(VAR x y)
(RULES
g(s(x),s(y)) -> if(and(f(s(x)),f(s(y))),t(g(k(minus(m(x,y),n(x,y)),s(s(0))),k(n(s(x),s(y)),s(s(0))))),g(minus(m(x,y),n(x,y)),n(s(x),s(y))))

n(0, y) ->  0
n(x, 0) -> 0
n(s(x), s(y)) -> s(n(x,y))

m(0, y) -> y
m(x, 0) -> x
m(s(x), s(y)) -> s(m(x,y))

k(0, s(y)) -> 0
k(s(x), s(y)) -> s(k(minus(x,y), s(y)))

t(x) -> p(x,x)

p(s(x), s(y)) -> s(s(p(if(gt(x,y),x,y), if(not(gt(x,y)),id(x),id(y)))))
p(s(x), x) -> p(if(gt(x,x), id(x), id(x)), s(x))
p(0, y) -> y
p(id(x), s(y)) -> s(p(x,if(gt(s(y),y), y, s(y))))

minus(x,0) -> x
minus(s(x), s(y)) -> minus(x,y)

id(x) -> x

if(true, x, y) -> x
if(false, x, y) -> y

not(x) -> if(x, false, true)

and(x, false) -> false
and(true, true) -> true

f(0) -> true
f(s(x)) -> h(x)

h(0) -> false
h(s(x)) -> f(x)

gt(s(x), 0) -> true
gt(0, y) -> false
gt(s(x), s(y)) -> gt(x, y)
)