mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
iso : K ≅ L
ψ : L ⟶ M
i j k : ι
inst✝¹ : K.HasHomology i
hi : c.prev j = i
inst✝ : K.HasHomology j
⊢ IsColimit (CokernelCofork.ofπ (K.pOpcycles j) ⋯)

Tactic:
by
  obtain rfl := hi
  exact (K.sc j).opcyclesIsCokernel

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
ι : Type u_2
c : ComplexShape ι
K L M : HomologicalComplex C c
φ : K ⟶ L
ψ : L ⟶ M
i : ι
inst✝² : K.HasHomology i
inst✝¹ : L.HasHomology i
inst✝ : M.HasHomology i
⊢ opcyclesMap (φ ≫ ψ) i = opcyclesMap φ i ≫ opcyclesMap ψ i

Tactic:
by
  dsimp [opcyclesMap]
  rw [Functor.map_comp, ShortComplex.opcyclesMap_comp]

Goal state after:
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: R : Type u
S : Type v
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra A B
inst✝⁴ : Algebra S B
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
⊢ Function.Surjective ⇑(map R S A B)

Tactic:
rw [← LinearMap.range_eq_top, _root_.eq_top_iff,
    ← @Submodule.restrictScalars_top A B, ← span_range_derivation,
    Submodule.restrictScalars_span _ _ h, Submodule.span_le]

Goal state after:
R : Type u
S : Type v
inst✝¹¹ : CommRing R
inst✝¹⁰ : CommRing S
inst✝⁹ : Algebra R S
A : Type u_2
B : Type u_3
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra A B
inst✝⁴ : Algebra S B
inst✝³ : Algebra R B
inst✝² : IsScalarTower R A B
inst✝¹ : IsScalarTower R S B
inst✝ : SMulCommClass S A B
h : Function.Surjective ⇑(algebraMap A B)
⊢ Set.range ⇑(D S B) ⊆ ↑(map R S A B).range
Module: Mathlib.RingTheory.Kaehler.Basic

Goal state before:
pp: case a.convert_2.inr
R : Type u
inst✝⁶ : CommRing R
A : Type u_2
B : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra A B
inst✝¹ : Algebra R B
inst✝ : IsScalarTower R A B
x : A
⊢ (fun x ↦ single ((algebraMap A B) x) 1) x ∈
    ↑(Submodule.comap (linearCombination B ⇑(D R B)) (mapBaseChange R A B).range)

Tactic:
use 1 ⊗ₜ D _ _ x; simp

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h✝ k : D ⟶ E
i✝ : ι
h : Homotopy f g
i : ι
⊢ (f - g).f i = (((dNext i) fun i j ↦ h.hom i j) + (prevD i) fun i j ↦ h.hom i j) + Hom.f 0 i

Tactic:
simp [h.comm]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
p q : ℤ
hpq : p + 1 = q
⊢ (inr φ).f p ≫ (↑(fst φ)).v p q hpq = 0

Tactic:
by
  simp [inr, fst]

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

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 0 = q
⊢ (lift φ α β eq).f p ≫ (snd φ).v p q hpq = β.v p q hpq

Tactic:
obtain rfl : q = p := by lia

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
q : ℤ
hpq : q + 0 = q
⊢ (lift φ α β eq).f q ≫ (snd φ).v q q hpq = β.v q q hpq
Module: Mathlib.Algebra.Homology.HomotopyCofiber

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i : ι
⊢ H.map (inrX φ i) ≫ (mapHomologicalComplexObjXIso φ H i).hom = inrX ((H.mapHomologicalComplex c).map φ) i

Tactic:
by
  rw [← inrX_mapHomologicalComplexObjXIso_inv_assoc, Iso.inv_hom_id, comp_id]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n m : ℤ
hm : n + 1 = m
p' q' : ℤ
hp' : p' + 1 = p
hq' : q + 1 = q'
p'' q'' : ℤ
hpq'' : p'' + m = q''
⊢ (δ n m (single f n)).v p'' q'' hpq'' =
    (single (f ≫ L.d q q') m + m.negOnePow • single (K.d p' p ≫ f) m).v p'' q'' hpq''

