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

Goal state before:
pp: case pos
C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i j : ι
hij : c.Rel i j
hj : c.Rel j (c.next j)
⊢ inlX ((H.mapHomologicalComplex c).map φ) j i hij ≫ (mapHomologicalComplexObjXIso φ H i).inv ≫ H.map (d φ i j) =
    inlX ((H.mapHomologicalComplex c).map φ) j i hij ≫
      d ((H.mapHomologicalComplex c).map φ) i j ≫ (mapHomologicalComplexObjXIso φ H j).inv

Tactic:
simp [← Functor.map_comp, inlX_d _ _ _ _ _ hj, inlX_d_assoc _ _ _ _ _ hj]

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

Goal state before:
pp: case left
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v, u_1} C
inst✝⁵ : Category.{v', u_2} D
inst✝⁴ : Preadditive C
inst✝³ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝² : HasHomotopyCofiber φ
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)
n : ℤ
⊢ (H.map ((↑(fst φ)).v n (n + 1) ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) n ⋯ +
        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫
      (-(↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n (n + 1) ⋯ ≫ H.map (F.d (n + 1) (n + 2))) =
    H.map ((mappingCone φ).d n (n + 1)) ≫
      (H.map ((↑(fst φ)).v (n + 1) (n + 2) ⋯) ≫
            (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 2) (n + 1) ⋯ +
          H.map ((snd φ).v (n + 1) (n + 1) ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f (n + 1)) ≫
        (↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v (n + 1) (n + 2)
          (CochainComplex.mappingCone.mapHomologicalComplexIso._proof_1 n)

Tactic:
simp only [Functor.mapHomologicalComplex_obj_X, comp_neg, add_comp, assoc, inl_v_fst_v_assoc,
        inr_f_fst_v_assoc, zero_comp, comp_zero, add_zero, inl_v_fst_v, comp_id, inr_f_fst_v,
        ← H.map_comp, d_fst_v φ n (n + 1) (n + 2) rfl (by lia), Functor.map_neg]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
x✝ : HomologicalComplex C (ComplexShape.up ℤ)
i✝ : ℤ
⊢ ((F.mapCochainComplexShiftIso 0).hom.app x✝).f i✝ =
    ((CommShift.isoZero (F.mapHomologicalComplex (ComplexShape.up ℤ)) ℤ).hom.app x✝).f i✝

Tactic:
rw [CommShift.isoZero_hom_app]

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
D : Type u'
inst✝² : Category.{v', u'} D
inst✝¹ : Preadditive D
F : C ⥤ D
inst✝ : F.Additive
x✝ : HomologicalComplex C (ComplexShape.up ℤ)
i✝ : ℤ
⊢ ((F.mapCochainComplexShiftIso 0).hom.app x✝).f i✝ =
    ((F.mapHomologicalComplex (ComplexShape.up ℤ)).map
            ((shiftFunctorZero (HomologicalComplex C (ComplexShape.up ℤ)) ℤ).hom.app x✝) ≫
          (shiftFunctorZero (HomologicalComplex D (ComplexShape.up ℤ)) ℤ).inv.app
            ((F.mapHomologicalComplex (ComplexShape.up ℤ)).obj x✝)).f
      i✝
Module: Mathlib.Algebra.Homology.HomotopyCategory.Shift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).Additive

Tactic:
have : ((quotient C (ComplexShape.up ℤ) ⋙ shiftFunctor _ n)).Additive :=
    Functor.additive_of_iso ((quotient C (ComplexShape.up ℤ)).commShiftIso n)

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : Preadditive D
n : ℤ
this : (quotient C (ComplexShape.up ℤ) ⋙ shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).Additive
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).Additive
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
K L M : CochainComplex C ℤ
n✝ : ℤ
γ✝ γ₁ γ₂ : Cochain K L n✝
n a n' : ℤ
hn' : n' + a = n
γ γ' : Cochain K L n
⊢ (γ + γ').rightShift a n' hn' = γ.rightShift a n' hn' + γ'.rightShift a n' hn'

