-- Lean 4 proof step from Mathlib.RepresentationTheory.Homological.FiniteCyclic (goal)
Goal:
k G : Type u
inst : CommRing k
inst_1 : CommGroup G
inst_2 : Fintype G
g : G
hg : ∀ (x : G), Membership.mem (γ := Subgroup G) (Subgroup.zpowers g) x
⊢ QuasiIsoAt.{u, u + 1, 0} (ι := ℕ) (C := Rep k G) (c := ComplexShape.down ℕ) (K :=
    Functor.obj (chainComplexFunctor k g) (leftRegular k G)) (L :=
    Functor.obj (ChainComplex.single₀ (Rep k G)) (trivial k G k))
    ((Equiv.symm
          (ChainComplex.toSingle₀Equiv (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) (trivial k G k)) :
        (a :
            { f : ((chainComplexFunctor k g).obj (leftRegular k G)).X 0 ⟶ trivial k G k //
              ((chainComplexFunctor k g).obj (leftRegular k G)).d 1 0 ≫ f = 0 }) →
          (chainComplexFunctor k g).obj (leftRegular k G) ⟶ (ChainComplex.single₀ (Rep k G)).obj (trivial k G k))
      (Subtype.mk (p :=
          fun
            (f :
              Quiver.Hom (V := Rep k G)
                (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0) (trivial k G k)) ↦
          Eq (α :=
            Quiver.Hom (V := Rep k G) (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1)
              (trivial k G k))
            (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0)
          (leftRegularHom (trivial k G k) 1) ⋯ :
        { f :
          Quiver.Hom (V := Rep k G) (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0)
            (trivial k G k) //
          Eq (α :=
            Quiver.Hom (V := Rep k G) (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1)
              (trivial k G k))
            (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0 }))
    0
have zero : ShortComplex.Exact (C := Rep k G)
    (ShortComplex.mk (X₁ :=
      ShortComplex.X₁
        (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
          (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
      (X₃ :=
      ShortComplex.X₂
        (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
          (Functor.obj (ChainComplex.single₀ (Rep k G)) (trivial k G k))))
      (ShortComplex.f
        (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
          (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
      (ShortComplex.Hom.τ₂
        (CategoryTheory.Functor.map (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
          ((Equiv.symm
                (ChainComplex.toSingle₀Equiv (Functor.obj (chainComplexFunctor k g) (leftRegular k G))
                  (trivial k G k)) :
              (a :
                  { f : ((chainComplexFunctor k g).obj (leftRegular k G)).X 0 ⟶ trivial k G k //
                    ((chainComplexFunctor k g).obj (leftRegular k G)).d 1 0 ≫ f = 0 }) →
                (chainComplexFunctor k g).obj (leftRegular k G) ⟶ (ChainComplex.single₀ (Rep k G)).obj (trivial k G k))
            (Subtype.mk (p :=
                fun
                  (f :
                    Quiver.Hom (V := Rep k G)
                      (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0)
                      (trivial k G k)) ↦
                Eq (α :=
                  Quiver.Hom (V := Rep k G)
                    (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1) (trivial k G k))
                  (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0)
                (leftRegularHom (trivial k G k) 1) sorry :
              { f :
                Quiver.Hom (V := Rep k G)
                  (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0) (trivial k G k) //
                Eq (α :=
                  Quiver.Hom (V := Rep k G)
                    (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1) (trivial k G k))
                  (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0 }))))
      sorry) ∧
  Epi.{u, u + 1} (C := Rep k G) (X :=
    ShortComplex.X₂
      (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
        (Functor.obj (chainComplexFunctor k g) (leftRegular k G))))
    (Y :=
    ShortComplex.X₂
      (Functor.obj (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
        (Functor.obj (ChainComplex.single₀ (Rep k G)) (trivial k G k))))
    (ShortComplex.Hom.τ₂
      (CategoryTheory.Functor.map (HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0)
        ((Equiv.symm
              (ChainComplex.toSingle₀Equiv (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) (trivial k G k)) :
            (a :
                { f : ((chainComplexFunctor k g).obj (leftRegular k G)).X 0 ⟶ trivial k G k //
                  ((chainComplexFunctor k g).obj (leftRegular k G)).d 1 0 ≫ f = 0 }) →
              (chainComplexFunctor k g).obj (leftRegular k G) ⟶ (ChainComplex.single₀ (Rep k G)).obj (trivial k G k))
          (Subtype.mk (p :=
              fun
                (f :
                  Quiver.Hom (V := Rep k G)
                    (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0)
                    (trivial k G k)) ↦
              Eq (α :=
                Quiver.Hom (V := Rep k G)
                  (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1) (trivial k G k))
                (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0)
              (leftRegularHom (trivial k G k) 1) sorry :
            { f :
              Quiver.Hom (V := Rep k G)
                (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 0) (trivial k G k) //
              Eq (α :=
                Quiver.Hom (V := Rep k G)
                  (HomologicalComplex.X (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1) (trivial k G k))
                (HomologicalComplex.d (Functor.obj (chainComplexFunctor k g) (leftRegular k G)) 1 0 ≫ f) 0 }))))
Tactic: rw [ChainComplex.quasiIsoAt₀_iff, ShortComplex.quasiIso_iff_of_zeros' _ rfl rfl rfl]