mathlib-initiative/mathlib-tactics huggingface dataset:
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
a : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a = c ≫ T₂.mor₃
⊢ ∃ b, T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂

Tactic:
obtain ⟨a, ⟨ha₁, ha₂⟩⟩ := complete_distinguished_triangle_morphism _ _
    (inv_rot_of_distTriang _ hT₁) (inv_rot_of_distTriang _ hT₂) (c⟦(-1 : ℤ)⟧') a (by
    dsimp
    simp only [neg_comp, comp_neg, ← Functor.map_comp_assoc, ← comm,
      Functor.map_comp, shift_shift_neg', Functor.id_obj, assoc, Iso.inv_hom_id_app, comp_id])

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
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
a✝ : T₁.obj₁ ⟶ T₂.obj₁
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃
a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃
ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂
ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃
⊢ ∃ b, T₁.mor₁ ≫ b = a✝ ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂
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
h₂ : IsZero T.obj₂
h₃ : IsZero T.obj₃
⊢ IsZero T.obj₁

Tactic:
rw [T.isZero₁_iff hT]
  exact ⟨h₂.eq_of_tgt _ _, h₃.eq_of_src _ _⟩

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

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

Tactic:
induction f with
  | nil => exact id
  | cons _ w h => exact comp _ w h

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

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X✝ Y✝ : Quotient (pathsHomRel C)
f : X✝.as ⟶ Y✝.as
⊢ (𝟭 (Quotient (pathsHomRel C))).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f) ≫
      (Quotient.casesOn (motive := fun t ↦
          Y✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj Y✝ ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj Y✝)) Y✝
          (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯).hom =
    (Quotient.casesOn (motive := fun t ↦
          X✝ = t → ((𝟭 (Quotient (pathsHomRel C))).obj X✝ ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj X✝)) X✝
          (fun as h ↦ ⋯ ▸ Iso.refl ((𝟭 (Quotient (pathsHomRel C))).obj { as := as })) ⋯).hom ≫
      (quotientPathsTo C ⋙ toQuotientPaths C).map (Quot.mk (HomRel.CompClosure (pathsHomRel C)) f)

Tactic:
by exact Quot.sound (HomRel.CompClosure.of (by simp))

Goal state after:
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_3
inst✝¹ : Category.{v_3, u_3} E
inst✝ : L.IsLocalization W
⊢ (whiskeringLeftFunctor' L W E).Faithful

Tactic:
rw [whiskeringLeftFunctor'_eq]
  apply @Functor.Faithful.comp _ _ _ _ _ _ _ _ ?_ ?_
  · infer_instance
  apply InducedCategory.faithful

Goal state after:
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
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ ∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f

Tactic:
suffices MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤ by
      intro X Y f
      simpa only [← this] using! MorphismProperty.top_apply f

Goal state after:
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
F₁ F₂ : W.Localization ⥤ D
τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂
⊢ MorphismProperty.naturalityProperty (NatTransExtension.app τ) = ⊤
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'
h : W.IsInvertedBy L'
F₁ : W.Localization ⥤ D := Construction.lift L ⋯
F₂ : W.Localization ⥤ E := Construction.lift L' h
e' : F₁ ⋙ eq.functor ≅ F₂ := liftNatIso W.Q W (L ⋙ eq.functor) L' (F₁ ⋙ eq.functor) F₂ e
⊢ L'.IsLocalization W

Tactic:
exact
    { inverts := h
      isEquivalence := Functor.isEquivalence_of_iso e' }

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
W : MorphismProperty C
X Y : C
φ : W.LeftFraction₂ X Y
F : C ⥤ D
hF : W.IsInvertedBy F
inst✝¹ : Preadditive D
inst✝ : F.Additive
this : IsIso (F.map φ.s)
⊢ φ.add.map F hF = φ.fst.map F hF + φ.snd.map F hF

Tactic:
rw [← cancel_mono (F.map φ.s), add_comp, LeftFraction.map_comp_map_s,
    LeftFraction.map_comp_map_s, LeftFraction.map_comp_map_s, F.map_add]

