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

Goal state before:
pp: case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (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'
j' : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j')
h : (j', 0) ∈ p ⁻¹' {j'}
⊢ (F.obj (X j')).map (𝟙 ((single 0).obj Y 0)) ≫ s.inj ⟨(j', 0), ⋯⟩ = s.inj ⟨(j', 0), h⟩

Tactic:
Functor.map_id,

Goal state after:
case pos
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (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'
j' : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j')
h : (j', 0) ∈ p ⁻¹' {j'}
⊢ 𝟙 ((F.obj (X j')).obj ((single 0).obj Y 0)) ≫ s.inj ⟨(j', 0), ⋯⟩ = s.inj ⟨(j', 0), h⟩
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✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (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'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j)
m : (mapBifunctorRightUnitorCofan F Y e p hp X j).pt ⟶ s.pt
hm : ∀ (j_1 : ↑(p ⁻¹' {j})), Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
⊢ m = e.inv.app (X j) ≫ (F.obj (X j)).map (𝟙 Y) ≫ e.hom.app (X j) ≫ m

Tactic:
Functor.map_id,

Goal state after:
C : Type u_1
D : Type u_2
I : Type u_3
J : Type u_4
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : Zero I
inst✝² : DecidableEq I
inst✝¹ : HasInitial C
F : D ⥤ C ⥤ D
Y : C
e : F.flip.obj Y ≅ 𝟭 D
inst✝ : ∀ (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'
j : J
s : Cofan ((((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).mapObjFun p j)
m : (mapBifunctorRightUnitorCofan F Y e p hp X j).pt ⟶ s.pt
hm : ∀ (j_1 : ↑(p ⁻¹' {j})), Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) j_1 ≫ m = s.inj j_1
⊢ m = e.inv.app (X j) ≫ 𝟙 ((F.obj (X j)).obj Y) ≫ e.hom.app (X j) ≫ m
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:
Functor.map_id,

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) ≫
          𝟙 ((F.obj (X' j)).obj ((single₀ I).obj Y 0)) ≫ ιMapBifunctorMapObj F p X' ((single₀ I).obj Y) j 0 j ⋯
Module: Mathlib.Algebra.Homology.SpectralObject.Basic

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_3, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f : i ⟶ j
g : j ⟶ k
h : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
eq : (X.H n₁).map (twoδ₂Toδ₁ g h (g ≫ h) ⋯) ≫ X.δ f (g ≫ h) n₁ n₂ ⋯ = X.δ f g n₁ n₂ ⋯ ≫ (X.H n₂).map (𝟙 (mk₁ f))
⊢ X.δ g h n₀ n₁ hn₁ ≫ X.δ f g n₁ n₂ hn₂ = 0

Tactic:
Functor.map_id,

Goal state after:
C : Type u_1
ι : Type u_2
inst✝² : Category.{u_4, u_1} C
inst✝¹ : Category.{u_3, u_2} ι
inst✝ : Abelian C
X : SpectralObject C ι
i j k l : ι
f : i ⟶ j
g : j ⟶ k
h : k ⟶ l
n₀ n₁ n₂ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
eq : (X.H n₁).map (twoδ₂Toδ₁ g h (g ≫ h) ⋯) ≫ X.δ f (g ≫ h) n₁ n₂ ⋯ = X.δ f g n₁ n₂ ⋯ ≫ 𝟙 ((X.H n₂).obj (mk₁ f))
⊢ X.δ g h n₀ n₁ hn₁ ≫ X.δ f g n₁ n₂ hn₂ = 0
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
n : ℤ
⊢ X.iCycles f g n ≫ (X.H n).map (𝟙 (mk₁ g)) = 𝟙 (X.cycles f g n) ≫ X.iCycles f g n

Tactic:
Functor.map_id,

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
n : ℤ
⊢ X.iCycles f g n ≫ 𝟙 ((X.H n).obj (mk₁ g)) = 𝟙 (X.cycles f g n) ≫ X.iCycles f g 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
n : ℤ
⊢ (X.H n).map (𝟙 (mk₁ f)) ≫ X.pOpcycles f g n = X.pOpcycles f g n ≫ 𝟙 (X.opcycles f g n)

Tactic:
Functor.map_id,

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
n : ℤ
⊢ 𝟙 ((X.H n).obj (mk₁ f)) ≫ X.pOpcycles f g n = X.pOpcycles f g n ≫ 𝟙 (X.opcycles f g n)
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
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
f₄₅ : i₃ ⟶ i₅
h₄₅ : f₄ ≫ f₅ = f₄₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.δ (f₁ ≫ f₂) f₃ n₁ n₂ hn₂ ≫
      (X.H n₂).map (𝟙 (mk₁ (f₁ ≫ f₂))) ≫ X.toCycles f₁ f₂ (f₁ ≫ f₂) ⋯ n₂ ≫ X.πE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    0

