YES

Problem:
 active(f(f(X))) -> mark(c(f(g(f(X)))))
 active(c(X)) -> mark(d(X))
 active(h(X)) -> mark(c(d(X)))
 active(f(X)) -> f(active(X))
 active(h(X)) -> h(active(X))
 f(mark(X)) -> mark(f(X))
 h(mark(X)) -> mark(h(X))
 proper(f(X)) -> f(proper(X))
 proper(c(X)) -> c(proper(X))
 proper(g(X)) -> g(proper(X))
 proper(d(X)) -> d(proper(X))
 proper(h(X)) -> h(proper(X))
 f(ok(X)) -> ok(f(X))
 c(ok(X)) -> ok(c(X))
 g(ok(X)) -> ok(g(X))
 d(ok(X)) -> ok(d(X))
 h(ok(X)) -> ok(h(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(f(X))) -> g#(f(X))
   active#(f(f(X))) -> f#(g(f(X)))
   active#(f(f(X))) -> c#(f(g(f(X))))
   active#(c(X)) -> d#(X)
   active#(h(X)) -> d#(X)
   active#(h(X)) -> c#(d(X))
   active#(f(X)) -> active#(X)
   active#(f(X)) -> f#(active(X))
   active#(h(X)) -> active#(X)
   active#(h(X)) -> h#(active(X))
   f#(mark(X)) -> f#(X)
   h#(mark(X)) -> h#(X)
   proper#(f(X)) -> proper#(X)
   proper#(f(X)) -> f#(proper(X))
   proper#(c(X)) -> proper#(X)
   proper#(c(X)) -> c#(proper(X))
   proper#(g(X)) -> proper#(X)
   proper#(g(X)) -> g#(proper(X))
   proper#(d(X)) -> proper#(X)
   proper#(d(X)) -> d#(proper(X))
   proper#(h(X)) -> proper#(X)
   proper#(h(X)) -> h#(proper(X))
   f#(ok(X)) -> f#(X)
   c#(ok(X)) -> c#(X)
   g#(ok(X)) -> g#(X)
   d#(ok(X)) -> d#(X)
   h#(ok(X)) -> h#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(f(f(X))) -> mark(c(f(g(f(X)))))
   active(c(X)) -> mark(d(X))
   active(h(X)) -> mark(c(d(X)))
   active(f(X)) -> f(active(X))
   active(h(X)) -> h(active(X))
   f(mark(X)) -> mark(f(X))
   h(mark(X)) -> mark(h(X))
   proper(f(X)) -> f(proper(X))
   proper(c(X)) -> c(proper(X))
   proper(g(X)) -> g(proper(X))
   proper(d(X)) -> d(proper(X))
   proper(h(X)) -> h(proper(X))
   f(ok(X)) -> ok(f(X))
   c(ok(X)) -> ok(c(X))
   g(ok(X)) -> ok(g(X))
   d(ok(X)) -> ok(d(X))
   h(ok(X)) -> ok(h(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Usable Rule Processor:
   DPs:
    active#(f(f(X))) -> g#(f(X))
    active#(f(f(X))) -> f#(g(f(X)))
    active#(f(f(X))) -> c#(f(g(f(X))))
    active#(c(X)) -> d#(X)
    active#(h(X)) -> d#(X)
    active#(h(X)) -> c#(d(X))
    active#(f(X)) -> active#(X)
    active#(f(X)) -> f#(active(X))
    active#(h(X)) -> active#(X)
    active#(h(X)) -> h#(active(X))
    f#(mark(X)) -> f#(X)
    h#(mark(X)) -> h#(X)
    proper#(f(X)) -> proper#(X)
    proper#(f(X)) -> f#(proper(X))
    proper#(c(X)) -> proper#(X)
    proper#(c(X)) -> c#(proper(X))
    proper#(g(X)) -> proper#(X)
    proper#(g(X)) -> g#(proper(X))
    proper#(d(X)) -> proper#(X)
    proper#(d(X)) -> d#(proper(X))
    proper#(h(X)) -> proper#(X)
    proper#(h(X)) -> h#(proper(X))
    f#(ok(X)) -> f#(X)
    c#(ok(X)) -> c#(X)
    g#(ok(X)) -> g#(X)
    d#(ok(X)) -> d#(X)
    h#(ok(X)) -> h#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    f(mark(X)) -> mark(f(X))
    f(ok(X)) -> ok(f(X))
    g(ok(X)) -> ok(g(X))
    d(ok(X)) -> ok(d(X))
    active(f(f(X))) -> mark(c(f(g(f(X)))))
    active(c(X)) -> mark(d(X))
    active(h(X)) -> mark(c(d(X)))
    active(f(X)) -> f(active(X))
    active(h(X)) -> h(active(X))
    c(ok(X)) -> ok(c(X))
    h(mark(X)) -> mark(h(X))
    h(ok(X)) -> ok(h(X))
    proper(f(X)) -> f(proper(X))
    proper(c(X)) -> c(proper(X))
    proper(g(X)) -> g(proper(X))
    proper(d(X)) -> d(proper(X))
    proper(h(X)) -> h(proper(X))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
     [top#](x0) = [1 1 1]x0,
     
     [proper#](x0) = [1 1 1]x0,
     
     [h#](x0) = [1 0 0]x0,
     
     [d#](x0) = [1 0 1]x0,
     
     [c#](x0) = [0 1 0]x0,
     
     [f#](x0) = [1 0 0]x0,
     
     [g#](x0) = [0 1 0]x0,
     
     [active#](x0) = [1 0 1]x0 + [1],
     
                [1 1 1]     [1]
     [ok](x0) = [0 1 0]x0 + [1]
                [0 0 1]     [1],
     
                      
     [proper](x0) = x0
                      ,
     
                    [1]
     [h](x0) = x0 + [0]
                    [0],
     
               [1 0 1]     [0]
     [d](x0) = [0 0 1]x0 + [0]
               [0 1 0]     [1],
     
                       [1]
     [mark](x0) = x0 + [0]
                       [0],
     
                    [1]
     [c](x0) = x0 + [0]
                    [0],
     
               [1 0 0]     [1]
     [g](x0) = [0 0 1]x0 + [0]
               [0 1 0]     [0],
     
                    [1 0 1]     [1]
     [active](x0) = [0 0 1]x0 + [0]
                    [0 1 0]     [1],
     
               [1 0 1]     [0]
     [f](x0) = [0 0 1]x0 + [0]
               [0 1 0]     [1]
    orientation:
     active#(f(f(X))) = [1 1 2]X + [3] >= [0 0 1]X = g#(f(X))
     
     active#(f(f(X))) = [1 1 2]X + [3] >= [1 0 1]X + [1] = f#(g(f(X)))
     
     active#(f(f(X))) = [1 1 2]X + [3] >= [0 0 1]X = c#(f(g(f(X))))
     
     active#(c(X)) = [1 0 1]X + [2] >= [1 0 1]X = d#(X)
     
     active#(h(X)) = [1 0 1]X + [2] >= [1 0 1]X = d#(X)
     
     active#(h(X)) = [1 0 1]X + [2] >= [0 0 1]X = c#(d(X))
     
     active#(f(X)) = [1 1 1]X + [2] >= [1 0 1]X + [1] = active#(X)
     
     active#(f(X)) = [1 1 1]X + [2] >= [1 0 1]X + [1] = f#(active(X))
     
     active#(h(X)) = [1 0 1]X + [2] >= [1 0 1]X + [1] = active#(X)
     
     active#(h(X)) = [1 0 1]X + [2] >= [1 0 1]X + [1] = h#(active(X))
     
     f#(mark(X)) = [1 0 0]X + [1] >= [1 0 0]X = f#(X)
     
     h#(mark(X)) = [1 0 0]X + [1] >= [1 0 0]X = h#(X)
     
     proper#(f(X)) = [1 1 2]X + [1] >= [1 1 1]X = proper#(X)
     
     proper#(f(X)) = [1 1 2]X + [1] >= [1 0 0]X = f#(proper(X))
     
     proper#(c(X)) = [1 1 1]X + [1] >= [1 1 1]X = proper#(X)
     
     proper#(c(X)) = [1 1 1]X + [1] >= [0 1 0]X = c#(proper(X))
     
     proper#(g(X)) = [1 1 1]X + [1] >= [1 1 1]X = proper#(X)
     
     proper#(g(X)) = [1 1 1]X + [1] >= [0 1 0]X = g#(proper(X))
     
     proper#(d(X)) = [1 1 2]X + [1] >= [1 1 1]X = proper#(X)
     
     proper#(d(X)) = [1 1 2]X + [1] >= [1 0 1]X = d#(proper(X))
     
     proper#(h(X)) = [1 1 1]X + [1] >= [1 1 1]X = proper#(X)
     
     proper#(h(X)) = [1 1 1]X + [1] >= [1 0 0]X = h#(proper(X))
     
     f#(ok(X)) = [1 1 1]X + [1] >= [1 0 0]X = f#(X)
     
     c#(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X = c#(X)
     
     g#(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X = g#(X)
     
     d#(ok(X)) = [1 1 2]X + [2] >= [1 0 1]X = d#(X)
     
     h#(ok(X)) = [1 1 1]X + [1] >= [1 0 0]X = h#(X)
     
     top#(mark(X)) = [1 1 1]X + [1] >= [1 1 1]X = proper#(X)
     
     top#(mark(X)) = [1 1 1]X + [1] >= [1 1 1]X = top#(proper(X))
     
     top#(ok(X)) = [1 2 2]X + [3] >= [1 0 1]X + [1] = active#(X)
     
     top#(ok(X)) = [1 2 2]X + [3] >= [1 1 2]X + [2] = top#(active(X))
     
                  [1 0 1]    [1]    [1 0 1]    [1]             
     f(mark(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = mark(f(X))
                  [0 1 0]    [1]    [0 1 0]    [1]             
     
                [1 1 2]    [2]    [1 1 2]    [2]           
     f(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = ok(f(X))
                [0 1 0]    [2]    [0 1 0]    [2]           
     
                [1 1 1]    [2]    [1 1 1]    [2]           
     g(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = ok(g(X))
                [0 1 0]    [1]    [0 1 0]    [1]           
     
                [1 1 2]    [2]    [1 1 2]    [2]           
     d(ok(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = ok(d(X))
                [0 1 0]    [2]    [0 1 0]    [2]           
     
                       [1 1 2]    [3]    [1 0 2]    [3]                      
     active(f(f(X))) = [0 0 1]X + [1] >= [0 0 1]X + [0] = mark(c(f(g(f(X)))))
                       [0 1 0]    [2]    [0 1 0]    [2]                      
     
                    [1 0 1]    [2]    [1 0 1]    [1]             
     active(c(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = mark(d(X))
                    [0 1 0]    [1]    [0 1 0]    [1]             
     
                    [1 0 1]    [2]    [1 0 1]    [2]                
     active(h(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = mark(c(d(X)))
                    [0 1 0]    [1]    [0 1 0]    [1]                
     
                    [1 1 1]    [2]    [1 1 1]    [2]               
     active(f(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = f(active(X))
                    [0 0 1]    [1]    [0 0 1]    [1]               
     
                    [1 0 1]    [2]    [1 0 1]    [2]               
     active(h(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = h(active(X))
                    [0 1 0]    [1]    [0 1 0]    [1]               
     
                [1 1 1]    [2]    [1 1 1]    [2]           
     c(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(c(X))
                [0 0 1]    [1]    [0 0 1]    [1]           
     
                      [2]        [2]             
     h(mark(X)) = X + [0] >= X + [0] = mark(h(X))
                      [0]        [0]             
     
                [1 1 1]    [2]    [1 1 1]    [2]           
     h(ok(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = ok(h(X))
                [0 0 1]    [1]    [0 0 1]    [1]           
     
                    [1 0 1]    [0]    [1 0 1]    [0]               
     proper(f(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = f(proper(X))
                    [0 1 0]    [1]    [0 1 0]    [1]               
     
                        [1]        [1]               
     proper(c(X)) = X + [0] >= X + [0] = c(proper(X))
                        [0]        [0]               
     
                    [1 0 0]    [1]    [1 0 0]    [1]               
     proper(g(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = g(proper(X))
                    [0 1 0]    [0]    [0 1 0]    [0]               
     
                    [1 0 1]    [0]    [1 0 1]    [0]               
     proper(d(X)) = [0 0 1]X + [0] >= [0 0 1]X + [0] = d(proper(X))
                    [0 1 0]    [1]    [0 1 0]    [1]               
     
                        [1]        [1]               
     proper(h(X)) = X + [0] >= X + [0] = h(proper(X))
                        [0]        [0]               
    problem:
     DPs:
      
     TRS:
      f(mark(X)) -> mark(f(X))
      f(ok(X)) -> ok(f(X))
      g(ok(X)) -> ok(g(X))
      d(ok(X)) -> ok(d(X))
      active(f(f(X))) -> mark(c(f(g(f(X)))))
      active(c(X)) -> mark(d(X))
      active(h(X)) -> mark(c(d(X)))
      active(f(X)) -> f(active(X))
      active(h(X)) -> h(active(X))
      c(ok(X)) -> ok(c(X))
      h(mark(X)) -> mark(h(X))
      h(ok(X)) -> ok(h(X))
      proper(f(X)) -> f(proper(X))
      proper(c(X)) -> c(proper(X))
      proper(g(X)) -> g(proper(X))
      proper(d(X)) -> d(proper(X))
      proper(h(X)) -> h(proper(X))
    Qed