(* 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)).

Lemma alw_imp' A s (p : proc A) scs (P1 P2 : state -> proc A -> Prop) :
(forall s p, s \In coherent -> P1 s p -> P2 s p) ->
always_sc s p scs P1 -> always_sc s p scs P2.