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

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.