Corollary prev_taken_In : forall h, prev_taken h = vint 1 \/ In (AE (prev_taken h) Empty) h.

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'.