mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.DerivabilityStructure.OfLocalizedEquivalences

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹¹ : Category.{v_1, u_1} C₁
inst✝¹⁰ : Category.{v_2, u_2} C₂
inst✝⁹ : Category.{v_3, u_3} D₁
inst✝⁸ : Category.{v_4, u_4} D₂
W₁ : MorphismProperty C₁
W₁' : MorphismProperty D₁
W₂ : MorphismProperty C₂
W₂' : MorphismProperty D₂
T : LocalizerMorphism W₁ W₂
L : LocalizerMorphism W₁ W₁'
R : LocalizerMorphism W₂ W₂'
B : LocalizerMorphism W₁' W₂'
inst✝⁷ : W₂'.RespectsIso
inst✝⁶ : L.IsLocalizedEquivalence
inst✝⁵ : R.IsLocalizedEquivalence
inst✝⁴ : R.functor.EssSurj
inst✝³ : L.functor.EssSurj
inst✝² : R.functor.Full
inst✝¹ : R.IsInduced
iso : T.functor ⋙ R.functor ≅ L.functor ⋙ B.functor
inst✝ : TwoSquare.GuitartExact iso.inv
x✝ : B.IsLeftDerivabilityStructure
X₂ : C₂
⊢ Nonempty (T.LeftResolution X₂)

Tactic:
by
    let ρ : B.LeftResolution (R.functor.obj X₂) := Classical.arbitrary _
    exact ⟨{
      X₁ := L.functor.objPreimage ρ.X₁
      w :=
        R.functor.preimage (iso.hom.app _ ≫
          B.functor.map (L.functor.objObjPreimageIso ρ.X₁).hom ≫ ρ.w)
      hw := by
        simp only [← R.inverseImage_eq, Functor.comp_obj,
          MorphismProperty.inverseImage_iff, Functor.map_preimage]
        refine (W₂'.arrow_mk_iso_iff ?_).2 ρ.hw
        exact Arrow.isoMk (iso.app _ ≪≫ B.functor.mapIso (L.functor.objObjPreimageIso ρ.X₁))
          (Iso.refl _) }⟩

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

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
inst✝⁴ : HasExt C
K L : CochainComplex C ℤ
a b : ℤ
inst✝³ : K.IsGE a
inst✝² : K.IsLE a
inst✝¹ : L.IsGE b
inst✝ : L.IsLE b
this : (C : Type u) → [inst : Category.{v, u} C] → [inst_1 : Abelian C] → HasDerivedCategory C :=
  HasDerivedCategory.standard
X : C
eX : Q.obj K ≅ (singleFunctor C a).obj X
Y : C
eY : Q.obj L ≅ (singleFunctor C b).obj Y
p q : ℤ
φ₁ φ₂ : (shiftFunctor (DerivedCategory C) p).obj (Q.obj K) ⟶ (shiftFunctor (DerivedCategory C) q).obj (Q.obj L)
h :
  (fun φ ↦
        ((singleFunctors C).shiftIso p (a - p) a ⋯).inv.app X ≫
          (shiftFunctor (DerivedCategory C) p).map eX.inv ≫
            φ ≫ (shiftFunctor (DerivedCategory C) q).map eY.hom ≫ ((singleFunctors C).shiftIso q (b - q) b ⋯).hom.app Y)
      φ₁ =
    (fun φ ↦
        ((singleFunctors C).shiftIso p (a - p) a ⋯).inv.app X ≫
          (shiftFunctor (DerivedCategory C) p).map eX.inv ≫
            φ ≫ (shiftFunctor (DerivedCategory C) q).map eY.hom ≫ ((singleFunctors C).shiftIso q (b - q) b ⋯).hom.app Y)
      φ₂
⊢ φ₁ = φ₂