Tactic:
Functor.map_id,

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
f₁₂ : i₀ ⟶ i₂
h₁₂ : f₁ ≫ f₂ = f₁₂
f₂₃ : i₁ ⟶ i₃
h₂₃ : f₂ ≫ f₃ = f₂₃
f₃₄ : i₂ ⟶ i₄
h₃₄ : f₃ ≫ f₄ = f₃₄
f₄₅ : i₃ ⟶ i₅
h₄₅ : f₄ ≫ f₅ = f₄₅
n₀ n₁ n₂ n₃ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
⊢ X.δ (f₁ ≫ f₂) f₃ n₁ n₂ hn₂ ≫
      𝟙 ((X.H n₂).obj (mk₁ (f₁ ≫ f₂))) ≫ X.toCycles f₁ f₂ (f₁ ≫ f₂) ⋯ n₂ ≫ X.πE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ =
    0
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
⊢ X.δ f₁ f₂ n₀ n₁ (CategoryTheory.Abelian.SpectralObject.d._proof_1 n₀ n₁ hn₁) ≫
      (X.H n₁).map (𝟙 (mk₁ f₁)) ≫ X.pOpcycles f₁ f₂ n₁ =
    0

Tactic:
Functor.map_id,

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
⊢ X.δ f₁ f₂ n₀ n₁ (CategoryTheory.Abelian.SpectralObject.d._proof_1 n₀ n₁ hn₁) ≫
      𝟙 ((X.H n₁).obj (mk₁ f₁)) ≫ X.pOpcycles f₁ f₂ n₁ =
    0
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
⊢ f₁ ≫ f₂₃ = f₁ ≫ f₂ ≫ f₃

Tactic:
h₂₃

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
⊢ f₁ ≫ f₂₃ = f₁ ≫ f₂₃
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₁₂ : i ⟶ k
h₁₂ : f₁ ≫ f₂ = f₁₂
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ (X.H n₀).map (𝟙 (mk₁ f₃)) ≫ X.δ f₂ f₃ n₀ n₁ ⋯ = X.δ f₂ f₃ n₀ n₁ hn₁

Tactic:
Functor.map_id,

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
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ 𝟙 ((X.H n₀).obj (mk₁ f₃)) ≫ X.δ f₂ f₃ n₀ n₁ ⋯ = X.δ f₂ f₃ n₀ n₁ hn₁
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 l : ι
f₁ : i ⟶ j
f₂ : j ⟶ k
f₃ : k ⟶ l
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f₁ f₂ n₀ n₁ ⋯ ≫ (X.H n₁).map (𝟙 (mk₁ f₁)) = X.δ f₁ f₂ n₀ n₁ hn₁

Tactic:
Functor.map_id,

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
f₂₃ : j ⟶ l
h₂₃ : f₂ ≫ f₃ = f₂₃
n₀ n₁ : ℤ
hn₁ : n₀ + 1 = n₁
⊢ X.δ f₁ f₂ n₀ n₁ ⋯ ≫ 𝟙 ((X.H n₁).obj (mk₁ f₁)) = X.δ f₁ f₂ n₀ n₁ hn₁
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
⊢ z ≫ X.iCycles f₂ f₃ n₀ =
    X.liftCycles f₁₂ f₃ n₀ n₁ hn₁ (z ≫ X.iCycles f₂ f₃ n₀) ⋯ ≫ X.iCycles f₁₂ f₃ n₀ ≫ (X.H n₀).map (𝟙 (mk₁ f₃))

Tactic:
Functor.map_id,

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
⊢ z ≫ X.iCycles f₂ f₃ n₀ =
    X.liftCycles f₁₂ f₃ n₀ n₁ hn₁ (z ≫ X.iCycles f₂ f₃ n₀) ⋯ ≫ X.iCycles f₁₂ f₃ n₀ ≫ 𝟙 ((X.H n₀).obj (mk₁ f₃))
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
H :
  X.pOpcycles f₁ f₂ n₁ ≫ X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁ =
    (X.H n₁).map (𝟙 (mk₁ f₁)) ≫ X.pOpcycles f₁ f₂₃ n₁
⊢ z₁ ≫ X.pOpcycles f₁ f₂₃ n₁ = 0

Tactic:
Functor.map_id,

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
H :
  X.pOpcycles f₁ f₂ n₁ ≫ X.opcyclesMap f₁ f₂ f₁ f₂₃ (threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁ =
    𝟙 ((X.H n₁).obj (mk₁ f₁)) ≫ X.pOpcycles f₁ f₂₃ n₁
⊢ z₁ ≫ X.pOpcycles f₁ f₂₃ n₁ = 0
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.H n₁).map (𝟙 (mk₁ (f₃ ≫ f₄))) ≫ X.δ f₂ (f₃ ≫ f₄) n₁ n₂ hn₂ ≫ X.pOpcycles f₂ f₃ n₂ =
    X.δ f₂ (f₃ ≫ f₄) n₁ n₂ ⋯ ≫ X.pOpcycles f₂ f₃ n₂