Tactic:
by simp only [rightShift_add]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' n'' : ℤ
hn' : n' + a = n
hn'' : n + a = n''
⊢ ((a * n).negOnePow * (a * (a - 1) / 2).negOnePow) • γ.shift a =
    (1 * (a * n).negOnePow * (a * (a - 1) / 2).negOnePow) • γ.shift a

Tactic:
one_mul

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' n'' : ℤ
hn' : n' + a = n
hn'' : n + a = n''
⊢ ((a * n).negOnePow * (a * (a - 1) / 2).negOnePow) • γ.shift a =
    ((a * n).negOnePow * (a * (a - 1) / 2).negOnePow) • γ.shift a
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : Epi T.mor₂
⊢ T.mor₂ ≫ T.mor₃ = T.mor₂ ≫ 0

Tactic:
comp_distTriang_mor_zero₂₃ _ hT,

Goal state after:
case mpr
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
a✝ : Epi T.mor₂
⊢ 0 = T.mor₂ ≫ 0
Module: Mathlib.CategoryTheory.Triangulated.Triangulated

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
X₁' X₂' X₃' Z₁₂' Z₂₃' Z₁₃' : C
u₁₂' : X₁' ⟶ X₂'
u₂₃' : X₂' ⟶ X₃'
u₁₃' : X₁' ⟶ X₃'
comm' : u₁₂' ≫ u₂₃' = u₁₃'
e₁ : X₁ ≅ X₁'
e₂ : X₂ ≅ X₂'
e₃ : X₃ ≅ X₃'
comm₁₂ : u₁₂ ≫ e₂.hom = e₁.hom ≫ u₁₂'
comm₂₃ : u₂₃ ≫ e₃.hom = e₂.hom ≫ u₂₃'
v₁₂' : X₂' ⟶ Z₁₂'
w₁₂' : Z₁₂' ⟶ (shiftFunctor C 1).obj X₁'
h₁₂' : Triangle.mk u₁₂' v₁₂' w₁₂' ∈ distinguishedTriangles
v₂₃' : X₃' ⟶ Z₂₃'
w₂₃' : Z₂₃' ⟶ (shiftFunctor C 1).obj X₂'
h₂₃' : Triangle.mk u₂₃' v₂₃' w₂₃' ∈ distinguishedTriangles
v₁₃' : X₃' ⟶ Z₁₃'
w₁₃' : Z₁₃' ⟶ (shiftFunctor C 1).obj X₁'
h₁₃' : Triangle.mk u₁₃' v₁₃' w₁₃' ∈ distinguishedTriangles
H : Octahedron comm' h₁₂' h₂₃' h₁₃'
iso₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ≅ Triangle.mk u₁₂' v₁₂' w₁₂' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₁₂ v₁₂ w₁₂) (Triangle.mk u₁₂' v₁₂' w₁₂') h₁₂ h₁₂' e₁ e₂ comm₁₂
iso₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ≅ Triangle.mk u₂₃' v₂₃' w₂₃' :=
  isoTriangleOfIso₁₂ (Triangle.mk u₂₃ v₂₃ w₂₃) (Triangle.mk u₂₃' v₂₃' w₂₃') h₂₃ h₂₃' e₂ e₃ comm₂₃
⊢ (Triangle.mk u₁₃ v₁₃ w₁₃).mor₁ ≫ e₃.hom = e₁.hom ≫ (Triangle.mk u₁₃' v₁₃' w₁₃').mor₁

Tactic:
dsimp; rw [← comm, assoc, ← comm', ← reassoc_of% comm₁₂, comm₂₃]

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
hT₀ : T.mor₃ = 0
inr : T.obj₃ ⟶ T.obj₂
inr_snd : inr ≫ T.mor₂ = 𝟙 T.obj₃
fst : T.obj₂ ⟶ T.obj₁
total : fst ≫ T.mor₁ + T.mor₂ ≫ inr = 𝟙 T.obj₂
this : Mono T.mor₁
eq : fst ≫ T.mor₁ = 𝟙 T.obj₂ - T.mor₂ ≫ inr
⊢ inr ≫ fst = 0

