induction 1; intuition; simpl in *.
intuition.
intuition; subst.
eauto.
Qed.

(* Specifications: *)
Lemma rel_map_eq_inv : forall (A B : Type)(ls1 ls2 : list A)(rel1 rel2 : A -> B -> Prop) ls1' ls2',
ls1 = ls2 ->
(forall a b1 b2, rel1 a b1 -> rel2 a b2 -> b1 = b2) ->
rel_map rel1 ls1 ls1' ->
rel_map rel2 ls2 ls2' ->
ls1' = ls2'.