MAYBE
Trs:
 {                logarithm(x) -> logIter(x, 0()),
                      inc(0()) -> s(0()),
                     inc(s(x)) -> s(inc(x)),
                     half(0()) -> 0(),
                  half(s(0())) -> 0(),
                 half(s(s(x))) -> s(half(x)),
                    le(0(), y) -> true(),
                 le(s(x), 0()) -> false(),
                le(s(x), s(y)) -> le(x, y),
          if(false(), b, x, y) -> logZeroError(),
  if(true(), false(), x, s(y)) -> y,
      if(true(), true(), x, y) -> logIter(x, y),
                 logIter(x, y) -> if(le(s(0()), x), le(s(s(0())), x), half(x), inc(y)),
                           f() -> g(),
                           f() -> h()}
 Comment:
  We consider a duplicating trs.
  FAIL:
   Open