intros.
repeat rewrite Y_skipn.
repeat rewrite skipn_length.
omega.
Qed.
Definition I_e : BC :=
comp 2 1 parity_e nil
[comp 2 1 Y_e
[comp 2 0 (succ true) nil [proj 2 0 0]; proj 2 0 1]
[proj 2 1 2] ].

Lemma I_Y_property j z w y :
length z < length w ->
length w - S (length z) < length y ->
Y (j :: z) w y = [nth (length w - S (length z)) y false] ++ Y z w y.