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 singleton_join_inv: forall k a b c,
join (singleton k a) (singleton k b) c -> exists c', join a b c' /\ c = singleton k c'.