YES

TRS:

  ack_in(0(),n) -> ack_out(s(n))
  ack_in(s(m),0()) -> u11(ack_in(m,s(0())))
  u11(ack_out(n)) -> ack_out(n)
  ack_in(s(m),s(n)) -> u21(ack_in(s(m),n),m)
  u21(ack_out(n),m) -> u22(ack_in(m,n))
  u22(ack_out(n)) -> ack_out(n)

linear polynomial interpretations on N:

  ack_in_A(x1,x2) = x2 + 1
  ack_in#_A(x1,x2) = x1 + 1
  0_A = 1
  0#_A = 0
  ack_out_A(x1) = 1
  ack_out#_A(x1) = 1
  s_A(x1) = x1 + 2
  s#_A(x1) = 1
  u11_A(x1) = 1
  u11#_A(x1) = 0
  u21_A(x1,x2) = 1
  u21#_A(x1,x2) = x2 + 2
  u22_A(x1) = 1
  u22#_A(x1) = 0

precedence:

  u11 = u21 > ack_in > ack_out = s > 0 = u22