MAYBE

Problem:
 battle(H(0(),x),n) -> battle(x,s(n))
 battle(H(H(0(),x),y),n) -> battle(f(x,y,n),s(n))
 battle(H(H(H(0(),x),y),z),n) -> battle(H(f(x,y,n),z),s(n))
 f(x,y,o()) -> y
 f(x,y,s(n)) -> H(x,f(x,y,n))

Proof:
 Open