mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
s : Finset Ω[S⁄R] := Finset.image (fun s ↦ (D R S) s) (EssFiniteType.finset R S)
x✝ x : S
⊢ x ∈ adjoin R ↑(EssFiniteType.finset R S) → (D R S) x ∈ Submodule.span S ↑s

Tactic:
hx

Goal state after:
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
s : Finset Ω[S⁄R] := Finset.image (fun s ↦ (D R S) s) (EssFiniteType.finset R S)
x✝ x : S
hx : x ∈ adjoin R ↑(EssFiniteType.finset R S)
⊢ (D R S) x ∈ Submodule.span S ↑s
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case refine_4
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
s : Finset Ω[S⁄R] := Finset.image (fun s ↦ (D R S) s) (EssFiniteType.finset R S)
x✝¹ x✝ : S
hx : x✝ ∈ adjoin R ↑(EssFiniteType.finset R S)
x y : S
hx✝ : x ∈ adjoin R ↑(EssFiniteType.finset R S)
hy✝ : y ∈ adjoin R ↑(EssFiniteType.finset R S)
⊢ (D R S) x ∈ Submodule.span S ↑s → (D R S) y ∈ Submodule.span S ↑s → (D R S) (x * y) ∈ Submodule.span S ↑s

Tactic:
hx

Goal state after:
case refine_4
R : Type u
S : Type v
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module S M
inst✝¹ : IsScalarTower R S M
inst✝ : EssFiniteType R S
s : Finset Ω[S⁄R] := Finset.image (fun s ↦ (D R S) s) (EssFiniteType.finset R S)
x✝¹ x✝ : S
hx✝¹ : x✝ ∈ adjoin R ↑(EssFiniteType.finset R S)
x y : S
hx✝ : x ∈ adjoin R ↑(EssFiniteType.finset R S)
hy✝ : y ∈ adjoin R ↑(EssFiniteType.finset R S)
hx : (D R S) x ∈ Submodule.span S ↑s
⊢ (D R S) y ∈ Submodule.span S ↑s → (D R S) (x * y) ∈ Submodule.span S ↑s
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx✝ : k = x ≫ K.d i' i
hx : k = 0
h : ¬c.Rel i' i
⊢ k = 0

Tactic:
hx

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : A ⟶ K.X i
j : ι
hj : c.next i = j
i' : ι
x : A ⟶ K.X i'
hx✝ : k = x ≫ K.d i' i
hx : k = 0
h : ¬c.Rel i' i
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : K.X i ⟶ A
j : ι
hj : c.prev i = j
i' : ι
x : K.X i' ⟶ A
hx✝ : k = K.d i i' ≫ x
hx : k = 0
h : ¬c.Rel i i'
⊢ k = 0

Tactic:
hx

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
i : ι
inst✝ : K.HasHomology i
A : C
k : K.X i ⟶ A
j : ι
hj : c.prev i = j
i' : ι
x : K.X i' ⟶ A
hx✝ : k = K.d i i' ≫ x
hx : k = 0
h : ¬c.Rel i i'
⊢ 0 = 0
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)
g : D ⟶ E
⊢ nullHomotopicMap' hom ≫ g = nullHomotopicMap' fun i j hij ↦ hom i j hij ≫ g.f j

Tactic:
rw [nullHomotopicMap', nullHomotopicMap_comp]
  congr
  ext i j
  split_ifs
  · rfl
  · rw [zero_comp]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
i j : ι
hij : c.Rel i j
hj : ¬c.Rel j (c.next j)
⊢ (inlX φ j i hij ≫ d φ i j) ≫ sndX φ j = (φ.f j ≫ inrX φ j) ≫ sndX φ j

Tactic:
simp [d_sndX φ i j hij]

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
⊢ (f ≫ nullHomotopicMap fun i j ↦ dite (c.Rel j i) (hom i j) fun x ↦ 0) =
    nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij

Tactic:
comp_nullHomotopicMap

Goal state after:
ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f : C ⟶ D
hom : (i j : ι) → c.Rel j i → (D.X i ⟶ E.X j)
⊢ (nullHomotopicMap fun i j ↦ f.f i ≫ dite (c.Rel j i) (hom i j) fun x ↦ 0) =
    nullHomotopicMap' fun i j hij ↦ f.f i ≫ hom i j hij
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i : ι
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ e.f 0 = dFrom P 0 ≫ 0 + (zero ≫ (xPrevIso Q ⋯).inv) ≫ dTo Q 0

