induction ls; intuition; simpl in *.
inversion H; clear H; subst.
destruct(eqd a0 a); subst.
intuition.
simpl in *.
intuition; subst.
eapply IHls; eauto.
Qed.

Lemma removeFirst_NoDup : forall (A : Set)(eqd : eq_dec A)(ls : list A) a,
NoDup ls ->
NoDup (removeFirst eqd ls a).