(** * Properties of availability sets *)

(** These properties are not used in the semantic preservation proof,
    but establish some confidence in the availability analysis. *)

Definition avail_above (v: ident) (s: avail) : Prop :=
  forall v' i', In (v', i') s -> Plt v v'.Inductive wf_avail: avail -> Prop :=
  | wf_avail_nil:
      wf_avail nil
  | wf_avail_cons: forall v i s,
     avail_above v s ->
     wf_avail s ->
     wf_avail ((v, i) :: s).Inductive wf_avail: avail -> Prop :=
  | wf_avail_nil:
      wf_avail nil
  | wf_avail_cons: forall v i s,
     avail_above v s ->
     wf_avail s ->
     wf_avail ((v, i) :: s).Inductive wf_avail: avail -> Prop :=
  | wf_avail_nil:
      wf_avail nil
  | wf_avail_cons: forall v i s,
     avail_above v s ->
     wf_avail s ->
     wf_avail ((v, i) :: s).

Lemma join_2: forall v i s1, wf_avail s1 -> forall s2, wf_avail s2 -> In (v, i) (join s1 s2) -> In (v, i) s1 /\ In (v, i) s2.