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

Lemma make_shares_inv_In : forall i lasts x shs (Hx : 0 <= x < Zlength lasts) (Hi : Znth x lasts = i) (Hlen : Zlength shs >= Zlength lasts), In (Znth x shs) (make_shares_inv shs lasts i).