YES

Problem:
 active(f(f(a()))) -> mark(f(g(f(a()))))
 active(f(X)) -> f(active(X))
 f(mark(X)) -> mark(f(X))
 proper(f(X)) -> f(proper(X))
 proper(a()) -> ok(a())
 proper(g(X)) -> g(proper(X))
 f(ok(X)) -> ok(f(X))
 g(ok(X)) -> ok(g(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(f(f(a()))) -> g#(f(a()))
   active#(f(f(a()))) -> f#(g(f(a())))
   active#(f(X)) -> active#(X)
   active#(f(X)) -> f#(active(X))
   f#(mark(X)) -> f#(X)
   proper#(f(X)) -> proper#(X)
   proper#(f(X)) -> f#(proper(X))
   proper#(g(X)) -> proper#(X)
   proper#(g(X)) -> g#(proper(X))
   f#(ok(X)) -> f#(X)
   g#(ok(X)) -> g#(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(a()))) -> mark(f(g(f(a()))))
   active(f(X)) -> f(active(X))
   f(mark(X)) -> mark(f(X))
   proper(f(X)) -> f(proper(X))
   proper(a()) -> ok(a())
   proper(g(X)) -> g(proper(X))
   f(ok(X)) -> ok(f(X))
   g(ok(X)) -> ok(g(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  Matrix Interpretation Processor: dim=2
   
   interpretation:
    [top#](x0) = [1 1]x0,
    
    [proper#](x0) = [1 0]x0 + [1],
    
    [f#](x0) = [1 0]x0,
    
    [g#](x0) = [1 0]x0,
    
    [active#](x0) = [1 0]x0,
    
                [2 3]  
    [top](x0) = [0 2]x0,
    
                    [1]
    [ok](x0) = x0 + [0],
    
                        [1]
    [proper](x0) = x0 + [0],
    
                      [1]
    [mark](x0) = x0 + [1],
    
              [1 0]     [1]
    [g](x0) = [0 0]x0 + [0],
    
                     
    [active](x0) = x0,
    
              [1 1]     [1]
    [f](x0) = [0 2]x0 + [2],
    
          [1]
    [a] = [0]
   orientation:
    active#(f(f(a()))) = 5 >= 2 = g#(f(a()))
    
    active#(f(f(a()))) = 5 >= 3 = f#(g(f(a())))
    
    active#(f(X)) = [1 1]X + [1] >= [1 0]X = active#(X)
    
    active#(f(X)) = [1 1]X + [1] >= [1 0]X = f#(active(X))
    
    f#(mark(X)) = [1 0]X + [1] >= [1 0]X = f#(X)
    
    proper#(f(X)) = [1 1]X + [2] >= [1 0]X + [1] = proper#(X)
    
    proper#(f(X)) = [1 1]X + [2] >= [1 0]X + [1] = f#(proper(X))
    
    proper#(g(X)) = [1 0]X + [2] >= [1 0]X + [1] = proper#(X)
    
    proper#(g(X)) = [1 0]X + [2] >= [1 0]X + [1] = g#(proper(X))
    
    f#(ok(X)) = [1 0]X + [1] >= [1 0]X = f#(X)
    
    g#(ok(X)) = [1 0]X + [1] >= [1 0]X = g#(X)
    
    top#(mark(X)) = [1 1]X + [2] >= [1 0]X + [1] = proper#(X)
    
    top#(mark(X)) = [1 1]X + [2] >= [1 1]X + [1] = top#(proper(X))
    
    top#(ok(X)) = [1 1]X + [1] >= [1 0]X = active#(X)
    
    top#(ok(X)) = [1 1]X + [1] >= [1 1]X = top#(active(X))
    
                        [5]    [5]                     
    active(f(f(a()))) = [6] >= [3] = mark(f(g(f(a()))))
    
                   [1 1]    [1]    [1 1]    [1]               
    active(f(X)) = [0 2]X + [2] >= [0 2]X + [2] = f(active(X))
    
                 [1 1]    [3]    [1 1]    [2]             
    f(mark(X)) = [0 2]X + [4] >= [0 2]X + [3] = mark(f(X))
    
                   [1 1]    [2]    [1 1]    [2]               
    proper(f(X)) = [0 2]X + [2] >= [0 2]X + [2] = f(proper(X))
    
                  [2]    [2]          
    proper(a()) = [0] >= [0] = ok(a())
    
                   [1 0]    [2]    [1 0]    [2]               
    proper(g(X)) = [0 0]X + [0] >= [0 0]X + [0] = g(proper(X))
    
               [1 1]    [2]    [1 1]    [2]           
    f(ok(X)) = [0 2]X + [2] >= [0 2]X + [2] = ok(f(X))
    
               [1 0]    [2]    [1 0]    [2]           
    g(ok(X)) = [0 0]X + [0] >= [0 0]X + [0] = ok(g(X))
    
                   [2 3]    [5]    [2 3]    [2]                 
    top(mark(X)) = [0 2]X + [2] >= [0 2]X + [0] = top(proper(X))
    
                 [2 3]    [2]    [2 3]                  
    top(ok(X)) = [0 2]X + [0] >= [0 2]X = top(active(X))
   problem:
    DPs:
     
    TRS:
     active(f(f(a()))) -> mark(f(g(f(a()))))
     active(f(X)) -> f(active(X))
     f(mark(X)) -> mark(f(X))
     proper(f(X)) -> f(proper(X))
     proper(a()) -> ok(a())
     proper(g(X)) -> g(proper(X))
     f(ok(X)) -> ok(f(X))
     g(ok(X)) -> ok(g(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
   Qed