YES
O(n^3)
TRS:
 {
      g(ok(X)) -> ok(g(X)),
    h(mark(X)) -> mark(h(X)),
      h(ok(X)) -> ok(h(X)),
    f(mark(X)) -> mark(f(X)),
      f(ok(X)) -> ok(f(X)),
  active(h(X)) -> h(active(X)),
  active(f(X)) -> mark(g(h(f(X)))),
  active(f(X)) -> f(active(X)),
  proper(g(X)) -> g(proper(X)),
  proper(h(X)) -> h(proper(X)),
  proper(f(X)) -> f(proper(X)),
  top(mark(X)) -> top(proper(X)),
    top(ok(X)) -> top(active(X))
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
        g(ok(X)) -> ok(g(X)),
      h(mark(X)) -> mark(h(X)),
        h(ok(X)) -> ok(h(X)),
      f(mark(X)) -> mark(f(X)),
        f(ok(X)) -> ok(f(X)),
    active(h(X)) -> h(active(X)),
    active(f(X)) -> mark(g(h(f(X)))),
    active(f(X)) -> f(active(X)),
    proper(g(X)) -> g(proper(X)),
    proper(h(X)) -> h(proper(X)),
    proper(f(X)) -> f(proper(X)),
    top(mark(X)) -> top(proper(X)),
      top(ok(X)) -> top(active(X))
   }
  Matrix Interpretation:
   Interpretation class: triangular
         [X2]    [1 0 3][X2]   [0]
   [top]([X1]) = [0 0 0][X1] + [0]
         [X0]    [0 0 0][X0]   [0]
   
        [X2]    [1 3 3][X2]   [0]
   [ok]([X1]) = [0 1 0][X1] + [2]
        [X0]    [0 0 1][X0]   [3]
   
            [X2]    [1 1 0][X2]   [2]
   [proper]([X1]) = [0 1 0][X1] + [0]
            [X0]    [0 0 0][X0]   [0]
   
            [X2]    [1 3 3][X2]   [3]
   [active]([X1]) = [0 1 0][X1] + [0]
            [X0]    [0 0 1][X0]   [1]
   
       [X2]    [1 0 3][X2]   [0]
   [f]([X1]) = [0 1 0][X1] + [2]
       [X0]    [0 0 1][X0]   [0]
   
       [X2]    [1 0 2][X2]   [0]
   [h]([X1]) = [0 1 0][X1] + [1]
       [X0]    [0 0 1][X0]   [0]
   
       [X2]    [1 1 1][X2]   [0]
   [g]([X1]) = [0 1 0][X1] + [1]
       [X0]    [0 0 1][X0]   [0]
   
          [X2]    [1 1 0][X2]   [0]
   [mark]([X1]) = [0 0 0][X1] + [0]
          [X0]    [0 0 1][X0]   [1]
   
   
   Qed