(* 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_minus : forall i lasts sh0 shs sh' sh1 (Hsh' : sepalg_list.list_join sh0 shs sh')
(Hsh1 : sepalg_list.list_join sh0 (make_shares shs lasts i) sh1)
(Hlen : Zlength shs = Zlength lasts),
sepalg_list.list_join sh1 (make_shares_inv shs lasts i) sh'.