(VAR b x xs xs2 y ys ys2 zs zs2 )
(RULES 
        isEmpty(cons(x, xs)) -> false
        isEmpty(nil) -> true
        isZero(0) -> true
        isZero(s(x)) -> false
        head(cons(x, xs)) -> x
        tail(cons(x, xs)) -> xs
        tail(nil) -> nil
        append(nil, x) -> cons(x, nil)
        append(cons(y, ys), x) -> cons(y, append(ys, x))
        p(s(s(x))) -> s(p(s(x)))
        p(s(0)) -> 0
        p(0) -> 0
        inc(s(x)) -> s(inc(x))
        inc(0) -> s(0)
        addLists(xs, ys, zs) -> if(isEmpty(xs), isEmpty(ys), isZero(head(xs)), tail(xs), tail(ys), cons(p(head(xs)), tail(xs)), cons(inc(head(ys)), tail(ys)), zs, append(zs, head(ys)))
        if(true, true, b, xs, ys, xs2, ys2, zs, zs2) -> zs
        if(true, false, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
        if(false, true, b, xs, ys, xs2, ys2, zs, zs2) -> differentLengthError
        if(false, false, false, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs2, ys2, zs)
        if(false, false, true, xs, ys, xs2, ys2, zs, zs2) -> addLists(xs, ys, zs2)
        addList(xs, ys) -> addLists(xs, ys, nil)
        
)