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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K : CochainComplex C ℤ
a b p : ℤ
⊢ (XIsoOfEq K ⋯).hom ≫ (XIsoOfEq K ⋯).hom = (XIsoOfEq K ⋯).hom

Tactic:
simp only [XIsoOfEq, eqToIso.hom, eqToHom_trans]

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

Goal state before:
pp: case a
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
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i j : ℤ
hij : ¬(ComplexShape.up ℤ).Rel j i
⊢ ¬(ComplexShape.up ℤ).Rel (j + n) (i + n)

Tactic:
intro hij'

Goal state after:
case a
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
K L : CochainComplex C ℤ
φ₁ φ₂ : K ⟶ L
h : Homotopy φ₁ φ₂
n i j : ℤ
hij : ¬(ComplexShape.up ℤ).Rel j i
hij' : (ComplexShape.up ℤ).Rel (j + n) (i + n)
⊢ False
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:
by
  have : ((quotient C (ComplexShape.up ℤ) ⋙ shiftFunctor _ n)).Additive :=
    Functor.additive_of_iso ((quotient C (ComplexShape.up ℤ)).commShiftIso n)
  apply Functor.additive_of_full_essSurj_comp (quotient _ _)

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : Preadditive D
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
n : ℤ
X Y : HomologicalComplex C (ComplexShape.up ℤ)
f : { as := X } ⟶ { as := Y }
r : R
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map (r • f) =
    r • (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map f

Tactic:
obtain ⟨f, rfl⟩ := (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map_surjective f

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : Preadditive D
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
n : ℤ
X Y : HomologicalComplex C (ComplexShape.up ℤ)
r : R
f : X ⟶ Y
⊢ (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map (r • (quotient C (ComplexShape.up ℤ)).map f) =
    r • (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).map ((quotient C (ComplexShape.up ℤ)).map f)
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
n : A
⊢ s (n + 0) = s n ⋙ 𝟭 D

Tactic:
rw [add_zero]

Goal state after:
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
n : A
⊢ s n = s n ⋙ 𝟭 D
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 : (commShiftIso F 0).hom.app X = (CommShift.isoAdd' h (commShiftIso F a) (commShiftIso F b)).hom.app 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:
simp only [commShiftIso_zero, comp_obj, CommShift.isoZero_hom_app,
    CommShift.isoAdd'_hom_app] at eq

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).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)
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L M : CochainComplex C ℤ
n : ℤ
γ : Cochain K L n
a n' : ℤ
hn' : n + a = n'
m t t' : ℤ
γ' : Cochain L M m
h : n + m = t
ht' : t + a = t'
⊢ (γ.comp γ' h).leftShift a t' ht' = (a * m).negOnePow • (γ.leftShift a n' hn').comp γ' ⋯

Tactic:
ext p q hpq
  have h' : n' + m = t' := by lia
  dsimp
  simp only [Cochain.comp_v _ _ h' p (p + n') q rfl (by lia),
    γ.leftShift_v a n' hn' p (p + n') rfl (p + a) (by lia),
    (γ.comp γ' h).leftShift_v a t' (by lia) p q hpq (p + a) (by lia),
    smul_smul, Linear.units_smul_comp, assoc, Int.negOnePow_add, ← mul_assoc, ← h',
    comp_v _ _ h (p + a) (p + n') q (by lia) (by lia)]
  congr 2
  rw [add_comm n', mul_add, Int.negOnePow_add]

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
n : ℤ
⊢ (Cochain.ofHom
          (desc (inr φ) 0 (triangle φ).mor₃ ⋯ ≫
            lift (inr φ) (-(Cocycle.ofHom φ).leftShift 1 1 ⋯) (-(inl φ).leftShift 1 0 ⋯) ⋯)).v
      n n ⋯ =
    (δ (-1) 0 (-(snd (inr φ)).comp ((snd φ).comp (inl (inr φ)) ⋯) ⋯) + Cochain.ofHom (𝟙 (mappingCone (inr φ)))).v n n ⋯

