induction lsa; intuition; simpl in *.
destruct (in_dec (EqDec_dec eqda) a lsa);
destruct (in_dec (EqDec_dec eqdb) (inj a) (map inj lsa));
intuition.
exfalso.
apply n.
apply in_map_iff.
econstructor.
intuition.
apply in_map_iff in i.
destruct i.
intuition.
apply H in H1.
subst.
intuition.
Qed.

Theorem Permutation_hasDups :
forall (A : Set)(eqd : EqDec A)(ls1 ls2 : list A),
Permutation ls1 ls2 ->
hasDups eqd ls1 = hasDups eqd ls2.