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

Lemma make_shares_out : forall b lgood lasts shs (Hb : ~In b lasts) (Hlen1 : Zlength lgood = Zlength shs) (Hlen2 : Zlength lasts = Zlength shs), make_shares shs (combine lgood lasts) b = map snd (filter fst (combine lgood shs)).