induction ls; intuition; simpl in *.
destruct n; simpl in *;
trivial.
inversion H0; clear H0; subst.
destruct n; simpl in *.
eapply eqRat_symm.
eapply eqRat_trans.
eapply fold_add_eq_init_f; intuition.
rewrite (ratAdd_0_r init1).
eapply ratAdd_eqRat_compat; intuition.
eapply IHls.
eapply ratAdd_eqRat_compat; intuition.
trivial.
intuition.
eapply H1.
intuition.
intuition.
subst.
intuition.
Qed.

Lemma permutation_NoDup : forall (A : Type)(ls1 ls2 : list A),
Permutation ls1 ls2 ->
NoDup ls1 ->
NoDup ls2.