Definition upd {A} (m : nat -> A) x e :=
fun y => if eq_nat_dec x y then e else m y.
Definition updatemap (m : map) (x e : nat) : option map :=
match m x with Some _ => Some (upd m x (Some e)) | None => None end.
Definition assertmap (m : map) (x : nat) : Prop :=
match m x with Some (S _) => True | _ => False end.
Definition joincell (a b : option nat) : option (option nat) :=
match a, b with
| None, None => Some None
| Some a, None | None, Some a => Some (Some a)
| Some _, Some _ => None
end.
Definition join (m n o : map) : Prop := forall x, joincell (m x) (n x) = Some (o x).
Definition joins (m n : map) : Prop := exists o, join m n o.
Add Parametric Morphism : join with signature eqmap ==> eqmap ==> eqmap ==> iff as join_morphism.

Lemma join_empty m n : empty n -> join m n m.
Definition nonempty (m : map) : Prop := exists x, m x <> None.