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

Lemma make_shares_add : forall i i' lasts j shs (Hj : 0 <= j < Zlength lasts) (Hi : Znth j lasts = i) (Hi' : i' <> i) (Hlen : Zlength shs >= Zlength lasts), exists shs1 shs2, make_shares shs lasts i = shs1 ++ shs2 /\ make_shares shs (upd_Znth j lasts i') i = shs1 ++ Znth j shs :: shs2.