : ι) (i' : ι'), e.f i = i' → QuasiIsoAt φ i'
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → IsZero (K.X i)
i : ℤ
hi : ∀ (i_1 : ℕ), (embeddingUpIntLE n).f i_1 ≠ i
⊢ IsZero (K.X i)

Tactic:
notMem_range_embeddingUpIntLE_iff

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → IsZero (K.X i)
i : ℤ
hi : n < i
⊢ IsZero (K.X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → ExactAt K i
i : ℤ
hi : ∀ (i_1 : ℕ), (embeddingUpIntLE n).f i_1 ≠ i
⊢ ExactAt K i

Tactic:
notMem_range_embeddingUpIntLE_iff

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → ExactAt K i
i : ℤ
hi : n < i
⊢ ExactAt K i
Module: Mathlib.CategoryTheory.Localization.SmallShiftedHom

Goal state before:
pp: C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁷ : AddMonoid M
inst✝⁶ : HasShift C M
inst✝⁵ : HasShift D M
X Y : C
inst✝⁴ : HasSmallLocalizedShiftedHom W M X Y
inst✝³ : HasSmallLocalizedShiftedHom W M Y Y
inst✝² : HasSmallLocalizedShiftedHom W M Y X
inst✝¹ : W.IsCompatibleWithShift M
inst✝ : W.RespectsIso
m₀ : M
hm₀ : m₀ = 0
f : Y ⟶ X
hf : W f
⊢ m₀ + m₀ = m₀

Tactic:
by subst hm₀; simp

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

Goal state before:
pp: C : Type u₁
inst✝⁹ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁸ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁷ : AddMonoid M
inst✝⁶ : HasShift C M
inst✝⁵ : HasShift D M
X Y : C
inst✝⁴ : HasSmallLocalizedShiftedHom W M X Y
inst✝³ : HasSmallLocalizedShiftedHom W M X X
inst✝² : HasSmallLocalizedShiftedHom W M Y X
inst✝¹ : W.IsCompatibleWithShift M
inst✝ : W.RespectsIso
m₀ : M
hm₀ : m₀ = 0
f : Y ⟶ X
hf : W f
⊢ m₀ + m₀ = m₀

Tactic:
by subst hm₀; simp

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

Goal state before:
pp: C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁷ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
L : C ⥤ D
inst✝³ : L.IsLocalization W
inst✝² : L.CommShift M
X Y : C
m : M
inst✝¹ : HasSmallLocalizedShiftedHom W M X Y
inst✝ : HasSmallLocalizedShiftedHom W M X Y
e : SmallShiftedHom W X Y m
⊢ (SmallHom.equiv W L) (chgUniv e) ≫ (Functor.commShiftIso L m).hom.app Y =
    (SmallHom.equiv W L) e ≫ (Functor.commShiftIso L m).hom.app Y

Tactic:
congr

Goal state after:
case e_a
C : Type u₁
inst✝⁸ : Category.{v₁, u₁} C
D : Type u₂
inst✝⁷ : Category.{v₂, u₂} D
W : MorphismProperty C
M : Type w'
inst✝⁶ : AddMonoid M
inst✝⁵ : HasShift C M
inst✝⁴ : HasShift D M
L : C ⥤ D
inst✝³ : L.IsLocalization W
inst✝² : L.CommShift M
X Y : C
m : M
inst✝¹ : HasSmallLocalizedShiftedHom W M X Y
inst✝ : HasSmallLocalizedShiftedHom W M X Y
e : SmallShiftedHom W X Y m
⊢ (SmallHom.equiv W L) (chgUniv e) = (SmallHom.equiv W L) e
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)
⊢ Mono (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)

Tactic:
apply mono_of_cancel_zero

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)
⊢ ∀ {P : C} (g : P ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4),
    g ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4 = 0 → g = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mpr.inr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X Y : C
a b : ℤ
hab : b < a
x :
  (shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
    (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y))
⊢ ∀
    (b_1 :
      (shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
        (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y))),
    x = b_1

Tactic:
y

Goal state after:
case mpr.inr
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
h :
  ∀ (X Y : C) (n : ℤ),
    0 ≤ n →
      Small.{w, w'} ((singleFunctor C 0).obj X ⟶ (shiftFunctor (DerivedCategory C) n).obj ((singleFunctor C 0).obj Y))
X Y : C
a b : ℤ
hab : b < a
x y :
  (shiftFunctor (DerivedCategory C) a).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj X)) ⟶
    (shiftFunctor (DerivedCategory C) b).obj (Q.obj ((CochainComplex.singleFunctor C 0).obj Y))
⊢ x = y
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 Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp α'.hom ⋯ + (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp β'.hom ⋯ = α.hom + β.hom

Tactic:
congr

Goal state after:
case e_a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp α'.hom ⋯ = α.hom

case e_a
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
α β : Ext X Y n
α' : Ext (X ⊞ X) Y n := (mk₀ biprod.fst).comp α ⋯
β' : Ext (X ⊞ X) Y n := (mk₀ biprod.snd).comp β ⋯
eq₁ : α + β = (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).comp (α' + β') ⋯
eq₂ : α' + β' = homEquiv.symm (α'.hom + β'.hom)
⊢ (mk₀ (biprod.lift (𝟙 X) (𝟙 X))).hom.comp β'.hom ⋯ = β.hom
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
n : ℕ
R₁ R₂ : ComposableArrows C n
hR₁ : R₁.Exact
hR₂ : R₂.Exact
φ : R₁ ⟶ R₂
k₀ : ℕ
h : k₀ + 3 ≤ n
h₂ : Epi (app' ((natAddLEFunctor h).map φ) 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃ : Mono (app' ((natAddLEFunctor h).map φ) 3 mono_of_epi_of_mono_of_mono'._proof_7)
h₀ : Epi (app' ((natAddLEFunctor h).map φ) 0 mono_of_epi_of_mono_of_mono'._proof_13)
⊢ Epi (app' ((natAddLEFunctor h).map φ) 1 mono_of_epi_of_mono_of_mono'._proof_11)

Tactic:
exact epi_of_epi_of_epi_of_mono _ (natAddLEFunctor_obj_exact h hR₁)
    (natAddLEFunctor_obj_exact h hR₂) h₀ h₂ h₃

Goal state after:
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
m n p : ℤ
h : m + n = p
⊢ (opShiftFunctorEquivalence C p).unitIso.hom.app X =
    (opShiftFunctorEquivalence C n).unitIso.hom.app X ≫
      ((shiftFunctor C n).map ((opShiftFunctorEquivalence C m).unitIso.hom.app ((shiftFunctor Cᵒᵖ n).obj X)).unop).op ≫
        ((shiftFunctorAdd' C m n p h).hom.app
              (Opposite.unop ((opShiftFunctorEquivalence C m).functor.obj ((shiftFunctor Cᵒᵖ n).obj X)))).op ≫
          ((shiftFunctor C p).map ((shiftFunctorAdd' Cᵒᵖ n m p ⋯).inv.app X).unop).op

Tactic:
dsimp [opShiftFunctorEquivalence]
  simp only [shiftFunctorAdd'_op_inv_app _ n m p (by lia) _ _ _ (add_neg_cancel n)
    (add_neg_cancel m) (add_neg_cancel p), shiftFunctor_op_map _ m (-m),
    Category.assoc, Iso.inv_hom_id_app_assoc]
  erw [Functor.map_id, Functor.map_id, Functor.map_id, Functor.map_id,
    id_comp, id_comp, id_comp, comp_id, comp_id]
  dsimp
  rw [comp_id, shiftFunctorCompIsoId_add'_hom_app _ _ _ _ _ _
    (neg_add_cancel m) (neg_add_cancel n) (neg_add_cancel p) h]
  dsimp
  rw [Category.assoc, Category.assoc]
  rfl

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: case refine_2.refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact

pp: case refine_2.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact

Tactic:
· refine exact₂_mk _ (by simp) ?_
      rw [ShortComplex.exact_iff_mono _ (by simp)]
      infer_instance

Goal state after:
case refine_2.refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 2
φ : R₁ ⟶ R₂
hR₁ : R₁.Exact
hR₂ : R₂.Exact
hR₁' : Mono (R₁.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
hR₂' : Mono (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_6)
h₁ : IsIso (app' φ 1 mono_of_epi_of_mono_of_mono'._proof_6)
h₂ : Mono (app' φ 2 mono_of_epi_of_epi_mono'._proof_1)
ψ : mk₄ 0 0 (R₁.map' 0 1 ⋯ ⋯) (R₁.map' 1 2 ⋯ ⋯) ⟶ mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯) :=
  homMk₄ 0 0 (app' φ 0 ⋯) (app' φ 1 ⋯) (app' φ 2 ⋯) ⋯ ⋯ ⋯ ⋯
⊢ (mk₂ ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 0 1 isComplex₂_iff._proof_2 ⋯)
      ((mk₄ 0 0 (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).δ₀.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact