Section Always.
Variable this : nid.
Variable W : world.
Notation coherent := (Coh W).
Arguments proc [this W].
Fixpoint always_sc A (s1 : state) p scs (P : state -> proc A -> Prop) : Prop :=
s1 \In coherent /\
if scs is sc :: scs' then
forall s2, network_rely W this s1 s2 ->
[/\ safe p sc s2, P s2 p &
forall s3 q, @pstep this W A s2 p sc s3 q -> always_sc s3 q scs' P]
else forall s2, network_rely W this s1 s2 -> P s2 p.
Definition always A s (p : proc A) P := forall scs, always_sc s p scs P.
(* Some basic properties of the alwas predicate *)
(* The predicates, corresponding to weakest postconditions, necessary for defining safe executions of programs.  *)
Notation alwsafe_sc s p scs := (always_sc s p scs (fun _ _ => True)).
Notation alwsafe s p := (always s p (fun _ _ => True)).
Arguments alwA [A B s p P].
(* always commutes with implication, so we can weaken the postconditions *)
Arguments alwI [A s p P Q].

Lemma alw_bnd A B (p1 : proc A) (p12 : proc B) pp2 s1
(P : state -> B -> Prop) :
p12 \In pcat p1 pp2 ->
always s1 p1 (fun s2 p2 =>
forall p v, p2 = Ret v -> p \In pp2 v ->
always s2 p (fun s q => forall v, q = Ret v -> P s v)) ->
always s1 p12 (fun s p => forall v, p = Ret v -> P s v).