Tactic:
by
          simp only [← cancel_mono T.mor₁, assoc, eq, comp_sub, reassoc_of% inr_snd,
            comp_id, sub_self, zero_comp]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ G.map ((triangle φ).mor₃.f n) ≫ 𝟙 (G.obj (K.X (n + 1))) =
    (G.map ((↑(fst φ)).v n (n + 1) ⋯) ≫ (inl ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v (n + 1) n ⋯ +
        G.map ((snd φ).v n n ⋯) ≫ (inr ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫
      (triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃.f n

Tactic:
simp only [Functor.mapHomologicalComplex_obj_X, add_comp, assoc, inl_v_triangle_mor₃_f,
    shiftFunctor_obj_X, shiftFunctorObjXIso, HomologicalComplex.XIsoOfEq_rfl, Iso.refl_inv,
    comp_neg, comp_id, inr_f_triangle_mor₃_f, comp_zero, add_zero]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ G.map ((triangle φ).mor₃.f n) = -G.map ((↑(fst φ)).v n (n + 1) ⋯)
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T₁ T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT₁ : T₁ ∈ distinguishedTriangles C
hT₂ : T₂ ∈ distinguishedTriangles C
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
⊢ ∃ c,
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map a = c ≫ T₂.mor₃

Tactic:
by
  obtain ⟨K₁, L₁, φ₁, ⟨e₁⟩⟩ := hT₁
  obtain ⟨K₂, L₂, φ₂, ⟨e₂⟩⟩ := hT₂
  obtain ⟨a', ha'⟩ : ∃ (a' : (quotient _ _).obj K₁ ⟶ (quotient _ _).obj K₂),
    a' = e₁.inv.hom₁ ≫ a ≫ e₂.hom.hom₁ := ⟨_, rfl⟩
  obtain ⟨b', hb'⟩ : ∃ (b' : (quotient _ _).obj L₁ ⟶ (quotient _ _).obj L₂),
    b' = e₁.inv.hom₂ ≫ b ≫ e₂.hom.hom₂ := ⟨_, rfl⟩
  obtain ⟨a'', rfl⟩ := (quotient _ _).map_surjective a'
  obtain ⟨b'', rfl⟩ := (quotient _ _).map_surjective b'
  have H : Homotopy (φ₁ ≫ b'') (a'' ≫ φ₂) := homotopyOfEq _ _ (by
    have comm₁₁ := e₁.inv.comm₁
    have comm₁₂ := e₂.hom.comm₁
    dsimp at comm₁₁ comm₁₂
    simp only [Functor.map_comp, ha', hb', reassoc_of% comm₁₁,
      reassoc_of% fac, comm₁₂, assoc])
  let γ := e₁.hom ≫ trianglehMapOfHomotopy H ≫ e₂.inv
  have comm₂ := γ.comm₂
  have comm₃ := γ.comm₃
  dsimp [γ] at comm₂ comm₃
  simp only [ha', hb'] at comm₂ comm₃
  refine ⟨γ.hom₃, ?_, ?_⟩
  -- the following list of lemmas was obtained by doing simpa? [γ] using comm₂
  · simpa only [triangleCategory_comp, Functor.mapTriangle_obj, triangle_obj₁, triangle_obj₂,
      triangle_obj₃, triangle_mor₁, triangle_mor₂, TriangleMorphism.comp_hom₃, Triangle.mk_obj₃,
      trianglehMapOfHomotopy_hom₃, TriangleMorphism.comm₂_assoc, Triangle.mk_obj₂,
      Triangle.mk_mor₂, assoc, Iso.hom_inv_id_triangle_hom₂, comp_id,
      Iso.hom_inv_id_triangle_hom₂_assoc, γ] using comm₂
  -- the following list of lemmas was obtained by doing simpa? [γ] using comm₃
  · simpa only [triangleCategory_comp, Functor.mapTriangle_obj, triangle_obj₁, triangle_obj₂,
      triangle_obj₃, triangle_mor₁, triangle_mor₂, TriangleMorphism.comp_hom₃, Triangle.mk_obj₃,
      trianglehMapOfHomotopy_hom₃, assoc, Triangle.mk_obj₁, Iso.hom_inv_id_triangle_hom₁, comp_id,
      Iso.hom_inv_id_triangle_hom₁_assoc, γ] using comm₃