Tactic:
simpa using

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Abelian C
inst✝⁴ : HasExt C
K L : CochainComplex C ℤ
a b : ℤ
inst✝³ : K.IsGE a
inst✝² : K.IsLE a
inst✝¹ : L.IsGE b
inst✝ : L.IsLE b
this : (C : Type u) → [inst : Category.{v, u} C] → [inst_1 : Abelian C] → HasDerivedCategory C :=
  HasDerivedCategory.standard
X : C
eX : Q.obj K ≅ (singleFunctor C a).obj X
Y : C
eY : Q.obj L ≅ (singleFunctor C b).obj Y
p q : ℤ
φ₁ φ₂ : (shiftFunctor (DerivedCategory C) p).obj (Q.obj K) ⟶ (shiftFunctor (DerivedCategory C) q).obj (Q.obj L)
h :
  (fun φ ↦
        ((singleFunctors C).shiftIso p (a - p) a ⋯).inv.app X ≫
          (shiftFunctor (DerivedCategory C) p).map eX.inv ≫
            φ ≫ (shiftFunctor (DerivedCategory C) q).map eY.hom ≫ ((singleFunctors C).shiftIso q (b - q) b ⋯).hom.app Y)
      φ₁ =
    (fun φ ↦
        ((singleFunctors C).shiftIso p (a - p) a ⋯).inv.app X ≫
          (shiftFunctor (DerivedCategory C) p).map eX.inv ≫
            φ ≫ (shiftFunctor (DerivedCategory C) q).map eY.hom ≫ ((singleFunctors C).shiftIso q (b - q) b ⋯).hom.app Y)
      φ₂
⊢ φ₁ = φ₂
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective

Goal state before:
pp: case zero
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughProjectives C
inst✝ : F.PreservesProjectiveObjects
Y X : C
⊢ Function.Bijective ⇑(F.mapExtAddHom X Y 0)

Tactic:
simpa using

Goal state after:
case zero
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughProjectives C
inst✝ : F.PreservesProjectiveObjects
Y X : C
⊢ Function.Bijective F.map
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.MapBijective

Goal state before:
pp: case zero
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughInjectives C
inst✝ : F.PreservesInjectiveObjects
X Y : C
⊢ Function.Bijective ⇑(F.mapExtAddHom X Y 0)

Tactic:
simpa using

Goal state after:
case zero
C : Type u
inst✝¹² : Category.{v, u} C
inst✝¹¹ : Abelian C
D : Type u'
inst✝¹⁰ : Category.{v', u'} D
inst✝⁹ : Abelian D
F : C ⥤ D
inst✝⁸ : F.Additive
inst✝⁷ : PreservesFiniteLimits F
inst✝⁶ : PreservesFiniteColimits F
inst✝⁵ : F.Full
inst✝⁴ : F.Faithful
inst✝³ : HasExt C
inst✝² : HasExt D
inst✝¹ : EnoughInjectives C
inst✝ : F.PreservesInjectiveObjects
X Y : C
⊢ Function.Bijective F.map
Module: Mathlib.Algebra.Homology.DerivedCategory.DerivabilityStructureInjectives

Goal state before:
pp: C : Type u_1
H : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Abelian C
inst✝⁴ : Category.{v_2, u_2} H
inst✝³ : EnoughInjectives C
F : Plus C ⥤ H
inst✝² : HasDerivedCategory C
F' : DerivedCategory.Plus C ⥤ H
α : F ⟶ DerivedCategory.Plus.Qh ⋙ F'
inst✝¹ : F'.IsRightDerivedFunctor α (quasiIso C)
this : ∀ (Y : Plus (InjectiveObject C)), IsIso (α.app ((InjectiveObject.ι C).mapHomotopyCategoryPlus.obj Y))
K : HomologicalComplex C (ComplexShape.up ℤ)
hX : plus C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
inst✝ :
  ∀ (n : ℤ), Injective ({ obj := (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K, property := hX }.obj.as.X n)
⊢ CochainComplex.plus C K

Tactic:
simpa using

Goal state after:
C : Type u_1
H : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Abelian C
inst✝⁴ : Category.{v_2, u_2} H
inst✝³ : EnoughInjectives C
F : Plus C ⥤ H
inst✝² : HasDerivedCategory C
F' : DerivedCategory.Plus C ⥤ H
α : F ⟶ DerivedCategory.Plus.Qh ⋙ F'
inst✝¹ : F'.IsRightDerivedFunctor α (quasiIso C)
this : ∀ (Y : Plus (InjectiveObject C)), IsIso (α.app ((InjectiveObject.ι C).mapHomotopyCategoryPlus.obj Y))
K : HomologicalComplex C (ComplexShape.up ℤ)
hX : plus C ((HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K)
inst✝ :
  ∀ (n : ℤ), Injective ({ obj := (HomotopyCategory.quotient C (ComplexShape.up ℤ)).obj K, property := hX }.obj.as.X n)
⊢ CochainComplex.plus C K
Module: Mathlib.Algebra.Homology.Embedding.StupidTrunc

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✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝¹ : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), e.f i ≠ i'

Tactic:
simpa using

Goal state after:
case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroObject C
K L M : HomologicalComplex C c'
φ : K ⟶ L
φ' : L ⟶ M
e : c.Embedding c'
inst✝¹ : e.IsRelIff
ι'' : Type u_4
c'' : ComplexShape ι''
e' : c''.Embedding c'
inst✝ : K.IsStrictlySupported e'
i' : ι'
hi' : ∀ (i : ι''), e'.f i ≠ i'
hi'' : ¬∃ i, e.f i = i'
⊢ ∀ (i : ι), ¬e.f i = i'
Module: Mathlib.CategoryTheory.DifferentialObject

Goal state before:
pp: S : Type u_1
inst✝⁵ : AddMonoidWithOne S
C : Type (u + 1)
inst✝⁴ : LargeCategory C
inst✝³ : HasZeroMorphisms C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝¹ : ConcreteCategory C FC
inst✝ : HasShift C S
X✝ Y✝ : DifferentialObject S C
f : HomSubtype S C X✝ Y✝
⊢ X✝.d ≫ (shiftFunctor C 1).map (ConcreteCategory.ofHom ↑f) = ConcreteCategory.ofHom ↑f ≫ Y✝.d

Tactic:
simpa using

Goal state after:
S : Type u_1
inst✝⁵ : AddMonoidWithOne S
C : Type (u + 1)
inst✝⁴ : LargeCategory C
inst✝³ : HasZeroMorphisms C
FC : C → C → Type u_2
CC : C → Type u_3
inst✝² : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝¹ : ConcreteCategory C FC
inst✝ : HasShift C S
X✝ Y✝ : DifferentialObject S C
f : HomSubtype S C X✝ Y✝
⊢ X✝.d ≫ (shiftFunctor C 1).map (ConcreteCategory.ofHom ↑f) = ConcreteCategory.ofHom ↑f ≫ Y✝.d
Module: Mathlib.Algebra.Homology.Embedding.Connect

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n✝ m✝ : ℤ
n m : ℕ
hnm : Int.ofNat n + 1 ≠ Int.ofNat m
⊢ ¬(ComplexShape.up ℕ).Rel n m

Tactic:
by simp at hnm ⊢; lia

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
K : ChainComplex C ℕ
L : CochainComplex C ℕ
h : ConnectData K L
n✝ m✝ : ℤ
n m : ℕ
hnm : Int.negSucc n + 1 ≠ Int.negSucc m
⊢ ¬(ComplexShape.down ℕ).Rel n m

Tactic:
by simp at hnm ⊢; lia

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{?u.4, u_1} C
inst✝ : HasZeroMorphisms C
n : ℕ
S : ComposableArrows C (n + 3)
hS : S.IsComplex
k : ℕ
hk : k ≤ n
cc : CokernelCofork (S.map' k (k + 1) ⋯ ⋯)
kf : KernelFork (S.map' (k + 2) (k + 3) ⋯ ⋯)
hcc : IsColimit cc
hkf : IsLimit kf
⊢ (S.map' k (k + 1) ⋯ ⋯ ≫ hkf.lift (KernelFork.ofι (S.map' (k + 1) (k + 1 + 1) ⋯ ⋯) ⋯)) ≫ Fork.ι kf = 0 ≫ Fork.ι kf

