(* The complement of make_shares. *)
Fixpoint make_shares_inv shs (lasts : list Z) i {struct lasts} : list share :=
match lasts with
| [] => []
| b :: rest => if eq_dec b i then hd Share.bot shs :: make_shares_inv (tl shs) rest i
else make_shares_inv (tl shs) rest i
end.

Lemma make_shares_sub : forall i lasts shs sh0 sh1 sh2 (Hlen : Zlength shs >= Zlength lasts)
(Hsh1 : sepalg_list.list_join sh0 shs sh1) (Hsh2 : sepalg_list.list_join sh0 (make_shares shs lasts i) sh2),
sepalg.join_sub sh2 sh1.