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 Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
⊢ add' W f₁ f₂ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g)

Tactic:
obtain ⟨β, hβ⟩ := exists_leftFraction L W g

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
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
f₁ f₂ : L.obj X ⟶ L.obj Y
g : L.obj Y ⟶ L.obj Z
α : W.LeftFraction₂ X Y
h₁ : f₁ = α.fst.map L ⋯
h₂ : f₂ = α.snd.map L ⋯
β : W.LeftFraction Y Z
hβ : g = β.map L ⋯
⊢ add' W f₁ f₂ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g)
Module: Mathlib.CategoryTheory.Shift.Localization

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹⁸ : Category.{v₁, u₁} C
inst✝¹⁷ : Category.{v₂, u₂} D
E : Type u₃
inst✝¹⁶ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝¹⁵ : L.IsLocalization W
A : Type w
inst✝¹⁴ : AddMonoid A
inst✝¹³ : HasShift C A
C₁ : Type u_1
C₂ : Type u_2
inst✝¹² : Category.{v_1, u_1} C₁
inst✝¹¹ : Category.{v_2, u_2} C₂
W₁ : MorphismProperty C₁
W₂ : MorphismProperty C₂
Φ : LocalizerMorphism W₁ W₂
M : Type u_3
inst✝¹⁰ : AddMonoid M
inst✝⁹ : HasShift C₁ M
inst✝⁸ : HasShift C₂ M
inst✝⁷ : Φ.functor.CommShift M
D₁ : Type u_4
D₂ : Type u_5
inst✝⁶ : Category.{v_3, u_4} D₁
inst✝⁵ : Category.{v_4, u_5} D₂
L₁ : C₁ ⥤ D₁
inst✝⁴ : L₁.IsLocalization W₁
L₂ : C₂ ⥤ D₂
inst✝³ : HasShift D₁ M
inst✝² : HasShift D₂ M
inst✝¹ : L₁.CommShift M
inst✝ : L₂.CommShift M
G : D₁ ⥤ D₂
e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G
this : Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso := e.symm }
⊢ G.CommShift M

Tactic:
exact Functor.commShiftOfLocalization L₁ W₁ M (Φ.functor ⋙ L₂) G

Goal state after:
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
T₁ T₂ : Triangle D
e : T₂ ≅ T₁
h : T₁ ∈ L.essImageDistTriang
⊢ T₂ ∈ L.essImageDistTriang

Tactic:
by
  obtain ⟨T', e', hT'⟩ := h
  exact ⟨T', e ≪≫ e', hT'⟩

Goal state after:
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: case succ.inl
I : Type u
inst✝⁴ : CompleteLattice I
φ : I → I
i₀ : I
J : Type w
inst✝³ : LinearOrder J
inst✝² : OrderBot J
inst✝¹ : SuccOrder J
inst✝ : WellFoundedLT J
hφ : ∀ (i : I), i ≤ φ i
k k' : J
hk' : ¬IsMax k'
hkk' : k ≤ k' → (fun j ↦ transfiniteIterate φ j i₀) k ≤ (fun j ↦ transfiniteIterate φ j i₀) k'
hkj✝ : k ≤ Order.succ k'
hkj : k < Order.succ k'
⊢ transfiniteIterate φ k' i₀ ≤ φ (transfiniteIterate φ k' i₀)

Tactic:
apply hφ

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
P Q : ObjectProperty C
A : Type u_2
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
E : Type u_3
inst✝⁴ : Category.{v_2, u_3} E
inst✝³ : HasShift E A
inst✝² : P.IsClosedUnderIsomorphisms
G : Type u_4
inst✝¹ : AddGroup G
inst✝ : HasShift C G
a : G
X : C
b : G
hb : P.shift b X
⊢ (P.shift (-a + b)).shift a X

Tactic:
by rwa [P.shift_shift _ _ _ (add_neg_cancel_left a b)]

Goal state after:
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 Q : ObjectProperty C
⊢ P.isoClosure.extensionProduct Q = P.extensionProduct Q

