induction 1; intuition; simpl in *.
intuition.
intuition;
subst.
econstructor; eauto.
destruct (IHrel_map b0).
trivial.
intuition.
econstructor; eauto.
Qed.

(* Specifications: *)
Lemma ne_all_not_in : forall (A : Type)(ls : list A) a,
(forall a', In a' ls -> a <> a') ->
~In a ls.