YES

TRS:

  double(0()) -> 0()
  double(s(x)) -> s(s(double(x)))
  half(0()) -> 0()
  half(s(0())) -> 0()
  half(s(s(x))) -> s(half(x))
  -(x,0()) -> x
  -(s(x),s(y)) -> -(x,y)
  if(0(),y,z) -> y
  if(s(x),y,z) -> z
  half(double(x)) -> x

linear polynomial interpretations on N:

  double_A(x1) = x1 + 1
  double#_A(x1) = x1 + 1
  0_A = 1
  0#_A = 3
  s_A(x1) = x1
  s#_A(x1) = 0
  half_A(x1) = x1 + 1
  half#_A(x1) = 4
  -_A(x1,x2) = x1
  -#_A(x1,x2) = 0
  if_A(x1,x2,x3) = x2 + x3
  if#_A(x1,x2,x3) = x1 + x2 + x3

precedence:

  double > s > 0 = - = if > half