Tactic:
refine le_antisymm ?_ (monotone_extensionProduct_left Q P.le_isoClosure)
  intro X ⟨Y, Z, f, g, h, hT, ⟨Y', hP, ⟨i⟩⟩, hQ⟩
  refine ⟨Y', Z, i.inv ≫ f, g, h ≫ i.hom⟦1⟧', ?_, hP, hQ⟩
  exact isomorphic_distinguished _ hT _ <| Triangle.isoMk _ _ i.symm (Iso.refl _) (Iso.refl _)

Goal state after:
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 Q : ObjectProperty C
⊢ P.extensionProduct Q.isoClosure ≤ P.extensionProduct Q

Tactic:
intro X ⟨Y, Z, f, g, h, hT, hP, ⟨Z', hQ, ⟨i⟩⟩⟩

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 Q : ObjectProperty C
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
Z' : C
hQ : Q Z'
i : Z ≅ Z'
⊢ P.extensionProduct Q X
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 : ℕ
⊢ P.retractClosure.extensionProductIter n ≤ (P.extensionProductIter n).retractClosure

Tactic:
by
  induction n with
  | zero => simp
  | succ n H =>
    grw [extensionProductIter_succ, extensionProductIter_succ, monotone_extensionProduct_right _ H,
      extensionProduct_retractClosure_retractClosure_le]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
hf : n.negOnePow • f ≫ K.d (i + n) (j + n) = 0
⊢ (n.negOnePow * n.negOnePow) • (f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n) = 0

Tactic:
mul_smul,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : CategoryWithHomology C
K : CochainComplex C ℤ
A : C
n i : ℤ
f : A ⟶ ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj K).X i
j : ℤ
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
hf : n.negOnePow • f ≫ K.d (i + n) (j + n) = 0
⊢ n.negOnePow • n.negOnePow • (f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n) = 0
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
E : Type u_3
inst✝² : Category.{v_3, u_3} E
inst✝¹ : HasShift E ℤ
P : ObjectProperty C
inst✝ : P.IsTriangulated
J : Type
x✝ : Finite J
X₁✝ X₂✝ : J → C
X₁ : HasProduct X₁✝
X₂ : HasProduct X₂✝
f : (j : J) → X₁✝ j ⟶ X₂✝ j
hf : ∀ (j : J), P.isoClosure.trW (f j)
⊢ P.isoClosure.trW (Limits.Pi.map f)

Tactic:
exact trW.mk _ (productTriangle_distinguished _
      (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose))
      (P.isoClosure.prop_pi _
        (fun j => (hf j).choose_spec.choose_spec.choose_spec.choose_spec))

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✝ : HasHomotopyCofiber φ
M : CochainComplex C ℤ
α : M ⟶ K
β : Cochain M L (-1)
hαβ : δ (-1) 0 β + Cochain.ofHom (α ≫ φ) = 0
⊢ δ (-1) 0 β + (↑(Cocycle.ofHom α)).comp (Cochain.ofHom φ) ⋯ = 0

Tactic:
simpa [← Cochain.ofHom_comp]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
A : Type u_3
inst✝² : AddCommSemigroup A
inst✝¹ : IsRightCancelAdd A
inst✝ : One A
a b : A
⊢ ∀ (i₁ i₂ : A), (up' a).Rel i₁ i₂ ↔ (up' a).Rel (i₁ + b) (i₂ + b)

Tactic:
by dsimp; simp_rw [add_right_comm _ b a, add_right_cancel_iff, implies_true]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
A : Type u_3
inst✝² : AddCommSemigroup A
inst✝¹ : IsRightCancelAdd A
inst✝ : One A
a b : A
⊢ ∀ (i₁ i₂ : A), (down' a).Rel i₁ i₂ ↔ (down' a).Rel (i₁ + b) (i₂ + b)

Tactic:
by dsimp; simp_rw [add_right_comm _ b a, add_right_cancel_iff, implies_true]

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
inst✝ : e.IsRelIff
i j : ι
hij : c.prev j = i
hj : ¬e.BoundaryGE j
hij' : ¬c.Rel i j
⊢ j = i