Tactic:
Functor.map_id,

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
⊢ 𝟙 ((X.H n₁).obj (mk₁ (f₃ ≫ f₄))) ≫ X.δ f₂ (f₃ ≫ f₄) n₁ n₂ hn₂ ≫ X.pOpcycles f₂ f₃ n₂ =
    X.δ f₂ (f₃ ≫ f₄) n₁ n₂ ⋯ ≫ X.pOpcycles f₂ f₃ n₂
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
q : ℕ
hq : q < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) + 1
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (2 + ↑k, 1 - (2 + ↑k))).Rel (p, ⟨q, hq⟩) pq'
⊢ q ≤ k

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
q : ℕ
hq : q < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) + 1
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (2 + ↑k, 1 - (2 + ↑k))).Rel (p, ⟨q, hq⟩) pq'
this : k < q
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
n : ℤ
k : ℕ
hr : 2 ≤ 2 + ↑k
p : ℤ
t : ℕ
hq : k + 1 + t < l
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨k + 1 + t, hq⟩) + 1
this : k < k + 1 + t
hpq : ∀ (a : ℤ) (b : Fin l), p + (2 + ↑k) = a → ¬↑(k + 1 + t) + (1 - (2 + ↑k)) = ↑↑b
⊢ False

Tactic:
exact hpq _ ⟨t, by lia⟩ rfl (by simp; lia)

Goal state after:
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (r, 1 - r)).Rel pq' (p, ⟨q, hq⟩)
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
⊢ ↑l < ↑q + r

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝³ : Category.{v_1, u_1} C
inst✝² : Abelian C
inst✝¹ : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝ : X.HasSpectralSequence data
l : ℕ
E : SpectralObject C (Fin (l + 1))
r n : ℤ
hr : 2 ≤ r
p : ℤ
q : ℕ
hq : q < l
hpq : ∀ (pq' : ℤ × Fin l), ¬(ComplexShape.spectralSequenceFin l (r, 1 - r)).Rel pq' (p, ⟨q, hq⟩)
hn : n = (coreE₂CohomologicalFin l).deg (p, ⟨q, hq⟩) - 1
this : ↑q + r ≤ ↑l
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
⊢ ↑q - r + 2 ≤ 0

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
this : 0 < ↑q - r + 2
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel pq' (p, q)
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11
⊢ ↑p + ↑q - 1 < ↑q + r - 1

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsFirstQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (r, 1 - r)).Rel pq' (p, q)
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11
this : ↑q + r - 1 ≤ ↑p + ↑q - 1
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
⊢ -↑q - r + 2 ≤ -↑p - ↑q + 1

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel (p, q) pq'
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_3
this : -↑p - ↑q + 1 < -↑q - r + 2
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.HasSpectralSequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel pq' (p, q)
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11
⊢ 0 < -↑q + r - 1

Tactic:
by_contra!

Goal state after:
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : Preorder ι
c : ℤ → ComplexShape κ
r₀ : ℤ
X : SpectralObject C ι
data : SpectralSequenceDataCore ι c r₀
inst✝¹ : X.HasSpectralSequence data
Y : SpectralObject C EInt
inst✝ : Y.IsThirdQuadrant
r : ℤ
p q : ℕ
hpq : ∀ (pq' : ℕ × ℕ), ¬(ComplexShape.spectralSequenceNat (-r, r - 1)).Rel pq' (p, q)
hr : autoParam (2 ≤ r) HasSpectralSequence._auto_11
this : -↑q + r - 1 ≤ 0
⊢ False
Module: Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Goal state before:
pp: case refine_2
C : Type u_1
ι : Type u_2
κ : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : Preorder ι
X : SpectralObject C ι
c : ℤ → ComplexShape κ
r₀ : ℤ
data : SpectralSequenceDataCore ι c r₀
r : ℤ
hr : r₀ ≤ r
pq pq' pq'' : κ
hpq : (c r).Rel pq pq'
hpq' : (c r).Rel pq' pq''
n₀ n₁ n₂ n₃ n₄ : ℤ
hn₁ : n₀ + 1 = n₁
hn₂ : n₁ + 1 = n₂
hn₃ : n₂ + 1 = n₃
hn₄ : n₃ + 1 = n₄
hn₂' : n₂ = data.deg pq'
⊢ (pageXIso X data r hr pq' (data.i₂ pq'') (data.i₃ r pq'' ⋯) (data.i₀ r pq ⋯) (data.i₁ pq) ⋯ ⋯ ⋯ ⋯ n₁ n₂ n₃ hn₂' ⋯
          ⋯).hom ≫
      (X.dShortComplex (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) (homOfLE ⋯) n₀ n₁ n₂ n₃
          n₄ hn₁ hn₂ hn₃ hn₄).g =
    ((page X data r hr).sc' pq pq' pq'').g ≫
      (pageXIso X data r hr pq'' (data.i₀ r pq'' ⋯) (data.i₁ pq'') (data.i₂ pq'') (data.i₃ r pq'' ⋯) ⋯ ⋯ ⋯ ⋯ n₂ n₃ n₄ ⋯
          ⋯ ⋯).hom

Tactic:
simp only [← Iso.comp_inv_eq, Category.assoc]
    exact (pageD_eq X data r hr pq' pq'' hpq' _ _ _ _ _ rfl rfl ..).symm

Goal state after: