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_andl (A: Type) (P: A -> Prop) (l: list A):
Forall P l <-> andl (map P l).