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 plus_eq_compat_l x y1 y2 : y1 = y2 -> x + y1 = x + y2.