mathlib-initiative/mathlib-tactics huggingface dataset:
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 : CochainComplex C ℤ
n' a : ℤ
γ : Cochain ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj K) L n'
n : ℤ
hn : n + a = n'
x : Rˣ
⊢ (x • γ).leftUnshift n hn = x • γ.leftUnshift n hn

Tactic:
by
  apply leftUnshift_smul

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

Goal state before:
pp: case h₁
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₁ = (-(↑n.succ • f)).hom₁

Tactic:
apply SubNegMonoid.zsmul_neg'

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

Goal state before:
pp: case h₂
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₂ = (-(↑n.succ • f)).hom₂

Tactic:
apply SubNegMonoid.zsmul_neg'

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

Goal state before:
pp: case h₃
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasShift C ℤ
T₁ T₂ T₃ : Triangle C
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
n : ℕ
f : T₁ ⟶ T₂
⊢ (Int.negSucc n • f).hom₃ = (-(↑n.succ • f)).hom₃

Tactic:
apply SubNegMonoid.zsmul_neg'

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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : T.mor₁ = 0
h₃ : T.mor₃ = 0
this : Epi T.mor₂
f : T.obj₃ ⟶ T.obj₂
hf : 𝟙 T.obj₂ = T.mor₂ ≫ f
⊢ f ≫ T.mor₂ = 𝟙 T.obj₃

Tactic:
by rw [← cancel_epi T.mor₂, comp_id, ← reassoc_of% hf]

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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
n : ℤ
⊢ ∀ T ∈ distinguishedTriangles, (CategoryTheory.shiftFunctor (Triangle C) n).obj T ∈ distinguishedTriangles

Tactic:
let H : ℤ → Prop := fun n => ∀ (T : Triangle C) (_ : T ∈ distTriang C),
    (Triangle.shiftFunctor C n).obj T ∈ distTriang C

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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
n : ℤ
H : ℤ → Prop := fun n ↦ ∀ T ∈ distinguishedTriangles, (shiftFunctor C n).obj T ∈ distinguishedTriangles
⊢ ∀ T ∈ distinguishedTriangles, (CategoryTheory.shiftFunctor (Triangle C) n).obj T ∈ distinguishedTriangles
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
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : Epi f
⊢ IsSplitEpi f

Tactic:
obtain ⟨Z, g, h, hT⟩ := distinguished_cocone_triangle f

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
X✝ Y✝ : C
f : X✝ ⟶ Y✝
hf : Epi f
Z : C
g : Y✝ ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X✝
hT : Triangle.mk f g h ∈ distinguishedTriangles
⊢ IsSplitEpi f
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
K₁ L₁ K₂ L₂ K₃ L₃ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
φ₂ : K₂ ⟶ L₂
a : K₁ ⟶ K₂
b : L₁ ⟶ L₂
H : Homotopy (φ₁ ≫ b) (a ≫ φ₂)
⊢ (triangleh φ₁).mor₁ ≫ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map b =
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map a ≫ (triangleh φ₂).mor₁

Tactic:
by
    dsimp
    simp only [← Functor.map_comp]
    exact HomotopyCategory.eq_of_homotopy _ _ H

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
p : ℤ
r_f : (σ (p + 1 + 1)).r ≫ S.f.f (p + 1 + 1) = 𝟙 (S.X₂.X (p + 1 + 1)) - S.g.f (p + 1 + 1) ≫ (σ (p + 1 + 1)).s
s_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))
⊢ (mappingConeHomOfDegreewiseSplitXIso S σ p (p + 1) ⋯).hom ≫ (-1 • S.X₂.d (p + 1) (p + 1 + 1)) =
    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫
      (mappingConeHomOfDegreewiseSplitXIso S σ (p + 1) (p + 1 + 1) ⋯).hom