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (-(shiftFunctor C 1).map v₁₃) ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (-(shiftFunctor C 1).map v₂₃)
⊢ Octahedron' comm h₁₂ h₂₃ h₁₃

Tactic:
rw [neg_comp, comp_neg, neg_inj] at eq₄

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
inst✝ : IsTriangulated C
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Z₁₂ ⟶ X₁
w₁₂ : X₂ ⟶ (shiftFunctor C 1).obj Z₁₂
h₁₂ : Triangle.mk v₁₂ u₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Z₂₃ ⟶ X₂
w₂₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₂₃
h₂₃ : Triangle.mk v₂₃ u₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : Z₁₃ ⟶ X₁
w₁₃ : X₃ ⟶ (shiftFunctor C 1).obj Z₁₃
h₁₃ : Triangle.mk v₁₃ u₁₃ w₁₃ ∈ distinguishedTriangles
o : Octahedron comm ⋯ ⋯ ⋯ := someOctahedron comm ⋯ ⋯ ⋯
m₁ : Z₁₂ ⟶ Z₁₃ := (shiftShiftNeg Z₁₂ 1).inv ≫ (shiftFunctor C (-1)).map o.m₁ ≫ (shiftShiftNeg Z₁₃ 1).hom
m₃ : Z₁₃ ⟶ Z₂₃ := (shiftShiftNeg Z₁₃ 1).inv ≫ (shiftFunctor C (-1)).map o.m₃ ≫ (shiftShiftNeg Z₂₃ 1).hom
eq₁ : w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
eq₂ : o.m₁ ≫ (shiftFunctor C 1).map v₁₃ = (shiftFunctor C 1).map v₁₂
eq₃ : w₁₃ ≫ o.m₃ = w₂₃
eq₄ : (shiftFunctor C 1).map v₁₃ ≫ (shiftFunctor C 1).map u₁₂ = o.m₃ ≫ (shiftFunctor C 1).map v₂₃
⊢ Octahedron' comm h₁₂ h₂₃ h₁₃
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case cons
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
this : f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p
⊢ (Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
        Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g') ≫
      φ.map p =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g' ≫ φ.map p

Tactic:
Category.assoc

Goal state after:
case cons
V : Type u₁
inst✝¹ : Quiver V
C : Type ?u.6
inst✝ : Category.{v_1, ?u.6} C
φ : V ⥤q C
X✝ Y✝ Z✝ : Paths V
f : X✝ ⟶ Y✝
b✝ c✝ : Paths V
g' : Quiver.Path Y✝ b✝
p : b✝ ⟶ c✝
ih :
  Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g') =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g'
this : f ≫ g'.cons p = Quiver.Path.cons (f ≫ g') p
⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g' ≫ φ.map p =
    Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫
      Quiver.Path.rec (𝟙 (φ.obj Y✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) g' ≫ φ.map p
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
A : C
k : A ⟶ S.X₂.cycles i
hk : k ≫ cyclesMap S.g i = 0
⊢ (k ≫ S.X₂.iCycles i) ≫ (S.map (eval C c i)).g = 0

Tactic:
dsimp
      rw [assoc, ← cyclesMap_i, reassoc_of% hk, zero_comp]

Goal state after:
Module: Mathlib.CategoryTheory.PathCategory.MorphismProperty

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : (of V).obj u ⟶ (of V).obj v) (q : v ⟶ w), P p → P (p ≫ (of V).map q)
⊢ P = ⊤

Tactic:
ext; constructor
  · simp
  · exact fun _ ↦ induction (fun f ↦ P f) id comp _