Tactic:
simp only [Int.reduceNeg, Cochain.ofHom_comp, ofHom_desc, ofHom_lift, Cocycle.coe_neg,
        Cocycle.leftShift_coe, Cocycle.ofHom_coe, Cochain.comp_zero_cochain_v,
        shiftFunctor_obj_X', δ_neg, δ_zero_cochain_comp _ _ _ (neg_add_cancel 1), δ_inl,
        Int.negOnePow_neg, Int.negOnePow_one, δ_snd, Cochain.neg_comp,
        Cochain.comp_assoc_of_second_is_zero_cochain, smul_neg, Units.neg_smul, one_smul,
        neg_neg, Cochain.comp_add, inr_snd_assoc, neg_add_rev, Cochain.add_v, Cochain.neg_v,
        Cochain.comp_v _ _ (add_neg_cancel 1) n (n + 1) n rfl (by lia),
        Cochain.zero_cochain_comp_v, Cochain.ofHom_v, HomologicalComplex.id_f,
        ext_to_iff _ _ (n + 1) rfl, assoc, liftCochain_v_fst_v,
        (Cochain.ofHom φ).leftShift_v 1 1 (zero_add 1) n (n + 1) rfl (n + 1) (by lia),
        shiftFunctor_obj_X, mul_one, sub_self, mul_zero, Int.zero_ediv, add_zero,
        shiftFunctorObjXIso, HomologicalComplex.XIsoOfEq_rfl, Iso.refl_hom, id_comp,
        Preadditive.add_comp, Preadditive.neg_comp, inl_v_fst_v, comp_id, inr_f_fst_v, comp_zero,
        neg_zero, neg_add_cancel_comm, ext_from_iff _ (n + 1) _ rfl, inl_v_descCochain_v_assoc,
        Cochain.zero_v, zero_comp, Preadditive.comp_neg, inl_v_snd_v_assoc,
        inr_f_descCochain_v_assoc, inr_f_snd_v_assoc, inl_v_triangle_mor₃_f_assoc, triangle_obj₁,
        Iso.refl_inv, inl_v_fst_v_assoc, inr_f_triangle_mor₃_f_assoc, inr_f_fst_v_assoc, and_self,
        liftCochain_v_snd_v,
        (inl φ).leftShift_v 1 0 (neg_add_cancel 1) n n (add_zero n) (n + 1) (by lia),
        Int.negOnePow_zero, inl_v_snd_v, inr_f_snd_v, zero_add, inl_v_descCochain_v,
        inr_f_descCochain_v, inl_v_triangle_mor₃_f, inr_f_triangle_mor₃_f, neg_add_cancel]

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

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : Category.{v_2, u_2} D
inst✝¹² : HasShift C ℤ
inst✝¹¹ : HasShift D ℤ
inst✝¹⁰ : HasZeroObject C
inst✝⁹ : HasZeroObject D
inst✝⁸ : Preadditive C
inst✝⁷ : Preadditive D
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : Pretriangulated C
inst✝³ : Pretriangulated D
F₁ F₂ : C ⥤ D
e : F₁ ≅ F₂
inst✝² : F₁.CommShift ℤ
inst✝¹ : F₂.CommShift ℤ
inst✝ : NatTrans.CommShift e.hom ℤ
⊢ F₂.IsTriangulated → F₁.IsTriangulated

Tactic:
intro
    have : NatTrans.CommShift e.symm.hom ℤ := inferInstanceAs (NatTrans.CommShift e.inv ℤ)
    exact isTriangulated_of_iso e.symm

Goal state after:
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 : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
⊢ { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i, g := opcyclesMap S.g i,
      zero := ⋯ }.Exact

Tactic:
have hj := (hS.map (HomologicalComplex.eval C c i)).gIsCokernel

Goal state after:
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 : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Epi (S.map (eval C c i)).g
hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)
⊢ { X₁ := S.X₁.opcycles i, X₂ := S.X₂.opcycles i, X₃ := S.X₃.opcycles i, f := opcyclesMap S.f i, g := opcyclesMap S.g i,
      zero := ⋯ }.Exact
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{v_1, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y b✝ c✝ : Paths V
g : Quiver.Path X b✝
e : b✝ ⟶ c✝
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ F.map (g.cons e) = eqToHom ⋯ ≫ G.map (g.cons e) ≫ eqToHom ⋯

