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 Exists2_exists A (d: A) (R: A -> A -> Prop) (xs ys: list A) :
Exists2 R xs ys ->
exists i, i < length xs /\ R (nth i xs d) (nth i ys d).