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

Lemma make_shares_app : forall i l1 l2 shs, Zlength l1 + Zlength l2 <= Zlength shs -> make_shares shs (l1 ++ l2) i = make_shares shs l1 i ++ make_shares (sublist (Zlength l1) (Zlength shs) shs) l2 i.