Tactic:
rw [δ_v n m hm (single f n) p'' q'' (by lia) (q'' - 1) (p'' + 1) rfl (by lia),
    add_v, units_smul_v]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n m : ℤ
hm : n + 1 = m
p' q' : ℤ
hp' : p' + 1 = p
hq' : q + 1 = q'
p'' q'' : ℤ
hpq'' : p'' + m = q''
⊢ (single f n).v p'' (q'' - 1) ⋯ ≫ L.d (q'' - 1) q'' +
      m.negOnePow • K.d p'' (p'' + 1) ≫ (single f n).v (p'' + 1) q'' ⋯ =
    (single (f ≫ L.d q q') m).v p'' q'' hpq'' + m.negOnePow • (single (K.d p' p ≫ f) m).v p'' q'' hpq''
Module: Mathlib.CategoryTheory.Quotient

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
r✝ : HomRel C
G : Type u_2
inst✝ : Groupoid G
r : HomRel G
X✝ Y✝ : Quotient r
f✝ : X✝ ⟶ Y✝
X Y a✝ b✝ : G
f g : a✝ ⟶ b✝
hfg : r f g
a : X✝.as ⟶ a✝
b : b✝ ⟶ Y✝.as
⊢ Quot.mk (HomRel.CompClosure r) (Groupoid.inv (a ≫ f ≫ b)) = Quot.mk (HomRel.CompClosure r) (Groupoid.inv (a ≫ g ≫ b))

Tactic:
simpa using! (Quot.sound (HomRel.CompClosure.intro _ _
      (inv b ≫ inv g) _ _ (inv f ≫ inv a) hfg)).symm

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝³ : DecidableRel c.Rel
inst✝² : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝¹ : K.HasCylinder
hc : ∀ (j : ι), ∃ i, c.Rel i j
D : Type u_3
inst✝ : Category.{v_2, u_3} D
H : HomologicalComplex C c ⥤ D
hH : (homotopyEquivalences C c).IsInvertedBy H
this : IsIso (H.map (π K))
⊢ H.map (ι₀ K) = H.map (ι₁ K)

Tactic:
simp only [← cancel_mono (H.map (cylinder.π K)), ← H.map_comp, ι₀_π, H.map_id, ι₁_π]

Goal state after:
Module: Mathlib.CategoryTheory.Quotient

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
r : HomRel C
h : Congruence r
X Y : C
f f' : X ⟶ Y
⊢ (functor r).map f = (functor r).map f' ↔ r f f'

Tactic:
dsimp [functor]
  rw [Equivalence.quot_mk_eq_iff, HomRel.compClosure_eq_self r]
  simpa only [HomRel.compClosure_eq_self r] using h.equivalence

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
a b : ℤ
⊢ shiftFunctorAdd (CochainComplex C ℤ) a b = shiftFunctorAdd' C a b (a + b) ⋯

Tactic:
by
  rw [← CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd, shiftFunctorAdd'_eq]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A B : Triangle C
e : A ≅ B
⊢ (𝟙 A).hom₃ = 𝟙 A.obj₃

Tactic:
id_hom₃

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A B : Triangle C
e : A ≅ B
⊢ 𝟙 A.obj₃ = 𝟙 A.obj₃
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 K ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) a).obj L) n'
n : ℤ
hn : n' + a = n
x : R
⊢ (x • γ).rightUnshift n hn = x • γ.rightUnshift n hn

Tactic:
by
  change (rightShiftLinearEquiv R K L n a n' hn).symm (x • γ) = _
  apply map_smul

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A B : Triangle C
e : A ≅ B
⊢ (𝟙 B).hom₃ = 𝟙 B.obj₃

