induction ls1; destruct ls2; intuition; simpl in *.
discriminate.
inversion H; clear H; subst.
trivial.
Qed.

(* Specifications: *)
Theorem In_zip_strong :
forall (A B : Set)(ls : list A) f a (b : B),
In (a, b) (zip ls (map f ls)) ->
(In a ls /\ b = f a).