induction ls; intuition; simpl in *;
intuition; subst.
destruct (eqd a0 a0); try congruence.
eapply Permutation_refl.
destruct (eqd a0 a); subst.
eapply Permutation_refl.
eapply perm_trans; [idtac | eapply perm_swap].
eapply perm_skip.
eauto.
Qed.

(* Specifications: *)
Lemma removeFirst_not_in : forall (A : Set)(eqd : eq_dec A)(ls : list A) a1 a2,
~In a1 ls ->
~In a1 (removeFirst eqd ls a2).