End FORALL2.

Lemma list_forall2_imply:
forall (A B: Type) (P1: A -> B -> Prop) (l1: list A) (l2: list B),
list_forall2 P1 l1 l2 ->
forall (P2: A -> B -> Prop),
(forall v1 v2, In v1 l1 -> In v2 l2 -> P1 v1 v2 -> P2 v1 v2) ->
list_forall2 P2 l1 l2.