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))

linear polynomial interpretations on N:

  active_A(x1) = 2
  active#_A(x1) = 3
  f_A(x1,x2) = x1
  f#_A(x1,x2) = 2
  g_A(x1) = x1
  g#_A(x1) = 1
  mark_A(x1) = 2
  mark#_A(x1) = 0
  proper_A(x1) = 1
  proper#_A(x1) = 3
  ok_A(x1) = 3
  ok#_A(x1) = 0
  top_A(x1) = x1
  top#_A(x1) = x1 + 2

precedence:

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