Tactic:
simpa only [c.prev_eq_self j (by simpa only [hij] using hij')] using hij

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
e : c.Embedding c'
k' : ι'
j : ι
hj : c'.Rel (e.f j) k'
hk' : ∀ (i : ι), e.f i ≠ k'
⊢ e.BoundaryLE j

Tactic:
by
  constructor
  · simpa only [c'.next_eq' hj] using hj
  · intro k hk
    apply hk' k
    rw [← c'.next_eq' hj, c'.next_eq' hk]

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

Goal state before:
pp: case pos
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
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' j' : ι'
hij' : c'.Rel i' j'
hi' : ∃ i, e.f i = i'
⊢ f φ i' ≫ (L.extend e).d i' j' = K.d i' j' ≫ f φ j'

pp: case neg
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
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' j' : ι'
hij' : c'.Rel i' j'
hi' : ¬∃ i, e.f i = i'
⊢ f φ i' ≫ (L.extend e).d i' j' = K.d i' j' ≫ f φ j'

Tactic:
· obtain ⟨i, hi⟩ := hi'
      rw [f_eq φ hi]
      by_cases hj' : ∃ j, e.f j = j'
      · obtain ⟨j, hj⟩ := hj'
        rw [f_eq φ hj, L.extend_d_eq e hi hj]
        subst hi hj
        simp [HomologicalComplex.restrictionXIso]
      · apply (L.isZero_extend_X e j' (by simpa using hj')).eq_of_tgt

Goal state after:
case neg
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
φ : K.restriction e ⟶ L
hφ : e.HasLift φ
i' j' : ι'
hij' : c'.Rel i' j'
hi' : ¬∃ i, e.f i = i'
⊢ f φ i' ≫ (L.extend e).d i' j' = K.d i' j' ≫ f φ j'
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: 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 : ι
hi : e.BoundaryGE i
⊢ Epi ((K.restrictionToTruncGE' e).f i)

Tactic:
rw [K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e rfl hi]

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 : ι
hi : e.BoundaryGE i
⊢ Epi ((K.restrictionXIso e ⋯).hom ≫ K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv)
Module: Mathlib.Algebra.Homology.Embedding.TruncGE

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncGE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ IsIso ((K.restrictionToTruncGE' e).f i)

Tactic:
rw [K.restrictionToTruncGE'_f_eq_iso_hom_pOpcycles_iso_inv e rfl hi]

Goal state after:
case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁶ : Category.{v_1, u_3} C
inst✝⁵ : HasZeroMorphisms C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝⁴ : e.IsTruncGE
inst✝³ : ∀ (i' : ι'), K.HasHomology i'
inst✝² : ∀ (i' : ι'), L.HasHomology i'
inst✝¹ : ∀ (i' : ι'), M.HasHomology i'
inst✝ : K.IsStrictlySupported e
i : ι
hi : e.BoundaryGE i
⊢ IsIso ((K.restrictionXIso e ⋯).hom ≫ K.pOpcycles (e.f i) ≫ (K.truncGE'XIsoOpcycles e ⋯ hi).inv)
Module: Mathlib.Algebra.Homology.Embedding.AreComplementary

Goal state before:
pp: ι : Type u_1
ι₁ : Type u_2
ι₂ : Type u_3
c : ComplexShape ι
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
C : Type u_4
inst✝¹ : Category.{v_1, u_4} C
inst✝ : HasZeroMorphisms C
e₁ : c₁.Embedding c
e₂ : c₂.Embedding c
ac : e₁.AreComplementary e₂
K : HomologicalComplex C c
h : K.IsStrictlySupportedOutside e₁
i : ι
hi : ∀ (i_1 : ι₂), e₂.f i_1 ≠ i
⊢ IsZero (K.X i)

Tactic:
obtain ⟨i₁, rfl⟩ := ac.exists_i₁ i hi
      exact h.isZero i₁

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

Goal state before:
pp: C : Type u_2
inst✝³ : Category.{u_3, u_2} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : CochainComplex C ℤ
inst✝ : ∀ (i : ℤ), HasHomology K i
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ (HomologicalComplex.truncLE K (embeddingUpIntLE n₀)).Acyclic ↔ K.IsGE n₁

