induction 1; intuition; simpl in *.
f_equal.
intuition.
Qed.

Lemma list_pred_zip_l_eq :
forall (A B: Set)(lsa : list A)(lsb : list B)(P : A -> B -> Prop),
list_pred P lsa lsb ->
list_pred (fun p a => (fst p) = a /\ P (fst p) (snd p)) (zip lsa lsb) lsa.