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'.