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

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
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
A : C
x₃ : A ⟶ S.X₃.X i
hx₃ : x₃ ≫ S.X₃.d i j = 0
x₂ : A ⟶ S.X₂.X i
hx₂ : x₂ ≫ S.g.f i = x₃
x₁ : A ⟶ S.X₁.X j
hx₁ : x₁ ≫ S.f.f j = x₂ ≫ S.X₂.d i j
k : ι
hk : c.next j = k
this : Mono S.f
⊢ x₁ ≫ S.X₁.d j k = 0

Tactic:
← cancel_mono (S.f.f k),

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
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
A : C
x₃ : A ⟶ S.X₃.X i
hx₃ : x₃ ≫ S.X₃.d i j = 0
x₂ : A ⟶ S.X₂.X i
hx₂ : x₂ ≫ S.g.f i = x₃
x₁ : A ⟶ S.X₁.X j
hx₁ : x₁ ≫ S.f.f j = x₂ ≫ S.X₂.d i j
k : ι
hk : c.next j = k
this : Mono S.f
⊢ (x₁ ≫ S.X₁.d j k) ≫ S.f.f k = 0 ≫ S.f.f k
Module: Mathlib.CategoryTheory.Localization.Predicate

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
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
⊢ P = ⊤

Tactic:
let e := compUniqFunctor W.Q L W

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
L : C ⥤ D
W : MorphismProperty C
inst✝² : L.IsLocalization W
P : MorphismProperty D
inst✝¹ : P.RespectsIso
inst✝ : P.IsMultiplicative
h₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (L.map f)
h₂ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) (hf : W f), P (isoOfHom L W f hf).inv
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
⊢ P = ⊤
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_1
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂ : Hom W Y Z
a : W.LeftFraction X Y
b₁ b₂ : W.LeftFraction Y Z
U : C
t₁ : b₁.Y' ⟶ U
t₂ : b₂.Y' ⟶ U
hst : b₁.s ≫ t₁ = b₂.s ≫ t₂
hft : b₁.f ≫ t₁ = b₂.f ≫ t₂
ht : W (b₁.s ≫ t₁)
z₁ : W.LeftFraction a.Y' b₁.Y'
fac₁ : { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.f ≫ z₁.s = { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.s ≫ z₁.f
⊢ a.comp b₁ = a.comp b₂

Tactic:
obtain ⟨z₂, fac₂⟩ := exists_leftFraction (RightFraction.mk a.s a.hs b₂.f)

Goal state after:
case refine_1
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a : W.LeftFraction X Y
b₁ b₂ : W.LeftFraction Y Z
U : C
t₁ : b₁.Y' ⟶ U
t₂ : b₂.Y' ⟶ U
hst : b₁.s ≫ t₁ = b₂.s ≫ t₂
hft : b₁.f ≫ t₁ = b₂.f ≫ t₂
ht : W (b₁.s ≫ t₁)
z₁ : W.LeftFraction a.Y' b₁.Y'
fac₁ : { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.f ≫ z₁.s = { X' := Y, s := a.s, hs := ⋯, f := b₁.f }.s ≫ z₁.f
z₂ : W.LeftFraction a.Y' b₂.Y'
fac₂ : { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.f ≫ z₂.s = { X' := Y, s := a.s, hs := ⋯, f := b₂.f }.s ≫ z₂.f
⊢ a.comp b₁ = a.comp b₂
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

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

Tactic:
rw [h₁, hβ]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ a₁.s ≫ z₁.f ≫ w₁.s = b.f ≫ z₁.s ≫ w₁.s

Tactic:
reassoc_of% fac₁

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ a₁.s ≫ z₁.f ≫ w₁.s = a₁.s ≫ z₁.f ≫ w₁.s
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
a : α
⊢ (P.colimitsClosure J).colimitsOfShape (J a) ≤ P.colimitsClosure J

Tactic:
by
    rintro X ⟨hX⟩
    exact .of_colimitPresentation hX.toColimitPresentation hX.prop_diag_obj

Goal state after: