mathlib-initiative/mathlib-tactics huggingface dataset:
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)
⊢ inr φ₁ ≫ mapOfHomotopy H = b ≫ inr φ₂

Tactic:
simp [mapOfHomotopy]

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

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
inst✝⁶ : Preadditive C
inst✝⁵ : HasBinaryBiproducts C
inst✝⁴ : Preadditive D
inst✝³ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝² : HasZeroObject C
inst✝¹ : HasZeroObject D
G : C ⥤ D
inst✝ : G.Additive
⊢ ∀ T ∈ distinguishedTriangles, (G.mapHomotopyCategory (ComplexShape.up ℤ)).mapTriangle.obj T ∈ distinguishedTriangles

Tactic:
rintro T ⟨K, L, f, ⟨e⟩⟩

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
inst✝⁶ : Preadditive C
inst✝⁵ : HasBinaryBiproducts C
inst✝⁴ : Preadditive D
inst✝³ : HasBinaryBiproducts D
K✝ L✝ : CochainComplex C ℤ
φ : K✝ ⟶ L✝
inst✝² : HasZeroObject C
inst✝¹ : HasZeroObject D
G : C ⥤ D
inst✝ : G.Additive
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
K L : CochainComplex C ℤ
f : K ⟶ L
e : T ≅ CochainComplex.mappingCone.triangleh f
⊢ (G.mapHomotopyCategory (ComplexShape.up ℤ)).mapTriangle.obj T ∈ distinguishedTriangles
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁶ : Category.{v_1, u_1} C
inst✝¹⁵ : Category.{v_2, u_2} D
inst✝¹⁴ : HasShift C ℤ
inst✝¹³ : HasShift D ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : Preadditive C
inst✝⁹ : Preadditive D
inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁶ : Pretriangulated C
inst✝⁵ : Pretriangulated D
F : C ⥤ D
inst✝⁴ : F.CommShift ℤ
inst✝³ : F.IsTriangulated
inst✝² : F.Full
inst✝¹ : F.Faithful
inst✝ : IsTriangulated D
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
comm' : F.map u₁₂ ≫ F.map u₂₃ = F.map u₁₃
H : Octahedron comm' ⋯ ⋯ ⋯ := someOctahedron comm' ⋯ ⋯ ⋯
⊢ F.map (v₁₂ ≫ F.preimage H.m₁) = F.map (u₂₃ ≫ v₁₃)

Tactic:
simpa using H.comm₁

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

Goal state before:
pp: case h_obj
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 : Paths V
⊢ F.obj X = G.obj X

Tactic:
rw [h_obj]

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

Goal state before:
pp: case refine_2
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
⊢ w₁₂ ≫
      (shiftFunctorCompIsoId C (-1) 1 ⋯).inv.app ((shiftFunctor C 1).obj Z₁₂) ≫
        (shiftFunctor C 1).map ((shiftFunctor C (-1)).map o.m₁) ≫
          (shiftFunctorCompIsoId C (-1) 1 ⋯).hom.app ((shiftFunctor C 1).obj Z₁₃) =
    u₂₃ ≫ w₁₃

Tactic:
shiftFunctorCompIsoId_naturality_1,

Goal state after:
case refine_2
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
⊢ w₁₂ ≫ o.m₁ = u₂₃ ≫ w₁₃
Module: Mathlib.CategoryTheory.PathCategory.MorphismProperty

Goal state before:
pp: case mpr
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)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ ⊤ f✝ → P f✝

Tactic:
exact fun _ ↦ induction (fun f ↦ P f) id comp _

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

Goal state before:
pp: case refine_3
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
⊢ w₁₃ ≫
      (shiftFunctorCompIsoId C (-1) 1 ⋯).inv.app ((shiftFunctor C 1).obj Z₁₃) ≫
        (shiftFunctor C 1).map ((shiftFunctor C (-1)).map o.m₃) ≫
          (shiftFunctorCompIsoId C (-1) 1 ⋯).hom.app ((shiftFunctor C 1).obj Z₂₃) =
    w₂₃