Tactic:
simp only [mappingConeHomOfDegreewiseSplitXIso, mappingCone.ext_from_iff _ _ _ rfl,
      mappingCone.inl_v_d_assoc _ (p + 1) _ (p + 1 + 1) (by linarith) (by lia),
      cocycleOfDegreewiseSplit, r_f, Int.reduceNeg, Cochain.ofHom_v, sub_comp, assoc,
      Hom.comm, comp_sub, mappingCone.inl_v_fst_v_assoc, mappingCone.inl_v_snd_v_assoc,
      shiftFunctor_obj_X', zero_comp, sub_zero, homOfDegreewiseSplit_f,
      mappingCone.inr_f_fst_v_assoc, comp_zero, zero_sub, mappingCone.inr_f_snd_v_assoc,
      neg_neg, mappingCone.inr_f_d_assoc, shiftFunctor_obj_d',
      Int.negOnePow_one, neg_comp, sub_neg_eq_add, zero_add, and_true,
      Units.neg_smul, one_smul, comp_neg, ShortComplex.map_X₂, eval_obj, Cocycle.mk_coe,
      Cochain.mk_v]

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
p : ℤ
r_f : (σ (p + 1 + 1)).r ≫ S.f.f (p + 1 + 1) = 𝟙 (S.X₂.X (p + 1 + 1)) - S.g.f (p + 1 + 1) ≫ (σ (p + 1 + 1)).s
s_g : (σ (p + 1)).s ≫ S.g.f (p + 1) = 𝟙 (S.X₃.X (p + 1))
⊢ -(σ (p + 1)).s ≫ S.X₂.d (p + 1) (p + 1 + 1) =
    -S.X₃.d (p + 1) (p + 1 + 1) ≫ (σ (p + 1 + 1)).s -
      ((σ (p + 1)).s ≫ S.X₂.d (p + 1) (p + 1 + 1) ≫ 𝟙 (S.X₂.X (p + 1 + 1)) -
        (σ (p + 1)).s ≫ S.X₂.d (p + 1) (p + 1 + 1) ≫ S.g.f (p + 1 + 1) ≫ (σ (p + 1 + 1)).s)
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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e₁ : T₁.obj₁ ≅ T₂.obj₁
e₂ : T₁.obj₂ ≅ T₂.obj₂
comm : T₁.mor₁ ≫ e₂.hom = e₁.hom ≫ T₂.mor₁
h :
  ∃ e', e'.hom.hom₁ = Arrow.Hom.left (Arrow.isoMk e₁ e₂ ⋯).hom ∧ e'.hom.hom₂ = Arrow.Hom.right (Arrow.isoMk e₁ e₂ ⋯).hom
⊢ T₁.mor₂ ≫ (Triangle.π₃.mapIso h.choose).hom = e₂.hom ≫ T₂.mor₂

Tactic:
have eq := h.choose_spec.2
    dsimp at eq ⊢
    conv_rhs => rw [← eq, ← TriangleMorphism.comm₂]

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

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasZeroObject C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
⊢ T ∈ distinguishedTriangles → ∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)

Tactic:
intro hT
    obtain ⟨K, L, φ, ⟨e⟩⟩ := inv_rot_of_distTriang _ hT
    exact ⟨_, _, ⟨(triangleRotation _).counitIso.symm.app _ ≪≫ (rotate _).mapIso e ≪≫
      CochainComplex.mappingCone.trianglehRotateIsoTrianglehOfDegreewiseSplit φ⟩⟩

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
c : ComplexShape ι
K : HomologicalComplex C c
i j : ι
inst✝¹ : K.HasHomology i
inst✝ : K.HasHomology j
⊢ K.homologyι i ≫ K.opcyclesToCycles i j = 0

Tactic:
simp only [← cancel_mono (K.iCycles j), assoc, opcyclesToCycles_iCycles,
    homologyι_comp_fromOpcycles, zero_comp]

Goal state after:
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 ≫ g'.cons 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'.cons p)

Tactic:
this

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) (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'.cons p)
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: case refine_2
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
H : { l // l ≫ S.f.f i = k ≫ S.X₂.iCycles i }
⊢ S.X₁.liftCycles (↑H) (c.next i) ⋯ ⋯ ≫ cyclesMap S.f i = k

Tactic:
· rw [← cancel_mono (S.X₂.iCycles i), liftCycles_comp_cyclesMap, liftCycles_i, H.2]

