mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.GradedObject.Unitor

Goal state before:
pp: C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Zero I
inst✝⁴ : DecidableEq I
inst✝³ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝² : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
inst✝¹ : (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).HasMap p
inst✝ : (((mapBifunctor F J I).obj X').obj ((single₀ I).obj Y)).HasMap p
j : J
⊢ φ j ≫
      e.inv.app (X' j) ≫
        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv ≫ ιMapBifunctorMapObj F p X' ((single₀ I).obj Y) j 0 j ⋯ =
    e.inv.app (X j) ≫
      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv ≫
        (F.map (φ j)).app ((single₀ I).obj Y 0) ≫
          (F.obj (X' j)).map (𝟙 ((single₀ I).obj Y) 0) ≫ ιMapBifunctorMapObj F p X' ((single₀ I).obj Y) j 0 j ⋯

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Zero I
inst✝⁴ : DecidableEq I
inst✝³ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝² : ∀ (X : D), PreservesColimit (Functor.empty C) (F.obj X)
p : J × I → J
hp : ∀ (j : J), p (j, 0) = j
X X' : GradedObject J D
φ : X ⟶ X'
inst✝¹ : (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).HasMap p
inst✝ : (((mapBifunctor F J I).obj X').obj ((single₀ I).obj Y)).HasMap p
j : J
⊢ φ j ≫
      e.inv.app (X' j) ≫
        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv ≫ ιMapBifunctorMapObj F p X' ((single₀ I).obj Y) j 0 j ⋯ =
    e.inv.app (X j) ≫
      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv ≫
        (F.map (φ j)).app ((single₀ I).obj Y 0) ≫
          (F.obj (X' j)).map (𝟙 ((single₀ I).obj Y 0)) ≫ ιMapBifunctorMapObj F p X' ((single₀ I).obj Y) j 0 j ⋯
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : MonoidalCategory C
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : (curriedTensor C).Additive
inst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝⁸ : AddMonoid I
c : ComplexShape I
inst✝⁷ : c.TensorSigns
inst✝⁶ : ∀ (X₁ X₂ : GradedObject I C), X₁.HasTensor X₂
inst✝⁵ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
inst✝⁴ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
inst✝³ : ∀ (X₁ X₂ X₃ X₄ : GradedObject I C), X₁.HasTensor₄ObjExt X₂ X₃ X₄
inst✝² : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝¹ : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝ : DecidableEq I
K : HomologicalComplex C c
⊢ (forget C c).map (λ_ K).hom =
    (((Iso.refl ((forget C c).obj (𝟙_ (HomologicalComplex C c)) ⊗ (forget C c).obj K)).symm ≪≫
          ((tensorUnitIso C c).symm ⊗ᵢ Iso.refl ((forget C c).obj K))) ≪≫
        λ_ ((forget C c).obj K)).hom

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : MonoidalCategory C
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : (curriedTensor C).Additive
inst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝⁸ : AddMonoid I
c : ComplexShape I
inst✝⁷ : c.TensorSigns
inst✝⁶ : ∀ (X₁ X₂ : GradedObject I C), X₁.HasTensor X₂
inst✝⁵ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
inst✝⁴ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
inst✝³ : ∀ (X₁ X₂ X₃ X₄ : GradedObject I C), X₁.HasTensor₄ObjExt X₂ X₃ X₄
inst✝² : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝¹ : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝ : DecidableEq I
K : HomologicalComplex C c
⊢ (forget C c).map (λ_ K).hom =
    (𝟙 ((forget C c).obj (𝟙_ (HomologicalComplex C c)) ⊗ (forget C c).obj K) ≫
        ((tensorUnitIso C c).inv ⊗ₘ 𝟙 ((forget C c).obj K))) ≫
      (λ_ ((forget C c).obj K)).hom
Module: Mathlib.Algebra.Homology.Monoidal

Goal state before:
pp: C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : MonoidalCategory C
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : (curriedTensor C).Additive
inst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝⁸ : AddMonoid I
c : ComplexShape I
inst✝⁷ : c.TensorSigns
inst✝⁶ : ∀ (X₁ X₂ : GradedObject I C), X₁.HasTensor X₂
inst✝⁵ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
inst✝⁴ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
inst✝³ : ∀ (X₁ X₂ X₃ X₄ : GradedObject I C), X₁.HasTensor₄ObjExt X₂ X₃ X₄
inst✝² : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝¹ : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝ : DecidableEq I
K : HomologicalComplex C c
⊢ (forget C c).map (ρ_ K).hom =
    (((Iso.refl ((forget C c).obj K ⊗ (forget C c).obj (𝟙_ (HomologicalComplex C c)))).symm ≪≫
          (Iso.refl ((forget C c).obj K) ⊗ᵢ (tensorUnitIso C c).symm)) ≪≫
        ρ_ ((forget C c).obj K)).hom

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹⁴ : Category.{v_1, u_1} C
inst✝¹³ : MonoidalCategory C
inst✝¹² : Preadditive C
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : (curriedTensor C).Additive
inst✝⁹ : ∀ (X₁ : C), ((curriedTensor C).obj X₁).Additive
I : Type u_2
inst✝⁸ : AddMonoid I
c : ComplexShape I
inst✝⁷ : c.TensorSigns
inst✝⁶ : ∀ (X₁ X₂ : GradedObject I C), X₁.HasTensor X₂
inst✝⁵ : ∀ (X₁ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).obj X₁)
inst✝⁴ : ∀ (X₂ : C), PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)
inst✝³ : ∀ (X₁ X₂ X₃ X₄ : GradedObject I C), X₁.HasTensor₄ObjExt X₂ X₃ X₄
inst✝² : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensor₁₂Tensor X₂ X₃
inst✝¹ : ∀ (X₁ X₂ X₃ : GradedObject I C), X₁.HasGoodTensorTensor₂₃ X₂ X₃
inst✝ : DecidableEq I
K : HomologicalComplex C c
⊢ (forget C c).map (ρ_ K).hom =
    (𝟙 ((forget C c).obj K ⊗ (forget C c).obj (𝟙_ (HomologicalComplex C c))) ≫
        (𝟙 ((forget C c).obj K) ⊗ₘ (tensorUnitIso C c).inv)) ≫
      (ρ_ ((forget C c).obj K)).hom
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

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
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ Mono (X.kernelSequenceCycles f g n₀ n₁ hn₁).f

Tactic:
dsimp

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
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ Mono (X.iCycles f g n₀)
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

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
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ Epi (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g

Tactic:
dsimp

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
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ Epi (X.pOpcycles f g n₁)
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

Goal state before:
pp: case h₀
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ app' (twoδ₁Toδ₀ f g fg h ≫ homMk₁ (α.app 1) (α.app 2) ⋯) 0 homMk₁._proof_4 =
    app' (β ≫ twoδ₁Toδ₀ f' g' fg' h') 0 homMk₁._proof_4

Tactic:
dsimp

Goal state after:
case h₀
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ f ≫ α.app 1 = β.app 0 ≫ f'
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

Goal state before:
pp: case h₁
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ app' (twoδ₁Toδ₀ f g fg h ≫ homMk₁ (α.app 1) (α.app 2) ⋯) 1 homMk₁._proof_5 =
    app' (β ≫ twoδ₁Toδ₀ f' g' fg' h') 1 homMk₁._proof_5

Tactic:
dsimp

Goal state after:
case h₁
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ 𝟙 k ≫ α.app 2 = β.app 1 ≫ 𝟙 k'
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

Goal state before:
pp: case h₁
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ app' (homMk₁ (α.app 0) (α.app 1) ⋯ ≫ twoδ₂Toδ₁ f' g' fg' h') 1 homMk₁._proof_5 =
    app' (twoδ₂Toδ₁ f g fg h ≫ β) 1 homMk₁._proof_5

Tactic:
dsimp

Goal state after:
case h₁
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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
α : mk₂ f g ⟶ mk₂ f' g'
β : mk₁ fg ⟶ mk₁ fg'
n : ℤ
hβ₀ : β.app 0 = α.app 0
hβ₁ : β.app 1 = α.app 2
⊢ α.app 1 ≫ g' = g ≫ β.app 1
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
n : ℤ
⊢ Epi (X.cokernelSequenceCycles f g fg h n).g

Tactic:
dsimp

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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
n : ℤ
⊢ Epi (X.toCycles f g fg h n)
Module: Mathlib.Algebra.Homology.SpectralObject.Cycles

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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
n : ℤ
⊢ Mono (X.kernelSequenceOpcycles f g fg h n).f

Tactic:
dsimp

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
i' j' k' : ι
f' : i' ⟶ j'
g' : j' ⟶ k'
i'' j'' k'' : ι
f'' : i'' ⟶ j''
g'' : j'' ⟶ k''
fg : i ⟶ k
h : f ≫ g = fg
fg' : i' ⟶ k'
h' : f' ≫ g' = fg'
n : ℤ
⊢ Mono (X.fromOpcycles f g fg h n)
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: case refine_1
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A : C
z :
  A ⟶
    { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
        f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.X₂
hz :
  z ≫
      { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
          f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.g =
    0
⊢ (z ≫ X.iCycles f₂ f₃ n₀) ≫ X.δ f₁₂ f₃ n₀ n₁ hn₁ = 0

Tactic:
dsimp

Goal state after:
case refine_1
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A : C
z :
  A ⟶
    { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
        f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.X₂
hz :
  z ≫
      { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
          f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.g =
    0
⊢ (z ≫ X.iCycles f₂ f₃ n₀) ≫ X.δ f₁₂ f₃ n₀ n₁ hn₁ = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Differentials

Goal state before:
pp: case refine_2
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A : C
z :
  A ⟶
    { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
        f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.X₂
hz :
  z ≫
      { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
          f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.g =
    0
⊢ 𝟙 A ≫ z =
    X.liftCycles f₁₂ f₃ n₀ n₁ hn₁ (z ≫ X.iCycles f₂ f₃ n₀) ⋯ ≫
      { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
          f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.f

Tactic:
dsimp

Goal state after:
case refine_2
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A : C
z :
  A ⟶
    { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
        f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.X₂
hz :
  z ≫
      { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁,
          f := X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.g =
    0
⊢ 𝟙 A ≫ z =
    X.liftCycles f₁₂ f₃ n₀ n₁ hn₁ (z ≫ X.iCycles f₂ f₃ n₀) ⋯ ≫
      X.cyclesMap f₁₂ f₃ f₂ f₃ (threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀
Module: Mathlib.Algebra.Homology.SpectralObject.EpiMono

Goal state before:
pp: case h₂
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₁ n₂ n₃ : ℤ
h : (X.H n₁).map (twoδ₁Toδ₀ f₃ f₄ f₃₄ h₃₄) = 0
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ IsIso (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯).τ₂

Tactic:
dsimp

Goal state after:
case h₂
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₁ n₂ n₃ : ℤ
h : (X.H n₁).map (twoδ₁Toδ₀ f₃ f₄ f₃₄ h₃₄) = 0
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ IsIso ((X.H n₂).map (homMk₁ (𝟙 ((mk₃ f₁ f₂ f₃).obj 1)) (𝟙 ((mk₃ f₁ f₂ f₃).obj 2)) ⋯))
Module: Mathlib.Algebra.Homology.SpectralObject.EpiMono

Goal state before:
pp: case h₃
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₁ n₂ n₃ : ℤ
h : (X.H n₁).map (twoδ₁Toδ₀ f₃ f₄ f₃₄ h₃₄) = 0
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ Mono (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃₄ (fourδ₄Toδ₃ f₁ f₂ f₃ f₄ f₃₄ h₃₄) n₁ n₂ n₃ ⋯ ⋯).τ₃

Tactic:
dsimp

Goal state after:
case h₃
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₀ i₁ i₂ i₃ i₄ : ι
f₁ : i₀ ⟶ i₁
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
n₁ n₂ n₃ : ℤ
h : (X.H n₁).map (twoδ₁Toδ₀ f₃ f₄ f₃₄ h₃₄) = 0
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ Mono ((X.H n₃).map (homMk₁ (𝟙 ((mk₃ f₁ f₂ f₃).obj 0)) (𝟙 ((mk₃ f₁ f₂ f₃).obj 1)) ⋯))
Module: Mathlib.Algebra.Homology.SpectralObject.EpiMono

Goal state before:
pp: case h₁
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ Epi (X.shortComplexMap f₂₃ f₄ f₅ f₃ f₄ f₅ (fourδ₁Toδ₀ f₂ f₃ f₄ f₅ f₂₃ h₂₃) n₀ n₁ n₂ ⋯ ⋯).τ₁

Tactic:
dsimp

Goal state after:
case h₁
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ Epi ((X.H n₀).map (homMk₁ (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 2)) (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 3)) ⋯))
Module: Mathlib.Algebra.Homology.SpectralObject.EpiMono

Goal state before:
pp: case h₂
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso (X.shortComplexMap f₂₃ f₄ f₅ f₃ f₄ f₅ (fourδ₁Toδ₀ f₂ f₃ f₄ f₅ f₂₃ h₂₃) n₀ n₁ n₂ ⋯ ⋯).τ₂

Tactic:
dsimp

Goal state after:
case h₂
C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Category.{v_2, u_2} ι
X : SpectralObject C ι
i₁ i₂ i₃ i₄ i₅ : ι
f₂ : i₁ ⟶ i₂
f₃ : i₂ ⟶ i₃
f₄ : i₃ ⟶ i₄
f₅ : i₄ ⟶ i₅
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ n₂ : ℤ
h : (X.H n₂).map (twoδ₂Toδ₁ f₂ f₃ f₂₃ h₂₃) = 0
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ IsIso ((X.H n₁).map (homMk₁ (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 1)) (𝟙 ((mk₃ f₂₃ f₄ f₅).obj 2)) ⋯))
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
i j k l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A✝ : C
z₀ :
  A✝ ⟶
    { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
        g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.X₂
hz₀ :
  z₀ ≫
      { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
          g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.g =
    0
A₁ : C
π₁ : A₁ ⟶ A✝
w✝ : Epi π₁
z₁ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁)
hz₁ : π₁ ≫ z₀ = z₁ ≫ X.pOpcycles f₁ f₂ n₁
⊢ z₁ ≫ (X.cokernelSequenceOpcycles f₁ f₂₃ n₀ n₁ hn₁).g = 0

Tactic:
dsimp

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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
A✝ : C
z₀ :
  A✝ ⟶
    { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
        g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.X₂
hz₀ :
  z₀ ≫
      { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁,
          g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.g =
    0
A₁ : C
π₁ : A₁ ⟶ A✝
w✝ : Epi π₁
z₁ : A₁ ⟶ (X.H n₁).obj (mk₁ f₁)
hz₁ : π₁ ≫ z₀ = z₁ ≫ X.pOpcycles f₁ f₂ n₁
⊢ z₁ ≫ X.pOpcycles f₁ f₂₃ n₁ = 0
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case h₁
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃ (𝟙 (mk₃ f₁ f₂ f₃)) n₀ n₁ n₂ hn₁ hn₂).τ₁ =
    (𝟙 (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯)).τ₁

Tactic:
dsimp

Goal state after:
case h₁
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.H n₀).map (homMk₁ (𝟙 ((mk₃ f₁ f₂ f₃).obj 2)) (𝟙 ((mk₃ f₁ f₂ f₃).obj 3)) ⋯) = 𝟙 ((X.H n₀).obj (mk₁ f₃))
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case h₂
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃ (𝟙 (mk₃ f₁ f₂ f₃)) n₀ n₁ n₂ hn₁ hn₂).τ₂ =
    (𝟙 (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯)).τ₂

Tactic:
dsimp

Goal state after:
case h₂
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.H n₁).map (homMk₁ (𝟙 ((mk₃ f₁ f₂ f₃).obj 1)) (𝟙 ((mk₃ f₁ f₂ f₃).obj 2)) ⋯) = 𝟙 ((X.H n₁).obj (mk₁ f₂))
Module: Mathlib.Algebra.Homology.SpectralObject.Page

Goal state before:
pp: case h₃
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.shortComplexMap f₁ f₂ f₃ f₁ f₂ f₃ (𝟙 (mk₃ f₁ f₂ f₃)) n₀ n₁ n₂ hn₁ hn₂).τ₃ =
    (𝟙 (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯)).τ₃

Tactic:
dsimp

Goal state after:
case h₃
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
⊢ (X.H n₂).map (homMk₁ (𝟙 ((mk₃ f₁ f₂ f₃).obj 0)) (𝟙 ((mk₃ f₁ f₂ f₃).obj 1)) ⋯) = 𝟙 ((X.H n₂).obj (mk₁ f₁))