Tactic:
| cons g e ih =>

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

Goal state before:
pp: case cons
C : Type u₁
inst✝ : Category.{v₁, u₁} C
X Y Z : C
f : Path X Y
b✝ c✝ : C
g : Path Y b✝
e : b✝ ⟶ c✝
ih : composePath (f.comp g) = composePath f ≫ composePath g
⊢ composePath (f.comp (g.cons e)) = composePath f ≫ composePath (g.cons e)

Tactic:
| cons g e ih =>

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✝
hS : S.ShortExact
i j : ι
hij : c.Rel i j
⊢ Mono ((cyclesFunctor C c j).mapShortComplex.obj S).f

Tactic:
have := hS.mono_f
    dsimp
    infer_instance

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

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ (liftNatTrans L₂ W₂ (F ⋙ G') L₂ (F' ⋙ G') (𝟭 D₂) β.hom).app (L₂.obj X) =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X

Tactic:
rw [liftNatTrans_app]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_4
D₂ : Type u_5
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_4, u_4} D₁
inst✝⁴ : Category.{v_5, u_5} D₂
L₁ : C₁ ⥤ D₁
W₁ : MorphismProperty C₁
inst✝³ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
W₂ : MorphismProperty C₂
inst✝² : L₂.IsLocalization W₂
G : C₁ ⥤ D₂
G' : D₁ ⥤ D₂
inst✝¹ : Lifting L₁ W₁ G G'
F : C₂ ⥤ D₁
F' : D₂ ⥤ D₁
inst✝ : Lifting L₂ W₂ F F'
α : G ⋙ F' ≅ L₁
β : F ⋙ G' ≅ L₂
X : C₂
⊢ (Lifting.iso L₂ W₂ (F ⋙ G') (F' ⋙ G')).hom.app X ≫ β.hom.app X ≫ (Lifting.iso L₂ W₂ L₂ (𝟭 D₂)).inv.app X =
    G'.map ((Lifting.iso L₂ W₂ F F').hom.app X) ≫ β.hom.app X
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
⊢ 𝟭 (W.Localization ⥤ D) = functor W D ⋙ inverse W D

Tactic:
refine Functor.ext (fun G => ?_) fun G₁ G₂ τ => ?_
      · apply uniq
        simp [functor, inverse, fac]
      · apply natTrans_hcomp_injective
        ext X
        simp

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

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
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
f : L.obj X ⟶ L.obj Y
α : W.LeftFraction X Y
hα : f = α.map L ⋯
⊢ { Y' := α.Y', f := α.f + 0, s := α.s, hs := ⋯ }.map L ⋯ = α.map L ⋯

Tactic:
rw [add_zero]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
φ : W.RightFraction₂ X Y
inst✝ : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f
α : W.LeftFraction ψ₁.Y' ψ₂.Y'
hα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f
⊢ φ.f' ≫ { Y' := α.Y', f := ψ₁.f ≫ α.f, f' := ψ₂.f ≫ α.s, s := ψ₂.s ≫ α.s, hs := ⋯ }.s =
    φ.s ≫ { Y' := α.Y', f := ψ₁.f ≫ α.f, f' := ψ₂.f ≫ α.s, s := ψ₂.s ≫ α.s, hs := ⋯ }.f'

Tactic:
reassoc_of% hψ₂

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
φ : W.RightFraction₂ X Y
inst✝ : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f ≫ ψ₁.s = φ.s ≫ ψ₁.f
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' ≫ ψ₂.s = φ.s ≫ ψ₂.f
α : W.LeftFraction ψ₁.Y' ψ₂.Y'
hα : ψ₂.s ≫ α.s = ψ₁.s ≫ α.f
⊢ φ.s ≫ ψ₂.f ≫ α.s = φ.s ≫ { Y' := α.Y', f := ψ₁.f ≫ α.f, f' := ψ₂.f ≫ α.s, s := ψ₂.s ≫ α.s, hs := ⋯ }.f'
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : HasShift D ℤ
inst✝ : L.CommShift ℤ
H :
  ∀ (T₁' T₂' : Triangle C),
    T₁' ∈ distinguishedTriangles →
      T₂' ∈ distinguishedTriangles →
        ∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
          L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
T₁ T₂ : Triangle D
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
T₁' : Triangle C
e₁ : T₁ ≅ L.mapTriangle.obj T₁'
hT₁' : T₁' ∈ distinguishedTriangles
T₂' : Triangle C
e₂ : T₂ ≅ L.mapTriangle.obj T₂'
hT₂' : T₂' ∈ distinguishedTriangles
comm₁ : (L.mapTriangle.obj T₁').mor₁ ≫ e₁.inv.hom₂ = e₁.inv.hom₁ ≫ T₁.mor₁
comm₁' : T₂.mor₁ ≫ e₂.hom.hom₂ = e₂.hom.hom₁ ≫ (L.mapTriangle.obj T₂').mor₁
comm₂ : T₁.mor₂ ≫ e₁.hom.hom₃ = e₁.hom.hom₂ ≫ (L.mapTriangle.obj T₁').mor₂
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor D 1).map a = c ≫ T₂.mor₃