Tactic:
shiftFunctorCompIsoId_naturality_1,

Goal state after:
case refine_3
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
⊢ w₁₃ ≫ o.m₃ = w₂₃
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:
by
    have := hS.mono_f
    dsimp
    infer_instance

Goal state after:
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
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
X Y : C
f : X ⟶ Y
hf : W f
⊢ e.hom.app Y ≫ (isoOfHom L W f hf).inv = (uniq W.Q L W).functor.map (Construction.wInv f hf) ≫ e.hom.app X

Tactic:
rw [Construction.wInv_eq_isoOfHom_inv, ← cancel_mono (isoOfHom L W f hf).hom,
          assoc, assoc, Iso.inv_hom_id, comp_id, isoOfHom_hom, ← NatTrans.naturality,
          Functor.comp_map, ← Functor.map_comp_assoc,
          isoOfHom_inv_hom_id, map_id, id_comp]

Goal state after:
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
e : W.Q ⋙ (uniq W.Q L W).functor ≅ L := compUniqFunctor W.Q L W
⊢ P = ⊤

Tactic:
have hP : P.inverseImage (uniq W.Q L W).functor = ⊤ :=
    Construction.morphismProperty_eq_top _
      (fun _ _ f ↦ (P.arrow_mk_iso_iff
        (((Functor.mapArrowFunctor _ _).mapIso e).app (Arrow.mk f))).2 (h₁ f))
      (fun X Y f hf ↦ by
        refine (P.arrow_mk_iso_iff (Arrow.isoMk (e.app _) (e.app _) ?_)).2 (h₂ f hf)
        dsimp
        rw [Construction.wInv_eq_isoOfHom_inv, ← cancel_mono (isoOfHom L W f hf).hom,
          assoc, assoc, Iso.inv_hom_id, comp_id, isoOfHom_hom, ← NatTrans.naturality,
          Functor.comp_map, ← Functor.map_comp_assoc,
          isoOfHom_inv_hom_id, map_id, id_comp])

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
hP : P.inverseImage (uniq W.Q L W).functor = ⊤
⊢ P = ⊤
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⟩

Goal state after:
case mpr
X : Type u
Y : Type u'
κ : Cardinal.{w}
hκ : Cardinal.aleph0 ≤ κ
hX : HasCardinalLT X κ
hY : HasCardinalLT Y κ
⊢ HasCardinalLT (X ⊕ Y) κ
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 : C
s : X ⟶ Y
hs : W s
⊢ homMk (ofInv s hs) ≫ homMk (ofHom W s) = 𝟙 ((Q W).obj Y)

Tactic:
rw [homMk_comp_homMk (ofInv s hs) (ofHom W s) (ofHom W (𝟙 Y)) (by simp)]

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 : C
s : X ⟶ Y
hs : W s
⊢ homMk ((ofInv s hs).comp₀ (ofHom W s) (ofHom W (𝟙 Y))) = 𝟙 ((Q W).obj Y)
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
n✝ : P.ContainsZero
n m✝ m : ℕ
hnm : n ≤ m
h : P.extensionProductIter n ≤ P.extensionProductIter m
⊢ P.extensionProductIter n ≤ P.extensionProductIter (m + 1)

Tactic:
refine le_trans h ?_

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : Preadditive C
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
P : ObjectProperty C
n✝ : P.ContainsZero
n m✝ m : ℕ
hnm : n ≤ m
h : P.extensionProductIter n ≤ P.extensionProductIter m
⊢ P.extensionProductIter m ≤ P.extensionProductIter (m + 1)
Module: Mathlib.Algebra.Homology.QuasiIso

Goal state before:
pp: ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
inst✝ : CategoryWithHomology C
X✝ Y✝ Z✝ : HomologicalComplex C c
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : QuasiIso f
hfg : QuasiIso (f ≫ g)
⊢ QuasiIso g

Tactic:
← quasiIso_iff_comp_left f g

Goal state after:
ι : Type u_1
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
c : ComplexShape ι
K L M K' L' : HomologicalComplex C c
inst✝ : CategoryWithHomology C
X✝ Y✝ Z✝ : HomologicalComplex C c
f : X✝ ⟶ Y✝
g : Y✝ ⟶ Z✝
hf : QuasiIso f
hfg : QuasiIso (f ≫ g)
⊢ QuasiIso (f ≫ g)
Module: Mathlib.Algebra.Homology.Localization

