induction 1; intuition; simpl in *.
subst.
destruct a1.
remember (split ls1) as z.
destruct z.
simpl.
trivial.
Qed.

(* Specifications: *)
Theorem NoDup_snd_split_if :
forall (A B : Type)(ls : list (A * B)),
NoDup (snd (split ls)) ->
NoDup ls.