Goal state after:
Module: Mathlib.CategoryTheory.Category.Quiv

Goal state before:
pp: V W : Type u
inst✝¹ : Quiver V
inst✝ : Quiver W
e : V ≃ W
he : (X Y : V) → (X ⟶ Y) ≃ (e X ⟶ e Y)
X Y : V
f : X ⟶ Y
X' Y' : V
hX : X = X'
hY : Y = Y'
X'' Y'' : W
hX' : e X' = X''
hY' : e Y' = Y''
⊢ e Y = Y''

Tactic:
hY,

Goal state after:
V W : Type u
inst✝¹ : Quiver V
inst✝ : Quiver W
e : V ≃ W
he : (X Y : V) → (X ⟶ Y) ≃ (e X ⟶ e Y)
X Y : V
f : X ⟶ Y
X' Y' : V
hX : X = X'
hY : Y = Y'
X'' Y'' : W
hX' : e X' = X''
hY' : e Y' = Y''
⊢ e Y' = Y''
Module: Mathlib.CategoryTheory.Category.Quiv

Goal state before:
pp: V : Type u
C : Type u₁
inst✝¹ : Quiver V
inst✝ : Category.{v₁, u₁} C
F : Paths V ⥤ C
⊢ (fun G ↦ Cat.freeMap G ⋙ pathComposition C) ((fun F ↦ Paths.of V ⋙q F.toPrefunctor) F) = F

Tactic:
by
    dsimp
    rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]

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

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
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
this : Localization.Construction.lift W.Q ⋯ = 𝟭 W.Localization
⊢ (Localization.Construction.lift W.Q ⋯).IsEquivalence

Tactic:
this

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
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
this : Localization.Construction.lift W.Q ⋯ = 𝟭 W.Localization
⊢ (𝟭 W.Localization).IsEquivalence
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
E : Type u_4
inst✝¹ : Category.{v_4, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
⊢ L'.IsLocalization W

Tactic:
have h : W.IsInvertedBy L' := by
    rw [← MorphismProperty.IsInvertedBy.iff_of_iso W e]
    exact MorphismProperty.IsInvertedBy.of_comp W L (Localization.inverts L W) eq.functor
  let F₁ := Localization.Construction.lift L (Localization.inverts L W)
  let F₂ := Localization.Construction.lift L' h
  let e' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso W.Q W (L ⋙ eq.functor) L' _ _ e
  exact
    { inverts := h
      isEquivalence := Functor.isEquivalence_of_iso e' }

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

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
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
h : StrictUniversalPropertyFixedTarget L W Eᵒᵖ
F₁ F₂ : Dᵒᵖ ⥤ E
eq : L.op ⋙ F₁ = L.op ⋙ F₂
this : F₁.rightOp = F₂.rightOp
⊢ F₁.rightOp.leftOp = F₂.rightOp.leftOp

Tactic:
this

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
L : C ⥤ D
W : MorphismProperty C
E : Type u_3
inst✝ : Category.{v_3, u_3} E
h : StrictUniversalPropertyFixedTarget L W Eᵒᵖ
F₁ F₂ : Dᵒᵖ ⥤ E
eq : L.op ⋙ F₁ = L.op ⋙ F₂
this : F₁.rightOp = F₂.rightOp
⊢ F₂.rightOp.leftOp = F₂.rightOp.leftOp
Module: Mathlib.CategoryTheory.CatCommSq

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ⥤ C₂
L : C₁ ≌ C₃
R : C₂ ≌ C₄
B : C₃ ⥤ C₄
h : CatCommSq T L.functor R.functor B
⊢ vInv B L.symm R.symm T (vInv T L R B h) = h

Tactic:
by
  ext X
  rw [vInv_iso_hom_app]
  dsimp
  rw [vInv_iso_inv_app]
  rw [← cancel_mono (B.map (L.functor.map (NatTrans.app L.unitIso.hom X)))]
  rw [← Functor.comp_map]
  dsimp
  simp only [Functor.map_comp, Equivalence.fun_inv_map, Functor.comp_obj,
    Functor.id_obj, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app, comp_id]
  rw [← B.map_comp, L.counit_app_functor, ← L.functor.map_comp, ← NatTrans.comp_app,
    Iso.inv_hom_id, NatTrans.id_app, L.functor.map_id]
  simp

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

Goal state before:
pp: C₁ : Type u₁
C₂ : Type u₂
inst✝² : Category.{v₁, u₁} C₁
inst✝¹ : Category.{v₂, u₂} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
inst✝ : Φ.functor.IsEquivalence
h : W₂ ≤ W₁.map Φ.functor
⊢ Φ.IsLocalizedEquivalence

Tactic:
by
  have : Functor.IsLocalization (Φ.functor ⋙ MorphismProperty.Q W₂) W₁ := by
    refine Functor.IsLocalization.of_equivalence_source W₂.Q W₂ (Φ.functor ⋙ W₂.Q) W₁
      (asEquivalence Φ.functor).symm ?_ (Φ.inverts W₂.Q)
      ((associator _ _ _).symm ≪≫ isoWhiskerRight ((Equivalence.unitIso _).symm) _ ≪≫
        leftUnitor _)
    erw [W₁.isoClosure.inverseImage_equivalence_functor_eq_map_inverse]
    rw [MorphismProperty.map_isoClosure]
    exact h
  exact IsLocalizedEquivalence.of_isLocalization_of_isLocalization Φ W₂.Q

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

Goal state before:
pp: 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.fst.f ≫ ψ₁.s = φ.fst.s ≫ ψ₁.f
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.snd.f ≫ ψ₂.s = φ.snd.s ≫ ψ₂.f
α : W.LeftFraction ψ₁.Y' ψ₂.Y'
hα : { X' := Y, s := ψ₁.s, hs := ⋯, f := ψ₂.s }.f ≫ α.s = { X' := Y, s := ψ₁.s, hs := ⋯, f := ψ₂.s }.s ≫ α.f
⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'

Tactic:
dsimp at hψ₁ hψ₂ hα

Goal state after:
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
⊢ ∃ ψ, φ.f ≫ ψ.s = φ.s ≫ ψ.f ∧ φ.f' ≫ ψ.s = φ.s ≫ ψ.f'
Module: Mathlib.CategoryTheory.Localization.Triangulated

Goal state before:
pp: case mp
C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
L : C ⥤ D
inst✝⁹ : HasShift C ℤ
inst✝⁸ : Preadditive C
inst✝⁷ : HasZeroObject C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : HasShift D ℤ
inst✝³ : L.CommShift ℤ
inst✝² : Preadditive D
inst✝¹ : L.Additive
inst✝ : ∀ (n : ℤ), (shiftFunctor D n).Additive
T : Triangle D
T' : Triangle C
e' : T ≅ L.mapTriangle.obj T'
hT' : T' ∈ distinguishedTriangles
⊢ T.rotate ∈ L.essImageDistTriang

Tactic:
exact ⟨T'.rotate, (rotate D).mapIso e' ≪≫ L.mapTriangleRotateIso.app T',
      rot_of_distTriang T' hT'⟩

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

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝⁴ : (a : α) → Category.{v', u'} (J a)
β : Type w'
inst✝³ : LinearOrder β
inst✝² : OrderBot β
inst✝¹ : SuccOrder β
inst✝ : WellFoundedLT β
b : β
⊢ P.strictLimitsClosureIter J b ≤ P.limitsClosure J

Tactic:
by
  induction b using SuccOrder.limitRecOn with
  | isMin b hb =>
    obtain rfl := hb.eq_bot
    simp
  | succ b hb hb' =>
    rw [strictLimitsClosureIter, transfiniteIterate_succ _ _ _ hb,
      strictLimitsClosureStep, sup_le_iff, iSup_le_iff]
    exact ⟨hb', fun a ↦ ((strictLimitsOfShape_le_limitsOfShape _ _).trans
      (limitsOfShape_monotone _ hb')).trans (limitsOfShape_le _ _)⟩
  | isSuccLimit b hb hb' =>
    simp only [transfiniteIterate_limit _ _ _ hb,
      iSup_le_iff, Subtype.forall, Set.mem_Iio]
    intro c hc
    exact hb' _ hc

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
P : ObjectProperty C
inst✝¹ : P.ContainsZero
inst✝ : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete PEmpty.{1}) X
Z : C
hZ : IsZero Z
hZ₂ : P Z
⊢ P X

Tactic:
have hX : IsTerminal X :=
      (IsLimit.equivOfNatIsoOfIso p.diag.uniqueFromEmpty _ _
        (by exact Cone.ext (Iso.refl _) (by rintro ⟨⟨⟩⟩))).1 p.isLimit

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
P : ObjectProperty C
inst✝¹ : P.ContainsZero
inst✝ : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete PEmpty.{1}) X
Z : C
hZ : IsZero Z
hZ₂ : P Z
hX : IsTerminal X
⊢ P X
Module: Mathlib.CategoryTheory.Triangulated.HomologicalFunctor

Goal state before:
pp: C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
⊢ F.homologicalKernel.trW f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)

Tactic:
obtain ⟨Z, g, h, hT⟩ := distinguished_cocone_triangle f

Goal state after:
C : Type u_1
A : Type u_3
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Category.{v_3, u_3} A
F : C ⥤ A
inst✝⁶ : HasZeroObject C
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : Abelian A
inst✝¹ : F.IsHomological
inst✝ : F.ShiftSequence ℤ
X Y : C
f : X ⟶ Y
Z : C
g : Y ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj X
hT : Triangle.mk f g h ∈ distinguishedTriangles
⊢ F.homologicalKernel.trW f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case mp
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.HasRightCalculusOfFractions
Y Z : C
f₁ f₂ : Y ⟶ Z
⊢ L.map f₁ = L.map f₂ → ∃ X s, ∃ (_ : W s), s ≫ f₁ = s ≫ f₂

Tactic:
intro h
    rw [← RightFraction.map_ofHom W _ L (Localization.inverts _ _),
      ← RightFraction.map_ofHom W _ L (Localization.inverts _ _),
      RightFraction.map_eq_iff] at h
    obtain ⟨Z, t₁, t₂, hst, hft, ht⟩ := h
    dsimp at t₁ t₂ hst hft ht
    grind

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
M : Type u_4
inst✝¹ : AddMonoid M
inst✝ : HasShift C M
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h : 0 + 0 = 0
⊢ f ≫ mk₀ 0 ⋯ g = mk₀ 0 ⋯ (f ≫ g)

Tactic:
mk₀,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
M : Type u_4
inst✝¹ : AddMonoid M
inst✝ : HasShift C M
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h : 0 + 0 = 0
⊢ f ≫ g ≫ (shiftFunctorZero' C 0 ⋯).inv.app Z = mk₀ 0 ⋯ (f ≫ g)
Module: Mathlib.CategoryTheory.Shift.ShiftedHom

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
M : Type u_4
inst✝¹ : AddMonoid M
inst✝ : HasShift C M
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h : 0 + 0 = 0
⊢ f ≫ g ≫ (shiftFunctorZero' C 0 ⋯).inv.app Z = mk₀ 0 ⋯ (f ≫ g)

Tactic:
mk₀,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
M : Type u_4
inst✝¹ : AddMonoid M
inst✝ : HasShift C M
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h : 0 + 0 = 0
⊢ f ≫ g ≫ (shiftFunctorZero' C 0 ⋯).inv.app Z = (f ≫ g) ≫ (shiftFunctorZero' C 0 ⋯).inv.app Z
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : Preadditive C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive D
inst✝⁵ : HasZeroObject D
inst✝⁴ : HasShift D ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝² : Pretriangulated D
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : HasShift E ℤ
P Q R : ObjectProperty C
X X' : C
i : X ≅ X'
⊢ P.extensionProduct Q X → P.extensionProduct Q X'

Tactic:
⟨Y, Z, f, g, h, hT, hP, hQ⟩

Goal state after:
C : Type u_1
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : Preadditive C
inst✝⁹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁸ : Pretriangulated C
D : Type u_2
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Preadditive D
inst✝⁵ : HasZeroObject D
inst✝⁴ : HasShift D ℤ
inst✝³ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝² : Pretriangulated D
E : Type u_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : HasShift E ℤ
P Q R : ObjectProperty C
X X' : C
i : X ≅ X'
Y Z : C
f : Y ⟶ X
g : X ⟶ Z
h : Z ⟶ (shiftFunctor C 1).obj Y
hT : Triangle.mk f g h ∈ distinguishedTriangles
hP : P Y
hQ : Q Z
⊢ P.extensionProduct Q X'