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 _))).

(* Specifications: *)
Lemma share_of_Some: forall p: rshare * AV.kind, readable_share (share_of (Some p)).
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'.
Lemma pure_readable_glb_Rsh: forall sh, pure_readable_share sh -> Share.glb Share.Rsh sh = sh.
Lemma join_glb_Rsh: forall a b c : Share.t, join a b c -> join (Share.glb Share.Rsh a) (Share.glb Share.Rsh b) (Share.glb Share.Rsh c).
Lemma pure_readable_share_glb: forall a, pure_readable_share a -> Share.glb Share.Rsh a = a.
Lemma glb_Rsh_bot_unreadable: forall a, Share.glb Share.Rsh a = Share.bot -> ~readable_share a.
Lemma fixup_join : forall a (ac ad: address -> Share.t) z, (forall x, @join_sub _ Join_pk (a x) (z x)) -> (forall x, join (ac x) (ad x) (share_of (a x))) -> (forall x, @join _ Join_pk (fixup_splitting ac z x) (fixup_splitting ad z x) (a x)).