Tactic:
id_hom₃

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasShift C ℤ
A B : Triangle C
e : A ≅ B
⊢ 𝟙 B.obj₃ = 𝟙 B.obj₃
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝¹⁰ : Category.{v_1, u_1} C
inst✝⁹ : Category.{v_2, u_2} D
inst✝⁸ : Category.{v_3, u_3} E
F : C ⥤ D
G : D ⥤ E
H : C ⥤ E
e : F ⋙ G ≅ H
inst✝⁷ : G.Full
inst✝⁶ : G.Faithful
A : Type u_5
inst✝⁵ : AddMonoid A
inst✝⁴ : HasShift C A
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : G.CommShift A
inst✝ : H.CommShift A
a : A
X : C
⊢ G.map ((iso e a).hom.app X) =
    e.hom.app ((shiftFunctor C a).obj X) ≫
      (commShiftIso H a).hom.app X ≫ (shiftFunctor E a).map (e.inv.app X) ≫ (commShiftIso G a).inv.app (F.obj X)

Tactic:
by
  have h : ((whiskeringRight C D E).obj G).map (iso e a).hom = _ :=
    Functor.map_preimage _ _
  simpa using congr_app h X

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

Goal state before:
pp: V₁ : Type u₁
V₂ : Type u₂
V₃ : Type u₃
inst✝² : Quiver V₁
inst✝¹ : Quiver V₂
inst✝ : Quiver V₃
F : V₁ ⥤q V₂
G : V₂ ⥤q V₃
X✝ Y✝ : Paths V₁
f : X✝ ⟶ Y✝
⊢ (freeMap (F ⋙q G)).map f ≫ (Iso.refl ((freeMap (F ⋙q G)).obj Y✝)).hom =
    (Iso.refl ((freeMap (F ⋙q G)).obj X✝)).hom ≫ (freeMap F ⋙ freeMap G).map f

Tactic:
by
    dsimp
    simp only [Category.comp_id, Category.id_comp, Prefunctor.mapPath_comp_apply]

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_4
inst✝¹ : Category.{v_4, u_4} E
L' : C ⥤ E
eq : D ≌ E
inst✝ : L.IsLocalization W
e : L ⋙ eq.functor ≅ L'
⊢ W.IsInvertedBy (L ⋙ eq.functor)

Tactic:
exact MorphismProperty.IsInvertedBy.of_comp W L (Localization.inverts L W) eq.functor

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

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
a b : A
X : C
| F'.map
      ((commShiftIso L b).inv.app ((shiftFunctor C a).obj X) ≫
        L.map ((shiftFunctorAdd C a b).inv.app X) ≫ L.map ((shiftFunctorAdd C a b).hom.app X)) ≫
    (Lifting.iso L W F F').hom.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫
      (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
        (shiftFunctor E b).map ((commShiftIso F a).hom.app X) ≫
          (shiftFunctorAdd E a b).inv.app (F.obj X) ≫ (shiftFunctor E (a + b)).map ((Lifting.iso L W F F').inv.app X)

Tactic:
← map_comp,

Goal state after:
C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
a b : A
X : C
| F'.map
      ((commShiftIso L b).inv.app ((shiftFunctor C a).obj X) ≫
        L.map ((shiftFunctorAdd C a b).inv.app X ≫ (shiftFunctorAdd C a b).hom.app X)) ≫
    (Lifting.iso L W F F').hom.app ((shiftFunctor C b).obj ((shiftFunctor C a).obj X)) ≫
      (commShiftIso F b).hom.app ((shiftFunctor C a).obj X) ≫
        (shiftFunctor E b).map ((commShiftIso F a).hom.app X) ≫
          (shiftFunctorAdd E a b).inv.app (F.obj X) ≫ (shiftFunctor E (a + b)).map ((Lifting.iso L W F F').inv.app X)
Module: Mathlib.CategoryTheory.Shift.Localization