Tactic:
by simpa using! comm_zero

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

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
F G K L : CochainComplex C ℤ
n m x✝² x✝¹ : ℤ
hij : ¬(ComplexShape.up ℤ).Rel x✝² x✝¹
x✝ : ↑(AddCommGrpCat.of (Cochain F G x✝²))
p✝ q✝ : ℤ
hpq✝ : p✝ + x✝¹ = q✝
⊢ Cochain.v ((AddCommGrpCat.Hom.hom (AddCommGrpCat.ofHom ↑(δ_hom ℤ F G x✝² x✝¹))) x✝) p✝ q✝ hpq✝ =
    Cochain.v ((AddCommGrpCat.Hom.hom 0) x✝) p✝ q✝ hpq✝

Tactic:
simp [δ_shape _ _ hij]

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i : ι
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
⊢ e.f 0 = 0 ≫ dTo Q 0 + dFrom P 0 ≫ (xNextIso P ⋯).hom ≫ zero

Tactic:
by simpa using! comm_zero

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

Goal state before:
pp: 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 m : ℤ
hnm : n + 1 = m
⊢ mapHomologicalComplexXIso φ H n = mapHomologicalComplexXIso' φ H n m hnm

Tactic:
subst hnm
  rfl

Goal state after:
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
⊢ ∀ (X : CochainComplex C ℤ),
    { f := fun x ↦ (𝟙 X).f (x + n), comm' := ⋯ } =
      𝟙 { X := fun i ↦ X.X (i + n), d := fun x x_1 ↦ n.negOnePow • X.d (x + n) (x_1 + n), shape := ⋯, d_comp_d' := ⋯ }

Tactic:
by intros; rfl

Goal state after:
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
⊢ ∀ {X Y Z : CochainComplex C ℤ} (f : X ⟶ Y) (g : Y ⟶ Z),
    { f := fun x ↦ (f ≫ g).f (x + n), comm' := ⋯ } =
      { f := fun x ↦ f.f (x + n), comm' := ⋯ } ≫ { f := fun x ↦ g.f (x + n), comm' := ⋯ }

Tactic:
by intros; rfl

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

Goal state before:
pp: C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
⊢ F.CommShift A

Tactic:
by
  letI := HasShift.induced F A s i
  exact
    { commShiftIso := fun a => (i a).symm
      commShiftIso_zero := by
        ext X
        dsimp
        simp only [isoZero_hom_app, shiftFunctorZero_inv_app_obj_of_induced,
          ← F.map_comp_assoc, Iso.hom_inv_id_app, F.map_id, Category.id_comp]
      commShiftIso_add := fun a b => by
        ext X
        dsimp
        simp only [isoAdd_hom_app, Iso.symm_hom, shiftFunctorAdd_inv_app_obj_of_induced,
          shiftFunctor_of_induced]
        rw [← Functor.map_comp_assoc, Iso.inv_hom_id_app]
        dsimp
        rw [Functor.map_id, Category.id_comp, Iso.inv_hom_id_app_assoc,
          ← F.map_comp_assoc, Iso.hom_inv_id_app, F.map_id, Category.id_comp] }

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

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
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
X : C
a b : A
h : a + b = 0
eq :
  F.map ((shiftFunctorZero C A).hom.app X) ≫ (shiftFunctorZero D A).inv.app (F.obj X) =
    F.map ((shiftFunctorAdd' C a b 0 h).hom.app X) ≫
      (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
        (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫ (shiftFunctorAdd' D a b 0 h).inv.app (F.obj X)
⊢ F.map ((shiftFunctorAdd' C a b 0 h).inv.app X ≫ (shiftFunctorZero C A).hom.app X) =
    (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
      (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫
        (shiftFunctorAdd' D a b 0 h).inv.app (F.obj X) ≫ (shiftFunctorZero D A).hom.app (F.obj X)

Tactic:
rw [← cancel_epi (F.map ((shiftFunctorAdd' C a b 0 h).hom.app X)), ← reassoc_of% eq, F.map_comp]

