∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : StarRing R] [inst_2 : NonUnitalSemiring A]
  [inst_3 : StarRing A] [inst_4 : Module R A] [inst_5 : IsScalarTower R A A] [inst_6 : SMulCommClass R A A]
  [inst_7 : StarModule R A] (s : Set A),
  (NonUnitalStarAlgebra.adjoin R s).toSubmodule = Submodule.span R ↑(Subsemigroup.closure (s ∪ star s))