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
a b : A
X : C
| F'.map
      ((shiftFunctor D b).map ((commShiftIso L a).hom.app X) ≫ (shiftFunctor D b).map ((commShiftIso L a).inv.app X)) ≫
    (commShiftOfLocalization.iso L W F F' b).hom.app (L.obj ((shiftFunctor C a).obj X)) ≫
      (shiftFunctor E b).map ((Lifting.iso L W F F').hom.app ((shiftFunctor C a).obj X)) ≫
        (shiftFunctor E b).map ((commShiftIso F a).hom.app X) ≫
          (shiftFunctor E b).map ((shiftFunctor E a).map ((Lifting.iso L W F F').inv.app X)) ≫
            (shiftFunctorAdd E a b).inv.app (F'.obj (L.obj X))

Tactic:
← map_comp,

Goal state after:
C : Type u₁
D : Type u₂
inst✝¹⁰ : Category.{v₁, u₁} C
inst✝⁹ : Category.{v₂, u₂} D
E : Type u₃
inst✝⁸ : Category.{v₃, u₃} E
L : C ⥤ D
W : MorphismProperty C
inst✝⁷ : L.IsLocalization W
A : Type w
inst✝⁶ : AddMonoid A
inst✝⁵ : HasShift C A
F : C ⥤ E
F' : D ⥤ E
inst✝⁴ : Lifting L W F F'
inst✝³ : HasShift D A
inst✝² : HasShift E A
inst✝¹ : L.CommShift A
inst✝ : F.CommShift A
a b : A
X : C
| F'.map ((shiftFunctor D b).map ((commShiftIso L a).hom.app X ≫ (commShiftIso L a).inv.app X)) ≫
    (commShiftOfLocalization.iso L W F F' b).hom.app (L.obj ((shiftFunctor C a).obj X)) ≫
      (shiftFunctor E b).map ((Lifting.iso L W F F').hom.app ((shiftFunctor C a).obj X)) ≫
        (shiftFunctor E b).map ((commShiftIso F a).hom.app X) ≫
          (shiftFunctor E b).map ((shiftFunctor E a).map ((Lifting.iso L W F F').inv.app X)) ≫
            (shiftFunctorAdd E a b).inv.app (F'.obj (L.obj X))
Module: Mathlib.Order.TransfiniteIteration

Goal state before:
pp: 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 : J
hk : IsMin ⊥
hkj : k ≤ ⊥
⊢ k = ⊥

Tactic:
by simpa using hkj

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

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : ObjectProperty C
H : ∀ (J : Type w) [Finite J], P.IsClosedUnderColimitsOfShape (Discrete J)
J : Type
x✝ : Finite J
⊢ P.IsClosedUnderColimitsOfShape (Discrete J)

Tactic:
by
    rw [P.isClosedUnderColimitsOfShape_iff_of_equivalence
      (Discrete.equivalence (equivShrink.{w} _))]
    exact H _

Goal state after:
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
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ { X₁ := (F.shift n₀).obj T.obj₂, X₂ := (F.shift n₀).obj T.obj₃, X₃ := (F.shift n₁).obj T.obj₁,
      f := (F.shift n₀).map T.mor₂, g := F.homologySequenceδ T n₀ n₁ h, zero := ⋯ }.Exact

Tactic:
refine ShortComplex.exact_of_iso ?_ (F.homologySequence_exact₂ _ (rot_of_distTriang _ hT) n₀)

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
T : Triangle C
hT : T ∈ distinguishedTriangles
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ { X₁ := (F.shift n₀).obj T.rotate.obj₁, X₂ := (F.shift n₀).obj T.rotate.obj₂, X₃ := (F.shift n₀).obj T.rotate.obj₃,
      f := (F.shift n₀).map T.rotate.mor₁, g := (F.shift n₀).map T.rotate.mor₂, zero := ⋯ } ≅
    { X₁ := (F.shift n₀).obj T.obj₂, X₂ := (F.shift n₀).obj T.obj₃, X₃ := (F.shift n₁).obj T.obj₁,
      f := (F.shift n₀).map T.mor₂, g := F.homologySequenceδ T n₀ n₁ h, zero := ⋯ }
Module: Mathlib.CategoryTheory.Shift.ShiftedHom