Tactic:
acyclic_truncLE_iff_isSupportedOutside,

Goal state after:
C : Type u_2
inst✝³ : Category.{u_3, u_2} C
inst✝² : HasZeroMorphisms C
inst✝¹ : HasZeroObject C
K : CochainComplex C ℤ
inst✝ : ∀ (i : ℤ), HasHomology K i
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ IsSupportedOutside K (embeddingUpIntLE n₀) ↔ K.IsGE n₁
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₂.g
⊢ Epi (homologyMap φ.τ₃ i)

Tactic:
have eq := (homologyFunctor C _ i).congr_map φ.comm₂₃

Goal state after:
case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S₁ S₂ : ShortComplex (HomologicalComplex C c)
φ : S₁ ⟶ S₂
hS₁ : S₁.ShortExact
hS₂ : S₂.ShortExact
i : ι
h₁ : Epi (homologyMap φ.τ₂ i)
h₂ : ∀ (j : ι), c.Rel i j → Epi (homologyMap φ.τ₁ j)
h₃ : ∀ (j : ι), c.Rel i j → Mono (homologyMap φ.τ₂ j)
hi : ¬∃ j, c.Rel i j
this : Epi S₂.g
eq : (homologyFunctor C c i).map (φ.τ₂ ≫ S₂.g) = (homologyFunctor C c i).map (S₁.g ≫ φ.τ₃)
⊢ Epi (homologyMap φ.τ₃ i)
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y Z : C
n : ℕ
α₁ α₂ : Ext X Y n
m : ℕ
β : Ext Y Z m
p : ℕ
h : n + m = p
⊢ (α₁ + α₂).comp β h = α₁.comp β h + α₂.comp β h

Tactic:
let := HasDerivedCategory.standard C; ext; simp [this, add_hom']

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
X Y Z : C
n : ℕ
α : Ext X Y n
m : ℕ
β₁ β₂ : Ext Y Z m
p : ℕ
h : n + m = p
⊢ α.comp (β₁ + β₂) h = α.comp β₁ h + α.comp β₂ h

Tactic:
let := HasDerivedCategory.standard C; ext; simp [this, add_hom']

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
n : ℕ
X₁ X₂ X₃ : Cᵒᵖ
f : X₁ ⟶ X₂
f' : X₂ ⟶ X₃
⊢ { app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ (f ≫ f').unop).comp α ⋯) ⋯),
      naturality := ⋯ } =
    { app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f.unop).comp α ⋯) ⋯), naturality := ⋯ } ≫
      { app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f'.unop).comp α ⋯) ⋯), naturality := ⋯ }

Tactic:
ext Y α

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
n : ℕ
X₁ X₂ X₃ : Cᵒᵖ
f : X₁ ⟶ X₂
f' : X₂ ⟶ X₃
Y : C
α : ↑((extFunctorObj (Opposite.unop X₁) n).obj Y)
⊢ (AddCommGrpCat.Hom.hom
        ({ app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ (f ≫ f').unop).comp α ⋯) ⋯),
              naturality := ⋯ }.app
          Y))
      α =
    (AddCommGrpCat.Hom.hom
        (({ app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f.unop).comp α ⋯) ⋯),
                naturality := ⋯ } ≫
              { app := fun Y ↦ AddCommGrpCat.ofHom (AddMonoidHom.mk' (fun α ↦ (Ext.mk₀ f'.unop).comp α ⋯) ⋯),
                naturality := ⋯ }).app
          Y))
      α
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExtClass

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n : ℕ
γ : Ext S.X₁ Y n
n' : ℕ
h : 1 + n = n'
⊢ (Ext.mk₀ S.g).comp (hS.extClass.comp γ h) ⋯ = 0

Tactic:
rw [← Ext.comp_assoc (a₁₂ := 1) _ _ _ (by lia) (by lia) (by lia),
    comp_extClass, Ext.zero_comp]

Goal state after: