mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
ι : Type u_2
c : ComplexShape ι
inst✝ : Preadditive V
F G : HomologicalComplex V c
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
n : ι
⊢ φ₁.f n = ((((dNext n) fun i j ↦ (h.hom j i).op) + (prevD n) fun i j ↦ (h.hom j i).op) + (φ₂.f n).op).unop

Tactic:
h.comm n

Goal state after:
V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
ι : Type u_2
c : ComplexShape ι
inst✝ : Preadditive V
F G : HomologicalComplex V c
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
n : ι
⊢ (dNext n) h.hom + (prevD n) h.hom + φ₂.f n =
    ((((dNext n) fun i j ↦ (h.hom j i).op) + (prevD n) fun i j ↦ (h.hom j i).op) + (φ₂.f n).op).unop
Module: Mathlib.Algebra.Homology.Opposite

Goal state before:
pp: V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
ι : Type u_2
c : ComplexShape ι
inst✝ : Preadditive V
F G : HomologicalComplex Vᵒᵖ c
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
n : ι
⊢ φ₁.f n = ((((dNext n) fun i j ↦ (h.hom j i).unop) + (prevD n) fun i j ↦ (h.hom j i).unop) + (φ₂.f n).unop).op

Tactic:
h.comm n

Goal state after:
V : Type u_1
inst✝¹ : Category.{v_1, u_1} V
ι : Type u_2
c : ComplexShape ι
inst✝ : Preadditive V
F G : HomologicalComplex Vᵒᵖ c
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
n : ι
⊢ (dNext n) h.hom + (prevD n) h.hom + φ₂.f n =
    ((((dNext n) fun i j ↦ (h.hom j i).unop) + (prevD n) fun i j ↦ (h.hom j i).unop) + (φ₂.f n).unop).op
Module: Mathlib.Algebra.Homology.Embedding.Restriction

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 : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
i' : ι'
hi : e.f i = i'
⊢ (restrictionMap φ e).f i = (K.restrictionXIso e hi).hom ≫ φ.f i' ≫ (L.restrictionXIso e hi).inv

Tactic:
subst hi

Goal state after:
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
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
⊢ (restrictionMap φ e).f i = (K.restrictionXIso e ⋯).hom ≫ φ.f (e.f i) ≫ (L.restrictionXIso e ⋯).inv
Module: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
X₁✝ X₂✝ X₃✝ : CochainComplex C ℤ
f : X₁✝ ⟶ X₂✝
g : X₂✝ ⟶ X₃✝
inst✝ : HasZeroObject C
X₁ X₂ X₃ : CochainComplex C ℤ
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
α : mappingCone.triangle u₁₂ ⟶ mappingCone.triangle (u₁₂ ≫ u₂₃) := mappingCone.triangleMap u₁₂ (u₁₂ ≫ u₂₃) (𝟙 X₁) u₂₃ ⋯
β : mappingCone.triangle (u₁₂ ≫ u₂₃) ⟶ mappingCone.triangle u₂₃ := mappingCone.triangleMap (u₁₂ ≫ u₂₃) u₂₃ u₁₂ (𝟙 X₃) ⋯
⊢ (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
          (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
              (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
            (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
              ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).mor₁ ≫
      (Iso.refl
          (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
              (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
                  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
                (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
                  ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).obj₂).hom =
    (Iso.refl
          (Triangle.mk ((quotient C (ComplexShape.up ℤ)).map α.hom₃) ((quotient C (ComplexShape.up ℤ)).map β.hom₃)
              (((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₂₃).mor₃ ≫
                  (Functor.commShiftIso (quotient C (ComplexShape.up ℤ)) 1).hom.app (mappingCone.triangle u₂₃).obj₁) ≫
                (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
                  ((quotient C (ComplexShape.up ℤ)).map (mappingCone.triangle u₁₂).mor₂))).obj₁).hom ≫
      (mappingConeCompTriangleh u₁₂ u₂₃).mor₁

Tactic:
by dsimp [α, mappingConeCompTriangleh]; simp

Goal state after:
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
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
cone : KernelFork (K.d j k)
hcone : IsLimit cone
W : C
f' : K.X i ⟶ cone.pt
hf' : f' ≫ Fork.ι cone = K.d i j
f'' : (K.extend e).X i' ⟶ cone.pt
hf'' : f'' ≫ Fork.ι cone ≫ (K.extendXIso e hj').inv = (K.extend e).d i' j'
φ : cone.pt ⟶ W
hij : c.Rel i j
hi'' : e.f i = i'
this : (K.extendXIso e hi'').hom ≫ f' = f''
⊢ f' ≫ φ = 0 ↔ f'' ≫ φ = 0

Tactic:
← cancel_epi (K.extendXIso e hi'').hom,

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
i' j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hi' : c'.prev j' = i'
cone : KernelFork (K.d j k)
hcone : IsLimit cone
W : C
f' : K.X i ⟶ cone.pt
hf' : f' ≫ Fork.ι cone = K.d i j
f'' : (K.extend e).X i' ⟶ cone.pt
hf'' : f'' ≫ Fork.ι cone ≫ (K.extendXIso e hj').inv = (K.extend e).d i' j'
φ : cone.pt ⟶ W
hij : c.Rel i j
hi'' : e.f i = i'
this : (K.extendXIso e hi'').hom ≫ f' = f''
⊢ (K.extendXIso e hi'').hom ≫ f' ≫ φ = (K.extendXIso e hi'').hom ≫ 0 ↔ f'' ≫ φ = 0
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

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✝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 j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ (if hi : e.BoundaryGE i then
        (K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv
      else (K.truncGE'XIso e ⋯ hi).hom ≫ φ.f (e.f i) ≫ (L.truncGE'XIso e ⋯ hi).inv) ≫
      (L.truncGE' e).d i j =
    (K.truncGE' e).d i j ≫ (K.truncGE'XIso e ⋯ ⋯).hom ≫ φ.f (e.f j) ≫ (L.truncGE'XIso e ⋯ ⋯).inv

Tactic:
rw [dif_pos hi]

Goal state after:
case pos
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 j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ ((K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv) ≫
      (L.truncGE' e).d i j =
    (K.truncGE' e).d i j ≫ (K.truncGE'XIso e ⋯ ⋯).hom ≫ φ.f (e.f j) ≫ (L.truncGE'XIso e ⋯ ⋯).inv
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 : ι
i' : ι'
hi' : e.f i = i'
hi : e.BoundaryGE i
⊢ (if hi : e.BoundaryGE i then K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv
    else (K.truncGE'XIso e ⋯ hi).inv) =
    (K.restrictionXIso e hi').hom ≫ K.pOpcycles i' ≫ (K.truncGE'XIsoOpcycles e hi' hi).inv

Tactic:
rw [dif_pos hi]

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
i' : ι'
hi' : e.f i = i'
hi : e.BoundaryGE i
⊢ K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv =
    (K.restrictionXIso e hi').hom ≫ K.pOpcycles i' ≫ (K.truncGE'XIsoOpcycles e hi' hi).inv
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 : ι
i' : ι'
hi' : e.f i = i'
hi : e.BoundaryGE i
⊢ f K e i = (K.restrictionXIso e hi').hom ≫ K.pOpcycles i' ≫ (K.truncGE'XIsoOpcycles e hi' hi).inv

Tactic:
by
  dsimp [f]
  rw [dif_pos hi]
  subst hi'
  simp [restrictionXIso]

Goal state after:
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 : HomologicalComplex C c
e : c.Embedding c'
j : ι
j' : ι'
hj' : e.f j = j'
inst✝¹ : K.HasHomology j
inst✝ : (K.extend e).HasHomology j'
⊢ (extend.homologyData' K e hj' ⋯ ⋯ (K.sc j).homologyData).left.cyclesIso.hom ≫ (K.sc j).homologyData.left.π =
    ((extend.homologyData' K e hj' ⋯ ⋯ (K.sc j).homologyData).left.cyclesIso.hom ≫
        (K.sc j).homologyData.left.cyclesIso.inv) ≫
      (K.sc j).homologyData.left.cyclesIso.hom ≫ (K.sc j).homologyData.left.π

Tactic:
simp only [assoc, Iso.inv_hom_id_assoc]

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✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
⊢ IsIso ((K.πTruncGE e).f i')

Tactic:
by_cases! hn : ∃ i, e.f i = i'

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
hn : ∃ i, e.f i = i'
⊢ IsIso ((K.πTruncGE e).f i')

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁷ : Category.{v_1, u_3} C
inst✝⁶ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁵ : e.IsTruncGE
inst✝⁴ : ∀ (i' : ι'), K.HasHomology i'
inst✝³ : ∀ (i' : ι'), L.HasHomology i'
inst✝² : ∀ (i' : ι'), M.HasHomology i'
inst✝¹ : HasZeroObject C
inst✝ : K.IsStrictlySupported e
i' : ι'
hn : ∀ (i : ι), e.f i ≠ i'
⊢ IsIso ((K.πTruncGE e).f i')
Module: Mathlib.Algebra.Homology.DerivedCategory.Fractions

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
inst✝² : X.IsStrictlyGE a
inst✝¹ : X.IsStrictlyLE b
inst✝ : Y.IsStrictlyGE a
f : Q.obj X ⟶ Q.obj Y
X' : CochainComplex C ℤ
hX' : X'.IsStrictlyLE b
s : X' ⟶ X
hs : IsIso (Q.map s)
g : X' ⟶ Y
fac : f = inv (Q.map s) ≫ Q.map g
⊢ ∃ X',
    ∃ (_ : X'.IsStrictlyGE a) (_ : X'.IsStrictlyLE b), ∃ s, ∃ (x : IsIso (Q.map s)), ∃ g, f = inv (Q.map s) ≫ Q.map g

Tactic:
have : IsIso (Q.map (CochainComplex.truncGEMap s a)) := by
    rw [isIso_Q_map_iff_quasiIso] at hs
    rw [isIso_Q_map_iff_quasiIso, CochainComplex.quasiIso_truncGEMap_iff]
    infer_instance

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
inst✝³ : HasDerivedCategory C
X Y : CochainComplex C ℤ
a b : ℤ
inst✝² : X.IsStrictlyGE a
inst✝¹ : X.IsStrictlyLE b
inst✝ : Y.IsStrictlyGE a
f : Q.obj X ⟶ Q.obj Y
X' : CochainComplex C ℤ
hX' : X'.IsStrictlyLE b
s : X' ⟶ X
hs : IsIso (Q.map s)
g : X' ⟶ Y
fac : f = inv (Q.map s) ≫ Q.map g
this : IsIso (Q.map (CochainComplex.truncGEMap s a))
⊢ ∃ X',
    ∃ (_ : X'.IsStrictlyGE a) (_ : X'.IsStrictlyLE b), ∃ s, ∃ (x : IsIso (Q.map s)), ∃ g, f = inv (Q.map s) ≫ Q.map g
Module: Mathlib.CategoryTheory.Localization.SmallHom

Goal state before:
pp: C₁ : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁹ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
X Y Z : C₁
inst✝⁸ : HasSmallLocalizedHom W₁ X Y
inst✝⁷ : HasSmallLocalizedHom W₁ Y Z
inst✝⁶ : HasSmallLocalizedHom W₁ X Z
X' Y' Z' : C₂
inst✝⁵ : HasSmallLocalizedHom W₂ X' X'
inst✝⁴ : HasSmallLocalizedHom W₂ Y' Y'
inst✝³ : HasSmallLocalizedHom W₂ Z' Z'
inst✝² : HasSmallLocalizedHom W₂ X' Y'
inst✝¹ : HasSmallLocalizedHom W₂ Y' Z'
inst✝ : HasSmallLocalizedHom W₂ X' Z'
eX : Φ.functor.obj X ≅ X'
eY : Φ.functor.obj Y ≅ Y'
eZ : Φ.functor.obj Z ≅ Z'
f : SmallHom W₁ X Y
g : SmallHom W₁ Y Z
this✝⁵ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
this✝⁴ : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) Y'
this✝³ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) Z'
this✝² : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) (Φ.functor.obj Z)
this✝¹ : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) Z'
this✝ : HasSmallLocalizedHom W₂ Y' (Φ.functor.obj Y)
this : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) (Φ.functor.obj Y)
⊢ Φ.smallHomMap' eX eZ (f.comp g) = (Φ.smallHomMap' eX eY f).comp (Φ.smallHomMap' eY eZ g)

Tactic:
simp only [smallHomMap', smallHomMap_comp, SmallHom.comp_assoc]

Goal state after:
C₁ : Type u₁
inst✝¹⁰ : Category.{v₁, u₁} C₁
W₁ : MorphismProperty C₁
C₂ : Type u₂
inst✝⁹ : Category.{v₂, u₂} C₂
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
X Y Z : C₁
inst✝⁸ : HasSmallLocalizedHom W₁ X Y
inst✝⁷ : HasSmallLocalizedHom W₁ Y Z
inst✝⁶ : HasSmallLocalizedHom W₁ X Z
X' Y' Z' : C₂
inst✝⁵ : HasSmallLocalizedHom W₂ X' X'
inst✝⁴ : HasSmallLocalizedHom W₂ Y' Y'
inst✝³ : HasSmallLocalizedHom W₂ Z' Z'
inst✝² : HasSmallLocalizedHom W₂ X' Y'
inst✝¹ : HasSmallLocalizedHom W₂ Y' Z'
inst✝ : HasSmallLocalizedHom W₂ X' Z'
eX : Φ.functor.obj X ≅ X'
eY : Φ.functor.obj Y ≅ Y'
eZ : Φ.functor.obj Z ≅ Z'
f : SmallHom W₁ X Y
g : SmallHom W₁ Y Z
this✝⁵ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) (Φ.functor.obj Y)
this✝⁴ : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) Y'
this✝³ : HasSmallLocalizedHom W₂ (Φ.functor.obj X) Z'
this✝² : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) (Φ.functor.obj Z)
this✝¹ : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) Z'
this✝ : HasSmallLocalizedHom W₂ Y' (Φ.functor.obj Y)
this : HasSmallLocalizedHom W₂ (Φ.functor.obj Y) (Φ.functor.obj Y)
⊢ (SmallHom.mk W₂ eX.inv).comp ((Φ.smallHomMap f).comp ((Φ.smallHomMap g).comp (SmallHom.mk W₂ eZ.hom))) =
    (SmallHom.mk W₂ eX.inv).comp
      ((Φ.smallHomMap f).comp
        ((SmallHom.mk W₂ eY.hom).comp ((SmallHom.mk W₂ eY.inv).comp ((Φ.smallHomMap g).comp (SmallHom.mk W₂ eZ.hom)))))
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: case mpr
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 : ℤ
⊢ ∀ (b : ℤ),
    Small.{w, w'}
      ((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)))

Tactic:
b

Goal state after:
case mpr
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
⊢ Small.{w, w'}
    ((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)))
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: case pos
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
j : ι
h₁ : Mono (HomologicalComplex.homologyMap S.g j)
h₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)
hj : ∃ i, c.Rel i j
⊢ IsZero (S.X₁.homology j)

pp: case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
j : ι
h₁ : Mono (HomologicalComplex.homologyMap S.g j)
h₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)
hj : ∀ (i : ι), ¬c.Rel i j
⊢ IsZero (S.X₁.homology j)

Tactic:
· obtain ⟨i, hij⟩ := hj
    have := h₂ i hij
    apply (hS.homology_exact₁ i j hij).isZero_X₂
    · simp [← cancel_epi (HomologicalComplex.homologyMap S.g i)]
    · simp [← cancel_mono (HomologicalComplex.homologyMap S.g j),
        ← HomologicalComplex.homologyMap_comp]

Goal state after:
case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
j : ι
h₁ : Mono (HomologicalComplex.homologyMap S.g j)
h₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)
hj : ∀ (i : ι), ¬c.Rel i j
⊢ IsZero (S.X₁.homology j)
Module: Mathlib.Algebra.Homology.Refinements

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
K : HomologicalComplex C c
i j : ι
hi : c.prev j = i
A : C
z₂ : A ⟶ K.cycles j
⊢ z₂ ≫ K.homologyπ j = 0 ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = x₁ ≫ K.toCycles i j

Tactic:
subst hi

Goal state after:
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
K : HomologicalComplex C c
j : ι
A : C
z₂ : A ⟶ K.cycles j
⊢ z₂ ≫ K.homologyπ j = 0 ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = x₁ ≫ K.toCycles (c.prev j) j
Module: Mathlib.Algebra.Homology.Refinements

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
K : HomologicalComplex C c
i j : ι
hi : c.prev j = i
A : C
z₂ z₂' : A ⟶ K.cycles j
⊢ z₂ ≫ K.homologyπ j = z₂' ≫ K.homologyπ j ↔ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = π ≫ z₂' + x₁ ≫ K.toCycles i j

Tactic:
subst hi

Goal state after:
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
K : HomologicalComplex C c
j : ι
A : C
z₂ z₂' : A ⟶ K.cycles j
⊢ z₂ ≫ K.homologyπ j = z₂' ≫ K.homologyπ j ↔
    ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ z₂ = π ≫ z₂' + x₁ ≫ K.toCycles (c.prev j) j
Module: Mathlib.CategoryTheory.Shift.Adjunction

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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
⊢ (shiftFunctor C (a + b)).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C (a + b)).obj X) ≫
      G.map ((Functor.CommShift.isoAdd e₁ f₁).hom.app X) ≫ (Functor.CommShift.isoAdd e₂ f₂).hom.app (F.obj X)

Tactic:
have := h' (X⟦a⟧)

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
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝² : AddMonoid A
inst✝¹ : HasShift C A
inst✝ : HasShift D A
a b : A
e₁ : shiftFunctor C a ⋙ F ≅ F ⋙ shiftFunctor D a
f₁ : shiftFunctor C b ⋙ F ≅ F ⋙ shiftFunctor D b
e₂ : shiftFunctor D a ⋙ G ≅ G ⋙ shiftFunctor C a
f₂ : shiftFunctor D b ⋙ G ≅ G ⋙ shiftFunctor C b
h : CompatibilityUnit adj e₁ e₂
h' : CompatibilityUnit adj f₁ f₂
X : C
this :
  (shiftFunctor C b).map (adj.unit.app ((shiftFunctor C a).obj X)) =
    adj.unit.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫
      G.map (f₁.hom.app ((shiftFunctor C a).obj X)) ≫ f₂.hom.app (F.obj ((shiftFunctor C a).obj X))
⊢ (shiftFunctor C (a + b)).map (adj.unit.app X) =
    adj.unit.app ((shiftFunctor C (a + b)).obj X) ≫
      G.map ((Functor.CommShift.isoAdd e₁ f₁).hom.app X) ≫ (Functor.CommShift.isoAdd e₂ f₂).hom.app (F.obj 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
⊢ 0 = -a + a

Tactic:
neg_add_cancel

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 = 0
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
⊢ 0 = -a + a

Tactic:
neg_add_cancel

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 = 0