Goal state before:
pp: C : Type u_1
inst✝¹¹ : Category.{v_1, u_1} C
D : Type u_2
inst✝¹⁰ : Category.{v_2, u_2} D
M : Type u_4
inst✝⁹ : AddMonoid M
inst✝⁸ : HasShift C M
inst✝⁷ : HasShift D M
X Y : C
R : Type u_5
inst✝⁶ : Ring R
inst✝⁵ : Preadditive C
inst✝⁴ : Linear R C
inst✝³ : Preadditive D
inst✝² : Linear R D
r : R
a : M
α : ShiftedHom X Y a
F : C ⥤ D
inst✝¹ : F.CommShift M
inst✝ : Functor.Linear R F
⊢ (r • α).map F = r • α.map F

Tactic:
simp [ShiftedHom.map, F.map_smul]

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
⊢ 1 • (f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n) = 0

Tactic:
rw [← Int.units_mul_self n.negOnePow, mul_smul, comp_id, hf, smul_zero]

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

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0

Tactic:
have h₁ : f' = 0 := by
      apply Cofork.IsColimit.hom_ext hcocone
      simp only [hf', comp_zero, K.shape _ _ hjk]

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
h₁ : f' = 0
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomology

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
h₁ : f' = 0
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0

Tactic:
have h₂ : f'' = 0 := by
      apply Cofork.IsColimit.hom_ext hcocone
      rw [← cancel_epi (extendXIso K e hj').hom, hf'', comp_zero, comp_zero,
        K.extend_d_from_eq_zero e j' k' j hj']
      rw [hk]
      exact hjk

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝² : Category.{v_1, u_3} C
inst✝¹ : HasZeroMorphisms C
inst✝ : HasZeroObject C
K : HomologicalComplex C c
e : c.Embedding c'
i j k : ι
j' k' : ι'
hj' : e.f j = j'
hk : c.next j = k
hk' : c'.next j' = k'
cocone : CokernelCofork (K.d i j)
hcocone : IsColimit cocone
W : C
f' : cocone.pt ⟶ K.X k
hf' : Cofork.π cocone ≫ f' = K.d j k
f'' : cocone.pt ⟶ (K.extend e).X k'
hf'' : (K.extendXIso e hj').hom ≫ Cofork.π cocone ≫ f'' = (K.extend e).d j' k'
φ : W ⟶ cocone.pt
hjk : ¬c.Rel j k
h₁ : f' = 0
h₂ : f'' = 0
⊢ φ ≫ f' = 0 ↔ φ ≫ f'' = 0
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
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i

pp: 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 : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i

Tactic:
· simp [truncGE'Map_f_eq_opcyclesMap _ _ hi rfl, opcyclesMap_comp]

Goal state after:
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 : ι
hi : ¬e.BoundaryGE i
⊢ (truncGE'Map (φ ≫ φ') e).f i = (truncGE'Map φ e ≫ truncGE'Map φ' e).f i
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✝ : HasZeroObject C
i' : ι'
hi' : ∃ i, e.f i = i'
⊢ Epi ((K.πTruncGE e).f i')

pp: case neg
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✝ : HasZeroObject C
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ Epi ((K.πTruncGE e).f i')

Tactic:
· obtain ⟨i, hi⟩ := hi'
    dsimp [πTruncGE]
    rw [e.epi_liftExtend_f_iff _ _ hi]
    infer_instance

Goal state after:
case neg
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✝ : HasZeroObject C
i' : ι'
hi' : ¬∃ i, e.f i = i'
⊢ Epi ((K.πTruncGE e).f i')