Definition oboxopt Delta ret P :=
match ret with
| Some id => obox Delta id P
| _ => P
end.
Definition odiaopt Delta ret P :=
match ret with
| Some id => odia Delta id P
| _ => P
end.
Definition temp_guard_opt (Delta : tycontext) (i: option ident): Prop :=
match i with
| Some i => temp_guard Delta i
| None => True
end.

Lemma oboxopt_closed: forall Delta i P,
temp_guard_opt Delta i ->
closed_wrt_vars (fun id => isSome (match i with Some i' => insert_idset i' idset0 | None => idset0 end) ! id) P ->
oboxopt Delta i P = P.