Module: Mathlib.Algebra.Homology.ConcreteCategory

Goal state before:
pp: case refine_1
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
ι : Type u_2
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
x₃ : ↑((forget₂ C Ab).obj (S.X₃.X i))
hx₃ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.d i j))) x₃ = 0
x₂ : ↑((forget₂ C Ab).obj (S.X₂.X i))
hx₂ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.g.f i))) x₂ = x₃
x₁ : ↑((forget₂ C Ab).obj (S.X₁.X j))
hx₁ :
  (ConcreteCategory.hom ((forget₂ C Ab).map (S.f.f j))) x₁ = (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.d i j))) x₂
k : ι
hk : c.next j = k
⊢ (ConcreteCategory.hom ((forget₂ C Ab).map (HomologicalComplex.opcyclesMap S.g i)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.pOpcycles i))) x₂) =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyι i)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.homologyπ i))) (S.X₃.cyclesMk x₃ j ⋯ hx₃))

pp: case refine_2
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
ι : Type u_2
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
x₃ : ↑((forget₂ C Ab).obj (S.X₃.X i))
hx₃ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.d i j))) x₃ = 0
x₂ : ↑((forget₂ C Ab).obj (S.X₂.X i))
hx₂ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.g.f i))) x₂ = x₃
x₁ : ↑((forget₂ C Ab).obj (S.X₁.X j))
hx₁ :
  (ConcreteCategory.hom ((forget₂ C Ab).map (S.f.f j))) x₁ = (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.d i j))) x₂
k : ι
hk : c.next j = k
⊢ (ConcreteCategory.hom ((forget₂ C Ab).map (HomologicalComplex.cyclesMap S.f j))) (S.X₁.cyclesMk x₁ k hk ⋯) =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.opcyclesToCycles i j)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.pOpcycles i))) x₂)

Tactic:
· rw [← ConcreteCategory.forget₂_comp_apply, ← ConcreteCategory.forget₂_comp_apply,
      HomologicalComplex.p_opcyclesMap, Functor.map_comp, ConcreteCategory.comp_apply,
      HomologicalComplex.homology_π_ι, ConcreteCategory.forget₂_comp_apply, hx₂,
      HomologicalComplex.i_cyclesMk]

Goal state after:
case refine_2
C : Type u
inst✝⁶ : Category.{v, u} C
FC : C → C → Type u_1
CC : C → Type v
inst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)
inst✝⁴ : ConcreteCategory C FC
inst✝³ : HasForget₂ C Ab
inst✝² : Abelian C
inst✝¹ : (forget₂ C Ab).Additive
inst✝ : (forget₂ C Ab).PreservesHomology
ι : Type u_2
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
i j : ι
hij : c.Rel i j
x₃ : ↑((forget₂ C Ab).obj (S.X₃.X i))
hx₃ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₃.d i j))) x₃ = 0
x₂ : ↑((forget₂ C Ab).obj (S.X₂.X i))
hx₂ : (ConcreteCategory.hom ((forget₂ C Ab).map (S.g.f i))) x₂ = x₃
x₁ : ↑((forget₂ C Ab).obj (S.X₁.X j))
hx₁ :
  (ConcreteCategory.hom ((forget₂ C Ab).map (S.f.f j))) x₁ = (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.d i j))) x₂
k : ι
hk : c.next j = k
⊢ (ConcreteCategory.hom ((forget₂ C Ab).map (HomologicalComplex.cyclesMap S.f j))) (S.X₁.cyclesMk x₁ k hk ⋯) =
    (ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.opcyclesToCycles i j)))
      ((ConcreteCategory.hom ((forget₂ C Ab).map (S.X₂.pOpcycles i))) x₂)