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

Lemma make_shares_In : forall i lasts lgood x shs (Hx : 0 <= x < Zlength lasts) (Hi : Znth x lasts 0 <> i) (Hgood : Znth x lgood false = true) (Hlen : Zlength shs >= Zlength lasts), In (Znth x shs Tsh) (make_shares shs (combine lgood lasts) i).