Goal state before:
pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
ι : Type u_2
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : CategoryWithHomology C
inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization
K L : HomologicalComplex C c
f : K ⟶ L
⊢ HomologicalComplex.quasiIso C c f → IsIso (Q.map f)

Tactic:
· intro h
    exact Localization.inverts Q (HomologicalComplex.quasiIso C c) _ h

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X : C
i : ℤ
⊢ ((single C (ComplexShape.up ℤ) a' ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj X).X i =
    ((single C (ComplexShape.up ℤ) a).obj X).X i

Tactic:
obtain rfl : a' = a + n := by lia

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a : ℤ
X : C
i : ℤ
ha' : n + a = a + n
⊢ ((single C (ComplexShape.up ℤ) (a + n) ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj X).X i =
    ((single C (ComplexShape.up ℤ) a).obj X).X i
Module: Mathlib.CategoryTheory.Triangulated.Subcategory

Goal state before:
pp: C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : HasZeroObject C
inst✝¹² : HasShift C ℤ
inst✝¹¹ : Preadditive C
inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁹ : Pretriangulated C
D : Type u_2
inst✝⁸ : Category.{v_2, u_2} D
inst✝⁷ : Preadditive D
inst✝⁶ : HasZeroObject D
inst✝⁵ : HasShift D ℤ
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated D
P : ObjectProperty C
F : D ⥤ C
inst✝² : F.CommShift ℤ
inst✝¹ : F.IsTriangulated
inst✝ : P.IsClosedUnderIsomorphisms
X Y : D
s : X ⟶ Y
⊢ (P.inverseImage F).trW s ↔ P.trW (F.map s)

Tactic:
by
  obtain ⟨Z, g, h, hT⟩ := distinguished_cocone_triangle s
  have eq₁ := (P.inverseImage F).trW_iff_of_distinguished _ hT
  have eq₂ := P.trW_iff_of_distinguished _ (F.map_distinguished _ hT)
  dsimp at eq₁ eq₂
  rw [eq₁, prop_inverseImage_iff, eq₂]

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a a' : ℤ
ha' : n + a = a'
X Y : C
f : X ⟶ Y
⊢ (single C (ComplexShape.up ℤ) a' ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map f ≫
      (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom =
    (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom ≫ (single C (ComplexShape.up ℤ) a).map f

Tactic:
obtain rfl : a' = a + n := by lia

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
n a : ℤ
X Y : C
f : X ⟶ Y
ha' : n + a = a + n
⊢ (single C (ComplexShape.up ℤ) (a + n) ⋙ CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map f ≫
      (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom =
    (Hom.isoOfComponents (fun i ↦ eqToIso ⋯) ⋯).hom ≫ (single C (ComplexShape.up ℤ) a).map f
Module: Mathlib.Algebra.Homology.DerivedCategory.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasDerivedCategory C
⊢ Qh.IsLocalization (HomotopyCategory.quasiIso C (ComplexShape.up ℤ))

Tactic:
by
  dsimp [Qh, DerivedCategory]
  infer_instance

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
K L : CochainComplex C ℤ
φ : K ⟶ L
inst✝ : HasBinaryBiproducts C
⊢ (triangle φ).rotate ≅ mappingCone.triangle φ

Tactic:
by
  refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _)
    ((shiftFunctorCompIsoId _ (-1 : ℤ) 1 (by lia)).app _)
    (by simp) (by simp [triangle]) ?_
  dsimp
  ext n
  simp [fst, mappingCone.triangle, Cochain.leftShift_v _ _ _ _ _ _ _ _ rfl,
    Cochain.rightShift_v _ _ _ _ _ _ _ _ rfl,
    shiftFunctorCompIsoId, shiftFunctorAdd'_inv_app_f', shiftFunctorZero_hom_app_f]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.Boundary

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j : ι
hij : c.prev j = j
hij' : ¬c.Rel j j
hj' : c'.Rel (c'.prev (e.f j)) (e.f j)
hj : c'.Rel (c'.prev (e.f j)) (e.f j) → ∃ x, c'.Rel (e.f x) (e.f j)
⊢ False

