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

Lemma make_shares_ext : forall i l l' shs (Hlen : Zlength l = Zlength l') (Hi : forall j, 0 <= j < Zlength l -> Znth j l = i <-> Znth j l' = i), make_shares shs l i = make_shares shs l' i.