Tactic:
simpa using

Goal state after:
C : Type u_1
inst✝¹ : Category.{?u.4, u_1} C
inst✝ : HasZeroMorphisms C
n : ℕ
S : ComposableArrows C (n + 3)
hS : S.IsComplex
k : ℕ
hk : k ≤ n
cc : CokernelCofork (S.map' k (k + 1) ⋯ ⋯)
kf : KernelFork (S.map' (k + 2) (k + 3) ⋯ ⋯)
hcc : IsColimit cc
hkf : IsLimit kf
⊢ S.map (homOfLE ⋯) ≫ S.map (homOfLE ⋯) = 0
Module: Mathlib.Algebra.Homology.GrothendieckAbelian

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
ι : Type t
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : Small.{w, t} ι
K L : HomologicalComplex C c
emb : (K ⟶ L) → (i : Shrink.{w, t} ι) → Shrink.{w, v} (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i)) :=
  fun f i ↦ (equivShrink (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i))) (f.f ((equivShrink ι).symm i))
f g : K ⟶ L
h : emb f = emb g
i : Shrink.{w, t} ι
⊢ f.f ((equivShrink ι).symm i) = g.f ((equivShrink ι).symm i)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
ι : Type t
c : ComplexShape ι
inst✝² : HasZeroMorphisms C
inst✝¹ : LocallySmall.{w, v, u} C
inst✝ : Small.{w, t} ι
K L : HomologicalComplex C c
emb : (K ⟶ L) → (i : Shrink.{w, t} ι) → Shrink.{w, v} (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i)) :=
  fun f i ↦ (equivShrink (K.X ((equivShrink ι).symm i) ⟶ L.X ((equivShrink ι).symm i))) (f.f ((equivShrink ι).symm i))
f g : K ⟶ L
h : emb f = emb g
i : Shrink.{w, t} ι
⊢ f.f ((equivShrink ι).symm i) = g.f ((equivShrink ι).symm i)
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p n : ℤ
α : Cocycle ((singleFunctor C p).obj X) K n
q : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
f : X ⟶ K.X q
hf : Cochain.fromSingleMk f h = ↑α
hα : (Cochain.fromSingleMk (f ≫ K.d q q') ⋯).v p q' ⋯ = Cochain.v 0 p q' ⋯
⊢ f ≫ K.d q q' = 0

Tactic:
simpa using

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p n : ℤ
α : Cocycle ((singleFunctor C p).obj X) K n
q : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
f : X ⟶ K.X q
hf : Cochain.fromSingleMk f h = ↑α
hα : (Cochain.fromSingleMk (f ≫ K.d q q') ⋯).v p q' ⋯ = Cochain.v 0 p q' ⋯
⊢ f ≫ K.d q q' = 0
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexSingle

Goal state before:
pp: case mp
C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
K : CochainComplex C ℤ
p q : ℤ
f : X ⟶ K.X q
n : ℤ
h : p + n = q
q' : ℤ
hq' : q + 1 = q'
hf : f ≫ K.d q q' = 0
q'' : ℤ
hq'' : q'' + 1 = q
α : Cochain ((singleFunctor C p).obj X) K (n - 1)
g : X ⟶ K.X q''
hα : Cochain.fromSingleMk (g ≫ K.d q'' q) h = Cochain.fromSingleMk f h
hg : Cochain.fromSingleMk g ⋯ = α
⊢ g ≫ K.d q'' q = f

Tactic:
exact (Cochain.fromSingleEquiv h).symm.injective hα

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
P Q : ProjectiveResolution X
⊢ Homotopy (lift (𝟙 X ≫ 𝟙 X) P P) (𝟙 P.complex)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
P Q : ProjectiveResolution X
⊢ Homotopy (lift (𝟙 X) P P) (𝟙 P.complex)
Module: Mathlib.CategoryTheory.Abelian.Projective.Resolution

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
P Q : ProjectiveResolution X
⊢ Homotopy (lift (𝟙 X ≫ 𝟙 X) Q Q) (𝟙 Q.complex)

Tactic:
simpa using

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
X : C
P Q : ProjectiveResolution X
⊢ Homotopy (lift (𝟙 X) Q Q) (𝟙 Q.complex)
Module: Mathlib.CategoryTheory.Abelian.LeftDerived

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
n : ℕ
X : C
inst✝ : Projective X
⊢ IsZero ((F.leftDerived (n + 1)).obj X)

Tactic:
refine IsZero.of_iso ?_ ((ProjectiveResolution.self X).isoLeftDerivedObj F (n + 1))

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u_1
inst✝⁵ : Category.{v_1, u_1} D
inst✝⁴ : Abelian C
inst✝³ : HasProjectiveResolutions C
inst✝² : Abelian D
F : C ⥤ D
inst✝¹ : F.Additive
n : ℕ
X : C
inst✝ : Projective X
⊢ IsZero
    ((HomologicalComplex.homologyFunctor D (ComplexShape.down ℕ) (n + 1)).obj
      ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj (ProjectiveResolution.self X).complex))
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: C : Type u_1
inst✝⁹ : Category.{v_1, u_1} C
inst✝⁸ : MonoidalCategory C
inst✝⁷ : Preadditive C
inst✝⁶ : HasZeroObject C
inst✝⁵ : (curriedTensor C).Additive
inst✝⁴ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝³ : AddMonoid I
c : ComplexShape I
inst✝² : c.TensorSigns
K : HomologicalComplex C c
inst✝¹ : DecidableEq I
inst✝ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
i j : I
⊢ K.rightUnitor'.inv i ≫ (K.tensorObj (tensorUnit C c)).d i j = K.d i j ≫ K.rightUnitor'.inv j

Tactic:
by
  by_cases hij : c.Rel i j
  · simp only [rightUnitor'_inv, assoc, mapBifunctor.d_eq,
      Preadditive.comp_add, mapBifunctor.ι_D₁, mapBifunctor.ι_D₂,
      tensor_unit_d₂, comp_zero, add_zero]
    rw [mapBifunctor.d₁_eq _ _ _ _ hij _ _ (by simp)]
    dsimp
    simp only [one_smul, whisker_exchange_assoc, whiskerRight_id, assoc, Iso.inv_hom_id_assoc]
  · simp only [shape _ _ _ hij, comp_zero, zero_comp]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Monoidal

Goal state before:
pp: I : Type u
inst✝⁶ : AddMonoid I
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝³ : X₃.HasTensor X₄
inst✝² : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹ : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
j : I
A : C
f g : tensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) j ⟶ A
inst✝ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
H : X₁.HasTensor₄ObjExt X₂ X₃ X₄
h :
  ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j),
    ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g
i₁ i₂₃₄ : I
h' : i₁ + i₂₃₄ = j
i₂ i₃ i₄ : I
h'' : i₂ + i₃ + i₄ = i₂₃₄
hj : i₁ + i₂ + i₃ + i₄ = j
⊢ X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫ ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ f =
    X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫ ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ g

Tactic:
simpa using

Goal state after:
I : Type u
inst✝⁶ : AddMonoid I
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ X₄ : GradedObject I C
inst✝³ : X₃.HasTensor X₄
inst✝² : X₂.HasTensor (tensorObj X₃ X₄)
inst✝¹ : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))
j : I
A : C
f g : tensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) j ⟶ A
inst✝ : X₂.HasGoodTensorTensor₂₃ X₃ X₄
H : X₁.HasTensor₄ObjExt X₂ X₃ X₄
h :
  ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j),
    ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g
i₁ i₂₃₄ : I
h' : i₁ + i₂₃₄ = j
i₂ i₃ i₄ : I
h'' : i₂ + i₃ + i₄ = i₂₃₄
hj : i₁ + i₂ + i₃ + i₄ = j
⊢ X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫ ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ f =
    X₁ i₁ ◁ ιTensorObj₃ X₂ X₃ X₄ i₂ i₃ i₄ i₂₃₄ h'' ≫ ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ g
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
D : Type u_4
I₁ : Type u_5
I₂ : Type u_6
I₃ : Type u_7
J : Type u_8
inst✝¹⁵ : Category.{v_1, u_1} C₁
inst✝¹⁴ : Category.{v_2, u_2} C₂
inst✝¹³ : Category.{v_3, u_3} C₃
inst✝¹² : Category.{v_4, u_4} D
inst✝¹¹ : Zero I₂
inst✝¹⁰ : DecidableEq I₂
inst✝⁹ : HasInitial C₂
F₁ : C₁ ⥤ C₂ ⥤ C₁
F₂ : C₂ ⥤ C₃ ⥤ C₃
G : C₁ ⥤ C₃ ⥤ D
associator : bifunctorComp₁₂ F₁ G ≅ bifunctorComp₂₃ G F₂
X₂ : C₂
e₁ : F₁.flip.obj X₂ ≅ 𝟭 C₁
e₂ : F₂.obj X₂ ≅ 𝟭 C₃
inst✝⁸ : ∀ (X₁ : C₁), PreservesColimit (Functor.empty C₂) (F₁.obj X₁)
inst✝⁷ : ∀ (X₃ : C₃), PreservesColimit (Functor.empty C₂) (F₂.flip.obj X₃)
r : I₁ × I₂ × I₃ → J
π : I₁ × I₃ → J
τ : TriangleIndexData r π
X₁ : GradedObject I₁ C₁
X₃ : GradedObject I₃ C₃
inst✝⁶ : (((mapBifunctor F₁ I₁ I₂).obj X₁).obj ((single₀ I₂).obj X₂)).HasMap τ.p₁₂
inst✝⁵ : (((mapBifunctor G I₁ I₃).obj (mapBifunctorMapObj F₁ τ.p₁₂ X₁ ((single₀ I₂).obj X₂))).obj X₃).HasMap π
inst✝⁴ : (((mapBifunctor F₂ I₂ I₃).obj ((single₀ I₂).obj X₂)).obj X₃).HasMap τ.p₂₃
inst✝³ : (((mapBifunctor G I₁ I₃).obj X₁).obj (mapBifunctorMapObj F₂ τ.p₂₃ ((single₀ I₂).obj X₂) X₃)).HasMap π
inst✝² : HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((single₀ I₂).obj X₂) X₃
inst✝¹ : HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃
inst✝ : (((mapBifunctor G I₁ I₃).obj X₁).obj X₃).HasMap π
triangle :
  ∀ (X₁ : C₁) (X₃ : C₃),
    ((associator.hom.app X₁).app X₂).app X₃ ≫ (G.obj X₁).map (e₂.hom.app X₃) = (G.map (e₁.hom.app X₁)).app X₃
⊢ (mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁ ((single₀ I₂).obj X₂) X₃).hom ≫
      mapBifunctorMapMap G π (𝟙 X₁) (mapBifunctorLeftUnitor F₂ X₂ e₂ τ.p₂₃ ⋯ X₃).hom =
    mapBifunctorMapMap G π (mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ ⋯ X₁).hom (𝟙 X₃)

Tactic:
rw [← cancel_epi ((mapBifunctorMapMap G π
    (mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁).inv (𝟙 X₃)))]
  ext j i₁ i₃ hj
  simp only [categoryOfGradedObjects_comp, ι_mapBifunctorMapMap_assoc,
    mapBifunctorRightUnitor_inv_apply, Functor.id_obj, Functor.map_comp,
    NatTrans.comp_app, categoryOfGradedObjects_id, Functor.map_id, id_comp, assoc,
    ι_mapBifunctorMapMap]
  congr 2
  rw [← ιMapBifunctor₁₂BifunctorMapObj_eq_assoc F₁ G τ.ρ₁₂ _ _ _ i₁ 0 i₃ j
    (by rw [τ.r_zero, hj]) i₁ (by simp), ι_mapBifunctorAssociator_hom_assoc,
    ιMapBifunctorBifunctor₂₃MapObj_eq_assoc G F₂ τ.ρ₂₃ _ _ _ i₁ 0 i₃ j
    (by rw [τ.r_zero, hj]) i₃ (by simp), ι_mapBifunctorMapMap]
  dsimp
  rw [Functor.map_id, NatTrans.id_app, id_comp,
    ← Functor.map_comp_assoc, ← NatTrans.comp_app_assoc, ← Functor.map_comp,
    ι_mapBifunctorLeftUnitor_hom_apply F₂ X₂ e₂ τ.p₂₃ τ.h₃ X₃ i₃,
    ι_mapBifunctorRightUnitor_hom_apply F₁ X₂ e₁ τ.p₁₂ τ.h₁ X₁ i₁]
  dsimp
  simp only [Functor.map_comp, NatTrans.comp_app, ← triangle (X₁ i₁) (X₃ i₃), ← assoc]
  congr 2
  symm
  apply NatTrans.naturality_app (associator.hom.app (X₁ i₁))

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Monoidal

Goal state before:
pp: I : Type u
inst✝² : AddMonoid I
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : MonoidalCategory C
n fst✝¹ snd✝¹ : ℕ
property✝¹ : (fst✝¹, snd✝¹) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}
fst✝ snd✝ : ℕ
property✝ : (fst✝, snd✝) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}
h :
  (fun x ↦
        match x with
        | ⟨(i₁, i₂), hi⟩ => (⟨i₁, ⋯⟩, ⟨i₂, ⋯⟩))
      ⟨(fst✝¹, snd✝¹), property✝¹⟩ =
    (fun x ↦
        match x with
        | ⟨(i₁, i₂), hi⟩ => (⟨i₁, ⋯⟩, ⟨i₂, ⋯⟩))
      ⟨(fst✝, snd✝), property✝⟩
⊢ ⟨(fst✝¹, snd✝¹), property✝¹⟩ = ⟨(fst✝, snd✝), property✝⟩

Tactic:
simpa using

Goal state after:
I : Type u
inst✝² : AddMonoid I
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : MonoidalCategory C
n fst✝¹ snd✝¹ : ℕ
property✝¹ : (fst✝¹, snd✝¹) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}
fst✝ snd✝ : ℕ
property✝ : (fst✝, snd✝) ∈ (fun i ↦ i.1 + i.2) ⁻¹' {n}
h :
  (fun x ↦
        match x with
        | ⟨(i₁, i₂), hi⟩ => (⟨i₁, ⋯⟩, ⟨i₂, ⋯⟩))
      ⟨(fst✝¹, snd✝¹), property✝¹⟩ =
    (fun x ↦
        match x with
        | ⟨(i₁, i₂), hi⟩ => (⟨i₁, ⋯⟩, ⟨i₂, ⋯⟩))
      ⟨(fst✝, snd✝), property✝⟩
⊢ fst✝¹ = fst✝ ∧ snd✝¹ = snd✝
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    X.Ψ f₂ f₃ f₄ n₁ n₂ hn₂

Tactic:
← cancel_epi (X.toCycles f₃ f₄ _ rfl n₁),

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ i₅ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.toCycles f₃ f₄ (f₃ ≫ f₄) ⋯ n₁ ≫
      X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂ ≫ X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃ ≫ X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    X.toCycles f₃ f₄ (f₃ ≫ f₄) ⋯ n₁ ≫ X.Ψ f₂ f₃ f₄ n₁ n₂ hn₂
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
hf : IsIso f
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f g n₀ n₁ hn₁ = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
hf : IsIso f
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f g n₀ n₁ hn₁ = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
hg : IsIso g
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f g n₀ n₁ hn₁ = 0

Tactic:
simpa using

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k : ι
f : i ⟶ j
g : j ⟶ k
hg : IsIso g
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f g n₀ n₁ hn₁ = 0