YES

TRS:

  half(0()) -> 0()
  half(s(0())) -> 0()
  half(s(s(x))) -> s(half(x))
  inc(0()) -> 0()
  inc(s(x)) -> s(inc(x))
  zero(0()) -> true()
  zero(s(x)) -> false()
  p(0()) -> 0()
  p(s(x)) -> x
  bits(x) -> bitIter(x,0())
  bitIter(x,y) -> if(zero(x),x,inc(y))
  if(true(),x,y) -> p(y)
  if(false(),x,y) -> bitIter(half(x),y)

max/plus interpretations on N:

  half_A(x1) = max{0, -3 + x1}
  half#_A(x1) = max{8, 0}
  0_A = 0
  0#_A = 2
  s_A(x1) = max{1, 4 + x1}
  s#_A(x1) = max{1, 3}
  inc_A(x1) = max{0, x1}
  inc#_A(x1) = max{0, 4}
  zero_A(x1) = max{1, 3 + x1}
  zero#_A(x1) = max{9, 0}
  true_A = 3
  true#_A = 1
  false_A = 6
  false#_A = 8
  p_A(x1) = max{3, -4 + x1}
  p#_A(x1) = max{0, 12}
  bits_A(x1) = max{0, 3 + x1}
  bits#_A(x1) = max{9, 14 + x1}
  bitIter_A(x1,x2) = max{3, 1 + x1, 2 + x2}
  bitIter#_A(x1,x2) = max{13, 11 + x1, 10}
  if_A(x1,x2,x3) = max{3, -5 + x1, x2, 2 + x3}
  if#_A(x1,x2,x3) = max{12, 8 + x1, 11 + x2, 9}

precedence:

  bits > bitIter > zero = if > half = false > s > inc = p > 0 > true