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

max/plus interpretations on N:

  double_A(x1) = max{1, 1 + x1}
  double#_A(x1) = max{1, 1 + x1}
  0_A = 0
  0#_A = 0
  s_A(x1) = max{2, x1}
  s#_A(x1) = max{2, x1}
  half_A(x1) = max{2, x1}
  half#_A(x1) = max{2, x1}
  -_A(x1,x2) = max{0, x1, x2}
  -#_A(x1,x2) = max{0, x1, x2}
  if_A(x1,x2,x3) = max{0, x1, x2, x3}
  if#_A(x1,x2,x3) = max{0, x1, x2, x3}

precedence:

  0 = - = if > double > s = half