Inductive Exists2 {A B} (R: A -> B -> Prop) : list A -> list B -> Prop :=
| Exists2_cons_hd : forall x xs y ys, R x y -> Exists2 R (x::xs) (y::ys)
| Exists2_cons_tl : forall x xs y ys, Exists2 R xs ys -> Exists2 R (x::xs) (y::ys).
Hint Constructors Exists2.

Lemma forall_exists_list (A B : Type) (P : A -> Prop) (Q : A -> B -> Prop) (l : list A) :
(forall a, In a l -> P a -> exists b, Q a b) ->
(Forall P l -> exists l', Forall2 Q l l').