(VAR x y b xs xs2 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)
)
(COMMENT
addList(xs,ys) returns a new list where the elements of xs and ys are pairwise added
)