induction ls1; destruct ls2; intuition; simpl in *; eauto; intuition; subst.
inversion H; clear H; subst.
inversion H0; clear H0; subst.
destruct (eqd a a1); subst.
intuition.
right.
eapply (IHls1 (a1 :: removeFirst _ ls2 a)); intuition.
econstructor.
eapply removeFirst_not_in; intuition.
eapply removeFirst_NoDup; intuition.
simpl.
destruct (H1 a0); auto.
right.
eapply removeFirst_in; eauto.
intuition. subst.
intuition.
inversion H; clear H; subst.
inversion H0; clear H0; subst.
destruct (H1 a); intuition; subst.
destruct (eqd a a); try congruence.
right.
eapply (IHls1 ls2); intuition.
destruct (H1 a0); intuition; subst.
intuition.
destruct (eqd a a0); try congruence.
destruct (eqd a a1); subst; auto.
right.
eapply (IHls1 (a0 :: removeFirst _ ls2 a)); intuition.
econstructor.
eapply removeFirst_not_in; trivial.
eapply removeFirst_NoDup; trivial.
simpl.
destruct (H1 a2); eauto.
destruct (eqd a0 a2); intuition.
right.
eapply removeFirst_in; trivial.
intuition.
subst.
intuition.
simpl.
destruct (eqd a0 a1); intuition.
right.
eapply removeFirst_in; trivial.
intuition.
inversion H; clear H; subst.
inversion H0; clear H0; subst.
destruct (eqd a0 a1); subst; intuition.
right.
assert (In a1 (if eqd a a0 then ls2 else a0 :: removeFirst eqd ls2 a)).
eapply IHls1.
trivial.
destruct (eqd a a0).
trivial.
econstructor.
eapply removeFirst_not_in; eauto.
eapply removeFirst_NoDup; intuition.
intuition.
destruct (eqd a a0).
destruct (H1 a2).
intuition.
subst.
intuition.
trivial.
simpl.
destruct (eqd a0 a2); intuition.
right.
destruct (H1 a2).
destruct (eqd a a2); intuition.
subst.
intuition.
eapply removeFirst_in.
trivial.
intuition. subst.
intuition.
trivial.
destruct (eqd a a0); intuition.
simpl in *.
intuition.
eapply removeFirst_in_iff.
eauto.
Qed.

(* Specifications: *)
Lemma matchOrder_not_in_h : forall (A : Set)(eqd : eq_dec A)(ls1 ls2 : list A) a,
~In a ls1 ->
~In a ls2 ->
~In a (matchOrder eqd ls1 ls2).