Tactic:
have comm₂' := e₂.hom.comm₂

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝² : Pretriangulated C
inst✝¹ : HasShift D ℤ
inst✝ : L.CommShift ℤ
H :
  ∀ (T₁' T₂' : Triangle C),
    T₁' ∈ distinguishedTriangles →
      T₂' ∈ distinguishedTriangles →
        ∀ (a : L.obj T₁'.obj₁ ⟶ L.obj T₂'.obj₁) (b : L.obj T₁'.obj₂ ⟶ L.obj T₂'.obj₂),
          L.map T₁'.mor₁ ≫ b = a ≫ L.map T₂'.mor₁ → ∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b
T₁ T₂ : Triangle D
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
T₁' : Triangle C
e₁ : T₁ ≅ L.mapTriangle.obj T₁'
hT₁' : T₁' ∈ distinguishedTriangles
T₂' : Triangle C
e₂ : T₂ ≅ L.mapTriangle.obj T₂'
hT₂' : T₂' ∈ distinguishedTriangles
comm₁ : (L.mapTriangle.obj T₁').mor₁ ≫ e₁.inv.hom₂ = e₁.inv.hom₁ ≫ T₁.mor₁
comm₁' : T₂.mor₁ ≫ e₂.hom.hom₂ = e₂.hom.hom₁ ≫ (L.mapTriangle.obj T₂').mor₁
comm₂ : T₁.mor₂ ≫ e₁.hom.hom₃ = e₁.hom.hom₂ ≫ (L.mapTriangle.obj T₁').mor₂
comm₂' : T₂.mor₂ ≫ e₂.hom.hom₃ = e₂.hom.hom₂ ≫ (L.mapTriangle.obj T₂').mor₂
⊢ ∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor D 1).map a = c ≫ T₂.mor₃
Module: Mathlib.SetTheory.Cardinal.HasCardinalLT

Goal state before:
pp: case mpr
X : Type u
Y : Type u'
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
⊢ HasCardinalLT X κ ∧ HasCardinalLT Y κ → HasCardinalLT (X ⊕ Y) κ

Tactic:
· rintro ⟨hX, hY⟩
    dsimp [HasCardinalLT] at hX hY ⊢
    rw [← Cardinal.lift_lt.{_, u'}, Cardinal.lift_lift, Cardinal.lift_lift] at hX
    rw [← Cardinal.lift_lt.{_, u}, Cardinal.lift_lift, Cardinal.lift_lift] at hY
    simp only [Cardinal.mk_sum, Cardinal.lift_add, Cardinal.lift_lift]
    exact Cardinal.add_lt_of_lt (by simpa using hκ) hX hY

Goal state after:
Module: Mathlib.CategoryTheory.ObjectProperty.ColimitsClosure

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁵ : (a : α) → Category.{v', u'} (J a)
inst✝⁴ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : Small.{w, t} α
inst✝¹ : ∀ (a : α), Small.{w, u'} (J a)
inst✝ : ∀ (a : α), LocallySmall.{w, v', u'} (J a)
⊢ ObjectProperty.EssentiallySmall.{w, v, u} (P.colimitsClosure J)

