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).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).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).

Lemma app_eq_compat_l {A: Type} (xs ys zs: list A) : ys = zs -> xs ++ ys = xs ++ zs.