Definition gname := nat.
Definition own {RA: Ghost} (n: gname) (a: G) (pp: preds) :=
EX v : _, Own (singleton n (existT _ RA (exist _ a v), pp)).
Definition list_set {A} (m : list (option A)) k v : list (option A) :=
firstn k m ++ repeat None (k - length m) ++ Some v :: skipn (S k) m.

Lemma ghost_fmap_singleton_inv : forall f g a k v,
ghost_fmap f g a = singleton k v ->
exists v', a = singleton k v' /\ v = let (a, b) := v' in (a, preds_fmap f g b).