induction lsa; destruct lsb; intuition; simpl in *; try omega.
inversion H0; clear H0; subst.
destruct i; intuition.
Qed.

Lemma map_eq_nth :
forall (A B C : Set) i def1 def2 (f1 : A -> C)(f2 : B -> C)(lsa : list A)(lsb : list B),
length lsa = length lsb ->
map f1 lsa = map f2 lsb ->
i < length lsa ->
f1 (nth i lsa def1) = f2 (nth i lsb def2).