Section SA_SMASH.
Variable T : Type.
Variable J_T: Join T.
Variable PA_T : Perm_alg T.
Definition smashed : Type := option (lifted J_T).
Definition Perm_smash :  Perm_alg smashed  := Perm_lower (lifted J_T).
Definition Sep_smash : Sep_alg smashed := Sep_lower (lifted J_T).
End SA_SMASH.
Arguments smashed _  {J_T}.
Existing Instance Perm_smash. (* Must not be inside a Section *)
Existing Instance Sep_smash. (* Must not be inside a Section *)

Lemma smashed_lifted_None_identity {A}`{Perm_alg A}:
@identity (smashed A) _ None.