(COMMENT harder variant of AG01 3.4)
(VAR x y z)
(RULES
minus(minus(x,y),z) -> minus(x,plus(y,z))
minus(0,y) -> 0
minus(x,0) -> x
minus(s(x),s(y)) -> minus(x,y)
plus(0,y) -> y
plus(s(x),y) -> plus(x,s(y))
plus(s(x),y) -> s(plus(y,x))
zero(s(x)) -> false
zero(0) -> true
p(s(x)) -> x
p(0) -> 0
div(x,y) -> quot(x,y,0)
quot(s(x),s(y),z) -> quot(minus(p(ack(0,x)),y),s(y),s(z))
quot(0,s(y),z) -> z

ack(0,x) -> s(x)
ack(0,x) -> plus(x,s(0))
ack(s(x),0) -> ack(x,s(0))
ack(s(x),s(y)) -> ack(x,ack(s(x),y))
)