induction ls; intuition; simpl in *;
intuition; subst.
destruct (eqd a0 a0); subst; intuition.
destruct (eqd a0 a); subst.
omega.
eapply lt_n_S.
eapply IHls.
trivial.
Qed.

Lemma skipn_app :
forall (A : Set)(ls1 ls2 : list A),
skipn (length ls1) (ls1 ++ ls2) = ls2.