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
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
X : C
a b : A
h : a + b = 0
eq :
  F.map ((shiftFunctorZero C A).hom.app X) ≫ (shiftFunctorZero D A).inv.app (F.obj X) =
    F.map ((shiftFunctorAdd' C a b 0 h).hom.app X) ≫
      (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
        (shiftFunctor D b).map ((commShiftIso F a).hom.app X) ≫ (shiftFunctorAdd' D a b 0 h).inv.app (F.obj X)
⊢ F.map ((shiftFunctorAdd' C a b 0 h).hom.app X) ≫
      F.map ((shiftFunctorAdd' C a b 0 h).inv.app X) ≫ F.map ((shiftFunctorZero C A).hom.app X) =
    F.map ((shiftFunctorZero C A).hom.app X) ≫
      (shiftFunctorZero D A).inv.app (F.obj X) ≫ (shiftFunctorZero D A).hom.app (F.obj X)
Module: Mathlib.CategoryTheory.Shift.CommShift

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 G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
⊢ NatTrans.CommShift e.hom A

Tactic:
let := ofIso e A

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 G : C ⥤ D
e : F ≅ G
A : Type u_4
inst✝³ : AddMonoid A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : F.CommShift A
this : G.CommShift A := ofIso e A
⊢ NatTrans.CommShift e.hom A
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case refine_1
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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
b : T₁.obj₂ ⟶ T₂.obj₂
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₂ ≫ c = b ≫ T₂.mor₂
a : T₁.rotate.obj₃ ⟶ T₂.rotate.obj₃
ha₁ : T₁.rotate.mor₂ ≫ a = c ≫ T₂.rotate.mor₂
ha₂ : T₁.rotate.mor₃ ≫ (shiftFunctor C 1).map b = a ≫ T₂.rotate.mor₃
⊢ (shiftFunctor C 1).map (T₁.mor₁ ≫ b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a ≫ T₂.mor₁)

Tactic:
dsimp at ha₂

Goal state after:
case refine_1
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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
b : T₁.obj₂ ⟶ T₂.obj₂
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₂ ≫ c = b ≫ T₂.mor₂
a : T₁.rotate.obj₃ ⟶ T₂.rotate.obj₃
ha₁ : T₁.rotate.mor₂ ≫ a = c ≫ T₂.rotate.mor₂
ha₂ : (-(shiftFunctor C 1).map T₁.mor₁) ≫ (shiftFunctor C 1).map b = a ≫ (-(shiftFunctor C 1).map T₂.mor₁)
⊢ (shiftFunctor C 1).map (T₁.mor₁ ≫ b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a ≫ T₂.mor₁)
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 T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : T.invRotate.mor₁ ≫ φ.hom₁ = (shiftFunctor C (-1)).map φ.hom₃ ≫ T'.invRotate.mor₁
⊢ h ≫ inv ((shiftFunctor C (-1)).map φ.hom₃) ≫ (shiftFunctor C (-1)).map φ.hom₃ ≫ T'.invRotate.mor₁ = g ≫ φ.hom₁

Tactic:
IsIso.inv_hom_id_assoc,

Goal state after:
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 T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : T.invRotate.mor₁ ≫ φ.hom₁ = (shiftFunctor C (-1)).map φ.hom₃ ≫ T'.invRotate.mor₁
⊢ h ≫ T'.invRotate.mor₁ = g ≫ φ.hom₁
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
⊢ BinaryBiproductData T.obj₁ T.obj₃

Tactic:
have : Mono T.mor₁ := T.mono₁ hT hT₀
  have eq : fst ≫ T.mor₁ = 𝟙 T.obj₂ - T.mor₂ ≫ inr := by rw [← total, add_sub_cancel_right]
  exact
    { bicone :=
      { pt := T.obj₂
        fst := fst
        snd := T.mor₂
        inl := T.mor₁
        inr := inr
        inl_fst := by
          simp only [← cancel_mono T.mor₁, assoc, id_comp, eq, comp_sub, comp_id,
            comp_distTriang_mor_zero₁₂_assoc _ hT, zero_comp, sub_zero]
        inl_snd := comp_distTriang_mor_zero₁₂ _ hT
        inr_fst := by
          simp only [← cancel_mono T.mor₁, assoc, eq, comp_sub, reassoc_of% inr_snd,
            comp_id, sub_self, zero_comp]
        inr_snd := inr_snd }
      isBilimit := isBinaryBilimitOfTotal _ total }

Goal state after:
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
K₁ L₁ K₂ L₂ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ mapOfHomotopy H ≫ (triangle φ₂).mor₃ = (triangle φ₁).mor₃ ≫ (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map a

