YES
O(n^2)
TRS:
 {
         c(ok(X)) -> ok(c(X)),
       f(mark(X)) -> mark(f(X)),
         f(ok(X)) -> ok(f(X)),
         g(ok(X)) -> ok(g(X)),
     active(c(X)) -> mark(d(X)),
     active(f(X)) -> f(active(X)),
  active(f(f(X))) -> mark(c(f(g(f(X))))),
     active(h(X)) -> mark(c(d(X))),
     active(h(X)) -> h(active(X)),
         d(ok(X)) -> ok(d(X)),
       h(mark(X)) -> mark(h(X)),
         h(ok(X)) -> ok(h(X)),
     proper(c(X)) -> c(proper(X)),
     proper(f(X)) -> f(proper(X)),
     proper(g(X)) -> g(proper(X)),
     proper(d(X)) -> d(proper(X)),
     proper(h(X)) -> h(proper(X)),
     top(mark(X)) -> top(proper(X)),
       top(ok(X)) -> top(active(X))
 }
 DUP: We consider a non-duplicating system.
  Trs:
   {
           c(ok(X)) -> ok(c(X)),
         f(mark(X)) -> mark(f(X)),
           f(ok(X)) -> ok(f(X)),
           g(ok(X)) -> ok(g(X)),
       active(c(X)) -> mark(d(X)),
       active(f(X)) -> f(active(X)),
    active(f(f(X))) -> mark(c(f(g(f(X))))),
       active(h(X)) -> mark(c(d(X))),
       active(h(X)) -> h(active(X)),
           d(ok(X)) -> ok(d(X)),
         h(mark(X)) -> mark(h(X)),
           h(ok(X)) -> ok(h(X)),
       proper(c(X)) -> c(proper(X)),
       proper(f(X)) -> f(proper(X)),
       proper(g(X)) -> g(proper(X)),
       proper(d(X)) -> d(proper(X)),
       proper(h(X)) -> h(proper(X)),
       top(mark(X)) -> top(proper(X)),
         top(ok(X)) -> top(active(X))
   }
  Matrix Interpretation:
   Interpretation class: triangular
         [X1]    [1 0][X1]   [0]
   [top]([X0]) = [0 0][X0] + [0]
   
        [X1]    [1 6][X1]   [3]
   [ok]([X0]) = [0 1][X0] + [7]
   
            [X1]    [1 1][X1]   [0]
   [proper]([X0]) = [0 1][X0] + [0]
   
       [X1]    [1 7][X1]   [0]
   [h]([X0]) = [0 1][X0] + [7]
   
       [X1]    [1 6][X1]   [0]
   [d]([X0]) = [0 1][X0] + [4]
   
            [X1]    [1 6][X1]   [2]
   [active]([X0]) = [0 1][X0] + [5]
   
       [X1]    [1 1][X1]   [0]
   [g]([X0]) = [0 1][X0] + [1]
   
       [X1]    [1 2][X1]   [0]
   [f]([X0]) = [0 1][X0] + [2]
   
       [X1]    [1 2][X1]   [0]
   [c]([X0]) = [0 1][X0] + [2]
   
          [X1]    [1 1][X1]   [4]
   [mark]([X0]) = [0 1][X0] + [2]
   
   
   Qed