YES

TRS:

  f(x,0()) -> s(0())
  f(s(x),s(y)) -> s(f(x,y))
  g(0(),x) -> g(f(x,x),x)

linear polynomial interpretations on N:

  f_A(x1,x2) = 1
  f#_A(x1,x2) = 1
  0_A = 2
  0#_A = 0
  s_A(x1) = 1
  s#_A(x1) = 0
  g_A(x1,x2) = x1
  g#_A(x1,x2) = x1

precedence:

  g > f > 0 = s