Tactic:
colimitsClosure_eq_unop_limitsClosure

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁵ : (a : α) → Category.{v', u'} (J a)
inst✝⁴ : ObjectProperty.EssentiallySmall.{w, v, u} P
inst✝³ : LocallySmall.{w, v, u} C
inst✝² : Small.{w, t} α
inst✝¹ : ∀ (a : α), Small.{w, u'} (J a)
inst✝ : ∀ (a : α), LocallySmall.{w, v', u'} (J a)
⊢ ObjectProperty.EssentiallySmall.{w, v, u} (P.op.limitsClosure fun a ↦ (J a)ᵒᵖ).unop
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2.refine_3
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
⊢ LeftFractionRel p₂ (a₂.comp₀ b z₂)

Tactic:
have eq : a₂.s ≫ z₂.f ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f := by
        rw [← fac₂', reassoc_of% fac₂]
      obtain ⟨Z, u, hu, fac₄⟩ := HasLeftCalculusOfFractions.ext _ _ _ a₂.hs eq
      simp only [assoc] at fac₄
      refine ⟨Z, u, w₂.s ≫ u, ?_, ?_, ?_⟩
      · dsimp [p₁, p₂]
        simp only [assoc]
      · dsimp [p₁, p₂]
        simp only [assoc, fac₄]
      · dsimp [p₁, p₂]
        simp only [assoc]
        exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z₂.hs (W.comp_mem _ _ w₂.hs hu))

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

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁷ : Category.{v_1, u_1} C
inst✝¹⁶ : HasShift C ℤ
inst✝¹⁵ : Category.{v_2, u_2} D
inst✝¹⁴ : HasZeroObject D
inst✝¹³ : HasShift D ℤ
inst✝¹² : Preadditive D
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹⁰ : Pretriangulated D
inst✝⁹ : Category.{v_3, u_3} A
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Abelian A
L : C ⥤ D
inst✝³ : L.CommShift ℤ
inst✝² : L.IsTriangulated
inst✝¹ : L.mapArrow.EssSurj
F : D ⥤ A
G : C ⥤ A
e : L ⋙ F ≅ G
inst✝ : G.IsHomological
this✝ : F.PreservesZeroMorphisms
this : (L ⋙ F).IsHomological
T : Triangle D
hT : T ∈ distinguishedTriangles
⊢ ∃ T' e, ((shortComplexOfDistTriangle T' ⋯).map F).Exact

Tactic:
L.distTriang_iff

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝¹⁷ : Category.{v_1, u_1} C
inst✝¹⁶ : HasShift C ℤ
inst✝¹⁵ : Category.{v_2, u_2} D
inst✝¹⁴ : HasZeroObject D
inst✝¹³ : HasShift D ℤ
inst✝¹² : Preadditive D
inst✝¹¹ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝¹⁰ : Pretriangulated D
inst✝⁹ : Category.{v_3, u_3} A
inst✝⁸ : HasZeroObject C
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Abelian A
L : C ⥤ D
inst✝³ : L.CommShift ℤ
inst✝² : L.IsTriangulated
inst✝¹ : L.mapArrow.EssSurj
F : D ⥤ A
G : C ⥤ A
e : L ⋙ F ≅ G
inst✝ : G.IsHomological
this✝ : F.PreservesZeroMorphisms
this : (L ⋙ F).IsHomological
T : Triangle D
hT✝ : T ∈ distinguishedTriangles
hT : T ∈ L.essImageDistTriang
⊢ ∃ T' e, ((shortComplexOfDistTriangle T' ⋯).map F).Exact
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case mpr
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
⊢ LeftFractionRel φ ψ → φ.map L ⋯ = ψ.map L ⋯

Tactic:
intro h
    simp only [← Localization.Hom.map_mk _ L (Localization.inverts _ _)]
    congr 1
    exact Quot.sound h

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

Goal state before:
pp: case mp
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hL : IsZero (L.sc i).homology
⊢ IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f)) → IsZero (K.sc i).homology

pp: case mpr
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hL : IsZero (L.sc i).homology
⊢ IsZero (K.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))

Tactic:
· intro h
    exact IsZero.of_iso hL (@asIso _ _ _ _ _ h)

Goal state after:
case mpr
ι : Type u_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K L : HomologicalComplex C c
f : K ⟶ L
i : ι
inst✝¹ : K.HasHomology i
inst✝ : L.HasHomology i
hL : IsZero (L.sc i).homology
⊢ IsZero (K.sc i).homology → IsIso (ShortComplex.homologyMap ((shortComplexFunctor C c i).map f))