fun equal_if_1 p x [] = false
   |equal_if_1 p [] y = false
   |equal_if_1 p [] [] = true
   |equal_if_1 p (a::x) (b::y) = p a b andalso equal_if_1 p x y;


fun equal_if_2 p [] [] = true
   |equal_if_2 p x [] = false
   |equal_if_2 p [] y = false
   |equal_if_2 p (a::x) (b::y) = p a b andalso equal_if_2 p x y

fun equal_if_3 p x y =
    (x=[] andalso y=[]) orelse 
    (length x = length y) andalso
    (p (hd x) (hd y)) andalso
    (equal_if_3 p (tl x) (tl y));

fun equal_if_4 p x y =
    let fun equal_if p [] [] = true
           |equal_if p (a::x) (b::y) = p a b andalso equal_if p x y
    in (length x = length y) andalso equal_if p x y end;
     

fun gt x y = x-y>0;