induction ls ; intuition.
simpl in *.
intuition; subst.
simpl.
trivial.
unfold compMap in *.
fold compMap in *.
repeat simp_in_support.
simpl.
f_equal.
eauto.
Qed.

Lemma map_eq :
forall (A B C : Set)(lsa : list A)(lsb : list B)(f1 : A -> C)(f2 : B -> C),
list_pred (fun a b => f1 a = f2 b) lsa lsb ->
map f1 lsa = map f2 lsb.