YES

TRS:

  s(a()) -> a()
  s(s(x)) -> x
  s(f(x,y)) -> f(s(y),s(x))
  s(g(x,y)) -> g(s(x),s(y))
  f(x,a()) -> x
  f(a(),y) -> y
  f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v))
  g(a(),a()) -> a()

linear polynomial interpretations on N:

  s_A(x1) = x1
  s#_A(x1) = x1
  a_A = 1
  a#_A = 1
  f_A(x1,x2) = x1 + x2
  f#_A(x1,x2) = x1 + x2
  g_A(x1,x2) = x1 + x2
  g#_A(x1,x2) = x1 + x2

precedence:

  a > s > f > g