(*****   SUBLIST  ******)
Definition sublist {A} (lo hi: Z) (al: list A) : list A :=
firstn (Z.to_nat (hi-lo)) (skipn (Z.to_nat lo) al).
Definition upd_Znth {A} (i: Z) (al: list A) (x: A): list A :=
sublist 0 i al ++ x :: sublist (i+1) (Zlength al) al.

Lemma Zlength_sublist:
forall {A} lo hi (al: list A),
0 <= lo <= hi -> hi <= Zlength al ->
Zlength (sublist lo hi al) = hi-lo.