(* Build the logic over worlds.  We take tuples of
   a nat, a local env, and a memory.  The nat exists
   to let us define the |> operator and the Geodel Leob
   rule in the usual way.  We get a SA by the obvious
   combination of SAs on locals and mems with the trival
   SA on nats.
*)
Definition world := (nat * (locals * mem))%type.Definition world_sa : sepalg world :=
  sa_prod (sa_equiv nat) (sa_prod locals_sa mem_sa).Definition world_age : ageable world :=
  ag_prod nat (locals * mem) ag_nat.Definition world_asa : ASA world :=
 asa_prod nat (locals * mem) (sa_equiv nat) ag_nat (sa_prod locals_sa mem_sa) asa_nat.

(* Specifications: *)
Lemma world_nec_split : forall n x n' x', @necR world world_age (n,x) (n',x') <-> necR n n' /\ x = x'.
Lemma world_later_split : forall n x n' x', @laterR world world_age (n,x) (n',x') <-> laterR n n' /\ x = x'.