YES

TRS:

  active(f(g(X),Y)) -> mark(f(X,f(g(X),Y)))
  active(f(X1,X2)) -> f(active(X1),X2)
  active(g(X)) -> g(active(X))
  f(mark(X1),X2) -> mark(f(X1,X2))
  g(mark(X)) -> mark(g(X))
  proper(f(X1,X2)) -> f(proper(X1),proper(X2))
  proper(g(X)) -> g(proper(X))
  f(ok(X1),ok(X2)) -> ok(f(X1,X2))
  g(ok(X)) -> ok(g(X))
  top(mark(X)) -> top(proper(X))
  top(ok(X)) -> top(active(X))

max/plus interpretations on N:

  active_A(x1) = max{0, 12 + x1}
  active#_A(x1) = max{13, -1}
  f_A(x1,x2) = max{2, x1, -13}
  f#_A(x1,x2) = max{0, 10, -2}
  g_A(x1) = max{2, x1}
  g#_A(x1) = max{8, 6}
  mark_A(x1) = max{11, 12}
  mark#_A(x1) = max{5, 7}
  proper_A(x1) = max{2, 1}
  proper#_A(x1) = max{9, 11}
  ok_A(x1) = max{15, 13 + x1}
  ok#_A(x1) = max{7, 1}
  top_A(x1) = max{0, 1}
  top#_A(x1) = max{12, 2 + x1}

precedence:

  f = top > active > g > mark = ok > proper