Definition fixup_splitting
  (a:address -> Share.t) (z: address -> option (rshare * kind)) : address -> option (rshare * kind) :=
  fun l => 
    match z l with
    | Some (sh, k) =>
       match dec_readable (a l) with
       | left p => Some (readable_part p,  k)
       | right _ => None
       end
    | None => None
    end.Definition share_of (x: option (rshare * kind)) : Share.t :=
  match x with Some (p,_) => proj1_sig p | None => Share.bot end.Definition Join_pk := (Join_lower (Join_prod rshare _ kind (Join_equiv _))).

Lemma join_sub_same_k: forall {a a' : rshare} {k k': AV.kind}, @join_sub _ Join_pk (Some (a,k)) (Some (a',k')) -> k=k'.