induction ls; intuition; simpl in *.
discriminate.
destruct n; simpl in *.
inversion H; clear H; subst.
trivial.
eauto.
Qed.

Theorem nth_option_snd_split :
forall (A B : Set)(ls : list (A * B)) n a b,
nth_option ls n = Some (a, b) ->
nth_option (snd (split ls)) n = Some b.