MAYBE

Problem:
 check(0()) -> zero()
 check(s(0())) -> odd()
 check(s(s(0()))) -> even()
 check(s(s(s(x)))) -> check(s(x))
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 times(x,y) -> timesIter(x,y,0())
 timesIter(x,y,z) -> if(check(x),x,y,z,plus(z,y))
 p(s(x)) -> x
 p(0()) -> 0()
 if(zero(),x,y,z,u) -> z
 if(odd(),x,y,z,u) -> timesIter(p(x),y,u)
 if(even(),x,y,z,u) -> plus(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))

Proof:
 DP Processor:
  DPs:
   check#(s(s(s(x)))) -> check#(s(x))
   half#(s(s(x))) -> half#(x)
   plus#(s(x),y) -> plus#(x,y)
   times#(x,y) -> timesIter#(x,y,0())
   timesIter#(x,y,z) -> plus#(z,y)
   timesIter#(x,y,z) -> check#(x)
   timesIter#(x,y,z) -> if#(check(x),x,y,z,plus(z,y))
   if#(odd(),x,y,z,u) -> p#(x)
   if#(odd(),x,y,z,u) -> timesIter#(p(x),y,u)
   if#(even(),x,y,z,u) -> half#(s(z))
   if#(even(),x,y,z,u) -> timesIter#(half(x),y,half(s(z)))
   if#(even(),x,y,z,u) -> half#(z)
   if#(even(),x,y,z,u) -> half#(x)
   if#(even(),x,y,z,u) -> timesIter#(half(x),y,half(z))
   if#(even(),x,y,z,u) -> plus#(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))
  TRS:
   check(0()) -> zero()
   check(s(0())) -> odd()
   check(s(s(0()))) -> even()
   check(s(s(s(x)))) -> check(s(x))
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   times(x,y) -> timesIter(x,y,0())
   timesIter(x,y,z) -> if(check(x),x,y,z,plus(z,y))
   p(s(x)) -> x
   p(0()) -> 0()
   if(zero(),x,y,z,u) -> z
   if(odd(),x,y,z,u) -> timesIter(p(x),y,u)
   if(even(),x,y,z,u) -> plus(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))
  Usable Rule Processor:
   DPs:
    check#(s(s(s(x)))) -> check#(s(x))
    half#(s(s(x))) -> half#(x)
    plus#(s(x),y) -> plus#(x,y)
    times#(x,y) -> timesIter#(x,y,0())
    timesIter#(x,y,z) -> plus#(z,y)
    timesIter#(x,y,z) -> check#(x)
    timesIter#(x,y,z) -> if#(check(x),x,y,z,plus(z,y))
    if#(odd(),x,y,z,u) -> p#(x)
    if#(odd(),x,y,z,u) -> timesIter#(p(x),y,u)
    if#(even(),x,y,z,u) -> half#(s(z))
    if#(even(),x,y,z,u) -> timesIter#(half(x),y,half(s(z)))
    if#(even(),x,y,z,u) -> half#(z)
    if#(even(),x,y,z,u) -> half#(x)
    if#(even(),x,y,z,u) -> timesIter#(half(x),y,half(z))
    if#(even(),x,y,z,u) -> plus#(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))
   TRS:
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    check(0()) -> zero()
    check(s(0())) -> odd()
    check(s(s(0()))) -> even()
    check(s(s(s(x)))) -> check(s(x))
    p(s(x)) -> x
    p(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    half(0()) -> 0()
    timesIter(x,y,z) -> if(check(x),x,y,z,plus(z,y))
    if(zero(),x,y,z,u) -> z
    if(odd(),x,y,z,u) -> timesIter(p(x),y,u)
    if(even(),x,y,z,u) -> plus(timesIter(half(x),y,half(z)),timesIter(half(x),y,half(s(z))))
   Open