Proof using.
induction l'; intros; simpl in *.
- find_rewrite_lem remove_all_nil.
destruct l0; simpl in *; match goal with H: [] = _ |- _ => contradict H end; auto using nil_cons.
- invc_NoDup.
break_if; subst.
* rewrite remove_not_in; auto.
pose proof remove_all_cons l a l'.
break_or_hyp; break_and.
+ find_rewrite.
match goal with H0: NoDup _, H1: _ = remove_all _ _ |- _ => specialize (IHl' _ _ _ H0 (eq_sym H1)) end.
rewrite remove_not_in in IHl'; auto.
+ destruct l0; simpl in *; find_rewrite; find_injection; auto.
assert (In a (remove_all l l')).
match goal with H: _ = remove_all _ _ |- _ => rewrite <- H end.
apply in_or_app.
right; left; auto.
find_apply_lem_hyp in_remove_all_was_in.
intuition.
* pose proof remove_all_cons l a l'.
break_or_hyp; break_and; find_rewrite.
+ pose proof remove_all_cons l a (remove A_eq_dec a0 l').
break_or_hyp; break_and; intuition.
aggressive_rewrite_goal; auto.
+ destruct l0; simpl in *; find_injection; intuition.
match goal with H0: NoDup _, H1: _ = remove_all _ _ |- _ => specialize (IHl' _ _ _ H0 (eq_sym H1)) end.
rewrite <- IHl'.
pose proof remove_all_cons l a (remove A_eq_dec a0 l').
break_or_hyp; break_and; intuition.
Qed.

(* Specifications: *)
Lemma remove_all_app_l :
forall xs ys zs,
remove_all (xs ++ ys) zs = remove_all xs (remove_all ys zs).