Tactic:
by
  ext p
  dsimp [mapOfHomotopy, triangle]
  -- the following list of lemmas as been obtained by doing
  -- simp? [ext_from_iff _ _ _ rfl, Cochain.rightShift_v _ 1 0 _ p p _ (p + 1) rfl]
  simp only [Int.reduceNeg, Cochain.rightShift_neg, Cochain.neg_v, shiftFunctor_obj_X',
    Cochain.rightShift_v _ 1 0 _ p p _ (p + 1) rfl, shiftFunctor_obj_X, shiftFunctorObjXIso,
    HomologicalComplex.XIsoOfEq_rfl, Iso.refl_inv, comp_id, Preadditive.comp_neg,
    Preadditive.neg_comp, ext_from_iff _ _ _ rfl, inl_v_desc_f_assoc, Cochain.add_v,
    Cochain.zero_cochain_comp_v, Cochain.ofHom_v, Cochain.comp_zero_cochain_v, Preadditive.add_comp,
    assoc, inl_v_fst_v, inr_f_fst_v, comp_zero, add_zero, inl_v_fst_v_assoc, inr_f_desc_f_assoc,
    HomologicalComplex.comp_f, neg_zero, inr_f_fst_v_assoc, zero_comp, and_self]

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

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
inst✝³ : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (triangleh φ).rotate.mor₃ ≫
      (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
        (Iso.refl (triangleh φ).rotate.obj₁).hom =
    ((Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).symm.app K ≪≫
          HomotopyCategory.isoOfHomotopyEquiv (rotateHomotopyEquiv φ)).hom ≫
      (triangleh (inr φ)).mor₃

Tactic:
dsimp
        rw [CategoryTheory.Functor.map_id, comp_id, assoc, ← Functor.map_comp_assoc,
          rotateHomotopyEquiv_comm₃, Functor.map_neg, Preadditive.neg_comp,
          Functor.commShiftIso_hom_naturality, Preadditive.comp_neg,
          Iso.inv_hom_id_app_assoc]

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

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasShift D ℤ
inst✝³ : HasShift E ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ E
inst✝¹ : G.CommShift ℤ
inst✝ : F.Faithful
X Y : Triangle C
f g : X ⟶ Y
h : F.mapTriangle.map f = F.mapTriangle.map g
⊢ f = g

Tactic:
ext <;> apply F.map_injective

Goal state after:
case h₁
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasShift D ℤ
inst✝³ : HasShift E ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ E
inst✝¹ : G.CommShift ℤ
inst✝ : F.Faithful
X Y : Triangle C
f g : X ⟶ Y
h : F.mapTriangle.map f = F.mapTriangle.map g
⊢ F.map f.hom₁ = F.map g.hom₁

case h₂
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasShift D ℤ
inst✝³ : HasShift E ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ E
inst✝¹ : G.CommShift ℤ
inst✝ : F.Faithful
X Y : Triangle C
f g : X ⟶ Y
h : F.mapTriangle.map f = F.mapTriangle.map g
⊢ F.map f.hom₂ = F.map g.hom₂

case h₃
C : Type u_1
D : Type u_2
E : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} E
inst✝⁵ : HasShift C ℤ
inst✝⁴ : HasShift D ℤ
inst✝³ : HasShift E ℤ
F : C ⥤ D
inst✝² : F.CommShift ℤ
G : D ⥤ E
inst✝¹ : G.CommShift ℤ
inst✝ : F.Faithful
X Y : Triangle C
f g : X ⟶ Y
h : F.mapTriangle.map f = F.mapTriangle.map g
⊢ F.map f.hom₃ = F.map g.hom₃
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case refine_2
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
⊢ ∃ a_1, (fun x ↦ x ≫ φ'.hom₃) a_1 = a

Tactic:
obtain ⟨a', ha'⟩ : ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃ := by
      have zero : ((productTriangle T).mor₃) ≫ (shiftFunctor C 1).map T'.mor₁ = 0 := by
        rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
        intro j
        exact comp_distTriang_mor_zero₃₁ _ (hT j)
      have ⟨g, hg⟩ := T'.coyoneda_exact₁ hT' (a ≫ (productTriangle T).mor₃) (by
        rw [assoc, zero, comp_zero])
      exact ⟨g, hg.symm⟩

Goal state after:
case refine_2
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
⊢ ∃ a_1, (fun x ↦ x ≫ φ'.hom₃) a_1 = a