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 count_occ_cons A eq_dec (a a': A) l : count_occ eq_dec (a::l) a' = if eq_dec a a' then S (count_occ eq_dec l a') else count_occ eq_dec l a'.