-- Module: Mathlib.RepresentationTheory.Homological.Resolution
Goal:
k G : Type u
inst : CommRing k
inst_1 : Monoid G
x : Fin 1 → G
r : k
⊢ Eq (α := k)
    ((mapDomain
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
                  ((SimplicialObject.augment
                          (classifyingSpaceUniversalCover G ⋙
                            forget (Action (Type u) G) (FC := Action.HomSubtype (Type u) G (FV := TypeCat.Fun)) (CC :=
                              fun (X : Action (Type u) G) ↦ X.V))
                          (⊤_ Type u)
                          (terminal.from
                            ((classifyingSpaceUniversalCover G ⋙
                                  forget (Action (Type u) G) (FC := Action.HomSubtype (Type u) G (FV := TypeCat.Fun))
                                    (CC := fun (X : Action (Type u) G) ↦ X.V)).obj
                              (Opposite.op { len := 0 : SimplexCategory })))
                          ⋯).hom.app
                    (Opposite.op { len := 0 : SimplexCategory }))) :
              (a :
                  ((𝟭 (SimplicialObject (Type u))).obj
                        (SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G))
                            (⊤_ Type u)
                            (terminal.from
                              ((classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G)).obj
                                (Opposite.op { len := 0 })))
                            ⋯).left).obj
                    (Opposite.op { len := 0 })) →
                ((SimplicialObject.const (Type u)).obj
                      (SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G))
                          (⊤_ Type u)
                          (terminal.from
                            ((classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G)).obj
                              (Opposite.op { len := 0 })))
                          ⋯).right).obj
                  (Opposite.op { len := 0 }))
            (single x r) :
          (a :
              ((SimplicialObject.const (Type u)).obj
                    (SimplicialObject.augment (classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G))
                        (⊤_ Type u)
                        (terminal.from
                          ((classifyingSpaceUniversalCover G ⋙ forget (Action (Type u) G)).obj
                            (Opposite.op { len := 0 })))
                        ⋯).right).obj
                (Opposite.op { len := 0 })) →
            k)
        default :
      k)
    ((linearCombination k fun (x : Fin 1 → G) ↦ 1 :
          (a : (Fin 1 → G) →₀ k) →
            (↑(((ChainComplex.single₀ (ModuleCat k)).obj ((forget₂ (Rep k G) (ModuleCat k)).obj (trivial k G k))).X 0) :
              Type u))
        (single x r) :
      (↑(HomologicalComplex.X
            (Functor.obj (ChainComplex.single₀ (ModuleCat k))
              (Functor.obj
                (forget₂ (Rep k G) (FC := fun (A B : Rep k G) ↦ A.ρ.IntertwiningMap B.ρ) (CC := V) (ModuleCat k) (FD :=
                  fun (x1 x2 : ModuleCat k) ↦ (↑x1 : Type u) →ₗ[k] (↑x2 : Type u)) (CD := ModuleCat.carrier))
                (trivial k G k)))
            0) :
        Type u))

Tactic:
simp [Unique.eq_default (terminal.from _), single_apply, if_pos (Subsingleton.elim _ _)]

Premises used: CategoryTheory.Limits.terminal.from, if_pos, Unique.eq_default, Finsupp.single_apply, Subsingleton.elim