Tactic:
obtain ⟨i, hi⟩ := hj hj'

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
j : ι
hij : c.prev j = j
hij' : ¬c.Rel j j
hj' : c'.Rel (c'.prev (e.f j)) (e.f j)
hj : c'.Rel (c'.prev (e.f j)) (e.f j) → ∃ x, c'.Rel (e.f x) (e.f j)
i : ι
hi : c'.Rel (e.f i) (e.f j)
⊢ False
Module: Mathlib.Algebra.Homology.Embedding.HomEquiv

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : HomologicalComplex C c'
L : HomologicalComplex C c
inst✝ : e.IsRelIff
ψ : K ⟶ L.extend e
i : ι
⊢ f ψ i = (K.restrictionXIso e ⋯).hom ≫ ψ.f (e.f i) ≫ (L.extendXIso e ⋯).hom

Tactic:
simp [f, restrictionXIso]

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i j : ι
hij : c.Rel i j
⊢ (if hi : e.BoundaryGE i then
        (K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv
      else (K.truncGE'XIso e ⋯ hi).hom ≫ φ.f (e.f i) ≫ (L.truncGE'XIso e ⋯ hi).inv) ≫
      (L.truncGE' e).d i j =
    (K.truncGE' e).d i j ≫ (K.truncGE'XIso e ⋯ ⋯).hom ≫ φ.f (e.f j) ≫ (L.truncGE'XIso e ⋯ ⋯).inv

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i j : ι
hij : c.Rel i j
hi : e.BoundaryGE i
⊢ (if hi : e.BoundaryGE i then
        (K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv
      else (K.truncGE'XIso e ⋯ hi).hom ≫ φ.f (e.f i) ≫ (L.truncGE'XIso e ⋯ hi).inv) ≫
      (L.truncGE' e).d i j =
    (K.truncGE' e).d i j ≫ (K.truncGE'XIso e ⋯ ⋯).hom ≫ φ.f (e.f j) ≫ (L.truncGE'XIso e ⋯ ⋯).inv

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : ∀ (i' : ι'), M.HasHomology i'
i j : ι
hij : c.Rel i j
hi : ¬e.BoundaryGE i
⊢ (if hi : e.BoundaryGE i then
        (K.truncGE'XIsoOpcycles e ⋯ hi).hom ≫ opcyclesMap φ (e.f i) ≫ (L.truncGE'XIsoOpcycles e ⋯ hi).inv
      else (K.truncGE'XIso e ⋯ hi).hom ≫ φ.f (e.f i) ≫ (L.truncGE'XIso e ⋯ hi).inv) ≫
      (L.truncGE' e).d i j =
    (K.truncGE' e).d i j ≫ (K.truncGE'XIso e ⋯ ⋯).hom ≫ φ.f (e.f j) ≫ (L.truncGE'XIso e ⋯ ⋯).inv
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i : ι
⊢ (truncGE'Map (𝟙 K) e).f i = (𝟙 (K.truncGE' e)).f i

Tactic:
by_cases hi : e.BoundaryGE i

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i : ι
hi : e.BoundaryGE i
⊢ (truncGE'Map (𝟙 K) e).f i = (𝟙 (K.truncGE' e)).f i

case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
i : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (𝟙 K) e).f i = (𝟙 (K.truncGE' e)).f i
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝¹ : e.IsTruncGE
inst✝ : ∀ (i' : ι'), K.HasHomology i'
j k : ι
j' : ι'
hj' : e.f j = j'
hj : e.BoundaryGE j
⊢ (K.homologyι j' ≫ (K.truncGE'XIsoOpcycles e hj' hj).inv) ≫ (K.truncGE' e).d j k = 0

Tactic:
by_cases hjk : c.Rel j k
  · rw [K.truncGE'_d_eq_fromOpcycles e hjk hj' rfl hj, assoc, Iso.inv_hom_id_assoc,
    homologyι_comp_fromOpcycles_assoc, zero_comp]
  · rw [shape _ _ _ hjk, comp_zero]

Goal state after: