induction ls; intuition; simpl in *.
f_equal.
eapply H.
intuition.
eauto.
Qed.

(* Specifications: *)
Lemma nth_map_In :
forall (A B : Set)(ls : list A)(f : A -> B) i defa defb,
i < length ls ->
nth i (map f ls) defa = f (nth i ls defb).